# Dr. William Schwalm

Department of Physics and Astrophysics

UND

**Anderson Localization in Fractal Structures**

Crystalline materials have been studied since the beginning of quantum mechanics.
Traveling waves propagate in a periodic structure within certain allowed frequency
bands, and these Bloch waves permit electric charge transport. In the limit of an
infinitely large crystal structure, the allowed energy bands that permit conduction
comprise a continuum, i.e. the bands are continua, or continuous ranges of energy.
In real materials, even when they are crystalline, there is always disorder. Quite
some time ago, P. W. Anderson showed that in many cases at absolute zero temperature,
a sufficient amount of disorder can cause the wave functions of electrons, or other
wave-like disturbances, to localize and thus it can cut off conduction. This phenomenon
is called Anderson localization when it is due mostly to the incoherent elastic scattering
from disorder in the structure. Often the localization phenomena competes with superconductivity,
so that as *T* goes to zero, one phenomenon competes with the other, and the conductivity may either
go toward zero or else diverge. Recently, there was a colloquium by Alan Goldman on
a related topic.

In this talk, I review some aspects of Anderson localization, particularly the basic idea of wave localization by superposition of incommensurate reflections. The scaling theory of Abrahams, Anderson, Licciardello, and Ramakrishnan (Phys. Rev. Lett. 42, 673, 1979) predicts that dimensionality will have a controlling influence on localization. Based on a few simple assumptions predicts that for 1 or 2 dimensional systems at zero temperature, any degree of disorder will cause all the states to be localized on some length scale. The scaling theory will be reviewed and I’ll also review some work of Henn Soonpaa on thin crystals. (I will not have much to say about the competition between this prediction and the onset of superconductivity.)

The main portion of the talk will concern simple model calculations on fractal structures. A fractal is a geometrical or physical structure that is self-similar—meaning that it when broken into parts, the parts are geometrically similar to the original structure rescaled in size. An example would be a piece of broccoli. Broccoli is approximately fractal over a certain range of length scales. I will mention some work on fractal structure in nature by Harold Bale and Paul Schmidt, and will introduce some simple toy model systems. Some time ago Mizuho and I studied length scaling of conductance statistics in an ensemble of fractal structures. These systems are amenable to a renormalization treatment where one can do calculations more or less exactly. The technique is very simple and will be described fully. Brian Moritz and I found a regular fractal structure with a continuous spectrum. This is very unusual, and the connection to conductance scaling in this case is not yet clear. The connection between (or lack thereof) a continuous energy spectrum and conductivity will be discussed, and several calculations will be described and the resulting properties discussed. Albert Schmitz is currently working on some model calculations that may shed some more light on this relationship. Recent results will be presented for these as well.