Three-temperature model for predicting critical temperatures of charge density waves using Fermi smearing

Smearing methods have been used to compute temperature-dependent phonon dispersions and predict critical temperatures of charge density waves, but usually lead to a much higher result because of its ambiguous mechanism for modeling temperature effects. Here, a three-temperature model was developed to describe the energy transfer process between electrons, soft-mode and non-soft-mode phonons. In particular, mode-selective smearing induced soft-mode phonons were assigned a temperature to analyze its contributions to the relaxation between electrons and phonons. A relative standard was established to screen soft-mode phonons quantitatively for different materials. In addition, three smearing methods (Fermi–Dirac, Gaussian, and Methfessel–Paxton) and eight materials (monolayer or bulk TX₂, T = Ti, Nb, Ta and X = Se, S) were tested. Critical temperatures corrected by the three-temperature model were in great agreement with experimental results. This work provides new insights into correctly predicting critical temperatures of charge density waves, addressing the relaxation process of electrons and phonons using smearing method and determining phase transitions by phonon softening.