Quantum electrical transport in samples of limited dimensions

Abstract

The ability to make electrically conducting structures of ever smaller size by nanofabrication techniques (the playground of mesoscopic physics) has brought with it entry into a wonderful new range of unexpected quantum phenomena. Interpretation of these phenomena requires full recognition of the wave nature of electrons and requires keeping track of the phase coherence of the electron wave functions and/or the discreteness of electron energy levels in samples of interest. Happily, many of the phenomena can be observed through the use of very straightforward experimental probes—commonly the dc electrical conductivity or conductance, and the Hall effect. The phenomena are observed in samples with one or more dimensions comparable to either the electron wavelength (up to 40 nm for carriers at the Fermi energy in some semiconductors) or the inelastic scattering length of the carriers (as large as many microns in some systems at low temperatures). Ohm’s law is no longer a firm guide to current–voltage relationships, and the Drude–Sommerfeld picture of electrical conduction is superseded. Many of the interesting phenomena are seen in samples of either two-dimensional (i.e., a third dimension is of the order of or less than the electron wavelength) or one-dimensional nature (either a tight, short constriction in the conductor or a longer “quantum wire”). In certain one-dimensional structures, one may have ballistic transport between input and output connections, and the quantum character of the electron motion is fully displayed. Planck’s constant h appears in the characteristic quantum of electrical conductance, e2/h. In two dimensions, the addition of a large magnetic field produces the remarkably deep and still somewhat mysterious Quantum Hall Effect, characterized by the quantum of resistance, RK=h/e2=25812.8 Ω. Other examples of the observation of electron interference and diffraction phenomena within solid materials are briefly highlighted. This short tutorial treatment emphasizes observed phenomena rather than details of the theoretical structures used to interpret them.