Research Article  Open Access
Simulations of the Light Scattering Properties of Metal/Oxide Core/Shell Nanospheres
Abstract
Given the importance of the optical properties of metal/dielectric core/shell nanoparticles, in this work we focus our attention on the light scattering properties, within the Mie framework, of some specific categories of these noteworthy nanostructures. In particular, we report theoretical results of angledependent light scattering intensity and scattering efficiency for Ag/Ag_{2}O, Al/Al_{2}O_{2}, Cu/Cu_{2}O, Pd/PdO, and Ti/TiO_{2} core/shell nanoparticles as a function of the core radius/shell thickness ratio and on a relative comparison. The results highlight the light scattering characteristics of these systems as a function of the radius/shell thickness ratio, helping in the choice of the more suitable materials and sizes for specific applications (i.e., dynamic light scattering for biological and molecular recognition, increasing light trapping in thinfilm silicon, organic solar cells for achieving a higher photocurrent).
1. Introduction
Metal nanoparticles (NPs) have been extensively studied in recent years due to their wide range of potential applications [1–11]. In particular, the increase in the local electric field produced by the localized surface plasmon resonance (LSPR) [1–4] is useful for several linear [12] and nonlinear [13, 14] optical processes. In fact, metal NPs can exhibit extraordinary optical resonances: when excited by electromagnetic radiation they can exhibit LSPR due to the collective oscillations of their conduction electrons [1–4]. The resonant excitation of LSPR leads to selective photon absorption and enhancement of local electromagnetic fields near the NPs by orders of magnitude. The possibility of controllably tuning the LSPR wavelength through the visible to near infrared region makes metal NPs very important and promising for the technological applications. In addition to their LSPR absorption, light scattering is another optical property of metal NPs that is of great interest [1–4, 15–18]. When NPs are illuminated with a beam of light at a wavelength () that matches the plasmon absorption maxima (), the NPs can both absorb and scatter light outside of their physical crosssections. So, also, dynamic light scattering by metal NPs is, today, widely used as a powerful tool in biological and molecular recognition [6, 7] and to increase the light trapping in thinfilm silicon and organic solar cells for achieving a higher photocurrent [8–14] (Mie scattering [15–18] plays an essential role in increasing the optical path length). Scattering and plasmon optical properties, such as peak wavelength, full width at half maximum, and contrast, strongly depend on the material, size, shape, and structure of the NPs, as well as on the surrounding media. As a consequence, in addition to pure metal NPs, recently, also metal/dielectric core/shell NPs are widely studied for the improvement and manipulation of the scattering and plasmon resonances properties. For coated (coreshell) NPs, these characteristics are strongly influenced by the presence of the shell: the position of the LSPR and the angulardependent scattered light intensity can be tuned by varying the ratio between the core radius and the thickness of the oxide layer making these nanostructures technologically versatile [19–39]. Some works deal with the fabrication, by both physical and chemical methodologies, and study core/shell NPs formed by a metal core and its own oxide as a shell: [23–27], [28–31], [32–36], [37, 38], and [39]. These works triggered an increasing need for the study of the optical response of such nanostructures as a function of size of the core and of the shell [25–27, 30, 34, 35, 37]. For example, Santillán et al. [26] reported data on the optical extinction of core/shell NPs, Kuzma et al. [27] reported data about the influence of the surface oxidation on plasmon resonance in monolayer of Ag NPs, Peng et al. [30] reported data about the absorption spectra of core/shell NPs, Ghodselahi et al. [34] reported data about surface plasmon resonance of core/shell NPs, PeñaRodríguez and Umapada [35] reported data on the effects of surface oxidation on the linear optical properties of Cu NP, Xiong et al. [37] reported data on the size dependence surface plasmon resonance in oxidated Pd nanostructures. On the basis of the interest in the optical response of the core shell NPs, in this work we focus our attention, in particular, on the light scattering properties of Ag/Ag_{2}O, Al/Al_{2}O_{3}, Cu/Cu_{2}O, Pd/PdO, and Ti/TiO_{2} core/shell NPs as a function of the core radius/shell thickness ratio and on a relative comparison (Figure 1). In particular, we report theoretical results about the angledependent light scattering properties of spherical NPs (nanospheres, NSPs) on the basis of a generalized Mie approach. So, after a short recall, in Section 2, of the theoretical basic concepts to treat the light scattering phenomena from small homogeneous and layered spherical particles as known in the literature on the basis of the Mie approach [17, 18, 40–42], in Section 3 we will report the results for the simulation of the light scattering intensity, using the SCATLAB simulation software [43], for , , , , and core/shell NSPs increasing the particle radius from 30 to 110 nm and the shell thickness from 0 to 80 nm (Figure 1). Finally, a comparison between the scattering efficiency of these core/shell NSPs will be drawn.
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The results exposed in this work can be of help in the design of tunable efficiency light scattering devices (biological and molecular sensors, solar cells).
2. Theory Concepts
Interaction of electromagnetic radiation with particles is well studied [15–18, 40–42]. The classical theory by Mie [40] gives an analytical solution for scattering from homogenous particles. Theoretical studies on the optical properties of multilayer spherical [42] and planar [41] particles are reported in the literature. In particular, for our general discussion, in this section we expose the basic concepts of the theory for scattering of electromagnetic waves by coreshell materials following the considerations of Small et al. [42]. The solution under appropriate conditions gives the same expression as Mie theory for single sphere.
The important parameter to treat the scattering process of electromagnetic radiation from a particle (i.e., a welldefined aggregate of many atoms or molecules) embedded in an otherwise homogeneous medium (in the sense that the atomic or molecular heterogeneity is small compared with the wavelength of the incident light) is the scattering crosssection . It is the ratio of the total scattered power to the radiation intensity. The cross section (units of area) is proportional to the total amount of light scattered by a particle when illuminated by a plane wave. Supposing to illuminate a particle with a plane wave of intensity (energy/area/time), and by detectors to measure the power scattered in each direction, by integrating the measured over all directions, we obtain the total scattered power (energy/time). The scattering cross section (i.e., the ratio of the total radiant power scattered by a particle in all directions to the radiant power incident on the particle, or, also, the number of photons scattered by particle through any scattering angle) is The scattering cross section is converted to scattering efficiency () which is the normalized scattering cross section to the geometrical cross section: being the radius of the particle. has the advantage of being a dimensionless number so that it can be used to compare particles of different sizes. can exceed 1 since in addition to scattering light incident on their geometrical cross section, particles also diffract light at their edges, so a particle can behave larger than its geometrical cross section. By solving Maxwell’s equations for a plane wave incident on a layered spherical particle can be calculated (for a homogeneous particle the first exact analytical solution was given by Mie [40]). To a qualitatively understanding of the physics of scattering by a single particle, we can consider an arbitrary particle, as conceptually subdividing into small regions. An applied oscillating field (e.g., an incident electromagnetic wave) induces a dipole moment in each region. These dipoles oscillate at the frequency of the applied field and therefore scatter secondary radiation in all directions. In a particular direction, the total scattered field is obtained by superposing the scattered wavelets, where due account is taken of their phase differences: scattering by the dipoles is coherent. In general, these phase relations change for a different scattering direction; we therefore expect the scattered field to vary with scattering direction. If the particle is small compared with the wavelength, all the secondary waves are approximately in phase; for such a particle we do not expect much variation of scattering with direction. As the particle size is increased, however, the number of possibilities for mutual enhancement and cancellation of the scattered wavelets increases. Thus, the larger the particle, the more peaks and valleys in the scattering pattern [17]. For a layered particle it is possible to use the Mie theory in a form that will be useful for making analogies with waves in planar dielectric coatings [42]. We consider a spherical particle with layers, with layer 1 being the core and layer the outermost shell (Figure 2). Each layer has radius and refractive index (with the parameter being the manifestation of scattering by the many molecules that compose the medium). The particle is embedded in a background matrix of refractive index . Because of spherical symmetry, the incident, scattered, and internal fields can be expanded as a superposition of vector spherical harmonics. So we can solve the boundary value problem to determine the expansion coefficients of the vector spherical harmonics. At each boundary between two distinct media, the electromagnetic fields have to satisfy Maxwell’s boundary conditions. For a multilayered sphere, in principle, there are three different boundaries to take into in account: (i) the boundary between the core and the innermost shell, (ii) the boundary between two shells, and (iii) the boundary between the particle and the embedding medium. For the latter, they connect the electromagnetic fields of the incident wave and the scattered wave with the wave in the outermost shell. For the quantitative analyses, we call the function describing the waves radiating outwards from the origin and for waves converging inward toward the origin. The superscript * denotes complex conjugation, ( the light velocity in vacuum), is the refractive index of the medium that the wave is traveling in, and , , and are the usual spherical coordinates. refers to the polarization of the wave and is 1 for transverse magnetic waves and 2 for transverse electric waves. The subscript refers to the order of the Hankel function of the first kind governing the radial dependence of the vector spherical harmonic [42]. At large distances from the origin the Hankel functions are proportional to , where the sign of the exponent depends on whether the wave radiates outward from the origin (+) or converges inward toward the origin (−). When the thickness of a layer is much smaller than the distance from the origin we can approximate the radial dependence of the field in that layer with a sinusoidal function. The plane wave incident on the particle can be written as [42] where . The scattered field can be written as [42] where is proportional to a scattering amplitude. The coefficients completely determine the scattered field. So, their calculation is the main aim in studying electromagnetic scattering by a particle. Once the scattered field is known, can be calculated. The field in the th layer of the particle can be written in the following form: where and are coefficients of the outgoing and incoming fields. In (6) the first boundary condition is imposed: at the core of the particle . An incoming spherical wave front that converges toward the center of the particle will, upon reaching the center, diverge outward, and so the amplitudes of the incoming and outgoing fields must be equal at the center of the particle. The other boundary conditions are that the transverse components of the electric and magnetic fields are continuous across the boundary between layers and and between layer and the surrounding matrix (note that the magnetic field can be easily calculated from the Faraday equation once that is known). These conditions yield a set of equations that we can solve to obtain the set of coefficients , which can then be used to obtain for a given particle morphology.
This general methodology proposed by Small et al. [42] is very efficient in treating the light scattering problems from homogeneous and layered particles. It is in general very long and laborious but excellent algorithms and software are available to perform these calculations. In this paper, we used ScatLab 1.2 [43] for theoretical calculations. In this computer code the scattering cross sections () and angledependent intensities can be calculated for homogeneous and coreshell geometries.
3. Calculations and Discussion
ScatLab is a software developed by Bazhan to perform electromagnetic scattering simulations mainly based on classical Mie theory solution [43]. We used it to calculate the scattering properties of , , , , and spherical NPs of radius between 30 and 110 nm, and of , , , , and core/shell spherical NPs with radius of the metal core varying between 30 and 90 nm and thickness of the dielectric shell between 20 and 80 nm. The reported calculations are performed fixing, for example, a wavelength of nm of the incident radiation (the center of the visible spectrum). The input parameters in each simulation are the core radius , the shell thickness , the real part of the environment (air) refractive index (), the imaginary part of the environment (air) refractive index (), the real part of the particle metal core refractive index (), the imaginary part of the particle metal core refractive index (), the real part of the particle dielectric coat refractive index (), and the imaginary part of the particle dielectric coat refractive index (). For the chosen wavelength we use the values of , , and for , , , the values are reported in Table 1 and extracted from [44–46]. As output parameters we report the scattered intensity (arbitrary units) as a function of the scattering angle (degrees) in polar diagrams and the scattering efficiency . In the polar diagrams, the wave impacts from 180°. Note, obviously, that we are always in a condition .

In Figure 3 we report the calculated polar diagrams for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness increasing from 0 to 80 nm. In general the effect of increasing relies on an increasing of the intensity of the scattered light at 0° with respect to the intensity of the light scattered at 180°. Regarding the NSPs: for nm and nm, the intensity of the light scattered at 180° is slightly higher than that scattered at 0°; this is the opposite with respect to the case for nm and nm; very interesting is the situation obtained for nm and nm, where the intensity of the light scattered at 0° is about 3 times higher than the intensity of the light scattered at 180°; such a phenomenon is enhanced for nm and nm and for nm and nm. Concerning the NSPs: for nm and nm, the intensity of the light scattered at 180° is higher than that scattered at 0° (more than the case of the NSPs); about the same are these intensities when nm and nm; then increasing to 40, 60, and 80 nm, the intensity of the light scattered at 0° becomes higher than that scattered at 180° (particularly evident for nm). About the NSPs: very similar are the intensities of the light scattered at 0° and 180°; the intensity of the light scattered at 180° becomes higher than that scattered at 0° for , 40, 60, and 80 nm; this situation is more pronounced for nm and less pronounced for nm. Regarding the NSPs: while, for nm and nm, the intensity of the light scattered at 0° is lower than the intensity of the light scattered at 180°, the opposite situation is obtained for , 40, 60, and 80 nm, with particular enhancing for nm. Finally, concerning the NSPs: the behavior is similar to the behavior of the NSPs, apart the case for nm. In this case, for the NSPs the intensities of the light scattered at 0° and 180° are about the same.
In Figure 4 we report the calculated scattering efficiency (in semilog scale) for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness increasing from 0 to 80 nm. Apart from the case of the NSPs, for the other types of core/shell NSPs an almost monotonic increase of as a function of is obtained. For the NSPs, instead, a minimum value of is obtained for nm and a maximum value for nm.
In Figure 5 we report the calculated polar diagrams for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness increasing from 0 to 60 nm. Regarding the NSPs: for nm and nm, the intensity of the light scattered at 180° is slightly higher than that scattered at 0°, while it is about the same when nm and nm; for nm and nm, the intensity of the scattered light at 180° is much higher than that scattered at 0°, while the opposite situation is obtained for nm and nm. Concerning the NSPs: for nm and nm, the intensity of the light scattered at 180° is much higher than that scattered at 0°, while they are about the same when nm and nm; for nm and and 60 nm, the intensity of the scattered light at 0° is higher than that scattered at 180°. About the NSPs: for nm and nm, the intensities of the scattered light at 0° and 180° are comparable, while for nm and and 40 nm the intensity of the scattered light at 0° is much higher than that scattered at 180°; for nm and nm, the intensity of the light scattered at 180° is slightly higher than that scattered at 0°. Regarding the NSPs: for nm and nm, the intensity of the light scattered at 180° is higher than that scattered at 0°, while the opposite is the situation for nm and nm; for nm and and 60 nm, the intensity of the scattered light at 0° is much higher than that scattered at 180°. Finally, concerning the NSPs: while, for nm and nm the intensities of the scattered light at 0° and 180° are comparable, for nm and , 40, and 60 nm the intensity of the scattered light at 0° is much higher than that scattered at 180°.
In Figure 6 we report the calculated scattering efficiency (in semilog scale) for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness increasing from 0 to 60 nm. While for the case of the and NSPs, increases monotonically with , for the and NSPs it presents a maximum value for nm and a minimum one for nm; for the NSPs it presents a minimum value for nm and a maximum one for nm.
In Figure 7 we report the calculated polar diagrams for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness increasing from 0 to 40 nm. Regarding the NSPs: for nm and nm, the intensity of the light scattered at 180° is slightly higher than that scattered at 0°; for nm and and 40 nm it is much higher. Concerning the NSPs: for nm and nm, the intensity of the light scattered at 180° is higher than that scattered at 0°, while they are comparable for nm and nm; for nm and nm, the intensity of the light scattered at 180° is lower than that scattered at 0°. About the NSPs: for nm and nm, the intensity of the light scattered at 180° is slightly lower than that scattered at 0°, while for nm and , 40 nm it is much lower than that scattered at 0°. Regarding the NSPs: for nm and nm, the intensity of the light scattered at 180° is slightly higher than that scattered at 0°, while for nm and and 40 nm it is much lower than that scattered at 0°. Finally, concerning the NSPs: or nm and nm, the intensity of the light scattered at 180° is slightly lower than that scattered at 0°, while for nm and and 40 nm it is much lower than that scattered at 0°.
In Figure 8 we report the calculated scattering efficiency (in semilog scale) for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness increasing from 0 to 40 nm. Regarding the NSPs, decreases monotonically when increases. Concerning the NSPs it increases monotonically when increases. About the NSPs, decreases when passes from 0 to 20 nm and increases when passes from 20 to 40 nm. Regarding the and NSPs it is constant with .
In Figure 9 we report the calculated polar diagrams for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness increasing from 0 to 20 nm. Regarding the NSPs: for nm and nm, the intensities of the light scattered at 0° and 180° are comparable, while for nm and nm, the intensity of the light scattered at 0° is much higher than that scattered at 180°. Concerning the NSPs: for nm and nm, the intensity of the light scattered at 180° is higher than that scattered at 0°, while the opposite is the case for nm and nm. About the NSPs: for nm and nm, the intensity of the light scattered at 180° is higher than that scattered at 0°, and this difference is enhanced for nm and nm. A similar behavior for the and NSPs.
In Figure 10, we report the calculated scattering efficiency (in semilog scale) for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness increasing from 0 to 20 nm. is the highest and is constant with , for the NSPs. Lower values are obtained for the NSPs. Furtehr lower values are obtained for the and NSPs. For the NSPs, the value of is comparable to the one obtained for the and NSPs for nm, while its value is the minimum one for nm.
In Figure 11 we report the calculated polar diagrams for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness nm. In any case, the intensity of the scattered light at 0° is higher than that scattered at 180°.
In Figure 12 we report the calculated scattering efficiency (in semilog scale) for , , , , and core/shell spherical NPs with metal core radius nm and dielectric shell thickness nm. It is the highest for the NSPs. Then it is lower, in order, for the , , , and NSPs.
4. Conclusion
In conclusion, in this work we reported theoretical results of angledependent light scattering intensity and scattering efficiency for , , , , and core/shell NPs increasing the radius of the metal core from 30 to 110 nm and the thickness of the dielectric shell from 0 to 80 nm. For various geometrical conditions the diagram and scattering efficiency were calculated and they can be compared for the different core/shell NPs for the different combination of sizes of the core and of the shell. Combining the and information, the best geometry for the metal/dielectric core/shell NPs can be chosen for a specific application involving a particular scattering properties of the system. Furthermore, our approach is broadly applicable rather than a relic of the parameters that we investigated in this work. A similar simulation can be performed, within the Mie theoretical framework, for homogeneous or layered particles of any materials embedded in any medium.
Conflict of Interests
The authors declare that there is no conflict of interests regarding thepublication of this paper.
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Copyright
Copyright © 2014 F. Ruffino et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.