Ballistic transport of electrons in a long single-mode 1-D channel

The dependence of coherent transport through a very long 1-D single-mode quantum wire on the influence of a remote doped layer is determined for three cases: (i) the donors in the layer are completely ionised, unscreened and randomly (Poisson) distributed; (ii) incomplete ionisation of the donors resulting in a more regular distribution of the ions; and (iii) the ionised layer is screened by a conducting layer in close proximity. In (i) the random potential in the wire is shown to behave like a marginal fractal with fluctuations increasing without limit as the length of the wire increases, resulting in strong scattering and short mean-free-paths. By contrast, for both cases (ii) and (iii) the charge distributions are approximated well by random dipoles giving rise to an outer scale in the potential fluctuations in the wire. This greatly reduces the scattering in long wires resulting in mean-free-paths which are consistent with experiment. In all cases, in contrast with metals, the Born approximation is not generally applicable for ballistic transport along the wire since potential fluctuations can be comparable with, or even exceed, the kinetic energy of the electrons. A more appropriate estimation of the mean-free-path is to determine the distance along the wire for which the r.m.s. fluctuation in potential is equal to the electron's kinetic energy. This results in conductance behaviour near threshold that is qualitatively different from that given by the usual localisation theory appropriate to metals.