Jukka Vayrynen

A two-dimensional topological insulator has a gap for bulk excitations, but conducts on its boundaries via gapless edge modes. Time-reversal symmetry prohibits elastic backscattering of electrons propagating within the edge, leading to quantized conductance at zero temperature. Inelastic backscattering, present at finite temperature, breaks the quantization and  increases the edge resistance; the resistance of a long edge acquires a linear dependence on its length. A phenomenological theory that  introduces  the least irrelevant operators in the low-energy description  predicts a strong temperature-dependence for the edge resistivity. Such a prediction is at odds with the experimentally observed weak  temperature dependence of the resistance.

In the first part of this thesis, we attempt to resolve the issue by studying a realistic microscopic mechanism for  inelastic backscattering. The small band gaps in the existing putative two-dimensional topological insulators make them sensitive to static charge fluctuations created by randomness in the density of dopant.
Charge fluctuations may lead to the formation of electron and hole puddles. Such a puddle – a quantum dot – tunnel-coupled to the edge may significantly enhance the inelastic backscattering rate. The added resistance is especially strong for dots carrying an odd number of electrons, due to the Kondo effect. For the same reason, the temperature dependence of the added resistance becomes rather weak. We present a detailed theory of the puddles’ effect on the helical edge properties, including (i) linear conductance, (ii) non-linear $I$–$V$ characteristic, and (iii) current noise.

In the second part of this thesis, we study a superconducting setup, namely a topological Josephson junction. Such a junction can be formed by bringing an s-wave superconductor in contact with a topological insulator. The junction is expected to host exotic Majorana zero modes whose observation is still being contested. Most of the present schemes to detect Majorana modes rely on the fractional ac Josephson effect. However, extracting a robust signature from this complicated non-equilibrium effect is difficult. Instead, we propose a linear-response method – microwave spectroscopy – to probe the Majorana modes. These modes lend a signature low-frequency behavior to the junction’s microwave admittance $Y(\omega)$, which we calculate.