Double Bilayer Graphene Research Articles

Electronic structure of 30∘ twisted double bilayer graphene

In this paper, the electronic properties of 30{\deg} twisted double bilayer graphene, which loses the translational symmetry due to the incommensurate twist angle, are studied by means of the tight-binding approximation. We demonstrate the interlayer decoupling in the low-energy region from various electronic properties, such as the density of states, effective band structure, optical conductivity and Landau level spectrum. However, at Q points, the interlayer coupling results in the appearance of new Van Hove singularities in the density of states, new peaks in the optical conductivity and importantly the 12-fold-symmetry-like electronic states. The k-space tight-binding method is adopted to explain this phenomenon. The electronic states at Q points show the charge distribution patterns more complex than the 30{\deg} twisted bilayer graphene due to the symmetry decrease. These phenomena appear also in the 30{\deg} twisted interface between graphene monolayer and AB stacked bilayer.

Topological superconductivity, ferromagnetism, and valley-polarized phases in moiré systems: Renormalization group analysis for twisted double bilayer graphene

Recent experiments have observed possible spin- and valley-polarized insulators and spin-triplet superconductivity in twisted double bilayer graphene, a moire structure consisting of a pair of Bernal-stacked bilayer graphene. Besides the continuously tunable band widths controlled by an applied displacement field and twist angle, these moire bands also possess van Hove singularities near the Fermi surface and a field-dependent nesting which is far from perfect. Here we carry out a perturbative renormalization group analysis to unbiasedly study the competition among all possible instabilities in twisted double bilayer graphene and related systems with a similar van Hove fermiology in the presence of weak but finite repulsive interactions. Our key finding is that there are several competing magnetic, valley, charge, and superconducting instabilities arising from interactions in twisted double bilayer graphene, which can be tuned by controlling the displacement field and the twist angle. In particular, we show that spin- or valley-polarized uniform instabilities generically dominate under moderate interactions smaller than the band width, whereas $p$-wave spin-triplet topological superconductivity and exotic spin-singlet modulated paired state become important as the interactions decrease. Realization of our findings in general moire systems with a similar van Hove fermiology should open up new opportunities for manipulating topological superconductivity and spin- or valley-polarized states in highly tunable platforms.

Microscopic pairing mechanism, order parameter, and disorder sensitivity in moiré superlattices: Applications to twisted double-bilayer graphene

Starting from a continuum-model description, we develop a microscopic weak-coupling theory for superconductivity in twisted double-bilayer graphene. We study both electron-phonon and entirely electronic pairing mechanisms. In each case, the leading superconducting instability transforms under the trivial representation, $A$, of the point group $C_3$ of the system, while the subleading pairing phases belong to the $E$ channel. We explicitly compute the momentum dependence of the associated order parameters and find that the leading state has no nodal points for electron-phonon pairing but exhibits six sign changes on the Fermi surface if the Coulomb interaction dominates. On top of these system-specific considerations, we also present general results relevant to other correlated graphene-based moir\'e superlattice systems. We show that, irrespective of microscopic details, triplet pairing will be stabilized if the collective electronic fluctuations breaking the enhanced $\text{SU}(2)_+ \times \text{SU}(2)_-$ spin symmetry of these systems are odd under time reversal, even when the main $\text{SU}(2)_+ \times \text{SU}(2)_-$-symmetric part of the pairing glue is provided by phonons. Furthermore, we discuss the disorder sensitivity of the candidate pairing states and demonstrate that the triplet phase is protected against disorder on the moir\'e scale.

Hofstadter butterfly and the quantum Hall effect in twisted double bilayer graphene

We study the energy spectrum and quantum Hall effects of the twisted double bilayer graphene in uniform magnetic field. We investigate two different arrangements, AB-AB and AB-BA, which differ in the relative orientation but have very similar band structures in the absence of a magnetic field. For each system, we calculate the energy spectrum and quantized Hall conductivities at each spectral gap by using a continuum Hamiltonian that satisfies the magneto-translation condition. We show that the Hofstadter butterfly spectra of AB-AB and AB-BA stackings differ significantly, even though their zero magnetic field band structures closely resemble; the spectrum of AB-AB has valley degeneracy, which can be lifted by applying interlayer potential asymmetry, while the spectrum of AB-BA has no such degeneracy in any case. We explain the origin of the difference from the perspectives of lattice symmetry and band topology.

Pairing in graphene-based moiré superlattices

We present a systematic classification and analysis of possible pairing instabilities in graphene-based moir\'e superlattices. Motivated by recent experiments on twisted double-bilayer graphene showing signs of triplet superconductivity, we analyze both singlet and triplet pairing separately, and describe how these two channels behave close to the limit where the system is invariant under separate spin rotations in the two valleys, realizing an SU(2)$_+$ $\times$ SU(2)$_-$ symmetry. Further, we discuss the conditions under which singlet and triplet can mix via two nearly degenerate transitions, and how the different pairing states behave when an external magnetic field is applied. The consequences of the additional microscopic or emergent approximate symmetries relevant for superconductivity in twisted bilayer graphene and ABC trilayer graphene on hexagonal boron nitride are described in detail. We also analyze which of the pairing states can arise in mean-field theory and study the impact of corrections coming from ferromagnetic fluctuations. For instance, we show that, close to the parameters of mean-field theory, a nematic mixed singlet-triplet state emerges. Our study illustrates that graphene superlattices provide a rich platform for exotic superconducting states, and allow for the admixture of singlet and triplet pairing even in the absence of spin-orbit coupling.

Tunable spin-polarized correlated states in twisted double bilayer graphene.

Reducing the energy bandwidth of electrons in a lattice below the long-range Coulomb interaction energy promotes correlation effects. Moiré superlattices-which are created by stacking van der Waals heterostructures with a controlled twist angle1-3-enable the engineering of electron band structure. Exotic quantum phases can emerge in an engineered moiré flat band. The recent discovery of correlated insulator states, superconductivity and the quantum anomalous Hall effect in the flat band of magic-angle twisted bilayer graphene4-8 has sparked the exploration of correlated electron states in other moiré systems9-11. The electronic properties of van der Waals moiré superlattices can further be tuned by adjusting the interlayer coupling6 or the band structure of constituent layers9. Here, using van der Waals heterostructures of twisted double bilayer graphene (TDBG), we demonstrate a flat electron band that is tunable by perpendicular electric fields in a range of twist angles. Similarly to magic-angle twisted bilayer graphene, TDBG shows energy gaps at the half- and quarter-filled flat bands, indicating the emergence of correlated insulator states. We find that the gaps of these insulator states increase with in-plane magnetic field, suggesting a ferromagnetic order. On doping the half-filled insulator, a sudden drop in resistivity is observed with decreasing temperature. This critical behaviour is confined to a small area in the density-electric-field plane, and is attributed to a phase transition from a normal metal to a spin-polarized correlated state. The discovery of spin-polarized correlated states in electric-field-tunable TDBG provides a new route to engineering interaction-driven quantum phases.

Phonon scattering induced carrier resistivity in twisted double-bilayer graphene

In this work we carry out a theoretical study of the phonon-induced resistivity in twisted double bilayer graphene (TDBG), in which two Bernal-stacked bilayer graphene devices are rotated relative to each other by a small angle $\theta$. We show that at small twist angles ($\theta\sim 1^\circ$) the effective mass of the TDBG system is greatly enhanced, leading to a drastically increased phonon-induced resistivity in the high-temperature limit where phonon scattering leads to a linearly increasing resistivity with increasing temperature. We also discuss possible implications of our theory on superconductivity in such a system, and provide an order of magnitude estimation of the superconducting transition temperature.

Anomalous Hall effect, magneto-optical properties, and nonlinear optical properties of twisted graphene systems

We study the anomalous Hall effect, magneto-optical properties, and nonlinear optical properties of twisted bilayer graphene aligned with hexagonal boron nitride substrate, as well as twisted double bilayer graphene systems. We show that non-vanishing valley polarizations in twisted graphene systems would give rise to the anomalous Hall effect, which can be tuned by in-plane magnetic fields. The valley polarized states are also associated with giant Faraday and Kerr rotations in the terahertz frequency regime. Moreover, we propose that the twisted graphene systems exhibit colossal nonlinear optical responses by virtue of the inversion-symmetry breaking, the small bandwidth, and the small excitation gaps of the systems. In twisted double bilayer graphene, we find that certain components of the nonlinear photo-conductivity tensor are directly proportional to the orbital magnetization of the system, which would exhibit remarkable hysteresis behavior in response to perpendicular magnetic fields.

Ferromagnetism in Narrow Bands of Moiré Superlattices.

Many graphene moiré superlattices host narrow bands with nonzero valley Chern numbers. We provide analytical and numerical evidence for a robust spin and/or valley polarized insulator at total integer band filling in nearly flat bands of several different moiré materials. In the limit of a perfectly flat band, we present analytical arguments in favor of the ferromagnetic state substantiated by numerical calculations. Further, we numerically evaluate its stability for a finite bandwidth. We provide exact diagonalization results for models appropriate for ABC trilayer graphene aligned with hBN, twisted double bilayer graphene, and twisted bilayer graphene aligned with hBN. We also provide DMRG results for a honeycomb lattice with a quasiflat band and nonzero Chern number, which extend our results to larger system sizes. We find a maximally spin and valley polarized insulator at all integer fillings when the band is sufficiently flat. We also show that interactions may induce effective dispersive terms strong enough to destabilize this state. These results still hold in the case of zero valley Chern number (for example, trivial side of TLG/hBN). We give an intuitive picture based on extended Wannier orbitals, and emphasize the role of the quantum geometry of the band, whose microscopic details may enhance or weaken ferromagnetism in moiré materials.

Ferromagnetism and superconductivity in twisted double bilayer graphene

Twisted double bilayer graphene is a highly tunable moir\'e system to study strongly interacting physics. Here, the authors present a theory of competing ferromagnetic and superconducting orders in this system. They show that the single-particle moir\'e bands can be drastically modified by a perpendicular electric displacement field. As a result, the ferromagnetic insulating gap driven by Coulomb interaction has a dome shape dependence on the layer potential difference, as observed in several experiments. The stability of the ferromagnetic insulator against collective excitations, including spin and valley magnons, is theoretically verified. Furthermore, the authors investigate the possibility of phonon-mediated intervalley equal-spin pairing, using the property that the acoustic phonon mediated attraction respects an enlarged SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) symmetry.

Pressure-induced gap modulation and topological transitions in twisted bilayer and twisted double bilayer graphene

We study the electronic and topological properties of fully relaxed twisted bilayer (TBG) and double bilayer (TDBG) graphene under perpendicular pressure. An approach has been proposed to obtain the equilibrium in-plane structural deformation and out-of-plane corrugation in moir\'{e} superlattices under pressure. We find that the in-plane relaxation becomes much stronger under higher pressure, while the corrugation height in each layer is maintained. The comparison between band structures of relaxed and rigid structures demonstrates that not only the gaps on the electron and hole sides ($\Delta_e$ and $\Delta_h$) are significantly underestimated without relaxation but also the detailed dispersions of the middle bands of rigid structures are rather different from those of relaxed systems. $\Delta_e$ and $\Delta_h$ in TBG reach maximum values around critical pressures with narrowest middle bands. Topological transitions occur in TDBG under pressure with the middle valence and conduction bands in one valley touching and their Chern numbers transferred to each other. The pressure can also tune the gap at the neutrality point of TDBG, which becomes closed for a pressure range and reopened under higher pressure. The behavior of electronic structure of supertlattices under pressure is sensitive to the twist angle $\theta$ with the critical pressures generally increase with $\theta$.

Correlated states in twisted double bilayer graphene

Electron-electron interactions play an important role in graphene and related systems and can induce exotic quantum states, especially in a stacked bilayer with a small twist angle. For bilayer graphene where the two layers are twisted by a "magic angle", flat band and strong many-body effects lead to correlated insulating states and superconductivity. In contrast to monolayer graphene, the band structure of untwisted bilayer graphene can be further tuned by a displacement field, providing an extra degree of freedom to control the flat band that should appear when two bilayers are stacked on top of each other. Here, we report the discovery and characterization of such displacement-field tunable electronic phases in twisted double bilayer graphene. We observe insulating states at a half-filled conduction band in an intermediate range of displacement fields. Furthermore, the resistance gap in the correlated insulator increases with respect to the in-plane magnetic fields and we find that the g factor according to spin Zeeman effect is ~2, indicating spin polarization at half filling. These results establish the twisted double bilayer graphene as an easily tunable platform for exploring quantum many-body states.

Tunable bandwidths and gaps in twisted double bilayer graphene on the verge of correlations

We use temperature-dependent resistivity in small-angle twisted double bilayer graphene to measure bandwidths and gaps of the bands. This electron-hole asymmetric system has one set of non-dispersing bands that splits into two flat bands with the electric field - distinct from the twisted bilayer system. With electric field, the gap between two emergent flat bands increases monotonically and bandwidth is tuned from 1 meV to 15 meV. These two flat bands with gap result in a series of thermally induced insulator to metal transitions - we use a model, at charge neutrality, to measure the bandwidth using only transport measurements. Having two flat bands with tunable gap and bandwidth offers an opportunity to probe the emergence of correlations.

Moiré Flat Bands in Twisted Double Bilayer Graphene.

We investigate twisted double bilayer graphene (TDBG), a four-layer system composed of two AB-stacked graphene bilayers rotated with respect to each other by a small angle. Our ab initio band structure calculations reveal a considerable energy gap at the charge-neutrality point that we assign to the intrinsic symmetric polarization (ISP). We then introduce the ISP effect into the tight-binding parametrization and perform calculations on TDBG models that include lattice relaxation effects down to very small twist angles. We identify a narrow region around the magic angle characterized by a manifold of remarkably flat bands gapped out from other states even without external electric fields. To understand the fundamental origin of the magic angle in TDBG, we construct a continuum model that points to a hidden mathematical link to the twisted bilayer graphene model, thus indicating that the band flattening is a fundamental feature of TDBG and is not a result of external fields.

Tunneling and fluctuating electron-hole Cooper pairs in double bilayer graphene

A strong low-temperature enhancement of the tunneling conductance between graphene bilayers has been reported recently, and interpreted as a signature of equilibrium electron-hole pairing, first predicted in bilayers more than forty years ago but previously unobserved. Here we provide a detailed theory of conductance enhanced by fluctuating electron-hole Cooper pairs, which are a precursor to equilibrium pairing, that accounts for specific details of the multi-band double graphene bilayer system which supports several different pairing channels. Above the equilibrium condensation temperature, pairs have finite temporal coherence and do not support dissipationless tunneling. Instead, they strongly boost the tunneling conductivity via a fluctuational internal Josephson effect. Our theory makes predictions for the dependence of the zero bias peak in the differential tunneling conductance on temperature and electron-hole density imbalance that capture important aspects of the experimental observations. In our interpretation of the observations, cleaner samples with longer disorder scattering times would condense at temperatures $T_c$ up to $\sim 50 {\rm K}$, compared to the record $T_c \sim 1.5 $K achieved to date in the experiment.

Flat bands and gaps in twisted double bilayer graphene.

We present electronic structure calculations of twisted double bilayer graphene (TDBG): a tetralayer graphene structure composed of two AB-stacked graphene bilayers with a relative rotation angle between them. Using first-principles calculations, we find that TDBG is semiconducting with a band gap that depends on the twist angle, that can be tuned by an external electric field. The gap is consistent with TDBG symmetry and its magnitude is related to surface effects, driving electron transfer from outer to inner layers. The surface effect competes with an energy upshift of localized states at inner layers, giving rise to the peculiar angle dependence of the band gap, which reduces at low angles. For these low twist angles, the TDBG develops flat bands, in which electrons in the inner layers are localized at the AA regions, as in twisted bilayer graphene.

Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene

Two graphene monolayers twisted by a small magic angle exhibit nearly flat bands, leading to correlated electronic states. Here we study a related but different system with reduced symmetry - twisted double bilayer graphene (TDBG), consisting of two Bernal stacked bilayer graphenes, twisted with respect to one another. Unlike the monolayer case, we show that isolated flat bands only appear on application of a vertical displacement field. We construct a phase diagram as a function of twist angle and displacement field, incorporating interactions via a Hartree-Fock approximation. At half-filling, ferromagnetic insulators are stabilized with valley Chern number {C}_{{rm{v}}}=pm 2. Upon doping, ferromagnetic fluctuations are argued to lead to spin-triplet superconductivity from pairing between opposite valleys. We highlight a novel orbital effect arising from in-plane fields plays an important role in interpreting experiments. Combined with recent experimental findings, our results establish TDBG as a tunable platform to realize rare phases in conventional solids.

Engineering Crossed Andreev Reflection in Double-Bilayer Graphene.

Crossed Andreev reflection (CAR) is a nonlocal process that converts an incoming electron (hole) from one normal electrode to an out-going hole (electron) in another normal electrode through a superconductor (SC). CAR corresponds to the inverse process of Cooper pair splitting, which generates a quantum-entangled electron pair with spatial separation. Here, we fabricated vertically stacked double bilayer graphene (BLG) connected via a superconducting electrode and achieved a spacing between BLG sheets of ∼14 nm, which is far shorter than the superconducting coherence length. We confirm the highly efficient CAR effect by observing strong negative differential resistance in a nonlocal configuration and demonstrate that the competing processes against the CAR can be effectively suppressed by separately tuning the chemical potential of each BLG. The dependence of nonlocal signals on bias voltage, temperature, and chemical potential is consistent with the predicted CAR process. Our results provide a new pathway to a novel SC-based quantum entangler with the in situ tunability of the correlated-pair-splitting efficiency.

Intrinsic band gap and electrically tunable flat bands in twisted double bilayer graphene

We present atomistic calculations on structural and electronic properties of twisted double bilayer graphene (TDBG) consisting of two sets of rotationally misaligned Bernal-stacked bilayer graphene. Obtained equilibrium atomic structures exhibit in-plane strains and the modulation of the interlayer distances at the rotationally mismatched interface layers. We find that the electronic structure of TDBG can have an intrinsic band gap at the charge neutral point for a large range of the twist angle theta. Near theta = 1.25 degree, the intrinsic band gap disappears and TDBG hosts flat bands at the Fermi level that are energetically well separated from higher and lower energy bands. We also show that the flat bands are easily tunable by applying vertical electric fields, and extremely narrow bandwidths less than 10 meV can be achieved for the electron-side flat bands in a wide range of the twist angle. Our results serve as a theoretical guide for exploring emergent correlated electron physics in this versatile moire superlattice system.

Correlated Insulating States in Twisted Double Bilayer Graphene.

We present a combined experimental and theoretical study of twisted double bilayer graphene with twist angles between 1° and 1.35°. Consistent with moiré band structure calculations, we observe insulators at integer moiré band fillings one and three, but not two. An applied transverse electric field separates the first moiré conduction band from neighboring bands, and favors the appearance of correlated insulators at 1/4, 1/2, and 3/4 band filling. Insulating states at 1/4 and 3/4 band filling emerge only in a parallel magnetic field (B_{||}), whereas the resistivity at half band filling is weakly dependent on B_{||}. Our findings suggest that correlated insulators are favored when a moiré flat band is spectrally isolated, and are consistent with a mean-field picture in which insulating states are established by breaking both spin and valley symmetries at 1/4 and 3/4 band filling and valley polarization alone at 1/2 band filling.