Conductance of Charge Density Waves Model

This is the first project report for ECE 5390/MSE 5472, focusing on the models and analyzing the conductance of 1D charge density waves (CDW).

For normal conductors, the charge density in the conducting band is uniform (Fig.1(a)) but for certain materials, the charge density fluctuates with position. The CDW is the ground state of many quasi-1D materials, such as NbSe\(_3\) when cooled down below Peierls temperature \(T_P\) and undergoing Peierls Transition (Peierls 1956). Below \(T_P\), the distortion of lattice minimizes the total energy of conducting electrons and the elastic energy of the distortion, rendering it the energetically favorable state. The electrons pile up around the periodic distortion potentials and the charge density in the conducting band varies with the same period \(2k_F\) (Fig.1(b)).

Charge density and band structure of (a) normal conductors and (b) CDW materials. (Grüner 1988)

When applied an electric field along the quasi-1D direction on CDW materials, nonlinear motion of CDW appears and nonlinear current and conductance can be observed (Fleming 1979). There are two problems of interest in this project. (i) What is the conductance if applied a DC electric field? (ii) What is the frequency dependence of conductance for an AC electric field?

For CDW state, the band structure breaks apart by the band gap energy \(E_g\) at the Fermi surface in the k-space since the lattice is distorted in real space with lattice constant from \(a \rightarrow 2a\). The wavelength of the potential or the CDW \(\lambda = 2a =\frac{\pi}{k_F}\) where \(k_F = \frac{\pi}{2a}\) is the Fermi wave vector. Therefore we can write the expression for the charge density as a function of position x and time t:

\[\rho(x) = \rho_0 + \rho_1 e^{i [2k_F(x-vt)+\phi]},\]

The v in the expression denotes the velocity of the right moving CDW while \(\phi\) is the phase or the translational shift in x. And the distortion potential:

\[V(x) = V_0 cos(2k_{F}r+\phi)\]

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 \(\omega\),

\[E(t) = E_{dc} + E_{ac} cos(\omega t)\]

there will be DC and AC current contributions. The empirical DC conductance (\(E_{ac} = 0\)) is of the form(Bardeen 1979):

\[\sigma(E) = \sigma_a + \sigma_b exp(-E_0/E)\]

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 is 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 (Grüner 1981). Once depinned, the equation of motion for CDW particle can be written in terms of position x or the phase shift \(\phi\).

\[\frac{d^{2}\phi}{dt^{2}} + \Gamma \frac{d\phi}{dt} + 2k_F V_0 sin(2k_F \phi) =\frac{e}{m} (E_{dc} + E_{ac}cos(\omega t))\]

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 \(\omega\) when obtaining the equation of motion for applied AC field.

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. When applied an electric field, 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 likely electrons in CDW will tunnel through the gap, which is the analogy of Zener diode (Zener 1934). The tunneling probability can be expressed as

\[P \approx exp(-E_0/E)\]

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 \(P\).

\[\frac{d^{2}\phi}{dt^{2}} + \Gamma \frac{d\phi}{dt} + 2k_F V_0 sin(2k_F \phi) =\frac{e}{m} E_{dc} exp(-E_0/E_{dc})\]

The weak electric field, i.e. \(E_0 >> E_{dc}\), leads to vanishing of the force. Thus the conductance is negligible. For large field \(E_{dc} >> E_0\), the force is nonlinearly dependent upon \(E_{dc}\), thus the second term in equation (4) shows up. Moreover, 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 the applied field. This problem was approached by considering the classical single particle model in oscillation mode instead of sliding mode. 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 close to the frequency of CDW, \(\omega_0\). In this case from equation (1)

\[\omega_0 = 2k_F v\]

The resonance of the applied AC field and CDW occurs at \(\omega \approx \omega_0\), where the electric field shifts CDW by one distorted lattice spacing within one period. The resonance frequency is determined by the gap energy and also the period of the potential. This primitive result for AC conductance is shown in (Fig.2).

The normalized AC conductance for CDW with \(\omega_0 = 10^{11} Hz\), which is the typical frequency for NbSe\(_3\) under the first Peierls transition. The larger the damping coefficient, the narrower the bandwidth of conductance. The narrow band oscillation in conductance and current is also called the narrow band noise.

However, in addition to the resonance contribution in the AC conductance, I’ll try to use the tunneling model to introduce the frequency dependence and then to account for more complex characteristics such as threshold frequency (Bardeen 1980) and AC DC interference (Grüner 1988).


  1. R. E. Peierls, Louis D. Roberts. Quantum Theory of Solids. Phys. Today 9, 29 AIP Publishing, 1956. Link

  2. G. Grüner. The dynamics of charge-density waves. Reviews of Modern Physics 60, 1129–1181 American Physical Society (APS), 1988. Link

  3. R. M. Fleming, C. C. Grimes. Sliding-Mode Conductivity in Nb Se 3 : Observation of a Threshold Electric Field and Conduction Noise. Phys. Rev. Lett. 42, 1423–1426 American Physical Society (APS), 1979. Link

  4. John Bardeen. Theory of Non-Ohmic Conduction from Charge-Density Waves in Nb Se 3. Phys. Rev. Lett. 42, 1498–1500 American Physical Society (APS), 1979. Link

  5. G. Grüner, A. Zawadowski, P. M. Chaikin. Nonlinear Conductivity and Noise due to Charge-Density-Wave Depinning in Nb Se 3. Phys. Rev. Lett. 46, 511–515 American Physical Society (APS), 1981. Link

  6. C. Zener. A Theory of the Electrical Breakdown of Solid Dielectrics. Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences 145, 523–529 The Royal Society, 1934. Link

  7. John Bardeen. Tunneling Theory of Charge-Density-Wave Depinning. Phys. Rev. Lett. 45, 1978–1980 American Physical Society (APS), 1980. Link

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