We report phonon renormalisation in bilayer graphene as a function of doping. The Raman G peak stiffens and sharpens for both electron and hole doping, as a result of the non-adiabatic Kohn anomaly at the $Γ$ point. The bilayer has two conduction and valence subbands, with splitting dependent on the interlayer coupling. This results in a change of slope in the variation of G peak position with doping, which allows a direct measurement of the interlayer coupling strength.
Gate-induced magnetic switching in bilayer CrI$_3$ has opened new ways for the design of novel low-power magnetic memories based on van der Waals heterostructures. The proposed switching mechanism seems to be fully dominated by electrostatic doping. Here we explain, by first-principle calculations, the ferromagnetic transition in doped bilayer CrI$_3$. For the case of a very small electron doping, our calculations predict the formation of magnetic polarons ("ferrons", "fluctuons") where the electron is self-locked in a ferromagnetic droplet in an antiferromagnetic insulating matrix. The self-t…
We present a theory of the influence of band renormalization and excitonic electron-electron interaction effects on the optical conductivity $σ(ω)$ of doped bilayer graphene. Using the Keldysh formalism, we derive a kinetic equation from which we extract numerical and approximate analytic results for $σ(ω)$. Our calculations reveal a previously unrecognized mechanism which couples the Drude and interband response and renormalizes the plasmon frequency, and suggest that screening must play an essential role in explaining the weakly renormalized conductivity seen in recent experiments.
We study the electronic spin relaxation effect in the hole-doped monolayer and bilayer transition-metal dichalcogenides in the presence of the crystal defects. We consider realistic models of the lattice vacancy and actually estimate the spin relaxation rate using the multi-orbital tight-binding model. In the monolayer, the spin-relaxation time is found to be extremely long compared to the momentum relaxation time, and this is attributed to the fact that the spin hybridization in the band structure is suppressed by the mirror reflection symmetry. The bilayer TMD has a much shorter spin relaxat…
Injection or optical generation of electrons and holes in graphene bilayers (GBLs) can result in the interband population inversion enabling the terahertz (THz) radiation lasing. The intraband radiative processes compete with the interband transitions. We demonstrate that remote doping enhances the indirect interband generation of photons in the proposed GBL heterostructures. Therefore such remote doping helps surpassing the intraband (Drude) absorption and results in large absolute values of the negative dynamic THz conductivity in a wide range of frequencies at elevated (including room) temp…
The microscopic quantum theory of nonlinear stimulated scattering of chiral particles in doped $AB$ stacked bilayer graphene on Coulomb field of charged impurities in the presence of strong coherent electromagnetic radiation is presented. The Liouville-von Neumann equation for the density matrix is solved analytically. Here the interaction of electrons with the scattering potential is taken into account as a perturbation. The absorption rate of nonlinear inverse-bremsstrahlung for a grand canonical ensemble of fermionic chiral particles is calculated using the obtained solution. The analysis o…
A bilayer graphene based electrostatically doped tunnel field-effect transistor (BED-TFET) is proposed in this work. Unlike graphene nanoribbon TFETs in which the edge states deteriorate the OFF-state performance, BED-TFETs operate based on different bandgaps induced by vertical electric fields in the source, channel, and drain regions without any chemical doping. The performance of the transistor is evaluated by self-consistent quantum transport simulations. This device has several advantages: 1) ultra-low power (VDD=0.1V), 2) high performance (ION/IOFF>104), 3) steep subthreshold swing (SS<1…
Rotated graphene bilayers form an exotic class of nanomaterials with fascinating electronic properties governed by the rotation angle theta. For large rotation angles, the electron eigenstates are restricted to one layer and the bilayer behaves like two decoupled graphene layer. At intermediate angles, Dirac cones are preserved but with a lower velocity and van Hove singularities are induced at energies where the two Dirac cones intersect. At very small angles, eigenstates become localized in peculiar moire zones. We analyse here the effect of an asymmetric doping for a series of commensurate …
Two recent experiments from Cornell and Columbia have reported insulator-to-metal transitions in two-dimensional (2D) moiré transition metal dichalcogenides (mTMD) induced by doping around half-filling, where the system is a Mott insulator. In the current work, we consider the temperature dependent resistivity of this metallic phase in the doped situation away from half-filling, arguing that it arises from the strongly temperature dependent 2D Friedel oscillations (i.e. finite momentum screening) associated with random quenched charged impurities, leading to the observed strongly increasing li…
We compute the nonlinear optical response of doped mono- and bilayer graphene using the full dispersion based on tight-binding models. The response is derived with the density matrix formalism using the length gauge and is valid for any periodic system, with arbitrary doping. By collecting terms that define effective nonlinear response tensors, we identify all nonlinear Drude-like terms (up to third-order) and show that all additional spurious divergences present in the induced current vanish. The nonlinear response of graphene comprises a large Drude-like divergence and three resonances that …
Unlike single layer graphene, in the case of $AB$-stacked bilayer graphene (BLG) one can induce a non-zero energy gap by breaking the inversion symmetry between the two layers using a perpendicular electric field. This is an essential requirement in field-effect applications, particularly since the induced gap in BLG systems can be further tuned by the magnitude of the external electric field. Doping is another way to modify the electronic properties of graphene based systems. We investigate here BLG systems doped with boron and nitrogen in the presence of external electric field, in the frame…
The quantum theory of multiphoton stimulated bremsstrahlung of charged carriers on an arbitrary electrostatic potential of impurity ion in doped bilayer graphene at the presence of coherent electromagnetic radiation is developed. A terahertz wave field is considered exactly, while the electrostatic potential of doped ions as a perturbation. The essentially nonlinear response of bilayer graphene to a pump wave and significant differences from the case of a single layer graphene are shown, which can be associated to nonlinear parabolic dispersion. The latter opens new way to manipulate with the …
Generating spintronic devices has been a goal for the nano-science. We have used density function theory to determine magnetic phases of single layer and bilayer lithium doped graphene nanoflakes. We have introduced graphene flakes as single molecular magnets, spin on/off switches and spintronic memory devices. To aim this goal, adsorption energies, spin polarizations, electronic gaps, magnetic properties and robustness of spin-polarized states have been studied in the presence of dopants and second layers. We find that for bilayer SMMs with two layers of different sizes the highest occupied m…
We report the experimental observation of sublattice-resolved resonant scattering in bilayer graphene by performing simultaneous cryogenic atomic hydrogen doping and electron transport measurements in ultrahigh vacuum. This allows us to monitor the hydrogen adsorption on the different sublattices of bilayer graphene without atomic-scale microscopy. Specifically, we detect two distinct resonant scattering peaks in the gate-dependent resistance, which evolve as a function of atomic hydrogen dosage. Theoretical calculations show that one of the peaks originates from resonant scattering by hydroge…
Polaritonic materials that support epsilon-near-zero (ENZ) modes offer the opportunity to design light-matter interactions at the nanoscale through phenomena like resonant perfect absorption and extreme sub-wavelength light concentration. To date, the utility of ENZ modes is limited in propagating polaritonic systems by a relatively flat spectral dispersion, which gives ENZ modes small group velocities and short propagation lengths. Here we overcome this constraint by coupling ENZ modes to surface plasmon polariton (SPP) modes in doped cadmium oxide ENZ-on-SPP bilayers. What results is a stron…
We study plasmon modes in doped AA-stacked bilayer graphene (BLG) within the nearest-neighbor tight-binding and the random phase approximation. We obtain closed analytical expressions for the polarizability function which are used to obtain the low-energy dispersion relations of and the numerical results for both acoustic and optical plasmon modes. Our result reveal the potential of AA-stacked BLG to be used as a tunable plasmonic device. In particular we find that the long-wavelength acoustic plasmon disperse as $ω_{+}\approx\sqrt{max(|μ|,t_{1})q}$ with a phase space which shrinks and vanishe…
We use angle-resolved photoemission spectroscopy to investigate the electronic structure of bilayer graphene at high n-doping and extreme displacement fields, created by intercalating epitaxial monolayer graphene on silicon carbide with magnesium to form quasi-freestanding bilayer graphene on magnesium-terminated silicon carbide. Angle-resolved photoemission spectroscopy reveals that upon magnesium intercalation, the single massless Dirac band of epitaxial monolayer graphene is transformed into the characteristic massive double-band Dirac spectrum of quasi-freestanding bilayer graphene. Analys…
We combine electrostatic and magnetic confinement to define a quantum dot in bilayer graphene. The employed geometry couples $n$-doped reservoirs to a $p$-doped dot. At magnetic field values around $B = 2~$T, Coulomb blockade is observed. This demonstrates that the coupling of the co-propagating modes at the $p$-$n$ interface is weak enough to form a tunnel barrier, facilitating transport of single charge carriers onto the dot. This result may be of use for quantum Hall interferometry experiments.
We theoretically study the optical activity in a doped system and derive the optical activity tensor from a light wavevector-dependent linear optical conductivity. Although the light-matter interaction is introduced through the velocity gauge from a minimal coupling Hamiltonian, we find that the well-known ``false divergences'' problem can be avoided in practice if the electronic states are described by a finite band effective Hamiltonian, such as a few-band tight-binding model. The expression we obtain for the optical activity tensor is in good numerical agreement with a recent theory derived…
We report the nanoscale imaging of Ti-doped bilayer calcium ruthenates during the Mott metal-insulator transition by microwave impedance microscopy. Different from a typical first-order phase transition where coexistence of the two terminal phases takes place, a new metallic stripe phase oriented along the in-plane crystalline axes emerges inside both the G-type antiferromagnetic insulating state and paramagnetic metallic state. The effect of this electronic state can be observed in macroscopic measurements, allowing us to construct a phase diagram that takes into account the energetically com…
Weak-coupling theory of half-metals and other fractional metallic phases in biased doped Bernal bilayer graphene
Abrikosov Center for Theoretical Physics, MIPT（阿布里科索夫理论物理中心，莫斯科物理技术学院）；Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences（俄罗斯科学院理论与应用电动力学研究所）
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.
Using a weak-coupling renormalization group formalism, we study competing ordered phases for repulsively interacting fermions on the bilayer honeycomb lattice away from half-filling, which is realized experimentally as doped bilayer graphene. As electrons are added to the system, excitonic order is suppressed, and unconventional superconductivity appears generically in its place. In general it is found that the maximum critical temperature for superconductivity appears directly adjacent to the dome of particle-hole order, illustrating the importance of fluctuations in these channels for the fo…
We use magneto-conductivity to study the magnetic proximity effect on surface states of doped topological insulators. Our bilayers consist of a layer of Fe$_7$Se$_8$, which is a metallic ferrimagnet and a layer of Bi$_{0.8}$Sb$_{1.2}$Te$_{3}$ which is a highly hole-doped topological insulator. Using transport measurements and a modified Hikami-Larkin-Nagaoka model, we show that the ferromagnet shortens significantly the effective coherence length of the surface states, suggesting that a gap is opened at the Dirac point. We show that the magnetically induced gap persists on surface states which…
We study the effects of an attractive interaction between the boron (B) and the nitrogen (N) atoms doped in a bilayer graphene (BLG), BC$_{14}$N, on the electronic, the thermal and the optical properties for two different types of a doping process: First, both the B and the N atoms are doped in the same layer while the other layer is undoped. Second, the B and N atoms are doped in both layers. An attractive interaction between the B and N atoms does not influence the interlayer interaction in the first case, while it does in the second case. We find that the strong B-N attractive interaction i…
We study the flow of a pure spin current through zinc oxide by measuring the spin Hall magnetoresistance (SMR) in thin film trilayer samples consisting of bismuth-substituted yttrium iron garnet (Bi:YIG), gallium-doped zinc oxide (Ga:ZnO), and platinum. We investigate the dependence of the SMR magnitude on the thickness of the Ga:ZnO interlayer and compare to a Bi:YIG/Pt bilayer. We find that the SMR magnitude is reduced by almost one order of magnitude upon inserting a Ga:ZnO interlayer, and continuously decreases with increasing interlayer thickness. Nevertheless, the SMR stays finite even f…
The screening properties and collective excitations (plasmons) in AA-stacked bilayer graphene are studied within the random phase approximation (RPA). Whereas long lived plasmons in single layer graphene and in AB-stacked bilayer graphene can exist only in doped samples, we find that coherent plasmons can disperse in AA-stacked bilayer graphene {\it even in the absence of doping}. Moreover, we show that the characteristic low energy dispersion relation is unaffected by changes in the number of carriers, unless the chemical potential of the doped sample exceeds the inter-layer hopping energy. W…
We consider ground state of electron-hole graphene bilayer composed of two independently doped graphene layers when a condensate of spatially separated electron-hole pairs is formed. In the weak coupling regime the pairing affects only conduction band of electron-doped layer and valence band of hole-doped layer, thus the ground state is similar to ordinary BCS condensate. At strong coupling, an ultrarelativistic character of electron dynamics reveals and the bands which are remote from Fermi surfaces (valence band of electron-doped layer and conduction band of hole-doped layer) are also affect…
Plasma oscillations in doped graphene bilayer at zero temperature has been investigated. Bias voltage effect on the dispersion curve for plasmon in bigraphene has been studied in random phase approximation. The possibility of controlling of curvature of dispersion curve for plasmon by changing of bias voltage has been shown. The dependence of this curvature on the bias voltage has been predicted to have the nonmonotonous character. Namely the existence of breaking point for such dependence has been found out. The bias voltage corresponding to the breaking point is shown to increase with free c…
We devise a model to explain why twisted bi-layer graphene (TBLG) exhibits insulating behavior when $ν=2,3$ charges occupy a unit moiré cell, a feature attributed to Mottness, but not for $ν=1$, clearly inconsistent with Mott insulation. We compute $r_s=E_U/E_K$, where $E_U$ and $E_K$ are the potential and kinetic energies, respectively, and show that (i) the Mott criterion lies at a density $10^4$ higher than in the experiments and (ii) a transition to a series of Wigner crystalline states exists as a function of $ν$. We find, for $ν=1$, $r_s$ fails to cross the threshold ($r_s = 37$) for the…
Motivated by recent experiments [H. Zhou, \emph{et al.}, Science {\bf 375}, 774 (2022) and S. C. de la Barrera, \emph{et al.}, arXiv:2110.13907], here we propose a general mechanism for valley and/or spin degeneracy lifting of the electronic bands in doped Bernal bilayer graphene, subject to electric displacement ($D$) fields. A $D$-field induced layer polarization (LP), when accompanied by Hubbard repulsion driven layer antiferromagnet (LAF) and next-nearest-neighbor repulsion driven quantum anomalous Hall (QAH) orders, lifts the four-fold degeneracy of electronic bands, yielding a quarter me…
It is generally believed that a Wigner Crystal in single layer graphene can not form because the magnitudes of the Coulomb interaction and the kinetic energy scale similarly with decreasing electron density. However, this scaling argument does not hold for the low energy states in bilayer graphene. We consider the formation of a Wigner Crystal in weakly doped bilayer graphene with an energy gap opened by a perpendicular electric field. We argue that in this system the formation of the Wigner Crystal is not only possible, but different phases of the crystal with very peculiar properties may exi…
We theoretically study the many-body effects of electron electron interaction on the single particle spectral function of doped bilayer graphene. Using random phase approximation, we calculate the real and imaginary part of the self-energy and hence the spectral function. The spectral function near the Fermi surface shows the usual quasiparticle peak, establishing doped bilayer graphene, in contrast to the unstable neutral system, to be a Fermi liquid. Away from the Fermi surface, an additional broad plasmaron peak is visible in the spectral function. From the low energy behaviour of the self-…
In search of an improved strategy to form low resistance contacts to MoS2 and related semiconducting transition metal dichalcogenides, we use ab initio density functional electronic structure calculations in order to determine the equilibrium geometry and electronic structure of MoO3/MoS2 and MoO2/MoS2 bilayers. Our results indicate that, besides a rigid band shift associated with charge transfer, the presence of molybdenum oxide modifies the electronic structure of MoS2 very little. We find that the charge transfer in the bilayer provides a sufficient degree of hole doping to MoS2, resulting …
We present an effective (minimal) theory for chiral two-dimensional materials. These materials possess an electro-magnetic coupling without exhibiting a topological gap. As an example, we study the response of doped twisted bilayers, unveiling unusual phenomena in the zero frequency limit. An in-plane magnetic field induces a huge paramagnetic response at the neutrality point and, upon doping, also gives rise to a substantial longitudinal Hall response. The system also accommodates nontrivial longitudinal plasmonic modes which are associated with a longitudinal magnetic moment, thus endowing t…
Spintronics is an analogue to electronics where spin of the electron rather than its charge is functionally controlled for devices. The generation and detection of spin current without ferromagnetic or exotic/scarce materials are two the biggest challenges for spintronics devices. In this study, we report a solution to the two problems of spin current generation and detection in Si. Using non-local measurement, we experimentally demonstrate the generation of helical dissipationless spin current using spin-Hall effect. Contrary to the theoretical prediction, we observe the spin-Hall effect in b…
Using a simple LCAO model by Harrison, we have qualitatively studied the edge state of bilayer phosphorene, which is a unit structure of the layered crystal of black phosphorus. This model successfully reproduces the isolated edge state in the bulk gap in monolayer phosphorene. In bilayer phosphorene, however, it shows that edge states are almost buried in the valence band and there is no isolated midgap edge state at the zigzag edge. Since the buried edge state works as acceptor, holes are doped from the edge state into the bulk. This gives a possible explanation for p-type conduction in undo…
We investigate the intraband nonlinear dynamics in doped bilayer graphene in the presence of strong, linearly-polarized, in-plane terahertz fields. We perform degenerate pump-probe experiments with 3.4 THz fields on doped bilayer graphene at low temperature (12 K) and find that when the pump is co-polarized with the probe beam, the differential pump-probe signal is almost double that found in the cross-polarized case. We show that the origin of this pump-induced anisotropy is the difference in the average electron effective mass in the probe direction when carriers are displaced in k-space by …
Performing a leading-order renormalization group analysis, here we compute the effects of generic local or short-range electronic interactions in monolayer and Bernal bilayer graphene. Respectively in these two systems chiral quasiparticles display linear and biquadratic band touching, leading to linearly vanishing and constant DOS. Consequently, the former system remains stable for weak enough local interactions, and supports a variety of ordered phases only beyond a critical strength of interactions. By contrast, ordered phases can nucleate for sufficiently weak interactions in bilayer graph…
We present a theoretical study of the modification of the near-field radiative heat transfer due to phonon-plasmon coupling in bilayer systems made of a doped semiconductor and a polar dielectric. By tuning the surface-plasmon mode of the former material in resonance with the phonon modes of the latter one, the near-field spectral radiative heat transfer can be suppressed in a frequency band gap. We distinguish between the interlayer mode coupling within each bilayer and the intragap one between the two bilayers. We elucidate the role of each of them in the formation of the band gap and we det…
We investigate quantum pumping of massless Dirac fermions in an ideal (impurity free) double layer of graphene. The pumped current is generated by adiabatic variation of two gate voltages in the contact regions to a weakly doped double graphene sheet. At the Dirac point and for a wide bilayer with width W much larger then length L, we find that the pumped current scales linearly with the interlayer coupling length l_{perp} for L/l_{perp} << 1, is maximal for L/l_{perp} ~ 1, and crosses over to a log(L/l_{perp})/(L/l_{perp})-dependence for L/l_{perp} >> 1. We compare our results with the behavi…
We present a systematic theory of acoustic-phonon-mediated superconductivity, which incorporates Coulomb repulsion, explaining the recent experiment in Bernal bilayer graphene under a large displacement field. The acoustic-phonon mechanism predicts that $s$-wave spin-singlet and $f$-wave spin-triplet pairings are degenerate and dominant. Assuming a spin-polarized valley-unpolarized normal state, we obtain $f$-wave spin-triplet superconductivity with a $T_c\sim 20$ mK near $n_e=-0.6\times 10^{12}$ cm$^{-2}$ for hole doping, in approximate agreement with the experiment. We further predict the ex…
Motivated by Weyl semimetals and weakly doped semiconductors, we study transport in a weakly disordered semiconductor with a power-law quasiparticle dispersion $ξ_{\bf k}\propto k^α$. We show, that in $2α$ dimensions short-correlated disorder experiences logarithmic renormalisation from all energies in the band. We study the case of a general dimension $d$ using a renormalisation group, controlled by an $\varepsilon=2α-d$-expansion. Above the critical dimensions, conduction exhibits a localisation-delocalisation phase transition or a sharp crossover (depending on the symmetries of the Hamilton…
Proximity-induced magnetic effects on the surface Dirac spectra of topological insulators are investigated by scanning tunneling spectroscopic (STS) studies of bilayer structures consisting of undoped Bi2Se3 thin films on top of Cr-doped Bi2Se3 layers. For thickness of the top Bi2Se3 layer equal to or smaller than 3 quintuple layers (QL), a spatially inhomogeneous surface spectral gap Δopens up below T_c^{2D}, which is much higher than the bulk Curie temperature T_c^{3D}. The mean value and spatial homogeneity of the gap Δgenerally increase with increasing c-axis magnetic field (H) and increas…
In this paper, we propose a novel S/D engineering for dual-gated Bilayer Graphene (BLG) Field Effect Transistor (FET) using doped semiconductors (with a bandgap) as source and drain to obtain unipolar complementary transistors. To simulate the device, a self-consistent Non-Equilibrium Green's Function (NEGF) solver has been developed and validated against published experimental data. Using the simulator, we predict an on-off ratio in excess of $10^4$ and a subthreshold slope of ~110mV/decade with excellent scalability and current saturation, for a 20nm gate length unipolar BLG FET. However, th…
Using ab initio methods based on the density functional theory we show that rotated graphene bilayers at angles different from the magic ones can have an electronic spectrum similar to those by applying moderate external pressures. We find that for an angle of $5.08^{\circ}$ and a pressure of 2.19 GPa the spin restricted spectrum displays a flat band at the Fermi level similar to the one found at magic angles. In addition, the spin unrestricted calculations show a correlated ferromagnetic ground state with a total magnetic moment of 3.7 $μ_{B}$ per unit cell being mostly localized in the AA st…
We analyze the statistical characteristics of the quasi-nonequilibrium two-dimensional electron-hole plasma in graphene layers (GLs) and graphene bilayers (GBLs) and evaluate their heat capacity.The GL heat capacity of the weakly pumped intrinsic or weakly doped GLs normalized by the Boltzmann constant is equal to $c_{GL} \simeq 6.58$. With varying carrier temperature the intrinsic GBL carrier heat capacity $c_{GBL}$ changes from $c_{GBL} \simeq 2.37$ at $T \lesssim 300$~K to $c_{GBL} \simeq 6.58$ at elevated temperatures. These values are markedly differentfrom the heat capacity of classical …
In Bernal stacked bilayer graphene interlayer coupling significantly affects the electronic bandstructure compared to monolayer graphene. Here we present magnetotransport experiments on high-quality $n$-doped bilayer MoS$_{2}$. By measuring the evolution of the Landau levels as a function of electron density and applied magnetic field we are able to investigate the occupation of conduction band states, the interlayer coupling in pristine bilayer MoS$_{2}$, and how these effects are governed by electron-electron interactions. We find that the two layers of the bilayer MoS$_{2}$ behave as two in…
High mobility single and few-layer graphene sheets are in many ways attractive as nanoelectronic circuit hosts but lack energy gaps, which are essential to the operation of field-effect transistors. One of the methods used to create gaps in the spectrum of graphene systems is to form long period moire patterns by aligning the graphene and hexagonal boron nitride (h-BN) substrate lattices. Here, we use planar tunneling devices with thin h-BN barriers to obtain direct and accurate tunneling spectroscopy measurements of the energy gaps in single- and bi-layer graphene-h-BN superlattice structures…
Electron wavefunctions in twisted bilayer graphene may have a strong single layer character or be intrinsically delocalized between layers, with their nature often determined by how energetically close they are to the Dirac point. In this paper, we demonstrate that in magnetic fields, optical absorption (cyclotron resonance) spectra contain signatures which may be used to distinguish the nature of these wavefunctions at low energies, as well as to locate low energy critical points in the zero-field energy spectrum. Optical absorption for two different configurations -- electric field parallel …
We derive core-level spectra for doped free-standing bilayer graphene. Numerical results are presented for all nine combinations of the doping concentrations $10^{12}\rm{cm}^{-2}$, $10^{13}\rm{cm}^{-2}$, and $10^{14}\rm{cm}^{-2}$ in the two graphene sheets and we compare the results to the reference spectra for monolayer graphene. We furthermore discuss the spectrum of single-particle inter-band and intra-band excitations in the $ωq$-plane, and show how the dispersion curves of the collective modes are modified in the bilayer system.
