Dielectric function for a disordered one-dimensional conductor

Abstract

We present a calculation of the wave vector and frequency dependence of the longitudinal dielectric function of a disordered material which is collection of noninteracting parallel chains. We used a one-dimensional tight-binding model with nearest-neighbor interaction only, and randomness in the diagonal elements of the Hamiltonian. The model does not contain electron-phonon interactions, and is therefore valid only at zero temperature. The dielectric function is calculated from integral equations after a method developed by Halperin. The graphs of ${\epsilon }_2$ vs frequency resemble those for ${\epsilon }_2$ (crystalline), but broadened by $\frac{W}{L}$, where $W$ is the crystal bandwidth and $L$ is the decay length of the wave functions in the random system. The graphs of ${\epsilon }_1$ show a similar broadening. We find ${\epsilon }_1(\omega =0)\approx 1+0.993{\kappa }^2{[q^2+\frac{1.56}{{\mathrm{}\left(\mathrm{Lb}\right)\mathrm{}}^2]}}^{-1\mathrm{}}$ ($\kappa$ is the Fermi-Thomas screening factor, $b$ is the lattice constant) for $\mathrm{qb}\le 0.2\pi$. The peak in the conductivity at zero wave vector occurs at a frequency which is inversely proportional to $L$. The disorder also removes the singularity in ${\epsilon }_1$ at $q=2k_F$, $\omega =0$; the condition for a Peierls distortion in the presence of disorder is discussed. These results are interpreted in terms of the localized states in one-dimensional disordered systems.