deletions | additions       diff --git a/When_applied_an_electric_field__.tex b/When_applied_an_electric_field__.tex  index e08f373..9556f73 100644  --- a/When_applied_an_electric_field__.tex  +++ b/When_applied_an_electric_field__.tex 
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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_{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}:  
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\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 $\omega$. 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. Also, When applied an electric field,  thedifferent 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}  Where $E_0$ is the function of the gap energy.  The probability comes in the right hand side of equation (5), reducing the force felt by CDW by a fraction of P. \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} exp(-E_0/E_{dc})  \end{equation}  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, $E_0$ the different lattice spacing gives rise to different values of gap energy. The constant in the exponent, $E_0$,  depends upon the gap energy; therefore, the relationship between the lattice spacing and the threshold electric field will also be investigated. Secondly, this project will investigate the relationship between the AC conductance and the frequency of applied field. This problem was dealt with by considering the classical single particle model. For very high or very low frequency, the AC conductance vanishes since CDW is overdamped. The AC conductance plays an important role if the frequency is near the frequency of CDW. In this case from equation (1)