Quantum Interference in Disordered Electron Systems

Abstract

While resistance in three-dimensional metals is rather well described by the (classical) Boltzmann theory one finds clear deviation from this description in two dimensions. The anomalous behavior of the resistance in two dimensions was first pointed out by Abrahams et al. ⁽¹⁾ Anderson et al. ⁽²⁾ and Gorkov et al. ⁽³⁾ showed that at low but finite temperatures the conductance is not constant but has a temperature-dependent correction and Altshuler et al. ⁽⁴⁾ calculated a strong magnetoresistance. This effect, which is generally called weak localization, is more than just a new anomaly in the resistance of low-dimensional systems. Weak localization corresponds to a time of flight experiment with conduction electrons. As we will see below a magnetic field allows observation of the scattering of the electrons as a function of time. The correction to the conductance is generally calculated in the Kubo formalism and is given by the fan diagram shown in Figure 5.1a. The corresponding scattering process is described in Figure 5.1b in k-space. We consider at the time t = 0 an electron of momentum k which has the wave function exp(i kr). The electron k is scattered after the time τ₀ into a state k′₁, after 2τ₀ into the state k′₂, etc.