# Nonlinear Refractive Index in Rectangular Graphene Quantum Dots
## 2 Theory and Calculation
\right.. \tag{7}\]  
These boundary conditions were successfully used to work out the band structures of the graphene nanoribbons with different edges and the energy spectrum of RGQDs. They originate from the requirement that the electron probability amplitude at the hard wall around RGQDs must vanish [20]. Applying the above boundary conditions for a RGQD, we can solve Equation (3), the solution of which can be expressed as:  
\[\left\{ \begin{array}{c}\psi_{\rm A}=Ze^{ik_{x}x}\sin(k_{y}y)\\ \psi^{\prime}_{\rm A}=-Ze^{-ik_{x}x}\sin(k_{y}y)\end{array} \right., \tag{8}\]  
and  
\[\left\{ \begin{array}{c}\psi_{\rm B}=Ze^{ik_{x}x}\sin(\theta_{k}+k_{y}y)\\ \psi^{\prime}_{\rm B}=-Ze^{-ik_{x}x}\sin(\theta_{k}+k_{y}y)\end{array} \right., \tag{9}\]  
where \(\theta_{k}=\arctan(k_{y}/k_{x})\), and \(Z\) is the normalization constant determined by the normalization condition [15]:  
\[Z=\left[2L_{ZZ}L_{AC}+L_{ZZ}\sin(2\theta_{k})/k_{y}\right]^{-1/2}. \tag{10}\]  
It can be seen from Equation (5) that the energy has the same linear dispersion relation as graphene. The difference of energy dispersions between 2D graphene and RGQDs is that the wave vectors around the K point are discrete in RGQDs, since they should satisfy the quantized condition [15]:  
\[\left\{ \begin{array}{c}(K_{0}+k_{x})L_{ZZ}=m_{k}\pi\\ \theta_{k}+k_{y}L_{AC}=n_{k}\pi\end{array} \right., \tag{11}\]where \(m_{k}\) and \(n_{k}\) are integers. So, the energy levels are instead predicted to be discrete for the continuous energy band. We adopt the symbol (\(m\), \(n\)) to present every electronic state, where \(m\) and \(n\) are determined by the orders of \(k_{x}\) and \(k_{y}\), respectively. After the obtainment of energy states of RGQDs, we can derive the dipole transition matrix element, with the assumption that the incident light polarization is along the \(y\) direction:  
\[ \begin{array}{ll}\mu_{ij}&=\left\langle\psi_{i}\middle|e^{\frac{1}{r}} \middle|\psi_{j}\right\rangle\\ &=\int_{0}^{L_{ZZ}}\int_{0}^{L_{AC}}4 eZ_{i}Z_{j}\sin[(K_{0}+k_{xi})x]\sin[(K_ {0}+k_{xi})x]\sin(k_{yi}y)\sin(k_{yy}y)ydxdy\\ &+\int_{0}^{L_{ZZ}}\int_{0}^{L_{AC}}4 eZ_{i}Z_{j}\sin[(K_{0}+k_{xi})x]\sin[(K_ {0}+k_{xj})x]\sin(\theta_{ki}+k_{yi}y)\sin(\theta_{kj}+k_{yj}y)ydxdy\\ &=4 eZ_{i}Z_{j}A_{1}(B_{1}+B_{2})\end{array}  \tag{12}\]  
with  
\[A_{1}=1/2L_{ZZ}\delta_{m_{i},m_{j}}-1/2L_{ZZ}\delta_{m_{i},-m_{j}}, \tag{13}\]  
\[ \begin{array}{ll}B_{1}&=\frac{L_{AC}^{2}}{2(n_{i}\tau-\theta_{i}-n_{j}\pi+ \theta_{j})}[\sin(n_{i}\pi-\theta_{i}-n_{j}\pi+\theta_{j})+\frac{\cos(n_{i} \pi-\theta_{i}-n_{j}\pi+\theta_{j})-1}{n_{i}\pi-\theta_{i}-n_{j}\pi+\theta_{j }}]\\ &-\frac{L_{AC}^{2}}{2(n_{i}\pi-\theta_{i}+n_{j}\pi-\theta_{j})}[\sin(n_{i}\pi- \theta_{i}+n_{j}\pi-\theta_{j})+\frac{\cos(n_{i}\pi-\theta_{i}+n_{j}\pi- \theta_{j})-1}{n_{i}\pi-\theta_{i}+n_{j}\pi-\theta_{j}}]\end{array}  \tag{14}\]  
and  
\[ \begin{array}{ll}B_{2}&=\frac{L_{AC}^{2}}{2(n_{i}\pi-\theta_{i}-n_{j}\pi+ \theta_{j})^{2}}[\cos(n_{i}\pi-n_{j}\pi)-\cos(\theta_{i}-\theta_{j})]\\ &-\frac{L_{AC}^{2}}{2(n_{i}\pi-\theta_{i}+n_{j}\pi-\theta_{j})^{2}}[\cos(n_{i} \pi+n_{j}\pi)-\cos(\theta_{i}+\theta_{j})]\end{array}  \tag{15}\]  
The transitions can only occur when quantum numbers satisfy \(m_{i}=\pm m_{j}\).