Optical-transition-matrix elements between localized electronic states in disordered one-dimensional systems.

Abstract

We study the properties of optical matrix elements between localized states in one-dimensional disordered systems. The localized states are obtained either from a two-band Anderson Hamiltonian with diagonal disorder or from a Hamiltonian in real-space representation. In the latter case the disorder is described by a chain of Gaussian potentials with statistically varying amplitudes, where the Gaussians are centered at the points of a periodic lattice. The calculations of the localized eigenstates were performed for finite chains, where the chain length was chosen sufficiently large, such that the results are independent of the imposed boundary conditions. We show that the eigenfunctions are not only localized in r space, but also in k space. The latter property is found through the decomposition of the localized eigenfunctions in terms of the Bloch states of the corresponding virtual crystals. The center of localization in k space varies with energy similar to the E(k) relation in the virtual-crystal case. The behavior of the calculated optical matrix elements is strongly influenced by the above localization properties of the eigenfunctions. We find that optical transitions between localized eigenfunctions take place only if the involved wave functions overlap sufficiently in real space as well as in k space.