Abstract
In this paper, we study the effect of restoration force caused by the limited size of a small metallic nanoparticle (MNP) on its linear response to the electric field of incident light. In a semi-classical phenomenological Drude-like model for small MNP, we consider restoration force caused by the displacement of conduction electrons with respect to the ionic positive background taking into account a free coefficient as a function of diameter of nanoparticle (NP) in the force term obtained by the idealistic Thomson model in order to adjust the classical approach. All important mechanisms of the energy dissipation such as electron-electron, electron-phonon and electron-NP surface scatterings and radiation are included in the model. In addition a correction term added to the damping factor of mentioned mechanisms in order to rectify the deficiencies of theoretical approaches. For determining the free parameters of model, the experimental data of extinction cross section of gold NPs with different sizes doped in the glass host medium are used and a good agreement between experimental data and results of our model is observed. It is shown that by decreasing the diameter of NP, the restoration force becomes larger and classical confinement effect becomes more dominant in the interaction. According to experimental data, the best fitted parameter for the coefficient of restoration force is a third order negative powers function of diameter. The fitted function for the correction damping factor is proportional to the inverse squared wavelength and third order power series of NP diameter. Based on the model parameters, the real and imaginary parts of permittivity for different sizes of gold NPs are presented and it is seen that the imaginary part is more sensitive to the diameter variations. Increase in the NP diameter causes increase in the real part of permittivity (which is negative) and decrease in the imaginary part.
Highlights
In this paper, we study the effect of restoration force caused by the limited size of a small metallic nanoparticle (MNP) on its linear response to the electric field of incident light
For an individual MNP, beside the occurrence of additional surface scattering caused by the limitation of the free path of conduction electrons, restriction of size of particle should lead to the appearance of restoring force caused by the displacement of electrons with respect to the background positive charges via exerting electric field of Electromagnetic Waves (EMWs)
In some studies related to the interaction of EMWs with systems including MNPs, in classical momentum equation, the restoring force which can predict the resonance frequency is considered[53,54,55,56,57,58,59,60,61,62,63,64,65,66] but surprisingly it is absent in the dynamics of the most of the studies related to the Drude-based problems and as it is mentioned in the previous paragraph, they considered only the free path limitation on the dynamics of conduction electrons in MNPs
Summary
We use classical Thomson model[73] for describing interaction of EMW with spherical MNP Even though this model was unsuccessful for describing atomic structure but it is still convenient for constructing classical theories in the light-cluster interaction phenomena[2]. In this model, it is supposed that conduction electrons of N individual atoms are homogenously distributed inside a sphere with radius R and background positively charged ions which are distributed homogenously as well, are immobile. Using it in Eq (1) after doing summation on motion equations of the whole electrons existing inside the NP, one can reach to the following well-known damped harmonic oscillation equation for the center of mass displacement. Where the term related to the electron-electron interactions vanishes during summation process because of the action-reaction law
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