When quasi-ID materials such as NbSe\(_3\) undergo the Peierls transition \cite{Peierls_1956}, the charge density in the valence band fluctuates according to the lattice potential. With larger potential distortion in the lattice, say, the cations form dimers changing the lattice spacing from \(a\) to \(2a\), the electrons tend to pile up around the dimers, analogous to the bounding state in the molecules. If external DC voltage is applied, the CDW remains pinned near the lattice potential on the site until the external voltage exceeds the threshold voltage and CDW starts to move. The DC conductivity was given empirically as the following expression\cite{Bardeen_1979}. \[\sigma_{dc}(E) = \sigma_a + \sigma_b\space exp\left(-\frac{E_0}{E}\right)\]
The CDW transport (depinning) problem with the presence of applied voltage is modeled by a classical single particle sliding down the periodic lattice potential with displacement \(x\) \cite{Rice_1975}. \[m \frac{d^2 x}{d^2 t} + \Gamma \frac{dx}{dt} + \omega_p^2 x = e^* \left(\frac{V_{applied}}{l}\right)\] where \(m\) is the collective mass of CDW, \(\Gamma\) the damping coefficient, \(l\) the length of the 1D conducting channel in the crystal and \(\omega_p\) is the pinning characteristic frequency. Solution to equation (2) by replacing \(V_{applied}\) by \(V_{dc}+V_{ac} \space cos(\omega t)\) gives the resonance contribution and will be discussed in detail later.
The other model for CDW depinning is the semiconductor tunneling analogy\cite{Zener_1934}. The CDW motion was considered to be tunneling through the pinning potential and the tunneling probability was found to be \[P = exp \left( -\frac{E_0}{E} \right)\]
\[E_0 = \frac{\pi \mathcal{E}_g^2}{4 \hbar e^* v_F}\]
where \(\mathcal{E}_g\) is the pinning gap of CDW and \(v_F\) is the Fermi velocity of electrons. The \(e^*\) is the effective charge due to the shielding effect of CDW with \(e^*/e \approx 10^{-4}\). Equation (3) directly contributes to the exponential term in equation (1) by reducing the fraction of conducting CDW by a factor of \(P\).
Although the tunneling model explained the empirical expression, it didn’t take applied AC voltage into account and its DC prediction deviated from the experimental data\cite{Gr_ner_1981}. The problem now is threefold. If the applied voltage and the resulting conductivity are of the form \[V = V_{dc} + V_{ac} \space cos(\omega t)\] \[\sigma = \sigma_{dc}(V_{dc},V_{ac},\omega) + \sigma_{ac}(V_{dc},V_{ac},\omega)\] what is the expression of the
DC conductivity in the absence of AC voltage i.e. \(\sigma_{dc}(V_{dc},V_{ac}=0)\)?
AC conductivity in the absence of AC voltage i.e. \(\sigma_{ac}(V_{dc}=0,V_{ac},\omega)\)?
conductivity of the combined AC and DC voltage?