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Nonlinear Plasmonics Near the Dirac Point in Negative-Zero-Positive Index Metamaterials–Optical Simulations of Electron in Graphene

In 2004, graphene, a single layer of carbon atoms arranged in a hexagonal lattice, has been experimentally realized by A. K. Geim and K. S. Novoselov (Novoselov et al., 2004). In graphene, the conduction and valence bands touch each other at Dirac point (DP) with a double-cone structure (Novoselov et al., 2004). Near DP, the dispersion of electron is linear with two branches (Novoselov et al., 2004). The valence electron dynamics in such a truly two-dimensional (2D) material is governed by a massless Dirac equation. So graphene exhibits many unique electronic properties (Beenakker, 2008; Castro Neto et al., 2009), including half-integer and unconventional quantum Hall effect (Zhang et al., 2005), observation of minimum conductivity (Novoselov et al., 2005), and Klein tunneling (Katsnelson et al., 2006). The optical-like behaviors of electron waves in graphene have also drawn considerable attention recently, such as focusing (Cheianov et al., 2007), collimation (Park et al., 2008a), subwavelength optics (Darancet et al., 2009), Bragg reflection (Ghosh et al., 2009), and Goos-Hanchen effect (Beenakker et al., 2009; Zhao et al., 2010). In this regard, one of the recent work is to investigate the guided modes in monolayer graphene waveguide, by analogy of optical waveguides (Zhang et al., 2009). The exotic properties of the graphene waveguide are found in two different cases of classical motion and Klein tunneing (Zhang et al., 2009). Similar behaviors also happened to the transmission of Dirac-like electron in 2D monolayer graphene barrier at nonzero angle of incidence (Chen and Tao, 2009). The modulation of the transmission gap by the incidence angle, the height, and width of potential barrier may lead to potential applications in graphene-based electronic devices (Chen and Tao, 2009). The electronics waves in graphene can also be treated as the guided waves in an optical fibre, i.e., graphene based electronic fibre (Wu, 2011). The graphene analog of the optical device, the fibre optic, has also been demonstrated both experimentally and numerically in a p− n junction (Williams et al., 2011). On the other hand, the DP with double-cone structure in photonic crystals (PCs) for the Bloch states is found from the similarity of the photonic bands of the 2D PCs with the electronic bands of solids. Several novel optical transport properties near the DP have been investigated, such as conical diffraction (Peleg et al., 2007), "pseudodiffusive" scaling (Sepkhanov et al., 2007), photon’s Zitterbewegung (Zhang, 2008), and perfect Klein tunneling (Bahat-Treidel et al., 2010a). Up to now, the dynamics of Dirac-like excitations in 2D PCs (honeycomb lattices) have been well studied when the propagation equation is linear. 14