Electronic properties of fractal-glass models

Abstract

Analytic results are presented for four new fractal lattices. A basic similarity between fractals and homogeneous glassy networks is the fluctuating environments of the lattice sites. The fractals are classified in terms of their local bonding geometry by comparison with glassy networks: (1) amorphous graphite, (2) two-dimensional and (3) three-dimensional Zachariasen glasses of two components, and (4) a three-component, infinitely ramified glass model with fractal dimension d\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\approxeq}3 and spectral dimension d\ifmmode\bar\else\textasciimacron\fi{}\ifmmode\bar\else\textasciimacron\fi{}>2.5. Localization and size scaling of corner-to-corner propagation are investigated, and it is found that there is a transition from power-law to exponential dependence in the case of model (1) at a correlation length \ensuremath{\lambda}\ensuremath{\approxeq}64. Edge states appear to play an important role in propagation over distances longer than \ensuremath{\lambda}. Effects of ring closure and site variety are studied by making two modifications to model (1). In the first, branch cuts unroll the ring structure into a Bethe lattice. In the second, the rings are severed, resulting in a fractal tree. Results suggest that noncontinuum spectral structure is more closely related to site variety than to connectedness or ring closure in these fractal-glass models. This is similar to Anderson localization in the homogeneous random case.