1.

Introduction

In the last few years, many studies have been done on the electromagnetic properties of nanostructures, particularly on the interesting applications available from their use. The extraordinary optical properties of such structures derive from the strong local electromagnetic field enhancement: the so-called localized surface plasmon resonance (LSPR). Typically, the LSPR phenomenon can be described as the resonant collective oscillation of electrons, stimulated by an incident electromagnetic wave. The resonance condition is set up when the electromagnetic wave frequency matches the natural frequency of the oscillating electrons. In addition to the LSPR, a high localization of the electromagnetic field in the neighborhood of the nanostructure is obtained. As a result, the interaction between the electromagnetic field and the surrounding dielectric environment is restricted to a small area.¹ For this reason, these structures are very sensitive to any change in the local index of refraction, which causes the resonance conditions of the surface plasmon waves to modify. Such characteristics allow these structures to be suitable for several application fields, such as optical² and photonic ones,³ biochemical sensing and detection,^4,5 protein analysis,^6,7 cell membrane function,⁸ biomedical applications,^9,10 and food quality analysis.¹¹ They can also be used as contrast agents in cellular and biological imaging.^12,13 The nanoparticle effectiveness as a sensing or therapeutic tool depends on its optical properties: for instance, a high scattering cross-section is crucial for light-scattering microscopy-based imaging or sensing applications.¹⁴ On the other hand, photothermal therapy requires a high nanoparticle absorption cross-section with low scattering losses for superficial and deep tumor treatment.¹⁵

The existing literature has mainly focused on the experimental evaluation of the nanoparticle electromagnetic characteristics, but it is almost silent on the development of proper analytical models and design methods describing their resonant behavior. It is known that the analytical closed-form electromagnetic solution exists only for a restricted number of particle geometries, such as sphere and cylinder,¹⁶ cube,¹⁷ disc and needle.¹⁸ For any other arbitrary shapes, the electromagnetic solution is given by numerical approaches.¹⁹

To describe the electromagnetic behavior of the nanoparticles, it is useful to study the electromagnetic extinction cross-section properties in terms of scattering and absorption.

Separate evaluation of the absorption and scattering phenomenon is crucial to understand why certain structures are preferred to others for specific applications.

In order to describe the nanoparticle electromagnetic behavior using closed-form formulas, it is fundamental to correlate the electromagnetic properties, in terms of extinction cross-section, with the structural and geometrical parameters.

The aim of this article is to find out the analytical relation between the nanoparticle electromagnetic properties and its geometrical parameters. In particular, this research attempts to present a new design method for ellipsoidal nanoparticles with the desired electromagnetic properties, in order to satisfy specific requirements. In the fabrication of nanorod particles, we have chosen the ellipsoidal geometry, as it represents a good approximation for the realistically obtained shapes.

This article is structured as follows:

2.

Analytical Models and Electromagnetic Properties of Ellipsoidal Nanoparticles

The structure under study consists of a metallic resonating inclusion embedded in a dielectric environment. Such a nanoparticle has a specific resonant wavelength in terms of magnitude, amplitude width, and position, depending on its geometrical characteristics (such as shape and dimensions) and on the metal optical properties. An electromagnetic source excites the nanostructure, and its electromagnetic properties are revealed in terms of absorption and scattering cross-section. To be more precise, when an impinging electromagnetic wave with an appropriate incident wavelength illuminates a metallic nanoparticle, free electrons of metal structure start oscillating collectively. Such oscillations lead to the propagation of strong surface waves.^20,21 The resonant optical properties of nanoparticles can be studied, starting with their polarizability expressions. It is well known²² that the polarizability of an arbitrary shaped particle can be expressed as

Under such conditions, we can relate the nanoparticle macroscopic electromagnetic behavior to its polarizability. In this study, the structure is excited by an electromagnetic plane wave with the electric field parallel (and the propagation vector $\mathbf{k}$ perpendicular) to the nanoparticle principal axis, as depicted in Fig. 1, where the single particle is shown. In particular, in this study, the nanoparticles are triaxial ellipsoids with $a>b>c$, where $a$, $b$, and $c$ are the semi-major axes aligned along the coordinate axes ($x$, $y$, $z$) (Fig. 1).

Fig. 1

Triaxial ellipsoidal nanoparticles geometry.

In this paper, the following is true:

2.1.

Analytical Models of Ellipsoidal Nanoparticles

In this section, the analytical models of ellipsoidal nanoparticles that link the electromagnetic nanoparticle properties to their geometrical and structural parameters are presented in order to describe their resonant behavior. Starting from the findings of Ref. 24, it is possible to develop new analytical closed-form formulas for the scattering ($C_{\mathrm{sca}}$) and absorption ($C_{\mathrm{abs}}$) cross-section of the aforementioned particles. The general corresponding expressions read, respectively,

Eq. (3)

$$C_{\mathrm{abs}}=k\mathrm{Im}[\alpha ]$$

Eq. (4)

$$C_{\mathrm{sca}}=\frac{k^4}{6\pi }{|\alpha |}^2,$$

where $k=2\pi n/\lambda$ is the wavenumber, $\lambda$ is the wavelength, and $n=\sqrt{{\epsilon }_e}$ is the refractive index of the surrounding dielectric environment. For oblate and prolate ellipsoids, several closed-form formulas for the depolarization factor can be found, while for triaxial ellipsoids, a closed formula for the depolarization factor does not exist. Typically, such factors can be written in terms of tabulated functions.²⁵ In this study, a new closed-form analytical formula for the triaxial case is obtained. Considering the electric field polarization of the impinging plane wave and the particle geometry, the absorption and scattering cross-section formulas follow:

Eq. (5)

$$C_{\mathrm{abs}}=\frac{2\pi }{\lambda }\sqrt{{\epsilon }_e}\mathrm{Im}\left[\frac{\left(\frac{4\pi }{3}abc\right)({\epsilon }_i-{\epsilon }_e)}{[L_{\mathrm{tri}-\mathrm{axial}}({\epsilon }_i-{\epsilon }_e)+{\epsilon }_e]}\right]$$

Eq. (6)

$$C_{\mathrm{sca}}=\frac{{\left(\frac{2\pi }{\lambda }\sqrt{{\epsilon }_e}\right)}^4}{6\pi }\mathrm{Abs}{\left[\frac{\left(\frac{4\pi }{3}abc\right)({\epsilon }_i-{\epsilon }_e)}{[L_{\mathrm{tri}-\mathrm{axial}}({\epsilon }_i-{\epsilon }_e)+{\epsilon }_e]}\right]}^2,$$

where $L_{\mathrm{tri}-\mathrm{axial}}$ for triaxial ellipsoids ($a>b>c$) reads

Eq. (7)

$$L_{\mathrm{tri}-\mathrm{axial}}=\frac{7}{2\pi }(1-\frac{\pi }{\sqrt{\frac{4b^{2}E{(e^2)}^2}{a^2}+{\pi }^2}}),e=\sqrt{1-{\left(\frac{c}{b}\right)}^2},$$

where $E$ is the complete elliptic integral of the second kind. It is worth noting that if the triaxial ellipsoid degenerates into a prolate/oblate ellipsoid or sphere, the depolarization factor coincides with the classical formulas existing in the literature (e.g., Ref. 25). In Fig. 2, a comparison among analytical and numerical models for absorption and scattering cross-section spectra of gold triaxial ellipsoids is presented.

Fig. 2

Analytical-numerical model comparison for $a=40\mathrm{nm}$, $b=30\mathrm{nm}$, $c=20\mathrm{nm}$ of absorption cross-section spectra (a) and scattering cross-section spectra (b).

2.3.

Electromagnetic Properties Analysis and Sensitivity Properties

Starting from the proposed polarizability analytical model, it is possible to predict how the geometrical parameters, the metal electromagnetic properties, and the surrounding dielectric environment permittivity affect the nanoparticle resonant behavior in terms of position, magnitude, and bandwidth. The inclusion permittivity can be expressed as ${\epsilon }_i={\epsilon }_{\text{real}}+j{\epsilon }_{\mathrm{imm}}$.

As the resonant behavior of the nanoparticle is reached when the denominator goes to zero in both its real and imaginary part [see Eq. (2)] and starting from the general polarizability expression,²⁸ the resonant condition reads:

Eq. (8)

$$L^2[{\epsilon }_{\text{real}}-(\frac{(L-1){\epsilon }_e}{L}-{\epsilon }_{\mathrm{imm}})][{\epsilon }_{\text{real}}-(\frac{(L-1){\epsilon }_e}{L}+{\epsilon }_{imm})]=0.$$

By solving Eq. (8) with respect to ${\epsilon }_{\text{real}}$ and by inserting the proposed depolarization factor [Eq. (7)], we can find the corresponding solution for the inclusion permittivity for which the polarizability of the overall structure rises to infinity (which is related to the nanoparticle resonant behavior):

Eq. (9)

$${\epsilon }_{real}=-\frac{{\epsilon }_e[{\pi }^2{a}^2(\sqrt{\frac{4b^{2}E{(e^2)}^2}{a^2}+{\pi }^2}+\pi )+2(2\pi -7)b^{2}E{(e^2)}^2]}{14b^{2}E{(e^2)}^2}-{\epsilon }_{\mathrm{imm}}.$$

Exploiting Eq. (9), it is possible to tune the nanostructure resonance by changing its geometrical parameters in order to coincide with the spectral absorption characteristics of the sample under study. Such a condition is necessary for specific absorption measurements.

In addition, in this section, we analyze the designed nanoparticle sensitivity properties to optimize the single inclusion for sensing applications. The sensitivity is commonly defined as

Eq. (10)

$$S=\frac{\mathrm{\Delta }\lambda }{\mathrm{\Delta }n},$$

expressed in nm/RIU ($\mathrm{RIU}=\text{refractive index unit}$), where $\mathrm{\Delta }\lambda$ is the wavelength shift and $\mathrm{\Delta }n$ is the refractive index variation. In Table 2, we report the sensitivity values for the inclusions considered in this paper. In particular, analytical, numerical, and experimental values²⁹ of sensitivity are compared.

Table 2

Comparison of the sensitivity values for the inclusions considered.

Metal nanoparticle	Sensitivity (nm/RIU)
	Analytical sensitivity	Numerical sensitivity	Experimental sensitivity29
Silver29	450	400	425
$a=62.5\mathrm{nm}$
$b=c=12.5\mathrm{nm}$

A test material surrounding the nanoparticle with a varying refractive index $n$ in the range of 1 to 3 was used. As shown in Fig. 3, the sensitivity is sufficiently constant over the range considered, as confirmed by analytical results. A good agreement among analytical, numerical, and experimental sensitivity values was obtained.

Fig. 3

Variation of the resonant wavelength as a function of the refractive index of the surrounding medium for the nanoparticle dimensions reported in Table 2.

3.

Conclusions

In this paper, a study on metallic nanoparticles electromagnetic modeling in the infrared and optical frequency regime was presented. The considered structures were triaxial ellipsoidal nanoparticles, embedded in a surrounding dielectric environment. In this regard, the electromagnetic properties of such structures, in terms of extinction cross-section (absorption and scattering), were evaluated by the use of a new analytical model.

Then the proposed model was compared to the results obtained by full-wave simulations and to the experimental values. A good agreement among analytical, numerical, and experimental results was achieved.

The ability to manipulate and control the electromagnetic phenomenon by exploiting the proposed analytical model on the nanometer scale opens up the possibility of using such structures for sensing applications.

For this reason, the nanoparticle sensitivity was studied. Sensitivity values obtained from the proposed analytical model were verified by full-wave simulations and compared to the experimental results, showing good agreement.