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  • AC and DC Conductivity of Charge Density Waves

    Abstract

    In this final report for ECE 5390/ MSE 5472, the tunneling model for charge density waves (CDW) in the quasi-one-dimensional material will be used to derive both DC and AC conductivity. The DC conductivity in the absence of AC signals has a threshold voltage, above which the conductivity is non-zero. The AC conductivity follows the same behavior due to the photon-assisted tunneling of CDW but there is an additional resonance contribution. Finally, the conductivity for mixed DC/AC signals is shown with the photon-assisted conductivity at low field and resonant AC conductivity.

    Introduction

    When quasi-ID materials such as NbSe\(_3\) undergo the Peierls transition (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(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 \(x\) (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(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(Grüner 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

    1. DC conductivity in the absence of AC voltage i.e. \(\sigma_{dc}(V_{dc},V_{ac}=0)\)?

    2. AC conductivity in the absence of AC voltage i.e. \(\sigma_{ac}(V_{dc}=0,V_{ac},\omega)\)?

    3. conductivity of the combined AC and DC voltage?

    Results

    DC Conductivity

    The classical single particle model states that the conductivity is zero until the applied voltage exceeds certain threshold voltage \(V_t\). Thus the tunneling probability in equation (3) can be modified as \[P = \left( 1-\frac{V_t}{V_{dc}}\right) \space exp \left( -\frac{E_0}{E_{dc}} \right) = \left( 1-\frac{V_t}{V_{dc}}\right) \space exp \left( -\frac{V_0}{V_{dc}} \right)\] where \(V_0=E_0 l\) and \(l\) is the length of the crystal. The threshold electric field \(E_{t}=V_{t}/l\) and the tunneling contribution is significant if \(V_{dc}>V_{t}=\mathcal{E}_gl/e^*L\) where \(L\) is the correlation length(Bardeen 1979). \[\frac{V_0}{V_t}=\frac{\pi \mathcal{E}_g L}{4 \hbar v_F}\] The ratio in equation (8) ranges from 0.25 to 4 for different materials and impurity concentration. The normalized DC conductivity \((\sigma_0+\sigma_{dc})/\sigma_0\) is shown in Fig.1. In the case of short correlation length, \(V _0 \approx V_t\), the small applied voltage results in the strong field in the channel so the conductivity increases more rapidly and vice versa.