Holes in nanowires and quantum dots : spin qubits, majorana fermions, and hole and hyperfine interactions

In the last decades much scientific effort was spent on manufacturing and understanding the properties of smaller and smaller condensed matter systems. This eventually resulted in the reliable production of semiconductor quantum dots and nanowires, systems that are reduced to sizes of several (tens of) nanometers in all three or at least two dimensions. The confinement in such nanoscale systems is so strong that quantum effects play a crucial role. Furthermore, the exact energy level or band structure of these systems depends heavily on the material composition, the precise confinement geometry, the intrinsic strain distribution, the spin-orbit interaction, as well as the presence of intrinsic and externally applied electric and magnetic fields.
The feasibility of loading only a single charge carrier onto a quantum dot led to the proposal of the spin qubit, as a possible smallest building block of a semiconductor-based quantum computer. In a spin qubit the quantum information is stored in a superposition of Zeeman-split spin-up and spin-down states. Most crucial for its successful implementation is the reliable control and the proper understanding of the carrier spin dynamics. Solving this task has been a highly active research field ever since, both on the experimental and theoretical side. Most of the research conducted so far focused on electrons in the lowest conduction band states. However, at some point it was realized that holes in the states close to the valence band edge may sometimes offer a more advantageous behavior regarding qubit control and coherence. This is due to the p-type symmetry of the associated Bloch states which results in a strong spin-orbit interaction on the atomic level and in an anisotropic hyperfine interaction that is much weaker than the hyperfine interaction of electrons.
Semiconductor nanowires can serve both as hosts for quantum dots and as one-dimensional channels. Nanowire quantum dots are defined by putting additional closely spaced gates on the nanowire that allow for an electrically tunable longitudinal confinement. Otherwise, a one-dimensional electron or hole gas forms inside the nanowire. Additionally, it has been proposed that nanowires with strong spin-orbit interaction can be employed as hosts for Majorana fermions. This is of special interest because their non-abelian braiding statistics make Majorana fermions good candidates for topological qubits.
In the first part of this thesis, we focus mostly on holes and hole spins confined to self-assembled quantum dots and quantum dots defined in Ge/Si core/shell nanowires. We calculate the hole spin relaxation and decoherence times in these quantum dots due to hole-spin phonon interactions and hole-spin nuclear-spin interactions that are mediated by the spin-orbit interaction and the hyperfine interaction, respectively. Subsequently, we show how these times are affected by specific system parameters such as intrinsic and extrinsic strain, the confinement strength as well as the magnitude and direction of applied electric and magnetic fields. Furthermore, we determine the effective Zeeman splitting by investigating the anisotropy and tunability of the effective g factor of electrons and holes in the lowest energy levels in these systems. In addition, we investigate the effect of non-collinear terms in the hole-spin nuclear-spin hyperfine interaction which reduce the degree of nuclear spin polarization that can be obtained by optical pumping. Also, we propose an experimental setup that allows to detect and quantify this effect.
In the second part of this work we survey holes in Ge/Si core/shell nanowires by employing an effective one-dimensional microscopic model that includes a strong and electrically tunable Rashba-type spin-orbit interaction. Using a Luttinger liquid description, we show that a screened Coulomb interaction strongly influences the nanowire properties. The strength of the interactions is explicitly quantified by calculating the scaling exponents of correlation functions and by examining the effect of the interactions on a partial gap opened by a small magnetic field.
Finally, we consider the nanowires as hosts for Majorana fermions. This is possible when the nanowire is placed in close proximity to an s-wave superconductor and put into a helical regime by applying electric fields. Furthermore, a magnetic field is needed to open a gap in the spectrum. In this setup, we calculate the field dependence of the localization lengths of the associated Majorana fermion wave functions. In short nanowires the Majorana fermions hybridize and form a subgap fermion whose energy oscillates as a function of the applied fields. The oscillation period allows to measure the strength of the spin-orbit interaction and the g factor anisotropy of the nanowire.