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

Abstract

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 $h \nu$. The mathematical structure of the BTE scattering integrals reveals that $h \nu$ is a natural energy scale for describing the dynamics. Normalizing all energy quantities by $h \nu$ leads to a set of three unitless parameters -- $\beta / \delta$, $\gamma$, and $\alpha$ -- that control the relaxation dynamics: $\beta / \delta$ is the normalized ratio of electron-phonon to electron-electron scattering strengths, $\gamma$ is the normalized phonon (lattice) temperature, and $\alpha$ is the normalized absorbed energy density. Using this theory, we systematically investigate relaxation times for the high-energy part of the distribution ($\tau_H$), energy transfer to the phonon subsystem ($\tau_E$), and intracarrier thermalization ($\tau_{th}$). In the linear region of response (valid when $\alpha$ is sufficiently small), we offer heuristic descriptions of each of these relaxation times as functions of $\beta / \delta$ and $\gamma$. Our results as a function of excitation level $\alpha$ show that many ultrafast experimental investigations lie in a transition region between low excitation (where the relaxation times are independent of $\alpha$) 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.