# Dirac-Based Quantum Admittance of 2D Nanomaterials at Radio Frequencies
## 4 Definition of Current
### 4.2. R.F. Quantum Admittance  
The definition of admittance of the main harmonic is  
\[Y(\omega)=-\frac{J}{E}\hskip 28.452756 pt[\Omega^{-1}] \tag{35}\]  
In our case, \(E=E_{x}=-\partial A/\partial t-\partial\phi/\partial x\), \(\phi=v^{2}k_{x}A/\omega\) is the scalar R.F. potential. The current propagation losses are now introduced in the potential. We have \(\partial\phi/\partial x=(-2/L_{F}+ik_{x})\phi\) and  
\[ \begin{split} E=i\omega A-\bigg{(}-\frac{2}{L_{F}}+ik_{x}\bigg{)} \phi&=\frac{1}{i\omega}\bigg{(}v^{2}k_{x}^{2}-\omega^{2}+i \frac{2v^{2}k_{x}}{L_{F}}\bigg{)}A=\\ &=\frac{1}{i\omega}\bigg{(}v^{2}k_{x}^{2}-\omega^{2}+i\omega \frac{2v}{L_{F}}\frac{n_{e}}{n}\bigg{)}A\end{split}  \tag{36}\]  
In the last step, we used the fact that \(vk_{x}=n_{e}/n\omega\), where \(n_{e}\) is the effective refractive index \(k_{x}/k_{0}\) of the e.m. wave at the frequency in question and \(n=c/v\). From Equations (26) and (29), at \(x=L_{x}\), we find the following for the parabolic E-k dispersion:  
\[Y(\omega)=-i\omega\frac{|e|v_{F}n_{e}\Delta}{\pi\hbar^{2}}e^{-2L_{x}/L_{F}} \frac{1}{\tilde{A}}I_{1}\bigg{(}\frac{2\omega_{1}}{\omega}\bigg{)}e^{-2\omega _{1}/\omega}\bigg{[}v^{2}k_{x}^{2}-\omega^{2}+i\omega\frac{2v}{L_{F}}\frac{n_ {e}}{n}\bigg{]}^{-1} \tag{37}\]  
whereas for the linear dispersion (graphene), we found the following (Equation (38)):  
\[Y(\omega)=-i\omega\frac{|e|\Delta^{\prime}}{\pi\hbar^{2}v_{F}}e^{-2L_{x}/L_{F} }\frac{1}{\tilde{A}}I_{1}\bigg{(}\frac{2\omega_{1}}{\omega}\bigg{)}e^{-2\omega _{1}/\omega}\bigg{[}v^{2}k_{x}^{2}-\omega^{2}+i\omega\frac{2v}{L_{F}}\frac{n_ {e}}{n}\bigg{]}^{-1} \tag{38}\]  
Recalling that \(\omega_{1}\) is defined as in Equation(15), we find the following in the limit of the small argument:  
\[Y(\omega)=-i\frac{|e|^{2}\Delta^{\prime}}{\pi\hbar^{3}}e^{-2L_{x}/L_{F}}\frac {\omega-v^{2}k_{x}/v_{F}}{\omega-v_{F}k_{x}}\bigg{[}v^{2}k_{x}^{2}-\omega^{2} +i\omega\frac{2v}{L_{F}}\frac{n_{e}}{n}\bigg{]}^{-1} \tag{39}\]  
If \(n_{e}=n\), then \(vk_{x}=\omega\), and thus:  
\[Y(\omega)=-\frac{|e|^{2}\Delta^{\prime}}{\pi\hbar^{3}}e^{-2L_{x}/L_{F}}\frac {L_{F}}{2\omega v_{F}} \tag{40}\]  
Figure 2 shows the admittance Y in the linear dispersion case in Equation (40) as functions of the frequency for different values of \(L_{x}\), considering the parameters in Table 1. As is shown, as \(L_{x}\) approaches to the mean free path, the admittance decreases due to a transition between the ballistic and diffusive transport regimes. The admittance of Equations (37) or (38) above is the admittance of a series RLC circuit seen through a frequency-dependent ideal transformer \(n^{\prime}(\omega)\):  
\[Y(\omega)=\frac{j\omega Cn^{\prime 2}\omega}{1-\omega^{2}LC+j\omega RC} \tag{41}\]  
where \(j=-i\), \(LC=1/(v^{2}k_{x}^{2})\), \(RC=(2v/L_{F})(n_{e}/n)(1/(v^{2}k_{x}^{2}))\) and  
\[Cn^{\prime 2}(\omega)=\frac{|e|v_{TF}m_{e}\Delta}{\pi\hbar^{2}v^{2}k_{x}^{2}}e^{ -2L_{x}/L_{F}}\frac{1}{A}I_{1}\bigg{(}\frac{2\omega_{1}}{\omega}\bigg{)}e^{-2 \omega_{1}/\omega} \tag{42}\]  
\[Cn^{\prime 2}(\omega)=\frac{|e|\Delta^{\prime}}{\pi\hbar^{2}v^{2}k_{x}^{2}v_{F} }e^{-2L_{x}/L_{F}}\frac{1}{A}I_{1}\bigg{(}\frac{2\omega_{1}}{\omega}\bigg{)}e^{ -2\omega_{1}/\omega} \tag{43}\]  
for the parabolic and linear dispersion cases, respectively. In the small argument limit (weak field and high frequency), we have \(I_{1}(z)e^{-z}\sim z/2\), whereas for large arguments (high field and low frequency), we have \(I_{1}(z)e^{-z}\sim 1/\sqrt{2\pi z}\). In the former limit, recalling that \(\omega_{1}\) is defined as in Equation (15), we obtain  
\[Cn^{\prime 2}(\omega)=\frac{|e|^{2}v_{F}^{2}m_{e}\Delta}{\pi\hbar^{2}v^{2}k_{x}^ {2}}e^{-2L_{x}/L_{F}}\frac{1}{\hbar\omega}\frac{\omega-(v^{2}k_{x})/v_{F}}{ \omega-v_{F}k_{x}} \tag{44}\]  
\[Cn^{\prime 2}(\omega)=\frac{|e|^{2}v_{F}\Delta^{\prime}}{\pi\hbar^{2}v^{2}k_{x }^{2}v_{F}}e^{-2L_{x}/L_{F}}\frac{1}{\hbar\omega}\frac{\omega-(v^{2}k_{x})/v_{ F}}{\omega-v_{F}k_{x}} \tag{45}\]  
for the parabolic and linear dispersion cases, respectively.