Introduction

Metallodielectric layered nanoparticles and nanostructures have attracted increasing scientific and technological interest in recent years due to their importance in the fundamental physics and the potential applications in nano-electronics, biomedical imaging, nano-optical device, and optical sensing [1–5]. A special interesting structure is metal–dielectric–metal three-layered nanoparticle, whose optical properties highly depend on the couplings between the plasmons of inner core and outer shell [6–8]. Changing the internal geometry of this nanoparticle not only shifts its resonance frequencies, but also strongly modifies the relative magnitudes of the absorption and scattering cross sections [5]. Mukherjee et al. [9] have found that an asymmetric core in Au/SiO2/Au can lead to an additional high multipolar Fano resonance. A significant superscattering phenomenon also has been found in the plasmonic–dielectric–plasmonic layered nanorod [3] and nanosphere [4].

Among the metallodielectric layered nanoparticles, the spherical anisotropic (SA) material–metal nano-composite shows a growing interest because of the applications in optical devices, surface-enhanced Raman spectroscopy (SERS) and optical nonlinearity enhancement [10–12]. Spherical anisotropy indicates that the tensor for dielectric function is radially anisotropic, i.e., the dielectric function is uniaxial in spherical coordinate with ɛ r along the radial direction and ɛ t along the tangential direction. Gao et al. [10] have reported that the adjustment of the dielectric anisotropy in the core or shell could result in large enhancements of the second harmonic generation and induced third harmonic generation susceptibilities at surface plasmon resonant frequencies. Yin et al. [11] have found that the introduction of spherical anisotropy into the core or the shell provides a novel approach to tailor the surface plasmon resonant frequencies and enhanced SERS peaks. It is further found that the spherically anisotropy of inner core can strongly modulate the Fano resonance in the Ag nanoshell with a spherically anisotropic core [12]. However, the modulation of spherical anisotropy on the plasmon couplings in metallodielectric layered nanoparticles is seldom reported.

In this paper, we have investigated the optical properties of the sandwiched gold nanoshell with an SA middle layer (Au/SA/Au) by using a modified Mie theory. Spherical anisotropy was indeed found in phospholipid vesicle systems [13] and in cell membranes containing mobile charges [14]. Lucas et al. [15] have easily established the spherically anisotropic materials by using the graphitic multishells. We focus on the influence of the spherical anisotropy of the middle layer on the plasmon couplings between the inner core and outer shell. In addition, the dependence of the local electric field enhancement on the spherical anisotropy of the middle layer has been further discussed in detail.

Electromagnetic scattering model

Figure 1 shows the specific geometry of a sandwiched gold nanoshell. The nanoparticle consists of a gold core with a radius of r 1, an SA middle layer with a radius of r 2, and a gold shell with a radius of r 3. The dielectric constant of embedding medium is ɛ 4. The SA middle layer is characterized by constitutive tensors of permittivity

$$ {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}}{\varepsilon }}}_{2}}=\left( {\begin{array}{*{20}{c}} {{{\varepsilon }_{{2\text{r}}}}} & 0 & 0 \\ 0 & {{{\varepsilon }_{{2\text{t}}}}} & 0 \\ 0 & 0 & {{{\varepsilon }_{{2\text{t}}}}} \\ \end{array}} \right). $$
(1)
Fig. 1
figure 1

Geometry diagram of a sandwiched gold nanoshell

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ɛ 2r is along the radial direction and ɛ 2t is along the tangential direction. The dielectric functions of inner metal core ɛ 1 and outer metal shell ɛ 3 have real and imaginary frequency-dependent components, which are affected by the scattering of the conduction electrons in the particle surfaces. Thus, ɛ 1 and ɛ 3 are usually accounted by replacing the ideal Drude part in the dielectric function with a size-dependent one [8]. Light scattering by a spherical particle can be expressed through Debye potentials [16]. For spherical anisotropic material, the electric ψ TM and magnetic ψ TE Debye potentials are presented as [17]

$$ \frac{{{\varepsilon_{\mathrm{r}}}}}{{{\varepsilon_{\mathrm{t}}}}}\frac{{\partial^2 {\psi_{\mathrm{TM}}}}}{{\partial {r^2}}}+\frac{1}{{r^2 \sin \theta }}\frac{\partial }{{\partial \theta }}\left( {\sin \theta \frac{{\partial {\psi_{\mathrm{TM}}}}}{{\partial \theta }}} \right)+\frac{1}{{r^2 {\sin^2}\theta }}\frac{{\partial^2 {\psi_{\mathrm{TM}}}}}{{\partial {\varphi^2}}}+k_0^2{\varepsilon_{\mathrm{r}}}{\mu_{\mathrm{t}}}{\psi_{\mathrm{TM}}}=0, $$
(2)
$$ \frac{{{\mu_{\mathrm{r}}}}}{{{\mu_{\mathrm{t}}}}}\frac{{\partial^2 {\psi_{\mathrm{TE}}}}}{{\partial {r^2}}}+\frac{1}{{r^2 \sin \theta }}\frac{\partial }{{\partial \theta }}\left( {\sin \theta \frac{{\partial {\psi_{\mathrm{TE}}}}}{{\partial \theta }}} \right)+\frac{1}{{r^2 {\sin^2}\theta }}\frac{{\partial^2 {\psi_{\mathrm{TE}}}}}{{\partial {\varphi^2}}}+k_0^2{\varepsilon_{\mathrm{t}}}{\mu_{\mathrm{r}}}{\psi_{\mathrm{TE}}}=0. $$
(3)

Here, \( {k_0}={{{2\pi }} \left/ {\lambda } \right.} \) is the wave vector in vacuum and \( {\mu_{\mathrm{r}}}={\mu_{\mathrm{t}}}=1 \). The electromagnetic waves are expanded to spherical partial waves using vector spherical harmonics, and then, Maxwell's boundary conditions are applied to resolve the unknown expansion coefficients of the scattered and interior waves. According to the Mie scattering theory, the obtained extinction efficiency Q ext, scattering efficiency Q sca, and absorption efficiency Q abs can be expressed as [16]

$$ {Q_{\mathrm{ext}}}=\frac{2}{{{{{\left( {k_4 {r_3}} \right)}}^2}}}\sum\limits_{n=1}^{\infty } {\left( {2n+1} \right){\mathop{\text Re}\nolimits}\left( {a_n +{b_n}} \right)}, $$
(4)
$$ {Q_{\mathrm{sca}}}=\frac{2}{{{{{\left( {k_4 {r_3}} \right)}}^2}}}\sum\limits_{n=1}^{\infty } {\left( {2n+1} \right)\left( {{{{\left| {a_n } \right|}}^2}+{{{\left| {b_n } \right|}}^2}} \right)}, $$
(5)
$$ {Q_{\mathrm{abs}}}={Q_{\mathrm{ext}}}-{Q_{\mathrm{sca}}}. $$
(6)

Here, \( {k_4}={k_0}\sqrt{\varepsilon_4 } \), a n and b n are the scattering coefficients. The information about spherical anisotropic middle layer is presented by the order of spherical Bessel functions [17]. The order can be expressed as

$$ v={{\left[ {n\left( {n+1} \right)\frac{{{\varepsilon_{2\mathrm{t}}}}}{{{\varepsilon_{2\mathrm{r}}}}}+\frac{1}{4}} \right]}^{{{1 \left/ {2} \right.}}}}-\frac{1}{2}. $$
(7)

Results and discussion

Figure 2a shows the extinction spectra of the Au/SA/Au nanoshells for the conditions of ρ < 1. We assume ρ as the anisotropic value of the middle layer \( {{{{\varepsilon_{{2\mathrm{t}}}}}} \left/ {{{\varepsilon_{{2\mathrm{r}}}}}} \right.} \). Here, r 1, r 2, and r 3 are fixed at 35, 50, and 65 nm, respectively. To discuss the influence of ɛ 2r on the plasmon resonances, ɛ 2t value is fixed at 2.04. The embedding medium is assumed to be water (ɛ 4 = 1.7689). The plasmon resonances in the Au/SA/Au nanoshell can be considered as an interaction between the plasmons of a gold nanosphere and a gold nanoshell [5, 8, 18]. As shown in Fig. 2a, for \( {{{{\varepsilon_{{2\mathrm{t}}}}}} \left/ {{{\varepsilon_{{2\mathrm{r}}}}}} \right.}=1 \), one dipole peak appears at 838 nm, which corresponds to the antisymmetric coupling (ω −− mode) between the plasmons of inner Au core (ω s ) and the symmetric mode of outer Au nanoshell (ω −). Another dipole peak appears at 577 nm, which corresponds to the symmetric coupling (ω +− mode) between ω s and ω − modes. The decreased ρ value means the increase of ɛ 2r value. With increasing ɛ 2r value, the dipole peak of ω +− mode shows a red-shift from 577 nm at ρ = 1 to 625 nm at ρ = 1/5, while the dipole peak of ω −− mode shows a very large red-shift from 838 nm to 1,511 nm. The increased ɛ 2r value should decrease the induced charges in the inner and outer surfaces of the middle layer [19]. In this case, the plasmon resonance energies of ω s and ω − modes decrease, and hence, the red-shifts of the ω −− and ω +− modes. According to plasmon hybridization theory [19–21], the ω −− mode is sensitive to the ω − mode, while the ω +− mode depends on the ω s mode. The large red-shift of the ω −− mode indicates that the variation of ɛ 2r has more effect on the ω − mode than the ω s mode. Figure 2b shows the extinction spectra of the Au/SA/Au nanoshells for the conditions of ρ > 1. Here, the ɛ 2r value is fixed at 2.04. The increased ρ value means the increase of ɛ 2t value. With increasing ɛ 2t value, the dipole peak of ω +− mode shows a red-shift from 577 nm at ρ = 1 to 611 nm at ρ = 5, while the dipole peak of ω −− mode shows a red-shift from 838 to 859 nm. The small red-shifts of the ω −− and ω +− modes suggest that the increased ɛ 2t value has little effect on the decrease of the induced charges in both surfaces of the middle layer. The red-shift of the mode is larger than that of the ω −− mode, which indicates the suppressed coupling between the ω s and ω − modes due to the increased ɛ 2t value.

Fig. 2
figure 2

Extinction spectra of the Au/SA/Au nanoshells with various anisotropic values of the middle layer for a ɛ 2t/ɛ 2r < 1 and ɛ 2t = 2.04 and b ɛ 2t/ɛ 2r > 1 and ɛ 2r = 2.04. Here, r 1, r 2, and r 3 are fixed at 35, 50, and 65 nm, respectively

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Figure 3a, d shows the distributions of the electric field enhancement in the sandwiched nanoshell (ɛ 2t/ɛ 2r = 1) at wavelengths of 577 nm (ω +− ) and 838 nm (ω −− ), respectively. The distributions of the electric field enhancement show the typical dipole resonance properties [22]. The large electrical field on the shell occurs along the incident polarization and only locates within a few nanometer of the shell surface, while the inner core also exhibits a similar dipole pattern. In Fig. 3d, a large electric field is observed near the core mainly due to the antisymmetric coupling between the ω s and ω − modes, which leads to different kinds of charges induced in inner and outer surfaces of the middle layer [18]. Figure 3b, e shows the contour plots of the electric field enhancement in the Au/SA/Au with ρ = 1/1.1 (divided by the results of the Au/SA/Au with ρ = 1). The calculation wavelengths are fixed at 577 and 838 nm for subpanels b and e of Fig. 3, respectively. In Fig. 3b, the decrease of the induced charge in outer surface of the middle layer is larger than that in inner surface of the middle layer. For the ω −− mode, the increased ɛ 2r value induces the large electric field inside the nanoparticle. Figure 3c, f shows the contour plots of the electric field enhancement in the Au/SA/Au with ρ = 1.1 at wavelengths of 577 and 838 nm (divided by the results of the Au/SA/Au with ρ = 1), respectively. The increased ɛ 2t value should lead to the decrease of the induced charges on the inner and outer surfaces of the SA layer. It is obvious that the effects of the variation of ɛ 2t on the induced charges and the coupling between ω s and ω − modes are weaker than those due to the variation of ɛ 2r.

Fig. 3
figure 3

Contour plots of the electric field enhancements in Au/SA/Au nanoshells with a ρ = 1 (ɛ 2r = ɛ 2t = 2.04) at 577 nm, b ρ = 1/1.1 (ɛ 2t = 2.04) at 577 nm, c ρ = 1.1 (ɛ 2r = 2.04) at 577 nm, d ρ = 1 (ɛ 2r = ɛ 2t = 2.04) at 838 nm, e ρ = 1/1.1 (ɛ 2t = 2.04) at 838 nm, and f ρ = 1.1 (ɛ 2r = 2.04) at 838 nm. Here, r 1, r 2, and r 3 are fixed at 35, 50, and 65 nm, respectively

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To further clarify the role of anisotropy, we keep the geometric average of dielectric components \( {\varepsilon_{\mathrm{i}}}={{{{\varepsilon_{{2\mathrm{r}}}}}} \left/ {3} \right.}+{{{2{\varepsilon_{{3\mathrm{t}}}}}} \left/ {3} \right.} \) unchanged for the SA middle layer [12], while ɛ 2t/ɛ 2r value is varied. Figure 4 shows the extinction spectra of the Au/SA/Au nanoshells with various ρ values. Here, r 1, r 2, and r 3 are fixed at 35, 50, and 65 nm, respectively. The ɛ i value is fixed at 2.04. With an increasing ρ value, the dipole peak of the ω −− mode shows a distinct blue-shift from 1,589 nm at ρ = 1/5 to 722 nm at ρ = 5, while the strength of the peak increases. At the same time, the dipole peak of the ω +− mode shows a blue-shift from 606 nm at ρ = 1/5 to 503 nm at ρ = 5, but the strength of the peak decreases. The influence of the variation of anisotropic value on the plasmon resonances in the Au/SA/Au nanoshell with ρ < 1 is stronger than that for ρ > 1. Figure 5a shows the dependences of the E-field enhancement in the Au/SA/Au nanoshell as a function of ρ value, which are calculated at the dipole resonance wavelengths of ω +− mode. In Fig. 5a, the dashed circle line for right-hand scale represents the variation of E-field enhancement maximum outside the nanoshell (EMO), which often locates on the outer surface of the particle and at the poles along the incident polarization. The dashed square line for left-hand scale shows the variation of the E-field enhancement maximum inside the nanoshell (EMI), which can be obtained on the surface of inner gold core and at the poles along the incident polarization. In Fig. 5b, the dashed circle line for right-hand scale and dashed square line for left-hand scale represent the variations of EMO and EMI, respectively, which are calculated at the dipole resonance wavelengths of ω −− mode. It is found with the increase in ρ value that the EMO for the ω +− mode decreases from 4.97 at ρ = 1/10 to 0.68 at ρ = 10, while the EMI decreases from 31.15 at ρ = 1/10 to 1.92 at ρ = 10. At the same time, the EMO for the ω −− mode increases first from 4.13 at ρ = 1/10 to 6.68 at ρ = 2 and then decreases to 6.51 at ρ = 0, while the EMI decreases from 149.00 at ρ = 1/10 to 0.85 at ρ = 10. It is found that the smaller ρ value is helpful to obtain larger electric field inside the nanoparticle. The larger ρ value of the middle SA layer can induce the larger near-field outside the nanoparticle for the ω −− mode and the smaller ρ value makes larger near-field for the ω +− mode.

Fig. 4
figure 4

Extinction spectra of the Au/SA/Au nanoshells with various ρ values (ɛ i = 2.04). Here, r 1, r 2, and r 3 are fixed at 35, 50, and 65 nm, respectively

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Fig. 5
figure 5

Dependences of the E-field enhancement maximum for a ω +− mode and b ω −− mode in Au/SA/Au nanoshells as a function of ρ value. Here, ɛ i = 2.04, r 1 = 35 nm, r 2 = 50 nm, and r 3 = 65 nm. The dashed circle line for right-hand scale and dashed square line for left-hand scale represent the variations of EMO and EMI, respectively

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Conclusion

We have investigated the plasmon resonance properties of the Au/SA/Au nanoshells. The extinction spectra and the electric field enhancement of the Au/SA/Au nanoshells have been calculated based on a modified Mie scattering theory. We focus on the influence of the spherical anisotropy of the middle layer on the plasmon resonance couplings in the Au/SA/Au nanoshells. It is found that the permittivity along the radial direction plays a dominant role on the plasmon couplings in Au/SA/Au nanoshells, and the permittivity along the tangential direction leads to significant modulations. With the increase of ρ, both the dipole peaks of the ω −− and ω +− modes show blue-shifts. The variation of the ω −− mode is stronger than the ω +− mode. The large ρ value of the middle SA layer can induce the large near-field outside the nanoparticle for the ω −− mode and the small ρ value can induce the large near-field for the ω +− mode. Such enhanced near-field can be used to the enhanced Raman excitation and emission. Furthermore, the small ρ value is helpful to obtain larger electric field inside the nanoparticle, which may be helpful for their potential applications in nonlinear optics.