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.

One of the largest obstacles facing scanning probe microscopy is the constant need to correct flaws in the scanning probe in situ. This is currently a manual, time-consuming process that would benefit greatly from automation. Here we introduce a convolutional neural network protocol that enables automated recognition of a variety of desirable and undesirable scanning probe tip states on both metal and non-metal surfaces. By combining the best performing models into majority voting ensembles, we find that the desirable states of H:Si(100) can be distinguished with a mean precision of 0.89 and an average receiver-operator-characteristic curve area of 0.95. More generally, high and low-quality tips can be distinguished with a mean precision of 0.96 and near perfect area-under-curve of 0.98. With trivial modifications, we also successfully automatically identify undesirable, non-surface-specific states on surfaces of Au(111) and Cu(111). In these cases we find mean precisions of 0.95 and 0.75 and area-under-curves of 0.98 and 0.94, respectively.

The boundary states of topological insulators are thought not to depend on the precise atomic structure of the boundary. A recent theoretical study showed that, for topological crystalline insulators with given bond strengths, topological states should only emerge for certain edge geometries. We experimentally probe this effect by creating artificial Kekulé lattices with different atomically well-defined edge geometries and hopping ratios in a scanning tunneling microscope. Topological edge modes are found to only appear for specific combinations of edge geometry and hopping ratio.

Quantum corrals can be considered artificial atoms. By coupling many quantum corrals together, artificial matter can be created at will. The atomic scale precision with which the quantum corrals can be made grants the ability to tune parameters that are difficult to control in real materials, such as the symmetry of the states that couple, the on-site energy of these states, the hopping strength and the magnitude of the orbital overlap. Here, we systematically investigate the accessible parameter space for the CO on Cu(111) platform by constructing (coupled) quantum corrals of different sizes and shapes. By changing the configuration of the CO molecules that constitute the barrier between two quantum corrals, the hopping integral can be tuned between 0 and -0.3 eV for s- and p-like states, respectively. Incorporation of orbital overlap is essential to account for the experimental observations. Our results aid the design of future artificial lattices.

The band structure and electronic properties of a material are defined by the sort of elements, the atomic registry in the crystal, the dimensions, the presence of spin−orbit coupling, and the electronic interactions. In natural crystals, the interplay of these factors is difficult to unravel, since it is usually not possible to vary one of these factors in an independent way, keeping the others constant. In other words, a complete understanding of complex electronic materials remains challenging to date. The geometry of two-and one-dimensional crystals can be mimicked in artificial lattices. Moreover, geometries that do not exist in nature can be created for the sake of further insight. Such engineered artificial lattices can be better controlled and fine-tuned than natural crystals. This makes it easier to vary the lattice geometry, dimensions, spin−orbit coupling, and interactions independently from each other. Thus, engineering and characterization of artificial lattices can provide unique insights. In this Review, we focus on artificial lattices that are built atom-by-atom on atomically flat metals, using atomic manipulation in a scanning tunneling microscope. Cryogenic scanning tunneling microscopy allows for consecutive creation, microscopic characterization, and band-structure analysis by tunneling spectroscopy, amounting in the analogue quantum simulation of a given lattice type. We first review the physical elements of this method. We then discuss the creation and characterization of artificial atoms and molecules. For the lattices, we review works on honeycomb and Lieb lattices and lattices that result in crystalline topological insulators, such as the Kekuléand "breathing" kagome lattice. Geometric but nonperiodic structures such as electronic quasi-crystals and fractals are discussed as well. Finally, we consider the option to transfer the knowledge gained back to real materials, engineered by geometric patterning of semiconductor quantum wells.