Quantum dissipation and decoherence of collective excitations in metallic nanoparticles
Quantum dissipation and decoherence of collective excitations in metallic nanoparticles.
Thèses de doctorat,
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Metallic nanoparticles are an ideal laboratory for the study of electronic correlations in the transition regime between microscopic and macroscopic systems. Consequently, their optical properties which depend on those correlations are currently intensively studied. Furthermore, the proposed applications of metallic nanoparticles (electro-optical devices, biological markers, nonvolatile memories, etc.) render crucial the comprehension of the optical properties of those objects. Pump-probe experiments permit to address directly the electronic degrees of freedom and to study the dynamics of the relaxation after a strong excitation in those mesoscopic systems. 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 electronelectron 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. The treatment of the surface plasmon as a quantum particle provides a model system for the study of decoherence and quantum dissipation in confined nanoscopic systems, where the role of the electronic correlations is preponderant. 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. Indeed, writing the general evolution of the density matrix of the total system, one can, by eliminating the environmental degrees of freedom (the relative coordinates in our case), deduce the equations of the temporal evolution of the center-of-mass system. Within the Markovian approximation (where the memory effects are neglected), one is then able to solve analytically or numerically these equations. There are mainly two parameters which govern the surface plasmon dynamics: the decay rate of the plasmon (its inverse giving the lifetime of the collective excitation), 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 (the decay rate) 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 (TDLDA).1 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 of the peak follows Kawabata and Kubo’s law which predicts that is proportional to the inverse size of the nanoparticle. For sizes smaller than 1 nm, presents oscillations as a function of the size, consistently with existing experimental data. By means of a semiclassical formalism using Gutzwiller’s trace formula for the density of states, 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. If one considers a noble-metal nanoparticle (where one has to take into account the screening of the s-electrons by the d-electrons) in an inert matrix (for example a glass matrix), we have shown that a naive application of the Kubo formula for the surface plasmon linewidth fails to reproduce the TDLDA numerical results, which are however consistent with experimental results. We have modified the Kubo theory in order to solve this discrepancy. Indeed, it is necessary to take into account the details of the mean-field potential (that one can obtain from the LDA calculations), especially its slope at the nanoparticle-matrix interface. If the intensity of the exciting laser field is sufficiently strong, one can ask the question if it is possible to have an excitation of the second quantum level of the center of mass, that we call double plasmon. This is possible if this second excited state is well defined, i.e., its width is sufficiently small compared to the other energy scales of the system. Up to now it has not been possible to answer this question from the experimental point of view, although indirect observations could render imaginable the existence of such a collective state. We have shown, by extending our semiclassical theory to the nonlinear case, that the double plasmon is indeed well defined. In certain cases, the electronic ionization can result from the excitation of the double plasmon, and this is observed in experiments. We have calculated the lifetime of the double plasmon associated to this second-order effect, and the obtained values are in qualitative agreement with the existing experiments. 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 √3. 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, spillout 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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