Invariant-imbedding approach to resistance fluctuations in disordered one-dimensional conductors

The invariant-imbedding method provides a first-order differential equation for the complex reflection amplitude R(L) of a one-dimensional conductor of length L, in terms of the random potential V(L) at the edge of incidence for a particle of energy E. The Landauer formula is used to express the resistance {ρ}(L) and the conductance ρ^-1(L) in terms of R(L) and to derive an exact second-order differential equation for {ρ}(L), from the above equation for R(L). This equation for {ρ}(L) emphasizes important aspects of the problem but has not been solved explicitly. However, two types of explicit solutions, referred to as (a) and (b), have been derived, starting from the differential equation for R(L).Solution (a) is valid in the special case where typical fluctuations of V(L) about an average barrier V>0 are comparable with 2E-V. Solution (b) is obtained by using an intuitively justified averaging over the phase angle of R(L) to derive an approximate first-order differential equation for the resistance, valid for both signs of V and for arbitrary E. This equation yields an expression for {ρ}(L) in terms of V(L) which is formally similar to that obtained in case (a). Averaging solutions (a) and (b) for {ρ}(L) over Gaussian, {δ}-correlated variables V(L)-V yields generalized Landauer expressions for 〈{ρ}(L)〉 and infinite average conductances 〈ρ^-1(L)〉, for arbitrary L.Closed-form expressions for the higher moments 〈ρⁿ(L)〉 can also be given and reveal that {ρ}(L) has no central limit. The exact probability distributions of R(L) in case (a) and of both {ρ}(L) and ρ^-1(L) in cases (a) and (b) are obtained for arbitrary L, using a moment method. It is found that the variable ln[(1-iR)(1+iR)^-1] in case (a) (V>0) and the variables ln{ρ}(lnρ^-1), {ρ}{}1, for V<0 in case (b) have Gaussian distributions with mean values and variances that scale linearly with L. In particular, this confirms ln{ρ} as the correct scaling variable for large L in the case V<0, but shows that the analogous scaling variable for V>0 is the quantity ln[(1-iR)(1+iR)^-1].On the other hand, the distributions of ln{ρ}(lnρ^-1), {ρ}{}1, for V>0, for both solutions (a) and (b) are rapidly decaying exponentials, corresponding to weak Gaussian tails. Averages of {}R(L){} and of {}R(L)², which determine the average electron density outside the conductor, are also studied and their asymptotic, length-dependent rates of exponential growth are obtained. In an appendix solution (a) is formally generalized to include the effect of small deviations from the electron-energy range considered above.