Analytical Approach to the Surface Plasmon Resonance Characteristic of Metal Nanoparticle Dimer in Dipole-Dipole Approximation

Abstract

This theoretical study deals with the effect of bi-particle interaction on the surface plasmon resonance (SPR) in a dimer which includes two identical metal nanoparticles (NPs). Considering the dipole-dipole interaction in a Drude-like model, an appropriate equation is derived for the permittivity of each NP. The restoration force related to the classical confinement originating from the finite size of NPs is considered, and an appropriate adjustment coefficient is considered for this term through analyzing experimental data. Two different polarizations are considered for the laser beam electric field, and it is shown that the orientation of the electric field has an essential role in the linear optical properties of a dimer. Numerical investigation is accomplished for a dimer of gold NPs with two different diameters of 4 nm and 20 nm. For the parallel polarization, dipole-dipole interaction leads to the redshift of SPR wavelength and increase in its peak value, while for the perpendicular polarization, the absolute opposite results are derived. For all cases, it is shown that SPR wavelength functionality with respect to the geometric factor a/d (NP radius to the separation) can be presented by a cubic equation that fits better than an exponential one suggested by the earlier studies which demonstrates the dipole-dipole characteristic of the interaction. Qualitatively, our results are in good agreement with the other experimental studies.

Appendix A: Single Gold NP Permittivity

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. In this Appendix, we use the same approach in order to find permittivity of large gold NPs ranged from 20 to 90 nm whose extinction cross-section can be found in experimental investigations [45, 46]. The value of the free parameters ξ and γ in the permittivity of an NP with a given diameter are suggested through a trial and error process in order to by applying this permittivity in the extinction cross-section of Eq. (20), good agreement reveals between experimental and theoretical data. In Fig. 5, the extinction efficiency of a single gold NP suspended in a water medium has been plotted for different sizes of spherical NPs. The dotted curves are obtained experimentally [45] and the solid lines show our model results. For the model, we have used the phenomenological parameters as follows:

$$ \begin{array}{@{}rcl@{}} \xi = {\xi_{0}} + {c_{1}}\frac{1}{a} + {c_{2}}\frac{1}{{{a^{2}}}} + {c_{3}}\frac{1}{{{a^{3}}}}, \end{array} $$

(27)

Fig. 5

The calculated extinction efficiency (solid lines) and the experimentally measured ones (dotted line) in dependence of wavelength for NPs diameters of 20, 30, 40, 50, 60, 70, 80, and 90 nm (the order of diameter increase is from bottom to top)

where

$$ \begin{array}{@{}rcl@{}} &&{\xi_{0}} = - 0.950 \times {10^{- 3}}, {c_{1}} = 5.44~\text{nm}, \\ &&{c_{2}} = - 1.17n{m^{2}}, {c_{3}} = 0.08~\text{nm}{^{3}}, \end{array} $$

(28)

and a is in nm. For the damping factor γ of NP, we use the following expression:

$$ \begin{array}{@{}rcl@{}} \gamma = {\gamma_{0}} + {\gamma_{surf}}, \end{array} $$

(29)

where γ₀ = 0.7 × 10¹⁴s^− 1 and γ_surf = Av_F/a is the electron damping coefficient caused by the scattering of electron by the surface of NP which is called the mean free path limitation effect [47, 48]; here, A = 0.25 is a dimensionless parameter and v_F is the gold Fermi velocity [49]. It should be mentioned that there is a deviation between our modeled extinction cross-section and the experimental data. Such deviations commonly are existed in all theoretical works because of the fact that including all classical and quantum effects in one simple model like Drude model practically is not possible and all efforts are focused on getting approximately correct peak place in the extinction cross-section.