# Light scattering by two concentric gold cylindrical hollow nanoshell

- First Online:

- Received:
- Accepted:

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, *Au*_{2}*S* 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.

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

*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

*A*

_{i}and

*B*

_{i}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:

*r*, \(\textbf{E} = E_{0}\hat{x}\). In region 1,

*B*

_{1}= 0; and in region 5, far from two concentric GCHNS, we must recover the potential

*V*

_{5}= −

*E*

_{o}

*r*cos

*ϕ*, thus giving

*A*

_{5}= −

*E*

_{o}. Now we apply the boundary conditions to the general solution with

*A*

_{5}and

*B*

_{1}determined. The boundary conditions to (1) lead to a set of eight equations and eight unknowns that can be solved to obtain

*A*

_{1},

*A*

_{2},

*A*

_{3},

*A*

_{4},

*B*

_{2},

*B*

_{3},

*B*

_{4}, and

*B*

_{5}. Therefore we have:

*k*is wave vector and

*α*

_{tot}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 *d*_{1} = *R*_{2} − *R*_{1} and a dielectric function *ε*_{2}(*ω*). The outer gold shell has a thickness *d*_{2} = *R*_{4} − *R*_{3} and a dielectric function *ε*_{4}(*ω*). The core dielectric constant is *ε*_{1} and the surrounding medium of each shell has a constant dielectric function which we show them by *ε*_{3} and *ε*_{5}. We take dielectric constant *ε*_{1} = 1 for hollow core and *ε*_{3} = *ε*_{5} = 1 for dielectric of the surrounding media of two gold shells, and *R*_{1}, *R*_{2}, *R*_{3} and *R*_{4} 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 *ε*_{2}(*ω*) and *ε*_{4}(*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* = 10^{13} s^{ − 1} is the relaxation time for plasmon oscillations, and *ω*_{p} 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].

*d*

_{1}= 1 nm and

*d*

_{2}= 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.

*d*

_{1}= 10 nm) and (b) for small inner gold shell thickness (

*d*

_{1}= 1 nm). This figure shows that coupling between the inner and outer gold shell depends on their thicknesses. For large inner gold shell thickness (

*d*

_{1}) (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.

*d*

_{1}) for a given outer gold shell thickness (

*d*

_{2}= 10 nm) is shown. In this figure we calculate the cross section for large

*d*

_{1}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.

## 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.