Based on first-principles calculations, we study the dependence of topological phase on anisotropic interactions in Bi${}_2$Se${}_3$-type materials. By applying different strains in order to vary interactions, we reveal that the topological phase is insensitive to lateral interaction but can be effectively tuned by longitudinal interaction. Longitudinal strain is inhomogeneous in the studied systems. The interquintuple interaction plays a dominant role in determining the topological phase. We explain the puzzling band-topology difference between Sb${}_2$Se${}_3$ and Bi${}_2$Se${}_3$ and propose an approach to tuning the topological phase by strain. It is found that Sb${}_2$Se${}_3$ can be converted into a topological insulator by applying compressive longitudinal strain while the converse strain can turn Bi${}_2$Se${}_3$ into a normal insulator. We have studied thin films of Sb${}_2$Se${}_3$ and Bi${}_2$Se${}_3$ and also observed a strain-induced topological phase transition.

By using density-functional theory-based GW method, we studied the valley degree of freedom of Janus monolayer VSSe. The GW corrections lead to a doubling of the band gap and change the band dispersion considerably, indicating significant many-body effects. VSSe is confirmed to be ferromagnetic, which breaks the time-reversal symmetry and the odd parity of the Berry curvature in momentum space. The dissimilar magnitudes of Berry curvatures of the inequivalent valleys give rise to appreciable anomalous Hall conductivity (AHC). The calculated valley optical response of VSSe exhibits a clear valley-selective circular dichroism. The ferromagnetism induces large valley-Zeeman splitting, making it possible to realize the selective valley excitation even by unpolarized light. The Janus VSSe is more tunable by external fields because of symmetry breaking. Due to the relief of time-reversal symmetry, the valley-Zeeman splitting can be continuously tuned by varying the magnetization direction. The loss of mirror symmetry in VSSe enables a bidirection modulation of the band gap by changing the direction of electric field. The strain can linearly tune the valley gap in a considerable range. The Berry curvature and AHC can be effectively regulated in the external fields.

Based on van der Waals density functional calculations, we have studied few-quintuple-layer (QL) films of Bi${}_2$Se${}_3$ and Bi${}_2$Te${}_3$. The separation between the QLs near the surface is found to have a large increase after relaxation, whereas, the separation between the inner QLs is smaller and approaches the bulk value as the thickness grows, showing a two-dimensional to three-dimensional structural crossover. Accordingly, the surface Dirac cone of the Bi${}_2$Se${}_3$ film is evidently gapped for small thicknesses (two to four QLs), and the gap is reduced and, finally, is closed with the increasing thickness, agreeing well with the experiments. We further studied the substrate effect by investigating the Bi${}_2$Se${}_3$/graphene system. It is found that the underlying graphene induces a giant thickness-dependent Rashba splitting and Dirac point shift. Because Bi${}_2$Te${}_3$ films have smaller relative inter-QL expansion and stronger spin-orbit coupling, the topological features start to appear in the film as thin as two QLs in good accord with the experiments.

Valleytronics uses valleys, a novel quantum degree of freedom, to encode information. It combines other degrees of freedom, such as charge and spin, to produce a more comprehensive, stable, and efficient information processing system. Valleytronics has become an intriguing field in condensed matter physics due to the emergence of new two-dimensional materials in recent years. However, in nonmagnetic valleytronic materials, the valley polarization is transient and the depolarization occurs once the external excitation is withdrawn. Introduction of magnetic field is an effective approach to realizing the spontaneous valley polarization by breaking the time-reversal symmetry. In hexagonal magnetic valleytronic materials, the inequivalent valleys at the K and –K(K′) Dirac cones have asymmetric energy gaps and Berry curvatures. The time-reversal symmetry in nonmagnetic materials can be broken by applying an external magnetic field, adding a magnetic substrate or doping magnetic atoms. Recent theoretical studies have demonstrated that valleytronic materials with intrinsic ferromagnetism, now termed as ferrovalley materials, exhibit spontaneous valley polarization without the need for external fields to maintain the polarization. The coupling of the valley and spin degrees of freedom enables stable and unequal distribution of electrons in the two valleys and thus facilitating nonvolatile information storage. Hence, ferrovalley materials are promising materials for valleytronic devices. In this review, we first briefly overview valleytronics and its related properties, the ways to realize valley polarization in nonmagnetic valleytronic materials. Then we focus on the recent developments in two-dimensional ferrovalley materials, which can be classified according to their molecular formula and crystal structure: MX₂; M(XY)₂, M(XY₂) and M(XYZ)₂; M₂X₃, M₃X₈ and MNX₆; MNX₂Y₂, M₂X₂Y₆ and MNX₂Y₆; and the Janus structure ferrovalley materials. In the inequivalent valleys, the Berry curvatures have opposite signs with unequal absolute values, leading to anomalous valley Hall effect. When the valley polarization is large, the ferrovalleys can be selectively excited even with unpolarized light. Intrinsic valley polarization in two-dimensional ferrovalley materials is of great importance. It opens a new avenue for information-related applications and hence is under rapid development.

Valley pseudospin, a novel quantum degree of freedom, is expected to show valley Zeeman effect in analogy to real spin in magnetic field B. By performing first-principles calculations, we studied the magnetic effect on valley pseudospin in monolayer WSe₂. With the application of B, the time reversal symmetry is broken. Our calculation shows that the valley energy degeneracy is broken and the valley Zeeman splitting varies linearly with B, agreeing well with the experiments. It is found that the valley Zeeman splitting is contributed mainly from the atomic orbital magnetic moment of W atom, but the valley contribution is still appreciable. The Berry curvatures of the two inequivalent valleys of monolayer WSe₂ are opposite and their change induced by B also depends linearly on B. The calculated circular dichroism and dielectric function reveal that the optical valley-dependent selection rule is preserved and the original single peak in polarization-resolved photoluminescence spectrum will be split into two peaks after the application of B. Our studies demonstrate the possibility of magnetic manipulation of the valley pseudospin.