Anderson localization induced by gauge-invariant bond-sign disorder in square PbSe nanocrystal lattices

Abstract

We study theoretically the problem of electrons moving on a two-dimensional square lattice characterized by nearest-neighbor hopping terms of constant amplitude but random sign. The original motivation came from the discovery that this ``bond-sign'' disorder can be present in square lattices of epitaxially connected PbSe nanocrystals, which have been recently synthesized using colloidal routes. We investigate how this type of disorder tends to localize the electronic wave-functions and modifies the electronic structure. This is done via the calculation of the density-of-states, the participation ratio and the localization length. We show that, when the relative fraction $p$ of negative signs increases from 0% to 50%, the effect of the disorder on the wave functions saturates at a constant level when $p$ reaches values above $\ensuremath{\sim}25%$. This behavior reveals that the true disorder experienced by the electrons is not the nominal disorder defined by $p$ but a smaller part of it, which is irreducible due to frustrations. The amount of true disorder can be obtained by successive local gauge transformations as developed in the past to study models of spin glasses. In the thermodynamic limit, this irreducible gauge-invariant disorder induces localization of all electronic states, except at the center of the band where our calculations suggest that zero-energy states have a critical behavior. The particle-hole symmetry, which characterizes these disordered systems plays a crucial role in this behavior, as already found in lattices with random hopping or random magnetic flux, for example. In the case of lattices of PbSe nanocrystals, the effects of the bond-sign disorder are found to be weaker than those of more conventional types of disorder.