Single-layer ScI2: A paradigm for valley-related multiple Hall effect

Valley-related multiple Hall effect in a two-dimensional lattice is of notable interest both for its fundamental physics and for its potential applications. In this work, by means of a low energy k·p model analysis, a mechanism of producing valley-related multiple Hall effects in the hexagonal lattice via strain engineering is proposed, and a general picture of valley-contrasted band inversion is developed. Through first-principles calculations, this mechanism is further established in a ferromagnetic hexagonal lattice of single-layer ScI2. Single-layer ScI2 prefers in-plane magnetization and exhibits neither anomalous valley Hall effects nor valley-polarized quantum anomalous Hall effects in nature. Remarkably, these two Hall effects emerge simultaneously in this system under 4.705% tensile strain and disappear simultaneously when further increasing strain, suggesting the exotic valley-related multiple Hall effect. The underlying physical mechanism is revealed using a model analysis and is generally applicable. Our work greatly enriches the valley-related physics.