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 in 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.
We report on the energy spectrum of electrons in twisted bilayer graphene (tBLG) obtained by the band-unfolding method in the tight-binding model. We find the band-gap opening at particular points in the reciprocal space, that elucidates the drastic reduction of the Fermi-level velocity with the tiny twisted angles in tBLGs. We find that Moirè pattern caused by the twist of the two graphene layers generates interactions among Dirac cones, otherwise absent, and the resultant cone-cone interactions peculiar to each point in the reciprocal space causes the energy gap and thus reduced the Fermi-le…
Ultraflat bands in twisted bilayers of two-dimensional materials have potential to host strong correlations, including the Mott-insulating phase at half-filling of the band. Using first principles density functional theory calculations, we show the emergence of ultraflat bands at the valence band edge in twisted bilayer MoS$_2$, a prototypical transition metal dichalcogenide. The computed band widths, 5 meV and 23 meV for 56.5$^\circ$ and 3.5$^\circ$ twist angles respectively, are comparable to that of twisted bilayer graphene near 'magic' angles. Large structural transformations in the Moiré …
Besides moiré superlattice, twisting can also generate moiré magnetic exchange interactions (MMEIs) in van der Waals magnets. However, due to the extreme complexity and twist-angle-dependent sensitivity, all existing models fail to capture the MMEIs, preventing the understanding of MMEIs-induced new physics. Here, we develop a microscopic moiré spin Hamiltonian that enables the effective description of MMEIs via a sliding-mapping approach in twisted magnets, as demonstrated in twisted bilayer CrI3. Unexpectedly, we discover that the emergence of MMEIs can create an unprecedented magnetic skyrm…
An important step in understanding the exotic electronic, vibrational, and optical properties of the moiré lattices is the inclusion of the effects of structural relaxation of the un-relaxed moiré lattices. Here, we propose novel structures for twisted bilayer of transition metal dichalcogenides (TMDs). For $θ\gtrsim 58.4^{\circ}$, we show a dramatic reconstruction of the moiré lattices, leading to a trimerization of the unfavorable stackings. We show that the development of curved domain walls due to the three-fold symmetry of the stacking energy landscape is responsible for such lattice reco…
Twisted bilayers of two-dimensional (2D) materials are proving a fertile ground for investigating strongly correlated electron phases. This is because the moiré pattern introduced by the relative twist between layers introduces long-wavelength effective potentials which lead to electron localization. Here, we develop a generalized continuum model for the electronic structure of moiré patterns, based on first-principles calculations and tailored to capture the physics of twisted bilayer 2D semiconductors. We apply this model to a database of eighteen 2D crystals covering a range of atomic relax…
Disorder-induced symmetry breaking in moiré bands of marginally twisted bilayer MoS$_2$
Monash University（蒙纳士大学）；University of Pennsylvania（宾夕法尼亚大学）；University of Antwerp（安特卫普大学）；John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）；Nanyang Technological University Singapore（新加坡南洋理工大学）；Washington University（华盛顿大学）；Universidad Autónoma de Madrid（马德里自治大学）
Moire superlattices in twisted bilayer graphene (TBG) and its derived structures can host exotic correlated quantum phenomena because the narrow moire flat minibands in those systems effectively enhance the electron-electron interaction. Correlated phenomena are also observed in 2H-transitional metal dichalcogenides moire superlattices. However, the number of moire systems that have been explored in experiments are still very limited. Here we theoretically investigate a series of two-dimensional (2D) twisted bilayer hexagonal materials (TBHMs) beyond TBG at fixed angles of 7.34 and 67.34 degre…
In this paper a systematic examination of graphene/hexagonal boron nitride (g/hBN) bilayers is presented, through a recently developed two-dimensional phase field crystal model that incorporates out-of-plane deformations. The system parameters are determined by closely matching the stacking energies and heights of graphene/hBN bilayers to those obtained from existing quantum-mechanical density functional theory calculations. Out-of-plane deformations are shown to reduce the energies of inversion domain boundaries in hBN, and the coupling between graphene and hBN layers leads to a bilayer defec…
Moiré materials have become one of the most active fields in material science in recent years due to their high tunability, and their unique properties emerge from the Moiré-scale structure modulation. Here, we propose twisted bilayer Bi$_2$(Te$_{1-x}$Se$_x$)$_3$ as a new Moiré material where the Moiré-scale modulation induces a topological phase transition. We show, in twisted bilayer Bi$_2$(Te$_{1-x}$Se$_x$)$_3$, a topological insulator domain and a normal insulator domain coexist in the Moiré lattice structure, and edge states on the domain boundary make nearly flat bands that dominate the …
Moiré superlattices of van der Waals heterostructures provide a powerful new way to engineer the electronic structures of two-dimensional (2D) materials. Many novel quantum phenomena have emerged in different moiré heterostructures, such as correlated insulators, superconductors, and Chern insulators in graphene systems and moiré excitons in transition metal dichalcogenide (TMDC) systems. Twisted phosphorene offers another attractive system to explore moiré physics because phosphorene features an anisotropic rectangular lattice, different from the isotropic hexagonal lattice in graphene and TM…
Observation of square-like moire lattice and quasicrystalline order in twisted rock-salt nitrides
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所北京凝聚态物理国家实验室）；Department of Physics & Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences（中国科学院大学物理科学与光电技术学院材料科学与光学电子工程中心）；School of information Science and Engineering, Shandong Agricultural University（山东农业大学信息科学与工程学院）；Department of Physics, Southern University of Science and Technology（南方科技大学物理系）
In two-dimensional (2D) twisted bilayers, the van der Waals (vdW) interlayer interaction introduces atomic-scale reconstruction at interface by locally rotating lattice to form strain-field vortex networks in their moiré superlattice. However, direct imaging the tiny local lattice rotation of the strain-field vortex requires extremely high spatial resolution and is an outstanding challenge in experiment. Here, a topmost small-period graphene moiré pattern is introduced to magnify sub-Angstrom distortions of the lattice and tiny local lattice rotation in underlying twisted bilayer graphene (TBG…
Moiré excitons are emergent optical excitations in 2D semiconductors with deep moiré superlattice potentials. While these excitations have been realized in several platforms, a system with dynamically tunable moiré potential to tailor the moiré exciton properties is yet to be realized. Here, we present a continuously tunable moiré potential in a monolayer WSe2 that is enabled by its proximity to twisted bilayer graphene (TBG) near the magic-angle. Due to its flat electronic bands, charge distribution is highly localized and forms a triangular lattice in TBG. Tuning the local charge density via…
We explore the moiré magnon bands in twisted bilayer magnets with next-nearest neighboring Dzyaloshinskii-Moriya interactions, assuming that the out-of-plane collinear magnetic order is preserved under weak interlayer coupling. By calculating the magnonic band structures and the topological Chern numbers for four representative cases, we find that (i) the valley moiré bands are extremely flat over a wide range of continuous twist angles; (ii) the topological Chern numbers of the lowest few flat bands vary significantly with the twist angle; and (iii) the lowest few topological flat bands in bi…
Remote Moiré Modulation of Decoupled Dirac Subsystems in Twisted Trilayer Graphene
Daegu Gyeongbuk Institute of Science and Technology (DGIST)（大邱庆北科学技术院）；National Institute for Materials Science（国立材料研究所）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；Institute for Basic Science（基础科学研究院）
Moiré superlattice-induced sub-bands in twisted van der Waals homo- and hetero-structures govern their optical and electrical properties, rendering additional degrees of freedom such as twist angle. Here, we demonstrate the moiré superlattice effects on the intralayer excitons and trions in a twisted bilayer of MoS$_2$ of H-type stacking at marginal twist angles. We identify the emission from localized and multiple delocalized sub-bands of intralayer moiré excitons and show their electrical modulation by the corresponding trion formation. The electrical control of the oscillator strength of th…
Moiré superlattices in graphene supported on various substrates have opened a new avenue to engineer graphene's electronic properties. Yet, 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 exhi…
Artificially twisted heterostructures of semiconducting transition metal dichalcogenides (TMDs) offer unprecedented control over their electronic and optical properties via the spatial modulation of interlayer interactions and structural reconstruction. Here we study twisted MoS2 bilayers in a wide range of twist angles near 0° using Scanning Tunneling Microscopy/Spectroscopy. We investigate the twist angle-dependence of the moiré pattern which is dominated by lattice reconstruction for small angles (<2°) leading to large triangular domains with rhombohedral stacking. Local spectroscopy measur…
University of Michigan（密歇根大学）；ITMO University（ITMO大学）；The University of Texas at Dallas（德克萨斯大学达拉斯分校）；Georgia Institute of Technology（佐治亚理工学院）；University of Iceland（冰岛大学）
Fabricating van der Waals (vdW) bilayer heterostructures (BL-HS) by stacking the same or different two-dimensional (2D) layers, offers a unique physical system with rich electronic and optical properties. Twist-angle between component layers has emerged as a remarkable parameter that can control the period of lateral confinement, and nature of the exciton (Coulomb bound electron-hole pair) in reciprocal space thus creating exotic physical states including moiré excitons. In this review article, we focus on opto-electronic properties of excitons in transition metal dichalcogenide (TMD) semicond…
We present a comprehensive theory of the magnetic phases in twisted bilayer Cr-trihalides through a combination of first-principles calculations and atomistic simulations. We show that the stacking-dependent interlayer exchange leads to an effective moiré field that is mostly ferromagnetic with antiferromagnetic patches. A wide range of noncollinear magnetic phases can be stabilized as a function of the twist angle and Dzyaloshinskii-Moriya interaction as a result of the competing interlayer antiferromagnetic coupling and the energy cost for forming domain walls. In particular, we demonstrate …
Moiré superlattices serve as a playground for emerging phenomena, such as localization of band states, superconductivity, and localization of excitons. These superlattices are large and are often modeled in the zero angle limit, which obscures the effect of finite twist angles. Here, by means of first-principles calculations we quantify the twist-angle dependence of the moiré potential in the MoS$_2$ homobilayer and identify the contributions from the constituent elements of the moiré potential. Furthermore, by considering the zero-angle limit configurations, we show that the moiré potential i…
Moiré superlattices that consist of two or more layers of two-dimensional materials stacked together with a small twist angle have emerged as a tunable platform to realize various correlated and topological phases, such as Mott insulators, unconventional uperconductivity and quantum anomalous Hall effect. Recently, the magic-angle twisted trilayer graphene (MATTG) has shown both robust superconductivity similar to magic-angle twisted bilayer graphene (MATBG) and other unique properties, including the Pauli-limit violating and re-entrant superconductivity. These rich properties are deeply roote…
A double moiré superlattice can be realized by stacking three layers of atomically thin two-dimensional materials with designer interlayer twisting or lattice mismatches. In this novel structure, atomic reconstruction of constituent layers could introduce significant modifications to the lattice symmetry and electronic structure at small twist angles. Here, we employ conductive atomic force microscopy (cAFM) to investigate symmetry breaking and local electrical properties in twisted trilayer graphene. We observe clear double moiré superlattices with two distinct moire periods all over the samp…
By using the first-principles method based on density of functional theory, we study the electronic properties of twisted bilayer graphene with some specific twist angles and interlayer spacings. With the decrease of the twist angle(the unit cell becomes larger), the energy band becomes narrower and Coulomb repulsion increases, leading to the enhancement of electronic correlation; On the other hand, as the interlayer spacing decreases and the interlayer coupling becomes stronger, the correlation becomes stronger. By tuning the interlayer coupling, we can realize the strongly correlated state w…
We predict that twisted bilayers of 1T-ZrS$_2$ realize a novel and tunable platform to engineer two-dimensional topological quantum phases dominated by strong spin-orbit interactions. At small twist angles, ZrS$_2$ heterostructures give rise to an emergent and twist-controlled moiré Kagomé lattice, combining geometric frustration and strong spin-orbit coupling to give rise to a moiré quantum spin Hall insulator with highly controllable and nearly-dispersionless bands. We devise a generic pseudo-spin theory for group-IV transition metal dichalcogenides that relies on the two-component character…
In twisted bilayer graphene (TBG) a moiré pattern forms that introduces a new length scale to the material. At the 'magic' twist angle of 1.1°, this causes a flat band to form, yielding emergent properties such as correlated insulator behavior and superconductivity [1-4]. In general, the moiré structure in TBG varies spatially, influencing the local electronic properties [5-9] and hence the outcome of macroscopic charge transport experiments. In particular, to understand the wide variety observed in the phase diagrams and critical temperatures, a more detailed understanding of the local moiré …
