Thermal oscillations and resonance in electron–phonon interaction process

A recent theoretical study (Xu in Proc R Soc A Math Phys Eng Sci 477:20200913, 2021) has derived conditions on the coefficients of Jeffreys-type equation to predict thermal oscillations and resonance during phonon hydrodynamics in non-metallic solids. Thermal resonance, in which the temperature amplitude attains a maximum value (peak) in response to an external exciting frequency source, is a phenomenon pertinent to the presence of underdamped thermal oscillations and explicit finite speed for the thermal wave propagation. The present work investigates the occurrence condition for thermal resonance phenomenon during the electron–phonon interaction process in metals based on the hyperbolic two-temperature model. First, a sufficient condition for underdamped electron and lattice temperature oscillations is discussed by deriving a critical frequency (a material characteristic). It is shown that the critical frequency of thermal waves near room temperature, during electron–phonon interactions, may be on the order of terahertz ( \(10-20\) $10-20$ THz for Cu and Au, i.e., lying within the terahertz gap). It is found that whenever the natural frequency of metal temperature exceeds this frequency threshold, the temperature oscillations are of underdamped type. However, this condition is not necessary, since there is a small frequency domain, below this threshold, in which the underdamped thermal wave solution is available but not effective. Otherwise, the critical damping and the overdamping conditions of the temperature waves are determined numerically for a sample of pure metals. The thermal resonance conditions in both electron and lattice temperatures are investigated. The occurrence of resonance in both electron and lattice temperature is conditional on violating two distinct critical values of frequencies. When the natural frequency of the system becomes larger than these two critical values, an applied frequency equal to such a natural frequency can drive both electron and lattice temperatures to resonate together with different amplitudes and behaviors. However, the electron temperature resonates earlier than the lattice temperature.