Electronic states in ordered and disordered quantum networks: with applications to graphene and to boron nanotubes

The idea behind the original quantum network (QN) model is simple enough. One joins each atom to its nearest neighbours, and then treats electrons (though quantum mechanically of course) as though they flowed through one-dimensional wires as in an electrical circuit obeying Kirchhoff's Laws at every node. Here we will begin with two periodic systems: namely a single graphene layer, which has recently been produced experimentally, and a two-dimensional sheet of boron atoms. This will be followed by a discussion of B nanotubes, using the simplest QN model, supplemented by comparison of these results with very recent work of other authors using density functional theory. Then the disordered quantum network (DQN) model will be treated in some detail. First of all, the main, physically motivated, steps by which Dancz, Edwards and March passed from the DQN model to the Boltzmann equation will be set out. They will then be related to substantial progress made on the mathematical solution of the DQN model by a number of authors; again a substantial part of this work invoking the Boltzmann equation.