However, experimentally, no threshold frequency \(\omega_t\) was observed\cite{Gr_ner_1981}. The pinned CDW oscillation has two contributions to the AC conductivity. One is the photon-assisted tunneling of single electrons. The tunneling processes through a sinusoidal potential can be approximated by WKB methods. The other contributions involves the collective mode of CDW oscillation as a classical single particle. This is where the resonance comes in to play and results in a non-zero AC conductivity in the low frequency domain.

First, consider the CDW in a sinusoidal potential of period \(2a\) where \(a\) is the lattice constant without distortion. \[V(x) = 0.5 \space \mathcal{E}_g \left(1+sin\left( \frac{\pi x}{2 a} \right)\right) = 0.5 \space \hbar \omega_t \left(1+sin\left( \frac{\pi x}{2 a} \right)\right)\] If the applied AC voltage is \(V = V_{ac} \space cos(\omega t)\) without any DC bias. The energy gained by the electron is \(\hbar \omega\). Therefore, the tunneling probability is \[T(\omega) = exp\left( -2 \int_0^{2a} \kappa_b(x)dx \right)\] \[\kappa_b(x) = \frac{1}{\hbar} \sqrt{2m^*[V(x)-\hbar \omega]}\] The AC current density in one channel will be \(J_{1}(\omega) = N T(\omega) e v\) where \(N\) is the number of electrons injected toward the potential and \(v\) is the average velocity of the electrons through the potential. And here, all the time-dependent terms such as \(cos(\omega t)\) are dropped so that the focus is on the magnitude of AC current and conductivity.

Secondly, consider the classical equation of motion for single particle in equation (2) and replace \(V_{applied} = V = V_{ac} \space cos(\omega t)\). Equation (2) can be solved numerically and giving the frequency dependent velocity amplitude \(|v| = |\frac{dx}{dt}| = v(\omega)\). When the frequency of applied voltage \(\omega\) is approximately equal to the characteristic frequency \(\omega_p\), resonance takes place with the largest velocity amplitude. The current density in this case is \(J_{2}(\omega) = Nev(\omega)\). Note that the frequency dependent velocity \(v(\omega)\) has nothing to do with the average velocity \(v\) in the previous discussion.

As a result, the current density in one channel is the sum of these two contributions, i.e. \(J = J_1+J_2\) and so is the magnitude of AC conductivity. The AC conductivity for NbSe\(_3\) is shown in Fig.3 with no DC bias. The threshold frequency is no longer obvious because the resonant conductivity dominates in the low frequency domain below \(100MHz\).