# Nonlinear Refractive Index in Rectangular Graphene Quantum Dots
## 1 Introduction  
Graphene is comprised of a plane of carbon atoms arranged in honeycomb lattices, and has attracted intense attention due to its extraordinary physical and chemical properties [1; 2; 3]. The planar geometry of graphene is advantageous to the tailoring of various nanostructures, such as one-dimensional nanoribbons [4] and zero-dimensional quantum dots [5] with desired size, edge, and shape [6; 7; 8]. Due to the quantum size and edge effect, the tunable electronic and optical properties make the graphene-based nanostructures promising candidates for building blocks of future opto-electronic devices [9; 10]. To this end, the integrability of the graphene quantum dot combined with its large refraction nonlinearity makes it ideal for use in several applications in optical communication and signal-processing [11].  
Numerous experimental works have been conducted on the nonlinear refractive index in the graphene and related nanostructures. [PERSON] et al. showed that graphene possesses the giant nonlinear refractive index of \(n_{2}\simeq 10^{-7}\) cm\({}^{2}\) W\({}^{-1}\), almost nine orders of magnitude larger than bulk dielectrics, using the Z-scan technique on loosely stacked few-layer graphene [12]. The spatial self-phase modulation was observed directly by [PERSON] et al. when a focused He-Ne laser beam at 633 nm went through liquid-phase-exfoliated graphene dispersions [13]. They estimated the relative change of effective nonlinear refractive index by tuning the incident intensity or the temperature of the dispersions. By means of the ultrafast optical Kerr effect method coupled to optical heterodyne detection (OHD-OKE), [PERSON]'s group characterized the third-order nonlinear responseof graphene and they estimated a negative nonlinear refractive index for monolayer graphene (i.e., \(n_{2}\simeq-1.1\times 10^{-13}\) m\({}^{2}\)/W) [14].  
To our knowledge, there have so far not been any theoretical calculations on the nonlinear refractive index in graphene and related nanostructures. In our previous work, we explored the two-photon absorption properties of RGQDs from the second-order perturbation theory with respect to the electron-photon interaction [15]. In this work, we report a simple theoretical calculation of the nonlinear refractive index for monolayer graphene with confining boundaries in the orthogonal directions of zigzag- and armchair-edge, which is an ideal object for us to investigate which edge plays a more important role in nonlinear refraction. Our derivation originates from the relationship between nonlinear refractive index \(n_{2}\), and the real part of the third-order nonlinear optical susceptibility \(\chi^{(3)}\), which can be derived employing the electronic states obtained from the Dirac equation around the K point in RGQDs under hard-wall boundary conditions. The results revealed that the nonlinear refractive index \(n_{2}\) for these sized RGQDs was in the magnitude of \(10^{-14}\) m\({}^{2}\)/W in the visible region, which was nearly five orders larger than conventional semiconductor quantum dots. In the infrared band (\(\sim\)2500 nm), the nonlinear refractive index \(n_{2}\) was up to \(10^{-11}\) m\({}^{2}\)/W. We also found that the refractive index and the position of the refractive peak could be tuned by the sizes of both edges in RGQDs, especially the armchair-edge dimension. If we varied the size of the two edges, for non-3\(M_{0}\) sized RGQDs, the increase of the armchair-edge dimension brought a larger increment of refractive index than that of the zigzag-edge. There was a blue shift for the refractive peak when the size of zigzag-edge dimension increased while there was a distinct red shift when armchair-edge dimension increased. For RGQDs with \(M=3M_{0}\), the peak value of nonlinear refractive index for \(M=3M_{0}\) was nearly eight times of the case of \(M=3M_{0}\pm 1\), because the resonance enhancement of electron transition occurred within the transitions between (1, \(n\)) states. The refractive peak did not move.