Topologische Elektronik und Materialien
Topological Electronics and Materials

This module provides a detailed overview on a fascinating new class of solid-state materials and a fast growing research area: topological electronics and corresponding quantum materials. The topological classification of solid-state materials revolutionized condensed matter physics in recent years (Nobel prize 2016) and lead to the discovery of novel topological quantum materials and phases. Topologically non-trivial band structures give rise to material properties that are insensitive against external perturbations, which renders topological and quantum  materials a promising and robust platform for novel electronic, spin-electronic and quantum optical applications. This module will introduce the basic concepts of topology in solid-state physics and discuss corresponding electronic and spin-electronic phenomena in quantum materials. A particular focus will be placed on the experimental realization and characterization of atomistic quantum materials with only a few nanometer thickness. The following specific topics will be addressed:

• Historical and topical introduction to topological quantum phenomena and materials.
• Review of solid state-band theory and introduction to modern topological band theory tools, including Berry phase, Berry curvature and Chern numbers.
• Overview, classification, and characteristic properties of the main families of topological materials, including 3D and 2D topological insulators and Weyl semimetals.
• Introduction to topological electronic phenomena, such as the quantum Hall effect, the (quantum) anomalous Hall effect, and the (quantum) spin Hall effect. 
• Nanofabrication and preparation methods applicable to topological materials, such as Bi₂Se₃ or few-layer and monolayer WTe₂.
• Nanoanalytical methods specialized to topological materials. This will include polarization-resolved optoelectronic studies, magneto-electronic transport experiments, atomic force, scanning tunneling, scanning electron microscopy, enhanced X-ray methods resolving the non-trivial band structure.
• Discussion of potential device application for selected topological materials in the area of electronics, spintronics as well as quantum information processing.
• Focus topics to introduce peculiar properties of selected materials in more detail:

1. Crystal structures and space groups in topological materials.
2. Broken time- and space-symmetries and corresponding Hamiltonians.
3. Berry curvature induced anomalous velocity and its influence on the transversal electron conductivity and corresponding spin polarization.

Furthermore, the students will become familiar with selected recent research papers and review articles in high-impact research journals such as Science, Nature publishing group and further literature related to topological quantum materials. The students are being trained in how to access and extract the information from those articles.

After a successful participation of the module, the student is able to:

1. understand different classes of topological materials and to apply the classification scheme to further solid state materials.
2. evaluate the non-trivial band-structure of selected topological materials.
3. understand the preparation and nanofabrication methods for topological materials and to evaluate suitable methodologies for novel quantum materials.
4. understand optical and structural characterization methods for topological materials, to analyze related results in recent literature and to apply suitable methodologies for given problems related to topological materials.
5. remember and explain magnetotransport phenomena, such as the quantum spin Hall effect in topological materials.
6. evaluate polarization resolved optoelectronic transport phenomena.
7. understand and discuss applications of topological materials for electronic, optoelectronic, and spintronics devices.
8. access and evaluate the content of topical research articles focusing on selected topics related to quantum material research in high-impact journals.

No preconditions in addition to the requirements for the Master’s program in Physics.

The module consists of a thematically structured lecture and an exercise. Therein the learning content is presented. The different parts of the lectures are cross-linked and thereby, the main physical concepts explained. The link to current research activities will be provided by discussing related and topical research article in high-impact journals. The students are actively involved by direct question and answer periods to better develop their individual understanding and to learn the use of up-to date’s research literature. In the exercise, the learning content is deepened and exercised using problem examples, calculations, examples from recent literature, as well as programming exercies.

Power-point presentation together with handwritten lecture notes based on tablet-PC / beamer presentation (“e-chalk”). For selected topics interactive, numerical simulation tools (based on Python as a state-of-the-art scientific programming language) will be used to further illustrate and visualize mathematical concepts. Additional literature / research articles will be provided in pdf-format. For selected topics, video tutorial will be provided.

All Materials will be available for download until the completion of the repeat exam.

• B.A. Bernevig: Topological Insulators and Topological Superconductors, Princeton University Press, (2013).
• B.A. Bernevig, T.L. Hughes & S.-C. Zhang: Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells, Science 314, 1757 (2006).
• Di Xiao, Ming-Che Chang, and Qian Niu, Berry phase effects on electronic properties , Rev. Mod. Phys. 82, 1959 (2010).
• D. Vanderbilt : Berry phases in electronic structure theory electric polarization, orbital magnetization and topological insulators , Cambridge University Press, (2018).

There will be an oral exam of 25 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using comprehension questions and sample calculations.

For example an assignment in the exam might be:

• Define fabrication methods of topological materials.
• Explain characterization methods of topological materials.
• Explain the (polarized) optoelectronic properties of few-layer topological materials.
• Analyze the electronic fingerprint of (quantum) spin Hall effect in monolayer topological materials.
• Discuss the impact of the layer number on the optical, electronic and spintronic of layered topological materials.