# Dirac Equation-Based Formulation for the Quantum Conductivity in 2D-Nanomaterials
## 2 Dirac Equation-Based Model  
We consider the 2-D problem of Figure 2, where a constant electric field \(\overline{E}\) is applied in the negative x-direction over a strip of dimensions \(L_{x}\times L_{y}\). We assume motion in the \(x\)-direction and uniformity in \(y\). In writing Dirac equation applicable to the structure of Figure 2, Fermi velocity, \(v_{F}\), effective mass, \(m_{e}\), and lifetime in the band, \(\tau\), are given constants in the band of interest. The loss per unit length  
\[\frac{1}{v_{F}\tau}=\frac{1}{L_{F}}\qquad(L_{F}>L_{x})\]  
subsumes all mechanism, such as intraband transition, interaction with the substrate, etc. \(L_{f}\) indicates the compound mean free path due to all loss mechanisms, \(L_{X}\) is the actual sample length. The condition \(L_{F}>L_{x}\) means that the current, with some attenuation, reaches the output end of the sample. The condition \(L_{F}>L_{x}\) means that the current, with some attenuation, reaches the output end of the sample. Figure 1: Potential in the structure (Sample length \(L_{x}=100\) nm). The only potential appearing in the Dirac equation is the d.c. potential \(V(x)=V_{0}+E_{x}x\), where \(V_{0}\) is the constant bias. Our aim is to be able to compare with the results of current Schrodinger-based formulas at a given energy \(E^{\prime}\) in the interval \(k_{B}T\leq E^{\prime}\leq\mu\), where \(k_{B}\) is the Boltzmann constant, T the absolute temperature (\(k_{B}T=25\) meV) and \(\mu\) is the chemical potential, that is, the upper energy level we consider still occupied. The latter can be considered as the superposition of the initial chemical potential, say the intrinsic Fermi level \(\mu_{0}=2E_{F}\), and the applied external potential \(E_{V}\), leading to \(\mu=\mu_{0}+E_{V}\). Now, in the Dirac equation we separate out the energy \(m_{e}v_{F}^{2}=2E_{F}\) (\(E_{F}\) indicates the energy at the Fermi surface) and set  
\[i\hbar\partial t=m_{e}v^{2}+E^{\prime}\]  
in absence of losses (\(L_{F}\rightarrow\infty\)), Dirac equation is  
\[-i\hbar v_{F}\mathbf{x}_{x}\partial_{x}\overline{\psi}+\beta m_{e}v_{F}^{2} \overline{\psi}=[m_{e}v_{F}^{2}+E^{\prime}+|e|V]\overline{\psi} \tag{1}\]  
The \(4\times 4\) Dirac Matrices \(\mathbf{\alpha}\), \(\mathbf{\beta}\) appearing above are defined as  
\[\mathbf{\alpha}_{x}= \begin{bmatrix}\mathbf{0}&\mathbf{\sigma}_{x}\\ \mathbf{\sigma}_{x}&\mathbf{0}\end{bmatrix} ;\qquad\mathbf{\beta}_{x}= \begin{bmatrix}\mathbf{I }&\mathbf{0}_{x}\\ \mathbf{0}_{x}&-\mathbf{I}\end{bmatrix} ;\qquad\text{where}\quad\mathbf{\sigma}_{x}= \begin{bmatrix}0&1\\ 1&0\end{bmatrix} \]  
and are endowed with the following properties  
\[\mathbf{\alpha}\cdot\mathbf{\alpha}=3\mathbf{\beta}^{2}=3\mathbf{I}_{4\times 4}\qquad\mathbf{ \alpha}\mathbf{\beta}=-\mathbf{\beta}\mathbf{\alpha}\]  
After pre-multiplying (1) by \(\mathbf{\alpha}_{x}\), we re-arrange in the form of an homogeneus system of four firs order differential equations over the semi-infinite interval \(0\leq x\) with given boundary condition \(\overline{\psi}(0)\):  
\[\partial\overline{\psi}=i\mathbf{K}(x)\overline{\psi} \tag{2}\]  
with \(\mathbf{K}(x)\equiv\frac{1}{\hbar v_{F}}(2E_{F}+E^{\prime}+|e|V)\mathbf{\alpha}_{x}+ \frac{1}{\hbar v_{F}}2E_{F}\mathbf{\beta}\mathbf{\alpha}_{x}\). Considerable simplification in the following is obtained if \(|e|V\) is replaced by its average value over the interval \(x_{0}\) (\(x_{0}\) is the length of an appropriate segmentation of the sample length \(L_{x}\)), \(E_{V}=|e|(V_{0}+\overline{E}\frac{\mathbf{\alpha}}{2})\) so that \(\mathbf{K}\) is now a constant matrix. Using the specific properties of the Dirac matrices in the general solution of the linear system (2) (as given in ref. [35] for instance), the following useful result can be shown to hold.