Discontinuous Galerkin Time-Domain Modeling of Graphene Nanoribbon Incorporating the Spatial Dispersion Effects

Abstract

It is well known that graphene demonstrates spatial dispersion properties, i.e., its conductivity is nonlocal and a function of spectral wavenumber (momentum operator) $\mathbf {q}$ . In this paper, to account for effects of spatial dispersion on transmission of high-speed signals along graphene nanoribbon (GNR) interconnects, a discontinuous Galerkin time-domain (DGTD) algorithm is proposed. The atomically thick GNR is modeled using a nonlocal transparent surface impedance boundary condition (SIBC) incorporated into the DGTD scheme. Since the conductivity is a complicated function of $\mathbf {q}$ (and one cannot find an analytical Fourier transform pair between $\mathbf {q}$ and spatial differential operators), an exact time-domain SIBC model cannot be derived. To overcome this problem, the conductivity is approximated by its Taylor series in spectral domain under low- $\mathbf {q}$ assumption. This approach permits expressing the time-domain SIBC in the form of a second-order partial differential equation (PDE) in current density and electric field intensity. To permit easy incorporation of this PDE with the DGTD algorithm, three auxiliary variables, which degenerate the second-order (temporal and spatial) differential operators to first-order ones, are introduced. Regarding to the temporal dispersion effects, the auxiliary differential equation method is utilized to eliminate the expensive temporal convolutions. To demonstrate the applicability of the proposed scheme, numerical results, which involve characterization of spatial dispersion effects on the transfer-impedance matrix of GNR interconnects, are presented.