Observation of interband collective excitations in twisted bilayer graphene

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

How Magical Is Magic-Angle Graphene?

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.

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.

Recent research on twisted bilayer graphene (TBG) uncovered that its twist-angle-dependent electronic structure leads to a host of unique properties, such as superconductivity, correlated insulating states, and magnetism. The flat bands that emerge at low twist angles in TBG result in sharp features in the electronic density of states, affecting transport. Here we show that they lead to superior and tuneable thermoelectric (TE) performance. Combining an iterative Boltzmann transport equation solver and electronic structure from an exact continuum model, we calculate TE transport properties of TBG at different twist angles, carrier densities, and temperatures. Our simulations show the room-temperature TE power factor (PF) in TBG reaches 40 mW m−1 K−2, significantly higher than single-layer graphene and among the highest reported to date. The peak PF is observed near the magic angle, at a twist angle of ≈1.3∘, and near complete band filling. We elucidate that its dependence is driven by two opposing forces: the band gap between the flat and remote bands suppresses bipolar transport and improves the Seebeck coefficient but the gap decreases with twist angle; on the other hand, the Fermi velocity, which impacts conductivity, is smallest at the magic angle and rises with twist angle. We observed a further increase in the PF of TBG with decreasing temperature. The strong TE performance, along with the ability to fine-tune transport using twist angle, makes TBG an interesting candidate for future research and applications in energy conversion and thermal sensing and management.

Single layer graphene has shown promising applications in fast and sensitive optoelectronic applications such as optical modulators and photodetectors. Twisted bilayer graphene has the potential to further improve the performances of such graphene devices due to its modified electronic band structure. However, the lack of synthesis method for large-area isolated twisted bilayer graphene has limited device applications. After a brief review of recent development in single layer graphene optoelectronic property and photodetectors, we discuss unique optical properties of twisted bilayer graphene. We then show two methods that we have developed to fabricate large-size twisted bilayer graphene islands. The first method uses chemical vapor deposition; the other method utilizes mechanical stacking of two large-size single layer graphene hexagons. The realization of such twisted bilayer graphene samples enables new development of novel graphene-based devices.

Recent experiments on magic-angle twisted bi-layer graphene have attracted intensive attention due to exotic properties such as unconventional superconductivity and correlated insulation. These phenomena were often found at a magic angle less than 1.1°. However, the preparation of precisely controlled bi-layer graphene with a small magic angle is challenging. In this work, electronic properties of large-angle twisted bi-layer graphene (TBG) under pressure are investigated with density functional theory. We demonstrate that large-angle TBG can display flat bands nearby the Fermi level under pressure, which may also induce interesting properties such as superconductivity which have only been found in small-angle TBG at ambient pressure. The Fermi velocity is found to decrease monotonously with pressure for large twisted angles, e.g., 21.8°. Our work indicates that applying pressure provides opportunities for flat-band engineering in larger angle TBG and supports further exploration in related investigations.

<sec>Twisted bilayer graphene (TBG) is a two-dimensional material composed of two layers stacked at a certain angle. When the twisted angle decreases, the lattice mismatch between two layers produces moiré pattern at a long wavelength which significantly modifies the low-energy band structure. In particular, when the twisted angle is close to the so-called “magic angle”, two moiré flat bands are formed near a charge neutral point due to the strong interlayer coupling. These flat bands with high density of states are essential in realizing superconductivity and correlated insulating states. More recently, the magic angle TBG combining an hBN system has exhibited spin-valley polarization when 3/4 of flat bands are filled, thereby providing an ideal platform to achieve quantum anomalous Hall states. Whether it is TBG system or TBG-hBN system, the flat band becomes a crucial condition for discovering so rich physical connotations. Besides the twisted angle, the strain gives an alternative way to modulate flat bands. It has been reported that applying heterostrain in magic angle TBG can makes flat moiré band tunable; strain can also generate flat bands in non-magic angle TBG. Moreover, the reconstruction of TBG due to the strain gives rise to a serial of novel physical phenomena such as topological protected soliton and photonic crystal. Another reason for studying strain effect is that the strain is ubiquitous in the fabrication progress. The strain can also be controlled via piezoelectric substrate which makes possible the in situ modulation of correlated states, topology and quantum effect. </sec><sec>Our work is to study the heterostrain effect in TBG band structure and optical conductivity by using a continuum model. Although the resulting conduction band and valence bands keep connected through Dirac points protected by the <i>C</i><sub>2</sub> symmetry, their separation increases significantly when heterostrain is applied while the Dirac point is also shifted. The optical conductivity is characterized by a series of peaks associated with van Hove singularities, and the peak energies are systematically shifted with the strain amplitude. These changes show that the heterostrain exerts a great influence on electron property of TBG.</sec>

Growing twisted bilayer graphene at small angles

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.

Unconventional superconductivity often arises from Cooper pairing between neighboring atomic sites, stipulating a characteristic pairing symmetry in the reciprocal space. The twisted bilayer graphene (TBG) presents a new setting where superconductivity emerges on the flat bands whose Wannier wavefunctions spread over many graphene unit cells, forming the so-called Moir\'e pattern. To unravel how Wannier states form Cooper pairs, we study the interplay between electronic, structural, and pairing instabilities in TBG. For comparisons, we also study graphene on boron-nitride (GBN) possessing a different Moir\'e pattern, and single-layer graphene (SLG) without a Moir\'e pattern. For all cases, we compute the pairing eigenvalues and eigenfunctions by solving a linearized superconducting gap equation, where the spin-fluctuation mediated pairing potential is evaluated from materials specific tight-binding band structures. We find an extended $s$-wave as the leading pairing symmetry in TBG, in which the nearest-neighbor Wannier sites form Cooper pairs with same phase. In contrast, GBN assumes a $p+ip$-wave pairing between nearest-neighbor Wannier states with odd-parity phase, while SLG has the $d+id$-wave symmetry for inter-sublattice pairing with even-parity phase. Moreover, while $p+ip$, and $d+id$ pairings are chiral, and nodeless, but the extended $s$-wave channel possesses accidental {\it nodes}. The nodal pairing symmetry makes it easily distinguishable via power-law dependencies in thermodynamical entities, in addition to their direct visualization via spectroscopies.

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.

The optical properties of graphene in monolayer and bilayer structure is essential for the development of optical devices viz surface plasmon resonance (SPR) based bio-sensors. The band structure of the twisted bilayer graphene (BLG) is remarkably different than the normal AA or AB stacking. This provides an opportunity to control the optical and electrical properties of BLG by applying an in-plane twist to one of the layer relative to other in a BLG system. Here, we calculated the refractive index (RI) of AA and AB stacking of BLG system using density functional theory. Though the spectrum for AA stacking shows some similarity with that of monolayer graphene, the spectrum for AB stacking was found to be remarkably different. The spectrum of AB stacked layer is red-shifted and the absorption peaks in low energy regime increases nearly by three-folds. A large dependency of the twist angle on RI of twisted BLG were found. Based on the calculation, a schematic of phase diagram showing material behavior of such twisted BLG systems as a function of twist angle and photon energy was constructed. The twisted AA stacked BLG shows largely dielectric behavior whereas the twisted AB stacked BLG shows predominately semimetallic and semiconducting behavior. This study presents a RI landscape of twisted BLG dependent on important parameters viz photon energy and inplane relative twist angle. Our studies will be very useful for the design and development of optical devices employing BLG systems particularly SPR based bio-sensors which essentially measures change in RI due to adsorption of analytes.

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 $\mu_B$.

Interesting electronic properties of a-few-degree-twisted bilayer graphene are caused by the formation of a flat band due to the interlayer interaction and electronic correlation. The authors quantitatively investigated the band structure of wide 3${}^{\mathrm{deg}}$-twisted bilayer graphene with clean interface by angle-resolved photoelectron spectroscopy, and compare the results with those of a band calculation using a recently-developed band unfolding method. The observed band structure indicates strong interlayer coupling that renormalizes the band structure including partial flat band features and gap formation. The observed band structure is in good agreement with the calculated results, indicating the importance of the interlayer coupling for the band renormalization.