1 Introduction

The integrated devices plasmonic based in the nanoscale region have attracted attention of the scientists for development research in manipulating light [1,2,3]. Metallic nanosphere (MNS) is the crucial factor for developing plasmonic-based devices due to the ability in the absorption of visible light to generate localized surface plasmonic resonance (SPR) [4, 5]. The SPR absorption band is highly dependent on the size, shape, and the nanoparticle–matrix interface, which is identified from the intensity and energy position of the optical and dielectric properties [5, 6]. The metallic nanostructures with tunable localized SPR are potential applications for optical imaging [7], material characterization research [8], EM wave absorber material [9], drugs delivery [10], meta-material [11], magnetic recording, data storage [12], surface-enhanced Raman scattering [13], and biosensing [14].

The optical, electric, and magnetic properties of nanoparticle-matrix are the basic and essential knowledge for technology development [15,16,17,18]. The optical response of nanoparticle-matrix is necessary for tunable localized SPR phenomena [11, 16, 19,20,21]. This phenomenon is due to the electron at the surface nanoparticle moving and oscillating together to form a short single dipole moment [6, 22] when the EM wave is incident on the nanoparticles. Some SPR was reported by different matrices using configurations of core–shell nanoparticles: Au with Ag coating (SPR at 525–505 nm) [23], Au–ZnO nanocomposites (SPR at 505–615 nm), and gold nanoparticle embedded with silicon (SPR at 539–583 nm) [24].

The Mie scattering theory was used for identifying phenomena of SPR by determining the optical properties and dielectric function [6, 25]. This theory solves the scattering phenomena in many shapes of nanoparticles [6, 20, 26, 27], nanorings [6, 28], cubes [29], and nano-strip structures [30, 31]. For the experimental report of the solid form, Ref [32] analyzed the SPR of gold nanoparticles coated with pure metallic material, and Ref [9] was reported for nanoparticle Fe3O4 and analyzed their scattering phase. Reference [8] reported the absorption efficiency of gold nanofluids based on the Mie scattering theory.

The magnetic polarizability and electric polarizability are fundamental properties that influence the performance of the plasmonic-based device which is needed to understand well the efficient future experimental conditions. For the nanoparticle coated with fluid as a medium, Ref. [33] reported and analyzed the effective electric polarizability using Mie scattering theory with four models: virtual cavity, real cavity, hard-sphere, and finite-size model. Their results show that more than 90% agree well between the models with the exact solution for the gas molecule in the medium water. For solid as a medium, there are no references for analyzing electric polarizability of nanosphere using Mie scattering theory which is essential for plasmonic-based devices.

Some models can identify the SPR properties based on their polarizability characteristics, but very few for the finite-size model. Hence, in this study, we applied only the finite-size model [33] to identify EM waves' effect on the localized SPR of hard gold nanosphere for various radii with silver as the medium. This study's model and calculation methods are extinction and scattering efficiency, electric and magnetic polarizability, and absorption properties as a function of nanosphere radius. We found the maximum imaginary part peak position of electric polarizability at 1.17 eV. This peak corresponds to the energy plasmon longitudinal resonance in the conduction band of the gold nanosphere in the silver medium.

2 Model and Calculation Method

For illustration, how to excite SPR when the EM waves are incoming to the metallic nanoparticle by creating dipole charge like a spring-mass of the harmonic oscillator, is shown in Fig. 1 [5]. Figure 2 illustrates dipole electric polarizability at the nanoparticle's surface when the EM wave is incoming. The electron cloud on one side of the surface metallic nanosphere and the hole cloud on another side for a radius smaller than the propagation wavelength show different SPR colors (Fig. 2). The SPR, as a response to the electric field analysis using Mie scattering theory, can be expressed by the dipole electric polarizability [5]. Several references reported the models used to identify and describe the SPR properties of nanoparticles based on their polarizability characteristics, as shown in Table 1.

Fig. 1
figure 1

The SPR in the metallic nanosphere can be modeled like a spring-mass harmonic oscillator derived from Ref. [5]

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Fig. 2
figure 2

Schematic illustration of dipole-electric polarizability when the EM incoming to the gold nanosphere the blue color is electron cloud and red to yellow is hole cloud, and green is a medium

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Table 1 Mie theory-based model describing the SPR characteristics from the phenomena particle's polarizability of various theoretical studies. We also provided the physical phenomena and conclusion which are strongly dependent on the method or model and type of particle
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2.1 Extinction and Scattering Efficiency

The main concept to understand the resonance when the EM waves hit the nanosphere is by observing the optical properties in the form of scattering and extinction part. The efficiencies energy of these properties is defined from the main peak [27]:

$$ Q_{sca} = \frac{2}{{x^{2} }}\mathop \sum \limits_{l = 1}^{\infty } \left( {2l + 1} \right)\left( {\left| {t_{l}^{E} } \right|^{2} + \left| {t_{l}^{M} } \right|^{2} } \right) $$
(1)
$$ Q_{ext} = \frac{2}{{x^{2} }}\mathop \sum \limits_{l = 1}^{\infty } \left( {2l + 1} \right){\Re }\left\{ {t_{l}^{E} + t_{l}^{M} } \right\} $$
(2)

with \(x = \frac{2\pi R}{\lambda }\) [27], \(l\) is the orbital momentum number, and \(t_{l}^{E}\) and \(t_{l}^{M}\) are the Lorenz–Mie coefficients [6, 25]:

$$ t_{l}^{E} = \frac{{ - \varepsilon_{m} j_{l} \left( {\rho_{m} } \right)\left[ {j_{l} \left( \rho \right) + \rho j_{l}^{{\prime}} \left( \rho \right)} \right] + \varepsilon j_{l} \left( \rho \right)\left[ {j_{l} \left( {\rho_{m} } \right) + \rho_{m} j_{l}^{{\prime}} \left( {\rho_{m} } \right)} \right]}}{{\varepsilon_{m} h_{l}^{\left( + \right)} \left( {\rho_{m} } \right)\left[ {j_{l} \left( \rho \right) + \rho j_{l}^{{\prime}} \left( \rho \right)} \right] - \varepsilon j_{l} \left( \rho \right)\left[ {h_{l}^{\left( + \right)} \left( {\rho_{m} } \right) + \rho_{m} h_{l}^{\left( + \right) ^{{\prime}}} \left( {\rho_{m} } \right)} \right]}} $$
(3)
$$ t_{l}^{M} = \frac{{ - \rho j_{l} \left( {\rho_{m} } \right)j_{l}^{{\prime}} \left( \rho \right) + \rho_{m} j_{l}^{{\prime}} \left( {\rho_{m} } \right)j_{l} \left( \rho \right)}}{{\rho h_{l}^{\left( + \right)} \left( {\rho_{m} } \right)j_{l}^{{\prime}} \left( \rho \right) - \rho_{m} h_{l}^{\left( + \right) ^{{\prime}}} \left( {\rho_{m} } \right)j_{l} \left( \rho \right)}} $$
(4)

\(\rho = \left( {kR} \right)\sqrt \varepsilon\), \(\rho_{m} = \left( {kR} \right)\sqrt {\varepsilon_{m} }\), \(j_{l} \left( x \right)\) and \(h_{l}^{\left( + \right)} \left( x \right)\) are Bessel and Hankel function, respectively, R is particle radius, \(\varepsilon\) and \(\varepsilon_{m}\) are particle and medium permittivity, respectively, and quotation mark (ʹ) indicated the derivative of the function. For dipole case (\(l = 1)\), the Bessel and Henkel function becomes \(j_{1} \left( a \right) = \frac{\sin a}{{a^{2} }} - \frac{\cos a}{a}\) and \(h_{1}^{\left( + \right)} \left( a \right) = \left( {\frac{1}{{a^{2} }} - \frac{i}{a}} \right){\text{exp}}\left( {ia} \right)\), respectively [25]. Dielectric function of material can be approximate by using classical EM solution in the form of Drude dielectric function [6, 25]:

$$ \varepsilon \left( \omega \right) \approx 1 - \frac{{\omega_{p}^{2} }}{{\omega \left( {\omega + i\gamma } \right)}} $$
(5)

where \(\omega_{p}\) and \(\gamma\) is the phonon frequency and attenuation coefficient, respectively. \(\omega_{p}\) and \(\gamma\) for gold as input parameters were taken from ref. [6].

2.2 Electric and Magnetic Polarizability

Mie scattering theory is used to explain EM waves’ scattering by a nanospherical object [10]. The electric and magnetic polarizability of a nanospherical material in a vacuum is described as follows [6, 25]:

$$ \alpha_{E} = \frac{{3t_{l}^{E} }}{{2k^{3} }} $$
(6)
$$ \alpha_{M} = \frac{{3t_{l}^{M} }}{{2k^{3} }} $$
(7)

where \(k = 2\pi /\lambda\) and \(\lambda\) is the wavelength.

2.3 Electric Polarizability Models

The electric polarizability models for particles embedded in a medium are approximate. In this study, we use a finite-size model for determining the electric polarizability of a particle in a solid medium. This model combines electric polarizability in a vacuum and a hard sphere [12, 33]. The hard-sphere model describes the electric polarizability of the nano-hard-sphere object in a medium with input parameters such as hard sphere radius and permittivity, and medium permittivity, as denoted by R, ε, and εm, respectively. The relation between these parameters with Mie scattering coefficient αHS is as follows [33]:

$$ \alpha_{{{\text{HS}}}} = 4\pi \varepsilon_{0} \varepsilon_{m} R^{3} \frac{{\varepsilon - \varepsilon_{m} }}{{\varepsilon + 2\varepsilon_{m} }} $$
(8)

By using the model above, a model can be explained that has a three-layer system [33]. First is the polarizability of material in a vacuum, which in this study uses the electric polarizability from Mie scattering theory. The excess for the cavity is:

$$ \alpha_{C} = 4\pi \varepsilon_{0} \varepsilon_{m} R_{C}^{3} \frac{{1 - \varepsilon_{m} }}{{1 + 2\varepsilon_{m} }} $$
(9)

by substituting Eq. (6) and (9) in Eq. (8):

$$ \alpha_{fs} = \alpha_{C} + \alpha \left( {\frac{{3\varepsilon_{m} }}{{2\varepsilon_{m} + 1}}} \right)^{2} \frac{1}{{1 + \left( {\frac{{\alpha_{C} \alpha }}{{8\pi^{2} \varepsilon_{0}^{2} R_{C}^{6} \varepsilon_{m} }}} \right)}} $$
(10)

where it is called a finite-size model. The illustration of the model can be seen in Fig. 3, and the computational flowchart in this study is summarized in Fig. 4.

Fig. 3
figure 3

Finite-size model illustration of particle inside a solid medium for three-dimensional (left) and two-dimensional (right). ε and εm are the particle and medium permittivity, respectively, R is the radius of the particle, and Rc is the radius of the spherical vacuum cavity, modification from [33] for molecule in water as a medium

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Fig. 4
figure 4

Computational flowchart using finite-size model based on the Eqs. (1–10) for determining extinction and scattering efficiency (Qsca and Qext) and electric and magnetic polarizability (αE and αM), and electric polarizability from Mie scattering theory (αfs) for particle inside a solid medium where R is the radius of the particle, and Rc is the radius of the spherical vacuum cavity

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Figure 3 clearly shows the spherical vacuum cavity area between hard sphere gold nanoparticle (radius R) and the solid medium (radius RC). The input parameters for calculation are particle radius of solid gold nanoparticle R, radius of the spherical vacuum cavity RC, phonon frequency ωp, attenuation coefficient γ, and wavelength λ. By using Eqs. 1, 2, 6, and 7, the extinction efficiency, scattering efficiency, electric polarizability and magnetic polarizability were determined, respectively. For more clear details, the computational flowchart for calculation method in this study can be seen in Fig. 4.

2.4 Calculation Method

3 Results and Discussion of Computational Test

3.1 Scattering and Extinction efficiencies

As shown in Fig. 5, both optical efficiencies are in a positive region which indicates that the particle’s resonance corresponds to the dielectric resonance. The dielectric characteristic of a particle is described by the electric and magnetic properties (electric polarizability, magnetic absorption, plasmonic characteristics) [26]. The efficiency increases drastically for a small radius (less than the maximum peaks) as the nanosphere radius increases to the point where it reaches a maximum and then decays slowly. For the particle radius 8.7 nm and 13.7 nm, the metallic nanosphere will gain maximum scattering and extinction efficiencies, indicating the highest probability of electron clouds.

Fig. 5
figure 5

Scattering and extinction efficiencies of gold nanosphere in vacuum for dipole case (\(l = 1\)), where l is the orbital momentum number

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3.2 Real and Imaginary Part of Electric Polarizability

Mie scattering theory shows that the electric polarizability of gold nanosphere in vacuum for dipole case increases and moves to the lower energy with the increasing the nanosphere radius. Figure 6c, d shows the electric polarizability spectra of the gold nanosphere with main peaks value presented in Table 2. The particle radius varied from 10 to 50 nm with an interval value of 10 nm. The real and imaginary part peak data shifted to 0.09 ± 0.02 eV and 0.06 ± 0.01 eV, respectively, to the lower energy with increasing radius of the gold nanosphere. The resonance of the plasmon longitudinal is in the negative value of the real part electric polarizability (− Re{αE}) [25]. It indicates that the plasmon longitudinal resonance frequency of the gold nanosphere reaches a maximum for the radius ~ 30– ~ 40 nm as shown in Fig. 6c.

Fig. 6
figure 6

The real (Re(αE)) part (a, c) and imaginary (Im(αE)) part (b, d) of electric polarizability for gold nanosphere in vacuum. a, b The radius maximum value of extinction (RExt) and scattering (RSca) efficiencies, c, d for various radius values of gold nanosphere

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Table 2 The main peak for the real (Re(αE)) part and imaginary (Im(αE)) part of electric polarizability and the ratio between magnetic and electric polarizability of gold nanosphere as a function of energy
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Figure 7 shows the ratio between an imaginary part of magnetic and electric polarizability (Im{αM}/Im{αE}) of the various radii of gold nanosphere in a vacuum. The insert figure is for skin depth effect (\(\delta = \lambda /\left( {2\pi {\text{Im}} \left\{ {\varepsilon^{\frac{1}{2}} } \right\}} \right)\)) to the magnetic properties [34]. In this study, the volume of the gold nanosphere is fully affected by the incident EM for the radius <  ~ 22.5 nm. The magnetic absorption by a metallic nanosphere will produce an Eddy current leading to high imaginary electric polarizability (Im{αE}). Consequently, those properties (magnetic absorption and imaginary electric polarizability) have an opposite trend in the same energy position [6].

Fig. 7
figure 7

The ratio between imaginary magnetic and electric polarizability of gold in vacuum for the various gold nanosphere radius. Skin depth shows decrease in magnetic absorption with decreasing the nanosphere radius

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As shown in the previous explanation, the metallic nanosphere will gain maximum extinction and scattering efficiencies for the sphere’s specific radius, as shown in Fig. 7. Equations (1)–(4) show that the efficiencies depend on the size of the nanosphere (\(x\)). On the other hand, the electric and magnetic polarizability shows good linear depending on the radius, as shown in Figs. 6 and 7. In contrast, the polarizability depends on Henkel and Bessel function to describe the radius effect. At the same time, the polarizability is affected by the quantity of the plasmon surrounding the particle. It is linearly dependent on the nanosphere radius. The efficiency of the existing plasmon surrounds the particle at a specific radius indicated by the scattering and extinction efficiencies.

Figure 8 shows real and imaginary parts of electric polarizability of the gold nanosphere in a vacuum and silver medium. It indicated specific energy of incident EM waves to the nanosphere to gain the highest electric polarizability. It shows a slight shift of the peak to the higher energy with an increasing radius of gold nanosphere for the particle embedded in a metallic medium which affected the electric polarizability. This model also shows that the electric polarizability increases sharply when the nanosphere is embedded in a metallic medium. This phenomenon indicates the existence of plasmon in the vacuum layer between the nanosphere and the medium. It may occur due to the metallic medium also producing an electron cloud when the EM waves are applied.

Fig. 8
figure 8

The complex (Re (αE) and Im (αE)) electric polarizability of gold in vacuum (a, b), and in silver medium (c, d) for the radius ~ 30–~ 40 nm (at the maximum value of extinction (Ext) and scattering efficiencies (Sca))

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Figure 9 shows that the electric polarizability value reaches 1e−20 and 1e−19 for real and imaginary parts, respectively. The maximum position electric polarizability at the imaginary part is 1.17 eV which corresponds to the SPR energy. This energy position is lower than some reported references probably due to the different condition (vacuum) and matrices (solid silver). The Au with Ag coating shows SPR at 525 nm (2.36 eV) to 505 nm (2.46 eV) [23], Au–ZnO nanocomposites at 505 (2.46 eV) nm to 615 nm (2.02 eV), and gold nanoparticle embedded with silicon at 539 nm (2.30 eV) to 583 nm (2.13 eV) [24]. Figure 9a clearly shows that the real part decreases sharply from the maximum to the negative value due to the imaginary part of electric polarizability in a finite-size model. This phenomenon corresponds to the maximum value of the plasmon longitudinal resonance in the conduction band of the gold nanosphere. Besides that, Fig. 9 shows that electric polarizability gains less negative value for \(R \sim R_{c}\), and more negative value for \(R > R_{c}\) [33].

Fig. 9
figure 9

Electric polarizability of gold nanosphere for the various radius in the silver medium for a real part at linear scale and b imaginary part at logarithmic scale by using the finite-size model

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4 Conclusion, Limitation, and Future Prospects

The extinction and scattering efficiency (Qsca and Qext), electric and magnetic polarizability (αE and αM), and electric polarizability from Mie scattering theory (αfs) were successfully studied. The electric polarizability is used to analyze the plasmonic phenomena of a metallic nanosphere (gold) in a vacuum and embedded in a metallic medium (silver) using the finite-size model. The main peak of real and imaginary parts of electric polarizability shows shifted 0.09 ± 0.02 eV and 0.06 ± 0.01 eV, respectively, to the lower energy with an increasing radius of gold nanosphere for every 10 nm. The plasmon longitudinal resonance frequency of the gold nanosphere reaches its maximum at a radius of ~ 34 nm. The electric polarizability value reaches 1e−20 and 1e−19 for real and imaginary parts, respectively. The maximum peak position of electric polarizability at the imaginary part is 1.17 eV, indicating the maximum value of the plasmon longitudinal resonance. The resonance takes place in the conduction band of the gold nanosphere, also affected by the ratio of \(R \) and \(R_{c}\).

In this study, we perform only characteristics SPR generated from gold nanosphere embedded by solid silver as a medium. However, for the better application of SPR from gold nanospheres including optical imaging and drugs delivery, there are still some bottlenecks that need to be addressed. First, optical imaging and drugs delivery is a multi-complex system, and interference is still a big problem for the real applications. Functionalization of SPR from gold nanosphere is an effective strategy for the generation of highly stable optical imaging and drugs delivery with high accuracy and affinity for many targets. Nevertheless, much research focused SPR for the novel sensors as an optical imaging and drugs delivery rather than applicability to real samples. Second is impurity materials; therefore, the sensitivity of hard gold nanosphere-based SPR colorimetric sensor is of great importance for optical imaging and drugs delivery. Third, optical imaging drug delivery usually needs on-site operation, so generating SPR from gold nanosphere for the complex system is a realistic challenge and needs to be overcome. Finally, reproducible application of SPR generate from gold nanosphere requires further improvement. Overall, despite the realistic challenge, significant progress has been achieved by theoretical system for the SPR based on metal nanoparticles over the past years. Moreover, numerous methods for generating SPR for optical imaging and drugs delivery have been reported, thus demonstrating the great promise for the application of SPR from metals nanosphere in future.

In future, an increased number of collaborations with scientists from various disciplines to overcome through the collective efforts can be expected, and novel methods for generating SPR from metal nanoparticles. We hopefully by the collaboration will get excellent advantages for developing various applications of SPR from metal nanoparticle including optical imaging and drugs delivery with excellent performance.