# Dirac-Based Quantum Admittance of 2D Nanomaterials at Radio Frequencies
## 4 Definition of Current  
From knowledge of the Dirac spinor \(\Psi\), the general current density vector \(J(x,t;E^{\prime})\) is defined as  
\[J=-v_{F}|e|N(E^{\prime})\Psi^{\dagger}\alpha\Psi \tag{21}\]  
\(N(E^{\prime})\) above is the electron density at the energy \(E^{\prime}\), which is considered to be unaffected by the presence of the R.F. as is reasonable for moderate fields, as reported in [4]. Using Equation (18), the longitudinal component \(J_{x}\) is given by  
\[Jx=-|e|Nv_{F}\Psi^{\dagger} \begin{bmatrix}0&\sigma_{x}\\ \sigma_{x}&0\end{bmatrix} \Psi=-2|e|Nv_{F}|a|^{2}\left[\sqrt{(E^{\prime}+E_{V} )(E^{\prime}+E_{V}+2m_{e}v_{F}^{2})}\right]^{-1} \tag{22}\]  
However, Equation (19) gives  
\[|a|^{2}=exp\left[-2\frac{x}{L_{F}}+2\frac{\omega_{1}}{\omega}cos(k_{x}x- \omega t)-1)\right]|a(0,0;E^{\prime})|^{2} \tag{23}\]  
Adopting the same current normalization as in [1], where \(J(0,0;E^{\prime})=-|e|N(E^{\prime})v_{F}\), implies that \(|a(0,0;E^{\prime})|^{2}=\frac{1}{2}\sqrt{(E^{\prime}+E_{V})(E^{\prime}+E_{V} +2m_{e}v_{F}^{2})}\) and consequently  
\[J_{x}(x,t;E^{\prime})=-|e|N(E^{\prime})v_{F}\,exp\left[-2\frac{x}{L_{F}}+2 \frac{\omega_{1}}{\omega}cos(k_{x}x-\omega t)-1)\right] \tag{24}\]  
The first term in Equation (24) represents just d.c. losses in the absence of the R.F. signal. As a function of the frequency, the current shows high-pass characteristics. Moreover, there appear to be infinite harmonics of the argument \(\theta=k_{x}x-\omega t\), which is a feature of RF mixers. This fact is made explicit by the identity [2]  
\[exp\Big{[}2\frac{\omega_{1}}{\omega}cos\theta\Big{]}=\sum_{k}I_{k}\Big{(}2\frac{ \omega_{1}}{\omega}e^{i\theta}\Big{)}\qquad\qquad(k=0,\pm 1,\pm 2,\ldots) \tag{25}\]  
where \(I_{k}(z)=i^{-k}I_{k}(iz)\) are the Bessel functions. Retaining only the fundamental harmonic of the signal (\(k=1\)) yields the scalar admittance at the signal frequency. We consider now a ballistic regime with flow in the x direction and which is uniform in the y direction. For a band of energies, the electron density \(N(E^{\prime})\) is to be replaced by a density of states, weighted by the occupation probability and integrated from the thermodynamic energy limit \(K_{B}\)\(T\approx 25\) meV at \(300^{\circ}\) up to the chemical potential \(\mu\). By retaining just the fundamental harmonic in Equation (24), this leads to a current density of  
\[J=-2|e|v_{F}\frac{1}{L_{x}L_{y}}exp\bigg{(}\frac{2x}{L_{F}}+i\theta\bigg{)}I_{ 1}\bigg{(}\frac{2\omega_{1}}{\omega}\bigg{)}e^{-2\omega_{1}/\omega}\int_{K_{ B}T}^{\mu}D(E^{\prime})f(E^{\prime})dE^{\prime}\qquad\quad[Am^{-1}] \tag{26}\]  
The pre-factor 2, now appearing in Equation (26), is due to two electrons of opposite spins finding places in a state. Moreover, the density of the states is \(D(E^{\prime})=\frac{dN_{S}}{dk}\frac{dk}{dE^{\prime}}\). Since we are considering a ribbon configuration, we assume the density of the states of a two-dimensional system with parabolic E-k characteristics first. The particular case of graphene will be considered presently. Apearring in Equation (16) as well, \(N_{S}=(\pi k^{2})/(\frac{(2\pi)^{2}}{L_{x}L_{y}})\) is the number of states in the k-space and \(E^{\prime}=(\hbar k)^{2}/(2m_{e})\) such that:  
\[D(E^{\prime})=\frac{dN_{S}}{dk}\frac{dk}{dE^{\prime}}=\frac{k}{2\pi}L_{x}L_{y} \frac{m_{e}}{\hbar^{2}k}=L_{x}L_{y}\frac{m_{e}}{2\pi\hbar^{2}}=const. \tag{27}\]  
The probability of occupation, in the presence of the applied voltage \(V\), is  
\[f=\frac{1}{1+exp[(E^{\prime}-(\mu-\mu_{0}))/(K_{B}T)]} \tag{28}\]  
where \(\mu-\mu_{0}=E_{V}\). Consistent with previous assumptions, we disregard the effect of the R.F. field energy on the probability distribution.