A topological superconductor (TSC) is characterized by the topologically protected gapless surface state that is essentially an Andreev bound state consisting of Majorana fermions. While a TSC has not yet been discovered, the doped topological insulator ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, which superconducts below $\sim 3\mathrm{K}$, has been predicted to possess a topological superconducting state. We report that the point-contact spectra on the cleaved surface of superconducting ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ present a zero-bias conductance peak (ZBCP) which signifies unconventional superconductivity. Theoretical considerations of all possible superconducting states help us conclude that this ZBCP is due to Majorana fermions and gives evidence for a topological superconductivity in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$. In addition, we found an unusual pseudogap that develops below $\sim 20\mathrm{K}$ and coexists with the topological superconducting state.

Topological insulators are predicted to present interesting surface transport phenomena but their experimental studies have been hindered by a metallic bulk conduction that overwhelms the surface transport. We show that the topological insulator Bi2Te2Se presents a high resistivity exceeding 1$\hskip0.3em${}$\Omega${}\,cm and a variable-range hopping behavior, and yet presents Shubnikov-de Haas oscillations coming from the topological surface state. Furthermore, we have been able to clarify both the bulk and surface transport channels, establishing a comprehensive understanding of the transport in this material. Our results demonstrate that Bi2Te2Se is, to our knowledge, the best material to date for studying the surface quantum transport in a topological insulator.

The massless Dirac fermions residing on the surface of three-dimensional topological insulators are protected from backscattering and cannot be localized by disorder, but such protection can be lifted in ultrathin films when the three-dimensionality is lost. By measuring the Shubnikov\char21{}de~Haas oscillations in a series of high-quality Bi2Se3 thin films, we revealed a systematic evolution of the surface conductance as a function of thickness and found a striking manifestation of the topological protection: The metallic surface transport abruptly diminishes below the critical thickness of $\sim${}6\,\,nm, at which an energy gap opens in the surface state and the Dirac fermions become massive. At the same time, the weak antilocalization behavior is found to weaken in the gapped phase due to the loss of $\pi${} Berry phase.

We show that in the new topological-insulator compound Bi1.5Sb0.5Te1.7Se1.3 one can achieve a surfaced-dominated transport where the surface channel contributes up to 70% of the total conductance. Furthermore, it was found that in this material the transport properties sharply reflect the time dependence of the surface chemical potential, presenting a sign change in the Hall coefficient with time. We demonstrate that such an evolution makes us observe both Dirac holes and electrons on the surface, which allows us to reconstruct the surface band dispersion across the Dirac point.

To optimize the bulk-insulating behavior in the topological insulator materials having a tetradymite structure, we have synthesized and have characterized single-crystal samples of Bi2-xSbxTe3-ySey solid solution at various compositions. We have elucidated that there are a series of intrinsic compositions where the acceptors and donors compensate each other and present a maximal bulk-insulating behavior. At such compositions, the resistivity can become as large as several $\Omega${} cm at low temperatures, and one can infer the role of the surface-transport channel in the nonlinear Hall effect. In particular, the composition of Bi1.5Sb0.5Te1.7Se1.3 achieves the lowest bulk carrier density and appears to be best suited for surface-transport studies.

The superconductivity recently found in the doped topological insulator CuxBi2Se3 offers a great opportunity to search for a topological superconductor. We have successfully prepared a single-crystal sample with a large shielding fraction and measured the specific-heat anomaly associated with the superconductivity. The temperature dependence of the specific heat suggests a fully~gapped, strong-coupling superconducting state, but the BCS theory is not in full agreement with the data, which hints at a possible unconventional pairing in CuxBi2Se3. Also, the evaluated effective mass of 2.6me (me is the free electron mass) points to a large mass enhancement in this material.