deletions | additions       diff --git a/When_applied_an_electric_field__.tex b/When_applied_an_electric_field__.tex  index 5c0fb6f..536c6ad 100644  --- a/When_applied_an_electric_field__.tex  +++ b/When_applied_an_electric_field__.tex 
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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 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. By DC conductance can be obtained by setting $E_{ac} = 0$. In addition, similar approach is applied by  varying parameters in equation (5), such as E_{ac} $E_{ac}$  and , one can obtain the "diode-like" conductance of depinned CDW. $\omega$.  The In this project, the  approach of dealing with DC electric field is to consider thescattering and  tunneling of CDW when interacting with distorted lattice potentials, which produce a band gap at the Fermi surface. Also, the different lattice spacing gives rise to different values of gap energy.  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 \begin{equation}  P \approx exp(-E_0/E)  \end{equation}  The probability comes in the right hand side of equation (5), reducing the force felt by CDW by a fraction of P. The weak electric field, i.e. $E_0 >> E$ leads to vanishing of the force. Thus the conductance is negligible. For large field $E >> E_0$, the force is nonlinearly dependent upon E, thus the second term in equation (4) shows up. Moreover, $E_0$ depends upon the gap energy; therefore, the relationship between the lattice spacing and the threshold electric field will also be investigated.  For weak AC electric fields, CDW oscillates in the potential and remained pinned. There will be no mesurable conductance.