Valley-Contrasting Physics in Single-Layer CrSi₂N₄ and CrSi₂P₄

The exploration of valley-contrasting physics in two-dimensional materials with strong spin-orbit coupling is of great significance for both fundamental physics and advanced information technology. Here, using first-principles calculations, we report the identification of promising valley-contrasting physics in single-layer CrSi₂N₄ and CrSi₂P₄. Single-layer CrSi₂N₄ and CrSi₂P₄ are semiconductors with a direct band gap locating at the K/K' point, which forms a pair of degenerate but nonequivalent valleys in both the conduction and valence bands. These valleys display the intriguing valley spin splitting when considering spin-orbit coupling. Particularly for the valence bands, the valley spin splitting can reach up to 0.13/0.17 eV, giving rise to the robust spin-valley coupling and thus the coexistence of spin and valley Hall effects. The underlying physics are uncovered in detail. Moreover, strain is demonstrated to be an effective way for manipulating their coupled spin and valley physics.