  The equilibration of electronic carriers in metals after excitation by an
ultra-short laser pulse provides an important class of non-equilibrium
phenomena in metals and allows measuring the effective electron-phonon coupling
parameter. Since the observed decay of the electronic distribution is governed
by the interplay of both electron-electron and electron-phonon scattering, the
interpretation of experimental data must rely on models that ideally should be
easy to handle, yet accurate. In this work, an extended rate-equation model is
proposed that explicitly includes non-thermal electronic carriers while at the
same time incorporating data from first-principles calculations of the
electron-phonon coupling via Eliashberg-Migdal theory. The model is verified
against experimental data for thin Pb films grown on Si(111). Improved
agreement between theory and experiment at short times (<0.3ps) due to
non-thermal electron contributions is found. Moreover, the rate equations allow
for widely different coupling strength to different phonon subsystems.
Consequently, an indirect, electron-mediated energy transfer between strongly
and weakly coupled groups of phonons can be observed in the simulations that
leads to a retarded equilibration of the subsystems only after several
picoseconds.
