Recent recent results on artificial lattices
How do electrons behave in fractal dimensions?
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. Geometric fractals, structures with self-repeating patterns and fractional dimensions, are pervasive in the macroscopic world, but were not yet experimentally realized in electronic systems. 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 wave functions at different energies with scanning tunneling microscopy and spectroscopy and show that they inherit the fractional dimension. Wave functions 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 an STM. 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 coupled into a well-defined fractal geometry can readily be transferred to planar semiconductor electronics, allowing for the exploration of fractal electrons, including interactions and external fields.
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Design and characterization of electronic fractals S. N. Kempkes, M.R. Slot, S. E. Freeney, S. J. M. Zevenhuizen, D. Vanmaekelbergh, I. Swart, and C. Morais Smith arXiv, 2, 1372 (2017).Design and characterization of an electronic Lieb lattice
Geometry, whether on the atomic or nanoscale, is a key factor for the electronic band structure of materials. Some specific geometries give rise to novel and potentially useful electronic bands. For example, a honeycomb lattice leads to Dirac-type bands where the charge carriers behave as massless particles. Theoretical predictions are triggering the exploration of novel two-dimensional (2D) geometries, such as graphynes and the kagomé and Lieb lattices. The Lieb lattice is the 2D analogue of the 3D lattice exhibited by perovskites; it is a square-depleted lattice, which is characterized by a band structure featuring Dirac cones intersected by a flat band. Whereas photonic and cold-atom Lieb lattices have been demonstrated, an electronic equivalent in 2D is difficult to realize in an existing material. Here, we report an electronic Lieb lattice formed by the surface state electrons of Cu(111) confined by an array of carbon monoxide molecules positioned with a scanning tunnelling microscope. Using scanning tunnelling microscopy, spectroscopy and wavefunction mapping, we confirm the predicted characteristic electronic structure of the Lieb lattice. The experimental findings are corroborated by muffin-tin and tight-binding calculations. At higher energies, second-order electronic patterns are observed, which are equivalent to a super-Lieb lattice.
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Experimental realization and characterization of an electronic Lieb lattice M.R. Slot, T.S. Gardenier, P.H. Jacobse, G.C.P. van Miert, S.N. Kempkes, S.J.M. Zevenhuizen, C. Morais Smith, D. Vanmaekelbergh and I. SwartNature Physics, 13, 672 (2017).