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Spectral properties of electrons in fractal nanowires

In view of promising applications of fractal nanostructures, we analyze the spectra of quantum particles in the Sierpinski carpet and study the non-correlated electron gas in this geometry. We show that the spectrum exhibits scale invariance with almost arbitrary spacing between energy levels, including large energy gaps at high energies. These features disappear in the analogous random fractal---where Anderson localization dominates---and in the regular lattice of equally sized holes---where only two length scales are present. The fractal structure amplifies microscopic effects, resulting in the presence of quantum behavior of the electron gas even at high temperatures. Our results demonstrate the potential of fractal nanostructures to improve the light-matter interaction at any frequency, with possible applications, e.g., in the development of solar cells with a wide absorption spectrum, artificial photosynthesis, or nanometamaterials with tailored Fermi levels and band gaps, operating in a wide range of frequencies, and with extended operating temperature range.