Plasmonic Scattering by Metal Nanoparticles for Solar Cells
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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
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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 cellsFDTDIntroduction
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 [2–4]. 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 [5–9], 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].
Note that the efficiency parameter Q_{sca} 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.
Results and Discussion
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 (SiO_{2}), and silicon nitride (Si_{3}N_{4}).
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 |
Averaged scattering and absorption and \(\hat{f}_{{\rm sub}}\) in SiO_{2} 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 |
Averaged scattering and absorption, and \(\hat{f}_{{\rm sub}}\) in Si_{3}N_{4} 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 |
The optimal size for scattering is around 150 nm in air but tends to decrease toward 100 nm in SiO_{2} and Si_{3}N_{4}. 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 Si_{3}N_{4} 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.
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 × N_{y} × 120 cubic Yee [16] cell lattice, with an increasing number of cells along the y-axis: N_{y} = 120, 168, and 216. We need a varying N_{y} 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 /c_{0}, with c_{0} 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 f_{sub} 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 Q_{i} 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.
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 |
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.