In twisted bilayers of semiconducting transition metal dichalcogenides (TMDs), a combination of structural rippling and electronic coupling gives rise to periodic moiré potentials that can confine charged and neutral excitations. Here, we report experimental measurements of the structure and spectroscopic properties of twisted bilayers of WSe2 and MoSe2 in the H-stacking configuration using scanning tunneling microscopy (STM). Our experiments reveal that the moiré potential in these bilayers at small angles is unexpectedly large, reaching values of above 300 meV for the valence band and 150 meV for the conduction band - an order of magnitude larger than theoretical estimates based on interlayer coupling alone. We further demonstrate that the moiré potential is a non-monotonic function of moiré wavelength, reaching a maximum at around a 13nm moiré period. This non-monotonicity coincides with a drastic change in the structure of the moiré pattern from a continuous variation of stacking order at small moiré wavelengths to a one-dimensional soliton dominated structure at large moiré wavelengths. We show that the in-plane structure of the moiré pattern is captured well by a continuous mechanical relaxation model, and find that the moiré structure and internal strain rather than the interlayer coupling is the dominant factor in determining the moiré potential. Our results demonstrate the potential of using precision moiré structures to create deeply trapped carriers or excitations for quantum electronics and optoelectronics.
Flexible long period moir\' e superlattices form in two-dimensional van der Waals crystals containing layers that differ slightly in lattice constant or orientation. In this Letter we show theoretically that isolated flat moir\' e bands described by generalized triangular lattice Hubbard models are present in twisted transition metal dichalcogenide heterobilayers. The hopping and interaction strength parameters of the Hubbard model can be tuned by varying the twist angle and the three-dimensional dielectric environment. When the flat moiré bands are partially filled, candidate many-body ground…
We develop a minimal theory for the recently observed metal-insulator transition (MIT) in two-dimensional (2D) moiré multilayer transition metal dichalcogenides (mTMD) using Coulomb disorder in the environment as the underlying mechanism. In particular, carrier scattering by random charged impurities leads to an effective 2D MIT approximately controlled by the Ioffe-Regel criterion, which is qualitatively consistent with the experiments. We find the necessary disorder to be around $5$-$10\times10^{10}$cm$^{-2}$ random charged impurities in order to quantitatively explain much, but not all, of …
Transition-metal dichalcogenide heterostructures exhibit moiré patterns that spatially modulate the electronic structure across the material's plane. For certain material pairs, this modulation acts as a potential landscape with deep, trigonally symmetric wells capable of localizing interlayer excitons, forming periodic arrays of quantum emitters. Here, we study these moiré localized exciton states and their optical properties. By numerically solving the two-body problem for an interacting electron-hole pair confined by a trigonal potential, we compute the localized exciton spectra for differe…
We study the edge states of twisted bilayer graphene and their topological origin. We show that the twisted bilayer graphene has special edge states associated with the moiré pattern, and the emergence of these moiré edge states is linked with the sliding Chern number, which describes topological charge pumping caused by relative interlayer sliding. When one layer of the twisted bilayer is relatively slid with respect to the other layer, the edge states are transferred from a single band to another across the band gap, and the number of the edge states pumped in a sliding cycle is shown to be …
The spatial texture of internal degree of freedom of electrons has profound effects on the properties of materials. Such texture in real space can manifest as an emergent magnetic field (or Berry curvature), which is expected to induce interesting valley/spin-related transport phenomena. Moiré pattern, which emerges as a spatial variation at the interface of 2D atomic crystals, provides a natural platform for investigating such real space Berry curvature effects. Here we study moiré structures formed in homobilayer transition metal dichalcogenides (TMDs) due to twisting, various uniform strain…
Moiré superlattice created by twist stacking has multiple physical properties. These physical properties depend on the twist angle, hence investigation of the twist angle dependency is important for the deep understanding of physical phenomena in moiré superlattice. In this work, negative magnetoresistance owing to weak localization (WL) was investigated in twisted double bilayer graphene (TDBG) as a function of the twist angle. The ratio of the intervalley scattering time to the intravalley scattering time, estimated using the WL formula for bilayer graphene, tended to decrease as the twist a…
Umklapp-Enhanced Interlayer Valley Drag in Moiré Bilayers
Institute of Mathematical Sciences（数学科学研究所）；Tata Institute of Fundamental Research（塔塔基础研究所）；Indiana University（印第安纳大学）；Indian Institute of Technology - Kanpur（坎普尔印度理工学院）
Low-energy moiré flat bands in magic-angle twisted bilayer graphene (tBG) have demonstrated incredible potentials to exhibit rich exotic quantum phenomena. Theoretically, the moiré flat bands of tBG are based on the extended structures, i.e., the moiré patterns with periodic boundary conditions. However, a fundamental question of whether the flat bands can exist in the graphene moiré patterns with a reduced structure symmetry, such as sample edges, remains unanswered. Here, via scanning tunneling microscopy and spectroscopy, we study the local electronic properties of a magic-angle tBG near th…
Moiré superlattices (MSL) formed in angle-aligned bilayers of van der Waals materials have become a promising platform to realize novel two-dimensional electronic states. Angle-aligned trilayer structures can form two sets of MSLs which could potentially interfere with each other. In this work, we directly image the moiré patterns in both monolayer graphene aligned on hBN and twisted bilayer graphene aligned on hBN, using combined scanning microwave impedance microscopy and conductive atomic force microscopy. Correlation of the two techniques reveals the contrast mechanism for the achieved ult…
We construct an analytic continuum model to describe the electronic structure and the electron-phonon interaction in twisted bilayer graphenes with arbitrary lattice deformation. Starting from the tight-binding model, we derive the interlayer Hamiltonian in the presence of general lattice displacement, and obtain a long-wavelength continuum expression for smooth deformation. We show that the continuum model correctly describes the band structures of the lattice-relaxed twisted bilayer graphenes. We apply the formula to the phonon vibration, and derive an explicit expression of the electron-pho…
We study the in-plane acoustic phonons in twisted bilayer graphenes using the effective continuum approach. We calculate the phonon modes by solving the continuum equation of motion for infinitesimal vibration around the static relaxed state with triangular domain structure. We find that the moiré interlayer potential only affects the in-plane asymmetric modes, where the original linear dispersion is broken down into miniphonon bands separated by gaps, while the in-plane symmetric modes with their linear dispersion are hardly affected. The phonon wave functions of asymmetric modes are regarded…
Transition metal dichalcogenide (TMD) bilayers have recently emerged as a robust and tunable moiré system for studying and designing correlated electron physics. In this work, by combining large-scale first principle calculation and continuum model approach, we provide an electronic structure theory that maps long-period heterobilayer TMD superlattices onto diatomic crystals with cations and anions. We find that the interplay between moiré potential and Coulomb interaction leads to filling-dependent charge transfer between MM and MX regions several nanometers apart. We show that the insulating…
Recent experimental progresses have demonstrated the great potential of electronic and excitonic moiré superlattices in transition metal dichalcogenides (TMDs) for quantum many-body simulations and quantum optics applications. Here we reveal that the moiré potential landscapes in the TMDs heterostructures have an electrostatic origin from the spontaneous charge transfer across the heterointerfaces dependent on the atomic registry. This allows engineering tunable multi-chromatic superlattices through the interference of moiré potentials from independently configurable heterointerfaces in multil…
Small-twist-angle transition metal dichalcogenide (TMD) heterobilayers develop isolated flat moiré bands that are approximately described by triangular lattice generalized Hubbard models [PhysRevLett.121.026402]. In this article we explore the metallic and insulating states that appear under different control conditions at a density of one-electron per moiré period, and the transitions between them. By combining fully self-consistent Hartree-Fock theory calculations with strong-coupling expansions around the atomic limit, we identify four different magnetic states and one nonmagnetic state nea…
When using hexagonal boron-nitride (hBN) as a substrate for graphene, the resulting moiré pattern creates secondary Dirac points. By encapsulating a multilayer graphene within aligned hBN sheets the controlled moiré stacking may offer even richer benefits. Using advanced tight-binding simulations on atomistically-relaxed heterostructures, here we show that the gap at the secondary Dirac point can be opened in selected moiré-stacking configurations, and is independent of any additional vertical gating of the heterostructure. On the other hand, gating can broadly tune the gap at the principal Di…
We theoretically study the electronic structure of magic-angle twisted bilayer graphene with disordered moiré patterns. By using an extended continuum model incorporating non-uniform lattice distortion, we find that the local density of states of the flat band is hardly broadened, but splits into upper and lower subbands in most places. The spatial dependence of the splitting energy is almost exclusively determined by the local value of the effective vector potential induced by heterostrain, whereas the variation of local twist angle and local moiré period give relatively minor effects on the …
Geometrical moiré patterns, generic for almost aligned bilayers of two-dimensional (2D) crystals with similar lattice structure but slightly different lattice constants, lead to zone folding and miniband formation for electronic states. Here, we show that moiré superlattice (mSL) effects in $\mathrm{MoSe}_2/\mathrm{WS}_2$ and $\mathrm{MoTe}_2/\mathrm{MoSe}_2$ heterobilayers that feature alignment of the band edges are enhanced by resonant interlayer hybridization, and anticipate similar features in twisted homobilayers of TMDs, including examples of narrow minibands close to the actual band ed…
Recent measurements in several different laboratories report the observation of an approximately linear-in-temperature resistivity with a large twist-angle-dependent slope (or temperature coefficient) in moiré twisted bilayer graphene (tBLG) down to a few K and sometimes to much lower temperatures. In this note, we theoretically discuss this `strange metal' linear-in-temperature transport behavior from the perspective of resistive scattering by acoustic phonons, emphasizing the aspects of the transport data, which are and which are not consistent with the phonon scattering mechanism. Extensive…
While a typical material exhibits field induced currents only at the boundary, a uniform out-of-plane magnetic field applied to two mutually rotated layers of graphene is shown to result in an ordered array of permanent current loops throughout the material. Each current loop consists of an interlayer current flowing through the open AA stacked regions of the moiré created by rotation, which then flows back through the neighboring AB regions to form a circuit, with significant current strength even at small fields. Similar moiré ordered arrays of current loops are also shown to exist in non-eq…
The large surface-to-volume ratio in atomically thin 2D materials allows to efficiently tune their properties through modifications of their environment. Artificial stacking of two monolayers into a bilayer leads to an overlap of layer-localized wave functions giving rise to a twist angle-dependent hybridization of excitonic states. In this joint theory-experiment study, we demonstrate the impact of interlayer hybridization on bright and momentum-dark excitons in twisted WSe$_2$ bilayers. In particular, we show that the strong hybridization of electrons at the $Λ$ point leads to a drastic reds…
A new implementation of stochastic many-body perturbation theory for periodic 2D systems is presented. The method is used to compute quasiparticle excitations in twisted bilayer phosphorene. Excitation energies are studied using stochastic $G_0W_0$ and partially self-consistent $\bar ΔGW_0$ approaches. The approach is inexpensive; it is used to study twisted systems with unit cells containing $>2,700$ atoms ($>13,500$ valence electrons), which corresponds to a minimum twisting angle of $\approx 3.1^\circ$. Twisted bilayers exhibit band splitting, increased localization and formation of localiz…
We present a theory of optical absorption by interlayer excitons in a heterobilayer formed from transition metal dichalcogenides. The theory accounts for the presence of small relative rotations that produce a momentum shift between electron and hole bands located in different layers, and a moiré pattern in real space. Because of the momentum shift, the optically active interlayer excitons are located at the moiré Brillouin zone's corners, instead of at its center, and would have elliptical optical selection rules if the individual layers were translationally invariant. We show that the excito…
Spontaneous order of layer pseudospins in two-dimensional bilayers is common in quantum Hall systems, where it is responsible for hysteretic responses to gate fields in states with Ising order, and giant drag voltages in states with XY (spontaneous inter-layer phase coherence) order. In this article we predict that layer pseudospin order will also occur in double-moiré strongly correlated two-dimensional electron systems. We comment on similarities and differences in the competition between the two types of order in quantum Hall and double-moiré systems, and relate our findings to previous wor…
The diffusive motion of metal nanoparticles Au and Ag on monolayer and between bilayer heterostructures of transition metal dichalcogenides and graphene are investigated in the framework of density functional theory. We found that the minimum energy barriers for diffusion and the possibility of cluster formation depend strongly on both the type of nanoparticle and the type of monolayers and bilayers. Moreover, the tendency to form clusters of Ag and Au can be tuned by creating various bilayers. Tunability of the diffusion characteristics of adatoms in van der Waals heterostructures holds promi…
Moiré superlattices formed in van der Waals bilayers have enabled the creation and manipulation of new quantum states, as is exemplified by the discovery of superconducting and correlated insulating states in twisted bilayer graphene near the magic angle. Twisted bilayer semiconductors may lead to tunable exciton lattices and topological states, yet signatures of moiré excitons have been reported only in closely angularly-aligned bilayers. Here we report tuning of moiré lattice in WS$_{2}$ /MoSe$_{2}$ bilayers over a wide range of twist angles, leading to the continuous tuning of moiré lattice…
We show that moiré bands of twisted homobilayers can be topologically nontrivial, and illustrate the tendency by studying valence band states in $\pm K$ valleys of twisted bilayer transition metal dichalcogenides, in particular, bilayer MoTe$_2$. Because of the large spin-orbit splitting at the monolayer valence band maxima, the low energy valence states of the twisted bilayer MoTe$_2$ at $+K$ ($-K$) valley can be described using a two-band model with a layer-pseudospin magnetic field $\boldsymbolΔ(\boldsymbol{r})$ that has the moiré period. We show that $\boldsymbolΔ(\boldsymbol{r})$ has a to…
Intercalation of lithium atoms between layers of 2D materials can alter their atomic and electronic structure. We investigate effects of Li intercalation in twisted bilayers of the transition metal dichalcogenide MoS$_2$ through first-principles calculations, tight-binding parameterization based on the Wannier transformation, and analysis of moiré band structures through an effective continuum model. The energetic stability of different intercalation sites for Li between layers of MoS$_2$ are classified according to the local coordination type and the number of vertically aligned Mo atoms, sug…
We consider twisted bilayer and trilayer graphene in the presence of Rashba spin-orbit coupling and explore the physics of Moiré spintronics. The electronic charge density has a sharp step right at the magic angles $θ_m$. As a result, local spin observables (polarization and equilibrium spin currents) have sharp peaks (of width about a small fraction of 1$^\circ$) as a function of the twist angle $θ$, and abrupt sign reversals at $θ_m$. Thereby, the magic angle can be determined in an unprecedented accuracy. In the first chiral limit, the spin currents vanish, but the peculiar pattern of the p…
Lattice reconstruction in twisted transition-metal dichalcogenide (TMD) bilayers gives rise to piezo- and ferroelectric moiré potentials for electrons and holes, as well as a modulation of the hybridisation across the bilayer. Here, we develop hybrid $\mathbf{k}\cdot \mathbf{p}$ tight-binding models to describe electrons and holes in the relevant valleys of twisted TMD homobilayers with parallel (P) and anti-parallel (AP) orientations of the monolayer unit cells. We apply these models to describe moiré superlattice effects in twisted WSe${}_2$ bilayers, in conjunction with microscopic \emph{ab…
Recently, the topological flat bands and spin Hall effect have been experimentally observed in the AB-stacked MoTe$_2$/WSe$_2$ heterostructures. In this work, we systematically study the Berry curvature effects in moiré transition metal dichalcogenide (TMD) heterobilayers. We point out that the moiré potential of the remote conduction bands would induce a sizable periodic pseudo-magnetic field (PMF) on the valence band. This periodic PMF creates net Berry curvature flux in each valley of the moiré Brillouin zone. The combination of the effect of the Berry curvature and the spin-valley locking …
The valence band maxima of most group-VI transition metal dichalcogenide thin films remain at the $Γ$-point all the way from bulk to bilayer. In this paper we develop a continuum theory of the moirè minibands that are formed in the valence bands of $Γ$-valley homobilayers by a small relative twist. Our effective theory is benchmarked against large-scale ab initio electronic structure calculations that account for lattice relaxation. As a consequence of an emergent $D_6$ symmetry we find that low-energy $Γ$-valley moirè holes differ qualitatively from their K-valley counterparts addressed previ…
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…
Twisted bilayer graphene (tBLG) forms a quasicrystal whose structural and electronic properties depend on the angle of rotation between its layers. Here we present a scanning tunneling microscopy study of gate-tunable tBLG devices supported by atomically-smooth and chemically inert hexagonal boron nitride (BN). The high quality of these tBLG devices allows identification of coexisting moiré patterns and moiré super-superlattices produced by graphene-graphene and graphene-BN interlayer interactions. Furthermore, we examine additional tBLG spectroscopic features in the local density of states be…
The discovery of unconventional ferroelectric behavior in twisted bilayers has prompted the consideration of moiré heterostructures as polar materials. However, misconceptions about the nature and origin of the observed ferroelectricity indicate that a better theoretical understanding of the polar properties of moiré heterostructures is needed. In this paper, it is proposed, and verified with first-principles calculations, that all moiré heterostructures exhibit an out-of-plane moiré polar domain (MPD) structure. In transition metal dichalcogenide bilayers, an interlayer charge transfer occurs…
Moir{é} superlattice realized in two-dimensional heterostructures offers an exciting platform to access strongly-correlated electronic states. In this work, we study transition metal dichalcogenides (TMD) Moir{é} superlattices with time-reversal-symmetry and nontrivial spin{/valley}-Chern numbers. Utilizing realistic material parameters and the method of exact diagonalization, we find that at a certain twisting angle and fractional filling, gapped fractional topological states, i.e., fractional Chern insulators, are naturally {stabilized} by simply introducing the Coulomb repulsion. In contras…
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 moireé potential. Via low-temperature dual-gated magnetotransport, we demonstrate the hallmarks of a $2.4^\circ$ -twisted superlattice, including tunable regimes of interlayer coupling, reduced Ferm…
Spontaneous orbital magnetism observed in twisted bilayer graphene (tBG) on nearly aligned hexagonal boron nitride (BN) substrate builds on top of the electronic structure resulting from combined G/G and G/BN double moire interfaces. Here we show that tBG/BN commensurate double moire patterns can be classified into two types, each favoring the narrowing of either the conduction or valence bands on average, and obtain the evolution of the bands as a function of the interlayer sliding vectors and electric fields. Finite valley Chern numbers $\pm 1$ are found in a wide range of parameter space wh…
We report on the theoretical electronic spectra of twisted phosphorene bilayers exhibiting moiré patterns, as computed by means of a continuous approximation to the moiré superlattice Hamiltonian. Our model is constructed by interpolating between effective $Γ$-point conduction- and valence-band Hamiltonians for the different stacking configurations approximately realized across the moiré supercell, formulated on symmetry grounds. We predict the realization of three distinct regimes for $Γ$-point electrons and holes at different twist angle ranges: a Hubbard regime for small twist angles $θ< 2^…
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…
The formation of interfacial moiré patterns from angular and/or lattice mismatch has become a powerful approach to engineer a range of quantum phenomena in van der Waals heterostructures. For long-lived and valley-polarized interlayer excitons in transition-metal dichalcogenide (TMDC) heterobilayers, signatures of quantum confinement by the moiré landscape have been reported in recent experimental studies. Such moiré confinement has offered the exciting possibility to tailor new excitonic systems, such as ordered arrays of zero-dimensional (0D) quantum emitters and their coupling into topologi…
Stacking-order-dependent electronic properties of MoTe2/WSe2 moiré bilayers
Cornell University（康奈尔大学）；Max Planck Institute for the Structure and Dynamics of Matter（马克斯·普朗克物质结构动力学研究所）；Kavli Institute at Cornell for Nanoscale Science（康奈尔纳米科学卡弗里研究所）；National Institute for Materials Science（国立材料研究所）
