deletions | additions       diff --git a/When_applied_an_electric_field__.tex b/When_applied_an_electric_field__.tex  index 36fd408..7a39687 100644  --- a/When_applied_an_electric_field__.tex  +++ b/When_applied_an_electric_field__.tex 
...
Since the charge density is dictated by potential in the absence of electric fields, thermal fluctuations and scattering events, the CDW is "pinned" on the potential. When applied an oscillating electric field with frequency $\omega$,   \begin{equation}  E(t) = E_0 E_{dc} E_{ac}  cos(\omega t) \end{equation}  there will be DC and AC current contributions. The empirical DC conductance ($E_{ac} = 0$)  is of the form\cite{Bardeen_1979}: \begin{equation}  \sigma(E) = \sigma_a + \sigma_b exp(-E_0/E)  \end{equation}  CDW was modeled as a classical particle moving in a "wash board" potential, which was approximated by parabolic and sinusoidal potentials. The particle was overdamped and subjected to electric force $e E$. Only when the applied DC electric field exceeds some threshold value $E_{th}$ will the CDW "particle" start to "slide", also known as the "depinning" process of CDW \cite{Gr_ner_1981}. Once depinned, the equation of motion for CDW particle  can be written in terms of position x or the phase shift $\phi$. \begin{equation}  \frac{d^{2}\phi}{dt^{2}} + \Gamma \frac{d\phi}{dt} + 2k_F V_0 sin(2k_F \phi) =\frac{e E_0}{m}cos(\omega t) E_{ac}cos(\omega t)}{m}  \end{equation}  where $\Gamma$ is the damping coefficient determined by experiment and m and e are the collective mass and charge of CDW respectively. By changing varying  parameters in equation (5), such as E_{ac} and ,  one can obtain the "diode-like" conductance of depinned CDW. The approach of dealing with DC electric field is to consider the scattering and tunneling of CDW when interacting with distorted lattice potentials, which produce a band gap at the Fermi surface. The electrons in the lower band can tunnel through the gap to the conducting band, contributing to the conductance. It is clear that the larger the applied field, the more possible electrons in CDW will tunnel through the gap, which is the analogy of Zener diode \cite{Zener_1934}. The tunneling probability can be expressed as