Design and characterization of electrons in a fractal geometry

Abstract

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.

Data availability

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

References

  1. 1.

    Mandelbrot, B. B. The Fractal Geometry of Nature (W. H. Freeman, San Francisco, 1982).

  2. 2.

    Gomes, K. K., Mar, W., Ko, W., Guinea, F. & Manoharan, H. C. Designer Dirac fermions and topological phases in molecular graphene. Nature 483, 306–310 (2012).

    ADS  Article  Google Scholar 

  3. 3.

    Slot, M. R. et al. Experimental realization and characterization of an electronic Lieb lattice. Nat. Phys. 13, 672–676 (2017).

    Article  Google Scholar 

  4. 4.

    Collins, L. C., Witte, T. G., Silverman, R., Green, D. B. & Gomes, K. K. Imaging quasiperiodic electronic states in a synthetic Penrose tiling. Nat. Commun. 8, 15961 (2017).

    ADS  Article  Google Scholar 

  5. 5.

    Newkome, G. R. et al. Nanoassembly of a fractal polymer: A molecular ‘Sierpinski hexagonal gasket’. Science 312, 1782–1785 (2006).

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Yu, B. Analysis of flow in fractal porous media. Appl. Mech. Rev. 61, 050801 (2008).

    ADS  Article  Google Scholar 

  7. 7.

    Dubal, D. P., Ayyad, O., Ruiz, V. & Gomez-Romero, P. Hybrid energy storage: the merging of battery and supercapacitor chemistries. Chem. Soc. Rev. 44, 1777–1790 (2015).

    Article  Google Scholar 

  8. 8.

    Fan, J. A. et al. Fractal design concepts for stretchable electronics. Nat. Commun. 5, 3266 (2014).

    ADS  Article  Google Scholar 

  9. 9.

    Rothemund, P. W., Papadakis, N. & Winfree, E. Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol. 2, e424 (2004).

    Article  Google Scholar 

  10. 10.

    Shang, J. et al. Assembling molecular Sierpiński triangle fractals. Nat. Chem. 7, 389–393 (2015).

    Article  Google Scholar 

  11. 11.

    Li, C. et al. Construction of Sierpiński triangles up to the fifth order. J. Amer. Chem. Soc. 139, 13749–13753 (2017).

    Article  Google Scholar 

  12. 12.

    De Nicola, F. et al. Multiband plasmonic Sierpinski carpet fractal antennas. ACS Photon. 5, 2418–2425 (2018).

    Article  Google Scholar 

  13. 13.

    Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    ADS  Article  Google Scholar 

  14. 14.

    Pan, W. et al. Fractional quantum Hall effect of composite fermions. Phys. Rev. Lett. 90, 016801 (2003).

    ADS  Article  Google Scholar 

  15. 15.

    Goerbig, M. O., Lederer, P. & Smith, C. M. On the self-similarity in quantum Hall systems. Europhys. Lett. 68, 72–78 (2004).

    ADS  Article  Google Scholar 

  16. 16.

    Morgenstern, M., Klijn, J., Meyer, C. & Wiesendanger, R. Real-space observation of drift states in a two-dimensional electron system at high magnetic fields. Phys. Rev. Lett. 90, 056804 (2003).

    ADS  Article  Google Scholar 

  17. 17.

    Richardella, A. et al. Visualizing critical correlations near the metal–insulator transition in Ga1−xMnxAs. Science 327, 665–669 (2010).

    ADS  Article  Google Scholar 

  18. 18.

    Evers, F. & Mirlin, A. D. Anderson transitions. Rev. Mod. Phys. 80, 1355–1417 (2008).

    ADS  Article  Google Scholar 

  19. 19.

    Domany, E., Alexander, S., Bensimon, D. & Kadanoff, L. P. Solutions to the Schrödinger equation on some fractal lattices. Phys. Rev. B 28, 3110–3123 (1983).

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Ghez, J., Wang, Y. Y., Rammal, R., Pannetier, B. & Bellissard, J. Band spectrum for an electron on a Sierpinski gasket in a magnetic field. Solid State Commun. 64, 1291–1294 (1987).

    ADS  Article  Google Scholar 

  21. 21.

    Andrade, R. F. S. & Schellnhuber, H. J. Exact treatment of quantum states on a fractal. Europhys. Lett. 10, 73–78 (1989).

    ADS  Article  Google Scholar 

  22. 22.

    Wang, X. R. Localization in fractal spaces: Exact results on the Sierpinski gasket. Phys. Rev. B 51, 9310–9313 (1995).

    ADS  Article  Google Scholar 

  23. 23.

    van Veen, E., Yuan, S., Katsnelson, M. I., Polini, M. & Tomadin, A. Quantum transport in Sierpinski carpets. Phys. Rev. B 93, 115428 (2016).

    ADS  Article  Google Scholar 

  24. 24.

    Chakrabarti, A. & Bhattacharyya, B. Sierpinski gasket in a magnetic field: electron states and transmission characteristics. Phys. Rev. B 56, 13768–13773 (1997).

    ADS  Article  Google Scholar 

  25. 25.

    Liu, Y., Hou, Z., Hui, P. M. & Sritrakool, W. Electronic transport properties of Sierpinski lattices. Phys. Rev. B 60, 13444–13452 (1999).

    ADS  Article  Google Scholar 

  26. 26.

    Lin, Z., Cao, Y., Liu, Y. & Hui, P. M. Electronic transport properties of Sierpinski lattices in a magnetic field. Phys. Rev. B 66, 045311 (2002).

    ADS  Article  Google Scholar 

  27. 27.

    Crommie, M. F., Lutz, C. P. & Eigler, D. M. Confinement of electrons to quantum corrals on a metal. Surf. Sci. 262, 218–220 (1993).

    Google Scholar 

  28. 28.

    Drost, R., Ojanen, T., Harju, A. & Liljeroth, P. Topological states in engineered atomic lattices. Nat. Phys. 13, 668–671 (2017).

    Article  Google Scholar 

  29. 29.

    Girovsky, J. et al. Emergence of quasiparticle Bloch states in artificial crystals crafted atom-by-atom. SciPost Phys. 2, 020 (2017).

    ADS  Article  Google Scholar 

  30. 30.

    Oftadeh, R., Haghpanah, B., Vella, D., Boudaoud, A. & Vaziri, A. Optimal fractal-like hierarchical honeycombs. Phys. Rev. Lett. 113, 104301 (2014).

    ADS  Article  Google Scholar 

  31. 31.

    Brzezińska, M., Cook, A. M. & Neupert, T. Topology in the Sierpiński–Hofstadter problem. Preprint at https://arXiv.org/abs/1807.00367 (2018).

  32. 32.

    Meyer, G. et al. Controlled manipulation of atoms and small molecules with a low temperature scanning tunneling microscope. Single Mol. 1, 79–86 (2000).

    ADS  Article  Google Scholar 

  33. 33.

    Celotta, R. J. et al. Invited Article: Autonomous assembly of atomically perfect nanostructures using a scanning tunneling microscope. Rev. Sci. Instrum. 85, 121301 (2014).

    ADS  Article  Google Scholar 

  34. 34.

    Sierpiński, W. Sur une courbe dont tout point est un point de ramification. Comptes Rendus Acad. Sci. 160, 302–315 (1915).

    MATH  Google Scholar 

  35. 35.

    Bouligand, G. Sur la notion d’ordre de mesure d’un ensemble plan. Bull. Sci. Math 2, 185–192 (1929).

    MATH  Google Scholar 

  36. 36.

    Foroutan-pour, K., Dutilleul, P. & Smith, D. Advances in the implementation of the box-counting method of fractal dimension estimation. Appl. Math. Comput. 105, 195–210 (1999).

    MATH  Google Scholar 

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Acknowledgements

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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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.

Corresponding authors

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. Nature Phys 15, 127–131 (2019). https://doi.org/10.1038/s41567-018-0328-0

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