We report that the twisted few layer graphite (tFL-graphite) is a new family of moiré heterostructures (MHSs), which has richer and highly tunable moiré flat band structures entirely distinct from all the known MHSs. A tFL-graphite is composed of two few-layer graphite (Bernal stacked multilayer graphene), which are stacked on each other with a small twisted angle. The moiré band structure of the tFL-graphite strongly depends on the layer number of its composed two van der Waals layers. Near the magic angle, a tFL-graphite always has two nearly flat bands coexisting with a few pairs of narrowed dispersive (parabolic or linear) bands at the Fermi level, thus, enhances the DOS at $E_F$. This coexistence property may also enhance the possible superconductivity as been demonstrated in other multiband superconductivity systems. Therefore, we expect strong multiband correlation effects in tFL-graphite. Meanwhile, a proper perpendicular electric field can induce several isolated nearly flat bands with nonzero valley Chern number in some simple tFL-graphites, indicating that tFL-graphite is also a novel topological flat band system.
We address the origin of bead-like edge states observed by scanning tunneling microscopy (STM) in moiré patterns of graphite. Low-bias scanning tunneling spectroscopy measurements indicate these edge states are centered around AB stacking sites, contrarily to the common assumption of them being at AA sites. This shift of the intensity of the beads with respect to the bulk moiré pattern has been corroborated by a tight-binding calculation of the edge states in bilayer nanoribbons. Our results are valid not only for graphite but also for few-layer graphene, where these states have also been rece…
We have observed with STM moiré patterns corresponding to the rotation of one graphene layer on HOPG surface. The moiré patterns were characterized by rotation angle and extension in the plane. Additionally, by identifying border domains and defects we can discriminate between moiré patterns due to rotation on the surface or subsurface layer. For a better understanding of moiré patterns formation we have studied by first principles an array of three graphene layers where the top or the middle layer appears rotated around the stacking axis. We compare the experimental and theoretical results an…
Moiré patterns on Highly Oriented Pyrolytic Graphite surfaces due to dislocated graphene layers were studied. We observed that the apparent corrugations of the moiré patterns in scanning tunnelling microscopy images change as a function of tunnel junction bias. A simple geometric investigation of the atomic structure of the graphene layers generating moiré patterns revealed that different atomic arrangements due to different twist angles can result in similar geometric moiré periods such that, only a small fraction of the observed moiré periodicities may coincide with the real atomic periodici…
We study the effect of atomic relaxation on the structure of moiré patterns in twisted graphene on graphite and double layer graphene by large scale atomistic simulations. The reconstructed structure can be described as a superlattice of `hot spots' with vortex-like behaviour of in-plane atomic displacements and increasing out-of-plane displacements with decreasing angle. These lattice distortions affect both scalar and vector potential and the resulting electronic properties. At low misorientation angles (<$\sim$1$^\circ$) the optimized structures deviate drastically from the sinusoidal modul…
The direct experimental probing of locally varying lattice parameters and anisotropic lattice deformations in atomic multilayers is extremely challenging. Here, we develop a new combined numerical/graphical method for the analysis of irregular moiré superstructures measured by scanning tunneling microscopy (STM) on a small-twist-angle ($\sim$0.6$^{\circ}$) graphene on highly oriented pyrolytic graphite (gr/HOPG). We observe distorted moiré patterns with a spatially varying period in annealed gr/HOPG. The nanoscale modulation of the moiré period observed by STM reflects a locally strained (and …
We study the electronic structure of a double-twisted few layer graphite (DTFLG), which consists of three few layer graphite (FLG), i.e. ABA-stacked graphene multilayer, stacked with two twist angles. We consider two categories of DTFLG, alternately twisted case and chirally twisted one, according to the rotation direction of the two twist angles. We show that, once the middle FLG of DTFLG is not thinner than trilayer, both kinds of DTFLG can remarkably host two pairs of degenerate moiré flat bands (MFBs) at $E_f$, twice that of the magic angle twisted bilayer graphene (TBG). The doubled MFBs …
Layers of two-dimensional materials stacked with a small twist-angle give rise to beating periodic patterns on a scale much larger than the original lattice, referred to as a moiré superlattice. When the stacking involves more than two layers with independent twist angles between adjacent layers, it generates moiré of moiré superlattices, with multiple length scales that control the system's behavior. Here we demonstrate these effects of a high-order moiré superlattice in twisted trilayer graphene with two consecutive small twist angles. We report correlated insulating states near the half fil…
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 …
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…
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 me…
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…
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…
Moiré superlattices in twisted van der Waals materials constitute a promising platform for engineering electronic and optical properties. However, a major obstacle to fully understanding these systems and harnessing their potential is the limited ability to correlate the local moiré structure with optical properties. By using a recently developed scanning electron microscopy technique to image twisted WSe$_2$/WSe$_2$ bilayers, we directly correlate increasing moiré periodicity with the emergence of two distinct exciton species. These can be tuned individually through electrostatic gating, and …
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…
We study the thermal transport properties of twisted-layer graphite nanofibers. We show that in the presence of a twisted layer, the phonon thermal conductance of a graphite nanofiber varies remarkably with the twisted angle and can reach minimum values either at two critical angles $θ_1$ and $θ_2$ that conform to the rule $θ_1$ + $θ_1$ = $180^0$ or exactly at the angle $θ$ = $90^0$. A reduction of roughly 50% of the phonon thermal conductance can be achieved in some structures. We unveil that the twisting effect mainly influences the optical modes, leaving almost unaltered the acoustic ones. …
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…
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 …
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…
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…
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…
Stacking three monolayers of graphene with a twist generally produces two moiré patterns. A moiré of moiré structure then emerges at larger distance where the three layers periodically realign. We devise here an effective low-energy theory to describe the spectrum at distances larger than the moiré lengthscale. In each valley of the underlying graphene, the theory comprises one Dirac cone at the ${\bf Γ}_M$ point of the moiré Brillouin zone and two weakly gapped points at ${\bf K}_M$ and ${\bf K}'_M$. The velocities and small gaps exhibit a spatial dependence in the moiré-of-moiré unit cell, e…
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…
Twisted graphene multi-layers have been recently demonstrated to share several correlation-driven behaviours with twisted bilayer graphene. In general, the van Hove singularities (VHSs) can be used as a proxy of the tendency for correlated behaviours. In this paper, we adopt an atomistic method by combining tight-binding method with the semi-classical molecular dynamics to investigate the electronic structures of twisted trilayer graphene (TTG) with two independent twist angles. The two independent twist angles can lead to the interference of the moiré patterns forming a variety of commensurat…
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 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…
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^…
The edge physics of graphene based systems is well known to be highly sensitive to the atomic structure at the boundary, with localized zero mode edge states found only on the zigzag type termination of the lattice. Here we demonstrate that the graphene twist bilayer supports an additional class of edge states, that (i) are found for all edge geometries and thus are robust against edge roughness, (ii) occur at energies coinciding with twist induced van Hove singularities in the bulk and (iii) possess an electron density strongly modulated by the moiré lattice. Interestingly, these "moiré edge …
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 …
The quantum anomalous Hall (QAH) effect is sometimes observed in twisted bilayer graphene (tBG) when it is nearly aligned with an encapsulating hexagonal boron nitride (hBN) layer. We propose that the appearance or absence of the QAH effect in individual devices could be related to commensurability between the graphene/graphene and graphene/hBN moiré patterns. We identify a series of points in the $(θ_{\rm GG},θ_{\rm GBN})$ twist-angle space at which the two moiré patterns are commensurate, allowing moiré band theory to be applied, and show that the band Chern numbers are in this case sensitiv…
Very recently, unconventional superconductivity has been observed in the double twisted trilayer graphene (TLG), where three monolayer graphene (MLG) are stacked on top of each other with two twist angles [J. M. Park, et al., Nature 590, 249 (2021); Z. Hao, et al., Science 371, 1133 (2021); X. Zhang, et al., Phys. Rev. Lett.127, 166802 (2021)]. When some of MLGs in the double twisted TLG are replaced by bilayer graphene (BLG), we get a new family of double twisted moire heterostructure, namely double twisted few layer graphene (DTFLG). In this work, we theoretically investigate the moire band …
It is widely known that the twisted bilayer graphene (TBG) shows flat bands at magic angles, which can be well described by the effective continuum model derived by Bistritzer and MacDonald (BM). We propose in this paper a similar twisted bilayer system but defined on the square lattice with $π$-flux per plaquette, and study its spectrum using the BM Hamiltonian with a mass term which is originated from the staggered potential. The basic difference between the TBG and the present model is simply rotational symmetry, $C_3$ vs $C_4$, as well as a mass term. Nevertheless, the feature of the flat …
Aharonov-Bohm (AB) effect, the well-known archetype of electron-wave interference phenomena, has been explored extensively through transport measurements. However, these techniques lack spatial resolution that would be indispensable for studying the magnetic and electrostatic AB oscillations at the nanometer scale. Here, we demonstrated that scanning tunneling microscopy (STM) can be used as an AB interferometer operating on nanometer length scales and the magneto-electric Aharonov-Bohm effect in minimally twisted bilayer graphene (TBG) was directly measured by using STM. In the minimally TBG,…
A moire pattern is formed when two copies of a periodic pattern are overlaid with a relative twist. We address the electronic structure of a twisted two-layer graphene system, showing that in its continuum Dirac model the moire pattern periodicity leads to moire Bloch bands. The two layers become more strongly coupled and the Dirac velocity crosses zero several times as the twist angle is reduced. For a discrete set of magic angles the velocity vanishes, the lowest moire band flattens, and the Dirac-point density-of-states and the counterflow conductivity are strongly enhanced.
