Ballistic and Hopping Transport

Abstract

The picture of collision-limited (diffusive) transport fails in many cases, as we have seen on the example of electron transport in magnetic fields; see Chapter 10. The other examples considered in this chapter are i) the ballistic transport under the conditions when the size of the sample is comparable to the mean free path length so that electrons experience a few or any scattering events, and ii) the transport in mesoscopic samples at low temperatures, when the inelastic scattering is practically absent. In the absence of inelastic scattering, when the phase coherence takes place, the conductance of the sample is determined by the quantum-mechanical transmission probabilities and depends both on the geometry of the sample and on the scattering potential distribution. The phase memory leads to such phenomena as the localization of electrons in one-dimensional conductors and the Aharonov-Bohm oscillations. The low-temperature current between the regions separated by potential barriers is limited by the quantum-mechanical probability of tunneling transmission. In more complex cases, when the energy and momentum conservation requirements cannot be satisfied simultaneously without involvement of scattering, the tunneling becomes scattering-assisted, and the current is inversely proportional to the scattering time. Next, the tunneling current through small metallic islands appears to be sensitive to the electric charge quantization leading to Coulomb blockade phenomena. The electron transport in all these cases is conveniently treated by introducing the Hamiltonian of tunnel-coupled systems, which describes the low-probability hopping of electrons between the regions where the electrons are in local equilibrium. A similar Hamiltonian describes the localization of electrons in the crystal lattice in the presence of a strong electron-phonon interaction (the polaronic effect) and allows one to calculate the hopping current. This current demonstrates thermal-activation behavior, and its dependence on the strength and frequency of the applied electric field is essentially different from that considered in previous chapters.