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 sensitive to a rigid in-plane hBN displacement. Given this property, we argue that the QAH effect is likely only when i) the $(θ_{\rm GG},θ_{\rm GBN})$ twist-angle-pair is close enough to a commensurate point that the two moiré patterns yield a supermoiré pattern with a sufficiently long length scale, and ii) the supermoiré has a percolating topologically non-trivial QAH phase. For twist angles far from commensurability, the hBN layer acts as a source of disorder that can destroy the QAH effect. Our proposal can explain a number of current experimental observations. Further experimental studies that can test this proposal more directly are suggested.
Van der Waals heterostructures of graphene and hexagonal boron nitride feature a moiré superlattice for graphene's Dirac electrons. Here, we review the effects generated by this superlattice, including a specific miniband structure featuring gaps and secondary Dirac points, and a fractal spectrum of magnetic minibands known as Hofstadter's butterfly.
We theoretically study in-plane acoustic phonons of graphene/hexagonal boron nitride moiré superlattice by using a continuum model. We demonstrate that the original phonon bands of individual layers are strongly hybridized and reconstructed into moiré phonon bands consisting of dispersive bands and flat bands. The phonon band structure can be effectively described by a spring-mass network model to simulate the motion of moiré domain walls, where the flat-band modes are interpreted as vibrations of independent, decoupled strings. We also show that the moiré phonon has angular momentum due to th…
Nearly aligned graphene on hexagonal boron nitride (G/BN) can be accurately modeled by a Dirac Hamiltonian perturbed by smoothly varying moiré pattern pseudospin fields. Here, we present the moiré-band model of G/BN for arbitrary small twist angles under a framework that combines symmetry considerations with input from ab-initio calculations. Our analysis of the band gaps at the primary and secondary Dirac points highlights the role of inversion symmetry breaking contributions of the moiré patterns, leading to primary Dirac point gaps when the moiré strains give rise to a finite average mass, …
Interference of double moire patterns of graphene (G) encapsulated by hexagonal boron nitride (BN) can alter the electronic structure features near the primary/secondary Dirac points and the electron-hole symmetry introduced by a single G/BN moire pattern depending on the relative stacking arrangements of the top/bottom BN layers. We show that strong interference effects are found in nearly aligned BN/G/BN and BN/G/NB and obtain the evolution of the associated density of states as a function of moire superlattice twist angles. For equal moire periods and commensurate patterns with $Δϕ= 0^{\cir…
Rhombohedral $N = 3$ trilayer graphene on hexagonal boron nitride (TLG/BN) hosts gate-tunable, valley-contrasting, nearly flat topological bands that can trigger spontaneous quantum Hall phases under appropriate conditions of the valley and spin polarization. Recent experiments have shown signatures of C = 2 valley Chern bands at 1/4 hole filling, in contrast to the predicted value of C = 3. We discuss the low-energy model for rhombohedral N-layer graphene (N = 1, 2, 3) aligned with hexagonal boron nitride (hBN) subject to off-diagonal moire vector potential terms that can alter the valley Che…
Van der Waals heterostructures employing graphene and hexagonal boron nitride (hBN) crystals have emerged as a promising platform for plasmonics thanks to the tunability of their collective modes with carrier density and record values for plasmonics figures of merit. In this Article we investigate theoretically the role of moiré-pattern superlattices in nearly aligned graphene on hBN by using continuum-model Hamiltonians derived from ab initio calculations. We calculate the system's energy loss function for a variety of chemical potential values that are accessible in gated devices. Our calcul…
The spatial dependence of the van der Waals (vdW) energy between graphene and hexagonal boron-nitride (h-BN) is investigated using atomistic simulations. The van der Waals energy between graphene and h-BN shows a hexagonal superlattice structure identical to the observed Moiré pattern in the local density of states (LDOS) which depends on the lattice mismatch and misorientation angle between graphene and h-BN. Our results provide atomistic features of the weak van der Waals interaction between graphene and BN which are in agreement with experiment and provide an analytical expression for the s…
In heterostructures consisting of atomically thin crystals layered on top of one another, lattice mismatch or rotation between the layers results in long-wavelength moiré superlattices. These moiré patterns can drive significant band structure reconstruction of the composite material, leading to a wide range of emergent phenomena including superconductivity, magnetism, fractional Chern insulating states, and moiré excitons. Here, we investigate monolayer graphene encapsulated between two crystals of boron nitride (BN), where the rotational alignment between all three components can be varied. …
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 investigate the electronic structure of the flat bands induced by moiré superlattices and electric fields in nearly aligned ABC trilayer graphene-boron nitride interfaces where Coulomb effects can lead to correlated gapped phases. Our calculations indicate that valley-spin resolved isolated superlattice flat bands that carry a finite Chern number $C = 3$ proportional to layer number can appear near charge neutrality for appropriate perpendicular electric fields and twist angles. When the degeneracy of the bands is lifted by Coulomb interactions these topological bands can lead to anomalous …
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 theoretically investigate the electronic structures of moiré superlattices arising in monolayer / bilayer graphene stacked on hexagonal boron nitride (hBN) in presence and absence of magnetic field. We develop an effective continuum model from a microscopic tight-binding lattice Hamiltonian, and calculate the electronic structures of graphene-hBN systems with different rotation angles. Using the effective model, we explain the characteristic band properties such as the gap opening at the corners of the superlattice Brillouin zone (mini-Dirac point). We also investigate the energy spectrum a…
Commensuration torques in double-moiré twisted trilayer hexagonal boron nitride and graphene heterostructures
University of Seoul（首尔大学）；Sungkyunkwan University（成均馆大学）；Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST（加泰罗尼亚纳米科学与技术研究所（ICN2，西班牙高等科学研究委员会和巴塞罗那独立科学研究所））
We study the magneto-optical conductivity of a number of Van der Waals heterostructures, namely, twisted bilayer graphene, AB-AB and AB-BA stacked twisted double bilayer graphene and monolayer graphene and AB-stacked bilayer graphene on hexagonal boron nitride. As magnetic field increases, the absorption spectrum exhibits a self-similar recursive pattern reflecting the fractal nature of the energy spectrum. Whilst twisted bilayer graphene displays only weak circular dichroism, monolayer graphene and AB-stacked bilayer graphene on hexagonal boron nitride show specifically strong circular dichro…
The conductivity of graphene on a boron nitride substrate exhibits features in the terahertz (THz) and infrared (IR) frequency regimes that are associated with the periodic moiré pattern formed by the weakly coupled two-dimensional materials. The THz and IR features are strongest when the two honeycomb lattices are orientationally aligned, and in this case are Pauli blocked unless the Fermi level is close to $\pm 150$ meV relative to the graphene sheet Dirac point. Because the transition energies between moiré bands formed above the Dirac point are small, ac conductivity features in n-doped gr…
We model optical absorption of monolayer and bilayer graphene on hexagonal boron nitride for the case of closely-aligned crystal lattices. We show that perturbations with different spatial symmetry can lead to similar absorption spectra. We suggest that a study of the absorption spectra as a function of the doping for almost completely full first miniband is necessary to extract meaningful information about the moire characteristics from optical absorption measurements and to distinguish between various theoretical proposals for the physically realistic interaction. Also, for bilayer graphene,…
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…
Electrons in graphene are described by relativistic Dirac-Weyl spinors with a two-component pseudospin1-12. The unique pseudospin structure of Dirac electrons leads to emerging phenomena such as the massless Dirac cone2, anomalous quantum Hall effect2, 3, and Klein tunneling4, 5 in graphene. The capability to manipulate electron pseudospin is highly desirable for novel graphene electronics, and it requires precise control to differentiate the two graphene sub-lattices at atomic level. Graphene/boron nitride (graphene/BN) Moire superlattice, where a fast sub-lattice oscillation due to B-N atoms…
Tuning Thermal Conductivity and Electron-Phonon Interactions in Carbon and Boron Nitride Moiré Diamanes via Twist Angle Manipulation
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Digital Materials LLC（数字材料有限责任公司）；Emanuel Institute of Biochemical Physics RAS（俄罗斯科学院埃马纽埃尔生物物理化学研究所）
In fractionally filled Landau levels there is only a small energy difference between broken translational symmetry electron-crystal states and exotic correlated quantum fluid states. We show that the spatially periodic substrate interaction associated with the long period moiré patterns present in graphene on nearly aligned hexagonal boron nitride (hBN) tilts this close competition in favor of the former, explaining surprising recent experimental findings.
Intra-Landau level excitations in the fractional quantum Hall regime are not accessible via optical absorption measurements. We point out that optical probes are enabled by the periodic potentials produced by a moiré pattern. Our observation is motivated by the recent observations of fractional quantum Hall incompressible states in moiré-patterned graphene on a hexagonal boron nitride substrate, and is theoretically based on $f-$sum rule considerations supplemented by a perturbative analysis of the influence of the moiré potential on many-body states.
The moiré potential of graphene on hexagonal boron nitride (hBN) generates a supercell sufficiently large as to thread a full magnetic flux quantum $Φ_0$ for experimentally accessible magnetic field strengths. Close to rational fractions of $Φ_0$, $p/q \cdotΦ_0$, magnetotranslation invariance is restored giving rise to Brown-Zak fermions featuring the same dispersion relation as in the absence of the field. Employing a highly efficient numerical approach we have performed the first realistic simulation of the magnetoconductance for a 250 nm wide graphene ribbon on hexagonal boron nitride using…
Employing graphene as a template, we fabricate moiré superlattices by stacking bilayer or folded bilayer-bilayer graphene (BLG or fBBLG) and hexagonal boron nitride (hBN), i.e., hBN/BLG/hBN or hBN/fBBLG/hBN stacks, with a small twist angle between the graphene and one of the two hBN layers. Because of the modulation due to the hBN, higher-generation Dirac points can emerge with a narrow bandwidth and van Hove singularities. In the moiré superlattice devices, we can therefore access the higher-generation Dirac points by in-situ gate tuning. This study is based on our previous paper (Appl. Phys.…
Due to atomically thin structure, graphene/hexagonal boron nitride (G/hBN) heterostructures are intensively sensitive to the external mechanical forces and deformations being applied to their lattice structure. In particular, strain can lead to the modification of the electronic properties of G/hBN. Furthermore, moiré structures driven by misalignment of graphene and hBN layers introduce new features to the electronic behavior of G/hBN. Utilizing {\it ab initio} calculation, we study the strain-induced modification of the electronic properties of diverse stacking faults of G/hBN when applying …
Studies on two-dimensional electron systems in a strong magnetic field first revealed the quantum Hall (QH) effect, a topological state of matter featuring a finite Chern number (C) and chiral edge states. Haldane later theorized that Chern insulators with integer QH effects could appear in lattice models with complex hopping parameters even at zero magnetic field. The ABC-trilayer graphene/hexagonal boron nitride (TLG/hBN) moiré superlattice provides an attractive platform to explore Chern insulators because it features nearly flat moiré minibands with a valley-dependent electrically tunable …
Heterostructures of atomically-thin materials have attracted significant interest owing to their ability to host novel electronic properties fundamentally distinct from their constituent layers. In the case of graphene on boron nitride, the closely-matched lattices yield a moiré superlattice that modifies the graphene electron dispersion and opens gaps both at the primary Dirac point (DP) and the moiré-induced secondary Dirac point (SDP) in the valence band. While significant effort has focused on controlling the superlattice period via the rotational stacking order, the role played by the mag…
Strong band engineering in two-dimensional (2D) materials can be achieved by introducing moiré superlattices, leading to the emergence of various novel quantum phases with promising potential for future applications. Presented works to create moiré patterns have been focused on a twist embedded inside channel materials or between channel and substrate. However, the effects of a twist inside the substrate materials on the unaligned channel materials are much less explored. In this work, we report the realization of superlattice multi-Dirac cones with the coexistence of the main Dirac cone in a …
Observation of Unconventional Ferroelectricity in Non-Moir'\e Graphene on Hexagonal Boron Nitride Boundaries and Interfaces
The University of Hong Kong（香港大学）；HK Institute of Quantum Science & Technology, The University of Hong Kong（香港大学量子科学与技术研究所）；National Institute for Materials Science（物质材料研究机构）
The Schrödinger equation dictates that the propagation of nearly free electrons through a weak periodic potential results in the opening of band gaps near points of the reciprocal lattice known as Brillouin zone boundaries. However, in the case of massless Dirac fermions, it has been predicted that the chirality of the charge carriers prevents the opening of a band gap and instead new Dirac points appear in the electronic structure of the material. Graphene on hexagonal boron nitride (hBN) exhibits a rotation dependent Moiré pattern. In this letter, we show experimentally and theoretically tha…
Graphene superlattices have recently been attracting growing interest as an emergent class of quantum metamaterials. In this paper, we report the observation of nonlocal transport in bilayer graphene (BLG) superlattices encapsulated between two hexagonal boron nitride (hBN) layers, which formed hBN/BLG/hBN moiré superlattices. We then employed these superlattices to detect a long-range charge-neutral valley current using an all-electrical method. The moiré superlattice with broken inversion symmetry leads to a hot spot with Berry curvature accumulating at the charge neutral point (CNP), and it…
All local electronic properties of graphene on a hexagonal boron nitride (hBN) substrate exhibit spatial moiré patterns related to lattice constant and orientation differences between shared triangular Bravais lattices. We apply a previously derived effective Hamiltonian for the $π$-bands of graphene on h-BN to address the carrier-dependence of transport properties, concentrating on the conductivity features at four electrons and four holes per unit cell. These transport features measure the strength of Bragg scattering of $π$-electrons off the moiré pattern, and exhibit a striking particle-ho…
Emergent electronic insulating states in a one-dimensional moiré superlattice
Zhejiang University（浙江大学）；The University of Osaka（大阪大学）；Dalian University（大连大学）；Inha University（仁荷大学）；National Institute for Materials Science (NIMS)（国立材料研究所）；Stony Brook University（石溪大学）
Second-order nonlinear optical response allows to detect different properties of the system associated with the inversion symmetry breaking. Here, we use a second harmonic generation effect to investigate the alignment of a graphene/hexagonal Boron Nitride heterostructure. To achieve that, we activate a commensurate-incommensurate phase transition by a thermal annealing of the sample. We find that this structural change in the system can be directly observed through a strong modification of a nonlinear optical signal. This result reveals the potential of a second harmonic generation technique …
Negative friction coefficients, where friction is reduced upon increasing normal load, are predicted for superlubric graphite-hexagonal boron nitride heterojunctions. The origin of this counterintuitive behavior lies in the load-induced suppression of the moiré superstructure out-of-plane distortions leading to a less dissipative interfacial dynamics. Thermally induced enhancement of the out-of-plane fluctuations leads to an unusual increase of friction with temperature. The highlighted frictional mechanism is of a general nature and is expected to appear in many layered material heterojunctio…
We identify features in the angle-resolved photoemission spectra (ARPES) arising from the periodic pattern characteristic for graphene heterostructure with hexagonal boron nitride (hBN). For this, we model ARPES spectra and intensity maps for five microscopic models used previously to describe moire superlattice in graphene/hBN systems. We show that detailed analysis of these features can be used to pin down the microscopic mechanism of the interaction between graphene and hBN. We also analyze how the presence of a moire-periodic strain in graphene or scattering of photoemitted electrons off h…
Mott insulator plays a central role in strongly correlated physics, where the repulsive Coulomb interaction dominates over the electron kinetic energy and leads to insulating states with one electron occupying each unit cell. Doped Mott insulator is often described by the Hubbard model3, which can give rise to other correlated phenomena such as unusual magnetism and even high-temperature superconductivity. A tunable Mott insulator, where the competition between the Coulomb interaction and the kinetic energy can be varied in situ, can provide an invaluable model system for the study of Mott phy…
When atomically thin two-dimensional (2D) materials are layered they often form incommensurate non-crystalline structures that exhibit long-period moir{\' e} patterns when examined by scanning probes. In this paper we present an approach which uses information obtained from {\it ab initio} calculations performed on short-period crystalline structures to derive effective Hamiltonians that are able to efficiently describe the influence of the moir{\' e} pattern superlattices on electronic properties. We apply our approach to the cases of graphene on graphene (G/G) and graphene on hexagonal boron…
We study the commensuration torques and layer sliding energetics of alternating twist trilayer graphene (t3G) and twisted bilayer graphene on hexagonal boron nitride (t2G/BN) that have two superposed moire interfaces. Lattice relaxations for typical graphene twist angles of $\sim 1^{\circ}$ in t3G or t2G/BN are found to break the out-of-plane layer mirror symmetry, give rise to layer rotation energy local minima dips of the order of $\sim 10^{-1}$ meV/atom at double moire alignment angles, and have sliding energy landscape minima between top-bottom layers of comparable magnitude. Moire superlu…
We study the atomic and electronic structures of the commensurate double moiré superlattices in fully relaxed twisted bilayer graphene (TBG) nearly aligned with the hexagonal boron nitride (BN). The single-particle effective Hamiltonian ($\hat{H}^0$) taking into account the relaxation effect and the full moiré Hamiltonian introduced by BN has been built for TBG/BN. The mean-field (MF) band structures of the self-consistent Hartree-Fock (SCHF) ground states at different number ($ν$) of filled flat bands relative to the charge neutrality point (CNP) are obtained based on $\hat{H}^0$ in the plane…
A periodic spatial modulation, as created by a moiré pattern, has been extensively studied with the view to engineer and tune the properties of graphene. Graphene encapsulated by hexagonal boron nitride (hBN) when slightly misaligned with the top and bottom hBN layers experiences two interfering moiré patterns, resulting in a so-called super-moiré (SM). This leads to a lattice and electronic spectrum reconstruction. A geometrical construction of the non-relaxed SM patterns allows us to indicate qualitatively the induced changes in the electronic properties and to locate the SM features in the …
Graphene flakes placed on hexagonal boron nitride feature in the presence of a magnetic field a complex electronic structure due to a hexagonal moiré potential resulting from the van der Waals interaction with the substrate. The slight lattice mismatch gives rise to a periodic supercell potential. Zone folding is expected to create replica of the original Dirac cone and Hofstadter butterflies. Our large-scale tight binding simulation reveals an unexpected coexistence of a relativistic and non-relativistic Landau level structure. The presence of the zeroth Landau level and its associated butter…
We calculate the electronic structure in quasiperiodic double-moiré systems of graphene sandwiched by hexagonal boron nitride, and identify the topological invariants of energy gaps. We find that the electronic spectrum contains a number of minigaps, and they exhibit a recursive fractal structure similar to the Hofstadter butterfly when plotted against the twist angle. Each of the energy gaps can be characterized by a set of integers, which are associated with an area in the momentum space. The corresponding area is geometrically interpreted as a quasi Brillouin zone, which is a polygon enclos…
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…
Spatially periodic structures with a long range period, referred to as moiré pattern, can be obtained in van der Waals bilayers in the presence of a small stacking angle or of lattice mismatch between the monolayers. Theoretical predictions suggest that the resulting spatially periodic variation of the band structure modifies the optical properties of both intra and interlayer excitons of transition metal dichalcogenides heterostructures. Here, we report on the impact of the moiré pattern formed in a MoSe$_2$/MoS$_2$ heterobilayer encapsulated in hexagonal boron nitride. The periodic in-plane …
The electronic structure of a crystalline solid is largely determined by its lattice structure. Recent advances in van der Waals solids, artificial crystals with controlled stacking of two-dimensional (2D) atomic films, have enabled the creation of materials with novel electronic structures. In particular, stacking graphene on hexagonal boron nitride (hBN) introduces moiré superlattice that fundamentally modifies graphene's band structure and gives rise to secondary Dirac points (SDPs). Here we find that the formation of a moiré superlattice in graphene on hBN yields new, unexpected consequenc…
We present a low-energy model describing the reconstruction of the electronic spectrum in twisted bilayers of honeycomb crystals with broken sublattice symmetry. The resulting moiré patterns are classified into two families with different symmetry. In both cases, flat bands appear at relatively large angles, without any magic angle condition. Transitions between them give rise to sharp resonances in the optical absorption spectrum at frequencies well below the gap of the monolayer. Owing to their chiral symmetry, twisted bilayers display circular dichroism, i.e., different absorption of left a…
The relative twist angle in heterostructures of two-dimensional (2D) materials with similar lattice constants result in a dramatic alteration of the electronic properties. Here, we investigate the electrical and magnetotransport properties in bilayer graphene (BLG) encapsulated between two hexagonal boron nitride (hBN) crystals, where the top and bottom hBN are rotationally aligned with bilayer graphene with a twist angle $θ_t\sim 0^{\circ} \text{and}~ θ_b < 1^{\circ}$, respectively. This results in the formation of two moiré superlattices, with the appearance of satellite resistivity peaks at…
While Fabry-Pérot resonances and Moiré superlattices are intensively studied in graphene on hexagonal boron nitride (hBN), the two effects have not been discussed in their coexistence. Here we investigate the FP oscillations in a ballistic pnp-junctions in the presence and absence of a Moiré superlattice. First, we address the effect of the smoothness of the confining potential on the visibility of the FP resonances and carefully map the evolution of the FP cavity size as a function of densities in and outside the cavity in the absence of a superlattice, when the cavity is bound by regular pn-…
