Dielectric function for a disordered one-dimensional conductor

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 ε₂ vs frequency resemble those for ε₂ (crystalline), but broadened by 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 ε₁ show a similar broadening. We find ε₁(ω=0)≈1+0.993κ²[q²+1.56(Lb)²]^-1 (κ is the Fermi-Thomas screening factor, b is the lattice constant) for qb≤0.2π. 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 ε₁ at q=2kF, ω=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.