Phonon Mode Coupling and Charge Density Wave in Monolayer NbSe₂

We present a detailed study on the phonon mode coupling mechanisms that drive charge density wave (CDW) formation in monolayer 1H-NbSe₂. By employing density functional perturbation theory and electron-phonon coupling calculations, we systematically analyze the interactions between longitudinal and transverse phonon modes. The results reveal that the CDW instability originates from the softening of a longitudinal optical phonon mode, which undergoes anticrossing with other phonon branches. This coupling leads to a significant Kohn anomaly, contributing to the emergence of the CDW vector. The origin of the CDW mechanism is further explored through phonon self-energy calculations, which incorporate both susceptibility and electron-phonon interactions, offering deeper insight into the CDW state. Additionally, the evolution of the phonon anti-crossing is captured across several quasi-temperatures, shedding light on how temperature influences phonon mode interactions and lattice distortions. This study provides a comprehensive framework for understanding phase transitions through phonon mode coupling in NbSe₂ and other transition metal dichalcogenides.