Mesoscopic and nano-scale systems

Studying the properties of structures where at least one spatial dimension is on the nanometer scale is motivated by many factors from engineering (such as the semiconductor industry’s push towards higher density integrated circuits), chemistry (developing methods to produce particles of well controlled size and structure), biology (using DNA to control the self-assembly of small structures) and theoretical physics (the ability to perform computer calculations on finite-size systems [25]). Systems with sizes in the range of 1 nm to 100 nm often have "mesoscopic" properties because they are larger than individual atoms or small molecules, but not large enough to neglect the statistical fluctuations (typically characterized by 1/ √ N, where N is the number of constituent particles) we routinely neglect in macroscopic systems.1 Perhaps the very first "nanostructures" exploited to bring quantum effects to a macroscopic scale were the naturally forming oxide layers on metallic aluminum. It is relatively straightforward to make a uniform, insulating layer of aluminum oxide of nanometer thickness by exposing fresh aluminum to air. When such an insulator separates two metallic aluminum electrodes, electrons can pass from one electrode to the other by tunneling through this insulating barrier. (The point contact between a metallic needle and a semiconductor crystal, another example of an early "accidental" nano-scale device, was the principal way of detecting electromagnetic waves in the early days of broadcast radio.) Tunnel junctions still play a major role in the design and characterization of nano-devices. In the simplest and yet quite productive model the tunneling current is expressed in terms of the density of electronic states (g1 and g2) and Fermi functions ( f ) on the two sides of the tunnel barrier: