# 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 $$E_{ac}$$