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Recent progress in the design and fabrication of artificial two-dimensional(2D) materials paves the way for the experimental realization of electronsystems moving on plane fractals. In this work, we present the results ofcomputer simulations for the conductance and optical absorption spectrum of a2D electron gas roaming on a Sierpinski carpet, i.e. a plane fractal withHausdorff dimension intermediate between one and two. We find that theconductance is sensitive to the spatial location of the leads and that itdisplays fractal fluctuations whose dimension is compatible with the Hausdorffdimension of the sample. Very interestingly, electrons in this fractal displaya broadband optical absorption spectrum, which possesses sharp "molecular"peaks at low photon energies.

Transport and optical properties of an electron gas in a Sierpinski carpet