Kondo effect in twisted bilayer graphene

The recent discovery of superconductivity in magic-angle twisted bilayer graphene (TBLG) has sparked a renewed interest in the strongly-correlated physics of sp 2 carbons, in stark contrast to preliminary investigations which were dominated by the one-body physics of the massless Dirac fermions. We thus provide a self-contained, theoretical perspective of the journey of graphene from its single-particle physics-dominated regime to the strongly-correlated physics of the flat bands. Beginning from the origin of the Dirac points in condensed matter systems, we discuss the effect of the superlattice on the Fermi velocity and Van Hove singularities in graphene and how it leads naturally to investigations of the moiré pattern in van der Waals heterostructures exemplified by graphene-hexagonal boron-nitride and TBLG. Subsequently, we illuminate the origin of flat bands in TBLG at the magic angles by elaborating on a broad range of prominent theoretical works in a pedagogical way while linking them to available experimental support, where appropriate. We conclude by providing a list of topics in the study of the electronic properties of TBLG not covered by this review but may readily be approached with the help of this primer.

Bilayer graphene has been predicted to host a moiré miniband with flat dispersion if the layers are stacked at specific twist angles known as the ’magic angles’1,2. Recently, twisted bilayer graphene (tBLG) with a magic angle twist was reported to exhibit a correlated insulating state and superconductivity3,4, where the presence of the flat miniband in the system is thought to be essential for the emergence of these ordered phases in the transport measurements. Although tunnelling spectroscopy5,6,7,8,9 and electronic compressibility measurements10 in tBLG have found a van Hove singularity that is consistent with the presence of the flat miniband, a direct observation of the flat dispersion in the momentum space of such a moiré miniband in tBLG is still lacking. Here, we report the visualization of this flat moiré miniband by using angle-resolved photoemission spectroscopy with nanoscale resolution. The high spatial resolution of this technique enabled the measurement of the local electronic structure of the tBLG. The measurements demonstrate the existence of the flat moiré band near the charge neutrality for tBLG close to the magic angle at room temperature.

An uncertainty in studying twisted bilayer graphene (TBG) is the minimum-energy geometry, which strongly affects the electronic structure. The minimum-energy geometry is determined by the potential-energy surface, which is dominated by van der Waals (vdW) interactions. In this paper, large-scale diffusion quantum Monte Carlo (QMC) simulations are performed to evaluate the energy of bilayer graphene at various interlayer distances for four stacking registries. An accurate registry-dependent potential is fit to the QMC data and is used to describe interlayer interactions in the geometry of near-magic-angle TBG. The band structure for the optimized geometry is evaluated using the accurate local-environment tight-binding model. We find that compared to QMC, DFT-based vdW interactions can result in errors in the corrugation magnitude by a factor of 2 or more near the magic angle. The error in corrugation then propagates to the flat bands in twisted bilayer graphene, where the error in corrugation can affect the bandwidth by about 30% and can change the nature and degeneracy of the flat bands.

Magic-angle twisted bilayer graphene exhibits quasi-flat low-energy bands with Van Hove singularities close to the Fermi level. These singularities play an important role in the exotic phenomena observed in this material, such as superconductivity and magnetism, by amplifying electronic correlation effects. In this work, we study the correspondence of four-terminal conductance and the Fermi surface topology as a function of the twist angle, pressure, and energy in mesoscopic, ballistic samples of small-angle twisted bilayer graphene. We establish a correspondence between features in the wide-junction conductance and the presence of van Hove singularities in the density of states. Moreover, we identify additional transport features, such as a large, pressure-tunable minimal conductance, conductance peaks coinciding with non-singular band crossings, and unusually large conductance oscillations as a function of the system size. Our results suggest that twisted bilayer graphene close the magic angle is a unique system featuring simultaneously large conductance due to the quasi-flat bands, strong quantum nonlinearity due to the Van Hove singularities and high sensitivity to external parameters, which could be utilized in high-frequency device applications and sensitive detectors.

Twisted graphene bilayers (TGBs) have low-energy van Hove singularities (VHSs) that are strongly localized around AA-stacked regions of the moire pattern. Therefore, they exhibit novel many-body electronic states, such as Mott-like insulator and unconventional superconductivity. Unfortunately, these strongly correlated states were only observed in magic angle TGBs with the twist angle theta~1.1{\deg}, requiring a precisely tuned structure. Is it possible to realize exotic quantum phases in the TGBs not limited at the magic angle? Here we studied electronic properties of a TGB with theta~1.64{\deg} and demonstrated that a VHS splits into two spin-polarized states flanking the Fermi energy when the VHS is close to the Fermi level. Such a result indicates that localized magnetic moments emerge in the AA-stacked regions of the TGB. Since the low-energy VHSs are quite easy to be reached in slightly TGBs, our result therefore provides a facile direction to realize novel quantum phases in graphene system.

The discovery of different phases as a result of correlations, especially in low-dimensional materials, has been always an exciting and fundamental subject of research. Recent experiments on twisted bilayer graphene have revealed reentrant unconventional superconductivity as a function of doping as well as a Mott-like insulating phase when the two layers are twisted with respect to each other at certain "magic" angles for doping corresponding to two particles per moire unit cell. In this work we propose a microscopic model that takes into account interactions and the van Hove singularities in the density of states of the twisted bilayer graphene at doping corresponding to one particle ($\nu$ =1) per moir\'{e} unit cell and study how superconductivity emerges. We identify the possible symmetry of the order parameter as $s^{\pm}$, while if the inter-valley coupling is negligible the symmetry is $s^{++}$. In addition, we find and characterise the insulating region of the system, as a region with a uniform charge instability where there is coexistence of the metallic and insulating phases.

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.

The novel electronic properties of bilayer graphene can be fine-tuned via twisting, which may induce flat bands around the Fermi level with nontrivial topology. In general, the band structure of such twisted bilayer graphene (TBG) can be theoretically obtained by using first-principles calculations, tight-binding method, or continuum model, which are either computationally demanding or parameters dependent. In this work, by using the sure independence screening sparsifying operator method, we propose a physically interpretable three-dimensional (3D) descriptor which can be utilized to readily obtain the Γ-point gap of TBG at arbitrary twist angles and different interlayer spacings. The strong predictive power of the descriptor is demonstrated by a high Pearson coefficient of 99% for both the training and testing data. To go further, we adopt the neural network algorithm to accurately probe the flat bands of TBG at various twist angles, which can accelerate the study of strong correlation physics associated with such a fundamental characteristic, especially for those systems with a larger number of atoms in the unit cell.

Motivated by the recent discovery of the Mott insulating phase and unconventional superconductivity due to the flat bands in twisted bilayer graphene, we propose more generic ways of getting two-dimensional (2D) emergent flat-band lattices using either 2D Dirac materials or ordinary electron gas (2DEG) subject to moderate periodic orbital magnetic fields with zero spatial average. We find stark contrast between Schr\"odinger and Dirac electrons, i.e., the former show recurring ``magic'' values of the magnetic field when the lowest band becomes flat, whereas, for the latter, the zero-energy bands are asymptotically flat without magicness. By examining the Wannier functions localized by the smooth periodic magnetic fields, we are able to explain these nontrivial behaviors using minimal tight-binding models on a square lattice. In particular the magicness of the 2DEG can be understood in terms of destructive quantum interference similar to classic flat-band lattice models. The two cases can be interpolated by varying the $g$ factor or effective mass of a 2DEG and by taking into account the Zeeman coupling, which also leads to flat bands with nonzero Chern numbers for each spin. Our paper provides flexible platforms for exploring interaction-driven phases in 2D systems with on-demand superlattice symmetries.

We introduce twisted trilayer graphene (tTLG) with two independent twist angles as an ideal system for the precise tuning of the electronic interlayer coupling to maximize the effect of correlated behaviors. As established by experiment and theory in the related twisted bilayer graphene system, van Hove singularities (VHS) in the density of states can be used as a proxy of the tendency for correlated behaviors. To explore the evolution of VHS in the twist-angle phase space of tTLG, we present a general low-energy electronic structure model for any pair of twist angles. We show that the basis of the model has infinite dimensions even at a finite energy cutoff and that no Brillouin zone exists even in the continuum limit. Using this model, we demonstrate that the tTLG system exhibits a wide range of magic angles at which VHS merge and that the density of states has a sharp peak at the charge-neutrality point through two distinct mechanisms: the incommensurate perturbation of twisted bilayer graphene's flatbands or the equal hybridization between two bilayer moiré superlattices.

A variety of new and interesting electronic properties have been predicted in graphene monolayer doped to Van Hove singularities (VHSs) of its density of state. However, tuning the Fermi energy to reach a VHS of graphene by either gating or chemical doping is prohibitively difficult, owing to their large energy distance ($\ensuremath{\sim}3$ eV). This difficulty can be easily overcome in twisted bilayer graphene (TBG). By introducing a small twist angle between two adjacent graphene sheets, we are able to generate two low-energy VHSs arbitrarily approaching the Fermi energy. Here, we report experimental studies of electronic properties around the VHSs of a slightly TBG through scanning tunneling microscopy measurements. The split of the VHSs is observed and the spatial symmetry breaking of electronic states around the VHSs is directly visualized. These exotic results provide motivation for further theoretical and experimental studies of graphene systems around the VHSs.

Twisted multilayer structures are considered a helpful framework for exploring strongly correlated many-particle systems, where phenomena of physics correlation emerge. Here, we present the Trotter–Suzuki tight-binding time propagation method for computing the density of states (DOS) in a 30° twisted bilayer graphene (TBG). This method solves the time-dependent Schrödinger equation by decomposing Hamiltonian matrices to derive the correlation function. The Fourier transform of this correlation function yields the DOS of the system up to ≈ 1000 000 atoms. Our calculation proves that geometry makes an outstanding contribution to our final results. Additionally, applying additional gate voltage induces shifts in Van Hove singularities, potentially leading to the emergence of new states at the Fermi energy level. The results demonstrate that TBG systems can be easily adjusted and modified for further investigation of optoelectronic features.

Graphene, a one-atom-thick material, has been a wonder material since its discovery because of its superlative electronic, mechanical, and optical properties. When a layer of graphene is rotated over another layer, it exhibits many intriguing behaviors, ranging from superconductivity to the anomalous Hall effect to ferromagnetism at a magic angle of 1°, and hence the twisted bilayer graphene has been the subject of intense research in recent years. The surge in interest in this moiré structure can be attributed to the emergence of electronic flat minibands near the magic angle. Here, we studied the electronic and magnetic properties of twisted bilayer graphene nanoflakes using first-principles density functional theory. In my first project, we found that when a layer of graphene nanoflakes is rotated over another layer, the nonmagnetic energy gap of the flakes changes due to the change in interlayer interaction between the layers. When the energy gap approaches zero, an electronic instability occurs, and a ferromagnetic gap state emerges spontaneously to lower the energy. Unlike the observed ferromagnetism at a magic angle in the graphene bilayer, we noticed the ferromagnetic phase appearing aperiodically between 0 and 30° in the twisted nanoflakes. The origin of electronic instability at various twist angles is ascribed to the several higher-symmetry phases that are broken to lower the energy resulting from an aperiodic modulation of the interlayer interaction in the nanoflakes. Besides unraveling a spin-pairing mechanism for the reappearance of the nonmagnetic phase, we found orbitals at the boundary of nanoflakes contributing to ferromagnetism. Subsequently, we investigated the role of electronic pressure on the magic-twist angle. Here, using first-principles density functional theory, we demonstrated that the magic twist angle in a bilayer graphene nanoflake at which the transition from a nonmagnetic to a ferromagnetic phase occurs can be tuned by exerting uniaxial electronic pressure. Electronic pressure, which provides another route to control the inter-layer electronic coupling energy, is simulated by varying the interlayer spacing in the nanoflake. In my third project, we designed more than 4348 Fe-based bimetallic chalcogenides and studied their magnetic properties using density functional theory. Further, we designed and developed a stacked machine learning model to predict the magnetic moment in all possible configurations of Fe-based bimetallic chalcogenides made of transition metals Ni, Co, Cr, or Mn and chalcogen elements S, Se, or Te. The final stacked machine learning model predicts the magnetic moment with an R2 score of 92%.

We report the calculation of the Density of States (DOS) in Twisted Bilayer Graphene (TBG) system for low energy range near Fermi level using commensurate rotation of 1.16, 1.79 and 3.48. We obtain a logarithmic behaviour of DOS showing that van Hove Singularities (VHS) are occurred in this system and its position depends on twisted angle. This behaviour coming from the saddle points (SP) of band dispersion of TBG. The position of VHS will move away from Fermi level along with the increase of twisted angle. This result can be used to calculate others physical quantities such as optical conductivity or critical temperature of superconductivity that may occur in TBG.

Bilayer graphene, which forms moiré superlattices, possesses distinct electronic and optical properties owing to its hybridized energy band and the emergence of van Hove singularities depending on its twist angle. Extensive research has been conducted on the global characteristics of moiré superlattices induced by their long-range periodicity. However, the local properties, which differ owing to the variations in the three-dimensional atomic arrangement, within a moiré unit cell have been rarely explored. In this study, we demonstrate the highly localized excitation of carbon 1s electrons to unoccupied van Hove singularities in twisted bilayer graphene by electron energy loss spectroscopy using a monochromated transmission electron microscope. The core-level excitations associated with the van Hove singularities exhibit a systematic twist-angle dependence analogous to optical excitations. Furthermore, local variations in the core-level van Hove singularity peaks, which can originate from the core-exciton lifetimes and band modifications corresponding to the local stacking geometry within a moiré unit cell, are unambiguously corroborated.