# Dirac Equation-Based Formulation for the Quantum Conductivity in 2D-Nanomaterials
## 2 Dirac Equation-Based Model
\[\overline{\psi}(x)=\exp i\mathbf{K}x\overline{\psi}(0)=[\cos(kx)\mathbf{I}+i\frac{\sin (kx)}{k}\mathbf{K}]\overline{\psi}(0) \tag{3}\]  
Eigenvalues of \(K\) are given by:  
\[k=\pm\frac{1}{\hbar v_{F}}\sqrt{(2E_{F}+E^{\prime}+E_{V})^{2}-(2E_{F})^{2}}\]  
Figure 2: 2-D schematic representation of the problem. and \(\pm k\) represents the forward and backward propagation constants, respectively, with real \(k\) if \(E^{\prime}\ +\ E_{V}\ \geq\ 0\). The eigenvectors (4-spinors) of \(K\) corresponding to \(\pm k\) are of the form  
\[\mathbf{\psi}_{a}= \begin{bmatrix}\mathbf{a}\\ \eta\sigma_{x}\mathbf{a}\end{bmatrix} e^{jkx}\qquad\mathbf{\psi}_{b}= \begin{bmatrix} \mathbf{b}\\ -\eta\sigma_{x}\mathbf{b}\end{bmatrix} e^{-ikx} \tag{4}\]  
with  
\[\eta=\sqrt{\frac{E^{\prime}+E_{V}}{E^{\prime}+E_{V}+2E_{F}}}\]  
Since \(E^{\prime}\),\(E_{F}\) are real and positive, if \(E^{\prime}+E_{V}\geq 0\),\(\eta\) is real; it becomes purely imaginary if \(E^{\prime}+E_{V}\leq 0\) and \(E^{\prime}+E_{V}+4E_{F}>0\), becoming real again when also \(E^{\prime}+E_{V}+4E_{F}<0\), as summarized in Figure 3.  
When \(E^{\prime}+E_{V}<0\), k is pure imaginary and (5) is modified as follows:  
\[\mathbf{\psi}_{a}= \begin{bmatrix}\mathbf{a}\\ \eta\sigma_{x}\mathbf{a}\end{bmatrix} e^{-\gamma x}\qquad\mathbf{\psi}_{b}= \begin{bmatrix} \mathbf{b}\\ -\eta\sigma_{x}\mathbf{b}\end{bmatrix} e^{\gamma x} \tag{5}\]  
with  
\[\gamma=\frac{1}{\hbar v_{F}}\sqrt{(2E_{F})^{2}-(2E_{F}+E^{\prime}+E_{V})^{2}}\]  
real, positive and \(\eta\) either real or pure imaginary. Such modification is important in the definition of current. The above spinors are orthogonal with weight \(\mathbf{\alpha}_{x}\); \(a,b\) are constant 2D vectors determined by boundary conditions and normalization. In absence of y-dependence, these may be assumed as  
\[\mathbf{a}=\frac{a}{\sqrt{2}} \begin{bmatrix}1\\ 1\end{bmatrix} \qquad\mathbf{b}=\frac{b}{\sqrt{2}} \begin{bmatrix}1\\ -1\end{bmatrix} \]  
The general solution (3) may be expressed as a linear combination of the above eigenvectors as a travelling wave or a standing wave with boundary condition \(\overline{\mathbf{\psi}}(0)\). Let us now consider the case of constant, low loss that assumes negligible coupling due to loss between progressive and regressive waves. We may introduce this constant propagation loss in (2), by writing:  
\[\partial_{x}\mathbf{\psi}=\bigg{[}\mp\frac{1}{L_{F}}+i\mathbf{K}\bigg{]}\mathbf{\psi} \tag{6}\]  
The upper/lower sign refers to the progressive/regressive wave, respectively. We have  
\[\mathbf{\psi}=\exp\mp\frac{x}{L_{F}}\overline{\mathbf{\psi}}(x)\]  
The solution of (6) is their linear combination weighted by the exponentials:  
\[\mathbf{\psi}(x)=\exp-\frac{x}{L_{F}}\mathbf{\psi}_{a}+\exp\frac{x}{L_{F}}\mathbf{\psi}_{b}\]  
Figure 3: Intervals which determine whether \(\eta\) is real or imaginary  
It is noted that the above two spinors modified by the exponentials are still orthogonal with weight \(\alpha_{x}\), consistently with the original low loss assumption, that considers the original lossless wavefunction attenuated without altering their shape  
**3. Current: The Classical Drude Limit**  
At a fixed energy, the general definition of quantum current density from Dirac equation is  
\[J(x)=-|e|Nv_{F}\psi^{\dagger}\alpha_{x}\psi\]  
For \(E^{\prime}+E_{V}\geq 0\), we have  
\[J(x)=J_{a}-J_{b}=-2|e|Nv_{F}\sqrt{\frac{E^{\prime}+E_{V}}{E^{\prime}+E_{V}+4E_ {F}}}\bigg{(}|\mathbf{a}|^{2}\exp-\frac{2x}{L_{F}}-|\mathbf{b}|^{2}\exp\frac{2x}{L_{F}} \bigg{)} \tag{7}\]  
where \(N\) is the charge density per unit area.