deletions | additions       diff --git a/subsection_AC_Conductivity_If_the__.tex b/subsection_AC_Conductivity_If_the__.tex  index 00aa3d1..6833d23 100644  --- a/subsection_AC_Conductivity_If_the__.tex  +++ b/subsection_AC_Conductivity_If_the__.tex 
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For $V_{dc}=0$ and $\hbar\omega/e^*>V_t$, the expression for AC current is reduced to  \begin{equation}  I_{ac}(V_{dc},V_{ac},\omega)=\left( \frac{e^*V_{ac}}{\hbar\omega} \right) I_{dc}\left(\frac{\hbar\omega}{e^*}\right)  \end{equation} Since the function $I_{dc}$ is similar to $\sigma_{dc}$, there is a threshold value for the argument.   \begin{equation}  V_t = \frac{\mathcal{E}_g l}{e^*L}=\frac{\hbar\omega_t l}{e^*L}  \end{equation}  which gives $\mathcal{E}_g=\hbar\omega_t$. The CDW has to absorb the energy more than the gap energy to make tunneling effect significant. The appearance of small bias voltage $V_{dc}\ll V_t$ comes in to shift the threshold frequency to $\omega_t-e^*V_{dc}/\hbar$ as long as the bias voltage is still small compared with the threshold voltage $V_t$. The AC conductivity for NbSe$_3$ is show in Fig.2. In the case of small energy gap $\mathcal{E}_g$ or the short correlation length $L$, lower energy photons are eligible to assist the CDW tunneling, resulting in the larger conductivity. In addition, the threshold voltage $\omega_t=38MHz$ can be seen.