Moire Superlattice Research Articles

Geometric structure and electronic properties of bilayer graphene with a Moire superlattice by interlayer asymmetric tensile strain

Using the density functional theory combined with the effective screening medium method, we investigated the energetics and electronic properties of bilayer graphene, comprising graphene layers without and with tensile strain. An interlayer interaction does not affect the structural reconstruction of each graphene layer despite the bilayer structure possessing a Moire superlattice. The structural asymmetry between the graphene layers causes a potential difference across the layers, leading to electron and hole doping in the layers without and with the tensile strain. Accumulated carriers show unique lateral distribution in each graphene layer, which depends on the interlayer atomic arrangements.

Mapping charge excitations in generalized Wigner crystals.

Transition metal dichalcogenide-based moiré superlattices exhibit strong electron-electron correlations, thus giving rise to strongly correlated quantum phenomena such as generalized Wigner crystal states. Evidence of Wigner crystals in transition metal dichalcogenide moire superlattices has been widely reported from various optical spectroscopy and electrical conductivity measurements, while their microscopic nature has been limited to the basic lattice structure. Theoretical studies predict that unusual quasiparticle excitations across the correlated gap between upper and lower Hubbard bands can arise due to long-range Coulomb interactions in generalized Wigner crystal states. However, the microscopic proof of such quasiparticle excitations is challenging because of the low excitation energy of the Wigner crystal. Here we describe a scanning single-electron charging spectroscopy technique with nanometre spatial resolution and single-electron charge resolution that enables us to directly image electron and hole wavefunctions and to determine the thermodynamic gap of generalized Wigner crystal states in twisted WS2 moiré heterostructures. High-resolution scanning single-electron charging spectroscopy combines scanning tunnelling microscopy with a monolayer graphene sensing layer, thus enabling the generation of individual electron and hole quasiparticles in generalized Wigner crystals. We show that electron and hole quasiparticles have complementary wavefunction distributions and that thermodynamic gaps of ∼50 meV exist for the 1/3 and 2/3 generalized Wigner crystal states in twisted WS2.

Kagome Quantum Oscillations in Graphene Superlattices.

Electronic spectra of solids subjected to a magnetic field are often discussed in terms of Landau levels and Hofstadter-butterfly-style Brown-Zak minibands manifested by magneto-oscillations in two-dimensional electron systems. Here, we present the semiclassical precursors of these quantum magneto-oscillations which appear in graphene superlattices at low magnetic field near the Lifshitz transitions and persist at elevated temperatures. These oscillations originate from Aharonov-Bohm interference of electron waves following open trajectories that belong to a kagome-shaped network of paths characteristic for Lifshitz transitions in the moire superlattice minibands of twistronic graphenes.

Optical properties and Landau quantisations in twisted bilayer graphene

ABSTRACT The optical properties and Landau quantisations in twisted bilayer graphene are investigated using a generalised tight-binding model that takes into account all interlayer and intralayer atomic interactions in the Moire superlattice. The system is a zero-gap semiconductor with double-degenerate Dirac-cone structures, and saddle-point energy dispersions appear at low energies for small twisting angles. The magnetic quantisation phenomena in this system are rich and unique, with Landau-level subgroups specified owing to specific Moire zone folding through modulation of the stacking angles. The Landau-level spectrum shows hybridised characteristics associated with those in monolayer, as well as AA and AB stackings. The detailed analysis explores the complex relations among the different sublattices on the same and different graphene layers.

Look-alike Landau levels in locally biased twisted bilayer graphene

Large lattice constant of Moire superlattice formed in twisted bilayer graphene enables observing Landau levels splitting into Hofstadter butterflies. This is expected for homogeneous bias conditions but its robustness against spatial potential fluctuations is left open question. We study the energy structure of twisted bilayer system in dependence of both, homogeneous magnetic field and bias voltage applied in its central region. Although translational symmetry is broken, the energy states localized outside the biased region still form Landau levels revealing self-similarity feature. Moreover, besides generic Landau levels appearing when both layers are biased with the same voltage, the second look-alike energy branch can be developed by states largely localized in central biased region. Otherwise, for counter-biasing of layers, only generic branch exists but with lowest Landau levels flanked by either, hole-like and electron-like states localized at the top or at the bottom layer of central biased part of twisted bilayer system.

Unified lattice sums accommodating multiple sublattices for solutions of the Helmholtz equation in two and three dimensions

The authors derive expressions for lattice sums of solutions of the Helmholtz equation in two and three dimensions that can be efficiently evaluated in numerical implementations. The results are applicable to complex materials involving multiple sublattices, describing systems including metasurfaces and moire superlattices.

Tailoring Dirac Fermions by In-Situ Tunable High-Order Moiré Pattern in Graphene-Monolayer Xenon Heterostructure

We report an experimental study of a high-order moiré pattern formed in graphene-monolayer xenon heterostructure. The moiré period is insitu tuned from few nanometers to +∞, by adjusting the lattice constant of the xenon monolayer through annealing. Using angle-resolved photoemission spectroscopy, we observe that Dirac node replicas move closer and finally overlap with a gap opening, as the moiré pattern expands to +∞ and evolves into a Kekulé distortion. A moiré Hamiltonian coupling Dirac fermions from different valleys explains experimental results and indicates narrow moiré band. Our Letter demonstrates a platform to study continuous evolution of the moiré pattern, and provides an unprecedented approach for tailoring Dirac fermions with tunable intervalley coupling.

Symmetry origin of lattice vibration modes in twisted multilayer graphene: Phasons versus moiré phonons

Lattice dynamics play a crucial role in the physics of Moir\'e systems. In twisted bilayer graphene (TBG), it was shown that, in addition to the graphene phonons, there is another set of gapless excitations termed Moir\'e Phonons [Phys. Rev. B, 075416, 2019] reflecting the lattice dynamics at the Moire superlattice level. These modes were later suggested to be phasons due to the incommensurate stacking of the two graphene layers [Phys. Rev. B, 155426, 2019]. In this work, we elucidate the equivalence of these two seemingly distinct perspectives by identifying an underlying symmetry, which we dub mismatch symmetry, that exists for any twist angle. For commensurate angles, this is a discrete symmetry whereas for incommensurate angles, it is equivalent to a continuous phase symmetry giving rise to phason modes. In the small angle limit, such symmetry becomes a continuous local symmetry whose spontaneous breaking gives rise to Moir\'e phonons as its Goldstone mode. We derive an effective field theory for these collective modes in TBG in precise agreement with the full model and discuss their different properties. Our analysis is then generalized to twisted multilayer graphene (TMG) where we identify higher order mismatch and deduce the count of gapless modes including graphene phonons, Moir\'e phonons and phasons. Especially, we study twisted mirror-symmetric trilayer graphene with an alternating twist angle $\theta$ and find that it can be mapped to a TBG with the re-scaled twist angle $\sqrt{2/3}\theta$, hosting the same Moir\'e phonon modes in the even mirror sector with an additional set of gapped modes in the odd sector. Our work presents a systematic study of lattice symmetries in TMG providing insights into its unique lattice dynamics.

Tunable Sample-Wide Electronic Kagome Lattice in Low-Angle Twisted Bilayer Graphene.

Overlaying two graphene layers with a small twist angle θ can create a moiré superlattice to realize exotic phenomena that are entirely absent in a graphene monolayer. A representative example is the predicted formation of localized pseudo-Landau levels (PLLs) with kagome lattice in tiny-angle twisted bilayer graphene (TBG) with θ<0.3° when the graphene layers are subjected to different electrostatic potentials. However, this was shown only for the model of rigidly rotated TBG, which is not realized in reality due to an interfacial structural reconstruction. It is believed that the interfacial structural reconstruction strongly inhibits the formation of the PLLs. Here, we systematically study electronic properties of the TBG with 0.075°≤θ<1.2° and demonstrate, unexpectedly, that the PLLs are quite robust for all the studied TBG. The structural reconstruction suppresses the formation of the emergent kagome lattice in the tiny-angle TBG. However, for the TBG around the magic angle, the sample-wide electronic kagome lattices with tunable lattice constants are directly imaged by using a scanning tunneling microscope. Our observations open a new direction to explore exotic correlated phases in moiré systems.

Designing Ultra-flat Bands in Twisted Bilayer Materials at Large Twist Angles: Theory and Application to Two-Dimensional Indium Selenide.

Intertwisted bilayers of two-dimensional (2D) materials can host low-energy flat bands, which offer opportunity to investigate many intriguing physics associated with strong electron correlations. In the existing systems, ultra-flat bands only emerge at very small twist angles less than a few degrees, which poses a challenge for experimental studies and practical applications. Here, we propose a new design principle to achieve low-energy ultra-flat bands with increased twist angles. The key condition is to have a 2D semiconducting material with a large energy difference of band edges controlled by stacking. We show that the interlayer interaction leads to defect-like states under twisting, which forms a flat band in the semiconducting band gap with dispersion strongly suppressed by the large energy barriers in the moiré superlattice even for large twist angles. We explicitly demonstrate our idea in bilayer α-In2Se3 and bilayer InSe. For bilayer α-In2Se3, we show that a twist angle of ∼13.2° is sufficient to achieve the band flatness comparable to that of twist bilayer graphene at the magic angle ∼1.1°. In addition, the appearance of ultra-flat bands here is not sensitive to the twist angle as in bilayer graphene, and it can be further controlled by external gate fields. Our finding provides a new route to achieve ultra-flat bands other than reducing the twist angles and paves the way toward engineering such flat bands in a large family of 2D materials.

Formation of Moiré Superlattices via Surfactant/Nanosheet-Co-mediated Crystallization.

Moiré superlattices (MSLs) of two-dimensional (2D) van der Waals materials have attracted considerable attention in recent years; however, studies of bottom-up growth of twisted MSL structures via solution-processed crystallization are rarely reported. Through facile one-pot solvothermal synthesis, here we demonstrate a nonclassical surfactant/nanosheet-co-mediated crystallization pathway for formation of MSL structures with two models of SnS2 and SnSe2. Our experimental results reveal that attractive interactions between 2D inorganic building blocks and surfactant organic molecules during the initial stage of crystallization are crucial to drive surfactant-covered nanosheets to crystallize into molecule-intercalated nanosheet aggregates. Under the high-pressure condition, further crystallograpic fusion can occur if the reaction time is prolonged, which alters the interactions of adjacent layers during the coalescence of small-grain-size 2D domains due to insertion of foreign molecules, leading to interlayer rotations. This work uncovers an interesting organic-inorganic cocrystallization growth mode and provides a novel pathway for large-scale fabrication of MSLs.

Electronic and optical excitation properties of vanadium pentoxide V2O5

Vanadium pentoxide (V2O5) exhibits an unusual lattice symmetry (orthorhombic crystal). The theoretical framework based on the first-principles calculations is utilized to investigate the essential properties of the V2O5 compound. The complete picture of multi-hybridization in V-O bonds could be verified in the optimal Moire superlattice, the atom-dominated electronic energy spectrum, the spatial charge densities, and the atom- & orbital-decomposed van Hove singularities. The optical excitation properties reveal in the dielectric functions, energy loss function, reflectance spectra, and absorption coefficients. The diversified available channels could be verified by a specific orbital hybridization under the strong linking of electronic and optical properties. An optical plasmon mode, with ωp~>8 eV, is identified to be the coherent charge oscillations of V-4s,3dx2-y2,3dxy,3dyz,3dxz,3dz2 and O-[2s,2px,2py,2pz] orbitals. This theoretical analysis has been developed and can be generalized into other emergent systems in comprehending their essential characteristics.

Eshelby-twisted three-dimensional moiré superlattices

Author(s): Song, Z; Sun, X; Wang, LW | Abstract: Twisted bilayers of van der Waals materials have recently attracted great attention due to their tunable, strongly correlated phenomena. Here, we investigate the chirality-specific physics in three-dimensional moire superlattices induced by Eshelby twist. Our direct density functional theory calculations reveal helical rotation leads to optical circular dichroism and chirality-specific nonlinear Hall effect, even though there is no magnetization or magnetic field. Both these phenomena can be reversed by changing the structural chirality. This provides a path to constructing chirality-specific materials.

Long-range nontopological edge currents in charge-neutral graphene.

Van der Waals heterostructures display numerous unique electronic properties. Nonlocal measurements, wherein a voltage is measured at contacts placed far away from the expected classical flow of charge carriers, have been widely used in the search for novel transport mechanisms, including dissipationless spin and valley transport1-9, topological charge-neutral currents10-12, hydrodynamic flows13 and helical edge modes14-16. Monolayer1-5,10,15-19, bilayer9,11,14,20 and few-layer21 graphene, transition-metal dichalcogenides6,7 and moiré superlattices8,10,12 have been found to display pronounced nonlocal effects. However, the origin of these effects is hotly debated3,11,17,22-24. Graphene, in particular, exhibits giant nonlocality at charge neutrality1,15-19, a striking behaviour that has attracted competing explanations. Using a superconducting quantum interference device on a tip (SQUID-on-tip) for nanoscale thermal and scanning gate imaging25, here we demonstrate that the commonly occurring charge accumulation at graphene edges23,26-31 leads to giant nonlocality, producing narrow conductive channels that support long-range currents. Unexpectedly, although the edge conductance has little effect on the current flow in zero magnetic field, it leads to field-induced decoupling between edge and bulk transport at moderate fields. The resulting giant nonlocality at charge neutrality and away from it produces exotic flow patterns that are sensitive to edge disorder, in which charges can flow against the global electric field. The observed one-dimensional edge transport is generic and nontopological and is expected to support nonlocal transport in many electronic systems, offering insight into the numerous controversies and linking them to long-range guided electronic states at system edges.

Ultrahigh-temperature ferromagnetism in MoS2 Moiré superlattice/graphene hybrid heterostructures

Realizing high-temperature ferromagnetism in two-dimensional (2D) semiconductor nanosheets is significant for their applications in next-generation magnetic and electronic nanodevices. Herein, this goal could be achieved on a MoS2 Moire superlattice grown on the reduced graphene oxide (RGO) substrate by a hydrothermal approach. The as-synthesized bilayer MoS2 superlattice structure with rotating angle (ϕ = 13° ± 1°) of two hexagonal MoS2 lattices, possesses outstanding ferromagnetic property and an ultra-high Curie temperature of 990 K. The X-ray absorption near-edge structure and ultraviolet photoelectron spectroscopies combined with density functional theory calculation indicate that the covalent interactions between MoS2 Moire superlattice and RGO substrate lead to the formation of interfacial Mo-S-C bonds and complete spin polarization of Mo 4d electrons near the Fermi level. This design could be generalized and may open up a possibility for tailoring the magnetism of other 2D materials.

The importance of being buckled

An article in Nature Materials reports a detailed characterization of the atomic and electronic structure of WS2/WSe2, highlighting the influence of the 3D structural reconstruction of the moire superlattice on the correlated phenomena observed in this material.

Exciton-exciton interaction in transition metal dichalcogenide monolayers and van der Waals heterostructures

Due to a strong Coulomb interaction, excitons dominate the excitation kinetics in 2D materials. While Coulomb-scattering between electrons has been well studied, the interaction of excitons is more challenging and remains to be explored. As neutral composite bosons consisting of electrons and holes, excitons show a non-trivial scattering dynamics. Here, we study on microscopic footing exciton-exciton interaction in transition-metal dichalcogenides and related van der Waals heterostructures. We demonstrate that the crucial criterion for efficient scattering is a large electron/hole mass asymmetry giving rise to internal charge inhomogeneities of excitons and emphasizing their cobosonic substructure. Furthermore, both exchange and direct exciton-exciton interactions are boosted by enhanced exciton Bohr radii. We also predict an unexpected temperature dependence that is usually associated to phonon-driven scattering and we reveal an orders of magnitude stronger interaction of interlayer excitons due to their permanent dipole moment. The developed approach can be generalized to arbitrary material systems and will help to study strongly correlated exciton systems, such as moire super lattices.

Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene

In bilayer graphene rotationally faulted to theta=1.1 degrees, interlayer tunneling and rotational misalignment conspire to create a pair of low energy flat band that have been found to host various correlated phenomena at partial filling. Most work to date has focused on the zero magnetic field phase diagram, with magnetic field (B) used as a probe of the B=0 band structure. Here, we show that twisted bilayer graphene (tBLG) in a B as low as 2T hosts a cascade of ferromagnetic Chern insulators with Chern number |C|=1,2 and 3. We argue that the emergence of the Chern insulators is driven by the interplay of the moire superlattice with the B, which endow the flat bands with a substructure of topologically nontrivial subbands characteristic of the Hofstadter butterfly. The new phases can be accounted for in a Stoner picture in which exchange interactions favor polarization into one or more spin- and valley-isospin flavors; in contrast to conventional quantum Hall ferromagnets, however, electrons polarize into between one and four copies of a single Hofstadter subband with Chern number C=-1. In the case of the C=\pm3 insulators in particular, B catalyzes a first order phase transition from the spin- and valley-unpolarized B=0 state into the ferromagnetic state. Distinct from other moire heterostructures, tBLG realizes the strong-lattice limit of the Hofstadter problem and hosts Coulomb interactions that are comparable to the full bandwidth W and are consequently much stronger than the width of the individual Hofstadter subbands. In our experimental data, the dominance of Coulomb interactions manifests through the appearance of Chern insulating states with spontaneously broken superlattice symmetry at half filling of a C=-2 subband. Our experiments show that that tBLG may be an ideal venue to explore the strong interaction limit within partially filled Hofstadter bands.

Energy dissipation mechanism of commensurate graphene layers

The energy dissipation mechanism of graphene layers in commensurate contact is studied by molecular dynamics simulations. The commensurability between graphene layers is controlled by misfit angles between graphene layers or an in-plane strain applied to one of the graphene layers. We find that for small size graphene, the main energy dissipation is not caused by stick-slip. We propose two potential energy dissipation channels, the lattice distortion induced by the moire superlattice structure and variation of entropy of the contacting landscape. Our results provide an insight into the energy dissipation mechanism of commensurate contact graphene.

Renormalization Group Study of Hidden Symmetry in Twisted Bilayer Graphene with Coulomb Interactions.

We develop a two stage renormalization group which connects the continuum Hamiltonian for twisted bilayer graphene at length scales shorter than the moire superlattice period to the Hamiltonian for the active narrow bands only which is valid at distances much longer than the moire period. In the first stage, the Coulomb interaction renormalizes the Fermi velocity and the interlayer tunnelings in such a way as to suppress the ratio of the same sublattice to opposite sublatice tunneling, hence approaching the so-called chiral limit. In the second stage, the interlayer tunneling is treated nonperturbatively. Via a progressive numerical elimination of remote bands the relative strength of the one-particle-like dispersion and the interactions within the active narrow band Hamiltonian is determined, thus quantifying the residual correlations and justifying the strong coupling approach in the final step. We also calculate exactly the exciton energy spectrum from the Coloumb interactions projected onto the renormalized narrow bands. The resulting softening of the collective modes marks the propinquity of the enlarged ("hidden") U(4)×U(4) symmetry in the magic angle twisted bilayer graphene.