Anniversary of Hopping Paper

Thursday, January 14, 2010 is the 25^th anniversary of one of my first scientific papers, Hopping in Exponential Band Tails, in Physical Review Letters. It came out just as I arrived at Bell Labs.

It still surprises me that this paper has gotten nearly 200 citations, and that they continue to dribble in even now. Most papers are surpassed by new developments within a few years of publication. In this case, I stumbled on a useful but very accessible concept that people can easily wrap their head around. But I'd guess that most people that cite it have never read it.

The paper concerns motion of electrons in amorphous semiconductors, that is, semiconductors without a crystalline lattice. The best known example is the amorphous silicon alloys that are used for cheap solar cells.

Until the 1960s, some physicists questioned whether amorphous semiconductors could even exist (although they clearly did), because the quantum-mechanical understanding of semiconductors depended on the mathematical properties of wavelike electrons moving among the regularly-spaced atoms in a crystal. For electrons in some range of energies, the electron waves that diffract from the atoms destructively interfere, creating a bandgap with no electron states. At other energies, where there are electron states, they extend through the entire crystal. None of this mathematical framework for understanding semiconductors seemed to work unless the atoms were arranged in a regular crystal.

Phil Anderson, then at Bell Labs, showed in 1958 that if atoms were arranged in an irregular pattern, electronic states could exist, but be localized near particular atoms. Neville Mott and others suggested that in an amorphous semiconductors, electrons would be localized over some range of energies but extend infinite distances at higher or lower energies. The energies that separated the localized and extended states, which have the character of a phase transition, were called "mobility edges." If one conceptually replaces the band gap of crystalline semiconductors with the gap between mobility edges, then the mathematical treatment of amorphous semiconductors looks very familiar. Anderson and Mott shared the 1977 Physics Nobel with John van Vleck for their discoveries. Instead of being denigrated as "dirt physics," disorder is now a perennial topic in condensed matter physics

In Mark Kastner's group at MIT, we were studying what happened to optically generated electrons in the "band tails": the localized states near the mobility edge, whose number decreases exponentially into the gap. Based on some experiments I had done, I proposed that, at low temperatures, electrons would at first simply "hop" from one localized state to another, avoiding the extended states at the mobility edge altogether. Later on, as they moved to energies where the states where farther and farther apart, they would find it faster to hop up to where there were more states--but not all the way back to the mobility edge.

I called the energy to which electrons hopped--and where they could move easily--the "transport energy," and used a simple model to calculate how this energy varies with temperature.

If once conceptually replaces the band gap of crystalline semiconductors with the gap between transport energies, then the mathematical treatment of amorphous semiconductors looks very familiar. There are some important differences, though. For example, a magnetic field has a different effect on hopping electrons than on those that are freely moving. But although some details are different, the overall picture of amorphous semiconductors looks much like the pictures used by electrical engineers.

At the time, I was concerned that people would only remember Marc Kastner's name, so he graciously agreed to let me be sole author. I later regretted that selfishness, because anyone who knew Mark could see his style in it, and he certainly helped me to frame the ideas. Such are the follies of youth.