Superlocalization of Electrons and Waves in Fractal Media

Abstract

We show that impurity quantum states and Anderson localized states exhibit "superlocalization" in fractal media, that is their wave function decays faster than exponentially with distance. More precisely, the wave function decays at large distances like exp [- |x|a] with a > 1. In the specific case of the quantum percolation problem at the percolation concentration pc, we argue that a ≈ 1.9 for d = 3. Analogous results hold for waves. Applications are suggested to disordered and amorphous materials and to porous materials.