Bands Of Twisted Bilayer Graphene Research Articles

Engineering Flat Bands in Twisted-Bilayer Graphene away from the Magic Angle with Chiral Optical Cavities.

Twisted bilayer graphene (TBG) is a recently discovered two-dimensional superlattice structure which exhibits strongly correlated quantum many-body physics, including strange metallic behavior and unconventional superconductivity. Most of TBG exotic properties are connected to the emergence of a pair of isolated and topological flat electronic bands at the so-called magic angle, θ≈1.05°, which are nevertheless very fragile. In this work, we show that, by employing chiral optical cavities, the topological flat bands can be stabilized away from the magic angle in an interval of approximately 0.8°<θ<1.3°. As highlighted by a simplified theoretical model, time reversal symmetry breaking (TRSB), induced by the chiral nature of the cavity, plays a fundamental role in flattening the isolated bands and gapping out the rest of the spectrum. Additionally, TRSB suppresses the Berry curvature and induces a topological phase transition, with a gap closing at the Γ point, towards a band structure with two isolated flat bands with Chern number equal to 0. The efficiency of the cavity is discussed as a function of the twisting angle, the light-matter coupling and the optical cavity characteristic frequency. Our results demonstrate the possibility of engineering flat bands in TBG using optical devices, extending the onset of strongly correlated topological electronic phases in moiré superlattices to a wider range in the twisting angle.

Resonant enhancement of the 2G Raman band in twisted bilayer graphene

Raman spectroscopy is an extremely useful tool to characterize graphene systems. The strongest Raman features are the first-order G band and the second-order 2D and 2D′ bands, which are the overtones of the double resonance D and D’ bands. However, the 2G band, which is the overtone of the G band, is not usually observed in the spectra of monolayer graphene and of crystalline graphite. In this work, we present an experimental and theoretical investigation of the resonance Raman spectra in twisted bilayer graphene (TBG) with different twisting angles and using several laser excitation energies in the NIR and visible ranges. We observed that the 2G band is enhanced when the incident photons are in resonance with the transition between the van Hove singularities in the density of states of the TBG. We show that the 2G band has three contributions (2G1, 2G2 and 2G3), that are not dispersive by changing the laser excitation energy. We also present theoretical calculations showing that the 2G1 and 2G2 bands are related to combinations of the in-phase (IP) and out-of-phase (OP) vibrations of the atoms in the different layers. The Raman excitation profiles (REPs) of the 2G peaks are upshifted in comparison with the REP of the G band. This behavior was confirmed theoretically using a graphene tight binding model. We conclude that the different resonance behavior comes from the fact that the G band is a first-order process whereas the 2G band is second-order processes giving rise to overall different resonance conditions.

Effects of pressure and heterostrain on electronic bands of twisted bilayer graphene

We study the effects of pressure and uniaxial heterostrain on the energy bands and density of states in twisted bilayer graphene based on the effective continuum model. We find that extremely flat bands (one tenth of the bandwidth at magic angle 1.05°) can be achieved under pressure for a wide range of twist angles due to the enhancement of interlayer coupling strength. Under strain, pressure dramatically reduces the separation and widths of the flatband van Hove singularities at angles beyond the magic angle. Strain direction provides an addition parameter for tuning the flat bands to realize narrow bandwidth. Moreover, high-order van Hove singularities favoring strong electron correlations can be managed by applying pressure and strain. These effects may promote correlated electron phenomena at a broad range of twist angles.

Spin-orbit coupling-enhanced valley ordering of malleable bands in twisted bilayer graphene on WSe2

Recent experiments in magic-angle twisted bilayer graphene have revealed a wealth of novel electronic phases as a result of interaction-driven spin-valley flavour polarisation. In this work, we investigate correlated phases due to the combined effect of spin-orbit coupling-enhanced valley polarisation and the large density of states below half filling of the moiré band in twisted bilayer graphene coupled to tungsten diselenide. We observe an anomalous Hall effect, accompanied by a series of Lifshitz transitions that are highly tunable with carrier density and magnetic field. The magnetisation shows an abrupt change of sign near half-filling, confirming its orbital nature. While the Hall resistance is not quantised at zero magnetic fields—indicating a ground state with partial valley polarisation—perfect quantisation and complete valley polarisation are observed at finite fields. Our results illustrate that singularities in the flat bands in the presence of spin-orbit coupling can stabilise ordered phases even at non-integer moiré band fillings.

Kondo effect in twisted bilayer graphene

The emergence of flat bands in twisted bilayer graphene at the magic angle can be understood in terms of a vanishing Fermi velocity of the Dirac cone. This is associated with van Hove singularities approaching the Fermi energy and becoming higher-order. In the density of states, this is reflected by flanking logarithmic van Hove divergences pinching off the central Dirac cone in energy space. The low-energy pseudogap of the Dirac cone away from the magic angle is replaced by a power-law divergence due to the higher-order van Hove singularity at the magic angle. This plays an important role in the exotic phenomena observed in this material, such as superconductivity and magnetism, by amplifying electronic correlation effects. Here we investigate one such correlation effect---the Kondo effect due to a magnetic impurity embedded in twisted bilayer graphene. We use the Bistritzer-MacDonald model to extract the low-energy density of states of the material as a function of twist angle and study the resulting quantum impurity physics using perturbative and numerical renormalization group methods. Although at zero temperature the impurity is only Kondo screened precisely at the magic angle, we find highly nontrivial behavior at finite temperatures relevant to experiments, due to the complex interplay between Dirac, van Hove, and Kondo physics.

3/2 magic angle quantization rule of flat bands in twisted bilayer graphene and its relationship to the quantum Hall effect

3/2 magic angle quantization rule of flat bands in twisted bilayer graphene and its relationship to the quantum Hall effect

Effects of strain on the flat band in twisted bilayer graphene

Based on the effective continuum model, we systematically study the electronic band structures and density of states of twisted bilayer graphene near the magic angle under the influence of different types of strain, including shear strain, volume-preserving strain and biaxial strain. We find that the flat bands behave very differently under various types of strain. Volume-preserving strain generically leads to broader van Hove singularities associated with the flat bands compared with those under shear strain, with dissimilar strain direction dependence. The band structures and density of states under shear and volume-preserving strains change with the strain direction, while those under biaxial strain are independent of the direction of strain. In particular, the effect of biaxial strain on twisted bilayer graphene is geometrically and electronically similar to the influence of the twisted angle. Our results reveal the characteristic structures in the band structures and density of states under various types of strain, which can serve as fingerprints for exploring the effects of strain on the novel physics of this system.

Machine learning of the Γ-point gap and flat bands of twisted bilayer graphene at arbitrary angles

The novel electronic properties of bilayer graphene can be fine-tuned via twisting, which may induce flat bands around the Fermi level with nontrivial topology. In general, the band structure of such twisted bilayer graphene (TBG) can be theoretically obtained by using first-principles calculations, tight-binding method, or continuum model, which are either computationally demanding or parameters dependent. In this work, by using the sure independence screening sparsifying operator method, we propose a physically interpretable three-dimensional (3D) descriptor which can be utilized to readily obtain the Γ-point gap of TBG at arbitrary twist angles and different interlayer spacings. The strong predictive power of the descriptor is demonstrated by a high Pearson coefficient of 99% for both the training and testing data. To go further, we adopt the neural network algorithm to accurately probe the flat bands of TBG at various twist angles, which can accelerate the study of strong correlation physics associated with such a fundamental characteristic, especially for those systems with a larger number of atoms in the unit cell.

Twisted bilayers of thin film magnetic topological insulators

Twisted bilayer graphene (TBG) near ``magic angles'' has emerged as a rich platform for strongly correlated states of two-dimensional (2D) Dirac semimetals. Here we show that twisted bilayers of thin film magnetic topological insulators (MTIs) with large in-plane magnetization can realize flat bands near 2D Dirac nodes. Using a simple model for thin films of MTIs, we derive a continuum model for two such MTIs, twisted by a small angle with respect to each other. When the magnetization is in plane, we show that interlayer tunneling terms act as effective SU(2) vector potentials, which are known to lead to flat bands in TBG. We show that by changing the in-plane magnetization, it is possible to tune the twisted bilayer MTI band dispersion to quadratic band touching or to flat bands, similar to the TBG. If realized, this system can be a highly tunable platform for strongly correlated phases of two-dimensional Dirac semimetals.

Giant nonlinear Hall effect in strained twisted bilayer graphene

Recent studies have shown that moir\'e flat bands in twisted bilayer graphene (TBG) can acquire nontrivial Berry curvatures when aligned with a hexagonal boron nitride substrate, which can be manifested as a correlated Chern insulator near the 3/4 filling. In this Letter, we show that the large Berry curvatures in the moir\'e bands lead to a strong nonlinear Hall (NLH) effect in strained TBG with general filling factors. Under a weak uniaxial strain $\ensuremath{\sim}0.1%$, the Berry curvature dipole which characterizes the nonlinear Hall response can be as large as $\ensuremath{\sim}200\phantom{\rule{0.28em}{0ex}}\text{\AA{}}$, exceeding the values of previously known nonlinear Hall materials by two orders of magnitude. The dependence of the giant NLH effect as a function of electric gating, strain, and twist angle is further investigated systematically. Importantly, we point out that the giant NLH effect appears generically for a twist angle near the magic angle due to the strong susceptibility of nearly flat moir\'e bands to symmetry breaking induced by strains, which can even induce a topological band inversion. Our results establish TBG as a promising platform for investigating nonlinear effects such as the NLH effect, the nonlinear Nernst effect, and the nonlinear thermal Hall effect due to its giant Berry curvature dipole.

Spin-orbit-driven ferromagnetism at half moiré filling in magic-angle twisted bilayer graphene.

Strong electron correlation and spin-orbit coupling (SOC) can have a profound influence on the electronic properties of materials. We examine their combined influence on a 2-dimensional electronic system at the atomic interface between magic-angle twisted bilayer graphene and a tungsten diselenide crystal. Strong electron correlation within the moiré flatband stabilizes correlated insulating states at both quarter and half filling, and SOC transforms these Mott-like insulators into ferromagnets, evidenced by robust anomalous Hall effect with hysteretic switching behavior. The coupling between spin and valley degrees of freedom is demonstrated through the control of the magnetic order with an in-plane magnetic field, or a perpendicular electric field. Our findings establish an experimental knob to engineer topological properties of moiré bands in twisted bilayer graphene and related systems.

Flat band carrier confinement in magic-angle twisted bilayer graphene

Magic-angle twisted bilayer graphene has emerged as a powerful platform for studying strongly correlated electron physics, owing to its almost dispersionless low-energy bands and the ability to tune the band filling by electrostatic gating. Techniques to control the twist angle between graphene layers have led to rapid experimental progress but improving sample quality is essential for separating the delicate correlated electron physics from disorder effects. Owing to the 2D nature of the system and the relatively low carrier density, the samples are highly susceptible to small doping inhomogeneity which can drastically modify the local potential landscape. This potential disorder is distinct from the twist angle variation which has been studied elsewhere. Here, by using low temperature scanning tunneling spectroscopy and planar tunneling junction measurements, we demonstrate that flat bands in twisted bilayer graphene can amplify small doping inhomogeneity that surprisingly leads to carrier confinement, which in graphene could previously only be realized in the presence of a strong magnetic field.

Twisted bilayer graphene. II. Stable symmetry anomaly

We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with a lattice model. Previous works, e.g., [Phys. Rev. Lett. 123, 036401], [Phys. Rev. X 9, 021013], [Phys. Rev. B 99, 195455], and others [1-4] found that the two flat bands in TBG possess a fragile topology protected by the $C_{2z}T$ symmetry. [Phys. Rev. Lett. 123, 036401] also pointed out an approximate particle-hole symmetry ($\mathcal{P}$) in the continuous model of TBG. In this work, we numerically confirm that $\mathcal{P}$ is indeed a good approximation for TBG and show that the fragile topology of the two flat bands is enhanced to a $\mathcal{P}$-protected stable topology. This stable topology implies $4l+2$ ($l\in\mathbb{N}$) Dirac points between the middle two bands. The $\mathcal{P}$-protected stable topology is robust against arbitrary gap closings between the middle two bands the other bands. We further show that, remarkably, this $\mathcal{P}$-protected stable topology, as well as the corresponding $4l + 2$ Dirac points, cannot be realized in lattice models that preserve both $C_{2z}T$ and $\mathcal{P}$ symmetries. In other words, the continuous model of TBG is anomalous and cannot be realized on lattices. Two other topology related topics, with consequences for the interacting TBG problem, i.e., the choice of Chern band basis in the two flat bands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed.

Collective excitations and flat-band plasmon in twisted bilayer graphene near the magic angle

Twisted bilayer graphene with tiny rotation angles have drawn significant attention due to the observation of the unconventional superconducting and correlated insulating behaviors. In this paper, we employ a full tight-binding model to investigate collective excitations in twisted bilayer graphene near magic angle. The polarization function is obtained from the tight-binding propagation method without diagonalization of the Hamiltonian matrix. With the atomic relaxation considered in the simulation, damped and undamped interband plasmon modes are discovered near magic angle under both room temperature and superconductivity transition temperature. In particular, an undamped plasmon mode in narrow bands can be directly probed in magic angle twisted bilayer graphene at superconductivity transition temperature. The undamped plasmon mode is tunable with angles and gradually fades away with both temperature and chemical potential. In practice, the flat bands in twisted bilayer graphene can be detected by exploring the collective plasmons from the measured energy loss function.

Interplay of fractional Chern insulator and charge density wave phases in twisted bilayer graphene

We perform an extensive exact diagonalization study of interaction driven insulators in spin- and valley-polarized moir\'{e} flat bands of twisted bilayer graphene aligned with its hexagonal boron nitride substrate. In addition to previously reported fractional Chern insulator phases, we provide compelling evidence for competing charge-density-wave phases at multiple fractional fillings of a realistic single-band model. A thorough analysis at different interlayer hopping parameters, motivated by experimental variability, and the role of kinetic energy at various Coulomb interaction strengths highlight the competition between these phases. The interplay of the single-particle and the interaction induced hole dispersion with the inherent Berry curvature of the Chern bands is intuitively understood to be the driving mechanism for the ground-state selection. The resulting phase diagram features remarkable agreement with experimental findings in a related moir\'{e} heterostructure and affirms the relevance of our results beyond the scope of graphene based materials.

Nematic topological semimetal and insulator in magic-angle bilayer graphene at charge neutrality

We report on a fully self-consistent Hartree-Fock calculation of interaction effects on the Moir\'e flat bands of twisted bilayer graphene, assuming that valley U(1) symmetry is respected. We use realistic band structures and interactions and focus on the charge neutrality point, where experiments have variously reported either insulating or semimetallic behavior. Restricting the search to orders for which the valley U(1) symmetry remains unbroken, we find three types of self-consistent solutions with competitive ground state energy (i) insulators that break $C_2 {\mathcal T}$ symmetry, including valley Chern insulators (ii) spin or valley polarized insulators and (iii) rotation $C_3$ symmetry breaking semimetals whose gaplessness is protected by the topology of the Moir\'e flat bands. We find that the relative stability of these states can be tuned by weak strains that break $C_3$ rotation. The nematic semimetal and also, somewhat unexpectedly, the $C_2 {\mathcal T}$ breaking insulators, are stabilized by weak strain. These ground states may be related to the semi-metallic and insulating behaviors seen at charge neutrality, and the sample variability of their observation. We also compare with the results of STM measurements near charge neutrality.

Spin-Orbit-Induced Topological Flat Bands in Line and Split Graphs of Bipartite Lattices.

Topological flat bands, such as the band in twisted bilayer graphene, are becoming a promising platform to study topics such as correlation physics, superconductivity, and transport. In this Letter, we introduce a generic approach to construct two-dimensional (2D) topological quasiflat bands from line graphs and split graphs of bipartite lattices. A line graph or split graph of a bipartite lattice exhibits a set of flat bands and a set of dispersive bands. The flat band connects to the dispersive bands through a degenerate state at some momentum. We find that, with spin-orbit coupling (SOC), the flat band becomes quasiflat and gapped from the dispersive bands. By studying a series of specific line graphs and split graphs of bipartite lattices, we find that (i)if the flat band (without SOC) has inversion or C_{2} symmetry and is nondegenerate, then the resulting quasiflat band must be topologically nontrivial, and (ii)if the flat band (without SOC) is degenerate, then there exists a SOC potential such that the resulting quasiflat band is topologically nontrivial. This generic mechanism serves as a paradigm for finding topological quasiflat bands in 2D crystalline materials and metamaterials.

Flat-band ferromagnetism in twisted bilayer graphene

We discuss twisted bilayer graphene (TBG) based on a theorem of flat band ferromagnetism put forward by Mielke and Tasaki. According to this theorem, ferromagnetism occurs if the single particle density matrix of the flat band states is irreducible and we argue that this result can be applied to the quasi-flat bands of TBG that emerge around the charge-neutrality point for twist angles around the magic angle $\theta\sim1.05^\circ$. We show that the density matrix is irreducible in this case, thus predicting a ferromagnetic ground state for neutral TBG ($n=0$). We then show that the theorem can also be applied only to the flat conduction or valence bands, if the substrate induces a single-particle gap at charge neutrality. Also in this case, the corresponding density matrix turns out to be irreducible, leading to ferromagnetism at half filling ($n=\pm2$).

Floquet-engineered topological flat bands in irradiated twisted bilayer graphene

We propose a tunable optical setup to engineer topologically nontrivial flat bands in twisted bilayer graphene under circularly polarized light. Using both analytical and numerical calculations, we demonstrate that nearly flat bands can be engineered at small twist angles near the magic angles of the static system. The flatness and the gaps between these bands can be tuned optically by varying laser frequency and amplitude. We study the effects of interlayer hopping variations on Floquet flat bands and find that changes associated with lattice relaxation favor their formation. Furthermore, we find that, once formed, the flat bands carry nonzero Chern numbers. We show that at currently known values of parameters, such topological flat bands can be realized using circularly polarized UV laser light. Thus, our work opens the way to creating optically tunable, strongly correlated topological phases of electrons in moiré superlattices.Received 21 October 2019Revised 5 May 2020Accepted 2 November 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043275Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasChern insulatorsFractional quantum Hall effectPhysical SystemsFloquet systemsGrapheneTopological materialsCondensed Matter, Materials & Applied Physics

Obstruction and Interference in Low-Energy Models for Twisted Bilayer Graphene.

The electronic bands of twisted bilayer graphene (TBLG) with a large-period moiré superlattice fracture to form narrow Bloch minibands that are spectrally isolated by forbidden energy gaps from remote dispersive bands. When these gaps are sufficiently large, one can study a band-projected Hamiltonian that correctly represents the dynamics within the minibands. This inevitably introduces nontrivial geometrical constraints that arise from the assumed form of the projection. Here we show that this choice has a profound consequence in a low-energy experimentally observable signature that therefore can be used to tightly constrain the analytic form of the appropriate low-energy theory. We find that this can be accomplished by a careful analysis of the electron density produced by backscattering of Bloch waves from an impurity potential localized on the moiré superlattice scale. We provide numerical estimates of the effect that can guide experimental work to clearly discriminate between competing models for the low-energy band structure.