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Quantum dissipation and decoherence of collective excitations in metallic nanoparticles

Abstract : The excitation of a nanoparticle by a laser pulse creates a collective mode of the electrons, the so-called surface plasmon. It decays because of surface effects and electron-electron interactions, creating particle-hole excitations (Landau damping). The thermal equilibrium of the electronic system is reached after about hundred femtoseconds, and only on a much larger time scale, the electron-phonon interactions permit the relaxation of the electronic energy to the ionic lattice.

Throughout this work we treat the metallic nanoparticle in the jellium approximation where the ionic structure is replaced by a continuous and homogeneous positive charge. Such an approximation allows to decompose the electronic Hamiltonian into a part associated with the electronic center of mass, a part describing the relative coordinates (treated here in the mean-field approximation), and finally a coupling between the two subsystems. The external laser field puts the center of mass into a coherent superposition of its ground and first excited state and thus creates a surface plasmon. The coupling between the center of mass and the relative coordinates causes decoherence and dissipation of this collective excitation.

We have developed a theoretical formalism well adapted to the study of this dissipation, which is the reduced-density-matrix formalism. Within the Markovian approximation, one is then able to solve analytically or numerically the corresponding equations. There are mainly two parameters which govern the surface plasmon dynamics: the decay rate of the plasmon, and the resonance frequency.

An experimentally accessible quantity is the photoabsorption cross section of the metallic cluster, where the surface plasmon excitation appears as a broad resonance spectrum. The width of the plasmon resonance peak is a quantity that one can determine in different manners. A numerical approach consists of the resolution of the time-dependent Kohn-Sham equations in the local density approximation. This yields the absorption spectrum for a given nanoparticle size, and one can then deduce the lifetime of the surface plasmon excitation. For nanoparticle sizes larger than approximately 1 nm, the width gamma of the peak follows Kawabata and Kubo's law which predicts that gamma is proportional to the inverse size of the nanoparticle. For sizes smaller than 1 nm, gamma presents oscillations as a function of the size, consistently with existing experimental data. By means of a semiclassical formalism, we have shown that those oscillations are due to the correlations of the density of states of the particles and holes in the nanoparticle. The semiclassical theory reproduces quantitatively the numerical calculations.

In addition to the width, we have also addressed the value of the resonance frequency. The classical electromagnetic Mie theory gives for the resonance frequency of the surface plasmon the plasma frequency of the considered metal, divided by a geometrical factor. However, the experimentally observed frequency is redshifted relative to the classical frequency. One usually attributes this shift to the spill-out effect that we have calculated semiclassically. The electronic density of the ground state extends outside of the nanoparticle, resulting in the decrease of the electronic density inside the cluster compared to its bulk value. This has the consequence to redshift the resonance frequency. We have shown by means of perturbative calculations that the coupling to the electronic environment produces an additional redshift of the surface plasmon resonance. This phenomenon is analogous to the Lamb shift in atomic systems. Both effects, spill-out and Lamb shift, have to be taken into account in the description of the numerical and experimental results.

Furthermore, we have extended our semiclassical calculations of the linewidth of the surface plasmon peak, of the spill-out, and of the environment-induced shift to the case of finite temperatures. We have shown that when the temperature increases, there is a broadening of the lineshape of the surface plasmon, as well as an additional redshift of the resonance frequency compared to the zero-temperature case. Even though the effect of the temperature is weak, it is essential for the comprehension of the electronic thermalization in pump-probe experiments. The study of the effect of the temperature has allowed us to qualitatively explain the
differential transmission curves measured in time-resolved experiments.
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Submitted on : Wednesday, October 4, 2006 - 2:08:27 PM
Last modification on : Friday, October 23, 2020 - 4:38:13 PM
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Guillaume Weick. Quantum dissipation and decoherence of collective excitations in metallic nanoparticles. Condensed Matter [cond-mat]. Université Louis Pasteur - Strasbourg I; Universität Augsburg, 2006. English. ⟨tel-00103438⟩

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