The quantum spin Hall (QSH) effect is considered to be unstable to perturbations violating the time-reversal (TR) symmetry. We review some recent developments in the search of the QSH effect in the absence of the TR symmetry. The possibility to realize a robust QSH effect by artificial removal of the TR symmetry of the edge states is explored. As a useful tool to characterize topological phases without the TR symmetry, the spin-Chern number theory is introduced.

This paper presents an overview of growth of Bi₂Se₃, a prototypical three-dimensional topological insulator, by molecular-beam epitaxy on various substrates. Comparison is made between growth of Bi₂Se₃ (111) on van der Waals (vdW) and non-vdW substrates, with attention paid to twin suppression and strain. Growth along the [221] direction of Bi₂Se₃ on InP (001) and GaAs (001) substrates is also discussed.

The last several years have witnessed the rapid developments in the study and understanding of topological insulators. In this review, after a brief summary of the history of topological insulators, we focus on the recent progress made in transport experiments on topological insulator films and nanowires. Some quantum phenomena, including the weak antilocalization, the Aharonov-Bohm effect, and the Shubnikov-de Haas oscillations, observed in these nanostructures are described. In addition, the electronic transport evidence of the superconducting proximity effect as well as an anomalous resistance enhancement in topological insulator/superconductor hybrid structures is included.

Topological insulators as a new type of quantum matter materials are characterized by a full insulating gap in the bulk and gapless edge/surface states protected by the time-reversal symmetry. We propose that the interference patterns caused by the elastic scattering of defects or impurities are dominated by the surface states at the extremal points on the constant energy contour. Within such a formalism, we summarize our recent theoretical investigations on the elastic scattering of topological surface states by various imperfections, including non-magnetic impurities, magnetic impurities, step edges, and various other defects, in comparison with the recent related experiments in typical topological materials such as BiSb alloys, Bi₂Te₃, and Bi₂Se₃ crystals.

Providing the strong spin-orbital interaction, Bismuth is the key element in the family of three-dimensional topological insulators. At the same time, Bismuth itself also has very unusual behavior, existing from the thinnest unit to bulk crystals. Ultrathin Bi (111) bilayers have been theoretically proposed as a two-dimensional topological insulator. The related experimental realization achieved only recently, by growing Bi (111) ultrathin bilayers on topological insulator Bi₂Te₃ or Bi₂Se₃ substrates. In this review, we started from the growth mode of Bi (111) bilayers and reviewed our recent progress in the studies of the electronic structures and the one-dimensional topological edge states using scanning tunneling microscopy/spectroscopy (STM/STS), angle-resolved photoemission spectroscopy (ARPES), and first principles calculations.

Quantum Hall effect (QHE), as a class of quantum phenomena that occur in macroscopic scale, is one of the most important topics in condensed matter physics. It has long been expected that QHE may occur without Landau levels so that neither external magnetic field nor high sample mobility is required for its study and application. Such a QHE free of Landau levels, can appear in topological insulators (TIs) with ferromagnetism as the quantized version of the anomalous Hall effect, i.e., quantum anomalous Hall (QAH) effect. Here we review our recent work on experimental realization of the QAH effect in magnetically doped TIs. With molecular beam epitaxy, we prepare thin films of Cr-doped (Bi,Sb)₂Te₃ TIs with wellcontrolled chemical potential and long-range ferromagnetic order that can survive the insulating phase. In such thin films, we eventually observed the quantization of the Hall resistance at h/e² at zero field, accompanied by a considerable drop in the longitudinal resistance. Under a strong magnetic field, the longitudinal resistance vanishes, whereas the Hall resistance remains at the quantized value. The realization of the QAH effect provides a foundation for many other novel quantum phenomena predicted in TIs, and opens a route to practical applications of quantum Hall physics in low-power-consumption electronics.