deletions | additions       diff --git a/subsection_Combined_AC_and_DC__.tex b/subsection_Combined_AC_and_DC__.tex  index 6973c8f..9db7ca7 100644  --- a/subsection_Combined_AC_and_DC__.tex  +++ b/subsection_Combined_AC_and_DC__.tex 
...
\subsection{Combined AC and DC Conductivity}  Now, consider the full expression in Eq.(2) with none-zero $V_{dc}$, $V_{ac}$ and $\omega$. The introduction of DC voltage with the presence of AC voltage is changing the energy of CDW from merely photon energy to the photon energy plus the additional  DC contribution. contribution, i.e.  \begin{equation}  \kappa_b^{'}(x) = \frac{1}{\hbar} \sqrt{2m^*[V(x)-e^* E_{dc} x -\hbar \omega]}  \end{equation}  The integral of Eq.(14) is of the DC contribution $E_{dc}$ over a potential region $x$. As $E_{dc}$ increases, the integral deceases but the tunneling probability surges due to the exponential behavior. factor $\kappa_b^{'}(x)$.  This phenomenon corresponds to the CDW depinning with DC voltage larger than the threshold. On the other hand, the classical single particle contribution should also be modified since if modified. If  the CDW is depinned, depinned and can move more freely,  the damping coefficient $\Gamma$ in Eq.(2)  will drop, following the approximated expression. expression for low field.  \begin{equation}  \Gamma(V_{dc})\approx \frac{\Gamma_0}{V{dc}^2} \frac{\Gamma_0}{V_{dc}^2}  \end{equation}  The DC and AC conductivity are carried out in the similar fashion and are  shown in Fig.4 and Fig.5 as a function of $V_{dc}$ and $\frac{\omega}{2\pi}$ respectively.