Double Bilayer Graphene Research Articles

Faraday rotations, ellipticity, and circular dichroism in magneto-optical spectrum of moiré superlattices* *Project supported by the National Natural Science Foundation of China (Grant Nos. 12050410228 and 12074260) and the Fund from the Science and Technology Commission of Shanghai Municipality, China (Grant No. 19ZR1436400).

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 the 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, the other four structures display strong circular dichroism with monolayer graphene and AB-stacked bilayer graphene on hexagonal boron nitride being particularly pronounced owing to strong inversion symmetry breaking properties of the hexagonal boron nitride layer. As the left and right circularly polarized light interact with these structures differently, plane-polarized incident light undergoes a Faraday rotation and gains an ellipticity when transmitted. The size of the respective angles is on the order of a degree.

Electric-field-tunable electronic nematic order in twisted double-bilayer graphene

Graphene-based moiré systems have attracted considerable interest in recent years as they display a remarkable variety of correlated phenomena. Besides insulating and superconducting phases in the vicinity of integer fillings of the moiré unit cell, there is growing evidence for electronic nematic order both in twisted bilayer graphene and twisted double-bilayer graphene (tDBG), as signaled by the spontaneous breaking of the threefold rotational symmetry of the moiré superlattices. Here, we combine symmetry-based analysis with a microscopic continuum model to investigate the structure of the nematic phase of tDBG and its experimental manifestations. First, we perform a detailed comparison between the theoretically calculated local density of states and recent scanning tunneling microscopy data (arXiv:2009.11645) to resolve the internal structure of the nematic order parameter in terms of the layer, sublattice, spin, and valley degrees of freedom. We find strong evidence that the dominant contribution to the nematic order parameter comes from states at the moiré scale rather than at the microscopic scale of the individual graphene layers, which demonstrates the key role played by the moiré degrees of freedom and confirms the correlated nature of the nematic phase in tDBG. Secondly, our analysis reveals an unprecedented tunability of the orientation of the nematic director in tDBG by an externally applied electric field, allowing the director to rotate away from high-symmetry crystalline directions. We compute the expected fingerprints of this rotation in both STM and transport experiments, providing feasible ways to probe it. Rooted in the strong sensitivity of the flat bands of tDBG to the displacement field, this effect opens an interesting route to the electrostatic control of electronic nematicity in moiré systems.

Valley-selective Floquet Chern flat bands in twisted multilayer graphene

We show that Floquet engineering with circularly polarized light (CPL) can selectively split the valley degeneracy of a twisted multilayer graphene (TMG), and thus generate a controlled valley-polarized Floquet Chern flat band with tunable large Chern number. It offers a feasible optical way to manipulate the valley degree of freedom in moir\'{e} flat bands, and hence opens new opportunities to study the valleytronics of mori\'{e} flat band systems. We thus expect that many of the valley-related properties of TMG, e.g. orbital ferromagnetism, can be switched by CPL with proper doping. We reveal a Chern number hierarchy rule for the Floquet flat bands in a generic (M+N)-layer TMG. We also illustrate that the CPL effects on TMG strongly rely on the stacking chirality, which is an unique feature of TMG. All these phenomena could be tested in the twisted double bilayer graphene systems, which is the simplest example of TMG and has already been realized in experiment.

Effects of dynamical screening on the BCS-BEC crossover in double bilayer graphene: Density functional theory for exciton bilayers

We derive a gap equation for bilayer excitonic systems based on density functional theory and benchmark our results against quantum Monte Carlo simulations and recent experiments on double bilayer graphene. The gap equation has a mean-field form but includes a consistent treatment of dynamical screening. We show that the gap survives at much higher densities than previously thought from mean-field estimates which gives strong indications that the double-bilayer graphene systems at zero magnetic field can be used as model systems to investigate the BCS-BEC crossover. Furthermore, we show that Josephson-like transfer of pairs can be substantial for small band gaps and densities.

Spectroscopy of a tunable moir\xe9 system with a correlated and topological flat band

Moiré superlattices created by the twisted stacking of two-dimensional crystals can host electronic bands with flat energy dispersion in which enhanced interactions promote correlated electron states. The twisted double bilayer graphene (TDBG), where two Bernal bilayer graphene are stacked with a twist angle, is such a moiré system with tunable flat bands. Here, we use gate-tuned scanning tunneling spectroscopy to directly demonstrate the tunability of the band structure of TDBG with an electric field and to show spectroscopic signatures of electronic correlations and topology for its flat band. Our spectroscopic experiments are in agreement with a continuum model of TDBG band structure and reveal signatures of a correlated insulator gap at partial filling of its isolated flat band. The topological properties of this flat band are probed with the application of a magnetic field, which leads to valley polarization and the splitting of Chern bands with a large effective g-factor.

Visualizing delocalized correlated electronic states in twisted double bilayer graphene

The discovery of interaction-driven insulating and superconducting phases in moiré van der Waals heterostructures has sparked considerable interest in understanding the novel correlated physics of these systems. While a significant number of studies have focused on twisted bilayer graphene, correlated insulating states and a superconductivity-like transition up to 12 K have been reported in recent transport measurements of twisted double bilayer graphene. Here we present a scanning tunneling microscopy and spectroscopy study of gate-tunable twisted double bilayer graphene devices. We observe splitting of the van Hove singularity peak by ~20 meV at half-filling of the conduction flat band, with a corresponding reduction of the local density of states at the Fermi level. By mapping the tunneling differential conductance we show that this correlated system exhibits energetically split states that are spatially delocalized throughout the different regions in the moiré unit cell, inconsistent with order originating solely from onsite Coulomb repulsion within strongly-localized orbitals. We have performed self-consistent Hartree-Fock calculations that suggest exchange-driven spontaneous symmetry breaking in the degenerate conduction flat band is the origin of the observed correlated state. Our results provide new insight into the nature of electron-electron interactions in twisted double bilayer graphene and related moiré systems.

Trigonal warping, satellite Dirac points, and multiple field tuned topological transitions in twisted double bilayer graphene

We show that the valley Chern number of the low energy band in twisted double bilayer graphene can be tuned through two successive topological transitions, where the direct bandgap closes, by changing the electric field perpendicular to the plane of the graphene layers. The two transitions with Chern number changes of -3 and +1 can be explained by the formation of three satellite Dirac points around the central Dirac cone in the moir\'e Brillouin zone due to the presence of trigonal warping. The satellite cones have opposite chirality to the central Dirac cone. Considering the overlap of the bands in energy, which lead to metallic states, we construct the experimentally observable phase diagram of the system in terms of the indirect bandgap and the anomalous valley Hall conductivity. We show that while most of the intermediate phase becomes metallic, there is a narrow parameter regime where the transition through three insulating phases with different quantized valley Hall conductivity can be seen. We systematically study the effects of variations in the model parameters on the phase diagram of the system to reveal the importance of particle-hole asymmetry and trigonal warping in constructing the phase diagram. We also study the effect of changes in interlayer tunneling on this phase diagram.

Phase diagram and orbital Chern insulator in twisted double bilayer graphene

Compared with twisted bilayer graphene, twisted double bilayer graphene (TDBG) provides another important platform to realize the moir\'e flat bands. In this paper, we first calculate the valley Chern number phase diagram of TDBG in the parameter space spanned by the twist angle and the interlayer electric potential. To include the effects of interactions, we then phenomenologically introduce the spin-splitting and valley-splitting. We find that when the valley splitting is larger than the bandwidth of the first conduction band so that a gap is opened and the spin splitting is relatively weak, the orbital Chern insulator emerges at half-filling, associated with a large orbital magnetization (OM). Further calculations suggest that there is no sign reversal of the OM when the Fermi energy goes from the bottom to the top of the half-filling gap, as the OM remains negative in both AB-AB stacking and AB-BA stacking. The implications of our results for the ongoing experiments are also discussed.

Landau Levels as a Probe for Band Topology in Graphene Moiré Superlattices.

We propose Landau levels as a probe for the topological character of electronic bands in two-dimensional moiré superlattices. We consider two configurations of twisted double bilayer graphene (TDBG) that have very similar band structures, but show different valley Chern numbers of the flat bands. These differences between the AB-AB and AB-BA configurations of TDBG clearly manifest as different Landau level sequences in the Hofstadter butterfly spectra calculated using the tight-binding model. The Landau level sequences are explained from the point of view of the distribution of orbital magnetization in momentum space that is governed by the rotational C_{2} and time-reversal T symmetries. Our results can be readily extended to other twisted graphene multilayers and h-BN/graphene heterostructures thus establishing the Hofstadter butterfly spectra as a powerful tool for detecting the nontrivial valley band topology.

Moiréless correlations in ABCA graphene

Atomically thin van der Waals materials stacked with an interlayer twist have proven to be an excellent platform toward achieving gate-tunable correlated phenomena linked to the formation of flat electronic bands. In this work we demonstrate the formation of emergent correlated phases in multilayer rhombohedral graphene--a simple material that also exhibits a flat electronic band edge but without the need of having a moiré superlattice induced by twisted van der Waals layers. We show that two layers of bilayer graphene that are twisted by an arbitrary tiny angle host large (micrometer-scale) regions of uniform rhombohedral four-layer (ABCA) graphene that can be independently studied. Scanning tunneling spectroscopy reveals that ABCA graphene hosts an unprecedentedly sharp van Hove singularity of 3-5-meV half-width. We demonstrate that when this van Hove singularity straddles the Fermi level, a correlated many-body gap emerges with peak-to-peak value of 9.5 meV at charge neutrality. Mean-field theoretical calculations for model with short-ranged interactions indicate that two primary candidates for the appearance of this broken symmetry state are a charge-transfer excitonic insulator and a ferrimagnet. Finally, we show that ABCA graphene hosts surface topological helical edge states at natural interfaces with ABAB graphene which can be turned on and off with gate voltage, implying that small-angle twisted double-bilayer graphene is an ideal programmable topological quantum material.

Spin-polarized superconductivity: Order parameter topology, current dissipation, and multiple-period Josephson effect

We discuss transport properties of fully spin-polarized triplet superconductors, where only electrons of one spin component (along a certain axis) are paired. Due to the structure of the order parameter space, wherein phase and spin rotations are intertwined, a configuration where the superconducting phase winds by $4\pi$ in space is topologically equivalent to a configuration with no phase winding. This opens the possibility of supercurrent relaxation by a smooth deformation of the order parameter, where the order parameter remains non-zero at any point in space throughout the entire process. During the process, a spin texture is formed. We discuss the conditions for such processes to occur and their physical consequences. In particular, we show that when a voltage is applied, they lead to an unusual alternating-current Josephson effect whose period is an integer multiple of the usual Josephson period. These conclusions are substantiated in a simple time-dependent Ginzburg-Landau model for the dynamics of the order parameter. One of the potential applications of our analysis is for moir\'e systems, such as twisted bilayer and double bilayer graphene, where superconductivity is found in the vicinity of ferromagnetism.

Gate-Tunable Fractional Chern Insulators in Twisted Double Bilayer Graphene.

We predict twisted double bilayer graphene to be a versatile platform for the realization of fractional Chern insulators readily targeted by tuning the gate potential and the twist angle. Remarkably, these topologically ordered states of matter, including spin singlet Halperin states and spin polarized states in Chern number C=1 and C=2 bands, occur at high temperatures and without the need for an external magnetic field.

Moir\xe9 metrology of energy landscapes in van der Waals heterostructures

The emerging field of twistronics, which harnesses the twist angle between two-dimensional materials, represents a promising route for the design of quantum materials, as the twist-angle-induced superlattices offer means to control topology and strong correlations. At the small twist limit, and particularly under strain, as atomic relaxation prevails, the emergent moiré superlattice encodes elusive insights into the local interlayer interaction. Here we introduce moiré metrology as a combined experiment-theory framework to probe the stacking energy landscape of bilayer structures at the 0.1 meV/atom scale, outperforming the gold-standard of quantum chemistry. Through studying the shapes of moiré domains with numerous nano-imaging techniques, and correlating with multi-scale modelling, we assess and refine first-principle models for the interlayer interaction. We document the prowess of moiré metrology for three representative twisted systems: bilayer graphene, double bilayer graphene and H-stacked MoSe2/WSe2. Moiré metrology establishes sought after experimental benchmarks for interlayer interaction, thus enabling accurate modelling of twisted multilayers.

Bulk valley transport and Berry curvature spreading at the edge of flat bands

2D materials based superlattices have emerged as a promising platform to modulate band structure and its symmetries. In particular, moiré periodicity in twisted graphene systems produces flat Chern bands. The recent observation of anomalous Hall effect (AHE) and orbital magnetism in twisted bilayer graphene has been associated with spontaneous symmetry breaking of such Chern bands. However, the valley Hall state as a precursor of AHE state, when time-reversal symmetry is still protected, has not been observed. Our work probes this precursor state using the valley Hall effect. We show that broken inversion symmetry in twisted double bilayer graphene (TDBG) facilitates the generation of bulk valley current by reporting experimental evidence of nonlocal transport in a nearly flat band system. Despite the spread of Berry curvature hotspots and reduced quasiparticle velocities of the carriers in these flat bands, we observe large nonlocal voltage several micrometers away from the charge current path — this persists when the Fermi energy lies inside a gap with large Berry curvature. The high sensitivity of the nonlocal voltage to gate tunable carrier density and gap modulating perpendicular electric field makes TDBG an attractive platform for valley-twistronics based on flat bands.

Effect of bilayer stacking on the atomic and electronic structure of twisted double bilayer graphene

Twisted double bilayer graphene has recently emerged as an interesting moir\'e material that exhibits strong correlation phenomena that are tunable by an applied electric field. Here we study the atomic and electronic properties of three different graphene double bilayers: double bilayers composed of two AB stacked bilayers (AB/AB), double bilayers composed of two AA stacked bilayers (AA/AA) as well as heterosystems composed of one AB and one AA bilayer (AB/AA). The atomic structure is determined using classical force fields. We find that the inner layers of the double bilayer exhibit significant in-plane and out-of-plane relaxations, similar to twisted bilayer graphene. The relaxations of the outer layers depend on the stacking: atoms in AB bilayers follow the relaxations of the inner layers, while atoms in AA bilayers attempt to avoid higher-energy AA stacking. For the relaxed structures, we calculate the electronic band structures using the tight-binding method. All double bilayers exhibit flat bands at small twist angles, but the shape of the bands depends sensitively on the stacking of the outer layers. To gain further insight, we study the evolution of the band structure as the outer layers are rigidly moved away from the inner layers, while preserving their atomic relaxations. This reveals that the hybridization with the outer layers results in an additional flattening of the inner-layer flat band manifold. Our results establish AA/AA and AB/AA twisted double bilayers as interesting moir\'e materials with different flat band physics compared to the widely studied AB/AB system.

Combined Minivalley and Layer Control in Twisted Double Bilayer Graphene.

Control over minivalley polarization and interlayer coupling is demonstrated in double bilayer graphene twisted with an angle of 2.37°. This intermediate angle is small enough for the minibands to form and large enough such that the charge carrier gases in the layers can be tuned independently. Using a dual-gated geometry we identify and control all possible combinations of minivalley polarization via the population of the two bilayers. An applied displacement field opens a band gap in either of the two bilayers, allowing us to even obtain full minivalley polarization. In addition, the carriers, formerly separated by their minivalley character, are mixed by tuning through a Lifshitz transition, where the Fermi surface topology changes. The high degree of control over the minivalley character of the bulk charge transport in twisted double bilayer graphene offers new opportunities for realizing valleytronics devices such as valley valves, filters, and logic gates.

Topological flat bands in twisted trilayer graphene

Twisted trilayer graphene (TLG) may be the simplest realistic system so far, which has flat bands with nontrivial topology. Here, we give a comprehensive calculation about its band structures and the band topology, i.e., valley Chern number of the nearly flat bands, with the continuum model. With realistic parameters, the magic angle of twisted TLG is about 1.12°, at which two nearly flat bands appears. Unlike the twisted bilayer graphene, a small twist angle can induce a tiny gap at all the Dirac points, which can be enlarged further by a perpendicular electric field. The valley Chern numbers of the two nearly flat bands in the twisted TLG depends on the twist angle θ and the perpendicular electric field E⊥. Considering its topological flat bands, the twisted TLG should be an ideal experimental platform to study the strongly correlated physics in topologically nontrivial flat band systems. And, due to its reduced symmetry, the correlated states in twisted TLG should be quite different from that in twisted bilayer graphene and twisted double bilayer graphene.

Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene

Twisted van der Waals heterostructures with flat electronic bands have recently emerged as a platform for realizing correlated and topological states with an extraordinary degree of control and tunability. In graphene-based moir\'e heterostructures, the correlated phase diagram and band topology depend strongly on the number of graphene layers, their relative stacking arrangement, and details of the external environment from the encapsulating crystals. Here, we report that the system of twisted monolayer-bilayer graphene (tMBG) hosts a variety of correlated metallic and insulating states, as well as topological magnetic states. Because of its low symmetry, the phase diagram of tMBG approximates that of twisted bilayer graphene when an applied perpendicular electric field points from the bilayer towards the monolayer graphene, or twisted double bilayer graphene when the field is reversed. In the former case, we observe correlated states which undergo an orbitally driven insulating transition above a critical perpendicular magnetic field. In the latter case, we observe the emergence of electrically tunable ferromagnetism at one-quarter filling of the conduction band, with a large associated anomalous Hall effect. Uniquely, the magnetization direction can be switched purely with electrostatic doping at zero magnetic field. Our results establish tMBG as a highly tunable platform for investigating a wide array of tunable correlated and topological states.

Floquet engineering of twisted double bilayer graphene

Motivated by the recent experimental realization of twisted double bilayer graphene (TDBG) samples, we study, both analytically and numerically, the effects of circularly polarized light propagating in free space and confined in a waveguide on the band structure and topological properties of these systems. These two complementary Floquet protocols allow us to selectively tune different parameters of the system by varying the intensity and light frequency. For the drive protocol in free space, in the high-frequency regime, we find that in TDBG with AB/BA stacking, we can selectively close the zone-center quasienergy gaps around one valley while increasing the gaps near the opposite valley by tuning the parameters of the drive. In TDBG with AB/AB stacking, a similar effect can be obtained upon the application of a perpendicular static electric field. Furthermore, we study the topological properties of the driven system in different settings, provide accurate effective Floquet Hamiltonians, and show that relatively strong drives can generate flat bands. On the other hand, longitudinal light confined in a waveguide couples to the components of the interlayer hopping that are perpendicular to the TDBG sheet, allowing for selective engineering of the bandwidth of Floquet zone-center quasienergy bands without breaking the symmetries of the static system.5 MoreReceived 15 June 2020Accepted 1 September 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.033494Published 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 AreasElectronic structureTwist deformationValleytronicsPhysical SystemsFloquet systemsGrapheneTopological insulatorsTechniquesBand structure methodsCondensed Matter, Materials & Applied Physics

Symmetry breaking in twisted double bilayer graphene

A variety of correlated phases have recently emerged in select twisted van der Waals (vdW) heterostructures owing to their flat electronic dispersions. In particular, heterostructures of twisted double bilayer graphene (tDBG) manifest electric field-tunable correlated insulating (CI) states at all quarter fillings of the conduction band, accompanied by nearby states featuring signatures suggestive of superconductivity. Here, we report electrical transport measurements of tDBG in which we elucidate the fundamental role of spontaneous symmetry breaking within its correlated phase diagram. We observe abrupt resistivity drops upon lowering the temperature in the correlated metallic phases neighboring the CI states, along with associated nonlinear $I$-$V$ characteristics. Despite qualitative similarities to superconductivity, concomitant reversals in the sign of the Hall coefficient instead point to spontaneous symmetry breaking as the origin of the abrupt resistivity drops, while Joule heating appears to underlie the nonlinear transport. Our results suggest that similar mechanisms are likely relevant across a broader class of semiconducting flat band vdW heterostructures.