## 2 Band Structure
### Graphene Layer  
Figure 1a shows the Bravais lattice of the graphene. The unit cell, \(S_{\mathrm{g}}\), is spanned by the two vectors, \(\mathbf{a}_{1}\) and \(\mathbf{a}_{2}\), and contains two carbon atoms. The basis vectors (\(\mathbf{a}_{1}\),\(\mathbf{a}_{2}\)) have the same length, \(\left|\mathbf{a}_{1}\right|=\left|\mathbf{a}_{2}\right|=a=\sqrt{3}b\), and form an angle of \(\pi/3\), where \(b=1.42\) A is the interatomic distance. The components of the vectors, \(\mathbf{a}_{1}\) and \(\mathbf{a}_{2}\), with respect to the rectangular coordinate system, (0,\(x\),\(y\)) are, respectively, \(\left(\sqrt{3}a_{{}_{0}}\,/2,a_{{}_{0}}\,/2\right)\) and \(\left(\sqrt{3}a_{{}_{0}}\,/2,-a_{{}_{0}}\,/2\right)\). The area of the unit cell, \(S_{\mathrm{g}}\), is \(A_{g}=\sqrt{3}a_{0}^{2}\,/\,2\). In the reciprocal \(k\)-space depicted in Figure 1b, the graphene is characterized by the unit cell, \(\sum_{g}\), spanned by the two vectors, \(\mathbf{b}_{1}\) and \(\mathbf{b}_{2}\), which have the same length \(\left|\mathbf{b}_{1}\right|=\left|\mathbf{b}_{2}\right|=b_{0}=4\pi\sqrt{3}a_{0}\) and form an angle of \(2\pi/3\). The components of the vectors, \(\mathbf{b}_{1}\) and \(\mathbf{b}_{2}\), with respect to the rectangular coordinate system (Q,\(K_{x}\),\(k_{y}\)) are, respectively, \(\left(2\pi/\sqrt{3}a_{{}_{0}},2\pi\,/\,a_{{}_{0}}\right)\) and \(\left(2\pi/\sqrt{3}a_{{}_{0}},-2\pi\,/\,a_{{}_{0}}\right)\). The area of the unit cell, \(\sum_{g}\), is \(B_{g}=8\pi^{2}\sqrt{3}\cdot a_{0}^{2}\). The basis vectors of the direct space (\(\mathbf{a}_{1}\),\(\mathbf{a}_{2}\)) and the basis vectors of the reciprocal space (\(\mathbf{b}_{1}\),\(\mathbf{b}_{2}\)) are related by \(\mathbf{a}_{i}\cdot\mathbf{b}_{j}=2\pi\delta_{ij}\) with \(i\,j=1\),\(2\) and the areas, \(A_{\mathrm{g}}\) and \(B_{\mathrm{g}}\), are related by \(A_{\mathrm{g}}B_{\mathrm{g}}=\left(2\pi\right)^{2}\). The graphene possesses four valence electrons for each carbon atom. Three of these (the so-called \(\sigma\)-electrons) form tight bonds with the neighboring atoms in the plane and do not play a part in the conduction phenomenon. The fourth electron (the so-called \(\pi\)-electron), instead, may move freely between the positive ions of the lattice. In the nearest-neighbors tight-binding approximation, the \(\pi\)-electrons energy dispersion relation is  
\[E^{(\pm)}(\mathbf{k})\pm\gamma\left[1+4\cos\left(\frac{\sqrt{3}k_{x}a_{0}}{2} \right)\cos\left(\frac{k_{y}a_{0}}{2}\right)+4\cos^{2}\left(\frac{k_{y}a_{0}}{ 2}\right)\right]^{1/2} \tag{1}\]  
Figure 1: The structure of graphene. (**a**) Bravais lattice; (**b**) reciprocal lattice. where \(E^{(\pm)}\) denotes the energy, the \(+\) sign denotes the conduction band, the \(-\) sign denotes the valence band and \(\gamma=2.7\) eV is the carbon-carbon interaction energy. The electronic band structure is depicted in Figure 2: the valence and conduction bands touch each other at the six vertices of each unit cell, the so-called Fermi points. In the neighborhood of each Fermi point, the energy dispersion relation may be approximated as:  
\[E^{(\pm)}\approx\pm\hbar v_{F}\left|\mathbf{k}-\mathbf{k}_{0}\right| \tag{2}\]  
where \(\mathbf{k}_{0}\) is the wavenumber at a Fermi point, v\({}_{F}\approx 0.87\times 10^{6}\) m/s is the Fermi velocity of the \(\pi\)-electrons and \(\hbar\) is the Planck constant. In the ground state, the valence band of the graphene is completely filled by the \(\pi\)-electrons. In general, at equilibrium, the energy distribution function of \(\pi\)-electrons is given by the Dirac-Fermi function:  
\[F\left[E^{(\pm)}\right]=\frac{1}{e^{E^{(\pm)}/k_{B}T_{0}}+1} \tag{3}\]  
where \(k_{\mathrm{B}}\) is the Boltzmann constant and \(T_{0}\) is the graphene absolute temperature, the electrochemical potential of the graphene being null valued.