Conductance of an array of elastic scatterers: A scattering-matrix approach

In the past, the conductance of disordered systems has been extensively studied with use of the Anderson tight-binding Hamiltonian. In this paper we use a different model, which views the semiconductor as regions of free propagation with occasional elastic scattering by a random array of scatterers. Each impurity is characterized by a scattering matrix which can, in principle, be derived for any arbitrary scattering potential. The randomness is introduced through the impurity location. The overall scattering matrix of the device is calculated by combining (using the appropriate law of composition) the scattering matrices of successive sections. The conductance is then evaluated with use of the multichannel Landauer formula. One advantage of this approach is that the quantum conductance can be compared with the semiclassical conductance, which is determined by combining the probability scattering matrices obtained by replacing each element of the (amplitude) scattering matrices by its squared magnitude. This comparison allows us to see clearly the effects of quantum interference. Numerical examples illustrating the onset of weak and strong localization, as well as conductance fluctuations, are presented. Even for samples shorter than the electron elastic mean free path, the size of the conductance fluctuations is close to the universal value if the two-probe conductance formula is used, though it is much larger when the four-probe formula is used.