## Abstract

Enhanced absorption of near infrared light in silicon solar cells is important for achieving high conversion efficiencies while reducing the solar cell’s thickness. Hexagonal gratings on the rear side of solar cells can achieve such absorption enhancement. Our wave optical simulations show photocurrent density gains of up to 3 mA/cm^{2} for solar cells with a thickness of 40 µm and a planar front side. Hexagonal sphere gratings have been fabricated and optical measurements confirm the predicted absorption enhancement. The measured absorption enhancement corresponds to a photocurrent density gain of 1.04 mA/cm^{2} for planar wafers with a thickness of 250 µm and 1.49 mA/cm^{2} for 100 µm.

© 2013 Optical Society of America

## 1. Introduction

Silicon solar cells do not utilize a considerable fraction of light in the near infrared, close
to the band gap of silicon, due to weak absorption. The penetration depth of photons in the
wavelength range between 900 and 1100 nm is up to 3 mm and therefore exceeds typical crystalline
silicon solar cell thicknesses of about 180 µm. To increase absorption, the effective
path length within the solar cell has to be increased for the photons in this wavelength range.
This becomes even more important as the photovoltaic industry aims for thinner solar cells. The
effective path length is enhanced when the photons are directed into shallow angles within the
solar cell and are subsequently totally internally reflected. In present commercial crystalline
silicon solar cells a chemically etched pyramidal front side texture causes light paths
deviating from the direction perpendicular to the solar cell’s surface. The size of the
pyramids is typically in the range of several µm and hence the processing gets more and
more difficult for very thin solar cells. Also a planar front surface might be beneficial
concerning the electrical properties. The light trapping can also be achieved with randomizing
structures on the rear side of the cell. With such structures a maximum enhancement factor of
4*n*^{2}, with *n* being the refractive index of the
solar cell material, can be achieved in the limit of vanishing absorption [1, 2]. With periodic rear side structures
also significant path length enhancements can be reached, as they can diffract light into
certain directions that are very effectively trapped within the solar cell [3, 4]. Several analytical
and numerical investigations indicated that diffractive structures can achieve efficient light
trapping [3, 5–7]. The approach presented in this paper
are hexagonal gratings made of monodisperse, hexagonally ordered silica spheres embedded in a
high refractive index matrix followed by a reflecting metal layer. The sphere grating is
separated from the silicon bulk by a very thin passivation layer (for example 10 nm
Al_{2}O_{3}) [8], which has a negligible
influence on the optical properties of the rear side and maintains a high-quality surface
passivation reducing surface recombination at the silicon surface. A schematic sketch of the
concept is shown in Fig. 1.

The grating structures can be realized via spin coating processes, and should be compatible with solar cell processing. Important questions of interest that we intend to answer in this work are the extent of the potential absorption enhancement achievable with such structures, the effects leading to such an enhancement and the necessary design parameters to exploit the full potential.

## 2. Simulations using rigorous coupled wave analysis

First we calculated the absorption enhancement in crystalline silicon wafers with a flat front
surface due to the diffractive rear side structure using rigorous coupled wave analysis (RCWA)
[9]. Second we fabricated the structure. As an aggregated
number that is easily conceivable and allows comparison to other works, we calculated the
maximal achievable photocurrent density *J*_{ph}, which is possible under
illumination with the air mass 1.5 global solar spectrum AM1.5g [10] and the simulated absorption *A* when an internal quantum efficiency
of unity is assumed via:

*N*

_{AM1.5}denotes the number of photons per area and wavelength and

*e*

_{0}the charge of an electron. By comparison to a reference system with planar front side and planar rear side a photocurrent density gain can be calculated from a measured or simulated absorption. This approach neglects that the collection efficiencies of created free charge carriers depend on the exact position in the solar cell where the absorption occurs. However, for thin crystalline silicon solar cells these collection efficiencies are very high [11] and the absorption of infrared light is distributed quite evenly over the cell [12] due to the large penetration depth that exceeds the cell thickness. Hence the maximum photocurrent density, which is calculated based on the absorption data, is a reasonable measure to determine the potential of the light trapping schemes [13], such as the sphere gratings. The RCWA requires a segmentation of the grating structure into a stack of planar layers. As we calculated sphere gratings this segmentation is an approximation. We conducted convergence tests concerning the number of layers that are used to build up the spheres and found good convergence when more than 40 layers are used for a monolayer of spheres. This corresponds to a layer thickness of 25 nm (at a sphere diameter of 1000 nm). In addition to the segmentation also the number of Fourier orders considered means an approximation. We also conducted convergence tests for this parameter and found convergent behavior for more than six Fourier orders (also for a sphere diameter of 1000 nm) and did the final simulations with seven Fourier orders. In the near infrared for a sphere diameter of 1000 nm three to four orders can propagate in the far field in silicon, which means that we are also considering at least three evanescent orders. Nevertheless these approximations introduce a potential error on the final results that is difficult to quantify exactly. From convergence tests comprising both parameters we deduced an error concerning the integrated photo current density of $\Delta {J}_{ph}=\pm 0.2\text{mA}{\text{cm}}^{-\text{2}}$, which is considerably below the absolute values. After the convergence tests we investigated the influence of three parameters: the number of sphere layers, refractive index of the matrix material and the sphere diameter. A monolayer of hexagonally ordered spheres results into a photocurrent density gain of about 3 mA/cm

^{2}for a crystalline silicon solar cell thickness of 40 µm (assumed matrix material: amorphous silicon, assumed sphere diameter: 1100 nm) adding to the total photocurrent density without sphere grating of 34.3 mA/cm

^{2}, thus representing a 9% increase. Additional layers, which would build up a three dimensional opal, lead to a further small enhancement (up to 3.5 mA/cm

^{2}total gain). These are promisingly high potential enhancements, in comparison to literature: for optimized binary gratings a maximum enhancement of 1.8 mA/cm

^{2}has been reported [12]. For a cell thickness of 20 µm and a variety of different periodic rear side structures (but not spheres) a calculated potential photocurrent density gain between 5 and 7 mA/cm

^{2}has been reported recently [14]. Due to the different cell thickness and the different structures these results are not directly comparable to the results presented here. The big advantage of the hexagonal sphere grating is the potentially easy realization and integration into the solar cell processing. In this line, we focus our further analysis on monolayers of hexagonally ordered sphere layers, as these are the easiest to produce and should be most compatible to later electrical contact formation. The bigger the difference between the refractive indices of the spheres and the matrix material, the bigger is the absorption enhancement. First, the diffraction effects increase with increasing difference between the refractive indices of the spheres and the matrix material and second, the coupling to the silicon bulk gets more efficient if the refractive index of the matrix material approaches the one of silicon. Both effects lead to higher diffraction efficiencies. For all further optimization calculations we assumed monolayers in an amorphous silicon matrix. The diffractive properties of the hexagonal sphere grating depend also strongly on the sphere diameter. At first, it was considered to be ideal, when only one order of diffraction can propagate that appears under an angle of close to 90° to the normal. Then the light path would be close to parallel to the cell surface. For this case, a maximum enhancement factor of 853

*n*was calculated by considering conservation of etendue [3]. Therefore previous experimental work aimed for rear side reflectors with a period in the order of 300 nm [15–17] achieving potential gains of about 0.3 mA/cm

^{2}. New investigations proposed an optimum diameter for gratings in a range, where several orders of diffraction can propagate and the first order is still totally internally reflected at the planar front surface [12, 13]. We did a systematic investigation of the optimum sphere diameter using RCWA, shown in Fig. 2.For a diameter of about 300 nm there is only one order of diffraction that can propagate in the silicon bulk. Although the diffraction angle is close to 90°, Fig. 2(a) shows that the photocurrent gain is very small because the diffraction efficiency is quite small. Only a small fraction of the incoming light is diffracted into the first order while the biggest fraction is reflected (diffraction into zeroth order). If the sphere diameter is increased, more propagating orders occur. For sphere diameters of about 1000-1100 nm, the first order is still totally internally reflected. The absorption enhancement reaches a maximum. For even bigger diameters the number of diffraction orders increases and some can leave the solar cell at the front side. Because of the growing number of relevant orders the calculation effort grows for bigger periods. In the case of a periodicity in two dimensions it grows with the diameter to the power of 6. To avoid excessive calculation times and storage demand, we investigated the properties of a layer of parallel lying cylinders on the rear side instead of spheres. The cylinders have the same diffraction pattern in one dimension while they have no periodicity in the other dimension. Thus it is possible to calculate bigger lattice constants with higher accuracy. Figure 2(b) shows that the behavior for small diameters is quite similar to the results with spheres. A local maximum is achieved in the range of 1000 nm sphere/cylinder diameter, but a significant photocurrent density gain can also be achieved for diameters up to 4 µm. The overall maximum occurs for a diameter of about 2000 nm. The efficient light trapping for larger feature sizes can be attributed to the fact that for bigger diameters the structure acts more similarly to a diffuse reflecting structure because a high number of diffraction orders can propagate. Evaluating both calculations with spheres, and cylinders no clearly defined ideal diameter can be found. Significant current gains can be reached with diameters in the range of 1 µm, but also with considerably bigger diameters of a few µm. As electric contacts have to be made through the diffractive structures, an overall thinner structure might be beneficial.

Furthermore we investigated the absorption enhancement considering the near field data to
understand where the absorption enhancement occurs. To calculate the local absorption at a point
**r** from the local values of the electric field
**E**(**r**,*ω*) which are the result of the
RCWA-calculations, we used the theorem of Poynting [18]
and followed the method of [19]:

_{0}is the vacuum permittivity, Im(ε) the imaginary part of the spectrally and spatially dependent dielectric function and ω the frequency of the light. The local absorption depends on the frequency and hence on the wavelength of the light. Important in terms of light trapping for silicon solar cells is a spectral range between 950 and 1200 nm. We calculated the local absorption in this range in steps of 10 nm and subsequently averaged the values to generate an absorption distribution for this wavelength range, displayed in Fig. 3 for a 40 µm thick solar cell with a sphere grating at the rear side. It can be seen that the absorption enhancement occurs for all depth values (

*z*dimension). There are some lateral modulations that follow the periodicity of the sphere grating and some vertical modulations due to interference effects that are not relevant under illumination with sunlight. The important result is that there is no overall trend of absorption enhancement in a certain area of the solar cell, especially not near the grating. The reason is the large penetration depth of the photons in this wavelength range and the occurrence of several diffraction orders for the presented size of spheres. Furthermore, this finding legitimates the simplification that we treat each absorbed photon equally in its contribution to the maximum possible photo current density as defined in Eq. (1).

## 3. Fabrication and experimental verification

For experimental verification of the simulated absorption enhancement, we fabricated hexagonal sphere gratings. The fabrication was realized in two steps. In the first step, the monodisperse silica spheres (diameter 922 nm), which were suspended in a solution consisting of 2-propanol and water, were spin coated on the rear side of a silicon wafer. Due to self-organized growth, a dense, hexagonally ordered layer can be generated. A sphere concentration of 210 mg/ml and spin speed of 4000 rpm was used. For each 4 inch wafer 0.5 ml of the sphere suspension were dropped onto the wafer surface and the rotation was started immediately. Figure 4(a) shows a representative part of a 4 inch wafer after the spin coating process. A homogenous monolayer of spheres can be seen. There are still some line defects and discontinuities, but in the range of a few µm a hexagonal order exists. Any kind of defect represents a deviation from the ideal lattice structure assumed in the simulations and will result into scattering, which also leads to a light path length enhancement. Therefore the visible defects in Fig. 4 are not considered to be a significant change for the worse. In the second step, the voids between the spheres were infiltrated with amorphous titaniumoxide using atomic layer deposition as shown in Fig. 4(b). The deposition took place at 130° C and was done in a batch reactor from Beneq. The refractive index of the titaniumoxide is 2.38 for a wavelength of 1000 nm, compared to the refractive index of the spheres of 1.5, as determined by spectral ellipsometry.

By spectrophotometric measurements the potential absorption enhancement in solar cells was
estimated. We therefore used a Cary 5000i from Varian and mounted the samples inside an
integrating sphere. By such a “center mount” measurement setup all light that is
not absorbed inside the sample is detected. To determine just the absorption enhancement in the
silicon bulk, a metal rear reflector, which would be necessary in complete solar cells, has not
been deposited. We verified in several simulations that for systems without a mirror a very
similar absorption enhancement can be expected as for systems with a mirror. Only small
quantitative changes or small spectral shiftings of the absorption enhancement have been
observed in the outcome of the simulation. Thus the measurements can be seen as a strong
evidence for the light trapping properties of fully processed solar cells. The measurements show
a significant absorption enhancement in the near infrared due to the inverted monolayer of
spheres (Fig. 5). Under illumination with the AM 1.5 spectrum this absorption enhancement corresponds to a
photocurrent density gain of 1.04 mA/cm^{2} for a wafer thickness of 250 µm.
Given the overall total photocurrent density without sphere grating (and without antireflection
coating) of 26.8 mA/cm^{2} (with a single layer antireflection coating 37.4
mA/cm^{2}) this constitutes a relative increase of 3.9%. A photocurrent density gain of
1.49 mA/cm^{2} was achieved for a 100 µm thick wafer, which is equivalent to an
increase of 5.8% based on the 25.7 mA/cm^{2} photocurrent density without the structure.
Due to the absorption enhancement the photocurrent density in a wafer with a thickness of 100
µm with the sphere grating exceeds the photocurrent density in a planar wafer with a
thickness of 250 µm. This highlights the potential of the proposed light trapping concept
with regard to thinner solar cells. These values are comparable to results achieved with
nano-imprinted gratings where an increase of 1.6 mA/cm^{2} and 1.7 mA/cm^{2} was
determined for a wafer thickness of 200 µm for crossed gratings and planarized line
gratings, respectively [20]. In the case of crossed
gratings additional parasitic absorption in a metallic rear side reflector was considered. With
an optimum Lambertian scatterer an overall enhancement of about 3 mA/cm^{2} could be
achieved for a wafer thickness of 250 µm.

For direct comparison of the simulation and measurement, simulations of the produced structures
were conducted. In contrast to the optimization calculations presented above, where an amorphous
silicon matrix was assumed, for the matrix material the measured *n*-values (2.38
at 1000 nm) of the deposited TiO_{2} were used here. The wave optical simulations using
RCWA show a smaller absorption enhancement than the measurements (see Fig. 6). One possible explanation for this deviation is scattering. The produced rear side
structure is not perfectly regular but has some dislocations and irregularities. These lead to
scattering. Scattering also causes a light path length enhancement [21]. A wavelength-independent light path length enhancement can be simulated
by an increased wafer thickness. The relative weight of scattering and diffraction depends on
the regularity of the sphere layer and is difficult to deduce analytically from a given
structure. The factor by which the wafer thickness is enhanced due to scattering and the
relative weight was determined by comparison to the measurement. For our structure, a cell
thickness enhancement by a factor of four was found for both sample thicknesses, when it is
assumed that half the number of photons at each wavelength is subjected to diffraction and the
other half is subjected to scattering. The light path length enhancement due to diffraction has
its maximum at higher wavelengths than the enhancement corresponding to scattering. The
absorption enhancement observed in our measurement covers the complete spectral range that is
affected by both the enhancements due to diffraction and scattering.

Our simulation results presented in Fig. 6 (left side) show that for the 250 µm thick wafer, the measured absorption enhancement can be modeled by a combination of scattering and diffraction. This result could be reproduced with the same approach and the same parameters for the samples with a thickness of 100 µm as can be seen in Fig. 6 (right side). This demonstrates that the used model that includes both, scattering and diffraction, is a good description for the real structure.

## 4. Summary

In summary, a hexagonal sphere grating for the rear side of crystalline silicon solar cells has been optimized by wave optical simulations and realized by spin coating and atomic layer deposition. Simulation and experiment show that such structures enhance the absorption in the near infrared within the silicon, and that the observed effect is very likely caused by a combination of diffraction and scattering. Simulations showed possible photocurrent density gains of up to 3 mA/cm^{2} for planar wafers with a thickness of 40 µm corresponding to a 9% increase. Measured absorption enhancements can be converted to a possible photocurrent density gain of 1.49 mA/cm^{2} and 1.04 mA/cm^{2} for thicknesses of 100 µm and 250 µm, respectively, corresponding to 5.8% and 3.9% increase.

## Acknowledgment

The research leading to these results has received funding from the German Federal Ministry of Education and Research in the project “InfraVolt” (project number 03SF0401B) and from the German Federal Ministry for the Environment, Nature Conservation and Nuclear Safety under contract number 0325292 “ForTeS”. The authors also thank Beneq for atomic layer depositions. Jan Christoph Goldschmidt gratefully acknowledges the scholarship support from the German Academic Exchange Service (DAAD). Johannes Eisenlohr gratefully acknowledges the scholarship support from the Deutsche Bundesstiftung Umwelt DBU.

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