Absorption and scattering coefficients for metallic nanospheres near a substrate

Satvik. N. Wani; Ashok. S. Sangani

doi:10.1364/OSAC.2.000279

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,

(1)$${W_{abs}} = 4z\,{\mbox{Im}} \left( {\frac{{\alpha - 1}}{{\alpha + 2}}} \right)\,\pi {a^2}{I_0},\,\,\,\,{W_{sc}} = (8/3){z^4}\,{\left|\left|{\frac{{\alpha - 1}}{{\alpha + 2}}} \right|\right|^2}\,\pi {a^2}{I_0}$$

where $z = 2\pi a/\lambda $ is the non-dimensional wavenumber, $\alpha $ is the relative permittivity of the particle, which, in general, is a frequency-dependent complex quantity for metals, and ${I_0}$ is the intensity of the wave, i.e. the time-averaged electromagnetic energy flux [9]. The double vertical bars in the above expression refer to the magnitude of a complex quantity. The relative errors in the above expressions are $O({z^2})$. Accurate expressions for arbitrary values of $z$ are also available in the literature [9] but the expressions given by (1) are generally adequate for spheres with diameters of about 20 nm or smaller in the visible light sprectrum.

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

(2)$$\alpha = 1 - \frac{{\omega _p^2}}{{{\omega ^2} + \mbox{i}\omega /\tau }} = 1 - \frac{1}{{\omega _r^2 + {\delta ^2}}} + \mbox{i}\frac{\delta }{{(\omega _r^2 + {\delta ^2}){\omega _r}}}$$

where $\omega $ is the frequency of the wave, ${\omega _p}$ is the plasma frequency of the metal, $\tau $ is the relaxation time of the electrons, $\delta = 1/({\omega _p}\tau )$, $\mbox{i} = \sqrt { - 1} $, and ${\omega _r} = \omega /{\omega _p}$. The plasma frequency equals ${N^{1/2}}q/{({\varepsilon _0}{m_e})^{1/2}}$, where N is the number density of the electrons in the metal, q is the electronic charge, ${\varepsilon _0}$ is the permittivity of the vacuum, and ${m_e}$ is the mass of an electron. The plasma frequencies of silver and gold are of the order of ${10^{16}}\,{s^{ - 1}}$. The relaxation time for electrons is typically of the order of ${10^{ - 15}}$s and therefore $\delta $ is of order ${10^{ - 1}}$. The resonance, for which ${\mbox{Re}} (\alpha ) = - 2$, can therefore be expected at roughly ${\omega _r} = 1/\sqrt 3 $. For noble metals, this frequency typically falls within the visible light spectrum.

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

(3)$${\textbf{k}} = k({ - \cos {\theta_i}{{\textbf e}_{\textbf 1}} + \sin {\theta_i}{{\textbf e}_{\textbf 2}}} )$$

where k is the magnitude of the wave vector and ${{\textbf e}_{\textbf j}}$ are the unit vectors along the ${x_j}$-axis (Fig. 1). Let the electric field associated with the incident wave be given by

(4)$${{\textbf E}^{in}} = {E_0}[{\sin {\theta_i}\sin {\psi_i}{{\textbf e}_{\textbf 1}} + \cos {\theta_i}\sin {\psi_i}{{\textbf e}_{\textbf 2}} + \cos {\psi_i}{{\textbf e}_{\textbf 3}}} ]{e^{\textrm{i}({\textbf k} \bullet {\textbf x} - \omega t)}}$$

Here, ${E_0}$ is the magnitude of the electric field and ${\psi _i}$ is the angle between the incident wave electric field and the normal ${{\textbf e}_{\textbf 3}}$ to the plane $\left( {{x_1} - {x_2}} \right)$ formed by the wave vector and the normal to the substrate. We assume that the wave undergoes a specular reflection at the surface of the substrate with the angle of reflection equal to the angle of incidence. The electric field of the reflected field is therefore given by

(5)$${{\textbf E}^{\textbf r}} = {E_0}[{ - \sin {\theta_i}\sin {\psi_i}{R_{TE}}{{\textbf e}_{\textbf 1}} + \cos {\theta_i}\sin {\psi_i}{R_{TE}}{{\textbf e}_{\textbf 2}} + \cos {\psi_i}{R_{TM}}{{\textbf e}_{\textbf 3}}} ]\; {e^{\textrm{i}({{{\textbf k}^{\textbf r}} \bullet {\textbf x} - \omega t} )}}$$

where ${{\textbf k}^{\textbf r}} = k(\cos{\theta _i}\,{{\textbf e}_{\textbf 1}} + \sin {\theta _i}\,{{\textbf e}_{\textbf 2}})$ is the wave vector of the reflected wave, and ${R_{TE}}$ and ${R_{TM}}$ are the reflection coefficients for, respectively, the in- and out- of plane wave components of the electric field as given by [9]

(6)$${R_{TE}} = \frac{{\cos {\theta _t} - \alpha _s^{1/2}\cos {\theta _i}}}{{\cos {\theta _t} + \alpha _s^{1/2}\cos {\theta _i}}}\,\,\,\mbox{and}\,\,{R_{TM}} = \frac{{\cos {\theta _i} - \alpha _s^{1/2}\cos {\theta _t}}}{{\cos {\theta _i} + \alpha _s^{1/2}\cos {\theta _t}}}\,\,$$

Here, we have assumed that the magnetic permeability of the substrate is essentially the same as that of the vacuum and that the electric permittivity is ${\varepsilon _m}{\alpha _s}$. The angle ${\theta _t}$ between the transmitted wave and the normal to the surface of the substrate satisfies the Snell’s law: $\sin {\theta _i}/\sin {\theta _t} = \alpha _s^{1/2}$.

Fig. 1. A sketch showing the incident, reflected, and transmitted waves and the nomeclature used in the analysis. The substrate with an electric permittivity ${\alpha _s}{\varepsilon _m}$ occupies the space ${x_1} < 0$, the medium for ${x_1} > 0$ contains spheres of radius a with permittivity $\alpha {\varepsilon _m}$ and the surrounding medium with permittivity ${\varepsilon _m}$.

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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],

(7)$${I_0} = \frac{{E_0^2}}{{2c{\mu _0}}}\,\,\,\mbox{and}\,\,\,{I_t} = {I_0}[{{{\sin }^2}{\psi_i}T_{TE}^2 + {{\cos }^2}{\psi_i}T_{TM}^2} ]$$

with

(8)$${T_{TE}} = \frac{{2\cos {\theta _i}}}{{\cos {\theta _t} + \alpha _s^{1/2}\cos {\theta _i}}}\,\,\,\mbox{and}\,\,\,{T_{TM}} = \frac{{2\cos {\theta _i}}}{{\cos {\theta _i} + \alpha _s^{1/2}\cos {\theta _t}}}$$

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

(9)$$\nabla \times {\textbf E} = \mbox{i}z{\textbf H},\,\,\,\,\,\nabla \times {\textbf H} = - \mbox{i}z{\varepsilon _r}({\textbf x}){\textbf E},\,\,\,\,\nabla {\boldsymbol \cdot } {\textbf E} = 0,\,\,\,\,\nabla {\boldsymbol \cdot } {\textbf {H}} = 0$$

where ${\varepsilon _r}$ is the permittivity relative to the medium outside the particle, and equals ${\alpha _s}$ for the substrate and $\alpha $ for the particles. Since both the electric and magnetic fields are solenoidal, the scattered fields by a sphere can be represented in terms of two scalar functions:

(10)$${{\textbf E}^s} = \mbox{i}z\nabla \times ({\textbf r}\,\Psi ) + \nabla \times \nabla \times ({\textbf r}\,\Phi )$$

(11)$${{\textbf H}^s} = - \mbox{i}z{\varepsilon _r}\nabla \times ({\textbf r}\Phi ) + \nabla \times \nabla \times ({\textbf r}\Psi )$$

where ${\textbf r} = {\textbf x} - {{\textbf x}_c}$, ${{\textbf x}_c}$ being the center of the sphere. The scalar functions satisfy the Helmholtz equation, i.e. $({\nabla ^2} + {z^2})\Psi = ({\nabla ^2} + {z^2})\Phi = 0$. For $z \ll 1$, these equations for $\Psi $ and $\Phi $ can be approximated by the Laplace equation for distances that are $O(1)$, i.e. comparable to the radius of the spheres. Further, for the special case where the magnetic properties of the particles, the substrate, and the medium surrounding the particle are all essentially the same, it can be shown that $\Psi \equiv 0$. To account for the presence of the substrate, (10)–(11) are supplemented by an image system for the sphere as given by

(12)$${\textbf E}_{im}^s = \nabla \times \nabla \times ({{\textbf r}_{im}}{\Phi _{im}}),\,\,\,{\textbf H}_{im}^s = - \mbox{i}z\nabla \times ({{\textbf r}_{im}}{\Phi _{im}})\,\,\,({\mbox{for}}\;{x_1} \ge 0)$$

where ${{\textbf r}_{im}} = {\textbf x} - {{\textbf x}_{im}}$, with the position vector for the image being given by ${{\textbf x}_{im}} = {{\textbf x}_c} - 2{{\textbf e}_{\textbf 1}}h$. Here, h is the non-dimensional distance between the center of the sphere and the substrate (cf. Fig. 1). Equations (10)–(12) are valid only in the medium outside the sphere and for ${x_1} > 0$. In the substrate, the scattered field due to the sphere is expressed by the apparent scalars:

(13)$${\textbf E}_{app}^s = \nabla \times \nabla \times ({\textbf r}{\Phi _{app}}),\,\,\,{\textbf H}_{im}^s = - \mbox{i}{\alpha _s}z\nabla \times ({\textbf r}{\Phi _{app}})\,\,\,({\mbox{for}}\;{x_1} \le 0)$$

To determine these scalar functions, we write $\Phi $ in terms of decaying spherical harmonics expanded around the center of the sphere:

(14)$$\Phi = \sum\limits_{n = 1}^\infty {( - 1/n){\phi _{ - n - 1}}} \,\,\,\mbox{with}\,\,\,{\phi _{ - n - 1}} = {r^{ - n - 1}}\sum\limits_{m = 0}^n {[{{A_{nm}}\,{Y_{nm}}(\theta ,\varphi ) + {{\widetilde A}_{nm}}\,{{\tilde{Y}}_{nm}}(\theta ,\varphi )} ]}$$

where ${Y_{nm}}(\theta ,\varphi ) = P_n^m(\cos \theta )\cos (m\varphi )$ and ${\tilde{Y}_{nm}}(\theta ,\varphi ) = P_n^m(\cos \theta )\sin(m\varphi )$ are the surface harmonics, and ${A_{nm}}$ and ${\tilde{A}_{nm}}$ are the magnitudes of the ${2^n}$-multipoles. Here, $\theta $ and $\varphi $ are defined such that ${r_1} = r\cos \theta ,$ $\,{r_2} = r\sin \theta \cos \varphi $ and $\,{r_3} = r\sin \theta \sin \varphi $. Similar expressions apply to the image and the apparent fields. Thus, for example, ${A_{nm}}$ is replaced by, respectively, ${A_{im,nm}}$ and ${A_{app,nm}}$ in the expressions for the image and the apparent fields. Relations among the multipole coefficients and their image multipoles and apparent multipoles are obtained by requiring that the components of the scattered electric and magnetic fields parallel to the substrate surface must be continuous at the medium-substrate interface, i.e. at ${x_1} = 0$. The continuity of the magnetic field components can be equivalently satisfied by the requiring that the normal component of the electric field satisfies the jump condition: ${ {E_1^s + E_{1,im}^s} |_{{x_1} = {0_ + }}} = { {{\alpha_s}E_{1,app}^s} |_{{x_1} = {0_ - }}}$. These boundary conditions are easily satisfied by expressing the electric field in terms of the gradient of a scalar potential as given by

(15)$${{\textbf E}^s} = \nabla {\phi _E}\,\,\,\mbox{with}\,\,\,{\phi _E} = \sum\limits_{n = 1}^\infty {{\phi _{ - n - 1}}}$$

This simplification is possible because $\Phi $ satisfies the Laplace equation to the leading order for small z, and is only valid for distances that are comparable to the radius of the sphere. In deriving (15) from (12), we made use of the fact that ${\textbf r}{\boldsymbol \cdot } \nabla \nabla {\phi _{ - n - 1}} = ( - n - 2)\nabla {\phi _{ - n - 1}}$ since ${\phi _{ - n - 1}}$ is a homogeneous polynomial of degree $- n - 1$ in ${r_1}$, ${r_2}$, and ${r_3}$[15]. The boundary conditions at the substrate surface are therefore satisfied by taking

(16)$${A_{im,nm}} = {( - 1)^{n - m + 1}}\frac{{{\alpha _s} - 1}}{{{\alpha _s} + 1}}{A_{nm}},\,\,\,{A_{app,nm}} = \frac{2}{{{\alpha _s} + 1}}{A_{nm}}$$

The above relations also apply to the coefficients ${\tilde{A}_{nm}}$, ${\tilde{A}_{im,nm}}$ and ${\tilde{A}_{app,nm}}$.

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

(17)$${\textbf E} = \nabla {\phi _{total}}\,\mbox{with}\,\,\,{\phi _{total}} = \sum\limits_{n = 1}^\infty {\sum\limits_{m = 0}^n {[{({A_{nm}}\,{r^{ - n - 1}} + {C_{nm}}{r^n}){Y_{nm}} + ({{\tilde{A}}_{nm}}\,{r^{ - n - 1}} + {{\tilde{C}}_{nm}}{r^n}){{\tilde{Y}}_{nm}}} ]} }$$

The scalar electric potential of the combined incident and reflected waves expanded near the center of the sphere is given by $\phi _E^{i + r} = {\textbf G}{\boldsymbol \cdot } {\textbf r}\,\,$ with

(18)$$\,{G_1} = \sin {\theta _i}\sin {\psi _i}(1 - {R_{TE}}),\,\,{G_2} = \cos {\theta _i}\sin {\psi _i}(1 + {R_{TE}}),\,\,{G_3} = \cos {\psi _i}{R_{TM}}$$

The incident and reflected fields contribute only to the $n = 1$ coefficients ${C_{10}},\,{C_{11}},\,\,\mbox{and}\,{\tilde{C}_{11}}$ with the respective contributions being ${G_1},\, - {G_2},\,\,\mbox{and}\, - {G_3}$. Note that the negative signs arise here since ${Y_{10}} = {r_1}/r\,\,\mbox{while}\,{Y_{11}} = - {r_2}/r\,\,\mbox{and}\,{\tilde{Y}_{11}} = - {r_3}/r\,$.

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

(19)$${t_n}{A_{nm}} + {C_{nm}} = 0\,\,\mbox{with}\,\,{t_n} = \frac{{\alpha n + n + 1}}{{n(\alpha - 1)}}$$

The same relation also applies to the quantities with tilde, i.e. ${\tilde{A}_{nm}}$ and ${\tilde{C}_{nm}}$. Thus, we obtain an infinite set of linear equations for the coefficients ${A_{nm}}$ and ${\tilde{A}_{nm}}$ that can be suitably truncated and the resulting equations can be solved numerically to determine these coefficients in terms of ${\textbf G}$, $\alpha $, ${\alpha _s}$, and h.

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

(20)$${Q_{abs}} = - \frac{1}{\pi }\,{\mbox{Re}} \int\limits_{r = 1} {({\textbf E} \times {{\textbf H}^{{\ast} }}} ){\boldsymbol \cdot } {\textbf n}\,dA$$

where ${{\textbf H}^\ast }$ is the complex conjugate of the magnetic field amplitude and ${\textbf n}$ is the unit outward normal on the surface of the sphere, i.e. at $r = 1$. The expressions for the scattered fields due to the particles and their images were given earlier (cf. (12)); we must add the contributions from the planar incident and reflected electric and magnetic fields to those scattered fields and then carry out the integration. It can be shown that the planar magnetic field contribution vanishes so that it suffices to use the following expressions for the electric and magnetic fields for the purpose of evaluating the integral in (20):

(21)$${\textbf E} = {\textbf G} + \sum\limits_k {\nabla \times \nabla \times ({{\textbf r}_k}} {\Phi _k}),\,\,\,\,{{\textbf H}^\ast } = \mbox{i}z\left[ {(1/2)\nabla \times ({\textbf r}\,{\textbf G}{\boldsymbol \cdot } {\textbf r}) + \sum\limits_k {\nabla \times ({{\textbf r}_k}\Phi_k^\ast )} } \right]$$

Here, the summation must be carried out over each sphere and its image in the substrate. The summation index k therefore includes all particles and images, including the particle for which the energy absorbed is being calculated (${{\textbf r}_k} = {\textbf r}$ for that sphere). ${Q_{abs}}$ is therefore given by

(22)$${Q_{abs}} = {\mbox{Im}} \left[ {\frac{z}{\pi }\int\limits_{r = 1} {\left\{ {{\textbf G} \times \sum\limits_k {\nabla \times ({{\textbf r}_k}{\Phi_k})} + (1/2)\nabla \times ({\textbf r}\,{\textbf G}{\boldsymbol \cdot } {\textbf r}) \times \sum\limits_k {\nabla \times \nabla \times ({{\textbf r}_k}\Phi_k^\ast )} } \right\}{\boldsymbol \cdot } {\textbf n}dA} } \right]$$

The term involving the product of the scattered fields is omitted from the above expression as it can be shown that it does not contribute to the overall integral, at least to the leading $O(z)$ term being determined here. The product of the scattered fields does contribute to the scattered energy and this contribution of $O({z^4})$ will be determined shortly. The surface integral in the above expression is converted into a volume integral using the divergence theorem to yield

(23)$${Q_{abs}} = \frac{z}{\pi }\int\limits_{r < 1} {{\textbf G}{\boldsymbol \cdot } \sum\limits_k {{{\textbf r}_k}{\nabla ^2}{\mbox{Im}} ({\Phi _k})dV} }$$

Now noting that, since ${\Phi _k}$ satisfies the Laplace equation, only the dipole singularity of the sphere under consideration will contribute to the above integral. Therefore it is easy to show, using the method of generalized functions, that

(24)$${Q_{abs}} = 4z\,{\mbox{Im}} ({\textbf G}{\boldsymbol \cdot } {\textbf D})$$

with ${\textbf D} = - {A_{10}}{{\textbf e}_{\textbf 1}} + {A_{11}}{{\textbf e}_{\textbf 2}} + {\widetilde A_{11}}{{\textbf e}_{\textbf 3}}$ being the induced dipole moment. For the special case of a single sphere in the absence of the substrate with the electric field aligned along the ${x_1}$-axis, it is easy to show that ${A_{10}} = - (\alpha - 1)/(\alpha + 2)G$ and (24) reduces in that case to (1).

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 $H:\,r = R \gg 1,\,\,{x_1} < 0$.

(25)$${Q_{sc}} = {\mbox{Re}} \left[ {\frac{1}{\pi }\int\limits_H {({{\textbf E}^s} \times {{\textbf H}^{s\ast }}){\boldsymbol \cdot } {\textbf n}\,dA} } \right]$$

This integral will be evaluated by choosing $zR \gg 1$ since, for such great distances from the spheres, the contributions from the higher-order multipoles can be neglected and so it is necessary to only retain the dipole term in the scattered fields. For large distances the scattered field satisfies the Helmholtz equation instead of the Laplace equation and therefore $1/r$ must be replaced by ${e^{i{z_s}r}}/r$. Therefore, the scattered electric field at large distances from the spheres is approximated by

(26)$${E^s} = \nabla \times \nabla \times ({\textbf r}{\Phi _{ap}})\,\,\mbox{with}\,\,{\Phi _{ap}} = {{\textbf D}_{ap}}{\boldsymbol \cdot } \nabla \left( {\frac{{\exp (\mbox{i}{z_s}r)}}{r}} \right)\,\,\,\mbox{and}\,\,{{\textbf D}_{ap}} = \frac{2}{{{\alpha _s} + 1}}\sum\limits_k {{{\textbf D}_k}}$$

Here, the summation is over all the spheres. In other words, the scattered field at great distances from a cluster of spheres can be represented in terms of a single dipole that is equal to the sum of the dipoles induced in the individual spheres. From the practical point of view, this condition is likely to be satisfied only for a cluster of a few small spheres since in practice z may not be sufficiently small to ensure that the cluster size times z is also much smaller than unity. In the present study we shall explicitly present results for at most a pair of nearly touching spheres, and combining their dipoles into a single dipole should yield reasonably accurate estimates.

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

(27)$${Q_{sc}} = \frac{{16{\alpha _s}zz_s^3}}{{3{{({\alpha _s} + 1)}^2}}}{\textbf D}{\boldsymbol \cdot } {{\textbf D}^\ast } = \frac{{16{z^4}\alpha _s^{5/2}}}{{3{{({\alpha _s} + 1)}^2}}}{\textbf D}{\boldsymbol \cdot } {{\textbf D}^\ast }$$

where ${\textbf D}$ is the sum of the induced dipoles over a cluster of spheres. The above expression agrees with that given by (1) for the special case of a single sphere in the presence of the substrate with ${\alpha _s} = 1$.

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

(28)$${\textbf D} = {D_ \bot }{G_1}{{\textbf e}_{\textbf 1}} + {D_\parallel }({G_2}{{\textbf e}_{\textbf 2}} + {G_3}{{\textbf e}_{\textbf 3}})$$

and

(29)$${Q_{abs}} = 4z\left[{{Q_n}G_1^2 + {Q_t}({G_2^2 + G_3^2} )} \right]$$

with ${Q_n} = {\mbox{Im}} ({D_ \bot })$ and ${Q_t} = {\mbox{Im}} ({D_\parallel })$. Likewise, the energy scattered into the substrate is given by

(30)$${Q_{sc}} = \frac{{16{z^4}\alpha _s^{5/2}}}{{3{{({\alpha _s} + 1)}^2}}}\left[{{S_n}G_1^2 + {S_t}(G_2^2 + G_3^2)} \right]$$

with ${S_n} = {D_ \bot }D_ \bot ^\ast $ and ${S_t} = {D_\parallel }D_\parallel ^\ast $.

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

(31)$${t_n}{A_{n0}} + {\delta _{n1}} + {\beta _s}{\sum\limits_{p = 1}^\infty {\left( {\frac{{ - 1}}{{2h}}} \right)} ^{n + p + 1}}\frac{{(n + p)!}}{{n!p!}}{A_{p0}} = 0,\,\,\,n = 1,2,\ldots $$

where

(32)$${\beta _s} = \frac{{{\alpha _s} - 1}}{{{\alpha _s} + 1}}$$

The above set of equations is truncated to a set of finite number equations in a finite number of unknowns by considering only the equations with $n \le {N_s}$ and ${2^p}$-multipoles with $p \le {N_s}$. The resulting set of linear equations is solved for selected values of h, $\alpha $ and ${\alpha _s}$ to determine ${A_{n0}}$ and hence ${D_ \bot }$ which equals $- {A_{10}}$. Calculations are repeated for several values of ${N_s}$ to check if the result for ${D_ \bot }$ are converged. We found that the results for the dipole converge rapidly with the increasing ${N_s}$ for most values of h, $\alpha $, and ${\alpha _s}$ except when h is close to unity and near the resonance conditions. This is illustrated in Figs. 2 and 3 which show the computed values of ${Q_n}$ and ${S_n}$ as functions of ${N_s}$ for $\alpha = - 2 + \,0.1\,\mbox{i}$, ${\alpha _s} = 10$, and $h = 1$ and 1.1.

Fig. 2. The absorption coefficient ${Q_n}$ for a sphere with $\alpha = - 2 + 0.1\,\mbox{i}$ placed near a substrate with ${\alpha _s} = 10$ as a function of the number of multipoles ${N_s}$ used in the computations. (a) $h = 1$; (b)$h = 1.1.$ The open circles are the computed values and the stars are the estimates obtained by the repeated application of the Shanks transformation as explained in the text.

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Fig. 3. The scattering coefficient ${S_n}$ for a sphere with $\alpha = - 2 + \,0.1\,\mbox{i}$ placed near a substrate with ${\alpha _s} = 10$ as a function of the number of multipoles used in the computations. (a) $h = 1$; (b) $h = 1.1$. See Fig. 2 for the legends.

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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

(33)$${D_ \bot }(n,1) = \frac{{{D_ \bot }(n + 1,0){D_ \bot }(n - 1,0) - {{({D_ \bot }(n,0))}^2}}}{{{D_ \bot }(n + 1,0) + {D_ \bot }(n - 1,0) - 2{D_ \bot }(n,0)}}$$

This transformation can be applied $L = ({N_s} - 1)/2$ times when ${N_s}$ is odd to reduce the original sequence of ${N_s}$ numbers to a single number. This is illustrated in Table 1 for ${N_s} = 9$ where we see that each subsequent transformation produces a sequence with reduced oscillations.

Table 1. Shanks transformation applied four times to the sequence of estimates of the dipole generated using N_s up to 9.

View Table

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.

Fig. 4. Absorption and scattering coefficients for the Drude model with $\delta = 0.1$. The open circles are the results of the computations and the dashed line is obtained by using the asymptotic formula for the dipole as given by (34). $h = 1.2$ and ${\alpha _s} = 11.3$.

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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

(34)$${D_ \bot } = {\left[ {{t_1} - 2{\beta_s}{R^{ - 3}} - \frac{{9\beta_s^2{R^{ - 8}}}}{{{t_2} - 6{\beta_s}{R^{ - 5}}}} - \frac{{16\beta_s^2{R^{ - 10}}}}{{{t_3} - 20{\beta_s}{R^{ - 7}}}} + O({R^{ - 12}})} \right]^{ - 1}}\,\,\,\,\,(R \equiv 2h)$$

with ${t_n}$ given by (19). The above asymptotic formula can be used to obtain reasonably accurate estimates of ${Q_n}$ and ${S_n}$ for $h \ge 1.2$. Note that ignoring the interaction with the substrate altogether yields ${D_ \bot } = t_1^{ - 1} = (\alpha - 1)/(\alpha + 2)$. The effect of the presence of the substrate is therefore to shift the resonance frequency. This shift can be determined by setting the real part of the quantity inside the square brackets in (34) to zero. Keeping only the first two terms in the right-hand side of (34) and evaluating the second term using the estimate ${\mbox{Re}} (\alpha ) \approx - 2$ we find that the presence of the substrate shifts the peak such that ${\mbox{Re}} (\alpha ) = - 2 - 6{\beta _s}{R^{ - 3}} + O({R^{ - 8}})$. The resonance in the presence of the substrate therefore shifts to a smaller frequency or to a larger wavelength. The resonance peak for $h = R/2 = 1.2$ in Fig. 4 is seen to occur at approximately ${\omega _r} = 0.51$. In the absence of the substrate, the resonance occurs at ${\omega _r} = 1/\sqrt 3 = 0.577\ldots $ indicating that, in the presence of the substrate, the peak shifts to a frequency that is approximately 12 percent lower.

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.

Fig. 5. Absorption and scattering coefficients for $\delta = 0.05$, $h = 1.2$, and ${\alpha _s} = 11.3$. The solid lines are computed using the asymptotic result (34) while the dashed lines correspond to the estimates obtained by ignoring the sphere-substrate interaction altogether.

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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.

Fig. 6. Absorption and scattering coefficients for $h = 1$, $\delta = 0.1$, and ${\alpha _s} = 11.3$. The open circles correspond to the computed results while the dashed line corresponds to the asymptote (34).

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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

(35)$${t_n}{A_{n1}} - {\delta _{n1}} + {\beta _s}{\sum\limits_{p = 1}^\infty {\left( {\frac{{ - 1}}{{2h}}} \right)} ^{n + p + 1}}\frac{{(n + p)!}}{{(n + 1)!(p - 1)!}}{A_{p1}} = 0$$

with the dipole ${D_\parallel }$ being equal to ${A_{11}}$. Once again, this set of infinite equations in infinite unknowns is truncated as before and the resulting equations are solved until the results for the dipole converge. For $h \le 1.1$, it is necessary to use the Shanks transformation repeatedly to speed up the convergence.

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

(36)$${D_\parallel } = {\left[ {{t_1} - {\beta_s}{R^{ - 3}} - \frac{{3\beta_s^2{R^{ - 8}}}}{{{t_2} - 4{\beta_s}{R^{ - 5}}}} - \frac{{6\beta_s^2{R^{ - 10}}}}{{{t_3} - 15{\beta_s}{R^{ - 7}}}} + O({R^{ - 12}})} \right]^{ - 1}}\,\,\,\,\,(R \equiv 2h)$$

The absorption and scattering coefficients obtained using the using the above expression for the dipole are accurate for $h \ge 1.2$. The effect of the substrate is to shift the peak to a higher frequency. Using only the first two terms on the right-hand side of (36), we find that the peak will occur for ${\mbox{Re}} (\alpha ) = - 2 - 3{\beta _s}{R^{ - 3}} + O({R^{ - 8}})$, the shift being smaller than for the dipole component perpendicular to the substrate surface for which the leading correction was $- 6{\beta _s}{R^{ - 3}}$.

Fig. 7. Absorption and scattering coefficients for the electric field parallel to the substrate for the Drude model. The open circles correspond to the computed results while the dashed line corresponds to the asymptote (36).

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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.

Fig. 8. Absorption and scattering coefficients for the electric field parallel to the surface of the substrate for $h = 1$ and ${\alpha _s} = 11.3$. The open circles represent numerical results while the dashed lines represent the estimates based on the dipole moment estimated using (36).

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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.

Fig. 9. The real and imaginary parts of the complex relative permittivity $\alpha $ as functions of the wavelength for silver (solid); gold (dashed); and copper (dashed-dotted). Data taken from Ref. 18.

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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.

Fig. 10. Absorption and scattering coefficients as functions of the wavelength for a silver sphere near a substrate with $h = 1.2$ and ${\alpha _s} = 11.2$. The solid lines correspond to the electric field normal to the substrate while the dashed lines are for the field parallel to the substrate.

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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.

Fig. 11. The absorption and scattering coefficients for gold (solid line) and copper (dashed line) for $h = 1.2$ and ${\alpha _s} = 11.2$.

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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:

(37)$$E(\lambda ) = \frac{{{c_1}}}{{{\lambda ^5}\left[ {\exp \left( {\frac{{{c_2}}}{{\lambda T}}} \right) - 1} \right]}}$$

Expressions (29) and (30) for the energy absorbed and scattered are modified with z replaced by a fixed wavenumber ${z_0} = 2\pi a/{\lambda _0}$, and the coefficients are replaced by their overall, energy-spectrum weighted average, values as given by

(38)$$< {S_n} > = \frac{{\int\limits_{{\lambda _1}}^{{\lambda _2}} {{{({\lambda _0}/\lambda )}^4}{S_n}(\lambda )E(\lambda )d\lambda } }}{{\int\limits_{{\lambda _1}}^{{\lambda _2}} {E(\lambda )d\lambda } }},\,\,\,\,\,\,\,\,\,\,\, < {Q_n} > = \frac{{\int\limits_{{\lambda _1}}^{{\lambda _2}} {({\lambda _0}/\lambda ){Q_n}(\lambda )E(\lambda )d\lambda } }}{{\int\limits_{{\lambda _1}}^{{\lambda _2}} {E(\lambda )d\lambda } }}$$

We used ${\lambda _0} = 500$ nm, ${c_2} = 1.4388 \times {10^7}\,\mbox{nm}\,\mbox{K}$ and $T = 5800\,\mbox{K}$, an estimated solar temperature, to determine the the average coefficients.

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.

Fig. 12. The energy-spectrum weighted average absorption and scattering coefficients as functions of h for silver spheres. The open circles correspond to the electric field normal to the substrate and the stars correspond to the electric field parallel to the substrate surface.

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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.

Fig. 13. Absorption and scattering coefficients for gold (stars) and copper (open circles) spheres as functions of the distance from the substrate. The solid lines joining the computed results correspond to the electric field normal to the substrate.

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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),

(39)$${Q_{sc}} = \frac{{16{z^4}\alpha _s^{5/2}}}{{3{{({\alpha _s} + 1)}^2}}}[{{S_n}G_1^2 + {S_t}(G_t^2 - {{({\textbf G}{\boldsymbol \cdot } {\textbf m})}^2}) + {S_{ta}}{{({\textbf G}{\boldsymbol \cdot } {\textbf m})}^2}} ]$$

where $G_t^2 = G_2^2 + G_3^2$ and ${\textbf m}$ is the unit vector along the line joining the centers of the two spheres. The coefficient ${S_n}$ can be determined by setting ${G_1} = 1$ and ${G_t} = 0$; ${S_t}$ by taking ${G_2} = 1$, ${G_1} = {G_3} = 0$, and ${\textbf m} = {{\textbf e}_{\textbf 3}}$; and ${S_{ta}}$ by taking ${G_2} = 1$, ${G_1} = {G_3} = 0$ and ${\textbf m} = {{\textbf e}_{\textbf 2}}$.

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.

Fig. 14. Scattering coefficients ${S_n}$ (circles), ${S_t}$ (squares), and ${S_{ta}}$ (stars) for a pair of silver spheres near the substrate.

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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),

(40)$${D_t} = {\left[ {{t_1} - \frac{\lambda }{{{R^3}}} + O({R^{ - 8}})} \right]^{ - 1}}\,\,\,\mbox{where}\,\,\,\lambda = {\beta _s}(1 - \frac{1}{{4\sqrt 2 }}) + 2\,$$

The resonance occurs at ${\mbox{Re}} (\alpha ) \approx - 2 - 3\lambda /{R^3}$. For the comparison sake, we note that for the case of a single sphere, $\lambda = {\beta _s}$ when the electric field is parallel to the substrate (cf. (36)) and $2{\beta _s}$ (cf. (34)) when the field is perpendicular to the substrate. Similarly, ${\lambda _s} = {\beta _s}(1 + 1/2\sqrt 2 ) - 1$ for the case when the electric field is parallel to the substrate but normal to the orientation of the pair of spheres, and ${\lambda _s} = {\beta _s}(2 + 1/4\sqrt 2 ) - 1$ for the case when the field is normal to the substrate. This analysis then predicts the largest shift in the resonance wavelength for the ${S_{ta}}$ case and the least for the ${S_t}$ case, in agreement with the computed results shown in Fig. 14. Note also that the dipole of a sphere is greater in the presence of the second sphere when the latter is aligned with the electric field. This is not the case when the electric field is normal to the pair orientation vector.

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

(41)$${S_n} = 19.9,\,\,\,{S_t} = 16.8,\,\,\mbox{and}\,\,{S_{ta}} = 30.9$$

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

(42)$$\frac{{{I_{sc}}}}{{{I_t}}} \approx {\phi _a}\frac{{4\alpha _s^{5/2}{{(1 + \alpha _s^{1/2})}^2}}}{{3{{({\alpha _s} + 1)}^2}}}{\left( {\frac{{2\pi a}}{{{\lambda_0}}}} \right)^4} < S > $$

Here, ${\phi _a}$ is the fraction of the substrate area occupied by the spheres and ${\lambda _0}$ is the reference wavelength used in determining the energy-spectrum weighted average scattering coefficient $< S > $. For the sake of simplicity we have assumed a normal incidence for which ${T_{in}} = {T_{out}} = 2/(1 + \alpha _s^{1/2})$ (cf. (8)). Our calculations with ${\lambda _0} = 500$ nm and the energy spectrum typical of the sunlight gave $< S > $ of about 10 for the spheres. If the spheres are present as clusters then the estimate would be higher. For example, for a pair of spheres randomly oriented in the plane of the substrate, $< S > $ would be roughly twice. With ${\alpha _s} = 11.2$, a value typical for a silicon substrate in contact with air, we therefore estimate the fractional increase in the energy transmitted by scattering in the absence of clustering as given by

(43)$${I_{sc}}/{I_t} \approx 1.77 \times {10^{ - 5}}{\phi _a}\,{a^4}\,\,\,(a\,\,\mbox{in}\,\,\mbox{nm})$$

Therefore if we deposit, for example, silver spheres of radius 20 nm close to the surface of silica with say ${\phi _a} = 0.1$, we can expect to increase the photovoltaic energy received by the substrate by about 30 percent. This amounts to approximately 0.1 g of silver per 1 ${m^2}$ of a solar panel. At the current price of silver of about $ 500 USD/kg this amounts to the material cost of silver of approximately 5 cents per 1${m^2}$ surface area of the panel, indeed a negligible cost compared to the cost of the panel itself (more than $100 USD per ${m^2}$). This illustrates significant gains to be made in harvesting the photovoltaic energy using the deposited silver nanospheres.

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.

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n	$D_{⊥} (n, 0)$	$D_{⊥} (n, 1)$	$D_{⊥} (n, 2)$	$D_{⊥} (n, 3)$	$D_{⊥} (n, 4)$
1	-4.787 + 0.784i
2	-0.686 + 0.885i	-3.359 + 9.845i
3	3.040 + 2.782i	-2.121 + 2.881i	-1.217 + 5.591i
4	4.536 + 11.43i	0.317 + 6.352i	-1.875 + 4.929i	-1.387 + 7.241i
5	-8.359 + 10.72i	-5.040 + 5.905i	-3.055 + 4.317i	-2.332 + 4.486i	-2.171 + 4.467i
6	-7.443 + 3.951i	-3.574 + 2.760i	-1.507 + 4.222i	-2.170 + 4.458i
7	-5.175 + 2.021i	-1.336 + 2.593i	-2.539 + 4.871i
8	3.432 + 1.407i	0.255 + 4.469i
9	-1.949 + 1.305i

Absorption and scattering coefficients for metallic nanospheres near a substrate

Abstract

1. Introduction

2. Expressions for the absorption and scattering in a multiparticle-substrate

2.1 Incident wave interaction with a substrate

2.2 Scattered fields by the particles

2.3 Expressions for the absorbed and scattered energy

3. Results

3.1 Single sphere in the vicinity of a substrate

3.1.1 Drude model

3.1.2 Results for silver, gold, and copper

3.2 Scattering from a pair of particles

4. Conclusions

Acknowledgement

References

Cited By

Figures (14)

Tables (1)

Equations (43)

OSA Continuum