M5 Quantum Transport

Topic

When the dimension of this nanostructure—along the direction of electric current flow—is reduced to below the mean free path of its own conduction electrons, electron transport through it ceases to be diffusive (as described by the classical Ohmic conduction found in other material systems, such as metals or ceramics) and instead becomes quantum transport. In this regime, electrons no longer scatter randomly against defects and impurities while traversing the nanostructure; rather, they pass through it coherently, without losing that quantum coherence along the way. Further reduction of this dimension—well below that same mean free path—causes quantum transport to dominate even more completely over any residual classical Ohmic conduction.

The coherent passage of electrons through the nanostructure, without scattering along the path, constitutes a ballistic conduction channel; in this regime, this ballistic channel gives rise to quantized conductance. The total conductance of the nanostructure no longer varies continuously but instead proceeds in discrete steps—each an integer multiple of the conductance quantum—as each additional ballistic channel opens. The opening of an additional ballistic channel within the nanostructure causes the quantized conductance to rise abruptly by exactly one additional conductance quantum. Conversely, when this nanostructure’s continuity is interrupted by a thin barrier—such as an insulating section or a gap too narrow to completely halt these electrons—the WKB approximation is used to calculate the tunneling probability. This probability determines the likelihood that an electron will appear on the other side, despite lacking the energy to classically overcome the thin barrier; according to the Simmons model, this same probability dictates the tunneling current that actually passes through the barrier. If the barrier is made even thinner, the WKB approximation predicts a higher tunneling probability and, consequently, a higher tunneling current.

Thus, this model—characterized by the quantum transport and quantized conductance with which it was introduced—describes the ideal electrical behavior of the nanostructure. This stands in stark contrast to the classical Ohmic conduction found in macroscopic metals or ceramics: whereas conductance in those macroscopic materials depends continuously on the sample's geometry, here it depends on the integer number of open ballistic channels. However, if the nanostructure is sufficiently small and weakly coupled—such that the charging energy of a single electron becomes significant—this quantized conductance is anomalously interrupted; in that specific case, a different model within the framework describes "Coulomb blockade," a phenomenon not accounted for by the ideal quantum transport model alone.

ID:4460