Dynamics of excited electrons in copper: The role of Auger electrons

Abstract

Within a theoretical model based on the Boltzmann equation, we analyze in detail the structure of the unusual peak recently observed in the energy dependence of the relaxation time in Cu. In particular, we discuss the role of Auger electrons in the electron dynamics and its dependence on the d-hole lifetime, the optical transition matrix elements, and the laser-pulse duration. We find that the Auger contribution to the distribution is very sensitive to both the d-hole lifetime h and the laser-pulse duration l , and that it largely dominates the excited-electron distribution for realistic parameters. It is shown that it can be expressed as a monotonic function of l / h and that for a given h , it is significantly smaller for a short pulse duration than for a longer one. Our results indicate that the relaxation time at the peak depends linearly on the d-hole lifetime, but interestingly not on the amount of Auger electrons generated. We provide a simple expression for the relaxation time of excited electrons which shows that the shape of the peak can be understood by a phase-space argument and its amplitude is governed by the d-hole lifetime. We also find that the height of the peak depends on both the ratio of the optical transition matrix elements Rϭ͉M d→sp ͉ 2 /͉M sp→sp ͉ 2 and the laser-pulse duration. Assuming a reasonable value for the ratio, namely, Rϭ2, and a d-hole lifetime of h ϭ35 fs, we obtain for the calculated height of the peak ⌬ th ϭ14 fs, in fair agreement with ⌬ exp Ϸ17 fs measured for polycrystalline Cu.

Distinguishing attosecond electron–electron scattering and screening in transition metals

“…From the above, we conclude that atto-ARPES can extend measurements of photoexcitedelectron lifetimes to higher energies (>20 eV) to distinguish and quantify fundamental electron interactions such as electron scattering and screening, as well as the influence of resonant interband transitions. Compared with other approaches, atto-ARPES also has the unique ability to distinguish band-specific electron-electron scattering for direct comparison with theory and can exclude other contributions such as hot electrons and intermediate-state refilling (6,8,48).…”

Response theory for time-resolved second-harmonic generation and two-photon photoemission

“…Of course, due to the sum over bands there are usually several contributions of this type. Only if the superpositions decay very rapidly compared to the pure states is a description in terms of rate equations, as suggested by figure 2, applicable [15,38,39]. Also compare the discussion of SFG above; see figure 4.…”

“…It only depends on the energy relaxation rates. This is the case where rate equations are appropriate [15,38,39]. Due to the vanishing polarization there are no interference effects.…”

“…The response theory presented here goes beyond previous theoretical treatments of ultrafast processes 1 in SFG and 2PPE in metals, which mainly fall into four classes: (a) density functional theory and approaches based thereon [28,[32][33][34][35][36][37], (b) rate equations [15,38,39], (c) optical Bloch equations [9,12,42], and (d) perturbative methods [26,31,43,44]. At first, density functional theory was applied in the time-dependent local density approximation for jellium models [32][33][34][35].…”

“…To this end one may consider rate equations for the occupation of excited states, e.g., the Boltzmann equation [15,38,39]. This approach allows one to incorporate important effects such as secondary electrons due to relaxation from higher-energy states and to Auger processes as well as transport into the bulk [15,38,39]. However, rate equations neglect the electric polarization of the electron gas, its dephasing, and any quantum-mechanical interference effects, resulting from the superposition of the laser field and the induced fields.…”