Theory of dielectric loss in Graphene-on-substrate: A tight- binding model study

Graphene-on-substrate exhibits interesting dielectric behaviour due to screening of coulomb interaction induced by many body effects. In this communication we attempt to study the dielectric loss property of graphene within tight-binding model approach. The Hamiltonian consisting of electron hopping upto third-nearest-neighbour's with impurities in two in equivalent sub-lattices. The graphene-on-substrate raises the energy +Δ at one sub lattice and reduces energy -Δ at other sub lattice. Further we introduced coulomb interaction between π - electrons at the two sub lattices separately with the same effective coulomb interaction. We calculate polarization function Π(q, ω) which is a two particle Green's function arising due to charge-charge correlation by using Zubarev's Green's function technique. Finally we calculate dielectric function of graphene i.e. ε(q, ω) =1+Π(q,ω) at arbitrary wave vector q and frequency ra. The dielectric loss in graphene calculated from the imaginary part of dielectric function which is a measure of absorption spectrum. Only a few Fragmentary theoretical attempts have been made to utilize the full frequency and wave vector dependent dielectric function. We compute numerically the frequency dependent dielectric loss function for 100x100 momentum grid points. We observe a low energy Plasmon resonance peak and a high energy flat peak arising due to absorption of optical energy at substrate induced gap. With increase of small Plasmon wave vector, both low and high energy peaks approach each other. The dielectric loss at low energies exhibits a parabolic curve, but it exhibit a clear peak on introduction of higher order electron hopping's. The Coulomb interaction suppresses induced gap in graphene and decreases the optical energy absorption spectra. The increase of substrate induced gap shifts the high energy flat peak to higher energies and enhances the dielectric loss throughout the frequency range. Finally the effect of doping on dielectric loss is investigated and compared with the experimental results.

4A,Shree Vihar, C. S. Pur, PO-Patia , Bhubaneswar-751031, Odisha, India 1) siva1987@iopb.res.in , 2) skp@iopb.res.in , * Corresponding author: gcr@iopb.res.in Abstract. Graphene-on-substrate exhibits interesting dielectric behaviour due to screening of coulomb interaction induced by many body effects. In this communication we attempt to study the dielectric loss property of graphene within tight-binding model approach. The Hamiltonian consisting of electron hopping upto third-nearest-neighbour's with impurities in two in equivalent sub-lattices. The graphene-on-substrate raises the energy +∆ at one sub lattice and reduces energy -∆ at other sub lattice. Further we introduced coulomb interaction between πelectrons at the two sub lattices separately with the same effective coulomb interaction. We calculate polarization function Π(q, ω) which is a two particle Green's function arising due to charge-charge correlation by using Zubarev's Green's function technique. Finally we calculate dielectric function of graphene i.e. ε(q, ω) =1+Π(q ,ω) at arbitrary wave vector q and frequency ω. The dielectric loss in graphene calculated from the imaginary part of dielectric function which is a measure of absorption spectrum. Only a few Fragmentary theoretical attempts have been made to utilize the full frequency and wave vector dependent dielectric function. We compute numerically the frequency dependent dielectric loss function for 100x100 momentum grid points. We observe a low energy Plasmon resonance peak and a high energy flat peak arising due to absorption of optical energy at substrate induced gap. With increase of small Plasmon wave vector, both low and high energy peaks approach each other. The dielectric loss at low energies exhibits a parabolic curve, but it exhibit a clear peak on introduction of higher order electron hopping's. The Coulomb interaction suppresses induced gap in graphene and decreases the optical energy absorption spectra. The increase of substrate induced gap shifts the high energy flat peak to higher energies and enhances the dielectric loss throughout the frequency range. Finally the effect of doping on dielectric loss is investigated and compared with the experimental results.

Introduction
The graphene with a honey-comb lattice structure exhibits a gap less semiconductor with linear dispersion relation in electron momentum in the vicinity of the two in equivalent corners of the hexagonal brillouin zone [1]. Due to the lack of band gap in graphene, the pristine graphene cannot be used in spintronic device applications. Therefore, intense attempts have been made both experimentally and theoretically to induce a band gap near Dirac point by several techniques like introducing impurities, ad-atoms, defects, different substrates, Coulomb correlation etc. [2]. The optical spectroscopy has proven to be an excellent probe to study band gap opening in graphene system. When infrared radiation is incident on graphene, the π-electron Plasmon vibration sets in graphene with a wave vector dependent Plasmon frequency. The optical frequency response of graphene leads to the dielectric function. There are a very few theoretical studies on frequency and temperature dependent dielectric function taking into account the electronic dispersion throughout the brillouin [3,4].
Hwang et. al. [5] have reported theoretical study of dielectric constant and plasmon dispersion in two dimensions. More recently Abergel have reported DFT calculation for infrared absorption of some heterostructures of monolayer and bilayer graphene with hexagonal boron nitride. In this report, we have considered a tight-binding model containing electron and hole doping effects, substrate induced gap as well as on-site Coulomb interaction at two sub-lattices of graphene. We attempt here to calculate the dielectric response function and hence the dielectric loss from its imaginary part graphene system [6].

Theoretical Model
The tight-binding model Hamiltonian for honey-comb lattice of graphene is written as The first term in the Hamiltonian given in equation (1)

Calculation of dielectric loss function
The frequency ( ) and wave vector (q) dependent dielectric function is written as , where the polarization function which describes the charge-charge correlation is written as

Result and Discussion
In order to investigate the effect of Coulomb interaction in , we evaluate numerically and self- The imaginary part of dielectric function describes the dielectric loss i.e. Im in graphene in present calculation . The dielectric loss is plotted vs. optical energies for small plasmon wave vectors, in the long wave limit. The dielectric loss exhibits a divergent peak (p1) at energy . This singular structure arises from the characteristics of the inter band excitation energy in the wave vector space. near the peak points to isotropic and linear at very small and . Fig. 1 shows that the peak p1 increases with increase of wave vector and shifts to higher energies. The band model calculation here is used to study the singular structure. We observe here another flat singular structure at higher energy arising due to substrate induced gap . If    fig. 2). Again the dielectric loss remains constant at higher energies. It is note to further that the dielectric loss or absorption of optical energy increases with increase of the magnitude of the substrate induced gap. The experimental measurements of graphene on different substrates exhibits gaps of energy 100 meV for graphene on BN [15], 200 meV for gold on ruthenium [16 ] and 250 meV for graphene on SiC [17 ].

Conclusion
We have calculated the dielectric loss or optical energy absorption co-efficient from the imaginary part of the dielectric function for graphene-on-substrate by tight-binding approach using Green's function technique. We have studied the evolution of low energy Plasmon momentum transfer energy excitation peak and substrate induced gap excitation peak by varying Plasmon momentum, substrate induced gap and Coulomb energy.