On Complex Spin Textures, Majorana Modes, and Machida-Shibata States - Exploring Nano-Scale Systems with Tight-Binding Models

Abstract

The search for Majorana zero modes has been a major undertaking in the field of solid state physics in recent years, as they have potentially promising applications in fault-tolerant quantum computing. On the theory side, tight-binding models combining magnetism, Rashba spin-orbit coupling, and superconductivity have been on the forefront of this quest. They feature all ingredients necessary for experimental realization of Majorana zero modes. In this thesis, we take such a model, adapt and apply it to vastly different problems and geometries, while leaving its core intact. We explore its magnetic ground state in one and two dimensions, showing a surprisingly rich magnetic phase space. By self-consistently calculating the magnetic ground states before identifying electronic topological phases, we demonstrate in one dimension that, for a significant portion of the phase space, spin-spirals and topologically non-trivial states naturally coexist. We develop a computationally highly efficient approach to find the magnetic ground states of tight-binding models with a fitted classical spin model that is not inherently limited by assumptions, like RKKY being limited to weak magnetic couplings, and also grants additional insight into the driving magnetic forces of the system. With this method, we completely characterize the magnetic parameter space of our two-dimensional tight-binding model, showing that also 2D systems feature a rich magnetic phase diagram with many exotic magnetic phases despite the simplicity of the model at first glance. As the amount of data points is quite large and the magnetic phases are too complex to be classified by a simple algorithm, we employ an artificial neural network to classify the magnetic phases, thereby demonstrating that artificial neural networks can be a useful tool for the classification of magnetic ground states. Additionally, we provide theoretical support for the first experimental measurement of simultaneous zero-bias-peaks at both ends of an atomic chain, which marks a milestone in the search for Majoranas. In the experiment, an atomic Mn chain was constructed on Nb(110) in the [1¯10]-direction to build hybridizing Yu-Shiba-Rusinov states in a bottom-up approach. To model this, we adapt our tight-binding model to three dimensions. Replicating the experimental geometry, we are able to predict the chain length at which the found zero-modes are expected to evolve into isolated Majorana zero modes and demonstrate the crucial role of the strength of Rashba spin-orbit coupling. Finally, we provide a geometrically correct model for experiments on quantum dots caged by a box of Ag adatoms on a Ag island on superconducting Nb. With these experiments, the existence of energetically sharp non-magnetic in-gap states has been shown for the first time. This confirms theoretical predictions of Machida and Shibata on spin-degenerate Andreev bound states from 1972.