Statistical scattering theory, the supersymmetry method and universal conductance fluctuations

This paper describes a novel analytical approach to the problem of conductance fluctuations in mesoscopic systems which, in particular, gives account of the influence of the coupling to external leads. We consider the case of a linear disordered sample in the metallic regime, which is coupled to two ideally conducting external leads. Using the many-channel approximation to Landauer's formula, we relate the conductance to the total transmission probability through the sample. The microscopic Hamiltonian of the quasi-one-dimensional disordered sample is formulated in terms of a random matrix, and the elements of the associated scattering matrix which determine the transmission are constructed from statistical scattering theory. We show that in addition to the Thouless energy, E_c, and the mean level spacing, d, there exists in the theory, a third energy scale, Γ, determined by the number of channels in the leads and the strength of the coupling between disordered sample and leads. Related to Γ, is a new length scale, L₀. We find that for sample lengths L >L₀, the properties of the conductance depend only weakly on the coupling to the external leads and, for very large L, become identical with those of quasi-one-dimensional conductors in the weak localization limit. On the other hand, for L < L₀, the coupling to the leads strongly affects the behaviour of both the average and the variance of the conductance. The magnitude of L₀ is typically several magnitudes of ten times the elastic mean free path and thus comparable to the sizes of experimental devices. A further novel aspect of our work is the demonstration that the assumption of GOE statistics for the Hamiltonian is sufficient to yield universal conductance fluctuations.