Valley splitting in the van der Waals heterostructure WSe2/CrI3 : The role of atom superposition

A recent experiment shows that the ${K}^{\ensuremath{'}}K$ valley degeneracy can be lifted in monolayer ${\mathrm{WSe}}_{2}$ deposited on a layered ferromagnetic substrate of ${\mathrm{CrI}}_{3}$. In this work, we take a van der Waals heterostructure ${\mathrm{WSe}}_{2}/{\mathrm{CrI}}_{3}$ to model the monolayer ${\mathrm{WSe}}_{2}$ on the ${\mathrm{CrI}}_{3}$ substrate and investigate the effects underpinning the ${K}^{\ensuremath{'}}K$ valley splitting based on first-principles calculations. We demonstrate that the interfacial atom superposition plays an important role and a W-Cr superposition is essential for a relatively large valley splitting. The results indicate inevitable sample-to-sample variations in the ${K}^{\ensuremath{'}}K$ valley splitting in the ${\mathrm{WSe}}_{2}/{\mathrm{CrI}}_{3}$. Furthermore, we show that the ${K}^{\ensuremath{'}}K$ valley splitting can be tuned in the trilayer ${\mathrm{CrI}}_{3}/{\mathrm{WSe}}_{2}/{\mathrm{CrI}}_{3}$ from nearly zero to more than two times of that in the bilayer ${\mathrm{WSe}}_{2}/{\mathrm{CrI}}_{3}$ by manipulating the layer alignment.