deletions | additions       diff --git a/However_experimentally_no_threshold_voltage__.tex b/However_experimentally_no_threshold_voltage__.tex  index c3ed81d..a43ed22 100644  --- a/However_experimentally_no_threshold_voltage__.tex  +++ b/However_experimentally_no_threshold_voltage__.tex 
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However, experimentally, no threshold voltage was observed\cite{Gr_ner_1981}. The pinned CDW oscillation has two contributions to the AC conductivity. One is the photon-assisted tunneling of single electrons. The tunneling processes through a sinusoidal potential can be approximated by WKB methods. The other is the collective mode of CDW oscillation as a classical single particle. This is where the resonance comes in to play and results in a non-zero AC conductivity in the low frequency domain.  Consider First, consider  the CDW in a sinusoidal potential of period $2a$ where a is the lattice constant without distortion. \begin{equation}  V(x) = 0.5 \space \mathcal{E}_g \left(1+sin\left( \frac{\pi x}{2 a} \right)\right) = 0.5 \space \hbar \omega_t \left(1+Sin\left( \left(1+sin\left(  \frac{\pi x}{2 a} \right)\right) \end{equation}  If the applied AC voltage is $V = V_{ac} \space cos(\omega t)$ without any DC bias. The energy gained by the electron is $\hbar \omega$. Therefore, the tunneling probability is  \begin{equation} 
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\begin{equation}  \kappa_b(x) = \frac{1}{\hbar} \sqrt{2m^*[V(x)-\hbar \omega]}  \end{equation}  The current density in one channel will be $J(\omega) = N T(\omega) e v$ where $N$ is the number of electrons injected toward the potential and $v$ is the average velocity of the electrons through the potential. Secondly, consider the classical equation of motion for classical single particle in equation (2) and replace $V_{applied} = V = V_{ac} \space cos(\omega t)$.