Implementation of the Su-Schrieffer-Heeger Model in the Self-Assembly Si-In Atomic Chains on the Si(553)-Au Surface.

Indium-decorated Si atomic chains on a stepped Si(553)-Au substrate are proposed as an extended Su-Schrieffer-Heeger (SSH) model, revealing topological end states. An appropriate amount of In atoms on the Si(553)-Au surface induce the self-assembly formation of trimer SSH chains, where the chain unit cell comprises one In atom and two Si atoms, confirmed by scanning tunneling microscopy images and density functional calculations. The electronic structure of the system, examined through scanning tunneling spectroscopy, manifests three electron bands within the Si-In chain, accompanied by additional midgap topological states exclusively appearing at the chain's end atoms. To elucidate the emergence of these topological states, a tight-binding model for a finite-length-extended SSH chain is proposed. Analysis of the energy spectra, density of states functions, and eigenfunctions demonstrates the topological nature of these self-assembled atomic chains.

Distribution of Electron Density in Self-Assembled One-Dimensional Chains of Si Atoms.

Scanning tunneling microscopy measurements of height profiles, along the chains of Si atoms on the terrace edges of a perfectly ordered Si(553)-Au surface, reveal an STM bias-dependent mixed periodicity with periods of one, two and one and a half lattice constants. The simple linear chain model usually observed with STM cannot explain the unexpected fractional periodicity in the height profile. It was found that the edge Si chain stands for, in fact, a zigzag structure, which is composed of two neighboring rows of Si atoms and was detected in the STM experiments. Tight-binding calculations of the local density of states and charge occupancy along the chain explain the voltage-dependent modulations of the STM profiles and show that oscillation periods are determined mainly by the surface and STM tip Fermi energies.

Braid Plot-A Mixed Palette Plotting Method as an Extension of Contour Plot.

In this article, we propose a new method of three-dimensional data plotting based on the use of mixed hue palettes, which makes it possible to distinguish simultaneously both huge and subtle changes in the value of the presented quantity at the same plot. This method called "braid plot" is based on the alternating use of multiple palettes of colors (a kind of interlacing), which greatly increases the sharpness of the graph and allows us to define areas of equal values more accurately than using traditional graphs with a single palette or contour plot. We present here an algorithm of preparing braid plot composed of any number of initial color sets. As a result of using this type of plot, it was possible to detect, e.g., weak perturbation effects or subtle oscillations of the spectral density function, which is very hard to observe using classical plots.

Topological Atomic Chains on 2D Hybrid Structure.

Mid-gap 1D topological states and their electronic properties on different 2D hybrid structures are investigated using the tight binding Hamiltonian and the Green's function technique. There are considered straight armchair-edge and zig-zag Su-Schrieffer-Heeger (SSH) chains coupled with real 2D electrodes which density of states (DOS) are characterized by the van Hove singularities. In this work, it is shown that such 2D substrates substantially influence topological states end evoke strong asymmetry in their on-site energetic structures, as well as essential modifications of the spectral density function (local DOS) along the chain. In the presence of the surface singularities the SSH topological state is split, or it is strongly localized and becomes dispersionless (tends to the atomic limit). Additionally, in the vicinity of the surface DOS edges this state is asymmetrical and consists of a wide bulk part together with a sharp localized peak in its local DOS structure. Different zig-zag and armachair-edge configurations of the chain show the spatial asymmetry in the chain local DOS; thus, topological edge states at both chain ends can appear for different energies. These new effects cannot be observed for ideal wide band limit electrodes but they concern 1D topological states coupled with real 2D hybrid structures.

DOS cones along atomic chains.

The electron transport properties of a linear atomic chain are studied theoretically within the tight-binding Hamiltonian and the Green's function method. Variations of the local density of states (DOS) along the chain are investigated. They are crucial in scanning tunnelling experiments and give important insight into the electron transport mechanism and charge distribution inside chains. It is found that depending on the chain parity the local DOS at the Fermi level can form cone-like structures (DOS cones) along the chain. The general condition for the local DOS oscillations is obtained and the linear behaviour of the local density function is confirmed analytically. DOS cones are characterized by a linear decay towards the chain which is in contrast to the propagation properties of charge density waves, end states and Friedel oscillations in one-dimensional systems. We find that DOS cones can appear due to non-resonant electron transport, the spin-orbit scattering or for chains fabricated on a substrate with localized electrons. It is also shown that for imperfect chains (e.g. with a reduced coupling strength between two neighboring sites) a diamond-like structure of the local DOS along the chain appears.