## Appendix A Basic Concepts of Graphene  
_Appendix A.1. Basic Concepts of Graphene Band Structure_  
In the twentieth century, scientists confirmed [PERSON]'s Equation (A1) as a formal quantitative basis to calculate the energy levels for any confining potential as:  
\[i\hbar\frac{\partial\Psi(r,t)}{\partial t}=(-\frac{h^{2}}{2m}\
abla^{2}+U(r)) \Psi(r,t)\] (A1)  
where \(h\), \(m\), \(U(r)\), and \(\Psi\) show the Planck constant, the mass of an electron, confined potential, and wave function of the electron, respectively. After solving this equation, \(\Psi\) is used to extract other electrical parameters, such as \(\Psi^{*}\Psi\mathrm{d}\mathrm{v}\) gives the presence probability of electron in a volume of \(\mathrm{d}\mathrm{v}\) and adds it up for all the electrons is used to obtain average electron density \(n(x,t)\). Additionally, the current density probability is obtained as:  
\[J=i\hbar/2m(\Psi\frac{\partial\Psi^{*}}{\partial x}-\Psi^{*}\frac{\partial \Psi}{\partial x})\] (A2)  
[PERSON]'s equation can be solved when the self-consistent potential \(U(r)\) is yielded from Poisson's Equation (A3). \[-\
abla.(\varepsilon_{r}\
abla U(r))=q^{2}n/\varepsilon_{0}\] (A3)  
Analytical modeling of a device in equilibrium generally requires an iterative solution of Equations (A1) and (A3). These equations for simple material structures and boundaries can be solved analytically, but most practical problems require a numerical solution. In numerical models such as finite-difference, the Partial Differential Equation (PDE) is converted to matrix equation as:  
\[i\hbar\frac{d}{dt}\{\Psi_{1}(t)\;\Psi_{2}(t)\ldots\Psi_{n}(t)\}=[H]\{\Psi_{1}( t)\Psi_{2}(t)\ldots\Psi_{n}(t)\}\] (A4)  
where \(\Psi_{i}(t)\) is the wave function value around lattice points i at time t.  
Additionally, the second derivative must be turned into a difference equation:  
\[\frac{\partial^{2}\Psi}{\partial r^{2}}=\frac{1}{a^{2}}(\Psi_{n+1}(t)+\Psi_{n- 1}(t)-2\Psi_{n}(t))\] (A5)  
Therefore:  
\[H=\left[ \begin{array}{cccccc}u_{0}+2t_{0}&-t_{0}&0&0&\cdots&0\\ -t_{0}&u_{0}+2t_{0}&-t_{0}&0&\cdots&0\\ 0&-t_{0}&u_{0}+2t_{0}&-t_{0}&\cdots&0\\ &\cdot&\cdot&\cdot&\cdot\\ 0&0&0&-t_{0}&u_{0}+2t_{0}\end{array} \right]\] (A6)  
where \(t_{0}=\frac{h^{2}}{2 ma^{2}}\). The solution of Equation (4) becomes any superposition \(e^{-i\frac{E_{r}}{\hbar}}\{\varphi_{\gamma}\}\):  
\[\{\Psi_{1}(t)\Psi_{2}(t)\ldots\Psi_{n}(t)\}=\sum_{\gamma=1}^{n}c_{\gamma}e^{- i\frac{E_{r}}{\hbar}}\{\varphi_{\gamma}\}\] (A7)So the eigenvalues \(E_{\gamma}\), energy dispersion, and eigenvectors \(\{\varphi_{\alpha}(r)\}\) of \(H\) are extracted from:  
\[[H]\{\varphi_{\gamma}\}=E_{\gamma}\{\varphi_{\gamma}\} \tag{10}\]  
That is known as the time-independent Schrodinger equation. This approach can be used for calculating the band structure of any periodic solid, such as graphene, with an arbitrary number of atoms per unit cell. In the general procedure, the reciprocal lattice in the k-space is constructed. In this case, any point on the direct lattice can be shown as:  
\[\overset{\rightarrow}{R}=m\overset{\rightarrow}{a_{1}}+n\overset{ \rightarrow}{a_{2}}+p\overset{\rightarrow}{a_{3}} \tag{11}\]  
where \(m\), \(n\), and \(p\) are integers and \(\overset{\rightarrow}{a_{1}}\), \(\overset{\rightarrow}{a_{2}}\), and \(\overset{\rightarrow}{a_{3}}\) are lattice basis vectors. Then, the points on the reciprocal lattice can be written as Equation (10), where \(M\), \(N\), \(P\) are integers.