Journal of Optics

, Volume 41, Issue 4, pp 224–230

Light scattering by two concentric gold cylindrical hollow nanoshell

Research Article

DOI: 10.1007/s12596-012-0098-5

Cite this article as:
Bahari, A. & Gharibi, S. J Opt (2012) 41: 224. doi:10.1007/s12596-012-0098-5

Abstract

The scattering cross section of two concentric gold cylindrical hollow nanoshell (GCHNS) is obtained as a function of wavelength at different thicknesses of two gold shells and intershell spacing between them. Theoretical calculations show that both the intensity and position of the scattering peak depend on these parameters for two concentric GCHNS and therefore the scattering peak can be tuned by changing these parameters.

Keywords

Nanoshell Nanostructure Light scattering Cross section 

Introduction

Metal nanoshells are a new class of nanoparticles with highly tunable optical properties and plasmon resonance, allowing materials to be specifically designed to match the wavelength required for a particular application. Nanomaterials have a wide range of interesting and potentially useful optical properties. These optical properties are depended on many factors such as particle size, shape, structure and surrounding medium [1, 2, 3].

Hollow nanoshells made of metals are readily obtained by selectively removing the cores via calcination or chemical etching [4, 5, 6, 7]. It is also worth pointing out that the surfaces of the gold nanoshells obtained by these methods could be coated with silica layers (by using some well-established procedures) to further modify the surface plasmon resonance (SPR) properties [8]. By repeating the gold and silica deposition steps, Prodan and co-workers have successfully prepared nanoparticles with multiply, concentric shells of gold and silica [9]. In [10, 11] authors have demonstrated that gold nanoparticles could be coated with double shells made of silica and polymer. Sun and co-workers have demonstrated a different method based on the galvanic replacement reaction for preparing hollow nanostructures composed of metals such as Au, Pd, Pt, and their alloys [12, 13]. Hollow nanostructures prepared using this approach are characterized by well-defined void sizes, wall thicknesses, shapes, and highly crystalline walls with controllable surface porosities [14, 15]. Sun and Xia [16] have synthesized the triple-walled nanotubes made of Au-Ag alloy and double walled nanotubes with constituent material of inner wall being Au-Ag alloy and the outer wall being Pd-Ag alloy.

Mock et al. [17] and Barbic et al. [18] studied the scattering properties of homogeneous and multisegment nanowires of silver, gold, and nickel. Zhu [19] has investigated theoretical study of the effect of wall thickness on the light scattering from gold single-walled nanotubes. In [20] optical studies of gold-coated, Au2S nanoshells have been investigated. They also have discussed the effect of electron surface scattering on the reduction of the local-field enhancement associated with metal nanoshells. Schroter and Dereux [21] have reported the surface plasmon polaritons on metal cylinders with dielectric core. The interaction of light with nanometer scale diameter single and multiple silver nanowires, coated externally with a dielectric material, has been studied in [22]. In [23] the light scattering by two concentric double spherical nanoshells has been investigated.

We consider two concentric GCHNS. Geometry of this structure is shown in Fig. 1. We have investigated the scattering of two concentric GCHNS, composed of a hollow core and three alternating metallic, hollow and metallic shells. We show that two concentric GCHNS display two separated scattering peaks that the intensity and position of the scattering peaks depend on two gold shells thicknesses and hollow layer thickness between them.
Fig. 1

Geometry of the structure of two concentric GCHNS. ε1, ε2, ε3 , ε4 , ε5 are the dielectric functions for the core, inner gold shell, hollow shell, outer gold shell and surrounding medium and R1, R2, R3, R4 denote the radius of these regions, respectively

In this paper we study the light scattering of two concentric GCHNS. In Section “Theoretical model” we develop theory of the light scattering by two concentric GCHNS. In Section “Results of numerical calculations” the numerical results the cross section scattering have been reported. In the last section a brief summary of the result will be presented.

Theoretical model

We consider the theoretical expressions, for the electric fields and the polarizability of two concentric GCHNS and then we investigate the scattering cross section of metal nanoshells. In our analysis the gold shells thickness (d) is much smaller than the wavelength (d ≪ λ), then the quasi-static theory can be used in calculations. The analyses have been performed using quasistatic approximations in which retardation effects are assumed to be negligible. However, such effects may become significant when the particle size is no longer small compared with the incident wavelength. In quasi-static calculations the spatial variation of the electromagnetic field is neglected. The incident field is time dependent, but it does not vary spatially over the diameter of the metal shell. In this case, the electrostatic solution can be obtained by solution of Laplace’s equation for the potential. The potential in each region could be derived from Laplace’s equation [24]. The general solution for the potential in each region in cylindrical coordinates is given by
$$\label{equ1} V_{i}=A_{i}r \cos\phi+\frac{B_{i}}{r}\cos\phi $$
(1)
where Ai and Bi are the constants and i is the index of each medium and has a value from 1 to 5. The boundary conditions must be specified so that the potential in each medium can be determined. First, there must be continuous of the tangential component of the electric field:
$$ \left(\frac{\partial V_{i}}{\partial \phi}\right)_{r=R_{i}}=\left(\frac{\partial V_{i+1}}{\partial \phi}\right)_{r=R_{i}} $$
(2)
Second, there must be continuous of the normal component of the displacement field:
$$ \left( \varepsilon_{i} \frac{\partial V_{i}}{\partial r}\right)_{r=R_{i}}=\left(\varepsilon_{i+1}\frac{\partial V_{i+1}}{\partial r}\right)_{r=R_{i}} $$
(3)
We choose the x axis along the incident field so that at sufficiently large r, \(\textbf{E} = E_{0}\hat{x}\). In region 1, B1 = 0; and in region 5, far from two concentric GCHNS, we must recover the potential V5 = − Eor cosϕ, thus giving A5 = − Eo. Now we apply the boundary conditions to the general solution with A5 and B1 determined. The boundary conditions to (1) lead to a set of eight equations and eight unknowns that can be solved to obtain A1, A2, A3, A4, B2 ,B3 ,B4, and B5. Therefore we have:
$$\begin{array}{lll} A_{2}&=&\alpha_{0}B_{2}, \;\;\;A_{3}=\alpha_{1}B_{2},\;\;\; B_{3}=\alpha_{2}B_{2}, \\ A_{4}&=&\alpha_{3}B_{2},\;\;\; B_{4}=\alpha_{4}B_{2} \end{array} $$
(4)
$$ B_{2}=-\frac{2\varepsilon_{5}E_{0}R_{4}^{2}}{(\varepsilon_{4}+\varepsilon_{5})R_{4}^{2}\alpha_{3}+(\varepsilon_{5}-\varepsilon_{4})\alpha_{4}} $$
(5)
where
$$ \alpha_{0}=\frac{\varepsilon_{1}+\varepsilon_{2}}{(\varepsilon_{2}-\varepsilon_{1})R^{2}_{1}} $$
(6)
$$ \alpha_{1}=\frac{(\varepsilon_{2}+\varepsilon_{3})\alpha_{0}R^{2}_{2}+(\varepsilon_{3}-\varepsilon_{2})}{2\;\varepsilon_{3}R^{2}_{2}} $$
(7)
$$ \alpha_{2}=\frac{(\varepsilon_{3}-\varepsilon_{2})\alpha_{0}R^{2}_{2}+(\varepsilon_{3}+\varepsilon_{2})}{2\;\varepsilon_{3}} $$
(8)
$$ \alpha_{3}=\frac{(\varepsilon_{4}+\varepsilon_{3})\alpha_{1}R^{2}_{3}+(\varepsilon_{4}-\varepsilon_{3})\alpha_{2}}{2\;\varepsilon_{4}R^{2}_{3}} $$
(9)
$$ \alpha_{4}=\frac{(\varepsilon_{4}-\varepsilon_{3})\alpha_{1}R^{2}_{3}+(\varepsilon_{4}+\varepsilon_{3})\alpha_{2}}{2\;\varepsilon_{4}} $$
(10)
Then the electric field in each region can be obtained with \(\textbf{E}_{i}=-\nabla V_{i}(r,\phi)\). From these fields we can calculate the polarizability of two concentric GCHNS. Then, we can obtain the scattering cross section from the polarizability by using scattering theory [18, 20, 25]. The polarizability and therefore the scattering cross section for the infinitely long nanotubes is infinite, therefore we calculate the polarizability in a unit length of the nanotube. For the scattering cross section we obtain
$$\label{eq11} \sigma_{\rm sca}=\frac{k^{4}|\alpha_{\rm tot}|^{2}}{6\pi\varepsilon^{2}_{0}} $$
(11)
where k is wave vector and αtot is total polarizability. Equation (11) has a complex form that depends on different parameters such as wavelength, nanostructure size, shell thickness and shell separation. This function can be plotted using the proper numerical calculations for a given parameters of nanoshell.

Results of numerical calculations

We consider the theoretical calculation results, for two concentric GCHNS. Two concentric GCHNS is characterized by a composition of a hollow core and three metallic, hollow and metallic shells. The inner gold shell has a thickness d1 = R2 − R1 and a dielectric function ε2(ω). The outer gold shell has a thickness d2 = R4 − R3 and a dielectric function ε4(ω). The core dielectric constant is ε1 and the surrounding medium of each shell has a constant dielectric function which we show them by ε3 and ε5. We take dielectric constant ε1 = 1 for hollow core and ε3 = ε5 = 1 for dielectric of the surrounding media of two gold shells, and R1, R2, R3 and R4 denote the radius of these regions, respectively.

Calculations show how the scattering cross section changes as a function of wavelength, gold shells thickness and the distance between them. The intensity and position of the scattering peak are sensitive to the thickness of gold shells and intershell spacing between them. Increasing the gold shells thickness leads to the scattering peak blue shifting. We use the Drude model for gold shells permittivity ε2(ω) and ε4(ω) [26]. For metals, the dielectric function can be well-described by Drude formula \(\varepsilon(\omega)=1-\omega^{2}_{p}/\omega(\omega+i\gamma)\), where γ = 1/τ = 1013  s − 1 is the relaxation time for plasmon oscillations, and ωp is plasma frequency. Physically, the Drude and Lorentzian terms are related to intraband (free-electron) and interband (bound-electron) transitions respectively. The dielectric function of the metallic nanoshells is dominated by free-electron behavior at the wavelengths studied in this paper; therefore we would expect our data to conform the Drude model. If a more realistic model is used in our calculation the scattering may be slightly changed. In our analysis the gold shells thickness is much smaller than the wavelength (300–1000 nm). So the quasi-static approximation can be used in the calculation [27].

In Fig. 2 the scattering cross section of Gold single walled nanotube (GSWNT) at different thicknesses as a function of wavelength is shown. Calculations show that the scattering cross section of GSWNT depends on thickness of shell. Increasing gold shell thickness leads to blue shift in surface plasmon resonance. When the shell thickness is large enough and wider than the bulk electron mean free path, electron scattering will not depend on the shell thickness anymore [21]. In Fig. 3 the cross section of the GSWNT and two concentric GCHNS as a function of wavelength for comparison is shown. This figure shows that two concentric GCHNSs display two separated peaks. Small peak could be attributed to inner gold shell, while other peak is associated with outer gold shell. Intensity and position of these peaks are dependent on coupling between two gold shells, and this coupling is dependent on intershell spacing between two gold shells and their thicknesses. In two concentric GCHNSs the magnitude of the scattering cross-section is much larger than the GSWNTs and it has a blue shift.
Fig. 2

The scattering cross section of the GSWNT as a function of wavelength at different thicknesses for R1 = 10 nm and R2 = 20, 40, 50 nm

Fig. 3

The scattering cross section of GSWNT and two concentric GCHNS with R1 = 10 nm, R2 = 2R1, R3 = 3R1, R4 = 4R1 as a function of wavelength

In Fig. 4 we show the cross section of two concentric GCHNSs as a function of wavelength for d1 = 1 nm and d2 = 10  nm, when the intershell spacing between two gold shells increases. This figure shows that increasing the separation between two gold shells of two concentric GCHNSs leads to shift the peak position of the cross-section to longer wavelength. Both the magnitude and position of cross-section depend on the separation between two gold shells. The results in this figure show that coupling between inner and outer shell decreases by increasing their intershell spacing. Therefore for large intershell spacing the inner gold shell would be screened by outer gold shell and no surface plasmon resonance scattering would be excited in inner gold shell. Due to finite penetration depth of the light, for large intershell spacing the inner-nanoshell plasmon is not excited, therefore, by increasing intershell spacing, the intensity of first peak decreases. But in the second peak the situation is quite different. For this peak increasing the intershell spacing causes to increase the intensity. So we have noninteracting inner- and outer-shell plasmons for large intershell spacing. Two concentric GCHNSs display two separated peaks. First peak could be attributed to inner gold shell, while other peak is associated with outer gold shell and could be continuously tuned in the spectral range from red to near-infrared.
Fig. 4

The scattering cross section of two concentric GCHNS with R1 = 10 nm, R2 = 1.1R1 and d2 = 10 nm, at different separation between two gold shells, as a function of wavelength

In Fig. 5 we show the effect of increasing of outer gold shell thickness on cross section for a given inner shell thickness in two cases (a) for large inner gold shell thickness (d1 = 10 nm) and (b) for small inner gold shell thickness (d1 = 1 nm). This figure shows that coupling between the inner and outer gold shell depends on their thicknesses. For large inner gold shell thickness (d1) (Fig. 5a) increasing outer gold shell thickness leads to increase the intensity in the second peak but the position of the first peak is constant and its intensity slightly changes. For small inner gold shell thickness (Fig. 5b) the second peak drops to zero, but increasing the outer gold shell thickness from 10 to 40 nm leads to blue shifting in the first peak.
Fig. 5

Effect of increasing thickness of outer gold shell on the scattering cross section of two concentric GCHNS with R1 = 10 nm, R3 = 3R1 at different R4: (a) for d1 = 10 nm and (b) for d1 = 1  nm

In Fig. 6 the cross section at different magnitudes of inner gold shell thickness (d1) for a given outer gold shell thickness (d2 = 10 nm) is shown. In this figure we calculate the cross section for large d1 from 12.5 to 50 nm. Increasing the inner gold shell thickness leads to the scattering peak blue shifting. The wavelength maximum of scattering peak is plotted as a function of inner gold shell thickness in Fig. 7. This figure shows how the maximum intensity of scattering cross section varies with thickness. It is obvious that position of the scattering peak blue shift changes from 480 to 400 nm that is approach to the scattering peak of gold nanowire.
Fig. 6

Effect of increasing thickness of inner gold shell on the scattering cross section of two concentric GCHNS at different d1 for R1 = 10 nm, R3 = 9R1 and R4 = 10R1

Fig. 7

The wavelength maximum of scattering peak with d2 = 10 nm with R3 = 9R1 and R4 = 10R1 as a function of inner gold shell thickness

Conclusions

In conclusion, we have investigated the scattering properties of two concentric GCHNSs. These gold nanoshells have a tunable plasmon resonance that depends on gold shells thickness and the intershell spacing between them. Calculations show scattering in two concentric GCHNSs depends on the thickness of both gold shells and intershell spacing between them. For a given shell thickness the reduction of this separation leads to increase the scattering peak intensity and its blue shift. The intensity and position of the scattering peak depend on intershell spacing between two gold shells and their thicknesses; therefore the scattering peak may be tuned by changing these parameters. Depending on the size and composition of each layer of the GCHNSs, they can be designed either to absorb or scatter light over much of the visible and infrared regions of the electromagnetic spectrum, including near infrared region where penetration of light through tissue is maximal.

Copyright information

© Optical Society of India 2012

Authors and Affiliations

  1. 1.Department of PhysicsLorestan UniversityLorestanIran

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