Ultrafast relaxation dynamics of excited carriers in metals: Simplifying the intertwined dependencies upon scattering strengths, phonon temperature, photon energy, and excitation level

Using the Boltzmann transport equation (BTE), we study the evolution of nonequilibrium carrier distributions in simple $(sp)$ metals, assumed to have been instantaneously excited by an ultrafast laser pulse with photon energy hν. The mathematical structure of the BTE scattering integrals reveals that hν is a natural energy scale for describing the dynamics. Normalizing all energy quantities by hν leads to a set of three unitless parameters β/δ, γ, and α that control the relaxation dynamics: β/δ is the normalized ratio of electron-phonon to electron-electron scattering strengths, γ is the normalized phonon (lattice) temperature, and α is the normalized absorbed energy density. Using this theory, we systematically investigate relaxation times for the high-energy part of the distribution (τH), energy transfer to the phonon subsystem (τE), and intracarrier thermalization (τₜₕ). In the linear region of response (valid when α is sufficiently small), we offer heuristic descriptions of each of these relaxation times as functions of β/δ and γ. Our results as a function of excitation level α show that many ultrafast experimental investigations lie in a transition region between low excitation (where the relaxation times are independent of α) and high excitation (where the two-temperature model of carrier dynamics is valid). Approximate boundaries that separate these three regions are described by simple expressions involving the normalized parameters of our model.