Dispersion Analysis of Graphene Nanostrip Lines R. Araneo, G. Lovat D.I.A.E.E. - Electrical Engineering Area “Sapienza” University of Rome Via Eudossiana 18, 00184, Rome – Italy rodolfo.araneo,giampiero.lovat@uniroma1.it P. Burghignoli D.I.E.T “Sapienza” University of Rome Via Eudossiana 18, 00184, Rome – Italy paolo.burghignoli@uniroma1.it Abstract—Propagation features of the quasi-TEM mode sup- ported by a graphene nano-ribbon placed on top of a grounded dielectric substrate are studied numerically with the Method of Moments in the spectral domain. A dyadic conductivity is adopted to model the graphene ribbon at microwave frequencies, taking into account the possible presence of static biasing fields and/or spatial-dispersion effects. Dispersion and attenuation properties are reported for typical values of the relevant physical parameters. I. I NTRODUCTION Graphene monolayers exhibit exceptional mechanical, ther- mal, and electrical properties and have been the subject of intense research also in the electromagnetic community (see, e.g., [1]- [4]). In particular, the possible use of graphene nanoribbons (GNRs) in integrated circuits has recently been demonstrated [5] and the surface-plasmon modes supported by a GNR placed at the interface between two dielectric half spaces have been studied in [6]. In this contribution, dispersion properties of the quasi-TEM mode supported by a GNR placed on top of a grounded dielectric substrate (i.e., a graphene microstrip line, or bet- ter a nanostrip line) are studied with a full-wave method- of-moments (MoM) approach, adopting for the GNR an anisotropic surface-impedance boundary condition that takes into account the possible presence of static bias fields and, possibly, spatial-dispersion effects. II. DESCRIPTION OF THE PROBLEM AND ITS SOLUTION The electromagnetic problem under analysis is sketched in Fig. 1. It consists of a graphene sheet (i.e., a GNR) of width W deposited on a dielectric substrate (typically SiO 2 ) of thickness h and relative dielectric constant ε r , placed over a perfectly conductive (PEC) ground plane. The fundamental quasi-TEM mode supported by this structure is investigated here by assuming an ideal behavior of both the dielectric and the ground plane (i.e., without any loss) and neglecting edge effects of the graphene strip, in order to assess only the role of graphene conductivity on the dispersion and attenuation properties. A. Graphene Conductivity The graphene strip is modeled as an infinitesimally-thin conductive sheet of finite width W with a surface conductivity tensor σ =  σ D σ O σ O σ D  . (1) The anisotropic characteristic may arise from two different mechanisms, i.e., presence of static bias fields or spatial- dispersion effects (usually negligible below the THz regime) [1]. The dyadic elements depend on different parameters, e.g., frequency f = ω/(2π), temperature T , and a phe- nomenological scattering rate Γ [7]; moreover, in the presence of bias fields, they also depend on the applied electrostatic E bias = u z E bias and/or magnetostatic B bias = u z B bias bias. The mathematical expressions for the dyadic elements have been derived in [7] starting from the Kubo formula and can be found summarized in [1] or [4]. It should be noted that, in the absence of a bias field and neglecting spatial- dispersion effects, the graphene is characterized by an isotropic conductivity σ = σ D . B. Dispersion analysis Starting from the boundary condition E t (x, y, z = h)= σ -1 · J S (x, y) on the microstrip line surface, the Electric Field Integral Equation (EFIE) can may be expressed in the form +∞  -∞ + W 2  - W 2 G EJ (x - x ′ ,y - y ′ ,z = z ′ = h)·J S (x ′ ,y ′ )dxdy = = σ -1 · J S (x, y) , (2) where G EJ (·) is the EJ-type dyadic Green function for planar layered media. By introducing the spectral domain Green’s x y h ε r W z dielectric substrate graphene nanoribbon ground plane Fig. 1. Graphene nanoribbon (GNR) on a grounded dielectric substrate. 978-1-4673-0462-7/12/$31.00 ©2012 IEEE