Recent studies on the interplay between band topology and layer degree of freedom provide an effective way to realize exotic topological phases. Here we systematically study the $C_6$- and $C_3$-symmetric higher-order topological phases in bilayer spinless tight-binding lattice models. For concreteness, we consider bilayer phononic crystals as the realizations of these models. We find that for mirror-symmetric-stacking bilayer lattices, the interlayer couplings control the emergence and disappearance of the topological bound states in the continuum where we consider the corner states as possible bound states in the bulk continuum. For the bilayer phononic crystals formed by two different lattices with identical symmetry, the band topology is determined by both the band topology of each layer as well as their mutual couplings. The bilayer phononic crystals experience a phase transition from nontrivial to trivial band topology when the interlayer couplings are gradually increased. Our work unveils the rich physics and topological phases emerging in bilayer lattice systems that can be used to engineer interesting phenomena and induce emergent topological phases.
Here, we propose quantum spin Hall-like phononic states in van der Waals bilayer systems with antiferromagnetic ordering. Antiferromagnetic ordering in bilayer systems with small interlayer interaction makes the total Chern number zero, where the Chern numbers for each layer are opposite to each other. In bilayer CrI3 where antiferromagnetic ordering has been experimentally demonstrated, we show that the quantum spin Hall-like states appear with spatially separated chiral edge modes.
Thanks to the recent discovery on the magic-angle bilayer graphene, twistronics is quickly becom11 ing a burgeoning field in condensed matter physics. This letter expands the realm of twistronics to acoustics by introducing twisted bilayer phononic graphene, which remarkably also harbors the magic angle, evidenced by the associated ultra-flat bands. Beyond mimicking quantum mechanical behaviors of twisted bilayer graphene, we show that their acoustic counterpart offers a considerably more straightforward and robust way to alter the interlayer hopping strength, enabling us to unlock magic angle…
We study theoretically AB-stacked honeycomb bilayers driven by light in resonance with an infrared phonon within a tight-binding description. We characterize the phonon properties of honeycomb bilayers with group theory and construct an electronic time-dependent tight-binding model for the system following photo-excitation in resonance with an infrared phonon. We adopt an "atomically adiabatic" approximation, introduced by Mohantya and Heller PNAS 116, 18316 (2019) to describe classically vibrating nuclei, but obtain the Floquet quasienergy spectrum associated with the time-dependent model exa…
In order to achieve low Noise-Equivalent-Powers (NEP < 1 aW Hz$^{-1/2}$) Transition Edge Sensors (TES) require high levels of thermal isolation between the superconducting bilayer and the heat bath. We propose that short micro-machined acoustic interferometers can be used for low-noise thermal isolation, avoiding many of the difficulties inherent in conventional, long- legged TES designs. In this paper, we present a detailed elastic wave model of interferometric phononic legs, and demonstrate the successful fabrication of TESs that are thermally isolated by interferometric structures.
We analytically predict and numerically demonstrate the existence of a family of bound modes in the continuum (BICs) in bi-layered spring mass chains. A coupled array of such chains is then used to illustrate transversely bound waves propagating along a channel in a lattice. We start by considering the compact region formed by coupling two spring mass chains with defects and predict bound modes arising due to reflection symmetries in this region. Dispersion analysis of a waveguide consisting of an array of appropriately coupled bi-layered chains reveals the presence of a branch having bound mo…
Topological phases, including the conventional first-order and higher-order topological insulators and semimetals, have emerged as a thriving topic in the fields of condensed-matter physics and material science. Usually, a topological insulator is characterized by a fixed order topological invariant and exhibits associated bulk-boundary correspondence. Here, we realize a new type of topological insulator in a bilayer phononic crystal, which hosts simultaneously the first-order and second-order topologies, referred here as the hybrid-order topological insulator. The one-dimensional gapless heli…
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…
Anderson impurity problem is considered for a graphene bilayer subject to a gap-opening bias. In-gap localized states are produced even when the impurity level overlaps with the continuum of band electrons. The effect depends strongly on the polarity of the applied bias as long as hybridization with the impurity occurs within a single layer. For an impurity level inside the conduction band a positive bias creates the new localized in-gap state. A negative bias does not produce the same result and leads to a simple broadening of the impurity level. The implications for transport are discussed i…
The electric structure of twisted bilayer GeSe, which shows a rectangular moiré pattern, is analyzed using a $\bm{k}\cdot\bm{p}$ type effective continuum model. The effective model is constructed on the basis of the the local approximation method, where the local lattice structure of a twisted bilayer system is approximated by its untwisted bilayer with parallel displacement, and the required parameters are fixed with the help of the first-principles method. By inspecting the twist angle dependence of the physical properties, we reveal a relation between the effective potential under moiré pat…
We introduce a complete physical model for the single-particle electronic structure of twisted bilayer graphene (tBLG), which incorporates the crucial role of lattice relaxation. Our model, based on $k \cdot p$ perturbation theory, combines the accuracy of DFT calculations through effective tight-binding Hamiltonians with the computational efficiency and complete control of the twist angle offered by continuum models. The inclusion of relaxation significantly changes the bandstructure at the first magic-angle twist corresponding to flat bands near the Fermi level (the "low-energy" states), and…
Multi Moire Networks in Engineered Lateral Hetero-Bilayers: Programmable Phononic Reconfiguration and Second Harmonic Generation
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）；Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）；Aarhus University（奥胡斯大学）；Indian Institute of Technology Goa（印度理工学院果阿分校）；Indian Institute of Science Education and Research Tirupati（蒂鲁帕蒂科学教育研究所）；KIIT Deemed to be University（KIIT认定大学）
Topological states of quantum matter exhibit unique disorder-immune surface states protected by underlying nontrivial topological invariants of the bulk. Such immunity from backscattering makes topological surface or edge states ideal carriers for both classical and quantum information. So far, topological matters have been explored only in the realms of electronics and photonics, with limited range of bulk properties and largely immutable materials. These constraints thus impose severe performance trade-offs in experimentally realizable topologically ordered states. In contrast, phononic meta…
We present a linear response calculation for twisted bilayer graphene. The calculation is performed for both the continuum and tight-binding models, with the aim of assessing the validity of the former. All qualitatively important features previously reported by us [T. Stauber et al. Phys. Rev. Lett. 120, 046801 (2018)] for the Drude matrix in the continuum model are also present in the tight-binding calculation, with increasing quantitative agreement for decreasing twist angle. These features include the chiral longitudinal magnetic moment associated with plasmonic modes, and the anomalous co…
The introduction of 'twist' or relative rotation between two atomically thin van der Waals (vdW) membranes gives rise to periodic Moire potential, leading to a substantial altercation of the band structure of the planar assembly. While most of the recent experiments primarily focus on the electronic-band hybridization by probing in-plane transport properties, here we report out-of-plane thermoelectric measurements across the van der Waals gap in twisted bilayer graphene (tBLG), which exhibits an interplay of twist-dependent inter-layer electronic and phononic hybridization. We show that at a l…
We present a simple derivation of a continuum Hamiltonian for bilayer graphene with an arbitrary smooth lattice deformation -- technically in a fashion parametrized by displacement fields with small gradients. We show that this subsumes the continuum model of Bistritzer and Macdonald for twisted bilayer grapheneas well as many generalizations and extensions of it. The derivation is carried out entirely in real space.
We derive an {\em ab initio} $π$-band tight-binding model for $AB$ stacked bilayer graphene based on maximally localized Wannier wave functions (MLWFs) centered on the carbon sites, finding that both intralayer and interlayer hopping is longer in range than assumed in commonly used phenomenological tight-binding models. Starting from this full tight-binding model, we derive two effective models that are intended to provide a convenient starting point for theories of $π$-band electronic properties by achieving accuracy over the full width of the $π$-bands, and especially at the Dirac points, in…
The Continuum Model for Uniaxially Strained Bilayer Graphene Moiré Systems
Hong Kong University of Science and Technology（香港科技大学）；Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；Southern University of Science and Technology（南方科技大学）
We provide a systematic real space derivation of the continuum Hamiltonian for a graphene bilayer starting from a microscopic lattice theory, allowing for an arbitrary inhomogeneous smooth lattice deformation, including a twist. Two different microscopic models are analyzed: first, a Slater-Koster like model and second, ab-initio derived model. We envision that our effective Hamiltonian can be used in conjunction with an experimentally determined atomic lattice deformation in twisted bilayer graphene in a specific device to predict and compare the electronic spectra with scanning tunneling spe…
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…
Twisted bilayer systems with discrete magic angles, such as twisted bilayer graphene featuring moiré superlattices, provide a versatile platform for exploring novel physical properties. Here, we discover a class of superflat bands in general twisted bilayer systems beyond the low-energy physics of magic-angle twisted counterparts. By considering continuous lattice dislocation, we obtain intrinsic localized states, which are spectrally isolated at lowest and highest energies and spatially centered around the AA stacked region, governed by the macroscopic effective energy potential well. Such lo…
Using the method developed in the companion paper, we construct the effective continuum theories for two different microscopic tight binding models of the twisted bilayer graphene at the twist angle of $1.05^\circ$, one Slater-Koster based and the other ab-initio Wannier based. The energy spectra obtained from the continuum theory -- either for rigid twist or including lattice relaxation -- are found to be in nearly perfect agreement with the spectra from the tight binding models when the gradient expansion is carried out to second order, demonstrating the validity of the method. We also analy…
Bilayer graphene is a recently isolated and intriguing class of many-body systems with massive chiral quasiparticles. We present theoretical results for the electronic compressibility of bilayer graphene that are based on a four-band continuum band structure model combined with a random phase approximation treatment of electronic correlations. We find that the compressibility is strongly suppressed by electron-electron interactions at low carrier densities. Correlations do not lead to any qualitative new features, but are crucially important for a quantitative understanding of this fundamental…
The continuum model of the twisted graphene bilayer (Phys. Rev. Lett. 99, 256802, 2007) is extended to include all types of commensurate structures. The essential ingredient of the model, the Fourier components of the spatially modulated hopping amplitudes, can be calculated analytically, for any type of commensurate structures in the low twist angle limit. We show that the Fourier components that could give rise to a gap in the SE-even structures discussed by Mele (Phys. Rev. B 81, 161405 2010) vanish linearly with angle, whereas the amplitudes that saturate to finite values, as $θ\to0$, ensu…
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 demonstrate the existence of localized electron and hole states in a ring-shaped potential kink in biased bilayer graphene. Within the continuum description, we show that for sharp potential steps the Dirac equation describing carrier states close to the K (or K') point of the first Brillouin zone can be solved analytically for a circular kink/anti-kink dot. The solutions exhibit interfacial states which exhibit Aharonov-Bohm oscillations as functions of the height of the potential step and/or the radius of the ring.
We show that the insulating states of magic-angle twisted bilayer graphene support a series of collective modes corresponding to local particle-hole excitations on triangular lattice sites. Our theory is based on a continuum model of the magic angle flat bands. When the system is insulating at moiré band filling $ν=-3$, our calculations show that the ground state supports seven low-energy modes that lie well below the charge gap throughout the moiré Brillouin zone, one of which couples strongly to THz photons. The low-energy collective modes are faithfully described by a model with a local $SU…
Bound states in the continuum (BICs) in photonic crystals represent the unique solutions of wave equations possessing an infinite quality-factor. We design a type of bilayer photonic crystal and study the influence of symmetry and coupling between TE and TM polarizations on BICs. The BIC modes possess $C_{3v}$ symmetry in the x-y plane while the mirror-flip symmetry in the z-direction is broken, and they provide selective coupling into different layers by varying frequency. The enhanced TE-TM coupling due to broken mirror-flip symmetry in the z-direction gives rise to high-Q factor BIC states …
We evaluate the electronic transmission and conductance in bilayer graphene through a finite number of potential barriers. Further, we evaluate the dispersion relation in a bilayer graphene superlattice with a periodic potential applied to both layers. As a model we use the tight-binding Hamiltonian in the continuum approxi- mation. For zero bias the dispersion relation shows a finite gap for carriers with zero momentum in the direction parallel to the barriers. This is in contrast to single-layer graphene where no such gap was found. A gap also appears for a finite bias. Numerical results for…
We combine state-of-the-art large-scale first principles calculations with a low-energy continuum model to describe the nearly flat bands of twisted bilayer graphene at the first magic angle $θ=1.08^\circ$. We show that the energy width of the flat band manifold, as well as the energy gap separating it from the valence and conduction bands, can be obtained only if the out-of-plane relaxations are fully taken into account. The results agree both qualitatively and quantitatively with recent experimental outcomes.
We study the exciton condensate (EC) in a bilayer two-dimension-electron-gas (2DEG) adjacent to a type-II superconductor thin film with an array of pinned vortex lattices. By applying continuum low energy theory and carrying numerical simulations of lattice model within mean-field approximation, we find that if the order parameter of EC has a vortex profile, there are exact zero modes and associated \emph{rational} fractional charge for zero pseudospin potential ($μ$) and average chemical potential ($h$): $μ$=0 and $h$=0; while for $μ\mathtt{\neq}0$ and $h$=0, intervalley mixing splits the zer…
The Bistritzer-MacDonald continuum model (BM model) describes the low-energy moiré bands for twisted bilayer graphene (TBG) at small twist angles. We derive a generalized continuum model for TBG near any commensurate twist angle, which is characterized by complex interlayer hoppings at commensurate $AA$ stackings (rather than the real hoppings in the BM model), a real interlayer hopping at commensurate $AB/BA$ stackings, and a global energy shift. The complex phases of the $AA$ stacking hoppings and the twist angle together define a single angle parameter $ϕ_0$. We compute the model parameters…
Electronic states at domain walls in bilayer graphene are studied by analyzing their four and two band continuum models, by performing numerical calculations on the lattice, and by using quantum geometric arguments. The continuum theories explain the distinct electronic properties of boundary modes localized near domain walls formed by interlayer electric field reversal, by interlayer stacking reversal, and by simultaneous reversal of both quantities. Boundary mode properties are related to topological transitions and gap closures which occur in the bulk Hamiltonian parameter space. The import…
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 …
Emergent quantum phases driven by electronic interactions can manifest in materials with narrowly dispersing, i.e. "flat", energy bands. Recently, flat bands have been realized in a variety of graphene-based heterostructures using the tuning parameters of twist angle, layer stacking and pressure, and resulting in correlated insulator and superconducting states. Here we report the experimental observation of similar correlated phenomena in twisted bilayer tungsten diselenide (tWSe2), a semiconducting transition metal dichalcogenide (TMD). Unlike twisted bilayer graphene where the flat band appe…
We provide a method for the generation of effective continuum Hamiltonians that goes beyond the well known k.p method in being equally effective in both high, and low (or no) symmetry situations. Our approach is based on a surprising exact map of the two-centre tight-binding method onto a compact continuum Hamiltonian, with a precise condition given for the hermiticity of the latter object. We apply this method to a broad range of low dimensional systems of both high and low symmetry: graphene, graphdiyne, γ-graphyne, 6,6,12-graphyne, twist bilayer graphene, and partial dislocation networks in…
When two sheets of graphene stack in a twisted bilayer graphene (tBLG) configuration, the resulting constrained overlap between interplanar 2p orbitals produce angle-tunable electronic absorption resonances. Using a novel combination of multiphoton transient absorption (TA) microscopy and TEM, we resolve the resonant electronic structure, and ensuing electronic relaxation inside single tBLG domains. Strikingly, we find that the transient electronic population in resonantly excited tBLG domains is enhanced many fold, forming a major electronic relaxation bottleneck. 2-photon TA microscopy shows…
We describe the gated bilayer graphene system when it is subjected to intense terahertz frequency electromagnetic radiation. We examine the electron band structure and density of states via exact diagonalization methods within Floquet theory. We find that dynamical states are induced which lead to modification of the band structure. We first examine the situation where there is no external magnetic field. In the unbiased case, dynamical gaps appear in the spectrum which manifest as dips in the density of states. For finite interlayer bias (where a static gap is present in the band structure of…
We generalize the continuum model for Moiré structures made from twisted graphene layers, in order to include lattice relaxation and the formation of channels at very small (marginal) twist angles. We show that a precise description of the electronic structure at such small angles can be achieved by i) calculating first the relaxed atomic structure, ii) projecting the interlayer electronic hopping parameters using a suitable basis of Bloch states, and iii) increasing the number of harmonics in the continuum approximation to interlayer hopping. The results show a complex structure of quasi one …
We present a theoretical study of the effects of heterostrain and lattice relaxation on the optical conductivity of twisted bilayer graphene near the magic angle, based on the band structures obtained from a continuum model. We find that heterostrain, lattice relaxation and their combination give rise to very distinctive spectroscopic features in the optical conductivity, which can be used to probe and distinguish these effects. From the spectrum at various Fermi energies, important features in the strain- and relaxation-modified band structure such as the bandgap, bandwidth and van Hove singu…
We study the effect of twisting on bilayer graphene. The effect of lattice relaxation is included; we look at the electronic structure, piezo-electric charges and spontaneous polarisation. We show that the electronic structure without lattice relaxation shows a set of extremely flat in-gap states similar to Landau-levels, where the spacing scales with twist angle. With lattice relaxation we still have flat bands, but now the spectrum becomes independent of twist angle for sufficiently small angles. We describe in detail the nature of the bands, and study appropriate continuum models, at the sa…
We demonstrate that AA-stacked bilayer graphene (AA-BLG) encapsulated by dielectric materials can possess an energy gap due to the induced mass term. Using the four-band continuum model, we evaluate transmission and reflection probabilities along with the respective conductance. Considering interlayer mass-term difference opens a gap in the energy spectrum and also couples the two Dirac cones. This cone coupling induces an inter-cone transport that is asymmetric with respect to the normal incidence in the presence of asymmetric mass-term. The energy spectrum of the gapped AA-BLG exhibits elect…
AA-stacked bilayer graphene supports Fermi circles in its bonding and antibonding bands which coincide exactly, leading to symmetry-breaking in the presence of electron-electron interactions. We analyze a continuum model of this system in the Hartree-Fock approximation, using a self-consistently screened interaction that accounts for the gap in the spectrum in the broken symmetry state. The order parameter in the groundstate is shown to be of the Ising type, involving transfer of charge between the layers in opposite directions for different sublattices. We analyze the Ising phase transition f…
We investigate the chirality of phonon modes in twisted bilayer WSe2 and demonstrate distinct chiral behavior of the $K/K^\prime$ valley phonons for twist angles close to $0^{\circ}$ and close to $60^{\circ}$. In particular, multiple chiral non-degenerate $K/K^\prime$ valley phonons are found for twist angles near $60^{\circ}$ whereas no non-degenerate chiral modes are found for twist angles close to $0^\circ$. Moreover, we discover two sets of emergent chiral valley modes that originate from an inversion symmetry breaking at the moiré scale and find similar modes in moiré patterns of strain-e…
Twistronics is currently one of the most active research fields in condensed matter physics, following the discovery of correlated insulating and superconducting phases in twisted bilayer graphene (tBLG). Here, we present a magnonic analogue of tBLG. We study magnons in twisted ferromagnetic bilayers (tFBL), including exchange and Dzyaloshinskii-Moriya interactions (DMI). For negligible DMI, tFBL presents discrete magnon magic angles and flat moiré minibands analogous to tBLG. The DMI, however, changes the picture and renders the system much more exotic. The DMI in tFBL induces a rich topologi…
By comparing with recently available experimental data from several groups, we critically discuss the manifestation of continuum many body interaction effects in twisted bilayer graphene (tBLG) with small twist angles and low carrier densities, which arise naturally within the Dirac cone approximation for the non-interacting band structure. We provide two specific examples of such continuum many body theories: one involving electron-phonon interaction and one involving electron-electron interaction. In both cases, the experimental findings are only partially quantitatively consistent with rath…
Twisted bilayer graphene (TBG) has been experimentally observed to exhibit almost flat bands when the twisting occurs at certain magic angles. In this letter, we report new results on the continuum model of twisted bilayer graphene and its electronic band structure. Under we show that in the approximation of vanishing AA-coupling, the magic angles (at which there exist entirely flat bands) are given as the eigenvalues of a non-hermitian operator, and that all bands start squeezing exponentially fast as the angle $θ$ tends to $0$. In particular, as the interaction potential changes, the dynamic…
