Plasmonics

, Volume 7, Issue 3, pp 525–534

Plasmonic Scattering by Metal Nanoparticles for Solar Cells

Authors

    • Interdisciplinary Laboratory for Computational Science (LISC)FBK-CMM
  • Alessandro Vaccari
    • Renewable Energies and Environmental Technologies (REET)FBK-CMM
  • Antonino Calà Lesina
    • Renewable Energies and Environmental Technologies (REET)FBK-CMM
  • Enrico Serra
    • Interdisciplinary Laboratory for Computational Science (LISC)FBK-CMM
  • Lucia Calliari
    • Interdisciplinary Laboratory for Computational Science (LISC)FBK-CMM
Article

DOI: 10.1007/s11468-012-9338-4

Cite this article as:
Paris, A., Vaccari, A., Calà Lesina, A. et al. Plasmonics (2012) 7: 525. doi:10.1007/s11468-012-9338-4

Abstract

We investigate on absorption and scattering from metal nanoparticles in view of possible applications to photovoltaic cells. The analysis, accounting for most of the parameters involved in the physical mechanism of scattering, is split into two parts. In the first part, scattering from a metallic sphere is treated analytically to investigate the dependence on sphere size, sphere metal, and surrounding medium. In the second part, scattering from a metallic particle is investigated as a function of particle shape (spheroids, hemispheres, and cylinders) via numerical simulations based on the finite-difference time-domain method. The aim of the work is to provide a systematic study on scattering and absorption by metal nanoparticles, exploring several combinations of material and geometrical parameters in order to identify those combinations that could play a key role in solar cell efficiency improvement.

Keywords

PlasmonicsScatteringNanoparticlesSolar cellsFDTD

Introduction

In the last years, there has been a growing interest in the application of light-trapping techniques to photovoltaic cells. In particular, the presence of metal nanostructures on the front and rear surfaces or in the bulk of a cell is a promising tool to enhance the efficiency through surface plasmon resonances [1], especially in thin film (e.g., a-Si:H) solar cells and dye-sensitized cells. Plasmon resonances from metal nanoparticles are a wide and popular sector of investigations, with many applications [24]. In particular, in solar cell devices, light can be converted to electricity via plasmon resonances in nanoparticles, by a far-field effect which extends the optical path within the cell, by a near-field effect which locally enhances energy conversion, by a creation of energy-rich charged carriers which are transferred to the solar cell. Among the different physical mechanisms leading to better-performing photovoltaic cells, we discuss here the enhancement in the optical path length which is due to far-field scattering from nanoparticles of different sizes, shapes, and materials. Although this method has been experimentally confirmed many times [59], to the authors’ knowledge, a theoretical treatment of the topic, with comprehensive comparisons among the possible choices of parameters, is still lacking. So, it is not completely clear how each parameter influences efficiency, how they interplay with each other, and which combination determines the optimal cell design. Independently from the type of solar cell, metal nanoparticles should satisfy two general conditions in order to improve the efficiency in a significant way: (a) Their scattering cross section should be as large as possible, and, at the same time, their absorption cross section should be as small as possible. (b) They should scatter most of the radiation into the cell with the largest angular spread to maximize the optical path length. To achieve these features, we discuss the case of a single particle in a homogeneous, infinite, dielectric medium as a function of several parameters, namely the shape and size of the particles, the material they are composed of, and the optical characteristics of the surrounding environment. Since the number of parameters is very large, it is important to treat the scattering analytically whenever possible, before starting time-consuming and expensive numerical simulations or cell fabrication. As a matter of fact, the scattering problem is exactly solvable only for few particle shapes, namely ellipsoid (spheres, in particular) and infinitely long cylinders. Nanospheres have already been discussed in literature (for a recent example, see [10]). Our aim here is to provide a wider analysis to investigate the effects of changing the material, the dimensions, and the media which embed the spheres. The analytical discussion can go so far and can not extend to other shapes. Numerical simulations fill this gap; in particular, finite-difference time-domain (FDTD) method is a powerful tool to solve Maxwell’s equations and to simulate the evolution of the electromagnetic field during the extinction process. The method will be applied to study the response of hemispheres, cylinders, and spheroids of various sizes.

The paper is organized as follows: in section “Scattering by Spherical Nanoparticles,” we discuss the equations governing the extinction mechanism by a metallic sphere; then we use these equations to investigate the effect of different combinations of size, metal, and surrounding dielectric media on the scattering and absorption by the metal particle. This section also includes an analysis of the angular distribution of the scattered radiation. In section “Scattering by Spherical Nanoparticles,” we present the same results for cylinders, hemispheres, and spheroids after describing the main features of the numerical simulation. Finally, in section “Conclusions,” we report our conclusions and outline future developments.

Scattering by Spherical Nanoparticles

Theory of Extinction

Scattering and absorption of light by a sphere is a well-known problem whose solution is given by the Lorenz–Mie theory [11].

Within this theory, the extinction, scattering, and absorption cross sections by spherical particles are as follows:
$$ C_{{\rm ext}}= \frac{2 \pi}{k^2}\sum\limits_{n=1}^\infty (2 n + 1) Re\big(a_n + b_n\big),$$
(1)
$$C_{{\rm sca}}= \frac{2 \pi}{k^2}\sum\limits_{n=1}^\infty (2 n + 1)\big(|a_n|^2 + |b_n|^2\big), $$
(2)
$$ C_{{\rm abs}}= C_{{\rm ext}}-C_{{\rm sca}}, $$
(3)
where k = (2 πN)/λ, N is the refractive index of the medium around the particle, and λ is the wavelength of the incident radiation. an and bn are defined as
$$ a_n=\frac{m\psi_n(mka) \psi'_n(ka)-\psi_n(ka) \psi'_n(mka)}{m\psi_n(mka) \chi'_n(ka)-\chi_n(ka) \psi'_n(mka)}, $$
(4)
$$ b_n=\frac{\psi_n(mka) \psi'_n(ka)-m\psi_n(ka) \psi'_n(mka)}{\psi_n(mka) \chi'_n(ka)-m\chi_n(ka) \psi'_n(mka)}, $$
(5)
where ψ and χ are Riccati–Bessel functions of the first and third kind, respectively. a is the radius of the particle, and m = Np/N, where Np is the complex refractive index of the particle [11]. When dealing with the behavior of nanoparticles illuminated by light in the visible range, the small particle approximation is often taken for granted, leading to stop at the lowest order term of the previous equations, which amounts to treat the particle as a simple dipole. As shown in [11], the cross sections Eqs. 1, 2, and 3 can also be written as a power series in the adimensional parameter x = 2πa /λ; for wavelengths in the visible region and spheres with radius ≥ 100, x ∼ 1. The dipole approximation, which is then valid only for very small particles (x ≪ 1), is not appropriate and would neglect an important part of the scattering mechanism. Hence, in our analysis, we will truncate the series at n = 10, a choice that guarantees satisfactory convergence of the results.
An efficiency parameter is usually defined as the ratio of the particle cross section over a surface which is the geometrical projection of the particle on a plane perpendicular to the incoming light. For a sphere, it is as follows:
$$ Q_i=\frac{C_i}{\pi a^2} $$
(6)
where i = {ext, sca, abs}.

Note that the efficiency parameter Qsca defines a minimum distance between neighbor particles if they are arranged in an array. In fact, a sphere scatters the radiation contained in a circle of area \(C_{{\rm sca}}=\pi r^2 Q_{{\rm sca}}\); if we require the effective area of neighboring particles not to overlap, the distance d between them should satisfy the condition \(d>2 r \sqrt{Q_{{\rm sca}}}\). Examining our data presented below, where in most cases Q ∼ 3 − 4, we found a good agreement with the results reported in [12], where it is claimed that a 30% surface coverage is enough to scatter almost all of the incident radiation. Moreover, our results are not in conflict with those presented in [13], given the difference in the choice of the parameters.

Although the efficiency parameters contain all the dynamics of the extinction process, they are unrelated to the spectrum of the incoming light. In order to consider a realistic situation with a given incident radiation, we introduce an averaged efficiency defined as follows:
$$ \hat{Q}_{i}=\frac{\int I(\lambda) Q_i(\lambda) \; \textrm{d}\lambda}{\int I(\lambda) \; \textrm{d}\lambda}. $$
(7)
where I(λ) is the standard AM1.5G spectrum intensity [14] and the integration is extended from 300 to 1,100 nm. The new parameter \(\hat{Q}_{i}\) weights the scattering efficiency of a sphere at each wavelength with the intensity of the incident radiation. \(\hat{Q}_{i}\) is then a useful tool for comparing different configurations, and it represents a sort of global response of the particle to a given illumination spectrum (since we worked on a set of discrete data, we used the trapezoidal formula for the evaluation of the integral). As a reference, for an opaque particle, whose cross section corresponds simply to its geometrical size, we have Qext = 1 all over the entire spectrum and consequently \(\hat{Q}_{{\rm ext}}=1\). It is worth stressing that though \(\hat{Q}_{i}\) combines the effect of all parameters and of the solar spectrum features, thereby allowing a quick comparison among several configurations, the design of specific solar cells should nonetheless consider nonaveraged features like the position of the resonance peaks or their absolute magnitude.
Since scattering should be maximized and, at the same time, absorption should be minimized, a useful quantity to describe this feature is the albedo, defined as the fraction of the extinct light which is reemitted as radiation. Written in percentage form, the albedo is as follows:
$$ \alpha=\frac{C_{{\rm sca}}}{C_{{\rm ext}}}\times 100\;. $$
(8)
If the nanoparticles are placed on the front surface of a solar cell, we want the scattered radiation to be highly peaked in the forward direction. In principle, this feature could depend on the particle size, the surrounding medium, and the wavelength of the incident radiation. The fraction of light scattered in the forward direction is the following:
$$ f_{{\rm sub}}=\frac{\int_0^{2 \pi} \int_0^{\pi / 2} \vec{S}_s \cdot \hat{\vec{e}}_r r^2 \sin\theta \; \textrm{d}\theta \; \textrm{d}\phi} {\int_0^{2\pi}\int_0^{\pi} \vec{S}_s\cdot \hat{\vec{e}}_r r^2 \sin\theta \; \textrm{d}\theta \; \textrm{d}\phi}, $$
(9)
where \(\vec{S}_s\) is the Poynting vector of the scattered field. The coordinates (r, θ, and ϕ) form a spherical system of reference, centered at the particle and with the propagation vector of the incoming radiation aligned along the θ = 0 direction.
fsub varies between 0 and 1 by definition, and it should maximized for solar cell applications. For fsub < 0.5, light is mostly scattered backward. As done with the extinction efficiency, we calculated an averaged parameter as follows:
$$ \hat{f}_{{\rm sub}}= \frac{\int I(\lambda) f_{{\rm sub}}(\lambda) \; \textrm{d}\lambda}{\int I(\lambda) \; \textrm{d}\lambda}, $$
(10)
which represents the total amount of radiation scattered in the forward direction.

Results and Discussion

The quantities introduced previously are calculated for different combinations of the parameters:
  • The metals chosen are silver, gold, aluminum, and copper;

  • The radius of the spheres is 50, 100, and 150 nm;

  • Nanospheres are embedded in air, silica (SiO2), and silicon nitride (Si3N4).

The choice of different media could reflect the presence of a dielectric passivating layer in the solar cell. The complex dielectric functions of the metals are taken from Palik [15], while we assume a constant value for the refractive index of each medium, N = 1.05 for air, N = 1.54 for silica, and N = 2.02 for silicon nitride. Although the true refractive index of the three environments is not constant over the optical range, this approximation should give an error of only a few percent. We also neglect any absorption in the media surrounding the nanoparticles. Results for the extinction efficiency are shown in Figs. 1, 2, and 3, and in Tables 1, 2, and 3.
https://static-content.springer.com/image/art%3A10.1007%2Fs11468-012-9338-4/MediaObjects/11468_2012_9338_Fig1_HTML.gif
Fig. 1

Scattering efficiency Qsca as a function of the wavelength for different choices of parameters. Top row: air, middle row: SiO2, and bottom row: Si3N4. Left column: r = 50, central column: r = 100, and right column: r = 150. Blue line: silver, red dotted line: gold, yellow dashed line: aluminum, and green dashed dotted line: copper

https://static-content.springer.com/image/art%3A10.1007%2Fs11468-012-9338-4/MediaObjects/11468_2012_9338_Fig2_HTML.gif
Fig. 2

Absorption efficiency Qabs as a function of the wavelength for different choices of the parameters. Top row: air, middle row: SiO2, and bottom row: Si3N4. Left column: r = 50, central column: r = 100, and right column: r = 150. Blue line: silver, red dotted line: gold, yellow dashed line: aluminum, and green dashed dotted line: copper

https://static-content.springer.com/image/art%3A10.1007%2Fs11468-012-9338-4/MediaObjects/11468_2012_9338_Fig3_HTML.gif
Fig. 3

Albedo α as a function of the wavelength for different choices of the parameters. Top row: silver, second row: gold, and third row: aluminum, bottom line: copper. Left column: air, central column: SiO2, and right column: Si3N4. Blue line: 50 nm, red dotted line: 100 nm, and yellow dashed line: 150 nm

Table 1

Averaged scattering and absorption and \(\hat{f}_{{\rm sub}}\) in air for metal spheres as a function of the particle metal and size

Metal

r (nm)

\(\hat{Q}_{{\rm sca}}\)

\(\hat{Q}_{{\rm abs}}\)

\(\hat{f}_{{\rm sub}}\)

Ag

50

0.98

0.26

0.46

100

2.80

0.19

0.50

150

3.21

0.16

0.61

Au

50

0.49

0.67

0.48

100

2.44

0.56

0.52

150

2.84

0.50

0.63

Al

50

0.60

0.15

0.40

100

2.27

0.24

0.45

150

2.56

0.22

0.59

Cu

50

0.41

0.69

0.47

100

2.10

0.73

0.52

150

2.71

0.67

0.63

Table 2

Averaged scattering and absorption and \(\hat{f}_{{\rm sub}}\) in SiO2 for metal spheres as a function of the particle metal and size

Metal

r (nm)

\(\hat{Q}_{{\rm sca}}\)

\(\hat{Q}_{{\rm abs}}\)

\(\hat{f}_{{\rm sub}}\)

Ag

50

2.19

0.22

0.47

100

3.20

0.16

0.63

150

3.03

0.14

0.75

Au

50

1.74

0.66

0.49

100

2.85

0.49

0.66

150

2.72

0.42

0.77

Al

50

1.70

0.22

0.41

100

2.55

0.22

0.61

150

2.47

0.20

0.74

Cu

50

1.40

0.80

0.49

100

2.72

0.66

0.65

150

2.69

0.58

0.77

Table 3

Averaged scattering and absorption, and \(\hat{f}_{{\rm sub}}\) in Si3N4 for metal spheres as a function of the particle metal and size

Metal

r (nm)

\(\hat{Q}_{{\rm sca}}\)

\(\hat{Q}_{{\rm abs}}\)

\(\hat{f}_{{\rm sub}}\)

Ag

50

2.82

0.19

0.50

100

3.09

0.15

0.70

150

2.95

0.13

0.78

Au

50

2.45

0.56

0.52

100

2.76

0.45

0.72

150

2.67

0.39

0.80

Al

50

2.28

0.24

0.45

100

2.48

0.20

0.69

150

2.42

0.19

0.77

Cu

50

2.12

0.73

0.52

100

2.74

0.60

0.72

150

2.63

0.54

0.80

From the combination of tables and plots, we draw several conclusions:
  • The optimal size for scattering is around 150 nm in air but tends to decrease toward 100 nm in SiO2 and Si3N4. In general, more than one peak is observable for large spheres due to the multipole excitation. This leads to scattering of a larger part of the spectrum.

  • Embedding particles in a high-refractive environment slightly increases the averaged scattering efficiency, especially at long wavelengths, but it tends to reduce the maximum intensity and to widen the profile in the scattering efficiency as a function of λ, as shown in Fig. 1. This effect is due to the polarization of the medium, which tends to weaken the restoring force inside the polarized particles, in order to excite different oscillation modes and to shift the resonance peaks.

  • Increasing the particle size or the refractive index of the external medium reduces absorption, possibly by decreasing the damping of plasmon excitations. Copper and aluminum are partial exceptions to this rule. As a consequence, the albedo is maximum for Si3N4 and for r = 150 nm. Absorption profiles are less affected than scattering profiles by the change in the parameters, as shown in Fig. 2.

  • Silver is, by far, the best scattering materials both in absolute and integrated magnitude, followed by aluminum, gold, and copper. The spread among different materials is reduced, increasing the refractive index, as the polarization of the surrounding environment becomes more and more important.

  • Silver and aluminum in particular, are very poor absorbers, and in general, they have a very high albedo. On the contrary, gold and copper absorb more light, especially at the peak of the solar spectrum between 400 and 500 nm, as shown in Figs. 2 and 3.

Regarding the angular distribution of the scattered light, Fig. 4 shows how fsub varies with different combinations of sphere size, material, and surrounding medium. Values for \(\hat{f}_{{\rm sub}}\) are reported in the last column of Tables 13. As a general feature, short wavelength radiation is scattered in the forward direction more than long wavelength radiation. Spheres with larger size tend to scatter more in the forward direction, which is a well-known fact. Less appreciated is that the same behavior is observed if the dielectric constant of the surrounding medium is increased. The presence of a surrounding dielectric, in fact, helps to excite higher multipole scattering, for which the reemitted radiation is peaked in the forward direction more than that of a dipole. The angular distribution is more or less unaffected by the material of which the spheres are composed, although aluminum is less efficient than other metals. This angular spread could enhance the optical path in a solar cell. We assume the enhancement to behave like 1/ < cosθ >, where < cosθ > is the average cosine of the scattered distribution, and only the forward half plane is considered in the averaging process. In the most favorable cases, this quantity is ≈ 2, i.e., the path length is doubled. It turns out that fsub and 1/ < cosθ > cannot be maximized at the same time, so that a trade-off should be looked for in practical applications, e.g., solar cell design. From a naive combination of the two parameters, i.e., taking fsub times 1/ < cosθ >, fsub appears to be the dominant factor. As a consequence, configurations for which fsub is maximized should be preferred.
https://static-content.springer.com/image/art%3A10.1007%2Fs11468-012-9338-4/MediaObjects/11468_2012_9338_Fig4_HTML.gif
Fig. 4

Fraction of forward scattered light as a function of the wavelength for different choices of the parameters. Top row: air, middle row: SiO2, and bottom row: Si3N4. Left column: r = 50, central column: r = 100, and right column: r = 150. Blue line: silver, red dotted line: gold, yellow dashed line: aluminum, and green dashed dotted line: copper

Scattering by Nonspherical Nanoparticles

In order to investigate the effect of changing the nanoparticle shape, we resorted to numerical simulation. We developed our own FDTD code for the solution of Maxwell’s equations [16, 17]. We used a 120 × Ny × 120 cubic Yee [16] cell lattice, with an increasing number of cells along the y-axis: Ny = 120, 168, and 216. We need a varying Ny because we want to accommodate metal nanostructures of different dimensions along the incident beam propagation direction, which we assume to lie along the y-axis. The space step amounts to 2 nm for a fairly good representation of the geometrical details. The time step is t = 0.5 /c0, with c0 the vacuum light speed, to satisfy the Courant stability condition [17] in three spatial dimensions. We use a total field/scattered field (TFSF) source [17], placed eight cells inward from each face of the outer boundary of the FDTD lattice in order to create a plane wave linearly polarized electromagnetic pulse impinging on the nanostructures (polarization is along the z-axis). The FDTD lattice is completed with an extra layer, 15 cells thick, simulating an infinite surrounding medium. This layer acts as an absorbing boundary condition [17]. It is implemented using the recursive convolution as described in [18], known as convolutional perfectly matched layer. We use a compact pulse-exciting signal, i.e., of finite duration and with zero values outside a given time interval [19, 20]. The signal duration is suitably chosen to get spectral distribution results in the range of 300–1100 nm, as obtained by discrete Fourier transform, which is updated at every FDTD time iteration until the excitation is extinguished inside the whole numerical lattice. To correctly track the metal behavior in the frequency domain with a single run, however, we also need to include frequency dispersion in the time domain algorithm. This is accomplished by generalizing the recursive convolution approach described in [21] to the two critical point correction of a Drude dielectric function as given in [22]. To calculate fsub in (9), the angular distribution of the Poynting vector is obtained by implementing the Kirchhoff integral formula [23], which transforms the near fields, calculated with the FDTD algorithm around the nanoparticles, into plane scattered waves far away from them, with one for each sampled direction. Angular values were sampled at 1° steps. Before applying the method to variable shapes, we verified that for spherical nanoparticles FDTD results reproduce those obtained by the analytical approach. Choosing aluminum as the nanoparticle metal, we have evaluated the scattering and absorption efficiency Qi for spheroids, hemispheres, and cylinders. For each geometry, the projected surface on the plane normal to the incident radiation is a circle of radius 100 nm, so that results are comparable to those obtained above for spheres of the same radius. For spheroids, the height is 100 and 300 nm (corresponding to oblate and prolate spheroids, respectively, while h = 200 nm would correspond to the sphere as discussed). Cylinders are 100, 200, and 300 nm high; while the hemisphere has, clearly, r = h = 100 nm.

Figure 5 shows the distribution of the field intensity inside and outside the nanoparticles. Figure 6 compares Qsca, Qabs, and the albedo α for the different shapes to the results obtained for a sphere of radius r = 100 nm. As done with spheres, we can extract the integrated efficiency \(\hat{Q}_{i}\) and compare it to the previous results, as shown in Table 4. Spheroidal shapes, including spheres and hemispheres, show a similar behavior, with very similar extinction cross sections and albedo . The most remarkable effect of abandoning a spherical shape is a shift of the scattering peaks. Another interesting feature is the reduction in the absorption when the height is reduced. Cylinders show a different behavior: independently from their height, the scattering and absorption, efficiency is increased with respect to the sphere, both in absolute and integrated value. On the contrary, the albedo resembles very close to that of a sphere; cylinders seem then to be a better choice as scatterer.
https://static-content.springer.com/image/art%3A10.1007%2Fs11468-012-9338-4/MediaObjects/11468_2012_9338_Fig5_HTML.gif
Fig. 5

FDTD/TFSF simulation of the electric field intensity surrounding an aluminum ellipsoid, a hemisphere and a cylinder with h = 100 nm. For each case, the polarized radiation comes from the top side. Colors represent a logarithmic scale of the intensity, normalized to a reference value of 1 V/m. The wavelength is 500 nm

https://static-content.springer.com/image/art%3A10.1007%2Fs11468-012-9338-4/MediaObjects/11468_2012_9338_Fig6_HTML.gif
Fig. 6

Scattering efficiency (left), absorbing efficiency (middle), and albedo (right) as a function of the wavelength for aluminum spheroids and cylinders. In the first row, blue line: spheroid with h = 300 nm; red dotted line: spheroid with h = 100 nm; yellow dashed line: hemisphere with r = 100 nm; green dashed dotted line: sphere with r = 100 nm. In the second row, blue line: cylinder with h = 300 nm; red dotted line: cylinder with h = 200 nm; yellow dashed line: cylinder with h = 100 nm; green dashed dotted line: sphere with r = 100 nm. All particles are placed in air

Table 4

Extinction parameters and integrated \(\hat{f}_{{\rm sub}}\) for aluminum cylinders and ellipsoid of different heights and with r = 100 nm, compared with an aluminum sphere with r = 100 nm

 

h (nm)

\(\hat{Q}_{{\rm sca}}\)

\(\hat{Q}_{{\rm abs}}\)

\(\hat{f}_{{\rm sub}}\)

Cylinders

100

2.98

0.30

0.49

200

3.29

0.41

0.50

300

3.57

0.53

0.49

Ellipsoids

100

2.00

0.20

0.49

300

1.75

0.35

0.42

Semisphere

100

1.81

0.22

0.50

Sphere

200

2.27

0.24

0.45

All particles are placed in air

Finally, the angular distributions of scattered light is shown in Fig. 7. We see that cylinders and spheroids are less performing than spheres at short wavelengths, but, in most cases, they perform better at long wavelengths. As shown in Table 4, if the averaged values are considered, the sphere is outperformed by all the other shapes, excluding the spheroids with h = 300 nm.
https://static-content.springer.com/image/art%3A10.1007%2Fs11468-012-9338-4/MediaObjects/11468_2012_9338_Fig7_HTML.gif
Fig. 7

Fraction of light scattered into the substrate as a function of the wavelength for different choices of the parameters. On the left: spheroids, semisphere and sphere. Blue line: spheroid with h = 300 nm, red dotted line: h = 100 nm, yellow dashed line: hemisphere with r = 100 nm; green dashed dotted line: sphere with r = 100 nm. On the right: cylinders and sphere. Blue line: cylinder with h = 300 nm, red dotted line: h = 200 nm, yellow dashed line: h = 100 nm, green dashed dotted line: sphere with r = 100 nm. All particles are placed in air

Conclusions

Using analytical and numerical methods, we have investigated light scattering and absorption from small metal particles embedded in an infinite, homogeneous medium. The aim is to provide a guide for choosing particle size and composition, as well as characteristics of the surrounding dielectric when designing so-called plasmonic layers for solar cell applications. Beside calculating the wavelength dependence of the relevant cross sections in the proximity of the plasmon resonance frequency, we have introduced the averaged quantities \(\hat{Q}_i\) and \(\hat{f}_{{\rm sub}}\) which provide a quick means to evaluate the optical response of the nanoparticles under a given illumination. Since this work is meant to be functional to solar cell design, the AM1.5G spectrum was chosen. As a general conclusion, silver and aluminum are the best choice for metals, the former for showing high scattering efficiency, the latter having poor absorption and high albedo. We recognize that varying the metal has a very poor impact on the angular distribution of scattered light. For spheres, the particle radius should be confined to the range of 100–150 nm, where scattering dominates over absorption, and radiation is reemitted mostly into the cell. A high dielectric constant medium has a positive impact in many respects; in particular, it drives a bigger fraction of the scattered radiation in the half-space which would be occupied by the cell. It also tends to decrease the restoring force and the damping of the plasmon excitations inside the nanoparticle, leading to lower absolute magnitude for scattering and absorption; but it allows the excitation of higher multipoles and the efficient scattering of larger parts of the spectrum. Regarding shapes, we show that cylinders have the greatest scattering efficiency, especially if the height of the cylinder is increased, but they also exhibit high absorption. There is not a big difference among spheres and spheroids in general, although for spheroids, it is better to reduce the height in order to keep absorption low. We also show that albedo is almost independent from the chosen shape. The present analysis is independent from the choice of a specific cell design, and it neglects the presence of the air/cell interface. Including it might change some of the conclusions, without invalidating their general value. We leave this part for investigation in a future work, where we aim at the complete simulation of a realistic cell.

Acknowledgements

This work is supported by the Fondazione Caritro through the project Mistico. In particular, A. P. recognizes that he is funded by the Fondazione Caritro under the same project.

Copyright information

© Springer Science+Business Media, LLC 2012