This thesis describes the studies of topological superconductivity, which is predicted to emerge when pair correlations are induced into the surface states of 2D and 3D topolog- ical insulators (TIs). In this regard, experiments have been designed to investigate the theoretical ideas first pioneered by Fu and Kane that in such system Majorana bound states occur at vortices or edges of the system [Phys. Rev. Lett. 100, 096407 (2008), Phys. Rev. B 79, 161408 (2009)]. These states are of great interest as they constitute a new quasiparticle which is its own antiparticle and can be used as building blocks for fault tolerant topological quantum computing. After an introduction in chapter 1, chapter 2 of the thesis lays the foundation for the understanding of the field of topology in the context of condensed matter physics with a focus on topological band insulators and topological superconductors. Starting from a Chern insulator, the concepts of topological band theory and the bulk boundary corre- spondence are explained. It is then shown that the low energy Hamiltonian of mercury telluride (HgTe) quantum wells of an appropriate thickness can be written as two time reversal symmetric copies of a Chern insulator. This leads to the quantum spin Hall effect. In such a system, spin-polarized one dimensional conducting states form at the edges of the material, while the bulk is insulating. This concept is extended to 3D topological insulators with conducting 2D surface states. As a preliminary step to treating topological superconductivity, a short review of the microscopic theory of superconductivity, i.e. the theory of Bardeen, Cooper, and Shrieffer (BCS theory) is presented. The presence of Majorana end modes in a one dimensional superconducting chain is explained using the Kitaev model. Finally, topological band insulators and conventional superconductivity are combined to effectively engineer p-wave superconductivity. One way to investigate these states is by measuring the periodicity of the phase of the Josephson supercurrent in a topological Josephson junction. The signature is a 4π-periodicity compared to the 2π-periodicity in conventional Josephson junctions. The proof of the presence of this effect in HgTe based Josephson junction is the main goal of this thesis and is discussed in chapters 3 to 6. Chapter 3 describes in detail the transport of a 3D topological insulator based weak link under radio-frequency radiation. The chapter starts with a review of the state of research of (i) strained HgTe as 3D topological insulator and (ii) the progress of induc- ing superconducting correlations into the topological surface states and the theoretical predictions of 3D TI based Josephson junctions. Josephson junctions based on strained HgTe are successfully fabricated. Before studying the ac driven Josephson junctions, the dc transport of the devices is analysed. The critical current as a function of temperature is measured and it is possible to determine the induced superconducting gap. Under rf illumination Shapiro steps form in the current voltage characteristic. A missing first step at low frequencies and low powers is found in our devices. This is a signature of a 4π-periodic supercurrent. By studying the device in a wide parameter range - as a 147148 SUMMARY function of frequency, power, device geometry and magnetic field - it is shown that the results are in agreement with the presence of a single gapless Andreev doublet and several conventional modes. Chapter 4 gives results of the numerical modelling of the I −V dynamics in a Josephson junction where both a 2π- and a 4π-periodic supercurrents are present. This is done in the framework of an equivalent circuit representation, namely the resistively shunted Josephson junction model (RSJ-model). The numerical modelling is in agreement with the experimental results in chapter 3. First, the missing of odd Shapiro steps can be understood by a small 4π-periodic supercurrent contribution and a large number of modes which have a conventional 2π-periodicity. Second, the missing of odd Shapiro steps occurs at low frequency and low rf power. Third, it is shown that stochastic processes like Landau Zener tunnelling are most probably not responsible for the 4π contribution. In a next step the periodicity of Josephson junctions based on quantum spin Hall insulators using are investigated in chapter 5. A fabrication process of Josephson junctions based on inverted HgTe quantum wells was successfully developed. In order to achieve a good proximity effect the barrier material was removed and the superconductor deposited without exposing the structure to air. In a next step a gate electrode was fabricated which allows the chemical potential of the quantum well to be tuned. The measurement of the diffraction pattern of the critical current Ic due to a magnetic field applied perpendicular to the sample plane was conducted. In the vicinity to the expected quantum spin Hall phase, the pattern resembles that of a superconducting quantum interference device (SQUID). This shows that the current flows predominantly on the edges of the mesa. This observation is taken as a proof of the presence of edge currents. By irradiating the sample with rf, missing odd Shapiro steps up to step index n = 9 have been observed. This evidences the presence of a 4π-periodic contribution to the supercurrent. The experiment is repeated using a weak link based on a non-inverted HgTe quantum well. This material is expected to be a normal band insulator without helical edge channels. In this device, all the expected Shapiro steps are observed even at low frequencies and over the whole gate voltage range. This shows that the observed phenomena are directly connected to the topological band structure. Both features, namely the missing of odd Shapiro steps and the SQUID like diffraction pattern, appear strongest towards the quantum spin Hall regime, and thus provide evidence for induced topological superconductivity in the helical edge states. A more direct way to probe the periodicity of the Josephson supercurrent than using Shapiro steps is the measurement of the emitted radiation of a weak link. This experiment is presented in chapter 6. A conventional Josephson junction converts a dc bias V to an ac current with a characteristic Josephson frequency fJ = eV /h. In a topological Josephson junction a frequency at half the Josephson frequency fJ /2 is expected. A new measurement setup was developed in order to measure the emitted spectrum of a single Josephson junction. With this setup the spectrum of a HgTe quantum well based Josephson junction was measured and the emission at half the Josephson frequency fJ /2 was detected. In addition, fJ emission is also detected depending on the gate voltage and detection frequency. The spectrum is again dominated by half the Josephson emission at low voltages while the conventional emission is determines the spectrum at high voltages. A non-inverted quantum well shows only conventional emission over the whole gateSUMMARY 149 voltage and frequency range. The linewidth of the detected frequencies gives a measure on the lifetime of the bound states: From there, a coherence time of 0.3–4ns for the fJ /2 line has been deduced. This is generally shorter than for the fJ line (3–4ns). The last part of the thesis, chapter 7, reports on the induced superconducting state in a strained HgTe layer investigated by point-contact Andreev reflection spectroscopy. For the experiment, a HgTe mesa was fabricated with a small constriction. The diameter of the orifice was chosen to be smaller than the mean free path estimated from magne- totransport measurements. Thus one gets a ballistic point-contact which allows energy resolved spectroscopy. One part of the mesa is covered with a superconductor which induces superconducting correlations into the surface states of the topological insulator. This experiment therefore probes a single superconductor normal interface. In contrast to the Josephson junctions studied previously, the geometry allows the acquisition of energy resolved information of the induced superconducting state through the measurement of the differential conductance dI/dV as a function of applied dc bias for various gate voltages, temperatures and magnetic fields. An induced superconducting order parame- ter of about 70µeV was extracted but also signatures of the niobium gap at the expected value around Δ Nb ≈ 1.1meV have been found. Simulations using the theory developed by Blonder, Tinkham and Klapwijk and an extended model taking the topological surface states into account were used to fit the data. The simulations are in agreement with a small barrier at the topological insulator-induced topological superconductor interface and a high barrier at the Nb to topological insulator interface. To understand the full con- ductance curve as a function of applied voltage, a non-equilibrium driven transformation is suggested. The induced superconductivity is suppressed at a certain bias value due to local electron population. In accordance with this suppression, the relevant scattering regions change spatially as a function of applied bias. To conclude, it is emphasized that the experiments conducted in this thesis found clear signatures of induced topological superconductivity in HgTe based quantum well and bulk devices and opens up the avenue to many experiments. It would be interesting to apply the developed concepts to other topological matter-superconductor hybrid systems. The direct spectroscopy and manipulation of the Andreev bound states using circuit quantum electrodynamic techniques should be the next steps for HgTe based samples. This was already achieved in superconducting atomic break junctions by the group in Saclay [Science 2015, 349, 1199-1202 (2015)]. Another possible development would be the on-chip detection of the emitted spectrum as a function of the phase φ through the junction. In this connection, the topological junction needs to be shunted by a parallel ancillary junction. Such a setup would allow the current phase relation I(φ) directly and the lifetime of the bound states to be measured directly. By coupling this system to a spectrometer, which can be another Josephson junction, the energy dependence of the Andreev bound states E(φ) could be obtained. The experiments on the Andreev reflection spectroscopy described in this thesis could easily be extended to two dimensional topological insulators and to more complex geometries, like a phase bias loop or a tunable barrier at the point-contact. This work might also be useful for answering the question how and why Majorana bound states can be localized in quantum spin Hall systems.