Substrate interaction effects on optical and acoustical plasmons in bi-waveguides based on graphene

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Abstract

We theoretically study the dynamic dielectric response function of a gas of massless Fermions embedded in a coupled double quantum wire structure based on graphene. We write the dielectric function within the random phase approximation (RPA). We approach the system using the two-dimensional (2D) Dirac-like Hamiltonian in the first place, where a parameter β, accounting for the interaction between the substrate and the graphene sheet, is considered in an ad-hoc manner. We study the weak tunneling regime between the two ribbons and find the energy dispersion of the acoustical and optical plasmon modes. Our results show that different choices for the parameter β in the structure should induce spatial anisotropy effects on the plasmon modes.

Research Highlights

► Dielectric function of a massless-Fermions gas in a graphene bi-waveguide is studied. ► Acoustical and optical plasmon modes in the weak tunneling regime are found. ► Graphene-substrate interaction leads to a spatial anisotropy effects on the modes.

Introduction

There has been a great deal of interest in studying a single 2D layer of carbon atoms, the graphene, due to its intriguing electronic properties [1]. More recently, there has been also much experimental concern about substrate induced effects on the graphene sheets. Anisotropic effects have been observed [2] on the conductivity in such experiments. These effects were claimed to be induced by the interaction between the graphene sheet and the underlying substrate. In fact, the graphene–substrate interaction is causing important effects in solving the 2D Dirac-like Hamiltonian which describes the graphene within the low-lying energy approximation [3]. Furthermore, the improving experimental techniques of growing and controlling such samples leads to the fabrication of one-dimensional (1D) graphene nanoribbons (GNRs), which are of promising technological applications [4]. The effects induced by both the electron–electron (e–e) correlation and the screening in these doped 1D systems have been subject of intense theoretical study [5], [6]. The dispersion relation of intrasubband collective excitations (plasmons) in these ribbons has been obtained. But, these results turned out to be dependent on the kind of GNR obtained in the laboratory. The band structure of the GNRs depends strongly on the kind of the edge of the ribbon (armchair or zigzag edges). Such a dependence leads to a challenging realization in a laboratory of ribbons of identical atomic structures [7].

In this paper we theoretically propose a simple model to describe two coupled quantum wires (ribbons) based on the graphene which has eventually been deposited over some sort of substrate. The graphene–substrate interaction is considered here through a phenomenological parameter which is taken into account in an ad-hoc manner [3]. We consider the 1D spatial confinement, which forms the coupled ribbons, as being produced by electrostatic gate potentials, so we avoid the edge effects on the sample. We then study the dynamically screening properties of a (massless) Fermion gas which is laid in the structure. We calculate the dielectric response function within the RPA, which turned out to be a good approximation describing e–e correlations in graphene even for small charge densities [8]. The roots of this function provide us with the collective excitations in the system. Our results show that the graphene–substrate interaction induced effects play an important role in obtaining the bare Coulomb potential and the dielectric function of the system. Such a role manifests itself in the dispersion relation of the optical and acoustical plasmon modes. We found that the graphene–substrate interaction might eventually increase the damping effects on these modes.

The paper is organized as follows. In Section 2 we present the theoretical formulation to the problem. In Section 3 we show our numerical results and in Section 4 we conclude our work.

Section snippets

Theoretical formalism

In order to study collective excitations in our double quantum wire system based on graphene, we should find first the spectrum of the Hamiltonian H = H2D + U(x), whereH2D=γσ.k+βσzis the 2D Dirac-like Hamiltonian describing the low-energy band structure for a single valley (K lattice point) in graphene. Here, the wavevector k=(kx,kyk) and the 2D Pauli vector σ=(σx,σy). The parameter β models a more general diatomic system in which the graphene lattice sites A and B might have a different

Numerical results

We show in Fig. 2 the Coulomb matrix elements VA and VC, as a function of both the wavevectors q and k. The colors indicate the intensity of these elements in units of e2/ε0. The upper (downer) panels show the results for β = 0 (β = 30 meV). Here, we set k = 0.05 nm 1. These elements diverge as |ln(q)| in the long wavelength limit q  0 for a fixed wavevector k. This is a similar asymptotic behavior as in the usual 1D Coulomb interaction that depends on the transferred wavevector q only [17]. Notice

Conclusions

In conclusion, we have theoretically studied the acoustical and optical plasmon modes in coupled graphene quantum wires in the extremely weak tunneling regime. In particular, attention was devoted to the effects induced by the interaction between the graphene sheet and the substrate. This interaction has been considered through an ad-hoc parameter β in the 2D Dirac-like Hamiltonian modelling a more general diatomic system in which the graphene lattice sites A and B might have different number

Acknowledgement

The authors thank the FAPESP (Brazil) for financial support.

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Presented at the Diamond 2010, 21st European Conference on Diamond, Diamond- Like Materials, Carbon Nanotubes, and Nitrides, Budapest.

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