# Nonlinear Refractive Index in Rectangular Graphene Quantum Dots
## 4 Conclusions  
In this paper, we derived the nonlinear refractive index \(n_{2}\) which is proportional to the real part of the third-order nonlinear optical susceptibility \(\chi^{(3)}\), starting from solving the electronic Dirac equation in momentum space analytically under hard-wall boundary conditions for different edges. 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 and \(10^{-11}\) m\({}^{2}\)/W in the infrared region. For non-\(3M_{0}\) sized RGQDs, with the increase of the size--in either armchair-edge dimension or zigzag-edge dimension--the peak value of \(n_{2}\) increased. However, the increase of armchair-edge dimension contributed much more. There was also a blue shift for the refractive peak when \(M\) increased, while there was a distinct red shift when \(N\) increased. For an RGQD with \(M\) = \(3M_{0}\), the nonlinear refractive index was enhanced several times compared to that of \(M\) = \(3M_{0}\)\(\pm\) 1, and there was no peak shift. We gave explanations to these phenomena, including different peak shift directions, peak value increase, and so on, by the quantized energy level interval, as well as the size-dependence of transition matrix element in RGQDs. These theoretical analyses are of great importance to applications based on optical communication, as well as academic interest.  
All the authors have contributed equally to the conception and idea of the paper, implementing and analyzing the calculation methods, evaluating and discussing the calculation results, and writing and revising this manuscript.  
This work was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11304275, 11764047) and Applied Basic Research Foundation of Yunnan Province (Grant No. 2017 FB009). It was also supported in part by Candidate Talents Training Fund of Yunnan Province (Grant No. 2014 HB010).  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Materials**} & \multirow{2}{*}{**Size**} & **Wavelength** & \(n_{2}\) **(\(\times 10^{-12}\) m\({}^{2}\)/W)** \\ \cline{3-4}  & & **(nm)** & **Experiment** & **Theory** \\ \hline CdS QDs & 6.4 nm & 790 & \(10^{-7}\)[25] \\ CdS-Ag QDs & 10.1 nm & 790 & \(-2.3\times 10^{-5}\)[25] \\ CdTe QDs & 20 nm & 1060 & \(-6\times 10^{-5}\)[26] \\ Graphene & 1–6 layers & 1550 & 10[12] \\  & monolayer & 1600 & \(-1.1\times 10^{-1}\)[14] \\  & & 1050 & 1.7[11] \\  & & 1064 & \(-13.7\)[27] \\  & & 532 & \(-2.34\)[27] \\ MoS\({}_{2}\) & & 1064 & \(-(0.207\pm 0.021)\)[27] \\  & & 532 & \(-(2.5\pm 1.2)\)[27] \\ WS\({}_{2}\) & monolayer & 800 & 0.8[28] \\  & 0.75 nm thick & 1040 & \((1.28\pm 0.03)\times 10^{-2}\)[29] \\  & 18.8 nm thick & 1040 & \((-8.55\pm 0.63)\times 10^{-4}\)[29] \\ RGQDs & 6.4 nm \(\times\) 6.4 nm & 790 & \(-2.13\times 10^{-2}\) \\  & 5.2 nm \(\times\) 4.3 nm & 699 & \(-1.75\times 10^{-2}\) \\  & 5.2 nm \(\times\) 4.3 nm & 1064 & \(-6.33\times 10^{-2}\) \\  & 3.4 nm \(\times\) 4.3 nm & 2638 & 9.5 \\  & 4.2 nm \(\times\) 4.3 nm & 2594 & 11.4 \\  & 4.9 nm \(\times\) 4.3 nm & 2583 & 11.6 \\  & 4.9 nm \(\times\) 3.7 nm & 2175 & 5.24 \\  & 3.7 nm \(\times\) 4.3 nm & 2583 & 65.2 \\  & 4.4 nm \(\times\) 4.3 nm & 2583 & 75.9 \\  & 5.2 nm \(\times\) 4.3 nm & 2583 & 90.8 \\ \hline \end{tabular}
\end{table}
Table 1: Calculated \(n_{2}\) in RGQDs and experimental data in other related materials for comparison.