726 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 57, NO. 4, AUGUST 2015 Modal Propagation and Crosstalk Analysis in Coupled Graphene Nanoribbons Rodolfo Araneo, Senior Member, IEEE, Paolo Burghignoli, Senior Member, IEEE, Giampiero Lovat, Member, IEEE, and George W. Hanson, Fellow, IEEE Abstract—A full-wave analysis of the fundamental quasi-TEM modes supported by multiple graphene nanoribbons above a ground plane is presented, aimed at characterizing crosstalk in graphene multiconductor lines. A method-of-moments discretiza- tion of the relevant electric-field integral equation is performed. As- suming first a local scalar conductivity, an efficient spatial-domain approach with subsectional basis functions is assuming first a local scalar conductivity, a spatial-domain approach with subsectional basis functions is developed. This allows for the efficient treatment of nanoribbons with wide transverse separations, and can be ex- panded to include in the simulation model spatial nonuniformity of the graphene conductivity. This spatial-domain formulation is then extended to treat the case of weakly nonlocal conductivity, via an original integro-differential approach derived by approximat- ing a recent full spectral graphene conductivity model in the limit of low wavenumbers. Numerical results are provided for propa- gation constants and characteristic impedances of two identical coupled graphene nanoribbons; on this basis, a crosstalk analysis is performed by means of the modal decomposition method. Index Terms—Crosstalk, graphene, multiconductor transmis- sion lines, nanoribbon, spatial dispersion. I. INTRODUCTION S TRIP transmission lines with submicrometric transverse dimensions based on graphene [graphene nanoribbons, (GNRs)] are emerging as candidates for the realization of future interconnects in carbon-based nanoelectronic circuits [1], and various approaches have been adopted so far for the assessment of their propagation features [2]–[8]. It has been demonstrated that the performance of GNR in- terconnects on the scale of nanometers are comparable to even the most optimistic projections for copper interconnects at the same scale; however, their development for future large-scale integrated circuits is still challenging due to several difficul- ties in fabrication processes, which are less practical or more expensive. In addition, when moving to the terahertz, regime, the absence of cheap and convenient sources for radiation adds Manuscript received November 2, 2014; revised January 15, 2015; accepted February 19, 2015. Date of publication March 19, 2015; date of current ver- sion August 13, 2015. This paper is an expanded version from the 2014 IEEE International Symposium on EMC, Raleigh, NC, USA, August 3–8, 2014. R. Araneo and G. Lovat are with the Department of Astronautical, Electrical, and Energetic Engineering, Sapienza University of Rome, 00185 Rome, Italy (e-mail: rodolfo.araneo@uniroma1.it; giampiero.lovat@uniroma1.it). P. Burghignoli is with the Department of Information Engineering, Electronics and Telecommunications, Sapienza University of Rome, 00185 Rome, Italy (e-mail: burghignoli@die.uniroma1.it). G. W. Hanson is with the Department of Electrical Engineering and Com- puter Science, University of Wisconsin-Milwaukee, Milwaukee, WI 53211 USA (e-mail: george@uwm.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2015.2406072 Fig. 1. Coupled GNRs. further difficulties (quantum-cascade laser and traveling-wave tube amplifier are good candidates, but not fully developed yet). The method-of-moments (MoM) is adopted here for the modal characterization of coupled GNRs, adopting two alter- native models for the graphene conductivity. The simplest ap- proach is based on a scalar frequency-dependent but local con- ductivity; whereas nonlocal effects in GNRs may be nonneg- ligible, especially in the microwave and millimeter-wave fre- quency ranges [9], [10], a local model may be still adequate in the Terahertz range [11] and allows for a straightforward implementation in a spatial-domain MoM code. This approach is much better suited than its spectral-domain counterpart for studying multiconductor transmission lines with widely sepa- rated ribbons. Furthermore, a spatial-domain approach is the only suitable tool for future modeling of spatially nonuniform graphene conductivity. Nonlocal effects in graphene can be modeled through a spec- tral domain dyadic conductivity, recently obtained in a semi- classical framework using the Bhatnagar–Gross–Krook (BGK) approximation of the Boltzmann transport equation [10]. Start- ing from such a representation, an approximate space-domain integro-differential equation is derived for the current density on a GNR, valid in the limit of low wavenumbers. The equa- tion is then discretized with the method of moments adopting subsectional basis functions and enforcing the correct behavior of the transverse current at the edges. The resulting formulation can be used for modeling weakly nonlocal effects in coupled GNRs. It is worth noting that, in principle, the spatial-domain formulation may be extended to a strongly nonlocal case if one could formulate the suitable differential equation. In this paper results are presented, based on such spatial- domain formulations, for dispersion and attenuation properties of the two fundamental (even and odd) quasi-TEM modes sup- ported by two coupled GNRs (see Fig. 1); the case of identical GNRs on an air substrate is considered for simplicity, although 0018-9375 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.