deletions | additions       diff --git a/When_applied_an_electric_field__.tex b/When_applied_an_electric_field__.tex  index a8fdc35..559b948 100644  --- a/When_applied_an_electric_field__.tex  +++ b/When_applied_an_electric_field__.tex 
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\rho(x) = \rho_0 + \rho_1 e^{i [2k_F(x-vt)+\phi]},  \end{equation}  The v in the expression denotes the velocity of the right moving CDW while $\phi$ is the phase or the translational  shift in x. And the distortion potential: \begin{equation}  V(x) = V_0 cos(2k_{F}r+\phi) 
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\sigma(E) = \sigma_a + \sigma_b exp(-E_0/E)  \end{equation}  From previous work, CDW was modeled as a classical particle moving in a "wash board" potential, which was approximated by parabolic and sinusoidal potentials. The particle was is  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}{m} (E_{dc} + E_{ac}cos(\omega t))  \end{equation}  where $\Gamma$ is the damping coefficient determined by experiment and m and e are the collective mass and charge of CDW respectively. DC conductance can be obtained by setting $E_{ac} = 0$. Although there is no analytical solution to equation (5), numerical solution is available. In addition, similar approach is applied by varying $E_{ac}$ and $\omega$. $\omega$ when obtaining the equation of motion for applied AC field.  In this project, the approach of dealing with DC electric field is to consider the tunneling of CDW when interacting with distorted lattice potentials, which produce a band gap at the Fermi surface. When applied an electric field, 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