The magneto-optical absorption properties of graphene multilayers are theoretically studied. It is shown that the spectrum can be decomposed into sub-components effectively identical to the monolayer or bilayer graphene, allowing us to understand the spectrum systematically as a function of the layer number. Odd-layered graphenes always exhibit absorption peaks which shifts in proportion to sqrt(B), with B being the magnetic field, due to the existence of an effective monolayer-like subband. We propose a possibility of observing the monolayer-like spectrum even in a mixture of multilayer graphene films with various layers numbers.
We study the effect of vertical electric field (E-field) on the electronic properties of multilayer graphene. We show that the effective mass, electron velocity and density-of-state of a bilayer graphene are modified under the E-field. We also study the transformation of the band structure of multilayer graphenes. E-field induces finite (zero) bandgap in the even (odd)-layer ABA-stacking graphene. On the other hand, finite bandgap is induced in all ABC-stacking graphene. We also identify the optimum E-field to obtain the maximum bandgap in the multilayer graphenes. Finally we compare our resul…
Synthesis of multilayer graphene on copper wires by a chemical vapor deposition method is reported. After copper etching, the multilayer tube collapses forming stripes of graphitic films, their electrical conductance as a function of temperature indicate a semiconductor-like behavior. Using the multilayer graphene stripes, a cross junction is built and owing to its electrical behavior we propose that a tunneling process exists in the device.
Using the low-energy effective Hamiltonian of the ABC-stacked multilayer graphene, pseudo spin coupling to real orbital angular momentum of electron in multilayer graphene is investigated. We show that electron wave function in N-layer graphene mimics behavior of particle with spin of N/2. It is said that for N greater than 1 the low-energy effective Hamiltonian for ABC-stacked graphene is no longer used to describe pseudo spin 1/2-particle wave function. The wave function of electron in multilayer graphene may behave like fermionic (or bosonic) particle when N is odd (or even). This work prop…
We calculate the transport properties of multilayer graphene, considering the effect of multisubband scattering in a high density regime, where higher subbands are occupied by charge carriers. To calculate the conductivity of multilayer graphene, we use the coupled multiband Boltzmann transport theory while fully incorporating the multiband scattering effects. We show that the allowed scattering channels, screening effects, chiral nature of the electronic structure, and type of impurity scatterings determine the transport behavior of multilayer graphene. We find that the conductivity of multil…
The modulation of the optical transmittance in multilayer graphene by means of an electrical signal in the simple configuration of coplanar electrodes is reported. Besides the fundamental frequency of modulation, higher harmonics also appear in the transmitted signal. Modulation is also observed in the optical reflectance. The modulation of the optical properties in multilayer graphene by electrical signals may be useful for the transmission of information by optical means.
Interlayer coupling in rotationally faulted graphene multilayers breaks the local sublattice-symmetry of the individual layers. We present a theory of this mechanism, which reduces to an effective Dirac model with space-dependent mass in an important limit. It thus makes a wealth of existing knowledge available for the study of rotationally faulted graphene multilayers. We demonstrate quantitative agreement between our theory and a recent experiment.
Since the advent of graphene, a variety of studies have been performed to elucidate its fundamental physics, or to explore its practical applications. Gate-tunable resistance is one of the most important properties of graphene and has been studied in 1-3 layer graphene in a number of efforts to control the band gap to obtain a large on-off ratio. On the other hand, the transport property of multilayer graphene with more than three layers is less well understood. Here we show a new aspect of multilayer graphene. We found that four-layer graphene shows intrinsic peak structures in the gate volta…
Charge carriers in single and multilayered graphene systems behave as chiral particles due to the particular lattice symmetry of the crystal. We show that the interplay between the meta-material properties of graphene multilayers and the pseudospinorial properties of the charge carriers result in the occurrence of Klein and anti-Klein tunneling for rhombohedral stacked multilayers. We derive an algebraic formula predicting the angles at which these phenomena occur and support this with numerical calculations for systems up to four layers. We present a decomposition of an arbitrarily stacked mu…
This article reviews progress in the theoretical modelling of the electronic structure of rotationally faulted multilayer graphenes. In these systems the crystallographic axes of neighboring layers are misaligned so that the layer stacking does not occur in the Bernal structure observed in three dimensional graphite and frequently found in exfoliated bilayer graphene. Notably, rotationally faulted graphenes are commonly found in other forms of multilayer graphene including epitaxial graphenes thermally grown on ${\rm SiC \, (000 \bar 1)}$, graphenes grown by chemical vapor deposition, folded m…
Influence of the chiral symmetry on the many-body problem in multilayer graphene in magnetic fields is investigated. For a spinless electron model on the honeycomb lattice the many-body ground state is shown to be a doubly-degenerate chiral condensate irrespective of the number of layers. The energy spectrum calculated numerically with the exact diagonalization method reveals for ABC-stacked multilayer graphenes that the many-body gap decreases monotonically with the number of layers.
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.
An atomic force microscope is used to structure a film of multilayer graphene. The resistance of the sample was measured in-situ during nanomachining a narrow trench. We found a reversible behavior in the electrical resistance which we attribute to the movement of dislocations. After several attempts also permanent changes are observed. Two theoretical approaches are presented to approximate the measured resistance.
The thermodynamic properties of the electron gas in multilayer graphene depend strongly on the number of layers and the type of stacking. Here we analyse how those properties change when we vary the number of layers for rhombohedral stacked multilayer graphene and compare our results with those from a conventional two dimensional electron gas. We show that the highly degenerate zero energy Landau level which is partly filled with electrons and partly with holes has a strong influence on the value of the different thermodynamic quantities.
We study spatial symmetry in general ABA-stacked multilayer graphene to illustrate how electronic spectra at the two valleys are related in a magnetic field. We show that the lattice of multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial inversion symmetry, which rigorously guarantees valley degeneracy in the absence of time-reversal symmetry. A multilayer with an odd number of layers (three or more) lacks inversion symmetry, but there is another transformation imposing an approximate valley degeneracy, which arises because the low-energy Hamiltoni…
Graphene layers are known to stack in two stable configurations, namely ABA or ABC stacking, with drastically distinct electronic properties. Unlike the ABA stacking, little has been done to experimentally investigate the electronic properties of ABC graphene multilayers. Here, we report the first magneto optical study of a large ABC domain in a graphene multilayers flake, with ABC sequences exceeding 17 graphene sheets. The ABC-stacked multilayers can be fingerprinted with a characteristic electronic Raman scattering response, which persists even at room temperatures. Tracing the magnetic fie…
We study the effect of perpendicular electric fields on the band structures of ABA and ABC graphene multilayers, and find that the electronic screening effect is significantly different between them. In ABA multilayers, the field produces a band overlap and gives a linear screening, while in ABC multilayers, in contrast, it opens an energy gap in the surface-state band at low energy, leading to a strong screening effect essentially non-linear to the field amplitude. The energy gap of a large ABC stack sharply rises when the external field exceeds a certain critical value.
We report on a theoretical study of one-dimensional (1D) states localized at few-layer graphene system ribbon edges, and at interfaces between few-layer graphene systems with different valley Hall conductivities. These 1D states are topologically protected when valley mixing is neglected. We address the influence on their properties of stacking arrangement, interface structure, and external electric field perpendicular to the layers. We find that 1D states are generally absent at multilayer ribbon armchair direction edges, but present irrespective of crystallographic orientation at any interna…
We produce multilayer graphene by the Chemical Vapor Deposition (CVD) method at atmospheric pressure and 1000 °C, using flexible copper substrates as catalyst and liquid hexane as the source of carbon. We designed an optical device to measure the transmittance of the carbon films; with this information we calculate that the approximate number of layers is 11.
We study the effect of transversal electric-field (E-field) on the electronic properties of multilayer armchair-graphene-nanoribbon (AGNR). The bandgap in multilayer-AGNRs can be reversibly modulated with the application of E-field. At optimized widths, we obtain a semiconductor (SC) to metallic (M), as well as M-SC transition. The AGNR electronic bands undergo vivid transformations due to the E-field, leading to phenomena such as increase in electron velocity, change in the sign of the electron effective mass, and the formation of linear dispersion with massless Dirac fermions similar to 2D-g…
Using non-equilibrium molecular dynamics method(NEMD), we have found that the thermal conductivity of multilayer graphene nanoribbons monotonously decreases with the increase of the number of layers, such behavior can be attributed to the phonon resonance effect of out-of-plane phonon modes. The reduction of thermal conductivity is found to be proportional to the layer size, which is caused by the increase of phonon resonance. Our results are in agreement with recent experiment on dimensional evolution of thermal conductivity in few layer graphene.
We consider the optical properties of finite systems composed of a series of graphene sheets separated by thin dielectric layers. Because these systems respond as conductors to electric fields in the plane of the graphene sheets and as insulators to perpendicular electric fields, they can be expected to have properties similar to those of hyperbolic metamaterials. We show that under typical experimental conditions graphene/dielectric multilayers have enhanced Purcell factors, and enhanced photonic densities-of-states in both the THz and mid-IR frequency range. These behaviors can be traced to …
We demonstrate that charge density wave (CDW) phase transition occurs on the surface of electronically doped multilayer graphene when the Fermi level approaches the M points (also known as van Hove singularities where the density of states diverge) in the Brillouin zone of graphene band structure. The occurrence of such CDW phase transitions are supported by both the electrical transport measurement and optical measurements in electrostatically doped multilayer graphene. The CDW transition is accompanied with the sudden change of graphene channel resistance at T$_m$= 100K, as well as the split…
We show that valley degeneracy in rotationally faulted multilayer graphene may be broken in the presence of a magnetic field and interlayer commensurations. This happens due to a simultaneous breaking of both time-reversal and inversion symmetries leading to a splitting of Landau levels linear in the field. Our theoretical work is motivated by an experiment [Y. J. Song et al., Nature 467, 185 (2010)] on epitaxially grown multilayer graphene where such linear splitting of Landau levels was observed at moderate fields. We consider both bilayer and trilayer configurations and, although a linear s…
Interlayer coupling in rotationally faulted graphene multilayers breaks the local sublattice-symmetry of the individual layers. Earlier we have presented a theory of this mechanism, which reduces to an effective Dirac model with space-dependent mass in an important limit. It thus makes a wealth of existing knowledge available for the study of rotationally faulted graphene multilayers. Agreement of this theory with a recent experiment in a strong magnetic field was demonstrated. Here we explore some of the predictions of this theory for the system in zero magnetic field at large interlayer bias…
With the help of a simple model, we analyze Scanning Tunneling Microscopy images of simple and double moiré patterns resulting from misoriented bi- and tri-layers graphene stacks. It is found that the model reproduces surprisingly well non-trivial features observed in the Fast Fourier Transform of the images. We point out difficulties due to those features in interpreting the patterns seen on the FFT.
We study the interfaces between ABC and ABA regions in multilayer graphene, in particular we consider regions in which the transition between the ABA and ABC structure arises due to the local compression in one of the graphene layers for a zigzag interface, and to a local shear of the atoms for the armchair interface. When we consider an infinite ribbon configuration we find that interface bands form in these regions, more pronouncedly for the armchair-type interfaces. We note that these states are not topological. For fully-finite-size structures the continuous interface bands transform into …
The specifics of charge screening and electrostatic potential spatial distribution in multilayered graphene films placed in between charged substrates is theoretically analyzed. It is shown that by varying the areal charge densities on the substrates and/or the thickness of the graphene stack one may tune the doped carriers distribution over the system. When the charge densities on the substrates are weak, the carriers distribution and electrostatic potential profile agree with semimetallic properties of graphene. However, when the amount of the donated charge is sufficiently large the transit…
Monolayer graphene shows Seebeck coefficient several times and gas-flow-induced voltage twenty times higher than that of bulk graphite. Here we find that the Seebeck coefficient of multilayer graphene increases monotonically with increasing layer and reaches its peak value at hexa-layer ~77% higher than for monolayer and then decreases, although the electric resistance decreases monotonically with increasing layer. The flow-induced voltage is significantly higher in 2, 4, 5, 6, 7 layered graphene than in 1, 3, 8 layered one, against the prevailing view that it should be proportional to Seebeck…
Strong magnetoresistance effects are often observed in ferromagnet-nonmagnet multilayers, which are exploited in state-of-the-art magnetic field sensing and data storage technologies. In this work we report a novel current-perpendicular-to-plane magnetoresistance effect in multilayer graphene as grown on a catalytic nickel surface by chemical vapor deposition. A negative magnetoresistance effect of 10^4% has been observed, which persists even at room temperature. This effect is correlated with the shape of the 2D peak as well as with the occurrence of D peak in the Raman spectrum of the as-gro…
Domain walls, topological defects that define the frontier between regions of different stacking in multilayer graphene, have proved to host exciting physics. The ability of tuning these topological defects in-situ in an electronic transport experiment brings a wealth of possibilities in terms of fundamental understanding of domain walls as well as for electronic applications. Here, we demonstrate through a MEMS (micro-electromechanical system) actuator and magnetoresistance measurements the effect of domain walls in multilayer graphene quantum Hall effect. Reversible and controlled uniaxial s…
We identify qualitative trends in the stacking sequence dependence of carrier-carrier interaction phenomena in multilayer graphene. Our theory is based on a new approach which explicitly exhibits the important role in interaction phenomena of the momentum-direction dependent intersite phases determined by the stacking sequence. Using this method, we calculate and compare the self-energies, density--density response functions, collective modes, and ground-state energies of several different few layer graphene systems. The influence of electron--electron interactions on important electronic prop…
We study the effect of vertical electric-field (E-field) on the electronic properties of the multilayer armchair graphene nanoribbon (aGNR). Under E-field, the band structure of a bilayer aGNR undergoes interesting transformations such as change in the electron velocity, sign of the electron effective mass, band gap, the position of the band gap in the momentum space. Depending on the width of the aGNR and the applied E-field, the band gap of the aGNR may either be increased or decreased. When the applied E-field is above a critical value, the band gap of the bilayer aGNR is identical to that …
Interlayer magnetoresistance (ILMR) effect is explored in a vertical stack of weakly coupled multilayer graphene as grown by chemical vapor deposition (CVD). This effect has been characterized as a function of temperature and tilt angle of the magnetic field with respect to the interlayer current. To our knowledge, this is the first experimental report on angle dependent ILMR effect in graphitic systems. Our data agrees qualitatively with the existing theories of ILMR in multilayer massless Dirac Fermion systems. However, a sharper change in ILMR has been observed as the tilt angle of the magn…
We present a theoretical study on the orbital magnetism in multilayer graphenes within the effective mass approximation. The Hamiltonian and thus susceptibility can be decomposed into contributions from sub-systems equivalent to monolayer or bilayer graphene. The monolayer-type subband exists only in odd layers and exhibits a delta-function susceptibility at $E_F=0$. The bilayer-type subband appearing in every layer number gives a singular structure in the vicinity of $E_F=0$ due to the trigonal warping as well as a logarithmic tail away from $E_F=0$. The integral of the susceptibility over en…
We analyze the ordered phases of Bernal stacked multilayer graphene in the presence of interaction induced band gaps due to sublattice symmetry breaking potentials, whose solutions can be analyzed in terms of light-mass and heavy-mass pseudospin doublets which have the same Chern numbers but opposite charge polarization directions. The application of a perpendicular external electric field reveals an effective Hund's rule for the ordering of the sublattice pseudospin doublets in a tetralayer, while a similar but more complex phase diagram develops with increasing layer number.
We present low-frequency electrical resistance fluctuations, or noise, in graphene-based field-effect devices with varying number of layers. In single-layer devices the noise magnitude decreases with increasing carrier density, which behaved oppositely in the devices with two or larger number of layers accompanied by a suppression in noise magnitude by more than two orders in the latter case. This behavior can be explained from the influence of external electric field on graphene band structure, and provides a simple transport-based route to isolate single-layer graphene devices from those wit…
The so called quantum spin Hall phase is a topologically non trivial insulating phase that is predicted to appear in graphene and graphene-like systems. In this work we address the question of whether this topological property persists in multilayered systems. We consider two situations: purely multilayer graphene and heterostructures where graphene is encapsulated by trivial insulators with a strong spin-orbit coupling. We use a four orbital tight-binding model that includes the full atomic spin-orbit coupling and we calculate the $Z_{2}$ topological invariant of the bulk states as well as th…
We study interlayer transport of multilayer graphenes in magnetic field with various stacking structures (AB, ABC, and AA types) by calculating the Hall and longitudinal conductivities as functions of Fermi energy. Their behavior depends strongly on the stacking structures and selection of the layers. The Hall conductivity between different layers is no longer quantized. Moreover, for AB stacking, the interlayer conductivity vanishes around zero energy with increasing layer separation, and shows negative values in particular cases. The fact that longitudinal interlayer conductivity suppressed …
We study the effects of site dilution disorder on the electronic properties in graphene multilayers, in particular the bilayer and the infinite stack. The simplicity of the model allows for an easy implementation of the coherent potential approximation and some analytical results. Within the model we compute the self-energies, the density of states and the spectral functions. Moreover, we obtain the frequency and temperature dependence of the conductivity as well as the DC conductivity. The c-axis response is unconventional in the sense that impurities increase the response for low enough dopi…
Calculations of friction and heat transfer at the water-multilayer graphene interface using the theories of phononic and radiative friction and heat transfer are presented. The phononic contributions to friction and heat transfer are many orders of magnitude larger than the radiative contributions. Phononic friction and heat transfer slightly increase with an increase in the number of graphene layers $N$ and reach saturation at $N>5$, which is associated with an increase in the phonon transmission coefficient through the interface and a finite phonon mean free path in the direction perpendicul…
Vibrating membranes are the cornerstone of acoustic technology, forming the backbone of modern loudspeakers and microphones. Acoustic performance of condenser microphone is derived mainly from the membrane's size and achievable static tension. The widely studied and available nickel has been the one of dominant membrane material for several decades. In this paper we introduce multilayer graphene as membrane material for a condenser microphone. The graphene device outperforms a high end commercial nickel-based microphone over a significant part of the acoustic spectrum, with a larger than 10 dB…
The ground state of the two-dimensional electron systems in Bernal bilayer and ABC-stacked multilayer graphenes in the presence of a strong magnetic field is investigated with the Hartree-Fock approximation. Phase diagrams of the systems are obtained, focusing on charge density wave states including states with vortices of valley pseudospins (called a Skyrme crystal). The single-electron states in these stacked graphenes are given by two-component wave functions. That of the first excited Landau level has the same component as the lowest Landau level of the ordinary two-dimensional electrons. …
We study the electronic structures of ABA (Bernal) stacked multilayer graphenes in uniform perpendicular electric field, and show that the interplay of the trigonal warping and the potential asymmetry gives rise to a number of emergent Dirac cones nearly touching at zero energy. The band velocity and the energy region (typically a few tens of meV) of these gate-induced Dirac cones are tunable with the external electric field. In ABA trilayer graphene, in particular, applying an electric field induces a non-trivial valley Hall state, where the energy gap at the Dirac point is filled by chiral e…
The effect of different stacking order of graphene multilayers on the electric field induced band gap is investigated. We considered a positively charged top and a negatively charged back gate in order to independently tune the band gap and the Fermi energy of three and four layer graphene systems. A tight-binding approach within a self-consistent Hartree approximation is used to calculate the induced charges on the different graphene layers. We found that the gap for trilayer graphene with the ABC stacking is much larger than the corresponding gap for the ABA trilayer. Also we predict that fo…
Peeling, shearing, and sliding are important mechanical phenomena in van der Waals solids. However, theoretically they have been studied mostly using minimal periodic cells and in the context of accurate quantum simulations. Here, we investigate the peeling of large-scale multilayer graphene stacks with varying thicknesses, stackings, and peeling directions by using classical molecular dynamics simulations with a registry-dependent interlayer potential. Simulations show that, while at large scale the peeling proceeds smoothly, at small scale the registry shifts and sliding patterns of the laye…
While the experimental progress on three dimensional topological insulators is rapid, the development of their two dimensional counterparts has been comparatively slow, despite their technological promise. The main reason is materials challenges of the to date only realizations of two-dimensional topological insulators, in semiconductor quantum wells. Here we identify a two dimensional topological insulator in a material which does not face similar challenges and which is by now most widely available and well-charaterized: graphene. For certain commensurate interlayer twists graphene multilaye…
We develop a theory for the optical conductivity of doped multilayer graphene including the effects of electron-electron interactions. Applying the quantum kinetic formalism, we formulate a set of pseudospin Bloch equations that governs the dynamics of the nonequilibrium density matrix driven by an external \emph{a.c.} electric field under the influence of Coulomb interactions. These equations reveal a dynamical mechanism that couples the Drude and interband responses arising from the chirality of pseudospin textures in multilayer graphene systems. We demonstrate that this results in an intera…
The crystal structure of solid-state matter greatly affects its electronic properties. For example in multilayer graphene, precise knowledge of the lateral layer arrangement is crucial, since the most stable configurations, Bernal and rhombohedral stacking, exhibit very different electronic properties. Nevertheless, both stacking orders can coexist within one flake, separated by a strain soliton that can host topologically protected states. Clearly, accessing the transport properties of the two stackings and the soliton is of high interest. However, the stacking orders can transform into one a…
A Zener diode is a paradigmatic device in semiconductor-based electronics that consists of a pn junction where an external electric field induces a switching behavior in the current-voltage characteristics. We study Zener tunneling in HgTe quantum wells and graphene multilayers. We find that the tunneling transition probability depends asymmetrically on the parallel momentum of the carriers to the barrier. In HgTe quantum wells the asymmetry is the opposite for each spin, whereas for graphene multilayers it is the opposite for each valley degree of freedom. In both cases, a spin/valley current…
Rhombohedral multilayer graphene is a physical realization of the chiral two-dimensional electron gas that can host zero-line modes (ZLMs), also known as kink states, when the local ap opened by inversion symmetry breaking potential changes sign in real space. Here we study how the variations in the local stacking coordination of multilayer graphene affects the formation of the ZLMs. Our analysis indicates that the valley Hall effect develops whenever an interlayer potential difference is able to open up a band gap in stacking faulted multilayer graphene, and that ZLMs can appear at the domain…
Epitaxial graphene is grown on a non-polar n+ 6H-SiC m-plane substrate and studied using atomic scale scanning tunneling microscopy. Multilayer graphene is found throughout the surface and exhibits rotational disorder. Moiré patterns of different spatial periodicities are found, and we found that as the wavelength increases, so does the amplitude of the modulations. This relationship reveals information about the interplay between the energy required to bend graphene and the interaction energy, i.e. van der Waals energy, with the graphene layer below. Our experiments are supported by theoretic…
In this paper we review thermal properties of graphene and multilayer graphene and discuss the optothermal technique developed for the thermal conductivity measurements. We also outline different theoretical approaches used for the description of phonon transport in graphene and provide comparison with available experimental thermal conductivity data.
Suspended Bernal-stacked graphene multilayers up to an unexpectedly large thickness exhibit a broken-symmetry ground state, whose origin remains to be understood. Here we show that a finite-temperature second order phase transition occurs in multilayers whose critical temperature Tc increases from 12 K in bilayers to 100 K in heptalayers. A comparison of the data to a phenomenological model inspired by a mean field approach suggests that the transition is associated with the appearance of a self-consistent valley- and spin-dependent staggered potential changing sign from one layer to the next,…
We explore the electronic ground states of Bernal-stacked multilayer graphenes using the Hartree-Fock mean-field approximation and the full-parameter band model. We find that the electron-electron interaction tends to open a band gap in multilayer graphenes from bilayer to 8-layer, while the nature of the insulating ground state sensitively depends on the band parameter $γ_2$, which is responsible for the semimetallic nature of graphite. In 4-layer graphene, particularly, the ground state assumes an odd-spatial-parity staggered phase at $γ_2 = 0$, while an increasing, finite value of $γ_2$ sta…
We study the orbital effect of a strong magnetic field parallel to the layers on the energy spectrum of the Bernal-stacked graphene bilayer and multilayers, including graphite. We consider the minimal model with the electron tunneling between the nearest sites in the plane and out of the plane. Using the semiclassical analytical approximation and exact numerical diagonalization, we find that the energy spectrum consists of two domains. In the low- and high-energy domains, the semiclassical electron orbits are closed and open, so the spectra are discrete and continuous, correspondingly. The dis…
The Landau level spectra and the quantum Hall effect of ABA-stacked multilayer graphenes are studied in the effective mass approximation. The low-energy effective mass Hamiltonian may be partially diagonalized into an approximate block-diagonal form, with each diagonal block contributing parabolic bands except, in a multilayer with an odd number of layers, for an additional block describing Dirac-like bands with a linear dispersion. We fully include the band parameters and, taking into account the symmetry of the lattice, we analyze their affect on the block-diagonal Hamiltonian. Next-nearest …
