Disordered Systems and the Metal-Insulator Transition
The most significant situation in which disordered materials differ from crystals is in transport. Instead of wave-like Bloch states, we can divide the states into localised and extended. The former are confined to a finite volume of the system and normally do not participate in transport, whereas extended states stretch throgh the whole system. Thus at zero temperature there can only be transport when the Fermi energy lies in a region of extended states. This is an example of a quantum phase transition: a phase transition at T=0 driven by some other parameter, such as disorder or energy. This transition is often called the Metal-Insulator transition, or sometimes Anderson transition.
Nano Systems and Non-Equlibrium Green's Functions
Electronic transport through nano-scale systems usually takes the form of a small device attached to 2 or more metal;lic leads. The device may be a single molecule or some small semiconductor device, for example. One impportant characteristic of such systems is that the difference in Fermi energies beteween 2 leads may be larger than other energy scales in the problem and is hence not amenable to perturbation theory. Based on the non-equilibrium Green's function formalism, originally introduced by Keldysh it is possible to develop a consistent description of such systems.