Design and characterization of electronic fractals

The dimensionality of an electronic quantum system is decisive for its properties. In 1D electrons form a Luttinger liquid and in 2D they exhibit the quantum Hall effect. However, very little is known about the behavior of electrons in non-integer, i.e. fractional dimensions. Here, we show how arrays of artificial atoms can be defined by controlled positioning of CO molecules on a Cu(111) surface, and how these sites couple to form electronic Sierpinski fractals. We characterize the electron wavefunctions at different energies with scanning tunneling microscopy and spectroscopy and show that they inherit the fractional dimension. Wavefunctions delocalized over the Sierpinski structure decompose into self-similar parts at higher energy, and this scale invariance can also be retrieved in reciprocal space. Our results show that electronic quantum fractals can be man-made by atomic manipulation in a scanning tunneling microscope. The same methodology will allow to address fundamental questions on the effects of spin-orbit interaction and a magnetic field on electrons in non-integer dimensions. Moreover, the rational concept of artificial atoms can readily be transferred to planar semiconductor electronics, allowing for the exploration of electrons in a well-defined fractal geometry, including interactions and external fields.