Electron transmission through the impurity band of a mesoscopic semiconductor quantum wire

The overlap between the states of electrons bound to different shallow impurities randomly distributed along the center of a semiconductor quantum wire leads to an impurity band. The T=0 electron transmission probability through such a band is calculated for a finite length of this disordered quantum channel sandwiched between perfect conductors. For comparison, a density-of-states calculation is made for the finite portion of disordered wire. It is found that transmission probability, in the region of the impurity band, increases both with increasing concentration of donors and decreasing confinement, and decreases with increasing length. It exhibits typical conductance fluctuations, which are also studied as a function of concentration and the length and width of the confining well. In the metallic regime of large concentrations, the fluctuations become independent of impurity concentration and sample length, as the universal conductance fluctuations. It can be thought that a somewhat poorly defined metal-insulator crossover takes place with increasing concentration, for a given fixed length of the quantum wire. We estimate, for example, that a 300×300 A${\mathrm{̊}}^2$ wire of GaAs doped with 0.5×${10}^{18}$ Si atoms/${\mathrm{cm}}^3$ should show no impurity conductance when longer than 0.4 μm.