Quantum corrections to the classical Boltzmann results for transport coefficients in disordered conductors can be systematically studied in the expansion parameter ħ /pFl, the ratio of the Fermi wavelength and the impurity mean free path, which typically is small in metals and semiconductors. The quantum corrections due to disorder are of two kinds, one being the change in interactions effects due to disorder, and the other having its origin in the tendency to localization. When it comes to an indiscriminate probing of a system, such as the temperature dependence of its resistivity, both mechanisms are effective, whereas when it comes to the low-field magneto-resistance only the weak localization effect is operative, and it has therefore become an important diagnostic tool in material science. We start by discussing the phenomena of localization and (especially weak localization) before turning to study the influence of disorder on interaction effects.

Localization

In this section the quantum mechanical motion of a particle at zero temperature in a random potential is addressed. In a seminal paper of 1958, P. W. Anderson showed that a particle's motion in a sufficiently disordered three-dimensional system behaves quite differently from that predicted by classical physics according to the Boltzmann theory [71]. In fact, at zero temperature diffusion will be absent, as particle states are localized in space because of the random potential. A sufficiently disordered system therefore behaves as an insulator and not as a conductor. By changing the impurity concentration, a transition from metallic to insulating behavior occurs, the Anderson metal–insulator transition.