A model of a superlattice consisting of alternating strips of single-layer and bilayer graphene is proposed, whose parameters of the energy spectrum can be controlled by changing the external electric field perpendicular to the surface of the sample. Using the Kronig-Penney model, the dispersion equation is obtained, based on the analysis of which the energy spectrum of a graphene superlattice is studied depending on the ratio of the strip widths of single-layer and bilayer graphene. The results of the analytical solution are compared with the results of modeling by methods of the theory of the density functional. It is shown that the low-energy approximation used to derive the dispersion equation is valid when considering a superlattice with narrow strips of bilayer graphene and wide strips of single-layer graphene.
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.…
Energy band structure of the bilayer graphene superlattices with $δ$-function magnetic barriers and zero average magnetic flux is studied within the four-band continuum model, using the transfer matrix method. The periodic magnetic potential effects on the zero-energy Dirac point of pristine bilayer graphene are exactly analyzed. Magnetic potential is shown also to generate finite-energy Dirac points at the edges of Brillouin zone the positions of which and the related dispersions are determined in the case of symmetric potentials.
We report on transport properties in dual-gated hexagonal boron nitride (hBN)/bilayer-graphene (BLG) superlattices. Here, BLG is nontwisted, i.e., plain. This paper focuses on the charge neutrality point (CNP) for a plain BLG. Under a perpendicular magnetic field, transitions between two insulating phases at the CNP are detected by varying a displacement field with the study on the resistance-temperature characteristics and the magnetoresistance. This work opens avenues for exploring the global phase diagram of the hBN/BLG superlattices beyond the CNP.
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 formulate a low energy effective Hamiltonian to study superlattices in bilayer graphene (BLG) using a minimal model which supports quadratic band touching points. We show that a one dimensional (1D) periodic modulation of the chemical potential or the electric field perpendicular to the layers leads to the generation of zero energy anisotropic massless Dirac fermions and finite energy Dirac points with tunable velocities. The electric field superlattice maps onto a coupled chain model comprised of 'topological' edge modes. 2D superlattice modulations are shown to lead to gaps on the mini-Br…
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…
Motivated by recent experimental findings on the low-energy spectrum of Kekulé-patterned graphene, the optoelectronic signatures of graphene superlattices with a spatial modulation that triples the size of the unit cell and folds the valleys to the center of the Brillouin zone are studied. For superlattices like those visualized in recent experiments, the optoelectronic response reveals multiple species of carriers distinguished by their effective masses or Fermi velocities. Their signatures are similar to those of systems hosting multifold fermions in which different frequency intervals are d…
Twisted bilayer graphene (TwBLG) is one of the simplest van der Waals heterostructures, yet it yields a complex electronic system with intricate interplay between moiré physics and interlayer hybridization effects. We report on electronic transport measurements of high mobility small angle TwBLG devices showing clear evidence for insulating states at the superlattice band edges, with thermal activation gaps several times larger than theoretically predicted. Moreover, Shubnikov-de Haas oscillations and tight binding calculations reveal that the band structure consists of two intersecting Fermi …
Energy band structure of the bilayer graphene superlattices with zero-averaged periodic $δ$-function potentials are studied within the four-band continuum model. Using the transfer matrix method, studies are mainly focused on examining the touching points between adjacent minibands. For the zero-energy touching points the dispersion relation derived shows the Dirac-like double-cone shape with the group velocity which is periodic in the potential strength $P$ with the period of $π$ and becomes anisotropic at relatively large $P$. From the finite-energy touching points we have identified those l…
We review recent work on superlattices in monolayer and bilayer graphene. We highlight the role of the quasiparticle chirality in generating new Dirac fermion modes with tunable anisotropic velocities in one dimensional (1D) superlattices in both monolayer and bilayer graphene. We discuss the structure of the Landau levels and magnetotransport in such superlattices over a wide range of perpendicular (orbital) magnetic fields. In monolayer graphene, we show that an orbital magnetic field can reverse the anisotropy of the transport imposed by the superlattice potential, suggesting possible switc…
Novel SuperLattice Plasmon Mode in a Grating of 2D Electron Strips
Institute of Solid State Physics, RAS（俄罗斯科学院固体物理研究所）；Kotelnikov Institute of Radio-engineering and Electronics of the RAS（俄罗斯科学院科捷利尼科夫无线电电子学研究所）；Institute of Solid State Physics, Vienna University of Technology（维也纳工业大学固体物理研究所）
Stack of correlated insulating states in bilayer graphene kagome superlattice
State Key Laboratory of Quantum Functional Materials, School of Physical Science and Technology, ShanghaiTech University（上海科技大学物理科学与技术学院量子功能材料国家重点实验室）；ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University（上海科技大学拓扑物理实验室）；Department of Physics and Astronomy, University of California, Riverside（加州大学河滨分校物理与天文学系）；Department of Physics, University of Oxford（牛津大学物理系）；School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）
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…
Gapless bilayer graphene (GBG), like monolayer graphene, is a material system with unique properties, such as anti-Klein tunneling and intrinsic Fano resonances. These properties rely on the gapless parabolic dispersion relation and the chiral nature of bilayer graphene electrons. In addition, propagating and evanescent electron states coexist inherently in this material, giving rise to these exotic properties. In this sense, bilayer graphene is unique, since in most material systems in which Fano resonance phenomena are manifested an external source that provides extended states is required. …
Due to low dimensionality, the controlled stacking of the graphene films and their electronic properties are susceptible to environmental changes including strain. The strain-induced modification of the electronic properties such as the emergence and modulation of bandgaps crucially depends on the stacking of the graphene films. However, to date, only the impact of strain on electronic properties of Bernal and AA-stacked bilayer graphene has been extensively investigated in theoretical studies. Exploiting density functional theory and tight-binding calculation, we investigate the impacts of in…
Twistronic van der Waals heterostrutures offer exciting opportunities for engineering optoelectronic properties of nanomaterials. Here, we use multiscale modeling to study trapping of charge carriers and excitons by ferroelectric polarisation and piezoelectric charges by domain structures in twistronic WX$_2$/MoX$_2$ bilayers (X=S,Se). For almost aligned 2H-type bilayers, we find that holes and electrons are trapped in the opposite -- WMo and XX (tungsten over molybdenum {\it versus} overlaying chalcogens) -- corners of the honeycomb domain wall network, swapping their position at a twist angl…
We consider the electronic structure of a slightly twisted graphene bilayer and show the coexistence of van Hove singularities (VHSs) and superlattice Dirac points in a continuum approximation. The graphene-on-graphene moiré pattern gives rise to a periodic electronic potential, which leads to the emergence of the superlattice Dirac points due to the chiral nature of the charge carriers. Owning to the distinguishing real and reciprocal structures, the sublattice exchange even and odd structures of the twisted graphene bilayer (the two types of commensurate structures) result in two different s…
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…
A graphene-based superlattice formed due to the periodic modulation of the band gap has been investigated. Such a modulation is possible in graphene deposited on a strip substrate made of silicon oxide and hexagonal boron nitride. The advantages and some possible problems in the superlattice under consideration are discussed. A model describing such a superlattice is proposed and the dispersion relation between the energy and momentum of carriers has been obtained using the transfer matrix method within this model.
We review the energy spectrum and transport properties of several types of one- dimensional superlattices (SLs) on single-layer and bilayer graphene. In single-layer graphene, for certain SL parameters an electron beam incident on a SL is highly collimated. On the other hand there are extra Dirac points generated for other SL parameters. Using rectangular barriers allows us to find analytic expressions for the location of new Dirac points in the spectrum and for the renormalization of the electron velocities. The influence of these extra Dirac points on the conductivity is investigated. In the…
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…
The quantum Goos-Hänchen shifts of the transmitted electron beam through an AA-stacked bilayer graphene superlattices is investigated. We found that the band structures of graphene superlattices can have more than one Dirac point, their locations do not depend on the number of barriers. It was revealed that any $n$-barrier structure is perfectly transparent at normal incidence around the Dirac points created in the superlattices. We showed that the Goos-Hänchen shifts display sharp peaks inside the transmission gap around two Dirac points ($E= V_B + τ$, $E= V_W + τ$), which are equal to those …
In this review article we discuss the recent progress in studying ballistic transport for charge carriers in graphene through highly inhomogenous magnetic field known as magnetic barrier in combination with gate voltage induced electrostatic potential. Starting with cases for a single or double magnetic barrier we also review the progress in understanding electron transport through the superlattices created out of such electromagnetic potential barriers and discuss the possibility of experimental realization of such systems. The emphasis is particularly on the analogy of such transport with pr…
We investigate the electronic Bloch oscillation in bilayer graphene gradient superlattices using transfer matrix method. By introducing two kinds of gradient potentials of square barriers along electrons propagation direction, we find that Bloch oscillations up to terahertz can occur. Wannier-Stark ladders, as the counterpart of Bloch oscillation, are obtained as a series of equidistant transmission peaks, and the localization of the electronic wave function is also signature of Bloch oscillation. Forthermore, the period of Bloch oscillation decreases linearly with increasing gradient of barri…
High mobility single and few-layer graphene sheets are in many ways attractive as nanoelectronic circuit hosts but lack energy gaps, which are essential to the operation of field-effect transistors. One of the methods used to create gaps in the spectrum of graphene systems is to form long period moire patterns by aligning the graphene and hexagonal boron nitride (h-BN) substrate lattices. Here, we use planar tunneling devices with thin h-BN barriers to obtain direct and accurate tunneling spectroscopy measurements of the energy gaps in single- and bi-layer graphene-h-BN superlattice structures…
The electronic properties of bilayer graphene strongly depend on relative orientation of the two atomic lattices. Whereas Bernal-stacked graphene is most commonly studied, a rotational mismatch between layers opens up a whole new field of rich physics, especially at small interlayer twist. Here we report on magnetotransport measurements on twisted graphene bilayers, prepared by folding of single layers. These reveal a strong dependence on the twist angle, which can be estimated by means of sample geometry. At small rotation, superlattices with a wavelength in the order of 10 nm arise and are o…
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 …
Recently the small-twist-angle (< 2°) bilayer graphene has received extraordinary attentions due to its exciting physical properties. Compared with monolayer graphene, the Brillouin zone folding in twisted bilayer graphene (TBG) leads to the formation of superlattice bandgap and significant modification of density of states. However, these emerging properties have rarely been leveraged for the realization of new optoelectronic devices. Here we demonstrate the strong, gate-tunable photoresponse in mid-infrared wavelength range of 5 to 12 um. A maximum extrinsic photoresponsivity of 26 mA/W has …
The recently observed unconventional ferroelectricity in AB bilayer graphene sandwiched by hexagonal Boron Nitride (hBN) presents a new platform to manipulate correlated phases in multilayered van der Waals heterostructures. We present a low-energy continuum model for AB bilayer graphene encapsulated by the top and bottom layers of either hBN or graphene, with two independent twist angles. For the graphene/hBN heterostructures, we show that twist angle asymmetry leads to a layer polarization of the valence and conduction bands. We also show that an out-of-plane displacement field not only tune…
We show that a sliding motion between the two layers of a \moire superlattice induces an electric current and realizes a two-dimensional version of the topological Thouless pump when the Fermi energy lies in one of the minigaps. Interestingly, a chiral charge pump, namely, a transverse current induced by the sliding motion, is possible in twisted homobilayers. This result is confirmed by a concrete calculation of the adiabatic current in twisted bilayer graphene. Our work reveals an interesting link between mechanical motion and electricity unique to \moire superlattices, and may find applicat…
In two-dimensional (2D) twisted bilayers, the van der Waals (vdW) interlayer interaction introduces atomic-scale reconstruction at interface by locally rotating lattice to form strain-field vortex networks in their moiré superlattice. However, direct imaging the tiny local lattice rotation of the strain-field vortex requires extremely high spatial resolution and is an outstanding challenge in experiment. Here, a topmost small-period graphene moiré pattern is introduced to magnify sub-Angstrom distortions of the lattice and tiny local lattice rotation in underlying twisted bilayer graphene (TBG…
We investigate the phonon mechanism for the orientational pinning of Wigner crystals and stripes in two-dimensional electron layers in GaAs matrices. We find the orientation of bilayer Wigner crystals on the (001), (111), (0-11) and (311) interfaces versus the interlayer distance and determine the regions of parameters, where polydomain structures can emerge. For the stripe states in electron layers situated close to the (001) surface we show that the interference between the piezoelectric and deformation potential interaction may be responsible for the preferable orientation of the stripes al…
Stripe phases, in which the rotational symmetry of charge density is spontaneously broken, occur in many strongly correlated systems with competing interactions. One representative example is the copper-oxide superconductors, where stripe order is thought to be relevant to the mechanism of high-temperature superconductivity. Identifying and studying the stripe phases in conventional strongly correlated systems are, however, challenging due to the complexity and limited tunability of these materials. Here we uncover stripe phases in WSe2/WS2 moiré superlattices with continuously gate-tunable ch…
Twisted bilayer graphene (tBLG) provides us with a large rotational freedom to explore new physics and novel device applications, but many of its basic properties remain unresolved. Here we report the synthesis and systematic Raman study of tBLG. Chemical vapor deposition was used to synthesize hexagon- shaped tBLG with a rotation angle that can be conveniently determined by relative edge misalignment. Superlattice structures are revealed by the observation of two distinctive Raman features: folded optical phonons and enhanced intensity of the 2D-band. Both signatures are strongly correlated w…
Moiré structures in van der Waals heterostructures lead to emergent phenomena including superconductivity in twisted bilayer graphene and optically accessible strongly-correlated electron states in transition metal dichalcogenide heterobilayers. Dual periodicity moiré structures (DPMS) formed in layered structures with more than two layers have been shown to lead to ferromagnetism and multiple secondary Dirac points in TBG. Whilst in principle it is possible to obtain DPMS in bilayers there has not been clear experimental evidence of this yet. In this paper we present signatures of DPMS in a t…
We find clear signatures of spin-dependent negative differential resistance in compound systems comprising a graphene nanoribbon and a set of ferromagnetic insulator strips deposited on top of it. The periodic array of ferromagnetic strips induces a proximity exchange splitting of the electronic states in graphene, resulting in the appearance of a superlattice with a spin-dependent energy spectrum. The electric current through the device can be highly polarized and both the current and its polarization manifest non-monotonic dependence on the bias voltage. The device operates therefore as an E…
We perform a comprehensive first-principles study of the electronic properties of van der Waals (vdW) trilayers via intercalating a two-dimensional (2D) monolayer (ML = BN, MoSe2, WS2, or WSe2) between MoS2 bilayer to form various MoS2/ML/MoS2 sandwich trilayers. We find that the BN monolayer is the most effective sheet to decouple the interlayer vdW coupling of the MoS2 bilayer, and the resulting sandwich trilayer can recover the electronic structures of the MoS2 monolayer, particularly the direct-gap character. Further study of the MoS2/BN superlattices confirms the effectiveness of the BN m…
The electronic structure of the zig-zag bilayer strip is analyzed. The electronic spectra of the bilayer strip is computed. The dependence of the edge state band flatness on the bilayer width is found. The density of states at the Fermi level is analytically computed. It is shown that it has the singularity which depends on the width of the bilayer strip. There is also asymmetry in the density of states below and above the Fermi energy.
Monolayer WTe2 stripes are quantum spin Hall (QSH) insulators. Density functional theory was used for investigating the electronic properties of the stripes and steps in bilayer Td-WTe2. For the stripes oriented along the dimer chains of W atoms (x direction), the hybridization between the two layers suppresses the QSH states. However, the QSH nature can be recovered by forming a step, depending on the atomic structure of the step. Conversely, the stripes and steps along the y direction maintain the QSH states. These findings can expand the application range of the QSH states in WTe2.
We report on transport properties of monolayer graphene with a laterally modulated potential profile, employing striped top gate electrodes with spacings of 100 nm to 200 nm. Tuning of top and back gate voltages gives rise to local charge carrier density disparities, enabling the investigation of transport properties either in the unipolar (nn') or the bipolar (np') regime. In the latter pronounced single- and multibarrier Fabry-Perot (FP) resonances occur. We present measurements of different devices with different numbers of top gate stripes and spacings. The data are highly consistent with …
The formation of a superstructure - with a related Moiré pattern - plays a crucial role in the extraordinary optical and electronic properties of twisted bilayer graphene, including the recently observed unconventional superconductivity. Here we put forward a novel, interdisciplinary approach to determine the Moiré angle in twisted bilayer graphene based on the photonic spin Hall effect. We show that the photonic spin Hall effect exhibits clear fingerprints of the underlying Moiré pattern, and the associated light beam shifts are well beyond current experimental sensitivities in the near-infra…
