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 behavior 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) surface2–4, and how these sites couple to form electronic Sierpiński 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 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 tunneling microscope. The same methodology will allow future study to address fundamental questions about 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.

Hall conductivity of a Sierpiński carpet

“…In that time, studies of quantum properties of fractal structures was of purely fundamental interest. Recent progress in technologies can produce fractals by both nanofabrication methods and manipulations with individual molecules on metal surfaces [5][6][7][8]. This creates a boost to new researches in the field.…”

Symmetry breaking in molecular artificial graphene

“…The unique physics of Dirac quasiparticles can be mimicked in artificial graphene (AG) systems [1]. These AG systems include cold atom lattices [2][3][4][5][6], phononic crystals [7][8][9][10], photonic crystals [11][12][13][14][15][16], semiconductor nanopatterns [17][18][19][20][21][22] and molecular lattices assembled on metal surfaces, termed as molecular designers [23][24][25][26][27][28][29][30][31][32][33][34]. The tunability of the artificial systems makes them ideal playgrounds to exploit phenomena that are extremely challenging to access in real materials.…”

“…The tunability of the artificial systems makes them ideal playgrounds to exploit phenomena that are extremely challenging to access in real materials. For example, Kekulé structure, graphene pn junctions, deformed graphene, graphene nanoribbons, point and line defects, topological domain wall states, Lieb lattice, quasicrystalline structure, and fractal electronics have been realized in molecular designers [23][24][25][26][27][28][29][30]. These systems exhibit exciting physics such as Gauge field, edge states and flat band [35][36][37][38], to name a few.…”

Energy levels in a self-similar fractal cluster

“…The experimental techniques for the fabrication and analysis of fractal structured materials have evolved considerably since a fractal molecule was synthesized by Newkome et al using molecular self-assembly techniques [1]. Recently, several studies on atomic clusters with fractal geometry have been reported [2][3][4][5][6]. It is expected that further advances in nanotechnology will enable a variety of new fractal structured materials to be fabricated.…”