Ultra-flat Bands Research Articles

Ultra-flat bands at large twist angles in group-V twisted bilayer materials.

Flat bands in 2D twisted materials are key to the realization of correlation-related exotic phenomena. However, a flat band often was achieved in the large system with a very small twist angle, which enormously increases the computational and experimental complexity. In this work, we proposed group-V twisted bilayer materials, including P, As, and Sb in the β phase with large twist angles. The band structure of twisted bilayer materials up to 2524 atoms has been investigated by a deep learning method DeepH, which significantly reduces the computational time. Our results show that the bandgap and the flat bandwidth of twisted bilayer β-P, β-As, and β-Sb reduce gradually with the decreasing of twist angle, and the ultra-flat band with bandwidth approaching 0eV is achieved. Interestingly, we found that a twist angle of 9.43° is sufficient to achieve the band flatness for β-As comparable to that of twist bilayer graphene at the magic angle of 1.08°. Moreover, we also find that the bandgap reduces with decreasing interlayer distance while the flat band is still preserved, which suggests interlayer distance as an effective routine to tune the bandgap of flat band systems. Our research provides a feasible platform for exploring physical phenomena related to flat bands in twisted layered 2D materials.

First-principles prediction of moiré ultra-flat bands in twisted bilayer nitrogene

Based on first-principles density functional theory, we investigated the electronic structures of twisted bilayer nitrogene at various twist angles. This electron localization by the influence of the moiré superlattice with very small θ leads to flat bands, especially one of valence bands at the Fermi level on atoms in the AA zone whereas for a larger θ no strong localization occurs owning to a large proportion of the AB zone. Also, there is a linear relationship between the band gaps and the torsion angles of the twisted bilayer nitrogene. A smaller twist angle corresponds to a larger band gap, whereas a larger twist angle corresponds to a smaller band gap. This localization should also reduce considerably the mobility of the electrons and thus modify transport properties.

Nonlinear Transport due to Magnetic-Field-Induced Flat Bands in the Nodal-Line Semimetal ZrTe_{5}.

The Dirac material ZrTe_{5} at very low carrier density was recently found to be a nodal-line semimetal, where ultraflat bands are expected to emerge in magnetic fields parallel to the nodal-line plane. Here, we report that in very low carrier-density samples of ZrTe_{5}, when the current and the magnetic field are both along the crystallographic a axis, the current-voltage characteristics presents a pronounced nonlinearity which tends to saturate in the ultra quantum limit. The magnetic-field dependence of the nonlinear coefficient is well explained by the Boltzmann theory for flat-band transport, and we argue that this nonlinear transport is likely due to the combined effect of flat bands and charge puddles; the latter appear due to very low carrier densities.

Network of chiral one-dimensional channels and localized states emerging in a moiré system

Moiré systems provide a highly tunable platform for engineering band structures and exotic correlated phases. Here, we theoretically study a model for a single layer of graphene subject to a smooth moiré electrostatic potential, induced by an insulating substrate layer. For sufficiently large moiré unit cells, we find that ultra-flat bands coexist with a triangular network of chiral one-dimensional (1D) channels. These channels mediate an effective interaction between localized modes with spin-, orbital- and valley degrees of freedom emerging from the flat bands. The form of the interaction reflects the chirality and 1D nature of the network. We study this interacting model within an SU(4) mean-field theory, semi-classical Monte-Carlo simulations, and an SU(4) spin-wave theory, focusing on commensurate order stabilized by local two-site and chiral three-site interactions. By tuning a gate voltage, one can trigger a non-coplanar phase characterized by a peculiar coexistence of three different types of order: ferromagnetic spin order in one valley, non-coplanar chiral spin order in the other valley, and 120∘ order in the remaining spin and valley-mixed degrees of freedom. Quantum and classical fluctuations have qualitatively different effects on the observed phases and can, for example, create a finite spin-chirality purely via fluctuation effects.

Emergence of flat bands in twisted bilayer C3N induced by simple localization and destructive interference

Flat bands in twisted layered two-dimensional materials are the key to realizing the exotic phenomena related to electron correlation. The origin of the flat bands in various twisted two-dimensional materials remains elusive. In this work, we study the coexistence of flat bands driven by localization and interference in the twisted bilayer ${\mathrm{C}}_{3}\mathrm{N}$. One ultraflat band is induced by destructive interference in the kagome arrangement of localized states. Meanwhile, other flat bands induced by simple localization exist. The effective kagome lattice with parameters tuned by twist angle is deduced. The flat band induced by interference is more robust than that induced by simple localization under vertical strain. The kagome bands formed at the bottom of the conduction band provide a platform for studying the kagome Hamiltonian in twisted two-dimensional materials.

Evidence of high-temperature exciton condensation in a two-dimensional semimetal

Electrons and holes can spontaneously form excitons and condense in a semimetal or semiconductor, as predicted decades ago. This type of Bose condensation can happen at much higher temperatures in comparison with dilute atomic gases. Two-dimensional (2D) materials with reduced Coulomb screening around the Fermi level are promising for realizing such a system. Here we report a change in the band structure accompanied by a phase transition at about 180 K in single-layer ZrTe2 based on angle-resolved photoemission spectroscopy (ARPES) measurements. Below the transition temperature, gap opening and development of an ultra-flat band top around the zone center are observed. This gap and the phase transition are rapidly suppressed with extra carrier densities introduced by adding more layers or dopants on the surface. The results suggest the formation of an excitonic insulating ground state in single-layer ZrTe2, and the findings are rationalized by first-principles calculations and a self-consistent mean-field theory. Our study provides evidence for exciton condensation in a 2D semimetal and demonstrates strong dimensionality effects on the formation of intrinsic bound electron–hole pairs in solids.

Moiré potential renormalization and ultra-flat bands induced by quasiparticle-plasmon coupling

Moiré potential profile can form flat electronic bands and manifest correlated states of electrons, where carrier doping is essential for observing those correlations. In this work, we uncover a hidden but remarkable many-electron effect: doped carriers form a two-dimensional plasmon and strongly couple with quasiparticles to renormalize moiré potential and realize ultra-flat bands. Using many-body perturbation theory, we demonstrate this effect in twisted MoS2/WS2 heterobilayer. The moiré potential is significantly enhanced upon carrier doping, and the bandwidth is reduced by order of magnitude, leading to drastic quenching of electronic kinetic energy and stronger correlation. We further predict that the competition between correlated mechanisms can be effectively controlled via doping, giving hope to a quantum transition between Mott and charge-transfer insulating states. Our work reveals that the potential renormalization effect of doping is much more significant in determining and controlling many-electron electronic correlations than sole filling-factor tuning in semiconducting moiré crystals.

Strong gate-tunability of flat bands in bilayer graphene due to moiré encapsulation between hBN monolayers.

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 Dirac point, and may thereby strongly compress the first moiré mini-band in width against the moiré-induced gap at the secondary Dirac point. We reveal that in hBN-encapsulated bilayer graphene this novel mechanism can lead to isolated bands flatter than 10 meV under moderate gating, hence presenting a convenient pathway towards electronically-controlled strongly-correlated states on demand.

Quantum cascade of correlated phases in trigonally warped bilayer graphene.

Divergent density of states offersan opportunity to explore a wide variety of correlated electron physics. In the thinnest limit, this has been predicted and verified in the ultraflat bands of magic-angle twisted bilayer graphene1-5, the band touching points of few-layer rhombohedral graphite6-8 and the lightly doped rhombohedral trilayer graphene9-11. The simpler and seemingly better understood Bernal bilayer graphene is also susceptible to orbital magnetism at charge neutrality7 leading to layer antiferromagnetic states12 or quantum anomalous Hallstates13. Here we report the observation of a cascade of correlated phases in the vicinity of electric-field-controlled Lifshitz transitions14,15 and van Hove singularities16 in Bernal bilayer graphene. We provide evidence for the observation of Stoner ferromagnets in the form of half and quarter metals10,11. Furthermore, we identify signatures consistent with a topologically non-trivial Wigner-Hall crystal17 at zero magnetic field and its transition to a trivial Wigner crystal, as well as two correlated metals whose behaviour deviates from that of standard Fermi liquids. Our results in this reproducible, tunable, simple systemopen up new horizons for studying strongly correlated electrons.

Designing Ultra-flat Bands in Twisted Bilayer Materials at Large Twist Angles: Theory and Application to Two-Dimensional Indium Selenide.

Intertwisted bilayers of two-dimensional (2D) materials can host low-energy flat bands, which offer opportunity to investigate many intriguing physics associated with strong electron correlations. In the existing systems, ultra-flat bands only emerge at very small twist angles less than a few degrees, which poses a challenge for experimental studies and practical applications. Here, we propose a new design principle to achieve low-energy ultra-flat bands with increased twist angles. The key condition is to have a 2D semiconducting material with a large energy difference of band edges controlled by stacking. We show that the interlayer interaction leads to defect-like states under twisting, which forms a flat band in the semiconducting band gap with dispersion strongly suppressed by the large energy barriers in the moiré superlattice even for large twist angles. We explicitly demonstrate our idea in bilayer α-In2Se3 and bilayer InSe. For bilayer α-In2Se3, we show that a twist angle of ∼13.2° is sufficient to achieve the band flatness comparable to that of twist bilayer graphene at the magic angle ∼1.1°. In addition, the appearance of ultra-flat bands here is not sensitive to the twist angle as in bilayer graphene, and it can be further controlled by external gate fields. Our finding provides a new route to achieve ultra-flat bands other than reducing the twist angles and paves the way toward engineering such flat bands in a large family of 2D materials.

Moiré flat bands in twisted 2D hexagonal vdW materials

Moiré superlattices (MSL) in twisted bilayer graphene (TBG) and its derived structures can host exotic correlated quantum phenomena because the narrow moiré flat minibands in those systems effectively enhance the electron-electron interaction. Correlated phenomena are also observed in 2H-transitional metal dichalcogenides MSL. However, the number of moiré systems that have been explored in experiments are still very limited. Here we theoretically investigate a series of two-dimensional (2D) twisted bilayer hexagonal materials beyond TBG at fixed angles of 7.34∘ and 67.34∘ with 22 2D van der Waals layered materials that are commonly studied in experiments. First-principles calculations are employed to systemically study the moiré minibands in these systems. We find that flat bands with narrow bandwidth generally exist in these systems. Some of the systems such as twisted bilayer In2Se3, InSe, GaSe, GaS and PtS2 even host ultra-flat bands with bandwidth less than 20 meV even for such large angles, which makes them especially appealing for further experimental investigations. We further analysis the characters of moiré flat bands and provide guidance for further exploration of 2D MSL that could host strong electron correlations.

Lattice reconstruction induced multiple ultra-flat bands in twisted bilayer WSe2

Moiré superlattices in van der Waals heterostructures provide a tunable platform to study emergent properties that are absent in the natural crystal form. Twisted bilayer transition metal dichalcogenides (TB-TMDs) can host moiré flat bands over a wide range of twist angles. For twist angle close to 60°, it was predicted that TB-TMDs undergo a lattice reconstruction which causes the formation of ultra-flat bands. Here, by using scanning tunneling microscopy and spectroscopy, we show the emergence of multiple ultra-flat bands in twisted bilayer WSe2 when the twist angle is within 3° of 60°. The ultra-flat bands are manifested as narrow tunneling conductance peaks with estimated bandwidth less than 10 meV, which is only a fraction of the estimated on-site Coulomb repulsion energy. The number of these ultra-flat bands and spatial distribution of the wavefunctions match well with the theoretical predictions, strongly evidencing that the observed ultra-flat bands are induced by lattice reconstruction. Our work provides a foundation for further study of the exotic correlated phases in TB-TMDs.

Extremely flat band in antiferroelectric bilayer α-In2Se3 with large twist-angle

The moiré-pattern with slightly interlayer-twisted bilayer two-dimensional (2D) materials has recently been receiving substantial attention. One of the major characters for these intriguing structures is the appearance of low-energy ultra-flat bands and thus a package of new physics associated with strong electron correlation emerges. However, such new physics may become vague unless the twist-angle θ is sufficiently small such as θ ∼ 1°, making practical applications and control-flexibility hard to handle. In this work, we explore the possible flat-band moiré physics in recently concerned 2D bilayer α-In2Se3 which is antiferroelectric with sublayer out-of-plane (OP) polarizations, addressing the potential role of polarization-bound charges in modulation of electron–electron correlation and interlayer hybridization, based on the extensive first-principles calculations. On one hand, it is found that the low energy band becomes extremely flat in the bilayer α-In2Se3 moiré-pattern with relatively large twist-angle, e.g. θ = 13.17°, which is more easily accessible experimentally. On the other hand, the impact of the sublayer OP polarizations on the band structure is asymmetric, and the flattening effect is much more remarkable for the end-to-end polarization alignment but weaker for the head-to-head alignment. This work thus opens a broad roadmap for technological access to artificial fabrication of novel moiré-patterned 2D materials by means of low-dimensional ferroelectricity.

Photonic analog of bilayer graphene

Drawing inspiration from bilayer graphene, this paper introduces its photonic analog comprising two stacked graphene-like photonic crystals, that are coupled in the near-field through spoof surface plasmons. Beyond the twist degree of freedom that can radically alter the band structure of the bilayer photonic graphene, the photonic dispersion can be also tailored via the interlayer coupling which exhibits an exponential dependence on the distance between the two photonic crystals. We theoretically, numerically, and experimentally characterize the band structures of AA- and AB-stacked bilayer photonic graphene, as well as for twisted bilayer photonic graphene with even and odd sublattice exchange symmetries. Furthermore, we numerically predict the existence of magic angles in bilayer photonic graphene, which are associated with ultra-flat bands resulted from interlayer hybridization. Finally, we demonstrate that the bilayer photonic graphene at a particular twist angle satisfying even sublattice exchange symmetry is a high-order photonic topological insulator. The proposed bilayer photonic graphene could constitute a useful platform for identifying new quantum materials and inspiring next-generation photonic devices with new degrees of freedom and emerging functionality.

Lattice relaxation, mirror symmetry and magnetic field effects on ultraflat bands in twisted trilayer graphene

Twisted graphene multilayers exhibit strongly correlated insulating states and superconductivity due to the presence of ultraflat bands near the charge neutral point. In this paper, the response of ultraflat bands to lattice relaxation and a magnetic field in twisted trilayer graphene (tTLG) with different stacking arrangements is investigated by using a full tight-binding model. We show that lattice relaxations are indispensable for understanding the electronic properties of tTLG, in particular, of tTLG in the presence of mirror symmetry. Lattice relaxations renormalize the quasiparticle spectrum near the Fermi energy and change the localization of higher energy flat bands. Furthermore, different from the twisted bilayer graphene, the Hofstadter butterfly spectrum can be realized at laboratory accessible strengths of magnetic field. Our work verifies tTLG as a more tunable platform than the twisted bilayer graphene in strongly correlated phenomena.

Magic-angle bilayer phononic graphene

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 angles (> 3 degrees) inaccessible in classical twisted bilayer graphene. This study, not only establishes the acoustical analog of twisted (magic-angle) bilayer graphene, providing a testbed more easily accessible to probe the interaction and misalignment between stacked 2D materials, but also points out the direction to a new phononic crystal design paradigm that could benefit applications such as enhanced acoustic emission and sensing.

Charge density wave and spin 1/2 insulating state in single layer 1T-NbS2

In bulk samples and few layer flakes, the transition metal dichalcogenides NbS2 and NbSe2 assume the H polytype structure with trigonal prismatic coordination of the Nb atom. Recently, however, single and few layers of 1T-NbSe2 with octahedral coordination around the transition metal ion were synthesized. Motivated by these experiments and by using first-principles calculations, we investigate the structural, electronic and dynamical properties of single layer 1T-NbS2. We find that single-layer 1T-NbS2 undergoes a star-of-David charge density wave. Within the generalized gradient approximation, the weak interaction between the stars leads to an ultraflat band that results to be located at the Fermi level and isolated from all other bands. The spin-polarized generalized gradient approximation stabilizes a total spin 1/2 magnetic state with a magnetic moment localized on the central Nb in the star and the opening of a 0.15 eV band gap. In the GGA + U framework, the magnetic moment on the central Nb is enhanced to and a larger gap occurs. Most important, this approximation gives a small energy difference between the 1T and 1H polytypes (only +0.5 mRy/Nb), suggesting that the 1T-polytype can be synthesized in a similar way as done for single layer 1T-NbSe2. Finally we compute first and second nearest neighbors magnetic inter-star exchange interactions finding J1 = 9.5 K and J2 = 0.4 K ferromagnetic coupling constants.

Low dispersion slow light waveguide with high coupling efficiency

We propose theoretically that a slow light waveguide constructed by placing a ridge waveguide between two photonic crystals holds an ultraflat band with a quasi-constant group velocity and a very low group velocity dispersion. Meanwhile, the single mode feature is the best choice to facilitate coupling light into the slow light waveguide and avoid losses. Thus, an efficient and compact coupler with a length of 5a for the slow light waveguide is designed to achieve a high coupling efficiency of more than 80% corresponding to the group velocity of 0.0166c in the wide bandwidth.

Demonstration of Ultraslow Modes in Asymmetric Line-Defect Photonic Crystal Waveguides

We propose an asymmetric photonic crystal (PC) waveguide, formed by moving the PC cladding on one side of the line defect along the waveguide core. By adjusting the structural parameters, we flatten the slow-mode dispersion curve, exclude the extreme points from the flat band, and avoid the unacceptably large dispersion induced by extreme points. We obtain an ultra-flat band with an inflection point corresponding to a slow-light mode with reduced distortion. We utilize the finite-difference time-domain method to verify the inflection-point ultraslow wave.