We review theoretical and experimental highlights in transport in two-dimensional topological materials over the last five years. Topological materials comprise topological insulators, Weyl semimetals and topological superconductors. This review focuses on key developments in the understanding of transport phenomena involving surfaces and interfaces of two-dimensional topological materials that have not been covered elsewhere. The review is structured around the following general topics: (i) topological insulators are finding applications in magnetic devices, while controversy continues to surround Hall transport in doped samples and the general issue of topological protection; (ii) state-of-the-art experiments on transition metal dichalcogenides have revealed new valley-dependent electrical and optical phenomena which have spin-dependent counterparts in topological insulators; (iii) in Weyl semimetals the manifestations of Fermi arcs in transport are actively investigated as well as controversial; (iv) a new and growing field centres on the non-linear electrical and optical responses of topological materials, where fundamental questions emerge about the interplay of the Berry curvature and disorder scattering; and (v) topological superconductor research is dominated by the quest for chiral superconductivity, Majorana fermions and topological quantum computing. Prospects for future research in this vast field are outlined in the closing section.

Anderson localization, the absence of diffusive transport in disordered systems, has been manifested as hopping transport in numerous electronic systems, whereas in recently discovered topological insulators it has not been directly observed. Here, we report experimental demonstration of a crossover from diffusive transport in the weak antilocalization regime to variable range hopping transport in the Anderson localization regime with ultrathin $({\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x{)}_2{\mathrm{Te}}_3$ films. As disorder becomes stronger, negative magnetoconductivity due to the weak antilocalization is gradually suppressed, and eventually, positive magnetoconductivity emerges when the electron system becomes strongly localized. This work reveals the critical role of disorder in the quantum transport properties of ultrathin topological insulator films, in which theories have predicted rich physics related to topological phase transitions.

Atomically smooth, single crystalline ${({\text{Bi}}_{1-x}{\text{Sb}}_x)}_2{\text{Te}}_3$ films have been grown on ${\text{SrTiO}}_3$ (111) substrates by molecular beam epitaxy. A full range of Sb-Bi compositions have been studied in order to obtain the lowest possible bulk conductivity. For the samples with optimized Sb compositions ($x=0.5\pm 0.1$ ), the carrier type can be tuned from n-type to p-type across the whole thickness with the help of a back-gate. Linear magnetoresistance has been observed at gate voltages close to the maximum in the longitudinal resistance of a ${({\text{Bi}}_{0.5}{\text{Sb}}_{0.5})}_2{\text{Te}}_3$ sample. These highly tunable ${({\text{Bi}}_{1-x}{\text{Sb}}_x)}_2{\text{Te}}_3$ thin films provide an excellent platform to explore the intrinsic transport properties of the three-dimensional topological insulators.

We report the observation of colossal positive magnetoresistance (MR) in single crystalline, high mobility TaAs₂ semimetal. The excellent fit of MR by a single quadratic function of the magnetic field B over a wide temperature range (T = 2–300 K) suggests the semiclassical nature of the MR. The measurements of Hall effect and Shubnikov-de Haas oscillations, as well as band structure calculations, suggest that the giant MR originates from the nearly perfectly compensated electrons and holes in TaAs₂. The quadratic MR can even exceed 1 200 000% at B = 9 T and T = 2 K, which is one of the largest values among those of all known semi-metallic compounds, including the very recently discovered WTe₂ and NbSb₂. The giant positive magnetoresistance in TaAs₂ not only has a fundamentally different origin from the negative colossal MR observed in magnetic systems but also provides a nice complemental system that will be beneficial for applications in magnetoelectronic devices.

High quality ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ single crystals have been investigated by magnetization, electron transport, and Andreev reflection spectroscopy. In the ferromagnetic ground state, the saturation magnetic moment of each unit cell corresponds to an integer number of electron spins ($3{\mu }_B/{\mathrm{Cr}}^{3+}$), and the Hall effect measurements suggest $n$-type charge carriers. Spin polarizations as high as 97% were obtained from fits of the differential conductance spectra of ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$/Pb junctions with the modified Blonder-Tinkham-Klapwijk theory. The temperature and bias-voltage dependencies of the subgap conductance are consistent with recent theoretical calculations based on spin active scatterings at a superconductor–half-metal interface. Our results suggest that $n\text{−}{\mathrm{HgCr}}_2{\mathrm{Se}}_4$ is a half-metal, in agreement with theoretical calculations that also predict undoped ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ is a magnetic Weyl semimetal.