We introduce a stable, accurate method for calculating the quantum-mechanical transmittance of random media. A Hamiltonian is constructed for a system consisting of a sample with a few simple, semi-infinite leads. This Hamiltonian is transformed into a block-tridiagonal matrix. Three-term matrix recurrences are then used to find the scattering matrix for electron waves impinging on the sample from the leads. In calculations for narrow wires described by the Anderson model we observe nearly transparent resonances in the transmittance as a function of energy in nearly all cases examined; the mean of the logarithm of the transmittance scales linearly with system length even for very short length scales, where resonances dominate the distribution. We also find agreement with previous results, including the statistics of the transmittances of an ensemble of wires and analytically predicted localization lengths. These methods are easily applicable to two- and three-dimensional systems, as well as four-lead devices.

• Received 28 April 1988

DOI:https://doi.org/10.1103/PhysRevB.38.5237