Lattice relaxation and energy band modulation in twisted bilayer graphene

Moiré is More: Access to New Properties of Two-Dimensional Layered Materials

How Magical Is Magic-Angle Graphene?

The twisted graphene provides a unique structural model distinct from the traditional Bernal stacking. As a new degree of freedom, the twist angle in graphene leads to the moire superlattice together with the re-constructed electronic band structures, which received considerable attention. Most of the previous reports about twisted graphene focused on theoretical analysis. With the recent development of synthetic methods, the performance of twisted graphene has been heavily investigated. Specifically, the emergent superconductivity and ferromagnetism exhibited by the “magic-angle” twisted bilayer graphene has aroused tremendous attention since these properties have never been proved by carbon-based materials. Moreover, these unpredicted properties indicate challenging puzzles to be explored brought by the twisted graphene. This review majorly focuses on the synthesis methods, electronic structures, characterizations and properties of twisted graphene, especially “magic-angle” twisted bilayer graphene, aiming to provide a comprehensive summary about the progress of twisted graphene and its related research advances. In the early experimental research, twisted graphene is mostly obtained by folding single-layer graphene, or accidental discovered in the graphene samples synthesized by chemical vapor deposition. These uncontrollable methods are soon eliminated because of the rapid development of micro-nano machining related technology. By stacking a single layer graphene onto another at a target angle, the twisted angle between graphene layers can be controlled with a high precision. Then, the theoretical researches about electronic structures of twisted graphene are presented, showing great dependence on the twisted angles especially at low angles. Importantly, the twist in bilayer graphene leads to the re-construction of the Brillouin zone and decreasing of Fermi velocity ( v F), pointing out a series of “magic angles” as v F=0. At these specific angles, twisted bilayer graphene possess flat bands around Fermi level and localized density of states, which has been confirmed by experimental results. Considering the importance of twisted angles, some characterization methods are introduced in this review based on the moire superlattice, the energy gaps between van Hove singularities or the gap-induced insulation state in transport properties of twisted graphene. The latter is majorly applied for fabricating devices containing twisted bilayer graphene packaged with hexagonal boron nitride (hBN). Following, the emergent correlated insulating phase, superconducting phase, topological phases and ferromagnetism in “magic-angle” twisted bilayer graphene are systematically classified. The rich phase diagram of twisted graphene declares a desirable platform for the exploration of many-body physics, gaining lots of efforts to reveal the phase transition and the relationships between different phases. Some possible mechanisms about the emergent unconventional superconductivity and orbital-based ferromagnetism are summarized. The progress of related experiments, characterizations and simulations together with their discussions are interpreted in this article. The results indicate the package layers, mostly hBN, have significant influences on the electronic band structure and performances of twisted graphene. These influences are previously ignored, but now regarded as the key for unveiling the distinct properties of twisted graphene. By tuning the package layers, the superconductivity is achieved at lower twisted angles than other “magic-angle”. At the end of this review, we discuss about the existing challenges and prospects in the related fields, which may give rise to great changes in physics and materials science.

Received 19 February 2020Accepted 20 February 2020DOI:https://doi.org/10.1103/PhysRevB.101.099901©2020 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBand gapDomainsElasticityElectronic structurePhysical SystemsGrapheneTechniquesTight-binding modelCondensed Matter, Materials & Applied Physics

Growing twisted bilayer graphene at small angles

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.

Twisted bilayer graphene (TBG), which has drawn much attention in recent years, arises from van der Waals materials gathering each component together via van der Waals force. It is composed of two sheets of graphene rotated relatively to each other. Moiré potential, resulting from misorientation between layers, plays an essential role in determining the band structure of TBG, which directly relies on the twist angle. Once the twist angle approaches a certain critical value, flat bands will show up, indicating the suppression of kinetic energy, which significantly enhances the importance of Coulomb interaction between electrons. As a result, correlated states like correlated insulators emerge from TBG. Surprisingly, superconductivity in TBG is also reported in many experiments, which drags researchers into thinking about the underlying mechanism. Recently, the interest in the atomic reconstruction of TBG at small twist angles comes up and reinforces further understandings of properties of TBG. In addition, twisted multilayer graphene receives more and more attention, as they could likely outperform TBG although they are more difficult to handle experimentally. In this review, we mainly introduce theoretical and experimental progress on TBG. Besides the basic knowledge of TBG, we emphasize the essential role of atomic reconstruction in both experimental and theoretical investigations. The consideration of atomic reconstruction in small-twist situations can provide us with another aspect to have an insight into physical mechanism in TBG. In addition, we cover the recent hot topic, twisted multilayer graphene. While the bilayer situation can be relatively easy to resolve, multilayer situations can be really complicated, which could foster more unique and novel properties. Therefore, in the end of the review, we look forward to future development of twisted multilayer graphene.

Close to a magical angle, twisted bilayer graphene (TBLG) systems exhibit isolated flat electronic bands and, accordingly, strong electron localization. TBLGs have hence been ideal platforms to explore superconductivity, correlated insulating states, magnetism, and quantized anomalous Hall states in reduced dimension. Below a threshold twist angle (∼1.1∘), the TBLG superlattice undergoes lattice reconstruction, leading to a periodic moiré structure which presents a marked atomic corrugation. Using a tight-binding framework, this research demonstrates that superlattice reconstruction affects significantly the electronic structure of small-angle TBLGs. The first magic angle at ∼1.1∘ is found to be a critical case presenting globally maximized electron localization, thus separating reconstructed TBLGs into two classes with clearly distinct electronic properties. While low-energy Dirac fermions are still preserved at large twist angles >1.1∘ , small-angle ( ≲1.1° ) TBLG systems present common features such as large spatial variation and strong electron localization observed in unfavorable AA stacking regions. However, for small twist angles below 1.1∘, the relative contribution of the local AA regions is progressively reduced, thus precluding the emergence of further magic angles, in very good agreement with existing experimental evidence.

We have used ab initio density functional theory, incorporating van der Waals corrections, to study twisted bilayer graphene (TBLG) where Stone-Wales defects or monovacancies are introduced in one of the layers. We compare these results to those for defects in single layer graphene or Bernal stacked graphene. The energetics of defect formation is not very sensitive to the stacking of the layers or the specific site at which the defect is created, suggesting a weak interlayer coupling. However signatures of the interlayer coupling are manifested clearly in the electronic band structure. For the "$\gamma\gamma$" Stone Wales defect in TBLG, we observe two Dirac cones that are shifted in both momentum space and energy. This up/down shift in energy results from the combined effect of a charge transfer between the two graphene layers, and a chemical interaction between the layers, which mimics the effects of a transverse electric field. Charge density plots show that states near the Dirac points have significant admixture between the two layers. For Stone Wales defects at other sites in TBLG, this basic structure is modified by the creation of mini gaps at energy crossings. For a monovacancy, the Dirac cone of the pristine layer is shifted up in energy by $\sim0.25$ eV due to a combination of the requirements of the equilibration of Fermi energy between the two layers with different numbers of electrons, charge transfer, and chemical interactions. Both kinds of defects increase the density of states at the Fermi level. The monovacancy also results in spin polarization, with magnetic moments on the defect of 1.2 - 1.8 $\mu_B$.

When two two-dimensional (2D) materials with different lattice constants or with different rotation angles are superimposed, a moiré superlattice can be constructed. The electronic properties of the superlattice are strongly dependent on the stacking configuration, twist angle and substrate. For instance, theoretically, when the rotation angle of twisted bilayer graphene is reduced to a set of specific values, the so-called magic angles, flat bands appear near the charge neutrality, and the electron-electron interaction is significantly enhanced. The Mott insulator and unconventional superconductivity are detected in the twisted bilayer graphene with a twist angle around 1.1°. For a moiré pattern with a large enough periodicity, lattice relaxation caused by an interplay between van der Waals force and the in-plane elasticity force comes into being. The atomic relaxation forces atoms to deviate from their equilibrium positions, and thus making the system reconstructed. This review mainly focuses on the effects of the lattice relaxation and substrates on the electronic properties of the graphene superlattices. From both theoretical and experimental point of view, the lattice relaxation effects on the atomic structure and electronic properties of graphene-based superlattices, for example, the twisted bilayer graphene, twisted trilayer graphene, graphene-hexagonal boron nitride superlattice and twisted bilayer graphene-boron nitride superlattice are discussed. Finally, a summary and perspective of the investigation of the 2D material superlattice are presented.

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

It is now well established theoretically and experimentally that a Moiré pattern, due to a rotation of two atomic layers with respect to each other, creates low-energy flat bands. First discovered in twisted bilayer graphene, these new electronic states are at the origin of strong electronic correlations and even of unconventional superconductivity. Twisted bilayers (tb) of transition metal dichalcogenides (TMDs) also exhibit flat bands around their semiconductor gap at small rotation angles. In this paper, we present a DFT study to analyze the effect of the atomic relaxation on the low-energy bands of tb-MoS2 with a rotation angle of 5.09°. We show that in-plane atomic relaxation is not essential here, while out-of-plane relaxation dominates the electronic structure. We propose a simple and efficient atomic model to predict this relaxation.

Twisted bilayer graphene (tBLG) has received substantial attention in various research fields due to its unconventional physical properties originating from Moiré superlattices. The electronic band structure in tBLG modified by interlayer interactions enables the emergence of low-energy van Hove singularities in the density of states, allowing the observation of intriguing features such as increased optical conductivity and photocurrent at visible or near-infrared wavelengths. Here, we show that the third-order optical nonlinearity can be considerably modified depending on the stacking angle in tBLG. The third-harmonic generation (THG) efficiency is found to significantly increase when the energy gap at the van Hove singularity matches the three-photon resonance of incident light. Further study on electrically tuneable optical nonlinearity reveals that the gate-controlled THG enhancement varies with the twist angle in tBLG, resulting in a THG enhanced up to 60 times compared to neutral monolayer graphene. Our results prove that the twist angle opens up a new way to control and increase the optical nonlinearity of tBLG, suggesting rotation-induced tuneable nonlinear optics in stacked two-dimensional material systems.

The creation of van der Waals heterostructures based on a graphene monolayer and other two-dimensional crystals has attracted great interest because the atomic registry of the two-dimensional crystals can modify the electronic spectra and properties of graphene. A twisted graphene bilayer can be viewed as a special van der Waals structure composed of two mutually misoriented graphene layers, where the sublayer graphene not only plays the role of a substrate, but also acts in an equivalent role as the top graphene layer in the structure. Here we report the electronic spectra of slightly twisted graphene bilayers studied by scanning tunneling microscopy and spectroscopy. Our experiment demonstrates that twist-induced van Hove singularities are ubiquitously present for rotation angles \ensuremath{\theta} of less than about 3.5\ifmmode^\circ\else\textdegree\fi{}, corresponding to moir\'e-pattern periods $D$ longer than 4 nm. However, they totally vanish for \ensuremath{\theta}>5.5\ifmmode^\circ\else\textdegree\fi{} $(D<2.5\text{nm})$. Such a behavior indicates that the continuum models, which capture moir\'e-pattern periodicity more accurately at small rotation angles, are no longer applicable at large rotation angles.

Twisted bilayer graphene (tBLG) has emerged these years with the growing research interests in the strongly correlated insulating phase, unconventional superconducting transition, and their tunable electronic properties. Hydrostatic pressure can effectively modulate the interlayer interactions in tBLG, thereby altering its electronic band structure and related physical properties. Herein, a pressure-engineering strategy has been developed to regulate the carrier transport behavior of a 2D tBLG nanodevice, fabricated via our established approach for in situ electrical measurements under high pressure. The obtained electrical results revealed that the tBLG (twist angle 1.3 ± 0.1°) device presented semiconducting behavior under different pressures. The strongly correlated state emerged in tBLG under hole doping within a rather narrow pressure window (∼4.1 GPa). Additionally, the extracted activation energy reached a maximum around 2.0 GPa. These findings provide valuable insights into pressure-tuned effective magic angle and further delving into the electronic properties of 2D twisted electronic systems.