Out-of-plane displacement of quantum color centers in monolayer h-BN

Color centers exhibiting deep-level states within the wide bandgap h-BN monolayer possess substantial potential for quantum applications. Uncovering precise geometric characteristics at the atomic scale is crucial for understanding defect performance. In this study, first-principles calculations were performed on the most extensively investigated CBVN and NBVN color centers in h-BN, focusing on the out-of-plane displacement and their specific impacts on electronic, vibrational, and emission properties. We demonstrate the competition between the {\sigma}*-like antibonding state and the {\pi}-like bonding state, which determines the out-of-plane displacement. The overall effect of vibronic coupling on geometry is elucidated using a pseudo Jahn-Teller model. Local vibrational analysis reveals a series of distinct quasi-local phonon modes that could serve as fingerprints for experimental identification of specific point defects. The critical effects of out-of-plane displacement during the quantum emission process are carefully elucidated to answer the distinct observations in experiments, and these revelations are universal in quantum point defects in other layered materials.