In this work, we systematically studied the stoichiometry and thickness effects of low pressure chemical vapor deposited (LPCVD) SiNx bilayer stacks on the electrical property of AlGaN/GaN heterojunction-based metal-insulator-semiconductor high electron mobility transistors (MIS-HEMTs). A Si-rich SiNx single layer reduces threshold voltage shift and hysteresis under gate stress but gives rise to high gate leakage. A near-stoichiometric SiNx single layer suppresses gate leakage but causes poor gate stability. A bilayer SiNx stack with an optimized thickness ratio improves both the gate stability and on-resistance while keeping a low current leakage. The bilayer SiNx stack consisting 5 nm Si-rich SiNx interfacial layer and a 15 nm SiNx capping layer resulted in the lowest sheet resistance and the highest gate stability. Such enhanced gate stability is explained by the low density of trap states and weakened electric field at the Si-rich SiNx/GaN interface and an extra positive charge at the bilayer interface.