Unified theoretical description of out-of-equilibrium electron intraband dynamics in gold induced by femtosecond laser pulses

Stéphane Coudert, Stefan Dilhaire, Philippe Lalanne, and Guillaume Duchateau

Phys. Rev. B 106, 134308 – Published 25 October 2022

Abstract

The modeling of the intraband dynamics of conduction electrons in gold induced by a femtosecond laser pulse is addressed. Current approaches, based on the numerical resolution of the Boltzmann equation, are only able to describe electron excitation $or$ relaxation processes. In the present paper, within a single formalism, our kinetic model accounts quantitatively for both excitation and relaxation processes, i.e., photon absorption, thermal conductivity, and electron-phonon coupling coefficient. We suggest that such an approach can only be built by including Umklapp processes in the description of electron collisions. In addition to normal electronic collisional processes with phonons and other electrons, the present theoretical description further includes these mechanisms: (i) a conduction electron can be scattered in the second Brillouin zone after a collision, contributing significantly to the photon absorption, and (ii) the influence of the discrete and periodic nature of the lattice on electron collisions is considered, enabling in particular the coupling between electron and transverse phonons. A good agreement of predictions of this unified modeling with experimental observations for absorption and energy transfer from electrons to the lattice is obtained. We have investigated the influence of the Umklapp processes on the imaginary part of the dielectric function, the electron-phonon coupling parameter, and the transient shape of the electron energy distribution.

Received 22 July 2022
Revised 5 October 2022
Accepted 18 October 2022

DOI:https://doi.org/10.1103/PhysRevB.106.134308

Physics Subject Headings (PhySH)

Acoustic phonons Carrier dynamics Electron relaxation Heat transfer Photoinduced effect

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Stéphane Coudert^1,2, Stefan Dilhaire³, Philippe Lalanne⁴, and Guillaume Duchateau ^1,2,*

¹CEA-CESTA, 15 Avenue des Sablières, CS60001, 33116 Le Barp Cedex, France
²Université de Bordeaux-CNRS-CEA, Centre Lasers Intenses et Applications - CELIA, 33405 Talence, France
³Laboratoire Onde et Matière d'Aquitaine (LOMA), UMR 5798, CNRS-Université de Bordeaux, 33400 Talence, France
⁴Laboratoire Photonique, Numérique et Nanosciences (LP2N), UMR 5298, CNRS-IOGS-Université de Bordeaux, Institut d Optique d Aquitaine, 33400 Talence, France

^*Corresponding author: guillaume.duchateau@cea.fr

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Vol. 106, Iss. 13 — 1 October 2022

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Images

Figure 1
Schematic representation of the diffusion of an electron by phonon Umklapp processes. An electron in the initial state $k$ absorbs a phonon with crystalline momentum $q$ and reach the state $k^{'} = k + q$ . It is well known that such normal diffusion process is possible only if the phonon has a longitudinal polarization. However, phonons potential exhibits components in higher Brillouin zone than the first one. Thus, diffusion with transverse phonons may occurs and must be taken into account.
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Figure 2
Imaginary part of the intraband permittivity as a function of the photon energy. The dashed-black curve corresponds to experimental data of Babar and Weaver [36]. The other curves are obtained with the present modeling where various processes are step-by-step considered [Eqs. (18) and (19)]. The orange curve is obtained with N processes only. The green curve further includes the $U_{2}$ process. In addition to previous processes, the $U_{1}$ e-pt-ph is included to obtain the red curve. The whole modeling is used for the blue curve.
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Figure 3
Total electron-phonon collision frequencies with respect to electron energy accounting for $N$ processes only (red line), $N + U_{1}$ processes (green line), $N + U_{2}$ processes (orange line), $N + U_{1} + U_{2}$ processes (blue line). These collision frequencies have been calculated with $F = 0$ in Eq. (4). The inset shows the separate contribution of $S^{+}$ and $S^{-}$ for normal processes (units are the same as the main graph).
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Figure 4
Perturbations of electron distribution function with a same total absorbed energy (associated to an increase of electron temperature about 40 K) induced by an increase of the electron temperature (green line), electron-photon-phonon processes (orange line), and electron-photon-electron Umklapp processes (blue line). Note that these perturbations are obtained just after the excitation process, without any relaxation process.
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Figure 5
Perturbations of the electronic distribution function for a same total absorbed energy ( $100 J / {cm}^{3}$ deposited in $20 fs$ , see text) as predicted by the present modeling. Solid lines stand for the case where only $e - p t - p h$ processes have been considered for photon absorption while dashed lines stand for the case where only $e - p t - e$ processes have been considered for photon absorption. Only electron-electron relaxation processes has been considered [electron-phonon relaxation takes place on longer time scales ( $t > 500$ fs)]. Each color stands for a given simulation time.
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