ABC Trilayer Graphene Research Articles

Tunneling in ABC trilayer graphene superlattice

Tunneling in ABC trilayer graphene superlattice

Electron/infrared-phonon coupling in ABC trilayer graphene

Stacking order plays a crucial role in determining the crystal symmetry and has significant impacts on electronic, optical, magnetic, and topological properties. Electron-phonon coupling, which is central to a wide range of intriguing quantum phenomena, is expected to be intricately connected with stacking order. Understanding the stacking order-dependent electron-phonon coupling is essential for understanding peculiar physical phenomena associated with electron-phonon coupling, such as superconductivity and charge density waves. In this study, we investigate the effect of stacking order on electron-infrared phonon coupling in graphene trilayers. By using gate-tunable Raman spectroscopy and excitation frequency-dependent near-field infrared nanoscopy, we show that rhombohedral ABC-stacked trilayer graphene has a significant electron-infrared phonon coupling strength. Our findings provide novel insights into the superconductivity and other fundamental physical properties of rhombohedral ABC-stacked trilayer graphene, and can enable nondestructive and high-throughput imaging of trilayer graphene stacking order using Raman scattering.

Behavior of localized states in double twisted ABC trilayer graphene

We show that a double-twisted ABC trilayer graphene, formed by stacking two ABC trilayers with a rotation angle between them, is a semiconductor with a gap of about 30 meV. Importantly, flat bands in the electronic structure are observed even for large angles, and the localization of electrons follows the same pattern as in ABC trilayer graphene. Our first-principles calculations show that this behavior holds at least for rotation angles from 7 to 21 degrees. We further study the charge redistribution as a function of the angle. Below two degrees, the charge escapes from the rotated layers and AA zones, moving to the outer layers. This behavior is general for any ABA trilayer with one of the outer layers twisted (ABT). Our findings shed light on some peculiarities of rotated graphene, explaining the absence of a superconductivity phase on these double-trilayer and ABT systems.

Uniaxial Strain-Induced Stacking Order Change in Trilayer Graphene.

The layer stacking order in two-dimensional heterostructures, like graphene, affects their physical properties and potential applications. Trilayer graphene, specifically ABC-trilayer graphene, has captured significant interest due to its potential for correlated electronic states. However, achieving a stable ABC arrangement is challenging due to its lower thermodynamic stability compared to the more stable ABA stacking. Despite recent advancements in obtaining ABC graphene through external perturbations, such as strain, the stacking transition mechanism remains insufficiently explored. In this study, we unveil a universal mechanism to achieve ABC stacking, applicable for understanding ABA to ABC stacking changes induced by any mechanical perturbations. Our approach is based on a novel strain engineering technique that induces interlayer slippage and results in the formation of stable ABC domains. We investigate the underlying interfacial mechanisms of this stacking change through computational simulations and experiments. Our findings demonstrate a highly anisotropic and significant transformation of ABA stacking to large and stable ABC domains facilitated by interlayer slippage. Through atomistic simulations and local energy analysis, we systematically demonstrate the mechanism for this stacking transition, that is dependent on specific loading orientation. Understanding such a mechanism allows this material system to be engineered by design compatible with industrial techniques on a device-by-device level. We conduct Raman studies to validate and characterize the formed ABC stacking, highlighting its distinct features compared to the ABA region. Our results contribute to a clearer understanding of the stacking change mechanism and provide a robust and controllable method for achieving stable ABC domains, facilitating their use in developing advanced optoelectronic devices.

New local pseudopotential for multilayer carbon materials and its application in wave packet dynamics

Formerly we created a one-electron local pseudopotential which proved to be successful in studying transport phenomena in various sp2 carbon nanosystems, e.g. graphene grain boundaries and nanotube networks. In this work, we present an extended version of the local pseudopotential which correctly describes the electronic structure of van der Waals stacks of carbon sheets, e.g. AA, AB bi-layer graphene, ABC (rhombohedral) tri-layer graphene, as well as AA, AB, and ABC graphite, etc., even in case of external electric fields. We utilize this potential to study the hopping dynamics of wave packets between the graphene sheets in multilayer systems. The frequency of the hopping is proportional to the band splitting, caused by the interlayer coupling. For the case of AB and ABC graphene there is a backscattering near the Fermi level, causing a Zitterbewegung-like phenomenon, an interference between +kBloch and −kBloch states. For the rhombohedral tri-layer graphene the time dependence of the wave packet probability density is an aperiodic function for the first- and third layer (the two outer layers), but a quasi-periodic function for the second (inner) layer.

Role of Local Conductivities in the Plasmon Reflections at the Edges and Stacking Domain Boundaries of Trilayer Graphene.

We employed infrared scattering-type scanning near-field optical microscopy (IR-sSNOM) to study surface plasmon polaritons (SPPs) in trilayer graphene (TLG). Our study reveals systematic differences in near-field IR spectra and SPP wavelengths between Bernal (ABA) and rhombohedral (ABC) TLG domains on SiO2, which can be explained by stacking-dependent intraband conductivities. We also observed that the SPP reflection profiles at ABA-ABC boundaries could be mostly accounted for by an idealized domain boundary defined by the conductivity discontinuity. However, we identified distinct shapes in the SPP profiles at the edges of the ABA and ABC TLG, which cannot be solely attributed to idealized edges with stacking-dependent conductivities. Instead, this can be explained by the presence of various edge structures with local conductivities differing from those of bulk TLGs. Our findings unveil a new structural element that can control SPP, and provide insights into the structures and electronic states of the edges of few-layer graphene.

Spin and orbital metallic magnetism in rhombohedral trilayer graphene

We provide a complete theoretical interpretation of the metallic broken spin/valley symmetry states recently discovered in ABC trilayer graphene (ABC) perturbed by a large transverse displacement field. Our conclusions combine insights from ABC trilayer graphene electronic structure models and mean field theory, and are guided by precise magneto-oscillation Fermi-surface-area measurements. We conclude that the physics of ABC trilayer graphene is shaped by the principle of momentum-space condensation, which favors Fermi surface reconstructions enabled by broken spin/valley flavor symmetries when the single-particle bands imply thin annular Fermi seas. We find one large outer Fermi surface enclosed majority-flavor states and one or more small inner hole-like Fermi surfaces enclosed minority-flavor states that are primarily responsible for nematic order. The smaller surfaces can rotate along a ring of van-Hove singularities or reconstruct into multiple Fermi surfaces with little cost in energy. We propose that the latter property is responsible for the quantum oscillation frequency fractionalization seen experimentally in some regions of the carrier-density/displacement-field phase diagram.

Quick identification of ABC trilayer graphene at nanoscale resolution via a near-field optical route

ABC-stacked trilayer graphene has exhibited a variety of correlated phenomena owing to its relatively flat bands and gate-tunable bandgap. However, convenient methods are still lacking for identifying ABC graphene with nanometer-scale resolution. Here we demonstrate that the scanning near-field optical microscope working in ambient conditions can provide quick recognition of ABC trilayer graphene with no ambiguity and excellent resolution (∼20 nm). The recognition is based on the difference in their near-field infrared (IR) responses between the ABA and ABC trilayers. We show that in most frequencies, the response of the ABC trilayer is weaker than the ABA trilayer. However, near the graphene phonon frequency (∼1585 cm−1), ABC’s response increases dramatically when gated and exhibits a narrow and sharp Fano-shape resonant line, whereas the ABA trilayer is largely featherless. Consequently, the IR contrast between ABC and ABA becomes reversed and can even be striking (ABC/ABA ∼ 3) near the graphene phonon frequency. The observed near-field IR features can serve as a golden rule to quickly distinguish ABA and ABC trilayers with no ambiguity, which could largely advance the exploration of correlation physics in ABC-stacked trilayer graphene.

Magnetic-atom and electric-field induced magnetic phase transitions in ABC-trilayer graphene nanoribbons

Magnetic-atom and electric-field induced magnetic phase transitions in ABC-trilayer graphene nanoribbons

Mott correlations in ABC graphene trilayer aligned with hBN

ABC trilayer graphene aligned with hexagonal boron nitride (ABC/hBN) and other moir\'e systems have shown insulating phases at integer fillings of the moir\'e lattice. The role of Mott physics in these insulating states has been questioned by observations of correlated states in a nonaligned ABC trilayer, but recent photocurrent experiments in aligned samples are consistent with the existence of a Mott insulating state at half filling. Using dynamical mean-field theory we address the effect of Mott correlations in ABC/hBN. We show that, at experimentally relevant interaction values, the electronic states are strongly affected by a significant spectral weight transfer not present without alignment to hBN. This effect, which emerges at interactions considerably smaller than ${U}_{\mathrm{Mott}}$ (at which the Mott metal-insulator transition takes place) and does not require symmetry breaking, also impacts the electronic properties at temperatures above the ordering transitions producing anomalous temperature and doping dependencies. Close to the Mott transition, the intramoir\'e unit cell interactions promote an antiferromagnetic state, probably breaking the ${C}_{3}$ symmetry, that would compete with the ferromagnetic exchange interactions to determine the ground state.

Competing orders and cascade of degeneracy lifting in doped Bernal bilayer graphene

Motivated by recent experiments [H. Zhou et al., Science 375, 774 (2022) and S. C. de la Barrera et al., arXiv:2110.13907], here we propose a general mechanism for valley and/or spin degeneracy lifting of the electronic bands in doped Bernal bilayer graphene, subject to electric displacement ($D$) fields. A $D$-field induced layer polarization (LP), when accompanied by a Hubbard repulsion-driven layer antiferromagnet (LAF) and next-nearest-neighbor repulsion-driven quantum anomalous Hall (QAH) orders, lifts the fourfold degeneracy of electronic bands, yielding a quarter metal for small doping, as also observed in ABC trilayer graphene. With the disappearance of the QAH order, electronic bands recover twofold valley degeneracy, thereby forming a conventional or compensated (with majority and minority carriers) half metal at moderate doping, depending on the relative strength of LP and LAF. At even higher doping and for a weak $D$ field only LAF survives and the Fermi surface recovers fourfold degeneracy. We also show that a pure repulsive electronic interaction mediated triplet $f$-wave pairing emerges from a parent correlated nematic liquid or compensated half metal when an in-plane magnetic field is applied to the system.

Proximity spin-orbit and exchange coupling in ABA and ABC trilayer graphene van der Waals heterostructures

We investigate the proximity spin-orbit and exchange couplings in ABA and ABC trilayer graphene encapsulated within monolayers of semiconducting transition-metal dichalcogenides and the ferromagnetic semiconductor Cr$_2$Ge$_2$Te$_6$. Employing first-principles calculations we obtain the electronic structures of the multilayer stacks and extract the relevant proximity-induced orbital and spin interaction parameters by fitting the low-energy bands to model Hamiltonians. We also demonstrate the tunability of the proximity effects by a transverse electric field. Using the model Hamiltonians we also study mixed spin-orbit/exchange coupling encapsulation, which allows to tailor the spin interactions very efficiently by the applied field. We also summarize the spin-orbit physics of bare ABA, ABC, and ABB trilayers, and provide, along with the first-principles results of the electronic band structures, density of states, spin splittings, and electric-field tunabilities of the bands, qualitative understanding of the observed behavior and realistic model parameters as a resource for model simulations of transport and correlation physics in trilayer graphene.

Magnetic-Atom and Electric-Field Induced Magnetic Phase Transitions in Abc-Trilayer Graphene Nanoribbons

Magnetic-Atom and Electric-Field Induced Magnetic Phase Transitions in Abc-Trilayer Graphene Nanoribbons

Flat bands, electron interactions, and magnetic order in magic-angle mono-trilayer graphene

Starting with twisted bilayer graphene, graphene-based moir\'e materials have recently been established as a new platform for studying strong electron correlations. In this paper, we study twisted graphene monolayers on trilayer graphene and demonstrate that this system can host flat bands when the twist angle is close to the magic angle of $1.{16}^{\ensuremath{\circ}}$. When monolayer graphene is twisted on ABA trilayer graphene, the flat bands are not isolated, but are intersected by a Dirac cone with a large Fermi velocity. In contrast, graphene twisted on ABC trilayer graphene (denoted AtABC) exhibits a gap between flat and remote bands. Since ABC trilayer graphene and twisted bilayer graphene are known to host broken-symmetry phases, we further investigate the ostensibly similar magic-angle AtABC system. We study the effect of electron-electron interactions in AtABC using both Hartree theory and an atomic Hubbard theory to calculate the magnetic phase diagram as a function of doping, twist angle, and perpendicular electric field. Our analysis reveals a rich variety of magnetic orderings, including ferromagnetism and ferrimagnetism, and demonstrates that a perpendicular electric field makes AtABC more susceptible to magnetic ordering.

Evidence for Moiré Trions in Twisted MoSe2 Homobilayers.

Moiré superlattices of van der Waals structures offer a powerful platform for engineering band structure and quantum states. For instance, Moiré superlattices in magic-angle twisted bilayer graphene, ABC trilayer graphene have been shown to harbor correlated insulating and superconducting states, while in transition metal dichalcogenide (TMD) twisted bilayers, Moiré excitons have been identified. Here we show that the effects of a Moiré superlattice on the band structure are general: In TMD twisted bilayers, excitons and exciton complexes can be trapped in the superlattice in a manner analogous to ultracold bosonic or Fermionic atoms in optical lattices. Using twisted MoSe2 homobilayers as a model system, we present evidence for Moiré trions. Our results thus open possibilities for designer van der Waals structures hosting arrays of Fermionic or bosonic quasiparticles, which can be used to realize tunable many-body states crucial for quantum simulation and quantum information processing.

Quantum Hall spin liquids and their possible realization in moiré systems

Recently, Chern insulators with Chern numbers $C=1$ and 2 in zero (or very small) magnetic field have been observed in two moire graphene systems: twisted bilayer graphene and ABC trilayer graphene, both aligned with a hexagonal boron-nitride (h-BN) substrate. These Chern insulator states arise due to many-body effects in the Chern bands of these systems when they are partially filled to a total integer filling ${\ensuremath{\nu}}_{T}=1,3$ (including spin and valley degrees of freedom). A simple possible explanation is from Hartree-Fock mean-field theory which predicts valley and spin polarization in the zero bandwidth limit, similar to the ``quantum Hall ferromagnetism'' in Landau levels. Though valley polarization is implied by the existing experiments, the fate of the spin degree of freedom is not presently clear. In this paper, we propose alternative valley polarized---but not spin polarized---candidates for the observed QAH effect. For a valley polarized spinful Chern band at filling ${\ensuremath{\nu}}_{T}=1$, we describe a class of exotic Chern insulator phases through spin-charge separation: charge is in a conventional Chern insulator phase with quantized Hall conductivity, while the spin forms disordered spin liquid phase with fractionalization, which we dub quantum Hall spin liquids. We construct a simple class of ${Z}_{2}$ quantum Hall spin liquid as analogs of the familiar ${Z}_{2}$ spin liquid through slave fermion-Schwinger boson parton theory. Condensation of the spinon from the ${Z}_{2}$ quantum Hall spin liquid can lead to a quantum Hall antiferromagnet which is yet another, less exotic, candidate for the experimentally observed Chern insulator. We offer several experimental proposals to probe the quantum Hall spin liquid and the quantum Hall antiferromagnetic phases. We briefly comment on the generalization to the filling ${\ensuremath{\nu}}_{T}=2$ and propose possible quantum valley Hall spin liquid without full spin polarization. Finally, we also propose another class of QHSL using fermionic spinons, including phases supporting non-Abelian anyons.

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.

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.

Large-area epitaxial growth of curvature-stabilized ABC trilayer graphene

The properties of van der Waals (vdW) materials often vary dramatically with the atomic stacking order between layers, but this order can be difficult to control. Trilayer graphene (TLG) stacks in either a semimetallic ABA or a semiconducting ABC configuration with a gate-tunable band gap, but the latter has only been produced by exfoliation. Here we present a chemical vapor deposition approach to TLG growth that yields greatly enhanced fraction and size of ABC domains. The key insight is that substrate curvature can stabilize ABC domains. Controllable ABC yields ~59% were achieved by tailoring substrate curvature levels. ABC fractions remained high after transfer to device substrates, as confirmed by transport measurements revealing the expected tunable ABC band gap. Substrate topography engineering provides a path to large-scale synthesis of epitaxial ABC-TLG and other vdW materials.

Spin liquids and pseudogap metals in the SU(4) Hubbard model in a moiré superlattice

Motivated by the realization of spin-valley Hubbard on triangular moir\'e superlattice in ABC trilayer graphene aligned with hexagon boron nitride (hBN) and possibly also in twisted transition metal dichalcogenide homobilayers, we study possible Mott insulating phases and pseudogap metals based on symmetry constraint and parton mean field theories. First we show that Luttinger constraint allows two distinct symmetric and featureless Fermi liquids when there is an inter-valley Hund's term breaking $SU(4)$ spin rotation. Especially, there exists a symmetric and featureless "pseudogap metal" with small Fermi surfaces. Then we suggest to search for such an unconventional metallic state by doping the Mott insulator at $\nu_T=2$. For this purpose, we study the $\nu_T=2$ Mott insulator using $SO(6)$ Schwinger boson or Schwinger fermion parton. At the $SU(4)$ symmetric point, we find two symmetric $Z_2$ spin liquids. With a large anti inter-valley Hund's term, a featureless Mott insulator is natural. Next we show that doping the featureless Mott insulator or a $Z_2$ spin liquid can lead to featureless or orthogonal "pseudogap metal" with small Fermi surfaces proportional to the doping. Besides, we also provide one scenario for the evolution from "pseudogap metal" to the conventional Fermi liquid through an intermediate exotic "deconfined metal" phase. Last, we give brief comments on the possibility of $U(1)$ spinon fermi surface state or $Z_4$ spin liquid at $\nu_T=1$.