Random walks in an exponential band

The behavior of a single random walker on a cubic lattice with energetic disorder is simulated. Site energies are selected at random from an exponential distribution, simulating thermally activated hopping in the Urbach band tail of a disordered semiconductor. The apparent fracton dimension d of the system [from S(t)∝$t^{d/2\mathrm{}}$, where S(t) is the number of distinct sites visited] varies linearly with temperature for up to 3×${10}^5$ simulation time steps. A separate examination of site visitation as a function of successful hop attempts and the hopping rate reveals more complex behavior.