# Dirac-Based Quantum Admittance of 2D Nanomaterials at Radio Frequencies
## 3 Spinor Solution  
As a preliminary, \(E_{V}\) is assumed to be constant either in absence of the d.c. field or following a stepwise approximation of \(V\) for modest fields. This expedience allows us to eliminate the inconvenient, constant diagonal matrix on the RHS of Equation (9) by means of a similarity transformation:  
\[\Psi^{\prime}\equiv\begin{bmatrix}\sqrt{E^{\prime}+E_{V}}I&0\\ 0&\sqrt{E^{\prime}+E_{V}+2}m_{e}v_{F}^{2}I\end{bmatrix}\Psi \tag{10}\]  
This reduces to the form of a standard eigenvalue equation:  
\[\Big{[}\alpha_{x}\Big{(}\partial_{x}+i\frac{|e|}{\hbar}A\Big{)}-i\frac{|e|}{ \hbar}\frac{\phi}{v_{F}}+\frac{1}{v_{F}}\partial_{\tau}\Big{]}\Psi^{\prime}= \frac{i}{\lambda}\Psi^{\prime} \tag{11}\]  
where \(\lambda\equiv 1/(\hbar v_{F})\sqrt{(E^{\prime}+E_{V})(E^{\prime}+E_{V}+2m_{e}v_{ F}^{2})}\ m^{-1}\). The Dirac matrix \(\alpha_{x}\) is now the only one appearing in Equation (11). The progressive and regressive traveling wave solutions of \(\Psi^{\prime}\) are eigenvectors pertaining to the eigenvalues \(\pm\) 1. In [1], in fact, it is shown that the spinor \(\Psi^{\prime}\) is in the following form:  
\[\Psi^{\prime}=\begin{bmatrix}a+b\\ \sigma_{x}(a-b)\end{bmatrix} \tag{12}\]  
where  
\[a=\begin{bmatrix}a_{0}\\ a_{1}\end{bmatrix}\quad b=\begin{bmatrix}b_{0}\\ b_{1}\end{bmatrix} \tag{13}\]  
Setting \(b=0\) in Equation (12) above retains just the progressive wave, as is required in this case. In the absence of y variation, moreover, we may set \(a_{0}=a_{1}=a/\sqrt{2}\).  
Since we are now dealing with just one eigenvector, in Equation (11), we may replace the matrix \(\alpha_{x}\) with its eigenvalue \(+1\) and reduce Equation (11) to an ordinary differential equation for the scalar a:  
\[\Big{[}\partial_{x}+i\frac{|e|}{\hbar\omega}\hat{A}\Big{(}\omega-\frac{v^{2}} {v_{F}}k_{x}\Big{)}e^{i(k_{x}x-\omega_{\Gamma})}+\frac{1}{v_{F}}\partial_{ \tau}\Big{]}a=\bigg{[}\frac{i}{\lambda}-\frac{1}{L_{F}}\bigg{]}a \tag{14}\]  
It is expedient to introduce the \"angular frequency\"  
\[\omega_{1}\equiv v_{F}\frac{|e|}{\hbar}\hat{A}\frac{\omega-\frac{v^{2}k_{x}}{ v_{F}}}{v_{F}k_{x}} \tag{15}\]  
and the position  
\[\hat{a}\equiv exp\Big{[}-\omega_{1}/\omega e^{i(k_{x}x-\omega_{\Gamma})}\Big{]}a \tag{16}\]  
Figure 1: Representation of the domain under study, showing the adopted sign convention. Electric potential and current density present opposite directions, which explains the negative sign in the admittance definition in Equation (37).  
This reduces Equation (14) to the standard transmission line equation  
\[\left[\partial_{x}+\frac{1}{v_{F}}\partial_{\tau}\right]\bar{a}=\left[\frac{i}{ \lambda}-\frac{1}{L_{F}}\right]\bar{a} \tag{17}\]  
The progressive wave solution of Equation (17) is  
\[\bar{a}(x,\tau)=exp\left[-\frac{x}{L_{F}}+\frac{i}{\lambda}(x-v_{F}\tau)\right] \bar{a}(0,0) \tag{18}\]  
with \(\bar{a}(0,0)=exp(-\omega_{1}/\omega)a(0,0)\). Finally, by introducing Equation (18) into Equation (16), we obtain the solution to Equation (14) as follows:  
\[a(x,\tau;E^{\prime})=exp\left[-\frac{x}{L_{F}}+\frac{i}{\lambda}(x-v_{F}\tau)+ \frac{\omega_{1}}{\omega}\left(e^{i(k_{x}x-\omega\tau)}-1\right)\right]a(0,0;E ^{\prime}) \tag{19}\]  
In addition, the original spinor \(\Psi\) is recovered by means of the inverse transformation of Equation (12):  
\[\Psi=\begin{bmatrix}\frac{I}{\sqrt{E^{\prime}+E_{V}}}&0\\ 0&\frac{I}{\sqrt{E^{\prime}+E_{V}+2m_{e}v_{F}^{2}}}\end{bmatrix}\Psi^{\prime}= \frac{a}{\sqrt{2}}\begin{bmatrix}\frac{1}{\sqrt{E^{\prime}+E_{V}}}\\ \frac{1}{\sqrt{E^{\prime}+E_{V}}}\\ \frac{1}{\sqrt{E^{\prime}+E_{V}+2m_{e}+v_{F}^{2}}}\\ \frac{1}{\sqrt{E^{\prime}+E_{V}+2m_{e}+v_{F}^{2}}}\end{bmatrix} \tag{20}\]  
Equation (19) shows that the R.F. produces frequency modulation as well as a time-retarded traveling wave in the phase of the spinor.