Electronic Shot Noise in Fractal Conductors

By solving a master equation in the Sierpiński lattice and in a planar random-resistor network, we determine the scaling with size L of the shot noise power P due to elastic scattering in a fractal conductor. We find a power-law scaling P∝L^d_f-2-α, with an exponent depending on the fractal dimension d_f and the anomalous diffusion exponent α. This is the same scaling as the time-averaged current I̅ , which implies that the Fano factor F=P/2eI̅ is scale-independent. We obtain a value of F=1/3 for anomalous diffusion that is the same as for normal diffusion, even if there is no smallest length scale below which the normal diffusion equation holds. The fact that F remains fixed at 1/3 as one crosses the percolation threshold in a random-resistor network may explain recent measurements of a doping-independent Fano factor in a graphene flake.