Bond-photon-phonon thermal relaxation in the M(X, X₂) (M = Mo, Re, Ta, Ge, Sn; X = S, Se, and Te)

We systematically investigated the temperature-dependent bandgap energy and Raman shift on the bond length and bond energy, Debye temperature, and atomic cohesive energy for M(X, X₂) via bond relaxation methods. It is revealed that the thermal decay of both bandgap energy and phonon frequency arose from the thermal integration of the specific heat of Debye approximation. The results indicate that (i) the bandgap energy relaxation is due to the thermal excitation-induced weakening of the bond energy, and the phonon frequency was just a function of bond length and bond energy; (ii) the Debye temperature determines the nonlinear range at low temperatures; (iii) the reciprocal of the atomic cohesive energy governs the linear behavior at high temperatures. Thus, the outcomes of this study include fundamental information about photon, phonon, and the thermal properties of layered semiconductors, which are crucial to develop the new generations of thermal and electronic applications of devices based on layered semiconductors.