Introduction

Investigations dealing with the interaction of Electromagnetic Waves (EMWs) with metal Nanoparticles (NPs) and nanostructures are very attractive because of their fascinating applications in science and technology. The noble Metal Nanoparticles (MNPs) show a resonant interaction with EMWs in the visible spectrum which makes them ideal candidates for some special exotic applications in the industry and medicine. The origin of resonant interaction of MNPs is the collective oscillation of surface conduction free electrons with respect to the positive metallic lattice under interaction with light fields, i.e. surface plasmon. Occurrence of tremendous Electromagnetic (EM) fields enhancements in the resonance leads to the intense scattering and absorption of light1,2,3 for MNPs. This plasmon resonance can either cause the light radiation via Mie scattering4, or the rapid conversion to heat through resonant absorption where both of mentioned processes play great roles in some new cutting-edge technological applications. Among the numerous applications of surface plasmon resonance of noble MNPs in new fields of science and technology, few important applications can be named as: localized surface plasmon resonance sensing5, surface-enhanced Raman scattering spectroscopy6, surface-enhanced infrared absorption spectroscopy7,8, enhanced nonlinear wave mixing9,10, nano-scaled emission engineering, i.e. nano-antenna11, optoelectronic devices12, optical metamaterials13,14, solar cells15, frequency-sensitive photodetectors16, wavefront engineering of semiconductor lasers17, and molding light propagation at engineered interfaces18. Beside the mentioned feasibilities of applications for noble MNPs related to their enhanced absorption and scattering, additionally, compositions of gold NPs are more suitable in medicine due to good biocompatibility, easy production19 and ability to conjugate to a variety of biomolecular ligands and antibodies20 which make them very useful for NP-based cancer therapy21,22,23,24,25,26,27, biological sensing28,29, imaging of bio-materials30,31,32 and medical diagnostics33.

Theoretical or experimental determination of complex dielectric permittivity or equivalently refractive index of media containing MNPs is necessary for explanation of dynamics of a lot of fundamental and applied phenomena and effects including the above mentioned subjects related to the absorption and scattering of light. In 1908, in the framework of classical electrodynamics by solving Maxwell’s equations, Mie4 could obtain an analytical expression for extinction coefficient of a spherical particle describing the dissipation of light by absorption and scattering. Up to date, Mie’s calculations are the basis of the most of mathematical processes related to the interaction of NPs with the fields of EMWs. The crucial parameter needed for the extinction coefficient obtained by the Mie theory is the complex permittivity of MNPs which is calculated by the well-known phenomenological Drude theory. Theoretical considerations and experimental investigations have revealed that optical properties of NPs should dramatically depend on the size and shape of NPs. In several experimental studies, since 1958, Fragstein with his coworkers Roemer34,35 and Schoenes36 have investigated the complex refractive index of silver NPs dispersed in colloidal solutions and determined the considerable differences between them and permittivity of bulk medium when the NPs dimensions were smaller than the mean free path of the conduction electrons. Then by Gans and Happel37, the effect of particles shape on the linear optical constants of such MNPs solutions has been calculated via Mie theory, so that one could determine the contribution of size and shape changes in the refractive index, separately. They measured linear optical constants of gold hydrosols and discovered that the changes in these values with respect to the bulk state are considerable only for the NPs including few atoms then they made conclusion that this difference arises from the additional electron scattering mechanism caused by the collision of conduction electrons with the particle surface that reduces their effective mean free path which is called the “free path effect”. To date, the free path effect is recognized as the main responsible factor for interpreting changes in the optical constants of MNPs with respect to the bulk state. There are a lot of theoretical and experimental studies where the modified Drude model is used for showing size dependence of MNPs permittivity only by adding free path effect on the damping coefficient of conduction electrons, just as a few examples see38,39,40,41,42,43,44,45,46,47,48,49,50,51,52.

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 EMW. Such a linear restoring force reveals in atomic clusters using Thompson model2. Considering restoring force in dynamic equation of conduction electrons leads to the appearance of the resonance characteristic frequency of \({\omega }_{res}={\omega }_{p}/\sqrt{3}\), where ωp is the plasma frequency of conduction free electrons2. Sometimes ωres is called the classical surface-plasmon frequency. Considering Mie theory for absorption of light by small MNPs with permittivity ε doped in a background medium with permittivity εm, such a resonance frequency can be predicted when ε = −2εm4. In the simplifying limit of \(\Gamma \ll {\omega }_{p}\), where Γ is the damping factor related to the various mechanisms of electron scattering, this condition leads to the maximum absorption of light at the resonance frequency \({\omega }_{res}={\omega }_{p}/\sqrt{1+2{\varepsilon }_{m}}\) which reduces to \({\omega }_{p}/\sqrt{3}\) in the case of considering air as the surrounding medium, i.e. εm = 14. It would be interesting to know that in such resonance condition, the polarizability of an MNP exposed to an electric field shows resonant behavior as well2. All the mentioned facts in this paragraph, may emphasize on the importance of considering restoring force in dynamical problems related to the interaction of EMWs with MNPs in the classical regime. 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 considered53,54,55,56,57,58,59,60,61,62,63,64,65,66 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. In some studies related to the plasmonics of metallic nanostructures, for extraction of permittivity of system by Drude-based models, in the equation of motion of damped harmonic oscillation, the restoring force is considered and the related spring constant is determined by simulation67,68,69,70,71,72. In this article using a simple Drude-like model that considers the restoration force, the complex permittivity of an individual gold NP is studied. Important mechanisms of conduction electron scattering including electron-electron, electron-phonon, electron-surface and radiation as well are considered via well-known theoretical relations and a correction term is added in order to correct some shortages of theories. Also, a correction coefficient is considered for the restoring force in order to rectify the shortages of ideal model of Thompson. Estimation for free parameters of system is accomplished by considering experimental data for extinction coefficient of gold NPs with different sizes doped in the glass. Results show a good agreement between experiments and our model.

Thomson model for small MNP

We use classical Thomson model73 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 phenomena2. 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. If the average separation of atoms is d, then the density of atoms or equivalently the density of background ions is na = 1/d3 while we denote the density of conduction electrons as \({n}_{e}=ZNe/[(4/3)\pi {R}^{3}]\), where e is the magnitude of electron charge and Z is the number of conduction electrons for each atom. For a small NP exposed to the low-intensity EM fields where its radius is much less than the wavelength, \(R\ll \lambda \), one can neglect the spatial variation of EM fields inside the NP and suppose that the same forces act on all conduction electrons at a moment. The motion equation of electrons inside the NP confined to the radius R, can be written as

$$\begin{array}{c}m{\ddot{{\bf{x}}}}_{{\bf{i}}}=-e{{\bf{E}}}_{{\bf{0}}}({{\bf{x}}}_{{\bf{i}}},t)-e{{\bf{E}}}_{{\bf{p}}}({{\bf{x}}}_{{\bf{i}}},t)-m{\boldsymbol{\Gamma }}{\dot{{\bf{x}}}}_{{\bf{i}}}+\frac{1}{4\pi {\varepsilon }_{0}}\mathop{\sum }\limits_{j\ne i}^{ZN}\frac{{e}^{2}}{{|{{\bf{x}}}_{{\bf{i}}}-{{\bf{x}}}_{{\bf{j}}}|}^{3}}({{\bf{x}}}_{{\bf{i}}}-{{\bf{x}}}_{{\bf{j}}}),\,\\ \,i\mathrm{=1,2,...,}ZN,\end{array}$$
(1)

where m, xi, ε0 and Γ are electron mass, the ith electron position vector, permittivity of vacuum, and damping factor related to any kind of energy dissipation mechanisms, respectively. At the right side of Eq. (1), the first term is the force of external electric field, the second term describes the force caused by the background positive ions and the last term denotes the electron-electron interaction. Using Gauss’s law, one can easily obtain the following equation for the electric field of the background ions

$${{\bf{E}}}_{{\bf{p}}}=\frac{e{n}_{e}}{3{\varepsilon }_{0}}{{\bf{x}}}_{{\bf{i}}}=\frac{m{{\omega }_{p}}^{2}}{3e}{{\bf{x}}}_{{\bf{i}}},$$
(2)

where \({\omega }_{p}=\sqrt{{n}_{e}{e}^{2}/m{\varepsilon }_{0}}\) is the plasma frequency. Introducing the well-known concept of center of mass for conduction electrons as

$${\bf{X}}=\frac{1}{ZN}\mathop{\sum }\limits_{i=1}^{ZN}\,{{\bf{x}}}_{{\bf{i}}},$$
(3)

and 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

$$\ddot{{\bf{X}}}+\Gamma \dot{{\bf{X}}}+\frac{{{\omega }_{p}}^{2}}{3}{\bf{X}}=-\frac{e}{m}{{\bf{E}}}_{{\bf{0}}}({\bf{t}}),$$
(4)

where the term related to the electron-electron interactions vanishes during summation process because of the action-reaction law.

Modified Drude model

For a metal bulk system, considering free electrons via ignoring the third term in the left side of Eq. (4) and taking into account a monochromatic field oscillating with frequency ω, i.e. \({E}_{0} \sim {e}^{-i\omega t}\), one can obtain the following equation for the permittivity of bulk metal

$${\varepsilon }_{bulk}=1-\frac{{{\omega }_{p}}^{2}}{{\omega }^{2}+i{\Gamma }_{bulk}\omega },$$
(5)

where we used index “bulk” to distinguish bulk system with confined one. Considering the role of inner electrons in atoms, Eq. (5) reads

$${\varepsilon }_{bulk}={\varepsilon }_{\infty }-\frac{{{\omega }_{p}}^{2}}{{\omega }^{2}+i{\varGamma }_{bulk}\omega },$$
(6)

where ε∞ reflects interaction of inner electrons with light and itself can be written in the form4

$${\varepsilon }_{\infty }=1+\mathop{\sum }\limits_{j=1}^{{N}_{o}}\frac{{{\omega }_{pj}}^{2}}{({{\omega }_{j}}^{2}-{\omega }^{2})-i{\Gamma }_{j}\omega },$$
(7)

where No denotes the number of Lorentz oscillators, j presents the special kinds of electrons located at inner levels having similar behavior during interaction with light fields, ωpj, ωj and Γj are the plasma frequency related to the special kind of electrons population, their resonant frequency and their damping factor, respectively, which can be measured experimentally.

For an NP with limited size, in Eq. (4), considering the third term related to the restoration force leads to the special resonance frequency for free electrons at \({\omega }_{p}/\sqrt{3}\) which called plasmon frequency. As experimental measurements show, the place of plasmon frequency extremely differs from \({\omega }_{p}/\sqrt{3}\) which can be referred to the existence of electron damping. As mentioned in the introduction section, in the most of works related to the calculation of optical parameters of NPs, restoring force is ignored from dynamical models and the size effect is only considered in damping factor by introducing new so-called surface scattering mechanism which is caused by the limitation of mean free path of electrons confined in an NP. Even though, for NP whose radius is greater than or comparable with light wavelength, considering the idealistic model in which all conduction electrons treat in the same manner, cannot be correct, however effect of background ions on electrons which reflects the classical confinement characteristic of system cannot be ignored. Here, we consider the restoration force by multiplying it with a coefficient which is a function of radius and introduce the permittivity of an individual NP as

$${\varepsilon }_{n}={\varepsilon }_{bulk}+\frac{{{\omega }_{p}}^{2}}{{\omega }^{2}+i{\Gamma }_{bulk}\omega }-\frac{{{\omega }_{p}}^{2}}{{\omega }^{2}-\alpha {{\omega }_{p}}^{2}+i{\Gamma }_{n}\omega },$$
(8)

where the sum of two first terms of the right hand is ε∞ for the bulk metal, \({\Gamma }_{Bulk}=0.07\times {10}^{15}{s}^{-1}\)74 and the last term denotes the contribution of conduction electrons of limited NP. α is a function of NP radius which should vanish for large particles and in ideal case of zero radius should limit to the well-known value of \(\mathrm{1/3}\). Γn stands for the damping factor of electrons in a confined region of NP and will be calculated as

$${\varGamma }_{n}={\varGamma }_{e-e}+{\varGamma }_{e-ph}+{\varGamma }_{rad}+{\varGamma }_{surf}+{\varGamma }_{cor},$$
(9)

Γe−e is the damping factor related to the scattering of an electron by another one in a bulk lattice of metal which can be calculated by the well-known theoretical relationship derived by Lawrence and Wilkins75,76

$${\varGamma }_{e-e}=\frac{{\pi }^{2}}{24\hslash {E}_{F}}[{({k}_{B}T)}^{2}+{(\hslash \omega )}^{2}],$$
(10)

where \(\hslash =h/(2\pi ),\,h\) is the Planck’s constant, EF, kB and T are the Fermi energy, the Boltzmann’s constant and temperature, respectively.

Γe−ph is the damping factor concerned with the energy dissipation due to the interaction of conduction electrons with metallic lattice which is obtained using the theoretical relation derived by Holstein77,78 as following

$${\Gamma }_{e-ph}={\Gamma }_{0}\left(\frac{2}{5}+\frac{4{T}^{5}}{{\theta }_{D}^{5}}{\int }_{0}^{\frac{{\theta }_{D}}{T}}\frac{{z}^{4}}{{e}^{z}-1}dz\right),$$
(11)

where θD is the Debye temperature and Γ0 is a constant that can be achieved through the fitting procedure of the bulk permittivity for the frequency interval which is located below the interband transition threshold.

Γrad denotes the damping factor caused by the radiation of accelerated electrons and it can be derived by classical electrodynamic considerations using Abraham-Lorentz force as the following simple relation79

$${\Gamma }_{rad}=\frac{{\omega }_{p}^{2}V{\omega }^{2}}{6\pi {c}^{3}},$$
(12)

where \(V=(4/3)\pi {R}^{3}\) is the NP and c is the light speed in vacuum.

The damping factor Γsurf is related to the limitation of mean free path of electrons bounded inside an MNP. It can be calculated through the scattering mechanism of a free electron by the surface of MNP. We use the following formula for considering contribution of size limitation in the energy dissipation80

$${\Gamma }_{surf}=\frac{A{v}_{f}}{{L}_{eff}}.$$
(13)

where A is a dimensionless parameter whose value can be obtained considering some details of the scattering mechanism and has own scientific story which will be mentioned briefly, \({v}_{f}=1.4\times {10}^{6}m/s\) is the Fermi velocity for gold and Leff is the reduced mean free path of electrons. Employing a geometrical probability method, Coronado and Schatz81 extracted the following simple equation for the reduced mean free path Leff = 4 V/S, where V and S are the and surface area of an NP with arbitrary curved shape. This term is related to totally inelastic scattering of conduction electrons by the surface of MNP and states the mean chord distance of any two arbitrary points located on the particle surface.82 Even though we know that parameter A should empirically have relation with the shape of MNP, however determining its value is the place of argument. In ref. 81, a value near to unity is suggested to the parameter A, but value A = 0.33 is proposed by Berciaud et al.83 by fitting values of parameters related to the absorption of light by the single gold NP. In the case of interaction between NP and surrounding medium, the quality of surface scattering will be changed and therefore additional damping mechanism should be considered in the surface scattering. This type of scattering is referred as Chemical Interface Damping (CID). First, by comparing the plasmon line widths related to the scattering of light by silver NPs embedded in a SiO2 matrix, Kreibig and co-workers suggested that CID occurs due to the plasmon decay by coupling to interfacial electronic states.2,39,84 Another scenario for medium effect on the electron surface scattering is the electron transfer from NP to the host medium or vice versa from medium to the NP. Recently, Wu et al. proposed an interfacial charge transfer mechanism to justify the high efficiency of electron transfer from a gold NP to an adjacent CdSe nanorod85 under interaction with photons. Generally, coefficient A can be considered as the contribution of electron surface scattering in free space (Asize) and additional effect of surrounding bound on this scattering (Ainterface), i.e. A = Asize + Ainterface. For small gold NPs embedded in glass matrix, experimental results show that the best value of A is 1.486.

Γcor is a correction term which we add to the damping factor in order to adjust theoretical formulae by considering experimental data. Two first terms of Eq. (9) are related to the dominant dissipation mechanisms for the bulk metal system which should be corrected for the limited size of an NP because of the quantum confinement effects. On the other hand some mechanisms like existence of defects in the lattice does not take into account. Therefore, in order to match theoretical results with experimental ones we will determine this term by trial and error method which will be explained in numerical section in detail.

Numerical Analysis

To extract the free parameters of the model, i.e. α and Γcor, we use some experimental data related to the linear interaction of light with NPs. Extinction cross section of media including MNPs is the well-known experimental data that we can employ them to guess the model parameters. The extinction cross section is defined as the sum of the absorption cross section and the scattering one which can be expressed as following by using Mie theory4

$${C}_{ext}=\frac{2\pi }{{k}^{2}}\mathop{\sum }\limits_{n=1}^{\infty }\,(2n+1)\mathrm{Re}({a}_{n}+{b}_{n}),$$
(14)
$${a}_{n}=\frac{m{\psi }_{n}(mx){\psi {\prime} }_{n}(x)-{\psi }_{n}(x){\psi {\prime} }_{n}(mx)}{m{\psi }_{n}(mx){\xi {\prime} }_{n}(x)-{\xi }_{n}(x){\psi {\prime} }_{n}(mx)},$$
(15)
$${b}_{n}=\frac{{\psi }_{n}(mx){\psi {\prime} }_{n}(x)-m{\psi }_{n}(x){\psi {\prime} }_{n}(mx)}{{\psi }_{n}(mx){\xi {\prime} }_{n}(x)-m{\xi }_{n}(x){\psi {\prime} }_{n}(mx)},$$
(16)

where \(x=2\pi NR/\lambda \) is the size parameter, \(m={N}_{1}/N\) is the relative refractive index, where N1 and N are the refractive indices of particle and background medium, respectively, \({\psi }_{n}(x)\) and \({\xi }_{n}(x)\) are Riccati-Bessel functions.

For small particles, i.e. R/λ = 1, the extinction cross section reduces to

$$\begin{array}{c}{C}_{ext}=4x\pi {R}^{2}{\rm{Im}}\{\frac{{m}^{2}-1}{{m}^{2}+2}[1+\frac{{x}^{2}}{15}\left(\frac{{m}^{2}-1}{{m}^{2}+2}\right)\times \frac{{m}^{4}+27{m}^{2}+38}{2{m}^{2}+3}]\}\\ \,+\frac{8}{3}{x}^{4}{\rm{Re}}\left[{\left(\frac{{m}^{2}-1}{{m}^{2}+2}\right)}^{2}\right]\end{array}$$
(17)

which in the case of very small particles or linear regime of x, limits to the well-known relationship for Rayleigh scattering as following

$${C}_{ext}=4x\pi {R}^{2}{\rm{Im}}\left(\frac{\varepsilon -{\varepsilon }_{m}}{\varepsilon +2{\varepsilon }_{m}}\right).$$
(18)

In Fig. 1, the extinction cross section of an individual gold NP doped in a glass background medium has been plotted for different sizes of small spherical NPs (2R < 10 nm). The dotted lines are obtained experimentally by Kreibig and Vollmer2 and the solid ones show our model results. Trial and error procedure for obtaining best fit for the extinction cross section reveals that the best fitted functions for α and Γcor are as following

$$\alpha ={\alpha }_{0}+{c}_{1}\frac{1}{R}+{c}_{2}\frac{1}{{R}^{2}}+{c}_{3}\frac{1}{{R}^{3}},$$
(19)
$${\Gamma }_{cor}\times {10}^{-20}={d}_{1}\frac{1}{{\lambda }^{2}}+{d}_{2}\frac{{R}^{2}}{{\lambda }^{2}}+{d}_{3}\frac{{R}^{3}}{{\lambda }^{2}},$$
(20)

where

$${\alpha }_{0}=-\,0.275,\,{{\rm{c}}}_{1}=1.751\,nm,\,{{\rm{c}}}_{2}=-\,3.659\,n{m}^{2},\,{{\rm{c}}}_{3}=2.571\,n{m}^{3},$$
(21)
$${d}_{1}=-\,2.169\,n{m}^{2},\,{d}_{2}=7.638\times {10}^{-3},\,{d}_{3}=-\,1.252\times {10}^{-2}\,n{m}^{-1},$$
(22)

and all lengths are in nm.

Figure 1
figure 1

The calculated extinction cross section (solid line) and the experimentally measured one (dotted line) in dependence of wavelength for NP diameters of 4, 4.4, 4.7, 4.9, 5.2, 5.4 nm (from bottom to top).

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It should be better to mention that we could choose other functionalities for free parameters in order to exactly fit the model and experimental data, however we choose the above forms because of their simplicity and being physically meaningful as well. The first and third terms of Γcor have the same form of the electron-electron and radiation scattering terms with respect to the radius of NP and light wavelength. These terms are negative and it indicates that the total amounts of the mentioned scattering terms become smaller by considering experimental corrections. The second term in Eq. (20) which is positive, has the functionality form of (R/λ)2 and it can be interpreted as the contribution of other ignored mechanisms of scattering and quantum corrections as well. The parameter α is the function of negative powers of R and it is independent of wavelength. In Fig. 2, the parameter α is presented as a function of NP diameter in nanometer. Increase in the NP diameter causes the decrease in α which is a logical behavior. Its value decreases approximately from 0.008 to 0.0025 when NP diameter increases from 4 nm to 5.4 nm, respectively. As it is mentioned earlier, we expect that by growing the size of NP, classical confinement effect (or in other words, appearing the restoring force) fades out and for large size NPs, it vanishes. The largest value for α which can be predicted by primitive classical theories is 1/3.

Figure 2
figure 2

Variations of parameter α as a function of NP diameter.

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In Fig. 3a,b, we have plotted Γcor and Γn as a function of light wavelength for different NP sizes. For all cases, Γcor is negative and its absolute value is an increasing function of NP diameter which in turn causes that the parameter Γn becomes a decreasing function of NP size. It is clear that increase in the wavelength causes the increase in both parameters Γcor and Γn at a fixed NP diameter size. The average value of Γn for small size NPs varies from 0.3 × 1015 s−1 to 0.8 × 1015 s−1 when wavelength changes from 450 nm to 650 nm, respectively.

Figure 3
figure 3

Variations of (a) ΓCor and (b) Γn as a function of wavelength for nanosphere diameters of 4, 4.4, 4.7, 4.9, 5.2, 5.4 nm. The order of diameter increase is from top to bottom.

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In Fig. 4a,b, the real and imaginary parts of gold NP permittivity is presented for different small size NPs and bulk gold metal as well. The data of bulk medium have been taken from ref. 87 As an example for metal, the real part of permittivity is negative for all cases and in a fixed wavelength, increase in the diameter of NP causes the decrease in the absolute value of real part. It is clear from Fig. 4b that increase in the size of NP causes the decrease in the imaginary part of permittivity. Totally, variations of permittivity values with respect to the NP size variations are more considerable for the imaginary part which reflects the absorption characteristic of medium. Especially, dependence of imaginary part on the NP size is more evident at high wavelengths or low photon energies. There are no experimental data for direct measurement of gold NP permittivity and only the permittivity of gold thin films can be found in refs. 87,88,89,90,91,92,93,94 Recently, Karimi et al.95 have been proposed a size-dependent plasma frequency model for MNP permittivity in quantum regime and similar theoretical results are obtained for the real and imaginary parts of gold small NPs. They interpreted the intense dependence of imaginary part on the NP size at low photon energies as the result of the intense increase in the surface scattering of NP.

Figure 4
figure 4

Variations of (a) real and (b) imaginary parts of permittivity as a function of wavelength for nanosphere diameters of 4, 4.4, 4.7, 4.9, 5.2, 5.4 nm and bulk case as well. The order of diameter increase for graphs (a) is from bottom to top and for graphs (b) is inversely.

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In order to show the size-dependence of plasmon resonance predicted by our semi-classical phenomenological model and compare them with experiments, in Fig. 5, we have plotted the plasmonic peak wavelength (or surface plasmon resonance peak wavelength) as a function of NP diameters. Increase in the NP diameter increases the resonance wavelength, or in other words causes a redshift in resonance wavelength. One can see the good agreement between our model and experimental data. The dependence of peak place to the NP diameter can be expressed via the following function

$${\lambda }_{{\rm{\max }}}={\lambda }_{0}+{f}_{1}R+{f}_{2}{R}^{2}+{f}_{3}{R}^{3},$$
(23)

where

$${\lambda }_{0}=303.90\,nm,\,{f}_{1}=303.20\,n{m}^{-1},\,{f}_{2}=-\,146.65\,n{m}^{-2},\,{f}_{3}=24.28\,n{m}^{-3},$$
(24)

where λmax and R are in nm.

Figure 5
figure 5

Variations of surface plasmon resonance wavelength as a function of NP diameter. Dashed line is the fitted function.

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Conclusions

In a semi-classical phenomenological Drude-like model for determining the permittivity of an individual MNP, we proposed to consider restoration force term in the interaction of light with small NPs which can be called the classical confinement effect. For energy dissipation, we have considered all dominant damping mechanisms including electron-electron and electron-phonon scatterings, radiation and electron scattering by the NP surface. In addition, in order to correct the shortages of theoretical background of dissipation mechanisms and take into account quantum confinement effect as well, we have taken into account a correction term to the damping factor obtained by the well-known theoretical studies. Numerical analysis has been done for small gold NPs and the free parameters of system have been determined by studying the existing experimental data related to the extinction cross section. Results showed the good agreement between experiments and our model. Dynamical parameters obtained by this model can be very useful for future theoretical studies about the interaction of electromagnetic fields with MNPs in the linear and nonlinear optics of media containing such NPs and plasmonics field as well.