Abstract
We determine the amount of light absorbed and scattered by metallic spheres in the presence of a substrate. The analysis is restricted to the spheres whose radius is small compared to the wavelength of the light such that the substrate-particle interactions are adequately described by the electrostatics limit. Results are presented for the absorption and scattering coefficients for: (i) the case when the electric permittivity of the spheres is described by the Drude model and (ii) for specific metals (silver, gold, and copper) for which the data on electrical permittivity as a function of the wavelength are available in the literature. It is found that it is possible to significantly increase the photovoltaic energy collected by a silicon substrate by depositing silver nanospheres on its surface. Mechanisms responsible for this increase are explored in detail in the electrostatics limit. Numerical results for the scattering are also used to derive an approximate formula that can be used to estimate the fractional increase in the photovoltaic energy. The increase predicted by this formula is qualitatively consistent with the literature data on the measured increase in the photocurrent by deposited silver nanospheres.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetic interactions of the incident light and the electrons in the nanoparticles of noble metals, such as silver or gold, result in a resonant behavior that can be used to absorb or trap light in optical devices. A number of applications of this phenomenon have been proposed over the last two decades including its use in harvesting light for photovoltaic and photocatalytic applications and optical sensor technology, such as fluorescence spectroscopy used for detecting biomolecules [1–6]. Metallic nanostructures deposited onto thin Si wafers are shown to increase the amount of solar energy absorbed by the solar devices [7,8].
The energy absorbed and scattered by a spherical particle of radius a placed in a plane electromagnetic wave with wavelength $\lambda > > a$ are given by, respectively,
The term $(\alpha - 1)/(\alpha + 2)$ in the above expressions arises from the dipole moment induced by the electric field which affects the spatial distribution of the electrons in the metal. The resonance occurs at the frequency (or, equivalently, wavelength) at which the real part of $\alpha $ equals −2. The frequency response of the noble metals, such as silver, can be approximately described by the Drude model [10] in which the electrons in the metals move freely except for a small drag force that is proportional to their velocity. According to this model the relative permittivity is given by
The above expressions for the absorbed and scattered energies must be modified when the particle is near a substrate since these energies would then also depend on the relative permittivity of the substrate, its thickness, the nature of the substrate surface, the angle of incidence of the wave, and the distance between the sphere and the substrate. Detailed calculations accounting for the effect of these parameters are not available in the literature. Lermé et al. [11] present details of a method based on multipole expansions to compute particle-substrate interactions for arbitrary wavenumbers. These investigators also present numerical results for the absorption and extinction (sum of absorption and scattering) coefficients for selected test cases and compare the results obtained with that obtained in the electrostatics limit of vanishingly small z, a limit that we shall be exploring in detail in the present study. Salary and Mosallaei [12] also used a method of multipole expansions to examine particle-substrate interaction. Their main interest was in computing the optical force on a particle near a substrate and therefore no results were presented for the scattering and absorption losses. Other studies use a point-dipole approximation to explain experimental observations of plasmonic behavior of particles near a substrate [13,14].
Our primary interest is in determining how much additional energy can be harvested for the photovoltaic applications if, say, a thin transparent film containing metal nanoparticles is deposited on a Si substrate. If we ignore the particle-substrate and particle-particle interactions altogether, then the estimate of the fractional increase in the energy harvested by depositing particles near the surface will be simply given by ${\phi _a}{W_{sc}}/(2{I_0})$, where ${\phi _a}$ is the fraction of the surface area occupied by the particles. The factor $1/2$ is included in this estimate because only half of the total scattered energy in (1) is scattered in the half-space occupied by the substrate. For substrates such as Si, however, this will grossly underestimate the fractional increase because of a number of reasons. First, the intensity ${I_t}$ of the transmitted wave into the substrate, and thereby available for energy conversion, is smaller than ${I_0}$ by a factor that depends on the angle of incidence of the plane wave and the relative permittivity of the substrate ${\alpha _s}$, the latter being $O(10)$ for Si. The ratio ${I_t}/{I_0}$ roughly scales as $1/{\alpha _s}$. Second, as we shall see from the detailed analysis, the factor ${z^4}$ is modified to $z{\alpha _s}z_s^3$, where ${z_s} = z\alpha _s^{1/2}$ is the non-dimensional effective wavenumber of the light in the substrate. In other words, the factor ${z^4}$ underestimates the scattered energy roughly by a factor of ${\alpha ^{5/2}}$. On the other hand, the apparent dipole as “seen” by the substrate decreases by $({\alpha _s} + 1)/2$, and, since the scattered energy is proportional to the square of the induced dipole, the overall energy scattered into the substrate increases by $4\alpha _s^{5/2}/{({\alpha _s} + 1)^2}$ while the transmitted energy decreases by a factor that scales roughly as ${\alpha _s}$. The ratio of the energy harvested by the scattering of the particles to that transmitted therefore increases roughly by a factor of $4\alpha _s^{3/2}$, or about 126 for ${\alpha _s} \sim 10$, over the simple estimate ${\phi _a}{W_{sc}}/(2{I_0})$. For substrates with finite thickness, the ratio can be expected to be even greater since the transmitted energy will traverse, in general, a shorter path over which the energy is absorbed than the scattered energy which, depending on the scattering angle, may go through several internal reflections within the substrate resulting thereby in a greater fraction of the scattered energy being absorbed than transmitted.
In addition to the above two obvious effects of placing a particle near a substrate, the presence of the substrate also changes the nature of the resonance peak observed in the scattered energy-versus-the frequency curve. The substrate-particle interaction induces higher-order multipoles that resonate at frequencies that are different from that for the dipole resonance. A ${2^n}$-multipole resonates when the real part of $\alpha $ equals $- (n + 1)/n$. Therefore an infinite number of multipole resonances occur as ${\omega _r}$ is varied from $1/\sqrt 3 $ to $1/\sqrt 2 $. To be sure, the higher-order multipole resonances also occur in the case of an isolated sphere, but their magnitude is small when z is small (e. g. the magnitude of a ${2^n}$-multipole is $O({z^{n - 1}})$), and, moreover, they do not affect the nature of the dipole resonance which plays the principal role in determining the scattered energy. The particle-wall interactions, however, excite all the multipoles at $O(1)$ and cause the resonance frequencies to move to lower values by amounts that depend on the distance between the particle and the wall. We show that these interactions cause a broadening of the peak and contribute thereby to an increased scattered energy.
We should note the present study is limited to determining only the leading order estimate of the fractional increase in the energy received by the substrate when z is sufficiently small. In most practical applications the higher-order terms will be needed to obtain quantitiatively accurate estimates as the additional peaks that arise in the finite z analysis are suppressed in the electrostatics limit examined here [11]. One would expect that these additional peaks will most likely lead to even greater fractioanal increase in the energy received by the substrate than predicted by the present study.
In Sec. 2 we review the relevant results for the interaction of a plane wave with the substrate and then outline the method used for determining the particle-wall and particle-particle interactions. We also derive expressions for the scattered and absorbed energy. Since our analysis is restricted to small z, the interactions on the particle radius scale are governed by the Laplace equation for the electrostatics. In Sec. 3, we first present results for the case of a single particle near a wall with the permittivity given by the Drude model. This is followed by the calculations for silver, gold and copper spheres near a substrate for which we use the actual data on the permittivity as a function of the wavelength. We also carry out calculations for the total scattered and absorbed energies for the energy spectrum satisfying the Plank’s law to determine an energy spectrum-averaged scattering coefficient. Our analysis shows that the silver particles are more effective for harvesting energy than copper or gold particles. We next consider the case of a pair of two spherical particles near a substrate and show that the clustering of the particles can be exploited to harvest even greater energy. Finally, in Sec. 4 we make a brief comparison of the results obtained in the present study with an experimental study reported in the literature [8].
2. Expressions for the absorption and scattering in a multiparticle-substrate
2.1 Incident wave interaction with a substrate
Let us consider a plane wave with frequency $\omega $ traveling through a medium with permittivity ${\varepsilon _m}$ be incident upon a semi-infinite substrate $({x_1} < 0)$ at an angle of incident ${\theta _i}$ with the wave vector ${\textbf k}$ given by
The intensities (i.e. the time-averaged flux of the electromagnetic energy per unit time) of the incident and the transmitted waves are given by, respectively [9],
2.2 Scattered fields by the particles
After non-dimensionalizing the electric field by ${E_0}$, the magnetic field by ${E_0}/(c{\mu _0})$, and the distances by the radius a of the spheres, the governing equations for the electromagnetic wave reduce to
To satisfy the boundary condition of the continuity of the tangential components of the fields at the surface of the sphere, the electric fields due to the image of the sphere and the incident and reflected waves are expanded near the center of the sphere in terms of regular spherical harmonic functions. The overall electric potential just outside the sphere is therefore written as
The contribution to ${C_{nm}}\,\mbox{and}\,\,{\tilde{C}_{nm}}$ from the image multipoles can be determined by using the well-known formulas for translating the solutions of the Laplace equation that are singular at the image point to the regular solutions at the center of the sphere (see, e.g. Sangani and Mo [16]).
The scalar potential ${\phi _{total}}$ inside the sphere likewise can be expanded in terms of regular spherical harmonics. The boundary conditions at the surface of the sphere $(r = 1)$ then yield
Although we have described the method for a single sphere near the substrate, it can be readily extended to solve for the interactions among the arbitrary number of spheres provided that all the separation distances are $O(1)$ and z is sufficiently small so that the translation formulas for the Laplace equation can be used to expand the singular solutions due to other spheres and their images into the regular solutions near a sphere. In practice, of course, this will severly restrict the size of the spheres and the inparticle distances over which the results will be applicable.
2.3 Expressions for the absorbed and scattered energy
The time-averaged electromagnetic energy absorbed by a sphere can be determining by integrating the normal component of the Poynting vector on the surface of the sphere [9]. The result is given by ${W_{abs}} = \pi {a^2}{Q_{abs}}{I_0}$ with
To estimate the total energy scattered into the substrate ${I_{sc}} = \pi {a^2}{Q_{sc}}{I_0}$, it is convenient to determine the electromagnetic energy leaving the surface of a hemisphere $H:\,r = R \gg 1,\,\,{x_1} < 0$.
Now the remainder of the derivation is the same as for the scattering of a single sphere in the half-space, and the resulting expression is
3. Results
3.1 Single sphere in the vicinity of a substrate
Let us first consider the case of a single sphere placed with its center at distance h from the substrate, h being $O(1)$. The induced dipole will be related to the electric field by
The dipole ${D_ \bot }$ is determined by considering the special case, ${\textbf G} = {{\textbf e}_{\textbf 1}}$. In this case all the terms with $m \ne 0$ vanish in the set of equations given by (19) which, upon using the translation formulas for ${C_{n0}}$ in terms of the images of ${A_{n0}}$ into the substrate, reduces to
For $h = 1$, the calculations show oscillations in the scattering and absorption coefficients with decreasing amplitude as ${N_s}$ is increased. Therefore very high values of ${N_s}$ will be needed to achieve reasonably accurate estimates. We found, however, that the convergence can be accelerated with the use of Shanks transformation [17]. Let ${D_ \bot }(n,0)\,\,(n = 1,2,\ldots ,{N_s})$ be the sequence of the computed dipoles using only n multipoles. Then an application of the Shanks transformation produces a sequence ${D_ \bot }(n,1)\,\,(n = 2,3,\ldots ,{N_s} - 1)$ as given by
The results for the estimates of the absorption and scattering coefficients produced after applying the Shanks transformation $({N_s} - 1)/2$ times are shown by stars in Figs. 2 and 3. We see a very good convergence of the results with the increasing ${N_s}$. Similar behavior was also observed for the coefficients ${Q_t}$ and ${S_t}$. All subsequent calculations were done with ${N_s} = 25$ for $1 \le h < 1.1$. For $h \ge 1.1$, ${N_s} = 19$ or even smaller was used and the use of Shanks transformation was found unnecessary for $h > 2$.
3.1.1 Drude model
Figure 4 shows the results for the absorption and scattering coefficients for the case when the relative permittivity of the sphere is adequately described by the Drude model as given by (2). The substrate relative permittivity is ${\alpha _s} = 11.3$, a value representative for the silicon substrate in contact with air. It must be noted that the relative permittivity of silicon is actually a complex quantity and a function of the frequency. The expressions we have derived (e.g. cf. (31) and (32)) can be used in principle to determine the dipole as a function of frequency using the actual data for a frequency-dependent complex relative permittivity. Some modifications would be required, however, for the expressions for the scattered and absorbed energy coefficients if we allow ${\alpha _s}$ to be complex. In the optical frequency range, however, the imaginary part of ${\alpha _s}$ for silicon is typically small and the calculations based on a constant real permittivity should provide reasonably good estimates.
The dashed curves in the above figures are obtained using the estimate of the dipole obtained by considering only the terms up to $p = 3$ in (31) which yields
The third and fourth terms on the right-hand side of (34) arise from, respectively, the effect of the wall-induced quadrupole $(p = 2)$ and octupole $(p = 3)$ on the dipole. The quadrupole resonance frequency corresponding to ${\mbox{Re}} (\alpha ) = - 3/2$ shifts to the frequency for which ${\mbox{Re}} (\alpha ) = - 3/2 - 15{\beta _s}{R^{ - 5}}$ and, likewise, the octupole resonance shifts to the frequency for which ${\mbox{Re}} (\alpha ) = - 4/3 - 140/3{\beta _s}{R^{ - 7}}$. Thus, the sphere-substrate interaction induces resonance in all the multipoles and the frequency at which these resonances occur shift to lower frequencies as the distance between the sphere and the substrate decreases.
The magnitude of the peak depends strongly on the value of the damping constant $\delta $. This can be seen from Fig. 5 where we show the results obtained with $\delta = 0.05$, half the value used in computing the results shown in Fig. 4.
Comparing Figs. 4 and 5, we see that reducing the damping parameter by a factor of 2 roughly increases the absorption coefficient by a factor of 2 and the scattering coefficient by a factor of 4. For the comparison sake, we have also shown in Fig. 5 the results that one would obtain if the substrate-sphere interaction is completely neglected, i.e. if we simply use ${D_ \bot } = t_1^{ - 1}$ to estimate the absorption and scattering coefficients. Note that the substrate-sphere interaction shifts the peak to a lower frequency as mentioned earlier. The magnitude of the peaks, on the other hand, are approximately the same. This is true provided that $\delta $ is sufficiently small. For larger $\delta $, the imaginary part of the dipole is also influenced by the damping of the higher-order multipole modes and this has the effect of decreasing the magnitude of the peaks. Finally, the “kink” in the solid curve seen around ${\omega _r} = 0.6$ is the influence of the quadrupole resonance. Such “kinks” or irregular shape curves can be expected around each multipole resonances when $\delta \ll 1$.
The computed results for $h < 1.1$ begin to differ significantly from those obtained using the asymptotic formula (34) as seen in Fig. 6 which shows the comparison for $h = 1$. We see that the peak values computed with sufficiently high ${N_s}$ (we used ${N_s} = 35$ and the repeated Shanks transformation as described earlier) are significantly lower than those predicted from the asymptotic theory. The peak value of the scattering coefficient of about 10 for this case is approximately one-third of that obtained for $h = 1.2$ (cf. Fig. 4b). On the other hand, the peak is somewhat broader for $h = 1$ than for $h = 1.2$. Finally, it is interesting to note that the “kink” seen in the curves obtained from the asymptotic formula (34) are smoothened out when the computations are carried out with h close to unity and ${N_s}$ is sufficiently large.
We now present results for the absorption and scattering coefficients ${Q_t}$ and ${S_t}$. These are obtained by setting ${\textbf G} = {{\textbf e}_{\textbf 2}}$. For this case, only the multipole coefficients ${A_{nm}}$ with $m = 1$ are nonzero, and (19) reduces to
The results for $h = 1.2$ are shown in Fig. 7. The dashed lines are obtained using the expression for the dipole obtained by keeping only the first three terms which yields
The results for the scattering and absorption coefficients for a resting on a substrate are shown in Fig. 8. For this case, the dipole estimated using (37) over-predicts the peak value for the scattering coefficient by about 50 per cent and the absorption coefficient by about 25 per cent.
3.1.2 Results for silver, gold, and copper
The Drude model is a highly idealized model for predicting the relative permittivity of the metals and therefore, while it provides a good insight into how the negative permittivites arise in metal particles, it is important to use the actual data for the frequency-dependent permittivity for specific metals to get better estimates of the magnitude of the scattering. The data for the relative permittivity as a function of the wavelength for silver, gold and copper are shown in Fig. 9 [18]. We note that ${\mbox{Re}} (\alpha )$ is negative and similar for all the three metals for all wavelengths greater than 350 nm. There is, however, significant differences in the values of ${\mbox{Im}} (\alpha )$ which are significantly larger for gold and copper than for silver, especially for wavelengths at which the resonance $({\mbox{Re}} (\alpha ) \approx - 2)$ is expected.
Figure 10 shows the results for the absorption and scattering coefficients for silver. We see a behavior similar to that for the Drude model. The coefficients ${S_t}$ and ${Q_t}$ for the electric field parallel to the surface are slightly smaller than their counterparts, ${S_n}$ and ${Q_n}$, for the field normal to the substrate and their peaks occur at slightly smaller wavelengths.
The results for copper and gold are shown in Fig. 11. We see that the peaks for both the absorption and scattering coefficients for these two metals are much smaller than those for silver. The peak in the scattering coefficient for silver is nearly one order of magnitude larger than for gold or copper. The peaks in the scattering coefficients occur at wavelengths of about 520 nm for gold and 590 nm for copper compared with around 370 for silver. The results for the electric field parallel to the surface are similar and therefore not shown here.
The results presented up to now have focused on the peak magnitude as a function of the wavelength or the frequency. For use in photovoltaic applications, it will be of interest to determine the total scattered energy from a radiation that is composed of a continuous spectrum of wavelength. We consider the case when the energy spectrum of the incident radiation satisfies the Planck’s law:
Figure 12 shows the results for the average absorption and scattering coefficients as functions of $h$ for a silver spherical particle. The absorption coefficient for the field normal to the substrate is about 50 per cent greater than for the field parallel to the surface when the sphere is at the surface and the difference between the two is insignificant for h greater than about 2. Likewise, the scattering field for the electric field normal to the substrate is also greater than for the field parallel to the substrate except for $h = 1$. The scattering coefficient ${S_n}$ dropped sharply between $h = 1.1$ and $h = 1$. Interestingly, the peak in the scattering versus wavelength curve drops sharply without significantly broadening of the curve at $h = 1$ from the corresponding curve at $h = 1.1$ resulting in lower values for the scattering coefficients. The scattering coefficients at $h = 1.1$ are about 50 per cent higher than the values for large h indicating that the substrate-sphere interaction increases the scattering coefficient by roughly 50 per cent.
Figure 13 shows the results for the gold and copper spheres near a substrate. We note that these scattering coefficients are significantly smaller than those for silver. A silver sphere near the substrate scatters roughly three to four times that by gold or copper sphere and therefore better suited for the photovoltaic applications.
3.2 Scattering from a pair of particles
Nanoparticles often form clusters during their processing and therefore it is of some interest to examine the effect of particle clustering. A number of studies have analyzed the interaction effects arising from a pair of spheres (see, e.g., [19,20]. Romero et al. [19] used a boundary element method to compute interactions for arbitrary values of z and showed a singular behavior for the case when the spheres are touching and aligned in the direction of the electric field. Norlander et al. [20] used a novel plasmon hybridization method to compute the pair interactions. The effect of the presence of the substrate was not examined by these investigators.
Let us consider a pair of equi-sized spheres with their center-to-center separation vector parallel to the surface of the substrate. The scattered energy per sphere is then given by, in lieu of (30),
Figure 14 shows the results for the scattering coefficients for $h = 1.1$ and $S = 2.2$, S being the center-to-center distance between the spheres. Our computations for $h = S/2 = 1$ showed poor convergence and therefore we present results only for the case when the spheres are not touching each other. We see that the peaks are generally greater than those for the case of a single sphere. For example, the peak values for ${S_n}$ and ${S_t}$ for the case of a single silver sphere at a slightly higher value of h equal to 1.2 were, respectively, 41 and 31 (cf. Fig. 10b) compared to about 50 for the case of a pair of spheres. More significantly, the scattering coefficient ${S_{ta}}$, which corresponds to the case when the pair is aligned along the direction of the electric field, has the peak value of about 98. If the induced dipoles for the pair of spheres were exactly the same as for the single sphere, then the scattering coefficients for the cluster would have been exactly twice. This is so because the apparent dipole of the cluster would be twice that of a single sphere, and the scattering per sphere is proportional to the square of the apparent dipole divided by the number of spheres in the cluster. The presence of the second sphere, however, reduces the dipole and therefore the pair of spheres gives total scattering per sphere that is generally less than twice that for an isolated sphere. The result for ${S_{ta}}$ is therefore somewhat surprising. The peak value of 98 per sphere is more than three times that for the single sphere of about 31. The peak also shifts considerably more to higher wavelengths for the case when the pair of spheres is aligned normal to the electric field.
This surprising trend could be understood by considering the limiting case corresponding to $2h = S = R > > 1$ for which the spheres and their images may be replaced by their respective point dipoles. The dipole of each sphere can be shown to be given by, in lieu of (36),
As in the case of the single sphere, it is possible to determine the Planck’s law energy-spectrum weighted averaged scattering coefficients. For $h = S/2 = 1.1$, the computed values are given by
If we assume the orientation of the pair to be a random unit vector in the plane parallel to the substrate, then the averaged scattering coefficient for the field parallel to the substrate is $({S_t} + {S_{ta}})/2$ and equals 23.8. Comparing these results with that given in Fig. 12b, we see that a pair of two spheres will roughly scatter twice the energy per sphere compared to an isolated sphere. Thus, the presence of clusters will increase the overall scattered energy.4. Conclusions
We have carried out detailed calculations for the absorption and scattering coefficients for metal spheres in the presence a substrate. Results show that the silver spheres scatter significantly more than copper or gold spheres having the same radius. The ratio of the scattered energy to the transmitted energy can be estimated using
Israelowitz et al. [8] measured the enhancement in the photocurrent in thin film silicon-on-insulator (SOI) devices after depositing silver nanospheres on the cell surface by spin coating a suspension of silver nanospheres. They report results for the photocurrent enhancement as a function of the wavelength of a monochromatic light source for various area fractions and the average size of the deposited particles together with the morphology of the deposits. They also report overall gain by averaging over the wavelength (assuming a constant $E(\lambda )$). The silver nanoparticle solutions used in spin coating were: (i) a commercial silver nano-ink consisting of silver particles with an average radius of 20 nm; and (ii) an in-house synthesized suspension of glucose-capped silver spheres which allowed greater control on the size of the spheres. Using two different conditions, they produced suspensions with average radii of 15 and 35 nm. The spin coating process was carried out for different concentrations of solutions which resulted in a broad range of area fractions of the deposited particles. These investigators report enhancements of, respectively, 49% and 199% for suspensions of 15 and 35 nm. No mention was made of the area fraction for these suspensions. The area fractions as determined by Image J analyzer were reported for the experiments with the nano-ink solutions. The greatest enhancement for nano-ink was reported to be about 150% for a 0.1 w/v solution with an estimated area fraction of about 0.07. This may be compared to our simplified expression (34) for small ${\phi _a}$ which predicts only about 21% increase. Presence of clusters of pairs of two or three particles may lead to somewhat greater increase but perhaps not large enough to quantitatively agree with the enhancement seen in the experiment. The small z analysis presented here suppresses the peaks that depend on the finite phase differences among the particles or between the incident and reflected waves and it is possible that the simplification underestimates the scattered energy. Another reason for the discrepancy may be related to the SOI device used by these investigators, which consisted of thin layers (of the order 1-2 $\mu m$) of buried oxide and a negative type C-Si on the top of a Si. It is possible that this multilayer substrate of finite thickness may absorbs less of the transmitted wave compared with the scattered waves leading thereby to a greater fractioanl increase than the one predicted for a semi-infinite substrate.
Pillai et al. [21] have also reported enhancements in photocurrent for silver particles deposited on silicon-on-insulator devices by a thermal process. Their particles were disk-shaped with diameters of about 100 nm. They also report significant increase in the photocurrent for wavelengths ranging from 300 nm to 1200 nm. The photocurrent at least doubled in the range 300-900 nm. At higher wavelengths the enhancement was even greater, with as much as 16-fold enhancement for wavelengths in the range 1100-1200 nm.
In summary, while we have been unable to quantitatively verify the theory presented from the data available in the literature, it is clear that the presence of the deposited silver particles significantly enhances the light collected by the solar devices. The present study outlines some of the mechanisms responsible for the enhancement, particularly for sufficiently small spheres.
Acknowledgement
The authors thank Professor R. Sureshkumar for suggesting the problem and for a number of valuable suggestions.
References
1. H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296(4), 56–62 (2007). [CrossRef]
2. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef]
3. S. Linic, P. Christopher, and D. B. Ingram, “Plasmonic-metal nanostructures for efficient conversion of solar to chemical energy,” Nat. Mater. 10(12), 911–921 (2011). [CrossRef]
4. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010). [CrossRef]
5. R. Adato and H. Altug, “In-situ ultra-sensitive infrared absorption spectroscopy of biomolecule interactions in real time with plasmonic nanoantennas,” Nat. Commun. 4(1), 2154 (2013). [CrossRef]
6. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006). [CrossRef]
7. K. R. Catchpole and S. Pillai, “Absorption enhancement due to scattering by dipoles into silicon waveguides,” J. Appl. Phys. 100(4), 044504 (2006). [CrossRef]
8. M. Israelowitz, J. Amey, T. Cong, and R. Sureshkumar, “Spin coated plasmonic nanoparticle interfaces for photocurrent enhancement in thin film Si solar cells,” J. Nanomater. 2014, 1–9 (2014). [CrossRef]
9. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).
10. P. Drude, The Theory of Optics (Courier Corporation, 1925).
11. J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin, “Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects,” J. Phys. Chem. C 117(12), 6383–6398 (2013). [CrossRef]
12. M. M. Salary and H. Mosallaei, “Tailoring optical forces for nanoparticle manipulation on layered substrates,” Phys. Rev. B 94(3), 035410 (2016). [CrossRef]
13. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. 67(12), 1607–1615 (1977). [CrossRef]
14. A. Pinchuk, A. Hilger, G. von Plessen, and U. Kreibig, “Substrate effect on the optical response of silver nanoparticles,” Nanotechnology 15(12), 1890–1896 (2004). [CrossRef]
15. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (CUP Archive, 1931).
16. A. S. Sangani and G. Mo, “An O (N) algorithm for Stokes and Laplace interactions of particles,” Phys. Fluids 8(8), 1990–2010 (1996). [CrossRef]
17. D. Shanks, “Non-linear Transformations of Divergent and Slowly Convergent Sequences,” Stud. Appl. Math. 34(1-4), 1–42 (1955). [CrossRef]
18. M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info.
19. I. Romero, J. Aizpurua, G. W. Bryant, and F. J. G. De Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14(21), 9988–9999 (2006). [CrossRef]
20. P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004). [CrossRef]
21. S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells.,” J. Appl. Phys. 101(9), 093105 (2007). [CrossRef]