Quantum theory of local perturbation of the charge-density wave by an impurity: Friedel oscillations

The interaction of a single impurity with a charge-density wave (CDW) cannot be described by Ginzburg-Landau theory. In the present paper a one-dimensional microscopic quantum theory is presented considering only the backward scattering of the electrons by the impurity at zero temperature. This theory considers the strong perturbation of the CDW inside the amplitude coherence length, which perturbation is dominated by the Friedel oscillations at short distances.

It treats the CDW within the framework of the mean-field approximation, and sums up the backward scattering to all orders in perturbation theory. The main features of a self-consistent treatment of the mean field is briefly outlined and the modification can be embodied into the renormalization of the impurity scattering. The results obtained are sensitive to the impurity-scattering strength. In first order, the results of the rigid CDW are reproduced; in second order, the previous results by Barnes and Zawadowski are obtained. The largest effects are in the strong-scattering region. The following physical quantities are calculated: electron density, ground-state energy, density of states, and the force exerted by the impurity on the CDW as a function of the relative position of the impurity with respect to the CDW, and a solution of the equation of motion is found. Considering the electron density in the intermediate-coupling-strength case, the Friedel oscillations dominate at short distances well inside the amplitude coherence length.

In the charge density, the Friedel oscillations and the CDW are additive to a good approximation. Outside the amplitude coherence length, the Friedel oscillations tunnel into the CDW gap. In the density of states at the impurity site, the singularity at the gap edge is smeared out and a pair of bound states appears in the gap if the CDW and the Friedel oscillations are out of phase. Further bound states appear also outside the conduction band. The effective potential describing the interaction of the CDW with the impurity is very nonsinusoidal in the intermediate-coupling-strength region, but becomes more sinusoidal for very weak and very strong coupling. The effect of this nonsinusoidal potential in the equation of motion is in the enhancement of higher harmonics appearing in the narrow-band noise, but their intensities remain monotonically decreasing. Among the observable effects predicted are the following: the temperature dependence of the ratios of the intensities of the harmonics in the narrow-band noise, the effect of the nonsinusoidal potential in the Shapiro steps, and the appearance of Friedel oscillations in NMR and diffraction experiments.