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We report ab-initio calculations of phonon dispersion relationship on bulk and single layers of SnO and PbO. In agreement with experimental observations of Raman spectroscopy measurement, we find that A 1g mode is higher in frequency than E g mode. Moreover, the reason behind the shift of A 2u mode to higher frequency for monolayers of both SnO and PbO is clearly understood from our calculations. In addition to vibrational spectroscopy, we also find avoided crossing or Landau quasi-degeneracy between the longitudinal acoustics (LA) and low-energetic transverse optical (TO) modes in bulk form of both SnO and PbO and in monolayer SnO. The signature of such avoided crossing can be found in the variation of both specific heat and entropy with temperature. Although both specific heat and entropy decrease at the temperature corresponding to the crossing region, the specific heat value shows a kink in that region. In contrast, such lowering/kink in thermodynamic variables does not appear in monolayer PbO. The reason behind the crossing can be analysed using the concept of Born effective charge and dielectric tensor. To further verify the crossing, we calculate phonon group velocity for every band as a function of phonon wave vector for each of these oxides, where the role of avoided crossing has been clearly shown and explained.

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