Anomaly in the relaxation dynamics close to the surface plasmon resonance

We propose an explanation for the anomalous behaviour observed in the relaxation dynamics of the differential optical transmission of noble-metal nanoparticles. Using the temperature dependences of the position and the width of the surface plasmon resonance, we obtain a strong frequency dependence in the time evolution of the transmission close to the resonance. In particular, our approach accounts for the slowdown found below the plasmon frequency. This interpretation is independent of the presence of a nearby interband transition which has been invoked previously. We therefore argue that the anomaly should also appear for alkaline nanoparticles. Absorption and transmission of a laser beam by a metallic nanoparticle unveil the properties of its conduction electrons. The use of femtosecond lasers in pump-probe spectroscopy then gives access to the electron dynamics on extremely short time scales [1–6]. In particular, the time dependence of the transmission spectrum during and after the excitation by the pump laser allows to follow the details of the electron relaxation. In pump-probe experiments, a pump laser with a wavelength much larger than the diameter of the nanoparticle couples to the centre of mass of the electron gas, thereby exciting the surface plasmon mode. The collective excitation decays on a very short timescale of the order of 10 fs. Subsequently, electron-electron interactions lead quite rapidly to the thermalisation of the electronic system at an elevated temperature. Only on longer timescales, of the order of 100 fs up to picoseconds, the electron-phonon coupling leads to an equilibration of the electronic system with the lattice degrees of freedom. In the case of noblemetal nanoparticles, this picture has to be completed by the possibility of interband transitions. In ref. [2] the relaxation of the differential transmission spectrum of copper nanoparticles was found to exhibit a slowdown close to the plasmon frequency. This feature persists far beyond the lifetime of the plasmon excitation and therefore was attributed to the slowdown of the energy transfer from the electrons to the lattice for frequencies close to the plasmon resonance. These considerations lead to the following puzzle: How can the plasmon reso∗Present address: Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany nance frequency play a role for relaxation processes which occur on timescales much larger than the lifetime of this excitation? The s-d multiband transition, which for copper has its onset close to the plasmon resonance, has been invoked as a possible explanation of the unexpected slowdown. A many-body effect based on the interband resonance scattering of d holes with surface plasmons was proposed to lead to a strong frequency dependence of the relaxation [7]. Subsequent experiments with silver nanoparticles also found the slowdown in the transmission dynamics [4]. Firstly, these experiments demonstrated the generality of the effect. Secondly, they established that the slowdown is only found above an excitation threshold. Finally, because the interband transition in silver is far away from the plasmon frequency [6], the explanation put forward in refs. [7] does not apply. In the present paper, we propose an explanation of the observed frequency dependence of the transmission dynamics which does not rely on interband transitions and therefore applies to both, copper and silver nanoparticles. Our mechanism is based on recently derived results for the finite-temperature corrections to the width and the position of the plasmon absorption peak [8]. We show that the temperature dependencies of the two quantities lead to an anomaly in the evolution of the differential transmission spectrum with decreasing temperature. In particular, a slowdown in the relaxation of the optical transmission appears close to the plasmon resonance. The quantity that is measured experimentally is the frequency-resolved time-dependent transmission Ton after