Topological insulators and topological superconductors are novel states of matter. One of the most characteristic properties of topological insulators are the topologically protected edge states. While the bulk of the material stays insulating, the edge-state conductance is quantized and topologically protected from backscattering. In topological superconductors the edge states manifest themselves in the form of Majorana bound states: zero energy states inside the superconducting gap that are located at the end of a one-dimensional topological superconductor. Chapter 2 of this thesis contains a detailed review and a discussion of k.p-theory. The k.p-theory allows one to go beyond commonly used effective models and obtain much more detailed description of a semiconductor's band structure around its gap. Topological insulators are often semiconductor-based and topological superconductors can be realized in a hybrid structure that consists of a semiconductor and a conventional superconductor. Chapter 3 covers implementation details of the numerical methods used in this thesis. Quantum spin Hall effect is one example of a topological insulator. Band inversion in HgTe/CdTe or InAs/GaSb two-dimensional system leads to a topological phase that is characterized by topologically protected helical edge states which carry electric current with a quantized conductance. It was believed that in-plane magnetic field would break time reversal symmetry, suppress the conductance and open an energy gap in the edge-state dispersion. However, the experiment conducted by Du et al. reported robust helical edge transport in InAs/GaSb persisting up to a magnetic fields of 12 T. In Chapter 4 of this thesis we show that the burying of a Dirac point in the valence band, a feature of the system dispersion revealed only by the detailed k.p-simulation, explains this unexpected observation. Experimental group of L.P. Kouwenhoven investigated experimentally the details of spin-orbit interaction in InAs/GaSb system in both topological and trivial phases. In Chapter 5 we connect the results of this experiment with our band structure calculations: in the topological phase, a quenching of the spin-splitting is observed and attributed to a crossing of spin bands, whereas in the trivial regime, the Rashba coefficient changes linearly with electric field and the linear Dresselhaus coefficient is constant. In Chapter 6 we take a look into the spin texture of the inverted InAs/GaSb system close to the hybridization gap. Transport measurements conducted by the experimental group of C.M. Marcus in Copenhagen revealed a giant spin-orbit splitting inherent to this system. This leads to a unique situation in which the Fermi energy in InAs/GaSb crosses a single spin-resolved band, resulting in a full spin-orbit polarization. In the last chapter of this thesis we focus on semiconducting nanowires with induced superconductivity that are considered to be a promising platform for hosting Majorana bound states. In this theoretical research conducted together with physcists from ETH Zurich we show that the orbital contribution to the electron g-factor in higher subbands of small-effective-mass semiconducting nanowires can lead to the g-factors that are larger by an order of magnitude or more than a bulk value.@en

Low dimensional semiconducting structures with strong spin-orbit interaction (SOI) and induced superconductivity attracted great interest in the search for topological superconductors. Both the strong SOI and hard superconducting gap are directly related to the topological protection of the predicted Majorana bound states. Here we explore the one-dimensional hole gas in germanium silicon (Ge-Si) core-shell nanowires (NWs) as a new material candidate for creating a topological superconductor. Fitting multiple Andreev reflection measurements shows that the NW has two transport channels only, underlining its one-dimensionality. Furthermore, we find anisotropy of the Landé g-factor that, combined with band structure calculations, provides us qualitative evidence for the direct Rashba SOI and a strong orbital effect of the magnetic field. Finally, a hard superconducting gap is found in the tunneling regime and the open regime, where we use the Kondo peak as a new tool to gauge the quality of the superconducting gap.

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We show that edge-state transport in semiconductor-based quantum spin Hall systems is unexpectedly robust to magnetic fields. The origin for this robustness lies in an intrinsic suppression of the edge-state g-factor and the fact that the edge-state Dirac point is typically hidden in the valence band. A detailed k·p band-structure analysis reveals that both InAs/GaSb and HgTe/CdTe quantum wells exhibit such buried Dirac points for a wide range of well thicknesses. By simulating transport in a disordered system described within an effective model, we demonstrate that edge-state transport remains nearly quantized up to large magnetic fields, consistent with recent experiments.

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Transport measurements in inverted InAs/GaSb quantum wells reveal a giant spin-orbit splitting of the energy bands close to the hybridization gap. The splitting results from the interplay of electron-hole mixing and spin-orbit coupling, and can exceed the hybridization gap. We experimentally investigate the band splitting as a function of top gate voltage for both electronlike and holelike states. Unlike conventional, noninverted two-dimensional electron gases, the Fermi energy in InAs/GaSb can cross a single spin-resolved band, resulting in full spin-orbit polarization. In the fully polarized regime we observe exotic transport phenomena such as quantum Hall plateaus evolving in e2/h steps and a nontrivial Berry phase.

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The spin-orbit interaction is investigated in a dual gated InAs/GaSb quantum well. Using an electric field, the quantum well can be tuned between a single-carrier regime with exclusively electrons as carriers and a two-carrier regime where electrons and holes coexist. The spin-orbit interaction in both regimes manifests itself as a beating in the Shubnikov-de Haas oscillations. In the single-carrier regime the linear Dresselhaus strength is characterized by β=28.5 meV Å and the Rashba coefficient α is tuned from 75 to 53 meV Å by changing the electric field. In the two-carrier regime a quenching of the spin splitting is observed and attributed to a crossing of spin bands.

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Recent experiments on Majorana fermions in semiconductor nanowires [S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nygård, P. Krogstrup, and C. M. Marcus, Nature (London) 531, 206 (2016)NATUAS0028-083610.1038/nature17162] revealed a surprisingly large electronic Landé g factor, several times larger than the bulk value - contrary to the expectation that confinement reduces the g factor. Here we assess the role of orbital contributions to the electron g factor in nanowires and quantum dots. We show that an L·S coupling in higher subbands leads to an enhancement of the g factor of an order of magnitude or more for small effective mass semiconductors. We validate our theoretical finding with simulations of InAs and InSb, showing that the effect persists even if cylindrical symmetry is broken. A huge anisotropy of the enhanced g factors under magnetic field rotation allows for a straightforward experimental test of this theory.

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