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Design and characterization of electrons in a fractal geometry

An Author Correction to this article was published on 09 July 2021

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The dimensionality of an electronic quantum system is decisive for its properties. In one dimension, electrons form a Luttinger liquid, and in two dimensions, they exhibit the quantum Hall effect. However, very little is known about the behaviour of electrons in non-integer, or fractional dimensions1. Here, we show how arrays of artificial atoms can be defined by controlled positioning of CO molecules on a Cu (111) surface2,3,4, and how these sites couple to form electronic Sierpiński fractals. We characterize the electron wavefunctions at different energies with scanning tunnelling microscopy and spectroscopy, and show that they inherit the fractional dimension. Wavefunctions delocalized over the Sierpiński 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 artificially created by atomic manipulation in a scanning tunnelling microscope. The same methodology will allow future studies to address fundamental questions about the effects of spin–orbit interactions and magnetic fields 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.

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Fig. 1: Geometry of the Sierpiński triangle fractal.
Fig. 2: Wavefunction mapping.
Fig. 3: Fractal dimension of the Sierpiński wavefunction maps.
Fig. 4: Fourier analysis of wavefunction maps.

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Data availability

All data is available from the corresponding authors upon reasonable request. The experimental data can be accessed using open-source tools.

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We thank G.C.P. van Miert for the discussions. We acknowledge funding from NWO via grants 16PR3245 and DDC13, as well as an ERC Advanced Grant ‘FIRSTSTEP’ 692691.

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Authors and Affiliations



S.N.K. did the calculations under the supervision of C.M.S. The experiments were performed by M.R.S. with contributions from S.E.F. and S.J.M.Z. under the supervision of I.S. and D.V. All authors contributed to the interpretation of the data and to the manuscript.

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Correspondence to I. Swart or C. Morais Smith.

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Supplementary Information

Supplementary Figures 1–18, mathematical derivations, and Supplementary References 1–29

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Kempkes, S.N., Slot, M.R., Freeney, S.E. et al. Design and characterization of electrons in a fractal geometry. Nat. Phys. 15, 127–131 (2019).

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