Femtosecond spectroscopy of electron-electron and electron-phonon energy relaxation in Ag and Au

We show experimentally that the electron distribution of a laser-heated metal is a nonthermal distribution on the time scale of the electron-phonon (e-ph) energy relaxation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{E}}$. We measured ${\mathrm{\ensuremath{\tau}}}_{\mathit{E}}$ in 45-nm Ag and 30-nm Au thin films as a function of lattice temperature (${\mathit{T}}_{\mathit{i}}$=10--300 K) and laser-energy density (${\mathit{U}}_{\mathit{l}}$=0.3--1.3 J ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$), combining femtosecond optical transient-reflection techniques with the surface-plasmon polariton resonance. The experimental effective e-ph energy relaxation time decreased from 710--530 fs and 830--530 fs for Ag and Au, respectively, when temperature is lowered from 300 to 10 K. At various temperatures we varied ${\mathit{U}}_{\mathit{l}}$ between 0.3--1.3 J ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$ and observed that ${\mathrm{\ensuremath{\tau}}}_{\mathit{E}}$ is independent from ${\mathit{U}}_{\mathit{l}}$ within the given range. The results were first compared to theoretical predictions of the two-temperature model (TTM). The TTM is the generally accepted model for e-ph energy relaxation and is based on the assumption that electrons and lattice can be described by two different time-dependent temperatures ${\mathit{T}}_{\mathit{e}}$ and ${\mathit{T}}_{\mathit{i}}$, implying that the two subsystems each have a thermal distribution. The TTM predicts a quasiproportional relation between ${\mathrm{\ensuremath{\tau}}}_{\mathit{E}}$ and ${\mathit{T}}_{\mathit{i}}$ in the perturbative regime where ${\mathrm{\ensuremath{\tau}}}_{\mathit{E}}$ is not affected by ${\mathit{U}}_{\mathit{l}}$.Hence, it is shown that the measured dependencies of ${\mathrm{\ensuremath{\tau}}}_{\mathit{E}}$ on lattice temperature and energy density are incompatible with the TTM. It is proven that the TTM assumption of a thermal electron distribution does not hold especially under our experimental conditions of low laser power and lattice temperature. The electron distribution is a nonthermal distribution on the picosecond time scale of e-ph energy relaxation. We developed a new model, the nonthermal electron model (NEM), in which we account for the (finite) electron-electron (e-e) and electron-phonon dynamics simultaneously. It is demonstrated that incomplete electron thermalization yields a slower e-ph energy relaxation in comparison to the thermalized limit. With the NEM we are able to give a consistent description of our data and obtain values for the e-e scattering rate K=0.10\ifmmode\pm\else\textpm\fi{}0.05 ${\mathrm{fs}}^{\mathrm{\ensuremath{-}}1}$ ${\mathrm{eV}}^{\mathrm{\ensuremath{-}}2}$ for Ag and Au and for the e-ph coupling ${\mathit{g}}_{\mathrm{\ensuremath{\infty}}}$=3.5\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\times\else\texttimes\fi{}${10}^{16}$ ${\mathrm{Wm}}^{\mathrm{\ensuremath{-}}3}$ ${\mathrm{K}}^{\mathrm{\ensuremath{-}}1}$ for Ag and 3.0\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\times\else\texttimes\fi{}${10}^{16}$ ${\mathrm{Wm}}^{\mathrm{\ensuremath{-}}3}$ ${\mathrm{K}}^{\mathrm{\ensuremath{-}}1}$ for Au.