The anomalous valley Hall effect (AVHE) can be used to explore and utilize valley degrees of freedom in materials, which has potential applications in fields such as information storage, quantum computing and optoelectronics. AVHE exists in two-dimensional (2D) materials possessing valley polarization (VP), and such 2D materials usually belong to the hexagonal honeycomb lattice. Therefore, it is necessary to achieve valleytronic materials with VP that are more readily to be synthesized and applicated experimentally. In this topical review, we introduce recent developments on realizing VP as well as AVHE through different methods, i.e., doping transition metal atoms, building ferrovalley heterostructures and searching for ferrovalley materials. Moreover, 2D ferrovalley systems under external modulation are also discussed. 2D valleytronic materials with AVHE demonstrate excellent performance and potential applications, which offer the possibility of realizing novel low-energy-consuming devices, facilitating further development of device technology, realizing miniaturization and enhancing functionality of them.

Valleytronics, using valley degree of freedom to encode, process, and store information, may find practical applications in low-power-consumption devices. Recent theoretical and experimental studies have demonstrated that two-dimensional (2D) honeycomb lattice systems with inversion symmetry breaking, such as transition-metal dichalcogenides (TMDs), are ideal candidates for realizing valley polarization. In addition to the optical field, lifting the valley degeneracy of TMDs by introducing magnetism is an efficient way to manipulate the valley degree of freedom. In this paper, we first review the recent progress on valley polarization in various TMD-based systems, including magnetically doped TMDs, intrinsic TMDs with both inversion and time-reversal symmetry broken, and magnetic TMD heterostructures. When topologically nontrivial bands are empowered into valley-polarized systems, valley-polarized topological states, namely valley-polarized quantum anomalous Hall effect can be realized. Therefore, we have also reviewed the theoretical proposals for realizing valley-polarized topological states in 2D honeycomb lattices. Our paper can help readers quickly grasp the latest research developments in this field.

The electron's charge and spin degrees of freedom are at the core of modern electronic devices. With the in-depth investigation of two-dimensional materials, another degree of freedom, valley, has also attracted tremendous research interest. The intrinsic spontaneous valley polarization in two-dimensional magnetic systems, ferrovalley material, provides convenience for detecting and modulating the valley. In this review, we first introduce the development of valleytronics. Then, the valley polarization forms by the p-, d-, and f-orbit that are discussed. Following, we discuss the investigation progress of modulating the valley polarization of two-dimensional ferrovalley materials by multiple physical fields, such as electric, stacking mode, strain, and interface. Finally, we look forward to the future developments of valleytronics.

Valleytronics is an emergent discipline in condensed matter physics and offers a new way to encode and manipulate information based on the valley degree of freedom in materials. Among the various materials being studied, Kekulé distorted graphene has emerged as a promising material for valleytronics applications. Graphene can be artificially distorted to form the Kekulé structures rendering the valley-related interaction. In this work, we review the recent progress of research on Kekulé structures of graphene and focus on the modified electronic bands due to different Kekulé distortions as well as their effects on the transport properties of electrons. We systematically discuss how the valley-related interaction in the Kekulé structures was used to control and affect the valley transport including the valley generation, manipulation, and detection. This article summarizes the current challenges and prospects for further research on Kekulé distorted graphene and its potential applications in valleytronics.

The presence of two sublattices in hexagonal graphene brings two energetically degenerate extremes in the conduction and valence bands, which are identified by the valley quantum number. Recently, this valley degree of freedom has been suggested to encode and process information, which develops a new carbon-based electronics named graphene valleytronics. In this topical review, we present and discuss valley-related transport properties in bulk graphene monolayers, which are due to strain-induced pseudomagnetic fields and associated vector potential, sublattice-stagger potential, and the valley-Zeeman effect. These valley-related interactions can be utilized to obtain valley filtering, valley spatial separation, valley-resolved guiding modes, and valley-polarized collective modes such as edge or surface plasmons. The present challenges and the perspectives on graphene valleytronics are also provided in this review.