## Appendix A Basic Concepts of Graphene
### Two-Dimensional Energy Band Structure of Single-Layer Graphene  
Graphene is constructed from carbon atoms bonded in the hexagonal 2D plane as in the honeycomb lattice (Figure 11).  
The basis vectors in real space, \(\overset{\rightarrow}{a_{1}}/\overset{\rightarrow}{a_{2}}\), and reciprocal vectors, \(\overset{\rightarrow}{b_{1}}/\overset{\rightarrow}{b_{2}}\), are shown in Figure 11 and calculated as:  
\[\overset{\rightarrow}{a_{1}}=\left(\frac{\sqrt{3}a}{2},\frac{a}{2}\right), \ \overset{\rightarrow}{a_{2}}=\left(\frac{\sqrt{3}a}{2},-\frac{a}{2}\right) \tag{14}\]  
\[\overset{\rightarrow}{b_{1}}=\left(\frac{2\pi}{\sqrt{3}a},\frac{2\pi}{a} \right),\ \overset{\rightarrow}{b_{2}}=\left(\frac{2\pi}{\sqrt{3}a},-\frac{2\pi}{a}\right) \tag{15}\]  
where \(a=|\overset{\rightarrow}{a_{1}}|=|\overset{\rightarrow}{a_{2}}|=\sqrt{3}\ a_{c-c}=0.246\) nm, \(a_{c-c}=0.142\) nm is carbon-carbon atom distance, and diamond contains A and B shows the unit cell.  
Because we have two atoms in the unit cell, the Hamiltonian matrix is a 2 \(\times\) 2 matrix [16,17]:  
\[H=\left[\begin{array}{cc}H_{AA}&H_{AB}\\ H_{BA}&H_{BB}\end{array}\right]\]  
The unit cell, shown in Figure A2, is A1 and B1, so B2, B3, A3, and A2 are the nearest atoms of its neighbor cells.  
Where the Hamiltonian matrix is computed as the equation represented in the previous section, and for this case is formalized as below.  
\[H_{AA}=H_{BB}=\in_{2p_{z}}\] (A14a) \[H_{AB}=t\bigg{(}e^{i\vec{k}\cdot\vec{k}_{1}}+e^{i\vec{k}\cdot\vec{k}_{2}}+e^{ i\vec{k}\cdot\vec{k}_{3}}\bigg{)}=tf(k)\] (A14b)  
where  
\[R_{1}=R_{B1}-R_{A1}=(\frac{a}{\sqrt{3}},0)\] (A15a) \[R_{2}=R_{B2}-R_{A1}=(-\frac{a}{2\sqrt{3}},\ \frac{a}{2})\] (A15b) \[R_{3}=R_{B3}-R_{A1}=(-\frac{a}{2\sqrt{3}},-\frac{a}{2})\] (A15c)  
Thus, \(H\) is yielded as:  
\[H=\left[\begin{array}{cc}\in_{2p_{z}}&tf(k)\\ tf(k)^{*}&\in_{2p_{z}}\end{array}\right],\ \ s=\left[\begin{array}{cc}1&sf(k)\\ sf(k)^{*}&1\end{array}\right]\] (A16)  
where:  
\[f(k)=e^{ik_{x}a/\sqrt{3}}+2e^{-ik_{x}a/2\sqrt{3}}cos\bigg{(}\frac{k_{y}}{2}a \bigg{)}\] (A17)  
Additionally, \(H\) eigenvalues are computed for extracting energy dispersion, as:  
\[E_{g2D}\big{(}k_{x},k_{y}\big{)}=\frac{\in_{2p_{z}}\pm tw\stackrel{{ \rightarrow}}{{(k)}}}{1\pm sw\stackrel{{\rightarrow}}{{(k)}}}\] (A18)  
where  
\[w(\stackrel{{\rightarrow}}{{k}})=\sqrt{\left|f\bigg{(}\stackrel{{ \rightarrow}}{{k}}\bigg{)}\right|^{2}}=\sqrt{1+4 cos\frac{\sqrt{3}k_{x}a}{2} cos\frac{k_{y}a}{2}+4 cos^{2}\frac{k_{y}a}{2}}\] (A19)  
The energy dispersion for \(\in_{2p_{z}}=0\), \(t=-3.033\) ev, and s = 0.129 is shown in Figure A3.  
By Slater-Koster [16] approximation s = 0, the dispersion equation becomes:  
\[E_{g2D}\left(k_{x},k_{y}\right)=\pm tw\left(\overset{\rightarrow}{k}\right)\] (A20)  
If other orbitals such as \(2s\), \(2p_{x}\), \(2p_{y}\), and \(2p_{z}\) were considered, the Hamiltonian matrix becomes \(8\times 8\), so eight eigenvalues are shown in Figure A4.  
Figure A4: Energy dispersion for \(2s\), \(2 px\), \(2 py\), and \(2 pz\). (**a**) Different orbits of graphene; (**b**) valence and conductance band; (**c**) energy dispersion of different orbits of graphene.  
Additionally, for the three nearest neighbors, the dispersion energy is shown in Figure A5.