Valley-dependent Berry phase effects and related valleytronic applications in two-dimensional materials

The field of valleytronics, which exploits the valley degree of freedom in two-dimensional (2D) materials, has emerged as a promising frontier in next-generation electronic and photonic devices. In this context, a valley refers to an energy extremum in the electronic band structure of a material, which can serve as a discrete quantum number — akin to charge or spin — for information encoding. Because many 2D materials possess two inequivalent valleys in momentum space, they can naturally represent binary logic states, making them attractive for low-power electronics and quantum computing. This topical review explores the valley-dependent Berry phase effects and their significant role in various valleytronic applications. We begin by introducing the fundamentals of topology in solid-state physics, emphasizing the Berry phase — a geometric phase with profound quantum mechanical implications — and its associated Berry curvature, which acts as an effective magnetic field in momentum space. Next, we review key 2D materials, including monolayer and bilayer graphene as well as transition metal dichalcogenides, to illustrate how valley-dependent Berry curvature gives rise to orbital magnetic moments and the valley Hall effect. Advanced concepts, such as Chern numbers and valley Chern numbers, are discussed to quantify the topological order in these systems. The review also highlights the emergence of topological kink states and domain wall states in graphene, which are linked to robust edge conduction. Furthermore, we examine the valley Zeeman effect and the mechanisms underlying valley polarization (VP), wherein an imbalance of electrons between distinct momentum valleys enables novel device functionalities. Recent experimental breakthroughs — such as observations of the exciton Hall effect — demonstrate the technological potential of Berry phase effects in controlling VP and current. Finally, we underscore the critical role of these geometric and topological phenomena in driving future valleytronic innovations in 2D materials.