## Abstract

Abstract: Using Mie theory and Rigorous Coupled Wave Analysis (RCWA) we compare the properties of dielectric particle and void resonators. We show that void resonators—low refractive index inclusions within a high index embedding medium—exhibit larger bandwidth resonances, reduced peak scattering intensity, different polarization anisotropies, and enhanced forward scattering when compared to their particle (high index inclusions in a low index medium) counterparts. We evaluate amorphous silicon solar cell textures comprising either arrays of voids or particles. Both designs support substantial absorption enhancements (up to 45%) relative to a flat cell with anti-reflection coating, over a large range of cell thicknesses. By leveraging void-based textures 90% of above-bandgap photons are absorbed in cells with maximal vertical dimension of 100 nm.

©2011 Optical Society of America

## 1. Introduction

Solar cells utilize textured surfaces to increase light absorption by reducing reflections and increasing the effective optical path length of an incident light ray. In thick solar cells, increased absorption is realized through, for example, deterministic texturing of wavelength-scale pyramidal structures which can be described with geometric optics and ray-tracing techniques [1,2]. Due to texture sizes comparable to or larger than a wavelength, however, these techniques are not applicable to thin film solar cells, which are instead often fabricated on roughened surfaces [3,4]. More recently, researchers have investigated the possibility of coupling photovoltaic (PV) cells to plasmonic [5–7] or dielectric [8,9] resonators. By leveraging similar resonances within the photovoltaic active material, it is also possible to achieve nanoparticle-based textures without coupling to external resonators [10–12].

Typically, the individual scattering element in such designs is a high-index dielectric or negative permittivity plasmonic nanoparticle surrounded by a low-index embedding medium, typically air. In this *particle* resonator geometry, scattering occurs as a result of the dielectric contrast between air and a dielectric or plasmonic nanoparticle (Fig. 1(a)
). For a given choice of material, however, one can also construct a *void* resonator, wherein light scatters off nanoscale voids within a high-index dielectric (Fig. 1(b)) or plasmonic medium. Such resonators can form the constituent subunits in photonic crystals [13] and metamaterials [14]. Recently, solar cell textures based on inclusions of low index materials within dielectric [15] or plasmonic [16,17] nanovoids have been proposed and demonstrated [18].

In this paper we compare the generalized scattering properties of single particle and void resonators. Next we propose and evaluate solar cell textures comprising periodic arrays of particle (Fig. 1(c)) or void (Fig. 1(d)) resonators. In comparison with previous investigations regarding anti-reflection effects in dielectric gratings [19–21] we will demonstrate how properties of *single* resonators are manifested as properties of periodic resonator arrays. In particular, we exploit the scattering properties of particle and void resonators in thin film amorphous silicon (a-Si) solar cell textures. Ultimately, we show that these active material textures can enable significant absorption enhancements over equivalent thin film cells with anti-reflection coatings. We demonstrate absorption of 90% of incident solar photons in 100 nm thick a-Si solar cells that take advantage of void resonator geometries.

## 2. Scattering properties of single resonators

In both particle and void geometries the scattering properties of individual spherical and cylindrical resonators can be described with analytical Mie theory [22]. While this work will concentrate on cylindrical scatterers, results can be applied to spherical resonators as well. In Mie theory the incident, internal, and scattered fields are decomposed into a set of vector harmonics ${\stackrel{\rightharpoonup}{M}}_{m}$and ${\stackrel{\rightharpoonup}{N}}_{m}$ which are indexed by their azimuthal phase dependence *e ^{imφ}*. At normal incidence the scattered fields are purely transverse electric (TE) or transverse magnetic (TM) with excitation coefficients

*a*and

_{m}*b*, respectively:

_{m}*J*and

_{m}*H*are respectively Bessel and Hankel functions of the first kind and primes denote derivatives with respect to the argument. The relative refractive index

_{m}*n*and relative size

_{r}*x*are given by

Here *n _{emb}* and

*n*are the refractive index of the embedding medium and the resonator respectively,

_{int}*k*is the wavevector,

*r*is the cylinder radius, and

_{0}*λ*is the free space wavelength.

The scattering and extinction properties of single resonators can be determined easily once the Mie coefficients have been calculated. For particle resonators *n _{int} > n_{emb}* and

*n*, whereas for void resonators

_{r}> 1*n*and

_{int}< n_{emb}*n*. In Fig. 2 we plot the lowest order (

_{r}< 1*m*= 0 or 1) Mie coefficients for particle (red) and void (blue) resonators comprising air and a material with refractive index 4—a value that approximates the refractive index of e.g. silicon or germanium (for a discussion of Mie resonances in spherical Si or Ge particles see e.g [23,24].). A unique feature of the void configuration is a redshift in the TE

_{1}resonance relative to the TE

_{0}resonance. In all other cases resonances shift to higher frequency with increasing mode order (larger

*m*).

At resonance the Mie coefficients equal 1 independent of their geometry, which can be proven analytically [25]. Although the peak values are identical, the void geometry supports larger bandwidth (lower Q) resonances. While small bandwidths are required for applications such as filters and sensors, the large bandwidths available in the inverse resonator geometry may be useful for scattering radiation across the solar spectrum. A more detailed analysis of the resonance bandwidths is given in Appendix A.

#### 2.1 Simplified resonance conditions

In either geometry the various resonance frequencies can be determined analytically by setting the Mie coefficients equal to one [25], and this condition can be further simplified by taking Mie theory in the limit of large index contrast. In the particle configuration a large index contrast implies *n _{r}* ≫ 1 and the lowest-order resonances occur for values of

*x*≪ 1. In the void geometry a large index contrast implies

*n*≪ 1 and the lowest-order resonances occur for values of

_{r}*xn*≪ 1. By taking limits of the Mie coefficients in these respective regimes (see Appendix B for details) we derive simplified resonance conditions, shown in Table 1 .

_{r}In both configurations the resonance conditions are functions of the variable *2πr _{0}n_{H}/λ*, where

*n*is the refractive index of the high-index medium, and the relative resonance frequencies are entirely governed by the location of Bessel function zeroes. For instance, the first zeroes of the Bessel functions

_{H}*J*

_{0},

*J*

_{1},

*Y*

_{0},

*Y*

_{1}, and

*Y*

_{2}occur when the arguments equal 2.40, 3.83, 0.89, 2.20, and 3.38, respectively. From a comparison of these values the red-shift of the TE

_{1}void resonance relative to the TE

_{0}void follows directly. Similarly, the derived resonance conditions capture a feature that is evident in comparing the particle and void Mie coefficients in Fig. 2: for a given choice of two materials the void geometry will exhibit a red-shifted TE

_{1}but blue-shifted TE

_{0}, TM

_{0}, and TM

_{1}resonance frequency (

*r*).

_{0}λ^{−1}#### 2.2 Scattering cross section

The scattering cross section (per unit length) for a TE illuminated cylindrical resonator is [22]

*a*Mie coefficients with

_{n}*b*. The larger bandwidth of the void Mie coefficients directly leads to broader bandwidth scattering cross sections. However, the cross section is also inversely proportional to the embedding medium refractive index

_{n}*n*, thereby reducing the scattering magnitude. In Fig. 3(a) we plot the scattering cross sections for cylindrical particles and voids. Scattering from particles results from a sequence of narrow resonances, whereas voids have significantly smaller variations with normalized frequency due to strongly overlapping broadband resonant modes. Additionally, very small voids scatter more TE radiation than their particle counterparts, whereas the opposite is true for TM polarization. Regardless, the larger bandwidths of void resonators is offset by the reduction in scattering due to large

_{emb}*n*and the particle resonators have a slightly larger unpolarized scattering cross section over most normalized frequencies.

_{emb}#### 2.3 Forward scattering properties

Although the *total* cross section may be slightly weaker for voids, for the solar cell textures in Fig. 1 the principle interest is in *forward* scattering, where the effect of an interface on scattering properties is very important. Previously, researchers noted that classical dipoles at an interface emit preferentially into the high index material and demonstrated the importance of this phenomenon in plasmon enhanced solar cells [26]. In Fig. 3(b) we plot the fraction of light emitted into a substrate for both parallel (blue) and perpendicular (red) dipoles embedded within a low-index (*n* = 1, dashed) or high-index (*n* = 4, solid) medium as a function of distance from the interface. The fraction of forward (into the substrate) emitted radiation decreases rapidly with distance for dipoles sitting above a high-index substrate. This situation approximates scattering in the particle geometry and necessitates careful design of particle resonators that are located within the interface near-field [22] in order to suppress back-scattering. In contrast, a dipole embedded *within* the high-index substrate emits more than 98% of its radiation in the forward direction regardless of orientation or distance from an interface. This situation is akin to the void geometry, and suggests that void resonators may exhibit substantial preference for forward scattering.

#### 2.4 Single particle summary

We have demonstrated the possibility of designing resonators in either a void or particle configuration for any given choice of two materials. In addition to deriving resonance conditions for the lowest order Mie resonances we have demonstrated a systematic comparison of the two scattering geometries. Both resonator types display an increase in scattering with normalized frequency up to the first Mie resonance, after which the scattering is mostly constant for the void geometry while the scattering by particles is dominated by narrow resonance peaks. Compared to particles, ultra-small voids scatter TE radiation more efficiently, while the opposite is true for TM polarization. Voids tend to exhibit larger bandwidths, have smaller unpolarized cross-sections, but are likely to scatter a larger portion of light in the forward direction in the presence of an interface. In the subsequent section we demonstrate how these various scattering properties are evident in dielectric resonator array solar cell textures and evaluate the use of both void and particle resonators for enhancing light absorption in solar cells.

## 3. Resonator arrays

In Fig. 1 we present two different solar cell textures comprising arrays of rod-shaped particle (Fig. 1(c)) or void (Fig. 1(d)) resonators. Despite significant differences between single scatterers and resonator arrays—square rather than circular cross sections, the presence of an interface, and periodicity dependent coupling between resonators [27]—we will demonstrate that single resonator properties are preserved in solar cell textures. Based on previous investigations of absorption in particle rods, we anticipate that the essential properties of lowest order Mie resonances are unchanged despite the differing cross-sections [11]. The electromagnetic response of arrays was calculated with a rigorous coupled wave algorithm (RCWA) implemented via DiffractMod from RSoft [28]. In RCWA reflected and transmitted light is decomposed into a basis set of diffracted plane-wave components [29]:

*a*is the array periodicity,

*n*is the substrate refractive index,

_{s}*k*is the free space wave-vector, and the quantities

_{0}*R*and

_{j}*T*are determined computationally. In Fig. 4(a) we plot the scattering properties of textures based on the designs in Fig. 1 for a substrate index of

_{j}*n =*4 but without an ARC. Simulations are presented for a periodicity 40 times larger than the resonator diameter such that inter-particle coupling is weak and there is a near continuum of diffracted orders

*k*. The sum of the power contained in all diffracted orders is a good approximation to the single resonator scattering cross-sections, and allows for a direct comparison with Fig. 3(a).

_{xj}The general results derived previously for a single resonator can be observed in the case of resonator arrays as well. Both resonator geometries display an increasing scattering cross-section with normalized frequency up to the first resonance, after which the scattering cross-section plateaus. Ultra-small void-based arrays scatter TE radiation more efficiently than the particle counterparts, while the reverse is true for TM polarization. Void arrays have larger bandwidths but less efficient scattering, especially for TM polarization.

By dividing the power in the transmitted orders by the total scattering we can determine the proportion of forward scattering (Fig. 4(b), bottom) and compare with dipole calculations (Fig. 3(b)). The comparison is most valid at small normalized frequencies, where the scattering arises from single dipolar resonances. As particles get larger, and their center moves away from the interface, the proportion of forward scattered light is reduced dramatically.

Furthermore, voids are superior forward scatterers, as expected from our dipole calculations. In the top of Fig. 4(b), however, we show that back-scattering can be reduced for particles by adding an anti-reflection coating; light scattered into oblique angles is totally internally reflected at the boundary of the air-ARC boundary and is redirected into the substrate.

While this analysis was presented for large periodicity arrays a practical solar cell texture requires a substantial fill fraction to maximize light scattering. In Fig. 5 we plot the normalized scattering intensity for arrays of 100nm diameter particles, as a function of fill fraction (i.e. periodicity). While the scattering amplitude depends on fill fraction (Figs. 5(c) and 5(d)), the frequency dependence (Figs. 5(a) and 5(b)) is nearly independent of periodicity for fill fractions less than approximately 45%. These results indicate that the properties of dielectric textures are primarily governed by the Mie resonances of individual resonators located within an inhomogeneous environment.

## 4. Thin film absorption enhancements

In previous sections we detailed the optical response of single particle and void resonators and showed that the response of resonator arrays is primarily determined by single resonator properties (at fill fractions less than 50%). In real solar cells, with finite thickness, scattering off surface texture enhances light absorption through light trapping. Light trapping effects are particularly important in thin-film cells where the single-pass absorption of a normal incident light ray is weak. In this section we will demonstrate significant absorption enhancements in resonator arrays atop very thin a-Si layers. Absorption enhancements may arise from anti-reflection effects, excitation of lossy Mie resonances, and coupling to waveguide modes. As these mechanisms are interrelated, finding an optimal cell design is non-trivial.

We use a numerical optimization algorithm, Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [30], to efficiently search large areas of parameter space for the global maximum in absorbed photons. We calculate the total fraction of absorbed photons by integrating the product of the solar spectrum, *b _{AM1.5}(λ)*, and the simulated absorption spectrum of the cell,

*A(λ)*, for photons with energy larger than the a-Si mobility gap (730nm):

We calculate the absorption in the amorphous silicon (a-Si) layer using a dielectric function from [31], and we model the back contact as a perfect electrical conductor (PEC) to avoid complexities associated with small but finite absorption effects in real metals. For the ARC we assume an index of 2, which approximates the refractive index of ZnO, ITO and Si_{3}N_{4} in the optical range. For every value of the *maximum* a-Si thickness *t*, we allow the square rod diameter *d*, the periodicity *a*, and the ARC coating thickness *t _{ARC}* to vary (Fig. 6(c)
). The optimized absorption values as a function of

*t*are compared to an equivalent

*maximum*thickness flat cell with optimized ARC and plotted in Figs. 6(a) and 6(b).

Both particle and void resonator arrays absorb significantly more solar photons than the untextured cell for all thicknesses, with enhancement factors as large as 45%. For thin cells between 0 and 50 nm and 75-112 nm the void geometry has superior absorption properties. In particular, 100 nm thick solar cells with an array of void resonators already absorb over 90% of the incoming solar photons, more than the absorption of a 300 nm thick flat cell. For cells thicker than ~140 nm the particle geometry absorbs more light than the void geometry.

For these thin cells the maximal possible rod diameter is small and we expect the absorption properties to be similar to the scattering properties for small resonators described in sections 2 and 3. As expected, in this regime the void (particle) resonator geometries absorb TE (TM) light more efficiently (Fig. 6(d)), especially at long wavelengths. While future investigations will focus on detailing the interplay of Mie resonances and waveguide modes in these structures, these results clearly demonstrate that leveraging Mie resonance effects within patterned grating structures can enhance light absorption in thin solar cells of varying thicknesses.

## 5. Conclusion

In this study we demonstrate a number of significant differences in the optical properties of dielectric particle and void resonators. Using Mie theory we derive simplified resonance conditions for the lowest order resonances in both configurations in the limit of large index contrast. Voids resonate at larger normalized frequencies than particles, with the exception of the TE_{1} resonance, which shows an unusual redshift, and have higher bandwidth resonances except for the TM_{0} mode. Due to the early onset of the TE_{1} mode ultrasmall voids are more efficient TE scatterers than ultrasmall particles, while for TM polarization the opposite is true. In general, however, the advantages from larger bandwidth resonances in the void configuration are partially offset by an inverse relationship between peak scattering cross-section and embedding medium refractive index.

We suggest two distinct active medium solar cell textures which rely on void or particle resonances to enhance light absorption. Using RCWA calculations we show that void geometries have preferential scattering into the active medium as expected from calculations of single dipole emitters. We show that the particle and void solar cell textures preserve many of the general features evident from Mie theory and calculate optimized absorption values for a-Si solar cells of varying thicknesses which exploit these textures. Enhancement factors as large as 45% are observed, and void-based textures allow for 90% absorption of incident above-bandgap solar photons in cells whose maximal thickness is 100 nm, a three-fold reduction in active layer thickness compared to untextured cells with equivalent absorption. We anticipate that single particle calculations will bring new understanding of basic properties of void resonators and resonator arrays. The results of this study may lead to novel solar cell textures and may be used to refine understanding of other void-based solar cell textures [18] (see Appendix C).

## Appendix A: Bandwidths

For solar cell applications, where the illuminating radiation is broadband, the bandwidth of resonators is a very important quantity. An examination of the Mie coefficients in Fig. 2 suggests that the peak values of the Mie coefficients are identical, but the void geometry supports larger bandwidth (lower *Q*) resonances. This trend becomes more evident as the index contrast between the particle and embedding medium is increased.

In the particle geometry, as the index contrast increases the resonator size gets smaller and *Q* increases exponentially [25]. With voids, however, the *Q* factor is independent of *n _{r}*. This fact can be proven by deriving simplified Mie coefficients in the limit of small

*n*, as we do in Appendix B. The Mie coefficients depend only on the normalized frequency

_{r}*x,*thus the resonance lineshape of a void resonator looks identical for all values of index contrast where the approximation holds. The possibility to have very small voids with large bandwidths is markedly different than what occurs in the particle geometry for the same choice of materials. In both cases the resonator size

*relative to the free space wavelength*gets smaller with increasing index contrast. However, in the void geometry the size

*relative to the embedding medium wavelength*is constant and it is this normalized size that determines the bandwidth.

Differences in bandwidth between the two scattering geometries also increase markedly with higher azimuthal mode index *n*. In the void configuration the *Q* factor is nearly constant with *m* whereas in the particle configuration the *Q* factor increases exponentially with *m* (Fig. 7
) a scaling which has been observed in low order whispering gallery microcavities [32]. In the particle geometry a light-ray within a high index medium encounters a low-index medium, and at high *m* the cylindrical boundary looks approximately planar leading to near-total internal reflection and high *Q*s. In voids, the light-ray approaches the boundary from the low-index side, there is finite transmission even for a planar interface, and bandwidths remain large.

## Appendix B: Simplified Mie coefficients and resonance conditions

Below, we derive the large index contrast limits for particle and void resonators presented in Table 1. For void resonators we can take the limit of the Mie coefficients as *n _{r}* → 0:

The Mie coefficients become functions solely of the variable *x = 2πr _{0}n_{emb}/λ* and can be solved directly to determine the resonance values in Table 1. The simplified Mie coefficients are reasonably accurate starting from index contrast of approximately 3, and are accurate even up to very high values of the mode index

*m*. These simplifications highlight the fact that in the high index contrast limit the void resonance lineshapes, and thus bandwidths, are independent of the specific value of

*n*and remain quite large even for resonant voids that are significantly smaller than the freespace wavelength.

_{int}These properties of voids are in stark contrast with particle resonators, where the bandwidth depends strongly on the particle refractive index *n _{int}*. For particles, we were unable to find a simplified limit of the Mie coefficient as

*n*→ ∞. To determine the resonance conditions we set the Mie coefficients equal to unity [25], which implies that

_{r}The Bessel functions have arguments of both *x* and *n _{r}x* and the resonances cannot be solved for arbitrary values of

*n*. However in the large index contrast limit (

_{r}*n*→ ∞) the resonance condition for TE polarization simplifies to

_{r}*J*= 0. For the lowest order TM resonances, the limit of large index contrast also implies that

_{m}(n_{r}x)*x*→ 0 and with suitable Taylor expansions about

*x*near 0 we can derive simplified resonance conditions:

We would like to refer the reader to chapters 10 and 15 in [33] for an interesting discussion of the resonant condition in the limit of very large *n _{r}*.

## Appendix C: Electric field intensity within void resonators

In reference [16] Yu et al. suggest enhancing light absorption in low index materials by embedding them within a high index medium. For deeply subwavelength spherical voids of low refractive index (*n _{L}*) surrounded by a high refractive index medium (

*n*) they derive a maximal absorption enhancement factor (Eq. (20)) of

_{H}This equation can be rewritten as

Note that this generalized form applies to planar inclusions as well (Eq. (17)). The enhancement factor depends directly on the average electric field intensity within the void. For a deeply subwavelength inclusion the internal field can be derived from electrostatic considerations. In Fig. 8
we show the average squared electric field enhancement for a spherical void of index *n _{int}*=1 surrounded by a medium of index

*n*=4 as a function of its normalized frequency (its size relative to the incident wavelength). In addition to recovering the electrostatic result, $\overrightarrow{E}\cdot {\overrightarrow{E}}^{*}~2.1{\left|{E}_{0}\right|}^{2}$, this result highlights the fact that the internal field drops off quickly as the particle size increases. To fully exploit the spherical void geometries it is important to construct inclusions that are significantly smaller than the wavelength of the illuminating radiation.

_{emb}## Acknowledgments

We would like to thank R. Zia for help with dipole emitter calculations. This material is based upon work supported as part of the Center for Re-Defining Photovoltaic Efficiency through Molecule Scale Control, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001085.

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