Plasmonic is an emerging branch of nanophotonics wherein the electromagnetic properties of nanoparticles are studied for variety of applications. The optics of nanoparticles is studied in terms of surface plasmon resonances and optical cross section. Initially the first principle approach has been used to study the plasmonic fundamentals known as electrostatic approach. Under this approach, various parameters are taken into account to observe the electromagnetic properties of plasmonic nanogeometries. This electrostatic model is only used to analyze the optical signature of smaller size plasmonic geometries. Therefore, for the estimation of optical properties of larger size nanoparticle numerical model (Discrete Dipole Approximation) has been used. The observed surface plasmon resonances could be useful in sensing field, SERS signal detection and thin film solar cell application.
- surface plasmon resonance
- optical cross section
Plasmonics is the emerging branch of nanophotonics which deals the coupling of light to the collective oscillation of electrons inside the metal nanoparticles. The coupling of light to the metal nanoparticles produces resonances under specific condition known as surface plasmon resonance (SPR) that has tremendous applications [1, 2, 3, 4, 5]. The resonant interaction between them will localize the electromagnetic field near the meal surface and drastically enhances the optical scattering phenomenon. When the light interacts with the metallic nanostructures two fundamental excitations are observed. These two fundamental excitations are surface plasmon polaritons (SPP) and localized surface plasmon resonance (LSPR). The surface plasmon polaritons is propagating, dispersive electromagnetic waves coupled to the electron plasma of a metal at a dielectric interface. The other excitation is localized surface plasmons which are non-propagating excitations of the conduction electrons of metallic nanostructures coupled to the electromagnetic field. Such modes arise due to the scattering of a sub-wavelength conductive nanoparticle in an oscillating electromagnetic field. An effective restoring force on the driven electrons is induced by the curved surface of the particle leading to resonance and field amplification both inside and in the near-field zone outside the particle. This resonance is called the localized surface plasmon or short localized plasmon resonance [6, 7, 8, 9, 10, 11]. Plasmon resonances can be excited by direct light illumination which do not require any phase-matching. In addition to solid particles, other nanostructures that support localized plasmons are dielectric inclusions in metal bodies or surfaces, and nanoshells [6, 12, 13, 14, 15]. For gold and silver nanoparticles, the resonance lies in the visible region of the electromagnetic spectrum leading to bright colors exhibited by particles both in transmitted and reflected light. This is direct consequence of resonantly enhanced absorption and scattering of light from these particles. From the viewpoint of electromagnetic and optics, a major consequence of the resonantly enhanced polarization is associated enhancement in the efficiency with which a metal nanoparticle scatters and absorbs light [6, 8, 16, 17, 18, 19].
An important properties exhibit by plasmonic elements like silver, gold, copper and aluminum is surface plasmons which concentrate the optical energy in nanoscale [20, 21, 22, 23, 24]. The existence plasmonic properties (like surface plasmon mode) in materials is entirely depends on their dielectric constants. Surface plasmon mode is an important property by which one can concentrate the optical energy in nanoscale domain [15, 25, 26, 27, 28]. As the dielectric constants are the complex quantities in which real part is the reflection and imaginary part represent the absorption or loss. Therefore, on the basis of dielectric constant value one can define the definition good and bad plasmonic materials and also decides the weather the material having plasmonic properties or not. Those materials are the plasmonic materials for which dielectric constant
has a negative real part, Re
The work furnishes the study of plasmonic properties of metal nanostructures within and beyond the electrostatic approximation. In the electrostatic approximation we have analyze the interaction of electromagnetic field to metal nanoparticle whose size is much smaller than the wavelength of incident light. In such situation particle experienced only constant or static field throughout the particle volume. Since this model is restricted to smaller size nanoparticles in which only dipolar analysis is taken into account. The dipolar analysis means, we have truncated the potential series upto two terms and apply Laplace equation to obtain the field, polarizability expression and resonance condition. As the size of nanoparticle increases it starts to experience the oscillation behavior of field and electrostatic approximation completely fails. Therefore we need to require techniques which describe the full wave analysis. Discrete Dipole Approximation (DDA) technique is one of them which is based on the dipole discretization concept and frequently used to simulate the plasmonic properties of arbitrary size, shape nanostructure. We have used this simulation technique to study the plasmonic properties of larger size metal nanostructure.
2. Theoretical model: scattering by small size metal nanosphere
Theoretical model discuss the plasmonic properties of metal nanoparticle within electrostatic approximation wherein particle size is smaller than the wavelength of light. Under such assumption Laplace equation has been solved with suitable boundary condition to find out the optical parameters [13, 32]. The Laplace equation is expressed as
and its solution in spherical polar coordinate is
Truncating the potential series only for dipolar term which correspond to one can find the value of potential inside and outside the spherical surface as
Once we have potential profile, the calculation of field and polarizability can be easily obtained by expression.
The applied field polarizes the metal nanostructure and its polarization is in the same direction of applied electric field. If the size of nanoparticle is small enough then polarization is in the direction of applied field while for larger size nanoparticle the oscillation of electron is no longer symmetric and the polarization mechanism is split into component form such as transverse and longitudinal. Here the study reveals the optical properties of smaller size nanoparticle therefore, polarization of particle is in the same direction of applied field. The polarization is simply the dipole moment per unit volume which is expressed as
where E0 is the applied electric field and is the polarizability of nanosphere expressed as
The symbol is the radius of metal nanosphere, are the dielectric constant of metal sphere and surrounding environment. The expression of polarizability is an important parameter in the discussion of resonance physics. The concept of resonance brought into the picture from the denominator part of polarizability expression. The polarizability gets resonantly enhanced when which is known as Fröhlich condition.
As we know when the incident electromagnetic field interacts with metallic nanostructure, some fraction of light gets absorbed and some of them gets scattered. The magnitude of the these absorbed and scattered light can be expressed in terms of scattering and absorption cross section as
where symbol is the incident light wave vector, are the dielectric constant of metal and surrounding medium.
A spherical shape metal nanoparticle is taken into account to observe the optical properties like scattering cross section and surface plasmon resonances. Figure 2 represent the wavelength dependent normalized cross section of three different radii of spherical shape silver metal nanoparticle. We observed that as the size of nanoparticle increases corresponding magnitude of scattering cross section increases. But the position of SPR peak position is same for all radii which is at 364 nm.
In Figure 3, we have discussed the wavelength dependent scattering cross section of gold metal nanoparticle for three different radii. Here we observed that, as the radii of gold nanoparticle increases corresponding cross section magnitude increases with red shifted SPR resonance. In case of silver the SPR peak position is fixed at one wavelength while for gold the peak positions are red shifted with the radii. The SPR peak position for gold nanosphere surrounded by air (N = 1) is at 554 nm for radius 5 nm. The additional advantages of gold nanoparticle over the silver are analyzed in terms of SPR peak positions and corresponding spectral width known as full width at half maxima (FWHM).
These two different types of metal nanoparticle are described within the electrostatic approximation which is only valid for smaller size. Therefore, for the description of lager size nanoparticle we have used the numerical method discrete dipole approximation (DDA) which is valid within and beyond the electrostatic approximation.
3. Numerical approach: scattering by large size metal nanosphere
The numerical technique that we have used to simulate the optical properties of large size nanoparticle is discrete dipole approximation method. In this method, target is discretized into large number of polarizable dipoles . Each dipole situated at the corner of a cubic lattice and the relation between them i.e., separation or distance of one dipole to other are managed by the lattice parameter. The chosen target in this technique is expressed in terms of volume as , where N is the number of discretized dipole and d is the lattice spacing and the size of target is also expressed in term of effective radius as . The technique is applicable only when the following criteria is satisfied
where k is wave vector, lattice parameter and m is the complex refractive index of chosen target material. The main input parameter in DDA technique is the dielectric constant of target and surrounding media in which target is embedded. The complex dielectric function of composite system is provided by input file which can be expressed as
Here, the target is assumed as a point dipole situated at the each corner of cube. Further, placement position and polarizability of the point dipoles need to be flexible for DDA calculation. Each entity is represented by a dipole moment as
where and is the applied and induced field respectively acting on the ith individual because of the radiation of all the others (N – 1) dipoles that set up the NPs. The field and is given by the following relation
where k = represents the incident wave vector.
Dipole moment symbolizes the optical behavior of the target geometry. Therefore, the extinction and absorption cross section can be achieved after calculating the set of dipole moments as given by
signifies the optical cross section,
The simulation effects could be visualized in terms of scattering cross section as shown in Figure 4. A 50 nm radius gold metal sphere is discretized into 4224 number of dipole to see the plasmonic properties like scattering cross section and surface plasmon resonance. The SRP wavelength of 50 nm gold nanosphere was observed at wavelength 560 nm.
The work described the optical properties of plasmonic nanogeometries in terms of optical cross section and surface plasmon resonance. Two different types of metals like silver and gold are taken into account to see the optical properties. The surface plasmon resonance corresponding to these metals lies in visible range of electromagnetic spectrum wherein most of the applications exist. Therefore, the work guides to plamonic community to simulate various types of metal nanostructure which exhibit SPR in different part of electromagnetic spectrum. These tunable nature of surface plasmon resonances can be used in many purposes such as sensing, photovoltaic and Raman spectroscopy.
The authors would like to thanks to Maharaja Agrasen College, University of Delhi, Delhi 110096, India.
Conflict of interest
The authors do not have any conflict of interest.
Catchpole KR, Polman A. plasmonic solar cell. Optics Express. 2008; 6:21793-21800
Pathak NK, Alok J, Sharma RP. Tunable properties of surface plasmon resonances: The influence of core–shell thickness and dielectric environment. Plasmonics. 2014; 9:651-657
Atwater HA, Polman A. Plasmonics for improved photovoltaic devices. Nature Mater. 2010; 9:205-213
Pathak NK, Chander N, Komarala VK, Sharma RP. Plasmonic perovskite solar cells utilizing Au@SiO2 core-shell nanoparticles. Plasmonics. 2017; 12:237-244
Runsheng W, Bingchu Y, Chujun Z, Yulan H, Yanxia C, Peng L, et al. Prominent efficiency enhancement in perovskite solar cells employing silica-coated gold nanorods. Journal of Physical Chemistry C. 2016; 120:6996-7004
Pandey GK, Pathak NK, Uma R, Sharma RP. Electromagnetic study of surface enhanced Raman scattering of plasmonicbiomolecule: An interaction between nanodimer and single biomolecule. Solid State Communications. 2017; 255:47-53
Verma M, Kedia A, Boazbou Newmaia M, Senthil Kumar P. Differential role of PVP on the synthesis of plasmonic gold nanostructures and their catalytic and SERS properties. RSC Advances. 2016; 6:80342-80353
Prodan E, Radloff C, Halas NJ, Nordlander P. A hybridization model for the plasmon response of complex nanostructures. Science. 2003; 302:419-422
Noguez C. Surface plasmons on metal nanoparticles: The influence of shape and physical environment. Journal of Physical Chemistry C. 2007; 111:3806
Zhang Y, Zhen YR, Neumann O, Day JK, Nordlander P, Halas NJ. Coherent anti-stokes raman scattering with single-molecule sensitivity using a plasmonic fano resonance. Nature Communications. 2014; 5:4424
Pathak NK, Sharma RP. Study of broadband tunable properties of surface plasmon resonances of noble metal nanoparticles using mie scattering theory: Plasmonic perovskite interaction. Plasmonics. 2016; 11:713-719
Wustholz KL, Brosseau CL, Casadio F, Van Duyne RP. Surface-enhanced Raman spectroscopy of dyes: from single molecules to the artists’ canvas. Physical Chemistry Chemical Physics. 2009; 11:7350
Maier S. Plasmonics: Fundamentals and Applications. Berlin: Springer; 2007
Hagfeldt A, Boschloo G, Sun L, Kloo L, Pettersson H. Dye-sensitized solar cells. Chemical Reviews. 2010; 110:6595-6663
Pathak NK, Ji A, Sharma RP. Study of efficiency enhancement in layered geometry of excitonic-plasmonic solar cell. Applied Physics A: Materials Science & Processing. 2014; 115:1445-1450
Kreibig U, Vollmer M. Optical Properties of Metal Clusters. Berlin: Springer; 1995
Palik ED. Handbook of Optical Constants of Solids. Orlando: Academic; 1985
Pandey GK, Pathak NK, Ji A, Pathak H, Sharma RP. Plasmonics. 2016; 11:1343
Wang H, Wang X, Yan C, Zhao H, Zhang J, Santschi C, et al. Full color generation using silver tandem nanodisks. ACS Nano. 2017; 11:4419-4427
Chandrasekhar PS, Chander N, Anjaneyulu O, Komaral VK. Plasmonic effect of Ag@TiO2 core–shell nanocubes on dye-sensitized solar cell performance based on reduced grapheme oxide–TiO2 nanotube composite. Thin Solid Films. 2015; 594:45-55
Thyagarajan K, Ch S, Langlet P, OJF M. Highly improved fabrication of Ag and Al nanostructures for UV and nonlinear plasmonics. Advanced Optical Materials. 2016; 4:871-876
Dong L, Yang X, Zhang C, Cerjan B, Zhou L, Tseng ML, et al. Nanogapped Au antennas for ultrasensitive surface-enhanced infrared absorption spectroscopy. Nano Letters. 2017; 17:5768-5774
Ji A, Sharma R, Pathak H, Pathak NK, Sharma RP. Numerical simulation of plasmonic light trapping in thin-film Si solar cells: Surface coverage effect. Journal of Physics D: Applied Physics. 2015; 48:275101-275107
Bellido EP, Zhang Y, Manjavacas A, Nordlander P, Botton GA. Plasmonic coupling of multipolar edge modes and the formation of gap modes. ACS Photonics. 2017; 4:1558-1565
Zhang D, Martin OJF. A universal law for plasmon resonance shift in biosensing. ACS Photonics. 2015; 2:144-150
Ye T, Ma S, Xi J, Wei L, Vijila C. Seeram Ramakrishna performance enhancement of tri-cation and dual-anion mixed perovskite solar cells by Au@SiO2 nanoparticles. Advanced Functional Materials. 2017; 27(13):1606545
Robatjazi H, Zhao H, Swearer DF, Hogan NJ, Zhou L, Alabastri A, et al. Plasmon-induced selective carbon dioxide conversion on earth-abundant aluminum-cuprous oxide antenna-reactor nanoparticles. Nature Communications. 2017; 8:27
Butet J, OJF M. Surface-enhanced hyper-Raman scattering: A new road to the observation of low energy molecular vibrations. Physical Chemistry C. 2015; 119:15547-15556
Mali SS, Chang SS, Hyungjin K, Pramod SP, Chang KH. In situ processed gold nanoparticle-embedded TiO2 nanofibers enabling plasmonic perovskite solar cells to exceed 14% conversion efficiency. Nanoscale. 2016; 8:2664-2677
Parashar PK, Sharma RP, Komarala VK. Double-layer antireflection from silver nanoparticle integrated SiO2 layer on silicon wafer: Effect of nanoparticle morphology and SiO2 film thickness. Journal of Physics D: Applied Physics. 2017; 50:035105
Rundong F, Ligang W, Yihua C, Guanhaojie Z, Li L, Lia Z, et al. Tailored Au@TiO2 nanostructures for the plasmonic effect in planar perovskite solar cells. Journal of Materials Chemistry A. 2017; 5:12034-12042
Bohren CF, Huffman DR. Absorption and Scattering of Light by Small Particles. New York: Willey; 1983
Draine BT, Flatau PJ. Discrete-dipole approximation for periodic targets: Theory and tests. Journal of the Optical Society of America. 2008; 25:2693-2703