# Dirac-Based Quantum Admittance of 2D Nanomaterials at Radio Frequencies
## 2 Dirac Equation  
Considering the problem in Figure 1, the Dirac equation in \(x\) and \(t\), with a finite effective mass \(m_{e}\) and Fermi velocity \(v_{F}\) in the band of interest, is  
\[\left[v_{F}\alpha_{x}(-i\hbar\partial_{x}+|e|\mathbf{A})+\beta m_{e}v_{F}^{2}\right] \Psi=[i\hbar\partial_{t}+|e|\Phi]\Psi \tag{1}\]  
where \(|e|\Phi\) stands for the energy related to the total scalar potential \(\Phi\), which is due to the d.c. electric field as well as that associated with the vector potential A. When dealing with the modulation of a time harmonic, namely a very high frequency carrier (e.g., optical) by a radio frequency (R.F.), it is expedient to separate the large but constant contribution of the carrier frequency \(\omega_{c}\) from that of the time modulation by writing  
\[\partial_{t}=-i\omega_{c}+\partial_{\tau}. \tag{2}\]  
In the present case, the \"rest\" energy is  
\[m_{e}v_{F}^{2}=2E_{F} \tag{3}\]  
where in the case of parabolic dispersion, in absence of doping and identifying \(m_{e}\) with the cyclotron mass, it is usually much larger than all other contributions.It is therefore appropriate to extract from the Hamiltonian on the RHS of Equation (1) its stationary terms by rewriting it as  
\[i\hbar\partial_{t}=m_{e}v_{F}^{2}+E^{\prime}+i\hbar\partial_{\tau} \tag{4}\]  
where \(\tau\) is the time scale of the modulation and \(E^{\prime}\) is the stationary state energy we are considering. Clearly, we have to integrate over the range of possible E' values. [PERSON]'s equation is actually recovered from Equation (1) by repeated application of the Hamiltonian operator in the limit \(m_{e}v_{F}^{2}\gg E^{\prime}\), neglecting the squares of the potentials and passing from a spinor to a scalar wave function by equating the currents. Moreover, we have in Equation (1) that  
\[V=V_{0}+E_{0}x \tag{5}\]  
\[\Phi=V+\phi(x,\tau) \tag{6}\]  
where \(V_{0}\) is the d.c. bias constant in space and time, \(E_{0}\) is the constant d.c. field, if any, and \(\phi\) is the time-dependent scalar potential associated with the vector potential \(\mathbf{A}\) such that the Lorentz gauge is satisfied:  
\[\
abla\cdot\mathbf{A}+\frac{1}{v^{2}}\partial_{\tau}\phi=0 \tag{7}\]  
In our case, by disregarding transverse variation over the ribbon width (\(2\pi/k\gg L_{y}\), where \(k\) is the wavenumber in the media) and boundary conditions of the e.m. field, we assume an x-directed vector potential  
\[\mathbf{A}=\hat{A}e^{i(k_{x}x-\omega\tau)}\hat{\mathbf{x}} \tag{8}\]  
that implies a scalar potential \(\phi=-\left(v^{2}/\omega\right)k_{x}\hat{A}e^{i(k_{x}x-\omega\tau)}=-n_{e}vA\), where \(n_{e}=k_{x}/k_{0}\) is the \"effective\" \"index\" of the propagating wave. The Dirac equation (Equation (1)) is now cast in the working form  
\[\left[\alpha_{x}\left(\partial_{x}+i\frac{|e|}{\hbar}\mathbf{A}\right)-i\frac{|e |}{\hbar}\frac{\phi}{v_{F}}+\frac{1}{v_{F}}\partial_{\tau}\right]\Psi=\frac{ i}{\hbar v_{F}}\left[\left(E^{\prime}+E_{v}+m_{e}v_{F}^{2}\right)I-m_{e}v_{F}^{2} \beta\right]\Psi \tag{9}\]  
where I is the \(4\times 4\) identity matrix and \(E_{v}=|e|V\) is the d.c. field energy. As mentioned, we also subsume constant small losses (e.g., due to interaction with the substrate) over a finite mean free path \(L_{F}\), leading to a propagation decay of the type \(e^{-x/L_{F}}\), simply by replacing \(\partial_{x}\) with \(\partial_{x}\pm 1/L_{F}\) according to whether the wave is progressive or regressive. This modification keeps the remainder of the equation Hermitian in nature. The solutionof the system of the partial differential equation (Equation (9)) is uniquely determined by the initial and boundary conditions. A proper definition of quantum admittance also requires no return wave and a proper input match at the origin of the computational domain (\(0\leq x\leq L_{x}\)) for a progressive wave traveling from left to right.