Coherent electron transport in nanowires with spin-orbit coupling

In this thesis I investigate the coherent electron transport in a nanowire with spin-orbit coupling. Spin orbit coupling is a relativistic effect resulting in a coupling between the spin of an electron and its motion in an electric field. While in atomic physics it explains the fine structure of the atomic spectrum, this effect is also relevant in various semiconductors (e.g. InSb or InAs), for spintronics applications. In particular, nanowires (NWs) with a strong Rashba spin orbit coupling (RSOC), are currently on the spotlight of the condensed matter community. Indeed the technological advances in NW gating allow to tune the RSOC. Moreover, when combined with a Zeeman magnetic field, the RSOC in NWs gives rise to one-dimensional helical electron states that are topologically protected. Inspired by these motivations, this thesis aims to model the electron transport properties of NWs with an inhomogeneous RSOC and exposed to a Zeeman magnetic field, in the low temperature mesoscopic regime, where decoherence is absent and the wavelike nature of electrons emerges. The Thesis is organized as follows. Chapter 1 reviews the origin of the spin-orbit coupling and its effects in semiconducting materials, Chapter 2 reviews the Scattering Matrix approach (SMA), i.e. the quantum approach used to analyze the transport properties in the quantum mesoscopic regime. The subsequent two chapters contain original research work. In particular: Chapter 3 applies the SMA to the case of NW. In particular, motivated by the advances in gating techniques, I focus on NWs with an inhomogeneous RSOC. After performing the analytical calculation of the Boundary Matrix in each NW portion, I have written a numerical Python code to compute the Scattering matrix of the NW, whence I derived its transport properties. Specifically, I have considered two configurations of the inhomogeneous RSOC that correspond to physically interesting situations, and I have shown that the conductance can be widely tuned both electrically and magnetically. Chapter 4 focusses on the regime where the RSOC is much bigger than the Zeeman energy. This is the case where the NWs exhibits one-dimensional helical electron states, i.e. states described by a massless Dirac equation where the helicity value (+/-1) encodes the locking between the direction of propagation and the spin orientation. In particular, I investigated the so called Dirac paradox, which emerges at the interface between two regions of opposite helicity: An electron impinging from one side can seemingly neither be transmitted nor reflected. While the Dirac paradox has been investigated in higher dimensions (e.g. in 3D topological insulators), its implementation in NWs is particularly interesting since the helical states are actual 1D channels, preventing electrons from escaping along the interface of the two regions. While purely massless Dirac models predict that the solution of the Dirac paradox does not exist or is trivial, in a NW the paradox has a non trivial solution, due to the role played by additional massive Dirac modes. Although these modes carry no current, they allow the wavefunction matching at the interface for the massless modes, and the electron transmission can be controlled electrically. These results are described in a research article that is currently under review: L. Gogin et. "The Dirac paradox in 1+1 dimensions and its realization with spin-orbit coupled nanowires", cond-mat arXiv:2109.07355