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The optical responses of metals are often dominated by plasmonic resonances, that is, the collective oscillations of interacting electron liquids. Here we unveil a new class of plasmons─quantum metric plasmons (QMPs)─that arise in a wide range of parity-violating magnetic metals. In these materials, a dipolar distribution of the quantum metric (a fundamental characteristic of Bloch wave functions) produces intrinsic nonreciprocal bulk plasmons. Strikingly, QMP nonreciprocity manifests even when the single-particle dispersion is symmetric: QMPs are sensitive to time-reversal and parity violations hidden in the Bloch wave function. In materials with asymmetric single-particle dispersions, quantum metric dipole induced nonreciprocity can continue to dominate at large frequencies. We anticipate that QMPs can be realized in a wide range of parity-violating magnets, including twisted bilayer graphene heterostructures, where quantum geometric quantities can achieve large values.

We study the properties of edge states for a selected (10,1)[(4,3)] twisted bilayer graphene (TBG) nanoribbon with minimal edges but a majority of zigzag edges. By using the tight-binding and Green's function methods, we find a remarkable rule of a local electronic transfer for the edge states. As the energy away from the Fermi level, the transfer is in the order of convex AB-, concave AB-, concave AA- and convex AA-stacked regions of the ribbon curve edges. We illustrate that this rule comes from the difference in interlayer couplings among the four types of local geometries at edges. Further, an in-plane transverse electric field can rearrange the edge bands and enlarge the energy regimes, leading to the lowest energy states modified from AB-stacked edge states to AA-stacked ones. The realignment of the edge bands results from the interplay between the interlayer coupling and the potential difference induced by the transverse electric field, which results in different bonding and antibonding edge states, i.e. the edge bands. In contrast, the total energy regime of the edge bands remain nearly unchanged under a relative strong off-plane perpendicular electric field, and the typical AA-stacked edge states are still maintained even the rotational symmetry of two layers is broken. Until a sufficiently strong value, the TBG nanoribbon tends to behave as two noninteracting monolayer ribbons except for a band distortion in low-energy regime. The conductance spectra reflects the edge bands well. We also discussed the influence of edge defects in the TBG nanoribbon on transport properties. It is found that the contributed conductance of each type of edge states shows different degrees of suppression for a monatomic vacancy in the corresponding region of edges., (© 2022 IOP Publishing Ltd.)

Moiré superlattices provide in-plane quantum restriction for light-matter interactions in twisted bilayer graphene (tBLG), leading to the exotic photon-Moiré physics and potential applications for light manipulation. Recently, our experiment identified a highly confined slow surface plasmons polaritons (SPPs) mode in tBLG. Here, we demonstrate that the propagation of the slow SPPs mode in tBLG is spatially tailored and steered at deep subwavelengths. Analysis by the perturbation theory indicates that the coupling between the slow SPPs mode and the Moiré system is greatly strengthened, which regulates the wavefront at the atomic scale and makes tBLG serve as a universal optical metamaterial. Consequently, the negative refraction is achieved at the interface of monolayer graphene and tBLG, by which a metalens with a controllable focal length and an extremely high resolution up to 1/150 of wavelength is devised. Our work paves the way for constructing optical metamaterial at the atomic scale and develops future photon-Moiré interaction systems.

Twisted bilayer graphene belong to van der Waals two-dimensional materials that demonstrate low-energy relativistic gapless dispersion relation. Although carries can not be permanently confined within electrostatically formed cavity for this reason (in pristine condition), we show that transitory states with lifetimes up to several microseconds may exist for twist angle θ = 1. 0 5 0 while changing the twist towards θ = 0. 5 0 lowers the lifetimes by few orders of magnitude. Generally, these resonances occur in density of states spectra grouped in small bunches which positions in energy depend on the magnitude of cavity's confinement potential. Analysis of local density of states corresponding to transitory states reveals that sequence of excitations in each bunch can have ascending as well as descending character. Moreover, density develop the superstructure composed of small density spots or closed-loops centered at AA sites and the overall superstructure may mimic the rotational symmetry of cavity or C 6 symmetry for higher excitations, even though all of them actually have C 3 symmetry inherited from underlying TBLG Moiré lattice. • Quantum cavity's resonance states in twisted bilayer graphene are described. • Lifetimes of cavity's resonance states may reach up to few microseconds. • Density of states reveals multiple bunches of resonances with subsequent ascending and descending excitations. • Positions of bunches of resonances in energy can be manipulated electrostatically. • Local density of states for cavity's resonances reveals density superstructure built on top of spot or closed-loop densities centered at AA sites. [ABSTRACT FROM AUTHOR]

• Suspended twisted bilayer graphene was prepared by using water assisted transfer approach. • A theoretical model for local strain analysis of twisted bilayer graphene was proposed. • Moiré pattern of twisted bilayer graphene was characterized by atomic force microscopy. • Local residual strain of twisted bilayer graphene was evaluated by deformation of moiré pattern. Twisted bilayer graphene with hexagonal moiré patterns have shown a variety of unusual physical and mechanical phenomena. Unfortunately, some residual strain is introduced during the fabrication process of twisted bilayer graphene, leading to the deformation of moiré pattern. In this work, we employed water assisted transfer approach to build the suspended twisted bilayer graphene over the microscale holes patterned SiO 2 /Si substrate. Atomic force microscopy is involved to characterize the moiré pattern. Based on reciprocal vectors analysis, a theoretical model was proposed to extract the local strain of twisted bilayer graphene sample. The local strain of twisted bilayer graphene induced by the transfer process as well as the biaxial tensile stretch was evaluated respectively. [ABSTRACT FROM AUTHOR]

Bilayer graphene with a twist angle θ between the layers generates a superlattice structure known as a Moiré pattern. This superlattice provides a θ-dependent q wavevector that activates phonons in the interior of the Brillouin zone. Here we show that this superlattice-induced Raman scattering can be used to probe the phonon dispersion in twisted bilayer graphene (tBLG). The effect reported here is different from the widely studied double-resonance in graphene-related materials in many aspects, and despite the absence of stacking order in tBLG, layer breathing vibrations (namely ZO' phonons) are observed. [Figure not available: see fulltext.] [ABSTRACT FROM AUTHOR]

Magic-angle twisted bilayer graphene (TBG) has attracted significant interest recently due to the discoveries of diverse correlated and topological states. In this work, we study the phonon properties in magic-angle TBG based on many-body classical potential and interatomic forces generated by a deep neural network trained with data from ab initio calculations. We have discovered a number of soft modes which can exhibit dipolar, quadrupolar, and octupolar vibrational patterns in real space, as well as some time-reversal breaking chiral phonon modes. We have further studied the phonon effects on the electronic structures by freezing certain soft phonon modes. We find that if a soft quadrupolar phonon mode is assumed to be frozen, the system would exhibit a charge order which is perfectly consistent with recent experiments. Moreover, once some low-frequency C 2 z -breaking modes get frozen, the Dirac points at the charge neutrality point would be gapped out, which provides an alternative perspective to the origin of correlated insulator state at charge neutrality point.

Twisted bilayer graphene (t-BLG) has recently been introduced as a rich physical platform displaying flat electronic bands, strongly correlated states, and unconventional superconductivity. Studies have hinted at an unusual Z topology of the moiré Dirac bands of t-BLG. However, direct experimental evidence of this moiré band topology and associated edge states is still lacking. Herein, using superconducting quantum interferometry, we reconstructed the spatial supercurrent distribution in t-BLG Josephson junctions and revealed the presence of edge states located in the superlattice band gaps. The absence of edge conduction in high resistance regions just outside the superlattice band gap confirms that the edge transport originates from the filling of electronic states located inside the band gap and further allows us to exclude several other edge conduction mechanisms. These results confirm the unusual moiré band topology of twisted bilayer graphene and will stimulate further research to explore its consequences.2 topology of the moiré Dirac bands of t-BLG. However, direct experimental evidence of this moiré band topology and associated edge states is still lacking. Herein, using superconducting quantum interferometry, we reconstructed the spatial supercurrent distribution in t-BLG Josephson junctions and revealed the presence of edge states located in the superlattice band gaps. The absence of edge conduction in high resistance regions just outside the superlattice band gap confirms that the edge transport originates from the filling of electronic states located inside the band gap and further allows us to exclude several other edge conduction mechanisms. These results confirm the unusual moiré band topology of twisted bilayer graphene and will stimulate further research to explore its consequences.

The giant light-matter interaction induced by van Hove singularities (vHSs) of twisted bilayer graphene (tBLG) is responsible for enhanced optical absorption and strong photoresponse. Here, we investigated the evolution of vHSs in tBLG under pressure by using Raman spectroscopy. Pressure not only induces a blue shift of the G/R band but also tunes the intensity of the G/R band. The blue shift of the G/R band is due to the reduction of the in-plane lattice constant, and the variation of the G/R band intensity is due to the vHSs' shift of tBLG. Moreover, the main band in the absorption spectrum of tBLG is attributed to multiple transitions from valence to conduction bands. Because the ratio of R to G band intensity increases under pressure and the origins of R and G bands are different, we claim that pressure enhances intervalley electron scattering. This study paves the way for pressure engineering of vHS and the corresponding photon-electron-phonon interaction in tBLG.

Quantum Hall (QH) interferometry provides an archetypal platform for the experimental realization of braiding statistics of fractional QH states. However, the complexity of observing fractional statistics requires phase coherence over the length of the interferometer, as well as suppression of Coulomb charging energy. Here, we demonstrate a new type of QH interferometer based on marginally twisted bilayer graphene (mtBLG), with a twist angle θ ≈ 0.16°. With the device operating in the QH regime, we observe distinct signatures of electronic Fabry-Pérot and Aharonov-Bohm oscillations of the magneto-thermopower in the density-magnetic field phase space, at Landau level filling factors ν = 4, 8. We find that QH interference effects are intrinsic to the triangular AB/BA domains in mtBLG that show diminished Coulomb charging effects. Our results demonstrate phase-coherent interference of QH edge modes without any additional gate-defined complex architecture, which may be beneficial in experimental realizations of non-Abelian braiding statistics.

Magic-angle twisted bilayer graphene (MATBG) is notable as a highly tunable platform for investigating strongly correlated phenomena such as unconventional superconductivity and quantum spin liquids, due to easy control of doping level through gating and sensitive dependence of the magic angle on hydrostatic pressure. Experimental observations of correlated insulating states, unconventional superconductivity and ferromagnetism in MATBG indicate that this system exhibits rich exotic phases. In this work, using density functional theory calculations in conjunction with the effective screening medium method, we find the MATBG under pressure at a twisting angle of 2.88 ∘ and simulate how its electronic states evolve when doping level and electric field perpendicular to plane are tuned by gating. Our calculations show that, at doping levels between two electrons and four holes per moiré unit cell, a ferromagnetic (FM) solution with spin density localized at AA stacking sites is lower in energy than the nonmagnetic solution. The magnetic moment of this FM state decreases with both electron and hole doping and vanishes at four electrons/holes doped per moiré unit cell. Hybridization between the flat bands at the Fermi level and the surrounding dispersive bands can take place at finite doping. On increasing the out-of-plane electric field at zero doping, a transition from the FM state to the nonmagnetic one is seen. An investigation of impurity effects shows that both absorption ofO2molecules and occurrence of Stone-Wales impurities suppress the FM state, and the mechanisms are understood from our calculations. We also analyze the interlayer bonding character due to flat bands via Wannier functions. Finally, we report trivial band topology of the flat bands in the FM state at a certain doping level., (© 2022 IOP Publishing Ltd.)

To realize the applicative potential of 2D twistronic devices, scalable synthesis and assembly techniques need to meet stringent requirements in terms of interface cleanness and twist-angle homogeneity. Here, we show that small-angle twisted bilayer graphene assembled from separated CVD-grown graphene single-crystals can ensure high-quality transport properties, determined by a device-scale-uniform moiré potential. Via low-temperature dual-gated magnetotransport, we demonstrate the hallmarks of a 2.4°-twisted superlattice, including tunable regimes of interlayer coupling, reduced Fermi velocity, large interlayer capacitance, and density-independent Brown-Zak oscillations. The observation of these moiré-induced electrical transport features establishes CVD-based twisted bilayer graphene as an alternative to "tear-and-stack" exfoliated flakes for fundamental studies, while serving as a proof-of-concept for future large-scale assembly.

We give a self contained review of a recently developed strong coupling theory of magic-angle graphene. An advantage of this approach is that a single formulation can capture both the insulating and superconducting states, and with a few simplifying assumptions, can be treated analytically. We begin by reviewing the electronic structure of magic angle graphene's flat bands, in a limit that exposes their peculiar band topology and geometry. We highlight how similarities between the flat bands and the lowest Landau level give insight into the effect of interactions. For example, at certain fractional fillings, we note the promise for realizing fractional Chern states. At integer fillings, this approach points to flavor ordered insulators, which can be captured by a sigma-model in its ordered phase. Unexpectedly, topological textures of the sigma model carry electric charge which allows us to extend the same theory to describe the doped phases away from integer filling. We show how this approach can lead to superconductivity through the proliferation of charged topological textures, and estimate the T c for the superconductor. We highlight the important role played by an effective super-exchange coupling both in pairing and in setting the effective mass of Cooper pairs. Seeking to enhance this coupling helps predict new superconducting platforms, including the recently discovered alternating twist trilayer platform. We also contrast our proposal from strong coupling theories for other superconductors. • Reviews theory of twisted bilayer graphene developed by authors and collaborators. • Discusses the band structure, topology, and geometry of twisted bilayer graphene. • Discusses insulating phases and orders at integer filling. • Discusses superconductivity. [ABSTRACT FROM AUTHOR]

• Relaxation causes a change in the charge density symmetry of low twist angle TBG. • Our computational framework can predict charge density for low twist angle TBG. • Our framework incorporates inclusion of Bragg and satellite peaks. • Satellite peaks in Fourier representation reveals symmetry change in charge density. Electronic properties such as band structure and Fermi velocity in low-angle twisted bilayer graphene (TBG) are intrinsically dependent on the atomic structure. Rigid rotation between individual graphene layers provides an approximate description of the bilayer symmetry. Upon relaxation, in-plane displacement of the atoms in low angle TBG causes a change in the symmetry through the enlargement of the AB stacking regions and the reduction in size of AA and SP stacking regions. However, the effect of this in-plane relaxation on the charge density remains unexplored, because the necessary electronic structure calculations of such large supercells of low twist angle TBG are computationally infeasible. Therefore, we develop a computationally efficient framework that enables the exploration of the charge density symmetry of the low twist angle TBG. This framework is based on the Fourier representation of the charge density which presents high intensity Bragg peaks. We find that with the decrease of twist angle, low intensity satellite peaks also become apparent. Our framework incorporates these satellite peaks which reveals transformation of symmetry in the charge density distribution from high to low twist angle TBG. One striking outcome is the demonstration of the electron localization in the AA region of low twist angle TBG. Our framework helps to explain the effect of the atomistic relaxation on the charge density distribution and thus, it provides information about exotic electronic properties of low twist angle TBG at a low computational expense. [ABSTRACT FROM AUTHOR]

Graphene moiré superlattice formed by rotating two graphene sheets can host strongly correlated and topological states when flat bands form at so-called magic angles. Here, we report that, for a twisting angle far away from the magic angle, the heterostrain induced during stacking heterostructures can also create flat bands. Combining a direct visualization of strain effect in twisted bilayer graphene moiré superlattices and transport measurements, features of correlated states appear at "non-magic" angles in twisted bilayer graphene under the heterostrain. Observing correlated states in these "non-standard" conditions can enrich the understanding of the possible origins of the correlated states and widen the freedom in tuning the moiré heterostructures and the scope of exploring the correlated physics in moiré superlattices.

In stacks of two-dimensional crystals, mismatch of their lattice constants and misalignment of crystallographic axes lead to formation of moiré patterns. We show that moiré superlattice effects persist in twisted bilayer graphene (tBLG) with large twists and short moiré periods. Using angle-resolved photoemission, we observe dramatic changes in valence band topology across large regions of the Brillouin zone, including the vicinity of the saddle point at M and across 3 eV from the Dirac points. In this energy range, we resolve several moiré minibands and detect signatures of secondary Dirac points in the reconstructed dispersions. For twists θ > 21.8°, the low-energy minigaps are not due to cone anticrossing as is the case at smaller twist angles but rather due to moiré scattering of electrons in one graphene layer on the potential of the other which generates intervalley coupling. Our work demonstrates the robustness of the mechanisms which enable engineering of electronic dispersions of stacks of two-dimensional crystals by tuning the interface twist angles. It also shows that large-angle tBLG hosts electronic minigaps and van Hove singularities of different origin which, given recent progress in extreme doping of graphene, could be explored experimentally.

In this work, we investigate the role of an external electric field in modulating the spectrum and electronic structure behavior of twisted bilayer graphene (TBG) and its physical mechanisms. Through theoretical studies, it is found that the external electric field can drive the relative positions of the conduction band and valence band to some extent. The difference of electric field strength and direction can reduce the original conduction band, and through the Fermi energy level, the band is significantly influenced by the tunable electric field and also increases the density of states of the valence band passing through the Fermi level. Under these two effects, the valence and conduction bands can alternately fold, causing drastic changes in spectrum behavior. In turn, the plasmon spectrum of TBG varies from semiconductor to metal. The dielectric function of TBG can exhibit plasmon resonance in a certain range of infrared. [ABSTRACT FROM AUTHOR]

In recent years, the discovery of 'magic angle' graphene has given new inspiration to the formation of heterojunctions. Similarly, the use of hexagonal boron nitride, known as white graphene, as a substrate for graphene devices has more aroused great interest in the graphene/hexagonal boron nitride heterostructure system. Based on the first principles method of density functional theory, the band structure, density of states, Mulliken population, and differential charge density of a tightly packed model of twisted graphene/hexagonal boron nitride/graphene sandwich structure have been studied. Through the establishment of heterostructure models twisted bilayer-graphene inserting hBN with different twisted angles, it was found that the band gap, Mulliken population, and charge density, exhibited specific evolution regulars with the rotation angle of the upper graphene, showing novel electronic properties and realizing metal-insulator phase transition. We find that the particular value of the twist angle at which the metal-insulator phase transition occurs and propose a rotational regulation mechanism with angular periodicity. Our results have guiding significance for the practical application of heterojunction electronic devices., (© 2022 IOP Publishing Ltd.)

We studied the interlayer coupling and decoupling of bilayer graphene (BLG) using spatially resolved electron energy loss spectroscopy with a monochromated electron source. We correlated the twist-angle-dependent energy band hybridization with Moiré superlattices and the corresponding optical absorption peaks. The optical absorption peak originates from the excitonic transition between the hybridized van Hove singularities (vHSs), which shifts systematically with the twist angle. We then proved that the BLG decouples when a monolayer of metal chloride is intercalated in its van der Waals gap and results in the elimination of the vHS peak.

• Hollow iron microparticles are synthesized in underwater arc discharge experiments. • Twisted bilayer graphene was found in graphitic layers covering iron microparticles. • Raman spectra in twisted bilayer domains are indicative of linear electronic bands. • We obtained robust and thick graphitic layers with the electronic properties of ideal graphene. The electronic properties of graphite with rotational stacking faults are strongly modified with respect to Bernal-stacked graphite, with the recovery of the Dirac cones for sufficiently high misorientation angles. A particular case is represented by twisted bilayer graphene (TBLG), in which twist angles are univocally related to the frequency of rotation modes observed in Raman spectra. The electronic properties determined by Dirac fermions make highly desirable to develop simple routes to synthesize highly decoupled turbostratic graphite and TBLG, as an alternative to single-layer graphene (SLG) for graphene-based electronic devices. In this work, we present a synthesis of hollow iron microspheres wrapped by graphitic layers in water-submerged arc discharge experiments between graphite electrodes, with an iron wire inserted into the anode. The microspheres were studied by scanning electron microscopy, microprobe energy-dispersive X-ray spectroscopy and Raman spectroscopy. We observed domains showing rotation and combination Raman modes typical of TBLG with different rotation angles, together with domains made of turbostratic graphite. The twist angles found in TBLG domains are indicative of values of the Fermi velocity close to the value of SLG. Our results open a new road for the synthesis of robust graphitic materials with the electronic properties of SLG. [ABSTRACT FROM AUTHOR]

• Twisted bilayer graphene synthesized on nickel substrate by CVD. • Raman and STM characterization of tBLG. • Determining the main features caused by the strain on the bilayer structure. • Simple models help to probe the main interrelations among STM features in CVD-tBLG. In this work, we have synthesised twisted bilayer graphene on nickel substrate by chemical vapour deposition. The Moiré superlattice generated by a misorientation in the stacking of two layers of graphene was studied. Raman spectroscopy was used to identify regions of interest on the films, in which scanning tunnelling microscopy (STM) was implemented to characterise the system in real space with atomic resolution, while also determining the main features caused by the strain on the bilayer structure. With the aid of simple models, we have probed relevant interrelations between features observed by STM and the heterostrain undergone by the film, which could be seen to suggest extra degrees of freedom in the sense of altering the properties of materials. [ABSTRACT FROM AUTHOR]

Graphene is a good candidate for protective material owing to its extremely high stiffness and high strength-to-weight ratio. However, the impact performance of twisted bilayer graphene is still obscure. Herein we have investigated the ballistic resistance capacity of twisted bilayer graphene compared to that of AA-stacked bilayer graphene using molecular dynamic simulations. The energy propagation processes are identical, while the ballistic resistance capacity of the twisted bilayer graphene is almost two times larger than the AA-bilayer graphene. The enhanced capacity of the twisted bilayer graphene is assumed to be caused by the mismatch between the two sheets of graphene, which results in earlier fracture of the first graphene layer and reduces the possibility of penetration. [ABSTRACT FROM AUTHOR]

The optical properties of graphene in monolayer and bilayer structure is essential for the development of optical devices viz surface plasmon resonance (SPR) based bio-sensors. The band structure of the twisted bilayer graphene (BLG) is remarkably different than the normal AA or AB stacking. This provides an opportunity to control the optical and electrical properties of BLG by applying an in-plane twist to one of the layer relative to other in a BLG system. Here, we calculated the refractive index (RI) of AA and AB stacking of BLG system using density functional theory. Though the spectrum for AA stacking shows some similarity with that of monolayer graphene, the spectrum for AB stacking was found to be remarkably different. The spectrum of AB stacked layer is red-shifted and the absorption peaks in low energy regime increases nearly by three-folds. A large dependency of the twist angle on RI of twisted BLG were found. Based on the calculation, a schematic of phase diagram showing material behavior of such twisted BLG systems as a function of twist angle and photon energy was constructed. The twisted AA stacked BLG shows largely dielectric behavior whereas the twisted AB stacked BLG shows predominately semimetallic and semiconducting behavior. This study presents a RI landscape of twisted BLG dependent on important parameters viz photon energy and inplane relative twist angle. Our studies will be very useful for the design and development of optical devices employing BLG systems particularly SPR based bio-sensors which essentially measures change in RI due to adsorption of analytes., (© 2021 IOP Publishing Ltd.)

In the Purcell effect, the efficiency of optical emitters is enhanced by reducing the optical mode volume. Here we predict an analogous enhancement for electron-phonon (el-ph) scattering, achieved by compressing the electronic Wannier orbitals. Reshaped Wannier orbitals are a prominent attribute of graphene moiré superlattices, where the orbital size is tunable by the twist angle. A reduction in the orbital size leads to an enhancement in the el-ph interaction strength, yielding the values considerably larger than those in pristine monolayer graphene. The enhanced coupling boosts the el-ph scattering rates, pushing them above the values expected for the flat-band-enhanced density of electronic states. The enhanced phonon emission and scattering rates are manifested through the observables such as the electron-lattice cooling and the linear-temperature ( T ) resistivity, both of which are directly tunable by the moiré twist angle.

By employing a linearised Boltzmann equation, we calculate the magneto-optical properties of twisted bilayer graphene using non-magnetic wave functions. Both transverse and longitudinal responses are calculated up to the second order in applied magnetic field with their twist angle and Fermi level dependence examined. We find that increasing the twist angle increases the transverse metallic response so long as the Fermi level remains below the upper conduction band. Interlayer transitions provide an appreciable enhancement when the Fermi level traverses the gap between the two conduction bands. Interlayer transitions are also responsible for a nonzero anomalous Hall conductivity in this model. As the Fermi level moves towards zero, the longitudinal response begins to dominate and a highly anisotropic negative magneto-resistance is observed., (© 2021 IOP Publishing Ltd.)

We describe how the out-of-plane dielectric polarizability of monolayer graphene influences the electrostatics of bilayer graphene-both Bernal (BLG) and twisted (tBLG). We compare the polarizability value computed using density functional theory with the output from previously published experimental data on the electrostatically controlled interlayer asymmetry potential in BLG and data on the on-layer density distribution in tBLG. We show that monolayers in tBLG are described well by polarizability α exp = 10.8 Å 3 and effective out-of-plane dielectric susceptibility ϵ z = 2.5, including their on-layer electron density distribution at zero magnetic field and the interlayer Landau level pinning at quantizing magnetic fields.

The convergent beam electron diffraction (CBED) patterns of twisted bilayer samples exhibit interference patterns in their CBED spots. Such interference patterns can be treated as off-axis holograms and the phase of the scattered waves, meaning the interlayer distance can be reconstructed. A detailed protocol of the reconstruction procedure is provided in this study. In addition, we derive an exact formula for reconstructing the interlayer distance from the recovered phase distribution, which takes into account the different chemical compositions of the individual monolayers. It is shown that one interference fringe in a CBED spot is sufficient to reconstruct the distance between the layers, which can be practical for imaging samples with a relatively small twist angle or when probing small sample regions. The quality of the reconstructed interlayer distance is studied as a function of the twist angle. At smaller twist angles, the reconstructed interlayer distance distribution is more precise and artefact free. At larger twist angles, artefacts due to the moiré structure appear in the reconstruction. A method for the reconstruction of the average interlayer distance is presented. As for resolution, the interlayer distance can be reconstructed by the holographic approach at an accuracy of ± 0. 5 Å, which is a few hundred times better than the intrinsic z -resolution of diffraction limited resolution, as expressed through the spread of the measured k -values. Moreover, we show that holographic CBED imaging can detect variations as small as 0. 1 Å in the interlayer distance, though the quantitative reconstruction of such variations suffers from large errors. • CBED patterns of bilayer 2D crystal samples are investigated. • Theory and protocol of the reconstruction of the interlayer distance is provided. • The interlayer distance can be reconstructed at accuracy of ± 0.5 Å. • Holographic CBED can pick as small as 0.1 Å variations in the interlayer distance. [ABSTRACT FROM AUTHOR]