A Microscopic Theory of the Pinning Effect in Peierls Systems with Dilute Impurities

Effect of dilute impurities on Fröhlich's collective mode is investigated by solving the Dyson equation for the collective mode Green's function which includes self-consistently an infinite series of impurity scattering processes occuring through electronic bubbles. The Dyson equation is shown to reduce to a simple algebraic equation in the low frequency region of interest. The electrical conductivity σ(ω) is examined and related to the collective mode Green's function. An analytic solution to the reduced Dyson equation is obtained. The resulting expression for conductivity is very simple: \(σ(ω){∝}ω⁻³√{ω²-(1/2)ωT²}\) where ω T is a constant. the analytic expression for σ(ω) describes the pinning effect fairly well: it gives a sharp asymmetric peak at \((√3/2)ωT\) with width ∼ω T , and it satisfies exactly the conductivity sum rule for the collective mode.