Abstract: Conductivity properties of disordered graphene
Active carbon is a common material---think of charcoal or of water filters. It was recently realized that it is mostly made of graphene sheets. Graphene is a two-dimensional carbon crystal with hexagonal symmetry. In active carbon, curved pieces of graphene and, occasionally, some other atoms, are combined in a disordered way to form a three-dimensional material which has curious electrical and magnetic properties. In order to explain them, we need to understand how disordered graphene conducts electricity. I will give an introduction to this problem, first talking about electron movement in a disordered material and about its mathematical model—-random Schrödinger equation. I will then modify it to account for special properties of graphene to obtain the two-dimensional random Dirac equation, the main mathematical object of the talk. Its one-dimensional version leads to simply stated analytical problems which will be helpful in understanding the two-dimensional case. The latter needs to be studied numerically. The goal is to understand curious properties of active carbon and related disordered systems. The project offers interesting research questions for graduate students. This is a joint project with Marek Dudyński, a physicist based in Warsaw, Poland.