Modified Drude model for small gold nanoparticles surface plasmon resonance based on the role of classical confinement

Analytical approach to the surface plasmon resonance characteristic of metal nanoparticle dimer in dipole-dipole approximation

“…where For R = 4nm, the data of single particle permittivity parameters, i.e. γ and ξ, have been taken from our previous work [32] and for R = 20nm these parameter are determined in the appendix. For comparison, we have plotted the extinction efficiency of a single NP when it does not interact with other NPs.…”

“…In a recent study [32], we have shown that based on a phenomenological Drude-like model which includes the role of classical confinement related to the appearance of restoration force caused by the displacement of conduction electrons with respect to the positive ionic background, the permittivity of a single NP can be determined properly through extracting free phenomenological parameters of ξ and γ by experimental data of extinction cross section of individual NPs. For small NPs these parameters are considered as a function of wavelength and NP radius.…”

“…Because of the symmetry of problem, dynamics of both NPs is the same, therefore, first, we consider only the dynamics of one of them. One can show that the total internal electric field related to the conduction electrons of MNP vanishes, thus we write the following momentum equation for the displacement of center of mass of conduction electrons of the first NP, r 1 , which in the rigid body approximation is equivalent to the displacement of electronic cloud of each NP from the equilibrium state [32]…”

“…where m e , e, γ and ξ are the electron mass, the magnitude of electron charge, damping coefficient and a function of NP radius obtained from the experimental data [32], respectively, ω p = n 0 e 2 /(m e ε 0 ) is plasma frequency of conduction electrons, n 0 is the density of conduction electrons and ε 0 is the vacuum permittivity. It is better to mention that the third term of the left side in Eq.…”

“…( 2) is related to the restoration force caused by the separation of electrons from the positive background ions and parameter ξ is for adjustment of experimental data with the theory. In most of theoretical studies related to the interaction of laser beam with MNPs [33][34][35][36][37][38][39][40], in an ideal case of rigid body approximation in which all conduction electrons are replaced equally, its value is 1/3, however in the real situation, its value depending on the diameter of NP is less than 1/3 and it vanishes for the large particles which shows the transition to bulk medium situation [32]. E 1 is the electric field at the place of the first NP which is the sum of the incident wave electric field E L and interaction term E 2→1 where E 2→1 is the electric field at the place of first NP r 01 caused by the electric dipole moment of the second NP (p 2 = −Zer 2 , where Z = n 0 V is the total conduction electrons of each NP and V is the volume of NP) that can be represented as…”