# Nonlinear Refractive Index in Rectangular Graphene Quantum Dots
## 2 Theory and Calculation  
In a system showing a negligible linear absorption (\(\alpha_{0}\approx 0\)), the nonlinear refractive index \(n_{2}\) is proportional to the real part of the third-order nonlinear optical susceptibility \(\chi^{(3)}\) through the following expression in SI units [16]:  
\[\chi^{(3)}_{R}=(4/3)n_{0}^{2}\varepsilon_{0}c\pi_{2}, \tag{1}\]  
where \(n_{0}\), \(\varepsilon_{0}\), and \(c\) denote, respectively, the linear refractive index of the material, the electric permittivity of free space, and the speed of light in vacuum. For the third harmonic generation effect, the resonant third-order susceptibility \(\chi^{(3)}(3\omega)\) is given by [17; 18]:  
\[\chi^{(3)}(3\omega)= \sum_{\varepsilon_{0}}\mu_{03}\mu_{32}\mu_{21}\mu_{10} \tag{2}\] \[\times\frac{1}{(\hbar\omega_{30}-3\hbar\omega-i\Gamma_{30})( \hbar\omega_{20}-2\hbar\omega-i\Gamma_{20})(\hbar\omega_{10}-\hbar\omega-i \Gamma_{10})}\]  
where \(\mu_{ij}\) the dipole transition matrix element transited from \(\psi_{j}\) state to \(\psi_{i}\) state and \(\mu_{ij}=\left\langle\psi_{i}\middle|e\overset{\rightarrow}{r}\middle|\psi_{j}\right\rangle\), the transition frequency \(\omega_{ij}=(E_{i}-E_{j})/\hbar_{i}\); D is the density of RGQDs, and \(\Gamma_{ij}\) is the relaxation energy. From Equations (1) and (2), it is found that the electronic energy states are the key for the calculation of nonlinear refractive index. We consider an isolated 2D confined graphene nanosheet with two orthogonal boundaries, which is shown in Figure 1a. The zigzag- and armchair-edge are referred to as the \(x\)- and \(y\)-axis, respectively, and thus the length of the edges can be calculated using the units \(M\) and \(N\), respectively, by \(L_{ZZ}=Ma\), \(L_{AC}=(N+1/3)\sqrt{3}a/2\), where \(a=2.46\) A is the lattice constant. Within the effective mass approximation, the Dirac equation of the electrons around the K = (\(4\pi/3a\), 0) and K \({}^{\prime}=(-4\pi/3a\), 0) points in momentum space can be expressed as [19; 20]:\[H\Psi=\hbar v_{F}\left( \begin{array}{cccc}0&k_{x}-ik_{y}&0&0\\ k_{x}+ik_{y}&0&0&0\\ 0&0&0&k_{x}+ik_{y}\\ 0&0&k_{x}-ik_{y}&0\end{array} \right)\left( \begin{array}{c}\psi_{\rm A}\\ \psi_{\rm B}\\ \psi^{\prime}_{\rm A}\\ \psi^{\prime}_{\rm B}\end{array} \right)=E\left( \begin{array}{c}\psi_{\rm A }\\ \psi_{\rm B}\\ \psi^{\prime}_{\rm A}\\ \psi^{\prime}_{\rm B}\end{array} \right), \tag{3}\]  
where \(v_{F}\approx 1\times 10^{6}\) m/s is the Fermi velocity at the Dirac points [15; 21], and \(k_{x(y)}=-i\hbar\partial_{x(y)}\) is the small wavevector perturbations from K and K' points. The electron band structure of the RGQD is studied by the effective mass model, which is reasonable to adopt since we are discussing the low-energy carrier states near the K and K' Dirac points [22; 23]. The total wave function \(\Psi({\bf r})\) of the sublattice A and B mixed with wavefunctions of valley K and K' is written as [23]:  
\[\left\{ \begin{array}{c}\Psi_{\rm A}({\bf r})=e^{i{\bf K}\cdot{\bf r}}\psi_{ \rm A}({\bf r})+e^{i{\bf K}\cdot{\bf r}}\psi^{\prime}_{\rm A}({\bf r})\\ \Psi_{\rm B}({\bf r})=e^{i{\bf K}\cdot{\bf r}}\psi_{\rm B}({\bf r})+e^{i{\bf K }\cdot{\bf r}}\psi^{\prime}_{\rm B}({\bf r})\end{array} \right.. \tag{4}\]  
The corresponding eigenenergy is given as:  
\[E=\pm\hbar v_{F}|k|=\pm\hbar v_{F}\sqrt{k_{x}^{2}+k_{y}^{2}}. \tag{5}\]  
As for the RGQD structure mentioned above, the electron obeys the same Dirac equation, but different boundary conditions for zigzag-edge and armchair-edge [23; 24]. For the zigzag edge along the \(x\) axis, the boundary conditions are given as:  
\[\left\{ \begin{array}{c}\Psi_{\rm A}(y=0)=0\\ \Psi_{\rm B}(y=L_{AC})=0\end{array} \right., \tag{6}\]  
while for armchair-edge along the \(y\) axis, the boundary conditions are given as:  
\[\left\{ \begin{array}{c}\Psi_{\rm A}(x=0)=\Psi_{\rm A}(x=L_{ZZ})=0\\ \Psi_{\rm B}(x=0)=\Psi_{\rm B}(x=L_{ZZ})=0\end{array}