We theoretically argue that, in doped AB bilayer graphene, the electron-electron coupling can give rise to the spontaneous formation of fractional metal phases. These states, being generalizations of a more common half-metal, have a Fermi surface that is perfectly polarized not only in terms of a spin-related quantum number, but also in terms of the valley index. The proposed mechanism assumes that the ground state of undoped bilayer graphene is a spin density wave insulator, with a finite gap in the single-electron spectrum. Upon doping, the insulator is destroyed, and replaced by a fractional metal phase. As doping increases, transitions between various types of fractional metal (half-metal, quarter-metal, etc.) are triggered. Our findings are consistent with recent experiments on doped AB bilayer graphene, in which a cascade of phase transitions between different isospin states was observed.
We study the localized magnetic states of impurity in biased bilayer and trilayer graphene. It is found that the magnetic boundary for bilayer and trilayer graphene presents the mixing features of Dirac and conventional fermion. For zero gate bias, as the impurity energy approaches the Dirac point, the impurity magnetization region diminishes for bilayer and trilayer graphene. When a gate bias is applied, the dependence of impurity magnetic states on the impurity energy exhibits a different behavior for bilayer and trilayer graphene due to the opening of a gap between the valence and the condu…
We present a simple derivation of a continuum Hamiltonian for bilayer graphene with an arbitrary smooth lattice deformation -- technically in a fashion parametrized by displacement fields with small gradients. We show that this subsumes the continuum model of Bistritzer and Macdonald for twisted bilayer grapheneas well as many generalizations and extensions of it. The derivation is carried out entirely in real space.
We show that the fractional statistics of quasiparticles in dimerized graphene, in recent proposals for charge and statistics fractionalization, can have two realizations depending whether elementary objects can be considered as point-like or extended objects. Therefore, there are two phases of proposed excitations and we give their topological descriptions with their respective statistics. We propose that a natural setting for fractionalization are certain systems with excitonic instabilities and demonstrate this by an example of graphene bilayer.
We have investigated the Coulomb screening properties and collective excitations in a graphene bilayer. The static screening effect is anisotropic and is much stronger in the undoped graphene bilayer than in a monolayer graphene [1]. The dynamic screening shows the properties of a Dirac gas in the low frequency and that of a Fermi gas in the high frequency. The transition from the Dirac to the Fermi gas is also observed in the plasmon spectrum. Finally, we find that an electron gas in a doped graphene bilayer has quite similar properties as those of a Fermi gas in materials containing two ener…
We discuss t-J-U model on a honeycomb monolayer that has the same low-energy description of the kinetic term as graphene bilayer, and in particular study coexistence of antiferromagnetism and superconducting correlations that originate from Cooper pairs without phase coherence. We show that the model is relevant for the description of graphene bilayer and that the presence of the d + i d superconducting correlations with antiferromagnetism can lead to quadratic dependence in small magnetic fields of the gap of the effective monolayer consistent with the transport measurements of Velasco et al.…
In single-layer graphene sheets non-local interband exchange leads to a renormalized Fermi-surface effective mass which vanishes in the low carrier-density limit. We report on a comparative study of Fermi surface effective mass renormalization in single-layer and AB-stacked bilayer graphene. We explain why the mass does not approach zero in the bilayer case, although its value is still strongly suppressed.
We provide a systematic real space derivation of the continuum Hamiltonian for a graphene bilayer starting from a microscopic lattice theory, allowing for an arbitrary inhomogeneous smooth lattice deformation, including a twist. Two different microscopic models are analyzed: first, a Slater-Koster like model and second, ab-initio derived model. We envision that our effective Hamiltonian can be used in conjunction with an experimentally determined atomic lattice deformation in twisted bilayer graphene in a specific device to predict and compare the electronic spectra with scanning tunneling spe…
Bilayer graphene is susceptible to a family of unusual broken symmetry states with spin and valley dependent layer polarization. We report on a microscopic study of the domain walls in these systems, demonstrating that they have interesting microscopic structures related to order-induced topological characters. We use our results to estimate Ginzburg-Landau model parameters and transition temperatures for the ordered states of bilayer graphene.
We demonstrate that single layer graphene exhibits the electronic structure of a bilayer when it is connected to two gated bilayers. The energy gap characteristic for gated bilayer is induced in the single layer and it persists for distances separating the bilayers up to 20 nm. Most importantly, we report on the appearance of topologically protected gapless state localized in a single graphene layer when the distant bilayers differ in the stacking order. The one-dimensional current in the topological state flows in a single layer graphene, and this finding suggests a new exploitation of graphe…
Additional component in multi-layer epitaxial graphene grown on the C-terminated surface of SiC, which exhibits the characteristic electronic properties of a AB-stacked graphene bilayer, is identified in magneto-optical response of this material. We show that these inclusions represent a well-defined platform for accurate magneto-spectroscopy of unperturbed graphene bilayers.
Nanoindentation of bilayer graphene is studied using molecular dynamics simulations. We compared our simulation results with those from elasticity theory as based on the nonlinear Föppl-Hencky equations with rigid boundary condition. The force deflection values of bilayer graphene are compered to those of monolayer graphene. Young's modulus of bilayer graphene is estimated to be 0.8 TPa which is close to the value for graphite. Moreover, an almost flat bilayer membrane at low temperature under central load has a 14$%$ smaller Young's modulus as compared to the one at room temperature.
Graphene bilayers can condense into a state with spontaneous interlayer phase coherence that supports dissipationless counterflow supercurrents. Here we address the influence of disorder on the graphene bilayer mean-field and Kosterlitz-Thouless critical temperatures and report on a simple criteria for the survival of pair condensation.
We demonstrate that the low-frequency resistance fluctuations, or noise, in bilayer graphene is strongly connected to its band structure, and displays a minimum when the gap between the conduction and valence band is zero. Using double-gated bilayer graphene devices we have tuned the zero gap and charge neutrality points independently, which offers a versatile mechanism to investigate the low-energy band structure, charge localization and screening properties of bilayer graphene.
We show that bilayer graphene exhibits magneto-electric effects that are formally similar to those commonly seen in band insulators with broken inversion and time-reversal symmetries. Three unusual features characterize the magneto-electric responses exhibited by bilayer graphene: (i) unlike most other magneto-electric media, bilayer graphene is a conductor, (ii) bilayer graphene has a non-quantized magneto-electric coupling even though its electronic structure does not break parity and time-reversal symmetry, and (iii) the magnitude of the magneto-electric coupling in bilayer graphene is dete…
We reveal that there exists a class of graphene structures (a sub-class of bilayer graphene nanoribbons) which has unusually strong optical response in the terahertz (THz) and far infrared (FIR) regime. The peak conductance of terahertz/FIR active bilayer ribbons is around two orders of magnitude higher than the universal conductance of $e^2/4\hbar$ observed in graphene sheets. The criterion for the terahertz/FIR active sub-class is a bilayer graphene nanoribbon with one-dimensional massless Dirac Fermion energy dispersion near the $Γ$ point. Our results overcome a significant obstacle that hi…
Quantum manipulation of valleys in bilayer graphene is investigated. We establish an effective Schrodinger model, and identify two key mechanisms for valley manipulation - band structure warping and generalized valley-orbit interaction. Specifically, we implement valley qubits / FETs in bilayer graphene, as prospective quantum devices to build valley-based quantum / classical information processing.
Heterostructures formed between two different van der Waals materials enable interactions and functionalities absent in each component. In this work we show that vicinity to an atomically thin WSe$_2$ sheet dramatically impacts the energies of the symmetry-broken low Landau levels of bilayer graphene, possibly due to screening. We present a systematic study of the magnetic field and electrical displacement field dependences of the Landau level gaps at filling factor $ν$ = 1, 2, 3, and compare to BN encapsulated pristine bilayer graphene. The exchange-dominated energy splitting between the N = …
Twisted bilayer graphenes with magical angle exhibit strongly correlated electronic properties because of the isolated flat band at the Fermi level. We studied the twisted bilayer graphene with substrates on both layers. The substrate induce Kekul$\acute{e}$ distortion for each graphene layer. The systems are investigated by both continuous Dirac Fermion model and tight binding model. The two investigations give similar conclusion that isolated flat band with ultra-narrow bandwidth could appear.
The momentum transport in ultraclean bilayer graphene is characterized by the viscous transport. In quantizing magnetic field the momentum current passes through the guiding centers of cyclotron orbits. In this study we derive the quantized Hall viscosity of bilayer graphene that is the next topological feature after the quantum Hall effect. This can be detected in the non-local magnetoresistivity measurements that varies with the quantized steps of magnetoresistivity.
Magnetotransport measurements on Hall bar devices fabricated on purely monolayer epitaxial graphene on Silicon Carbide (SiC/G) show a very tight spread in carrier concentration and mobility across wafer- size dimensions. In contrast, SiC/G devices containing bilayer graphene domains display variations in their electronic properties linked to the amount of bilayer content.
High-harmonic generation (HHG) in solids is an emerging method to probe ultrafast electron dynamics in solids at attosecond timescale. In this work, we study HHG from monolayer and bilayer graphene. Bilayer graphenes with AA and AB stacking are considered in this work. It is found that the monolayer and bilayer graphenes exhibit significantly different harmonic spectra. The difference in the spectra is attributed to the interlayer coupling between the two layers. Also, the intraband and interband contributions to the total harmonic spectrum play a significant role. Moreover, interesting polari…
We discuss the physics of the tunable bandgap in bilayer graphene with the gate voltage and doping. A comparison with experimental data obtained by Kuzmenko et al [Phys. Rev. B \textbf{80}, 165406 (2009)] demonstrates the good agreement.
Electron's tunneling through potential barrier in monolayer and bilayer graphene lattices is investigated by using full tight-binding model. Emphasis is placed on the resonance tunneling feature and inter-valley scattering probability. It is shown that normal incidence transmission probabilities for monolayer and bilayer graphene exhibit different properties. Our calculation indicates that valleytronics in graphene systems may be detected, generated and controlled by changing the structure parameters of the external electric potential.
Using electrical transport experiments and shot noise thermometry, we investigate electron-phonon heat transfer rate in a suspended bilayer graphene. Contrary to monolayer graphene with heat flow via three-body supercollision scattering, we find that regular electron - optical phonon scattering in bilayer graphene provides the dominant scattering process at electron energies $ \gtrsim 0.15$ eV. We determine the strength of these intrinsic heat flow processes of bilayer graphene and find good agreement with theoretical estimates when both zone edge and zone center optical phonons are taken into…
We theoretically investigate a folded bilayer graphene structure as an experimentally realizable platform to produce the one-dimensional topological zero-line modes. We demonstrate that the folded bilayer graphene under an external gate potential enables tunable topologically conducting channels to be formed in the folded region, and that a perpendicular magnetic field can be used to enhance the conducting when external impurities are present. We also show experimentally that our proposed folded bilayer graphene structure can be fabricated in a controllable manner. Our proposed system greatly …
In this paper, we study the quantum properties of a bilayer graphene with (asymmetry) line defects. The localized states are found around the line defects. Thus, the line defects on one certain layer of the bilayer graphene can lead to an electric transport channel. By adding a bias potential along the direction of the line defects, we calculate the electric conductivity of bilayer graphene with line defects using Landauer-Büttiker theory, and show that the channel affects the electric conductivity remarkably by comparing the results with those in a perfect bilayer graphene. This one-dimension…
This article reviews the theoretical and experimental work related to the electronic properties of bilayer graphene systems. Three types of bilayer stackings are discussed: the AA, AB, and twisted bilayer graphene. This review covers single-electron properties, effects of static electric and magnetic fields, bilayer-based mesoscopic systems, spin-orbit coupling, dc transport and optical response, as well as spontaneous symmetry violation and other interaction effects. The selection of the material aims to introduce the reader to the most commonly studied topics of theoretical and experimental …
We compare the main feature of the measured Raman scattering spectra from single layer graphene with a bilayer in which the two layers are arbitrarily misoriented. The profiles of the 2D bands are very similar having only one component, contrary to the four found for commensurate Bernal bilayers. These results agree with recent theoretical calculations and point to the similarity of the electronic structures of single layer graphene and misoriented bilayer graphene. Another new aspect is that the dependance of the 2D frequency on the laser excitation energy is different in these two latter sys…
We present a theoretical study of Weiss oscillations in magnetoconductivity of bilayer graphene. Bilayer graphene in the presence of a perpendicular magnetic field and a unidirectional weak electric modulation is considered.We determine the $σ_{yy}$ component of the magnetoconductivity tensor for this system which is shown to exhibit Weiss oscillations. We show that Weiss oscillations in the magnetoconductivity of bilayer graphene are enhanced and more robust with temperature compared to those in conventional two-dimensional electron gas systems whereas they are less robust with temperature co…
The electromagnetic response of bilayer graphene in a magnetic field is studied in comparison with that of monolayer graphene. Both types of graphene turn out to be qualitatively quite similar in dielectric and screening characteristics, especially those deriving from vacuum fluctuations, but the effect is generally much more sizable for bilayers. The presence of the zero-(energy-)mode Landau levels is a feature specific to graphene. In bilayers, unlike in monolayers, the effect of the zero-mode levels becomes visible and even dominant in density response as an externally-controllable band gap…
Bilayer graphene is a recently isolated and intriguing class of many-body systems with massive chiral quasiparticles. We present theoretical results for the electronic compressibility of bilayer graphene that are based on a four-band continuum band structure model combined with a random phase approximation treatment of electronic correlations. We find that the compressibility is strongly suppressed by electron-electron interactions at low carrier densities. Correlations do not lead to any qualitative new features, but are crucially important for a quantitative understanding of this fundamental…
The intrinsic spin-orbit interactions in bilayer graphene and in graphite are studied, using a tight binding model, and an intraatomic LS coupling. The spin-orbit interactions in bilayer graphene and graphite are larger, by about one order of magnitude, than the interactions in single layer graphene, due to the mixing of pi and sigma bands by interlayer hopping. Their value is in the range 0.1 - 1K. The spin-orbit coupling opens a gap in bilayer graphene, and it also gives rise to two edge modes. The spin-orbit couplings are largest, 1-4K, in orthorhombic graphite, which does not have a center…
Charge neutral bilayer graphene has a gapped ground state as transport experiments demonstrate. One of the plausible such ground states is layered antiferromagnetic spin density wave (LAF) state, where the spins in top and bottom layers have same magnitude with opposite directions. We propose that lightly charged bilayer graphene in an electric field perpendicular to the graphene plane may be a half metal as a consequence of the inversion and particle-hole symmetry broken in the LAF state. We show this explicitly by using a mean field theory on a 2-layer Hubbard model for the bilayer graphene.
We propose a non-magnetic, pseudospin-based version of a spin valve, in which the pseudospin polarization in neighboring regions of a graphene bilayer is controlled by external gates. Numerical calculations demonstrate a large on-off ratio of such a device. This finding holds promise for the realization of pseudospintronics: a form of electronics based upon the manipulation of pseudospin analogous to the control of physical spin in spintronics applications.
Using the Landauer formula approach, it is proven that minimal conductivity of order of $e^{2}/h$ found experimentally in bilayer graphene is its intrinsic property. For the case of ideal crystals, the conductivity turns our to be equal to $e^{2}/2h$ per valley per spin. A zero-temperature shot noise in bilayer graphene is considered and the Fano factor is calculated. Its value $1-2/π$ is close to the value 1/3 found earlier for the single-layer graphene.
We study the electronic transmission through a graphene bilayer in the presence of an applied bias between layers. We consider different geometries involving interfaces between both a monolayer and a bilayer and between two bilayers. The applied bias opens a sizable gap in the spectrum inside the bilayer barrier region, thus leading to large changes in the transmission probability and electronic conductance that are controlled by the applied bias.
We show measurements on a bilayer graphene quantum dot with an integrated charge detector. The focus lies on enabling charge detection with a 30 nm wide bilayer graphene nanoribbon located approximately 35 nm next to a bilayer graphene quantum dot with an island diameter of about 100 nm. Local resonances in the nanoribbon can be successfully used to detect individual charging events in the dot even in regimes where the quantum dot Coulomb peaks cannot be measured by conventional techniques.
This work investigates the electronic properties of the energy spectrum of a hybrid system composed of (i) a circular quantum dot of monolayer graphene surrounded by an infinite sheet of AA-stacked bilayer graphene and (ii) a circular quantum dot of AA-stacked graphene bilayer surrounded by infinite monolayer graphene. We establish analytical findings for the related energy levels and wave functions using the continuum model and the zigzag boundary conditions at the graphene monolayer-bilayer interface. We investigate the effects of perpendicular magnetic, dot radius, and electrical fields on …
We show that in gapped bilayer graphene, quasiparticle tunneling and the corresponding Berry phase can be controlled such that it exhibits features of single layer graphene such as Klein tunneling. The Berry phase is detected by a high-quality Fabry-Pérot interferometer based on bilayer graphene. By raising the Fermi energy of the charge carriers, we find that the Berry phase can be continuously tuned from $2π$ down to $0.68π$ in gapped bilayer graphene, in contrast to the constant Berry phase of $2π$ in pristine bilayer graphene. Particularly, we observe a Berry phase of $π$, the standard val…
In this comment we show that some equations and results of the paper titled "Dielectric screening and plasmons in AA-stacked bilayer graphene" are not correct. Furthermore, we present our results which seems to be more correct.
We report experimental signatures of plasmonic effects due to electron tunneling between adjacent graphene layers. At sub-nanometer separation, such layers can form either a strongly coupled bilayer graphene with a Bernal stacking or a weakly coupled double-layer graphene with a random stacking order. Effects due to interlayer tunneling dominate in the former case but are negligible in the latter. We found through infrared nano-imaging that bilayer graphene supports plasmons with a higher degree of confinement compared to single- and double-layer graphene, a direct consequence of interlayer tu…
We consider bilayer graphene in the presence of spin orbit coupling, to assess its behavior as a topological insulator. The first Chern number $n$ for the energy bands of single and bilayer graphene is computed and compared. It is shown that for a given valley and spin, $n$ in a bilayer is doubled with respect to the monolayer. This implies that bilayer graphene will have twice as many edge states as single layer graphene, which we confirm with numerical calculations and analytically in the case of an armchair terminated surface. Bilayer graphene is a weak topological insulator, whose surface …
We analyze the spectroscopic features of bilayer graphene determined by the formation of pairs of low-energy and split bands in this material. We show that the inter-Landau-level absorption spectrum in bilayer graphene at high magnetic field is much denser in the far-infrared range than that in monolayer material, and that the polarization dependence of its lowest energy peak can be used to test the form of the bilayer ground state in the quantum Hall-effect regime.
Electron transport in bilayer graphene is studied by using a first principles analysis and theMonte Carlo simulation under conditions relevant to potential applications. While the intrinsic properties are found to be much less desirable in bilayer than in monolayer graphene, with significantly reduced mobilities and saturation velocities, the calculation also reveals the dominant influence of extrinsic factors such as the substrate and impurities. Accordingly, the difference between two graphene forms are more muted in realistic settings although the velocity-field characteristics remain subst…
We present a study of the nonlinear optical response of twisted bilayer graphene. We discuss the contribution of the Berry phase to the non-linearity when inversion symmetry is broken, thus underlining the interplay between band and real space geometry, and nonlinear response. We also highlight an effect which is characteristic of extreme nonlinear optics: the generation of harmonics in disguise. This effect emerges in twisted bilayer graphene at relatively moderate fields strengths because of the much reduced band width. Our findings contribute to the understanding of the link between geometr…
The electron transmission between monolayer and bilayer graphene is theoretically studied for zigzag and armchair boundaries within an effective-mass scheme. Due to the presence of an evanescent wave in the bilayer graphene, traveling modes are well connected to each other. The transmission through the boundary is strongly dependent on the incident angle and the dependence is opposite between the K and K' points, leading to valley polarization of transmitted wave.
We study a family of globally smooth spatially varying two dimensional stacking textures in bilayer graphene. We find that the strain-minimizing stacking patterns connecting inequivalent ground states with local $AB$ and $BA$ interlayer registries are two dimensional twisted textures of an interlayer displacement field. We construct and display these topological stacking textures for bilayer graphene, examine their interactions and develop the composition rules that allow us to represent other more complex stacking textures, including globally twisted graphenes and extended one dimensional dom…
Bilayer graphene has the very interesting property of an energy gap tunable with the vertical electric field. We propose an analytical model for a bilayer-graphene field-effect transistor, suitable for exploring the design parameter space and to find a device structure with promising performance in terms of transistor operation. Our model, based on the effective mass approximation and ballistic transport assumptions, takes into account bilayer-graphene tunable gap and self polarization, and includes all band-to-band tunneling current components, which are shown to represent the major limitatio…
We study the conductance of a biased bilayer graphene flake with monolayer nanoribbon contacts. We find that the transmission through the bilayer ribbon strongly depends on the applied bias between the two layers and on the relative position of the monolayer contacts. Besides the opening of an energy gap on the bilayer, the bias allows to tune the electronic density on the bilayer flake, making possible the control of the electronic transmission by an external parameter.
Here we show that the Pfaffian state proposed for the $\frac52$ fractional quantum Hall states in conventional two-dimensional electron systems can be readily realized in a bilayer graphene at one of the Landau levels. The properties and stability of the Pfaffian state at this special Landau level strongly depend on the magnetic field strength. The graphene system shows a transition from the incompressible to a compressible state with increasing magnetic field. At a finite magnetic field of ~10 Tesla, the Pfaffian state in bilayer graphene becomes more stable than its counterpart in convention…
Magnetic impurity adsorbed on one of the carbon planes of a bilayer graphene is studied. The formation of the many-body SU(2) and SU(4) resonances close to the bandgap is analyzed within the mean field Kotliar-Ruckenstein slave boson approach. Impact of enhanced hybridization and magnetic instability of bilayer doped near the Van Hove singularity on the screening of magnetic moment is discussed.
We present an analytical description of pi electrons of a finite size bilayer graphene within a framework of the tight-binding model. The bilayered structures considered here are characterized by a rectangular geometry and have a finite size in one or both directions with armchair- and zigzag-shaped edges. We provide an exact analytical description of the spectrum of pi electrons in the zigzag and armchair bilayer graphene nanoribbons and nanotubes. We analyze the dispersion relations, the density of states, and the conductance quantization.
We study the electronic properties of commensurate faulted bilayer graphene by diagonalizing the one-particle Hamiltonian of the bilayer system in a complete basis of Bloch states of the individual graphene layers. Our novel approach is very general and can be easily extended to any commensurate graphene-based heterostructure. Here, we consider three cases: i) twisted bilayer graphene, ii) bilayer graphene where triaxial stress is applied to one layer, and iii) bilayer graphene where uniaxial stress is applied to one layer. We show that the resulting superstructures can be divided into distinc…
We show that the twisted graphene bilayer can reveal unusual topological properties at low energies, as a consequence of a Dirac-point splitting. These features rely on a symmetry analysis of the electron hopping between the two layers of graphene and we derive a simplified effective low-energy Hamiltonian which captures the essential topological properties of twisted bilayer graphene. The corresponding Landau levels peculiarly reveal a degenerate zero-energy mode which cannot be lifted by strong magnetic fields.
We investigate the transport properties of double-gated bilayer graphene nanoribbons at room temperature. The devices were fabricated using conventional CMOS-compatible processes. By analyzing the dependence of the resistance at the charge neutrality point as a function of the electric field applied perpendicular to the graphene surface, we show that a band gap in the density of states opens, reaching an effective value of ~sim50 meV. This demonstrates the potential of bilayer graphene as FET channel material in a conventional CMOS environment.
We calculate the density-density response function and electron self-energy for undoped bilayer graphene, within the Random Phase Approximation (RPA). We show that the quasiparticle decay rate scales linearly with the quasiparticle energy, and quasiparticle weight vanishes logarithmically in the low-energy limit, indicating non-Fermi liquid behavior. This is a consequence of the absence of a Fermi surface for neutral bilayer graphene and corresponding larger phase space available for scattering processes. Experimental consequences of our results as well as their differences from those of singl…
We present a self-consistent calculation of the interlayer asymmetry in bilayer graphene caused by an applied electric field in magnetic fields. We show how this asymmetry influences the Landau level spectrum in bilayer graphene and the observable inter-Landau level transitions when they are studied as a function of high magnetic field at fixed filling factor as measured experimentally by E.A. Henriksen et al., Phys. Rev. Lett. 100 (2008), 087403. We also analyze the magneto-optical spectra of bilayer flakes in the photon-energy range corresponding to transitions between degenerate and split b…
We predict the existence of transverse electric (TE) plasmons in bilayer graphene. We find that their plasmonic properties are much more pronounced in bilayer than in monolayer graphene, in a sense that they can get more localized at frequencies just below $\hbarω=0.4$~eV for adequate doping values. This is a consequence of the perfectly nested bands in bilayer graphene which are separated by $\sim 0.4$~eV.
We introduce a minimum tight-binding model with only three parameters extracted from graphene and untwisted bilayer graphene. This model reproduces quantitatively the electronic structure of not only these two systems and bulk graphite near the Fermi level, but also that of twisted bilayer graphene including the value of the magic angle, at which bands at $E_F$ flatten without overlap and two gaps open, one above and one below $E_F$. The Hamiltonian is sufficiently transparent and flexible to be adopted to other twisted layered systems.
