The effect of sublattice symmetry breaking on the electronic properties of a doped graphene

Motivated by a number of recent experimental studies, we have carried out the microscopic calculation of the quasiparticle self-energy and spectral function in a doped graphene when a symmetry breaking of the sublattices is occurred. Our systematic study is based on the many-body G$_0$W approach that is established on the random phase approximation and on graphene's massive Dirac equation continuum model. We report extensive calculations of both the real and imaginary parts of the quasiparticle self-energy in the presence of a gap opening. We also present results for spectral function, renormalized Fermi velocity and band gap renormalization of massive Dirac Fermions over a broad range of electron densities. We further show that the mass generating in graphene washes out the plasmaron peak in spectral weight.


I. INTRODUCTION
Graphene is a single atomic layer of crystalline carbon on the honeycomb lattice consists of two interpenetrating triangular sublattices A and B, has opened up a new field for fundamental studies and applications. [1,2,3,4] Peculiar electronic properties of graphene give rise possibility to come over silicon-based electronics limitations. [5] The single-particle energy spectrum in graphene contains two zero-energy at K + and K − points of the Brillouin zone which are called as valleys or Dirac points. Due to the presence of the two carbon atoms per unit cell, the quasiparticle (QP) need to be described by a two component wave function.
The charge carriers in a pristine graphene show linear and isotropic energy dispersion relation and massless chiral behavior for the energy scales up to 1 eV. Recently, graphene has revealed a variety of unusual transport phenomena characteristics of two-dimensional (2D) Dirac Fermions such as an anomalous integer quantum Hall effect at room temperature, a minimum quantum conductivity, Klein tunneling paradox, weak and anti-localization, an absence of Wigner crystallization phase and Shubnikov-deHaas oscillations that exhibit a phase shift of π due to Berry's phase. [6,7,8,9,10,11,12] One important difference between conventional electron gas and Dirac Fermion particle is that the contribution of exchange and correlation to the chemical potential is an increasing rather than a decreasing function of carrier-density. This property implies that exchange and correlation increase the effectiveness of screening, in contrast to the usual case in which exchange and correlation weakens screening. This unusual property follows from the difference in sublattice pseudospin chirality between the Dirac model's negative energy valence band states and its conduction band states.
The massless Dirac-like carriers in graphene have almost semi-ballistic transport behavior with small resistance due to the suppression of back-scattering process, and moreover graphene is a good thermal conductor. [13] The mobility of carriers in graphene is quite high [14,15,16,17] which is much higher than the electron mobility revealed on the semiconductor hetrostructures. [18,19] On the other hand, by measuring the stiffness of materials it is shown that graphene is the strongest material in two-dimension structures. [20,21] These properties as well as capability to control of the type and density of charge carriers by gate voltage or the chemical doping [22,23,24,25] make graphene an ideal candidate for superior nano-electronic devices operating at high frequencies.
Most electronic applications are based on the presence of a gap between the valence and conduction bands in the conventional semiconductors. The band gap is a measure of the threshold voltage and on-off ratio of the field effect transistors (FETs). [26,27] Therefore, for integrating graphene into semiconductor technology, it is crucial to induce a band gap in Dirac points to control the transport of carriers. Consequently, band gap engineering in graphene is a hot topic with fundamental and applied significance. [28] In the literature several routes have being proposed and applied to induce and control a gap in graphene. One of them is using quantum confined geometries such as quantum dots and nanoribbons. [29,30,31,32,33] It is shown that the gap values increases by decreasing of nanoribbon width.
Another alternative way is spin-orbit coupling whose origin is due to both intrinsic spin-orbit interactions and the Rashba interaction. [34,35,36,37] Another method to generate a gap in graphene sheets is an inversion symmetry breaking of the sublattices when the number of electrons on A and B atoms are different [38,39,40,41] or Kekulé [42] distortion, e.g.
Recently angle resolved photoemission spectroscopy ( ARPES) experiments on graphene epitaxially grown on SiC and ab initio simulations reported a gap opening in the band structure of graphene placed on proper substrates, and suggested that interactions between the graphene sheet and the substrate leads to symmetry breaking of the A and B sublattices and it consequences to induce a gap in the band structure. Experimenters [24,25,43,44,54] observed a gap of 260 meV in band structure of the epitaxial graphene on SiC substrate due to interaction with substrate. In addition, Zhou et al. [24] found a reversible metalinsulator transition and a fine tuning of the carriers from electron to hole by molecular doping in gapped graphene. A Density Functional Theory (DFT) calculation confirmed a substrate induced symmetry breaking. [55]. Their results showed a gap in the band spectra of graphene about 200 meV which is in agreement with recent experimental observation.
Their calculation determined that there is a 140 meV on-site energy difference between two sublattices. In addition, a band gap is observed in spectra of graphene on Ni (111) substrate [45,46] as well as a gap about of 10 meV in suspended graphene above a graphite substrate [47] due to sublattice symmetry breaking mechanism. Moreover, based on the ab initio calculations, it is suggested that boron nitride substrate induced a gap of 53 meV. [48] Note that the gap value calculated within DFT is in general underestimating the true band gap value.
In this paper we consider the sublattice symmetry breaking mechanism for a gap opening in a pristine doped graphene sheet and study the impact of gap upon some electronic properties of QPs. To investigate the influence of gap in the many-body properties of QP in graphene we use the random phase approximation (RPA) and the G 0 W approximation.
It should be noted that a detailed analysis provided a framework for the microscopic evaluation of the QP-QP interaction in the gapless graphene by means of the RPA was carried out by us in Ref. [49] At the beginning, we review briefly the results of the ground state thermodynamic properties that we have already presented elsewhere. [50] Our new results are based on the QP self-energy properties in the presence of a gap opening in the electronic spectrum. From the self-energy we then obtain the QP energies, renormalized Fermi velocity, spectral function which can be compared with ARPES spectra and finally the band gap renormalization of massive Dirac Fermions in doped graphene. We have shown that mass generating in graphene washes out a satellite band in the spectral function in agreement with recent experimental observations. [43] This paper is organized as followed. In Section II we introduce our model Hamiltonian and then review some ground state properties of gapped graphene. In Section III we focus on the properties of imaginary and real parts of self-energy for gapped graphene and then calculate QP spectral function, renormalized Fermi velocity and band gap renormalization.
Finally we conclude in Section IV.

II. GROUND STATE THERMODYNAMIC PROPERTIES
We consider the sublattice symmetry breaking mechanism in which the densities of particles associated to on-site energy µ a(b) , for A(B) sublattice are different. The electronic structure of graphene can be reasonably good described using a rather simple tight-binding Hamiltonian, leading to analytical solutions for their energy dispersion and related eigenstates. The noninteracting tight binding Hamiltonian for π band electrons is determined by [38,39,40,41] where the sums run over unit cells, t ≃ 2.7 eV denotes the nearest neighbor hopping pa- We consider the long-range Coulomb electron-electron interaction. We left out the intervalley scattering and use the two component Dirac Fermion model. Accordingly, the total interacting Hamiltonian in a continuum model at K + point is expressed as [57,58] where Ψ † k,σ = (ψ a +,σ (k), ψ b +,σ (k)) is two component pseudospinors of the noninteracting Hamiltonian, S is the sample area,N is the total number operator and V q = 2πe 2 /ǫq is the bare Coulomb interaction where ǫ is an average dielectric constant of the surrounding medium. The coupling constant in graphene is α gr = g s g v e 2 /ǫ v F where g s = g v = 2 being the spin and valley degeneracy, respectively. The coupling constant in graphene depends only on the substrate dielectric constant while in the conventional 2D electron systems is density dependent. The typical value of dimensionless coupling constant is 1 or 2 for graphene supported on a substrate such a SiC or SiO 2 .
A central quantity in the many-body techniques is the noninteracting dynamical polarizability function χ (0) (q, iω, µ) where µ is the chemical potential. The problem of linear density response is set up by considering a fluid described by the Hamiltonian,Ĥ, which is subject to an external potential. The external potential must be sufficiently weak for low-order perturbation theory to suffice. The induced density change has a linear relation to the external potential through the noninteracting dynamical polarizability function. This function is recently calculated along the imaginary frequency axis and it is given by [50] where The Fermi energy of a 2D massive Dirac Fermion system is given by which is density dependent at the Fermi surface. It should be noticed that D(E F ) equals to m/2π 2 in the conventional 2D electron gas system. Here, Θ(x) is Heaviside step function.
We now turn to present our first numerical results which are based on the noninteracting polarization function. The static polarization function as a function of wavevector for various gap values is shown in Fig. 1(a). The static polarization function in gapless case is a smooth function whereas a kink at q = 2k F occurs for gapped graphene and thus the derivatives of  -  We can calculate the total ground state energy of gapped graphene within RPA. [50,58] The ground-state energies can be calculated using the coupling constant integration technique, which has the contributions ε tot = ε kin + ε xc . The kinetic energy per particle is [50,58] we might subtract the vacuum energy contribution from the total energy, δε tot = ε tot (k F ) − ε tot (k F = 0). Due to the number of states in the Brillouin zone must be conserved, we do need a ultraviolet cut-off k c , which is approximated by where A 0 is the area of the unit cell. The dimensionless parameter Λ is defined as k c /k F ≃ (g s g v n −1 √ 3/9.09) 1/2 × 10 2 .
In Fig. 3, we have shown the exchange-correlation energy in units of ε F = v F k F , as a function of n −1/2 in units of 10 −6 cm for various ∆ value. The exchange energy arises entirely from the antisymmetry of the many-body wave function under exchange of two electrons is positive while the correlation energy, the difference between the ground state energy and the sum of the kinetic energy and the exchange energy is negative. This has important implications on the thermodynamic properties can be calculated from the derivative of the ground state energy with respect to the density. The compressibility can be calculated from its definition, κ −1 = n 2 ∂ 2 (nδε tot )/∂n 2 . Fig. 3(b) shows the ratio between the noninteractiong value, κ 0 = 2/nε F and the interaction value of compressibility as a function of n −1/2 . The exchange tends to reduce the compressibility while correlations tends to enhance it. At large ∆, a minimum structure occurs at the inverse of compressibility behavior and we expect that at very large ∆, it starts at κ 0 and reduces by increasing n −1/2 behaves like the compressibility of the conversional 2D electron gas.    -Imχ

III. THE QP SELF-ENERGY AND THE SPECTRAL FUNCTION
The generation of QPs in an electron liquid leads to two effects. First it induces a decay of a particle losing momentum via inelastic scattering which is determined by the imaginary part of self-energy and second is the renormalization of the dispersion relation of the carriers which is described by the real part of self-energy. ℜeΣ ret (k, ω) is defined as the difference between the measured carrier energy ω, and the energy of free particle, ξ sk = sE k − E F .
To satisfy causality, the real and imaginary parts of self-energy are related by a Hilbert transformation. In this section, we first derive the imaginary and the real part of QP self-energies and then calculate some important quantities such as a renormalized Fermi velocity, a spectral function and a band gap renormalization in the presence of a band gap value. These quantities are related to some important physical properties of both theoretical and practical applications like the band structure of ARPES, the energy dissipation rate of injected carriers and the width of the QP spectral function. [61,62] In the G 0 W approximation, the self-energy of gapped graphene is given by (β = 1/(k B T )) [63]: where W (q, iΩ m ) = V q /ǫ(q, iΩ m ) is the dynamical screened effective interaction and The overlap function for gapped graphene F ss ′ (k, k + q) arises from the graphene band structure is given by [50] F ss ′ (k, k + q) It should be noted that F s=−s ′ (q = 0) = 0. However, in gapless graphene, intraband backward scattering should not be allowed, namely F s=s ′ (q = −2k, ∆ = 0) = 0, as well as F s=−s ′ (q = 0, ∆ = 0) = 0. In Eq. (4), G s (k, iω) = 1/(iω − ξ sk / ) is the noninteracting Green's function. Notice that in typical density of carriers in graphene namely n > 10 12 cm −2 , the Fermi temperature is about T F = ε F /k B > 10 3 K, and we therefore can eliminate temperature parameter in our calculations. To evaluate the zero-temperature retarded self-energy we perform the line-residue decompositions, Σ ret s (k, ω) = Σ line s (k, ω) + Σ res s (k, ω), where Σ line is obtained by performing the analytic continuation before summing over the Matsubara frequencies, and Σ res is the correction which must be taken into account in the total selfenergy. [57] At zero temperature we have The line contribution of the self-energy is purely real. The imaginary part of the self-energy has two contributions where ℑmΣ ret + (k, ω) = ℑmΣ res +,intra (k, ω) + ℑmΣ res +,inter (k, ω), and real part of the self energy can be decomposed as ℜeΣ ret + (k, ω) = Σ line + (k, ω) + ℜeΣ res +,inter (k, ω) + ℜeΣ res +,intra (k, ω). For ω > 0 and fixed q, the RPA decay process represents scattering of an electron from momentum k and energy ω to k + q and ξ s ′ (k + q), with all energies in Eq. (7) measured from the Fermi energy of doped graphene. Since the Pauli exclusion principle requires that the final state is unoccupied, it must lie in the conduction band, i.e. s ′ = +1. Furthermore since the Fermi sea is initially in its ground state, the QP must lower its energy, i.e. ξ s ′ < ω, electrons decay by going down in energy. For ω < 0, the self-energy expresses the decay of holes inside the Fermi sea, which scatter to a final state, by exciting the Fermi sea.
In this case the final state must be occupied so both band indices are allowed for s ′ , and energy conservation requires that holes decay by moving up in energy. Since photoemission measures the properties of holes produced in the Fermi sea by photo ejection, only ω < 0 is relevant for this experimental probe.
In what follows, we calculate the intraband and interband contributions of self-energy.
We have found the intraband term of residue part of self-energy as following for various values of the frequencies, where On the other hand, the interband contribution of residue part of the self energy is determined by and eventually for the line contribution of self-energy we have where and φ denotes an angle between k and q. Note that the real part of self-energy is k c dependent.
Now we are in a situation that can calculate some important physical quantities. The QP lifetime or the single-particle relaxation time τ , is obtained by setting the frequency to the onshell energy in imaginary part of the self-energy, τ −1 s = Γ s (k, ξ sk / ) = 2 |ℑmΣ ret s (k, ξ sk / )| where Γ s (k, ξ sk / ) is the quantum level broadening of the momentum eigenstate |sk >. This quantity is identical with the Fermi's golden rule expression for the sum of the scattering rate of a QP and quasihole at wavevector k. [57] From Eqs. 8 and 9, one can conclude that total contribution of the imaginary part of the retarded self-energy on the energy shell comes from the intraband term, ℑmΣ ret + (k, ξ k / ) = ℑmΣ res intra (k, ξ k / ). [63] In the case of gapless graphene, scattering rate is a smooth function because of the absence of both plasmon emission and interband processes. [64,65] However, with generating a gap and increasing the amount of it, plasmon emission cause discontinuities in the scattering time, similar to conventional 2D electron gas. [66,67] We have thus two mechanisms for scattering of the QPs.
The excitation of electron-hole pairs which is dominant process at long wavelength regions and the excitation of plasmon appears in a specific wave vector. As discussed previously [63], in clean graphene sheets the inelastic mean free path reduces by increasing the gap whereas the mean free path is large enough in the range of the typical gap values 10-130 meV, and thus transport remains in the semi-ballistic regime.
where δΣ ret s (k, ω) = Σ ret s (k, ω) − Σ ret s (k F , 0), and then ARPES intensity can be described by I(k, ω) = A(k, ω)n(ω), where n(ω) is the Fermi-Dirac distribution. The spectral function is the Lorentzian function where ℜeΣ specifying the location of the peak of the distribution, and |ℑmΣ| is the linewidth. The amplitude of the the Lorentzian function is proportional to 1/|ℑmΣ|. This quantity is the distribution of energies ω, in the system when a QP with momentum k, is added or removed from that. For the noninteracting system we get A (0) (k, ω) = δ(ω − ξ(k)/ ). The Fermi liquid theory applies only when the spectral function at the Fermi momentum A (0) (k = k F , ω), behaves as a delta function, and has a broadened peak indicating damped QPs at k = k F .
To progress of the interband single particle excitation and plasmon effects on the ℑmΣ ret s , we must study the retarded self-energy on the off-shell frequency which is ω = ξ sk / . [49,68] This quantity gives the scattering rate of a QP with momentum k and kinetic energy ω+E F .
The scattering rate or the linewidth raising from electron-electron interactions is anisotropic and varies significantly via wavevector at a constant energy. The imaginary part of self energy shows the width of the QP spectral function.
In Fig. 4 we have shown the absolute value of the imaginary part of the self energy in unit of ε F for various gap values. It would be noticed that there is an area of frequency is associated to the gap value, 2∆ in which no QP could enter in. In this case, there is a gap in the ℑmΣ between ξ −,k=0 and ξ +,k=0 . We see that ℑmΣ + vanishes as ω 2 for ω tends to zero, a universal properties of normal Fermi liquid. Moreover, at large frequency, ℑmΣ + tend towards to ω linearly. Except from the Dirac point, the conduction band ℑmΣ + peaks broaden because of the dependence on scattering angle of ξ(k + q). For low energy, only intraband single particle excitation contributes to ℑmΣ up to E F and then the interband single particle excitation contribution increases sharply about E F . The interband contribution increases with increasing the gap values.
To evaluate the scattering rate in interband channel, we have shown ℑmΣ inter(intra) as a function of frequency in Fig. 5. The intraband contribution of the imaginary part of self energy associated to scattering rate of QP in the intraband contribution increases with increasing the gap values while the interband contribution reduces, as we physically expected.
Moreover, by increasing of the electrons in the conduction band the interband scattering rate reduces whereas the intraband scattering contribution increases. The gap value suppresses the scattering rate at ω = −ε F . In Fig. 6 we have plotted the real part of self-energy in unit of ε F as a function of the energy for various gap values. Notice again that the real part of residue self-energy has a gap which is associated the feature calculated in the imaginary part of self-energy. The line part of self energy is a continues curve and then we have a jump near to the boundary of gap values in the ℜeΣ for gapped graphene. A kink around E F is associated to the interband plasmon contribution and it is broaden due to the gap value. This feature affects noticeably in the interacting electron density of states.     As discussed before [49,68] in a zero temperature and disorder free gapless graphene, the peaks of the spectral function correspond to the nearly solutions of Dyson's equation in which the quasiparticle excitation energies are obtained by E = ξ + + ℜeδΣ ret + . The intersection of ℜeΣ and the lines E − ξ + indicates a satellite long wavelength plasmaron peak related to the electron-plasmon excitation due to the long-range electron-electron Coulomb interaction and the Dyson equation with ℑmΣ = 0 corresponds to a QP peak related to the single particle excitation. Importantly, in the presence of gap values, the plasmaron peak suppressed.
In Figs. 7(a) and (b) we have shown the energy distribution curves (EDC) and momentum distribution curves (MDC), respectively. In the presence of gap values, as shown in Fig. 7(a) there is only the single QP peak.
The valance band self-energy contributions are shown in Fig. 8. There is an area of ℜe Σ ℜe Σ   It is essential to note that the satellite band which is theoretically predicted [49] for gapless garphene has not been seen in experiments. There are several reasons that could wash out this feature. For example, the plasmon damping, disorder effects, electron interactions with   the buffer layer and importantly the effect of gap at Dirac point.
One of the important information which can be extracted from ARPES spectra is the renormalized Fermi velocity v * . A consequence of the interaction is a Fermi velocity renormalization from the backflow of the fluid around a moving particle.
where v s = sv F . It is found before [63,69,70,71,72] that electron-electron interaction increases the renormalized Fermi velocity in gapless graphene sheets which this behavior is in contrast to conventional 2DES. [66,73] Fig . 9 shows the renormalized Fermi velocity in unit of the bare Fermi velocity as a function of band gap for various carrier densities. The renormalized Fermi velocity decreasing with increasing the gap value. v * is density independent after ∆ = 0.8ε F which is in good agreement with recent experiment observation. [24,25] Finally we calculated a band gap renormalization (BGR). [74,75,76,77] The BGR for conductance band is given by the QP self-energy at the band edge, namely BGR = ℜeΣ ret + (k = 0, ω = (∆ − E F )/ ). Fig. 10 shows the BGR for the various gap values as a function of the electron density. The BGR decreases by increasing of the electron density and in the small energy gap values, it is less density dependent respect to large energy gap values. In gapless case, we have obtained a induced band gap or kink due to manybody electron-electron interactions and it tends to a constant with increasing the electron density. [49,68,71,72] This feature is in agreement with the results obtained within ab intio DFT calculation. [71]

IV. SUMMERY AND CONCLUSION
We have revisited the problem of the microscopic calculation of the QP self-energy and many-body effective velocity suppression in a gapped graphene when the conduction band is partially occupied. We have performed a systematic study is based on the many-body