Vibrational spectra of MO (M = Sn/Pb) in their bulk and single-layer forms: role of avoided crossing in their thermodynamic properties

We report ab-initio calculations of the phonon dispersion relation on the bulk and single layer of SnO and PbO. We identify Raman active modes and infrared active modes at the zone center Γ point. In agreement with experimental observations of Raman spectroscopy measurement, we find that A1g mode is higher in frequency than that of Eg mode. Moreover, the reason behind the shift of A2u mode to higher frequency for monolayer of both SnO and PbO is revealed from our calculations. We also find that long-range Coulomb interaction enhances the dielectric constant and Born effective charges in bulk SnO and bulk PbO, compared to their monolayer. Here, we observe avoided crossing or Landau degeneracy between longitudinal acoustics (LA) and low energetic transverse optical (TO) modes in bulk form of both SnO and PbO. Additionally, monolayer SnO also shows low energetic Raman modes (Eg and A1g) of same frequency as bulk. As a result, we notice avoided crossing between LA and TO modes in monolayer SnO. Interestingly, higher Born effective charge and low dielectric constant enhances self-force constants and the interatomic force constants (IFCs) between the M-O bonds. The enhanced force constants give rise to higher vibrational frequency of phonon modes for monolayer PbO. Our studies reveal that due to avoided crossing between two degenerate bands, the phonon dispersion near high symmetry X point lowers specific heat and vibrational entropy in bulk SnO, bulk PbO and only in monolayer SnO upto temperature 150 K. Moreover, the large mass difference between Pb and Oxygen atoms and absence of interlayer van der Waal interactions give rise to high phonon vibration which reduces the occurrence of band crossing between two degenerate energy levels. The absence of avoided crossing leads higher specific heat and vibrational entropy in monolayer PbO at low temperatures.