Abstract:

This thesis presents a study of several important aspects of quantum solids with a low-density of quenched (i.e. non-mobile) impurities, using lightly doped semiconductors as a paradigm. Considered first are the electronic states of impurities, which form two bands of single-particle states separated by a gap. Electron-electron interactions are neglected, and the one-electron states are obtained by exact diagonalization of a disordered tight binding Hamiltonian. We find that randomness in the positions of the impurities drastically alters the single-particle spectrum, evidenced by the density of states and inverse participation ratio characterizing the electronic states. We analyze features in the density of states by comparing exact diagonalization results to pair approximations and renormalization group methods, and find that while pair approximations are accurate at low densities, more complex effects captured by renormalization group techniques must be included to explain the asymmetry and quenched singularity in the density of states at intermediate and high density.

Next, we add electron-electron interactions, and the many-body states of small systems are found by exact solution of a generalized Hubbard model. The resulting wavefunctions are used to analyze the magnetic behavior of finite lattices and clusters, specifically with regard to ferromagnetism. If it were not for the positional randomness of the dopants, the Hubbard model with parameters appropriate for doped semiconductors would be expected to have a ferromagnetic ground state at low dopant concentration. In reality, experiments do not find any signs of ferromagnetism, and attribute this to disorder localizing would-be carriers. In this work, a variant of the Hubbard model that captures the intrinsic electron-hole asymmetry of (at least) the doped semiconductor problem is studied, and it is found that certain small systems above half-filling (more electrons than dopants) show ferromagnetic behavior even in the presence of various types and strengths of positional randomness. We analyze the dependence of the ground state's total spin on factors such as system size, density, and electron-/hole-doping.

Lastly, the transport properties of systems with randomly positioned impurities are studied. The systems considered are assumed to have a ferromagnetic ground state, a supposition motivated by preceding work, and one that allows the generalized Hubbard model to be simplified to a single-electron model. This model is then solved for large system sizes (thousands of sites) using exact diagonalization. The mobility edges, which mark the energies at which single particle states become delocalized, are identified and used to determine the gap to activated transport as a function of electron- and hole-doping. We find, due to the asymmetric shape of the impurity bands, that even in systems substantially below half-filling, the dominant carriers are electrons. Also, results are compared with experimental values and are found to give a much better prediction of the activation gap than results from the analogous problem but where the sites are assumed to be on a lattice.