# Nonlinear Refractive Index in Rectangular Graphene Quantum Dots
## 3 Results and Discussion  
Following the analytical expressions derived above, we performed calculations to predict the nonlinear refractive index for RGQDs associated with intra conduction band transitions near the K point. In our calculations the following parameters were adopted: \(D\) = 3 \(\times\) 10\({}^{24}\) m\({}^{-3}\), and \(\Gamma_{ij}\) = 10 meV [15]. Firstly, we displayed the energy spectra of electrons and holes as a function of \(k_{x}\) for RGQDs with different sizes in order to discuss how the size and edge influenced the energy levels. Figure 2a shows the energy levels of RGQDs for \(M\) = 17 and \(M\) = 20 with fixed \(N\) = 20. It could be seen that when we increased \(M\), that is, the length of the zigzag dimension, the corresponding allowed value of \(k_{x}\) changed and tended to be closer. Additionally, the energy levels moved to the  
Figure 1: **(a)** Illustration of a monolayer rectangular graphene quantum dot (RGQD). The black and red circles are two types of carbon atoms, A and B. **a** and **b** are unit vectors. **(b)** The reciprocal lattice. lower energy direction. However, the energy level intervals increased slightly, in the magnitude of \(10^{-2}\) eV. For \(N=17\) and \(N=20\) with fixed \(M=20\) in Figure 2b, \(k_{x}\) did not change. The energy levels also moved to the lower energy direction since \(k_{y}\) decreased, except for the lowest energy level in conduction band which remained the same, since \(k_{y}=0\) there. So, the density of states increased. Conversely, where \(M=17\) and \(M=20\) with fixed \(N=20\), the energy level intervals decreased in the magnitude of \(10^{-1}\) eV. The above two situations were both \(M=3M_{0}\pm 1\) (\(M_{0}\) is an integer), in which the lowest energy was non-zero and the quantum dot was a semiconductor. In the case of \(M=3M_{0}\), we plotted the energy spectra in Figure 3. It was very different from the case of \(M=3M_{0}\pm 1\). Firstly, the lowest energy of electron was zero, so it was metallic; secondly, the (\(1\), \(n\)) states, that is, the states where \(k_{x}=0\), did not change once \(N\) was fixed even if \(M\) varied. For these states, \(k_{y}=0\) or \((l-1/2)\pi/L_{AC}\) with \(l\) an integer, the neighboring energy difference between (\(1\), \(n\)) states kept the exact values: the energy difference between the lowest two states was \(\hbar v_{F}\pi/(2L_{\rm AC})\) while \(\hbar v_{F}\pi/L_{AC}\) for other neighboring states, and so, it provided the possibility for transition resonance. After getting the energy states of RGQDs, we calculated the nonlinear refractive index for RGQDs with different sizes in order to explore the influence of the size and edge. Figure 4a,b shows the size-dependent nonlinear refractive index spectra of RGQDs with non-\(3M_{0}\) sizes as a function of incident photon energy, calculated by the theoretical model presented in the previous section. We could  
Figure 3: Energy spectra of \(3M_{0}\) sized RGQDs as a function of \(k_{x}\). Figure 2: Energy spectra of \(M=3M_{0}\pm 1\) sized RGQDs as a function of \(k_{x}\). (**a**) RGQD with the sizes \(M=17\), \(20\), and fixed \(N=20\); (**b**) RGQD with the sizes \(N=17\), \(20\), and fixed \(M=20\). see that the peak value of nonlinear refractive index \(n_{2}\) for these sized RGQDs was in the magnitude of \(10^{-11}\) to \(10^{-12}\) m\({}^{2}\)/W. If we varied the sizes, either armchair-edge dimension or zigzag-edge dimension, the peak value of \(n_{2}\) increased. However, the amplitude of the increase contributed from the same three units was not the same. It was observed that the increase amplitude of \(n_{2}\) was much larger when \(N\) increased every three units than that of \(M\). This could be explained by the fact that the transition matrix element was proportional to \(L_{ZZ}\) while the square of \(L_{AC}\), which could be concluded from Equations (12)-(15). Additionally, the increase of the density of states for RGQDs with \(M=20\) and \(N=14\), 17, 20 that we mentioned previously was another important reason.