Resonant dielectric structures are a promising platform for addressing the key challenge of light trapping in thin-film solar cells. We experimentally and theoretically demonstrate efficiency enhancements in solar cells from dielectric nanosphere arrays. Two distinct amorphous silicon photovoltaic architectures were improved using this versatile light-trapping platform. In one structure, the colloidal monolayer couples light into the absorber in the near-field acting as a photonic crystal light-trapping element. In the other, it acts in the far-field as a graded index antireflection coating to further improve a cell which already included a state-of-the-art random light-trapping texture to achieve a conversion efficiency over 11%. For the near-field flat cell architecture, we directly fabricated the colloidal monolayer on the device through Langmuir–Blodgett deposition in a scalable process that does not degrade the active material. In addition, we present a novel transfer printing method, which utilizes chemical crosslinking of an optically thin adhesion layer to tether sphere arrays to the device surface. The minimally invasive processing conditions of this transfer method enable the application to a wide range of solar cells and other optoelectronic devices.



False-color SEM image of an amorphous silicon solar cell with resonant spheres on top

Atwater, Harry A.

Ballif, Christophe

Battaglia, Corsin

Bukowsky, Colton R.

Callahan, Dennis M.

Deceglie, Michael G.

Grandidier, Jonathan

Grubbs, Robert H.

Weitekamp, Raymond A.

Caltech Authors - Main

Phys. Status Solidi A (2012) – Supporting Information  1 
 
 
Supporting Information for: 
“Solar cell efficiency enhancement via light trapping in 
printable resonant dielectric nanosphere arrays” 
Jonathan Grandidier*,1, Raymond A. Weitekamp1,2, Michael G. Deceglie1, Dennis M. Callahan1, Corsin 
Battaglia3, Colton R. Bukowsky1, Christophe Ballif3, Robert H. Grubbs2, and Harry A. Atwater1 
1 Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA 
2 Arnold and Mabel Beckman Laboratories of Chemical Synthesis, Division of Chemistry and Chemical Engineering, California Institute 
of Technology, Pasadena, CA 91125, USA 
3 Ecole Polytechnique Fédérale de Lausanne (EPFL), Institute of Microengineering (IMT), Photovoltaics and Thin Film Electronics 
Laboratory, 2000 Neuchâtel, Switzerland 
Received 5 October 2012, revised 24 October 2012, accepted 24 October 2012 
Published online November 2012 
Keywords resonant dielectric structures, solar cells, nanospheres, photovoltaics, photonic crystal, amorphous silicon  
* Corresponding author: e-mail jgrandid@caltech.edu, Phone: +1 (626) 395-2193, Fax: +1 (626) 844-9320 
 
1 Silica nanosphere functionalization 
Silica spheres of 700 nm diameter were obtained from Polysciences Inc. as a 10% (by weight) suspension in water.  
This suspension was filtered on a fine filtration frit, rinsed with tetrahydrofuran and acetone. The powder of spheres was 
washed with 10 mL of 1:1 methanol/HCl, and rinsed again with acetone.  The mostly dried powder was then heated in an 
oven for 5 minutes at 110 °C and dried under vacuum overnight.  To 25 mL toluene in a 50 mL round-bottomed flask, 786 
mg of dry silica spheres were suspended and stirred.  To this suspension was added 1 mL 3-aminopropyl(diethoxy)methyl 
silane.  The suspension was stirred 72 hours, filtered on a fine frit, rinsed with toluene and dried in vacuo to yield 756 mg 
dry, amine-functionalized silica spheres. 
 
2 Langmuir-Blodgett deposition 
A ~1% (by weight) suspension for Langmuir-Blodgett deposition was prepared by suspending 235 mg of 
functionalized silica spheres in a solution of 4 mL ethanol and 17 mL methylene chloride. We first perform an isotherm 
measurement where we record the surface pressure of the water as a function of the surface area, which is reduced using the 
compression barriers of the LB trough. When the area of the trough is large, the surface pressure of the water is around 4 
mN/m. The spheres are freely spread on the surface of the water. This is the so-called “gaseous” state. While the LB trough’s 
barriers compress the spheres and reduce the area where the spheres stand on, the surface pressure slowly increases until 5 
mN/m. The slope abruptly increases until 10 mN/m. This is the “liquid” state corresponding to a dense and condensed 
monolayer of hexagonally close packed spheres at the surface of the water. Upon further compression, the slope of the curve 
decreases and the monolayer collapses into multilayer structures. For our purpose, the optimal point is at the middle of the 
“liquid” condensed state where the spheres are well close packed and still form a monolayer. This point is reached when the 
surface pressure is around 7.5 mN/m. In a second step, knowing the optimal surface pressure for the deposition, we perform 
a dipping experiment. While the spheres are on the surface of the water in the “gaseous” state, we immerse the substrate into 
the LB trough. We then close the LB’s barriers until the surface pressure reaches 7.5 mN/m. From that point, we slowly pull 
up the substrate at a rate of 1 mm/min while simultaneously keeping the surface pressure constant with a computer 
controlled feedback system between the electrobalance measuring of the surface pressure and the barrier moving 
mechanism. Consequently, the floating hexagonally close packed monolayer is adsorbed on the ITO surface. When the 
structure is totally removed from the water, the part that was initially immersed in the water is coated by a large area of 
nanoscale dielectric nanospheres on its entire surface. 
 
3 Transfer printing preparation 
Poly(vinyl alcohol) (avg. MW = 10,000 g/mol, 88% hydrolyzed, Sigma Aldrich) was spin cast from an aqueous 
solution containing 1 wt % PVA and 5 wt % gluteraldehyde onto the top glass surface of the cells with a thickness of 15-20 
Phys. Status Solidi A (2012) – Supporting Information  2 
 
 
nm, as measured by ellipsometry. Poly(dimethylsiloxane) stamps were prepared from Sylgard 184 (1:10 curing agent : 
elastomer base, Dow Corning), poured into petri dishes to a thickness of ~5 mm and baked for 85 minutes at 80 °C. A 2D 
colloidal crystal of 700 nm diameter, aminated silica spheres was deposited on a glass slide, as described above. Spheres 
were transferred to the PDMS by firmly pressing the stamp into the colloidal array and carefully peeling it away. The stamp, 
now “inked” with spheres, was pressed against the PVA-coated surface by hand. The cells were placed in a large glass jar, 
which was purged with argon, covered with a large crystallization dish, and baked for 2 hours at 100 °C. The atmosphere 
was again purged with argon after the first hour. The jar containing the cells was removed and allowed to cool to room 
temperature. The PDMS stamps were carefully peeled away to render high fidelity colloidal crystals adhered to the PVA-
coated glass surface, with good transfer yield. 
4 Effective index model  
We plot in Fig. S1 the simulation of EQE for the flat solar cell represented in Fig. 1 with a layer of 1.28 effective 
index corresponding to the equivalent occupied volume of the spheres and of the same thickness as the diameter of the 
spheres. Enhancement occurs in the blue part of the spectrum due to ARC effect. In the red part to the spectrum, EQE is 
equivalent to the solar cell without the additional layer. This confirms that the enhancement in the red part of the spectrum is 
due to resonant modes of the dielectric spheres as described in Fig. 3. 
 
Figure S1 Effective index corresponding to the index of the spheres accounting for air and comparison with the measured data with 
and without the spheres. 
 
5 Optical measurement on bare crystalline silicon 
We also considered optically thick bare crystalline silicon (c-Si) with spheres on top (Fig. S2). This simple structure 
can be easily understood since distinct features due to resonant dielectric structures can be observed. We accounted for the 
size distribution of the spheres as described in the manuscript.  
 
Figure S2 Schematic of the sphere array PV structure. 
Optical characterization is used to measure the influence of the dielectric sphere monolayer array on the absorption 
in the photovoltaic structure. We performed reflection measurements as a function of wavelength and incidence angle using 
a motorized integrating sphere apparatus. A supercontinuum laser (Fianium) was coupled to a monochromator to provide a 
collimated illumination beam for a wavelength range between λ=400 nm and λ=1000 nm. The transmitted light through the 
c-Si layer can be neglected within that range, due to its thickness. As a reference, a reflection measurement at normal 
incidence was compared with a FDTD simulation of a bare semi-infinite c-Si layer [1]. From the reflection, we calculated 
the absorption (A = 1 - R). This is a valid approximation on an integrating sphere, which effectively captures all angles of 
Phys. Status Solidi A (2012) – Supporting Information  3 
 
 
reflected light. We then measured the reflectance spectrum of the wafer with a sphere array on top, and simulated 21 FDTD 
cases for sphere diameters between 615 nm and 715 nm. Each simulation assumes an infinite array of spheres with periodic 
boundary conditions. In order to account for the experimentally measured distribution of the sphere size, we weighted the set 
of simulations by the Gaussian distribution.  This weighted average fits the measured absorption well, and its median 
diameter of 665 nm. The resonant features of the experimental spectrum and the weighted average spectrum are very well 
matched (Fig. S3b). We attribute the difference in the measured intensity to the fact that in the experiment, sizes of spheres 
are randomly distributed on a single sample whereas we combined a weighted set of simulations, each of a single sphere 
size. 
In order to analyze each of the features in the measured reflection curve, we also plot the result of the simulation of 
the median diameter of 665 nm in Fig. S3b. Its resonant response is similar to that of the simulated average and measured 
curves. Three sharp and distinct peaks are observed and labeled c, d, and e. For each of these peaks, we plot the electric field 
intensity for a cross section at the middle of a sphere in the same plane as the polarization of the normal incident plane wave. 
We verified experimentally and theoretically [2] that the absorbed power is independent of the polarization at normal 
incidence. The features in the absorption spectra are clearly associated with the excitation of resonant modes of the 
nanospheres [3]. Directly under the modal profile of the spherical cross section, we plot the electric field intensity in the first 
300 nm of c-Si both with and without the nanosphere array, at the corresponding wavelengths. It is clearly seen that the 
electric field intensity is greater in the c-Si layer with the resonant dielectric structures on top. This result demonstrates that 
the excitation of resonant modes within a dielectric structure can have a significant impact on the absorption enhancement of 
a photovoltaic structure. In our measurement, the total relative absorption enhancement measured is about 9%. 
 
Figure S3 (a) Map of the absorption as a function of the wavelength for the different sphere diameters across the measured size 
distribution. The Gaussian distribution of sphere diameters, used to weight each individual simulations, is plotted on top, with 
arbitrary amplitude. (b) Simulated and measured absorption for the photovoltaic structure with dielectric spheres on top of it. (c,d,e) 
For each labeled peak on b: electric field intensity for a cross section at the middle of a sphere. Below on a different color scale is the 
electric field intensity for a cross section within the first 300 nm of absorbing layer and below that, the equivalent case without 
spheres. 
 
Phys. Status Solidi A (2012) – Supporting Information  4 
 
 
 
Figure S4 Angle resolved measurement of the absorption for a (a) transverse electric and (b) transverse magnetic polarization. (c) 
Jsc generated by the active material for the case with and without spheres. TE and TM polarization were averaged to account for the 
isotropic radiation of the sun. The solar irradiance that was used to weight the measured absorption [4]. (d) Relative absorption 
enhancement over the bare structure as a function of the incident angle. 
We then measured the angle dependence of the absorbed light in both transverse electric (TE) and transverse 
magnetic (TM) polarizations (Fig. S4a and b). The observed absorption bands account for the variation of the resonant 
dielectric structure as a function of the angle of incidence. In order to evaluate the performance of the considered 
photovoltaic structure, we interpolated angle-resolved data [4] of the incident solar energy on a sunny day to match 
resolution of our absorption measurements represented in Fig. S4a and b. The TE and TM measurements are averaged and 
weighted by the interpolated data of the sun’s energy. We then calculated the relative absorption enhancement due to the 
colloidal crystal (Fig. S4d). The enhancement is relatively constant with an average of almost 10%. Interestingly, the 
strongest enhancement of 11.21% is measured for an angle of 55°. The angle-independent enhancement is a promising 
result; it demonstrates that these resonant dielectric structures can couple direct and indirect sunlight throughout the day. 
 
6 Graded-index model: transfer matrix simulations 
To support our proposed light-trapping mechanism via PC modes, we performed 1D transfer matrix simulations [5] 
to account for the graded index effects of the colloidal geometry. Because these 1D simulations cannot account for 3D 
modes, it provides a useful complement to our 3D FDTD simulations. For the transfer matrix simulations, the spheres were 
divided into 100 layers, each with an effective index corresponding to the volume fraction of glass in air. Two simulations 
(Fig. S5) demonstrate the calculated effects of these dielectric structures in the near- and far-field cases. 
Figure S5a shows the simulated absorption of an array of silica spheres on a c-Si wafer (Fig. S2) using each 
method, both averaged over 21 simulations as previously described, and for the mean sphere diameter. While the averaged 
simulations qualitatively share features, the absorption peak near 750 nm cannot be accounted for using the transfer matrix 
method. This peak corresponds to the strongest PC mode resonance, which has been aligned with the amorphous silicon 
absorption profile by design. This provides further evidence that the observed absorption enhancement is due to the 
proposed PC mode coupling, and not simply a graded index effect due to the sphere geometry. The FDTD simulations were 
supported by experiment (see the previous section). 
 
 
 
Phys. Status Solidi A (2012) – Supporting Information  5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure S5 (a) 3D FDTD simulations are compared to 1D transfer matrix simulations for a graded index array approximating a silica 
sphere with 665 nm diameter. The main resonance is not observed in the transfer matrix simulations, as expected. (b) A graded index array 
approximating a 700 nm diameter sphere is simulated, with various thicknesses of PVA surrounding the base of the sphere. 
To simulate the graded-index effect of the spheres for the ZnO/a-Si:H/ ZnO cell structure, where the spheres are 
separated from the active layer by 0.5 mm of SiO2, we simulated the modified transmission of light at the air/glass interface. 
The simulations reveal that the graded index profile of the silica sphere and PVA surrounding the base provide an 
antireflective coating for the air/glass interface. The PVA has an index very close to that of glass, and was taken as equal to 
glass (n = 1.466) for these simulations. The thickness of the PVA layer was nominally ~15 nm before the spheres were 
transfer printed onto the surface. Because the PVA is simultaneously being heated above its glass transition temperature and 
cross-linked with gluteraldehyde, the thickness of the PVA after the spheres are embedded is difficult to determine. If the 
density of the PVA remains unchanged, and the sphere is fully embedded in the layer, the corresponding height of the PVA 
layer would be 67 nm due to the excluded volume of the sphere. Thus, we take 15 and 67 nm as the upper and lower bounds 
for PVA thickness. We can see in the figure that transmission into the glass is improved at the wavelengths where we 
experimentally observed improvement in the EQE (Fig. 4c), between ~400-500 nm and ~600-700nm. 
 
 
7 Angle measurements on a flat solar cell 
We show in Fig. S6 the experimental measurements of the EQE. We also plot the simulated EQE using RCWA 
angle dependent characterization. For the simulations, we considered for both TE and TM polarizations, and kx and ky 
directions. These measurements and simulations were used as an input for Fig. 4a and b of the main manuscript. 
 
 
 
 
 
 
 
a 
b 
Phys. Status Solidi A (2012) – Supporting Information  6 
 
 
 
 
 
EXPERIMENT 
 
 
SIMULATION 
 
Figure S6 Angle resolved measurement of the EQE for a (a) transverse electric and (b) transverse magnetic polarization for a flat solar 
cell and for a solar cell with spheres on top (c,d). Angle resolved simulation using RCWA of the EQE for a (e) transverse electric and (f) 
transverse magnetic polarization for a flat solar cell and for a solar cell with spheres on top considering kx (g,h) and ky (i,j) directions. 
 
 
e f
g h
i j
Phys. Status Solidi A (2012) – Supporting Information  7 
 
 
 
8 Optical parameters 
As an input for the 3D FDTD and RCWA simulations, optical constants of all materials were measured by 
spectroscopic ellipsometry, except aluminum, for which they were taken from Palik [1]. The method employed here for 
calculating EQE from optical simulations assumes each absorbed photon creates a mobile electron hole pair, which is 
subsequently collected with unity efficiency. Since these assumptions may not hold for an experimental a-Si:H device, a 
wavelength independent collection efficiency of 0.8 was assumed to fit the experimental data (Fig. 2a and 3b). The 
thicknesses were adjusted slightly for each layer to reproduce the EQE of the flat cell without spheres. 
9 Electrical measurement 
A supercontinuum laser (Fianium) was coupled to a monochromator to provide collimated illumination between 
350 and 840 nm. A beam splitter was employed to direct part of the beam to a reference photodiode, to account for real-time 
fluctuations in source intensity. To calibrate the ratio of light incident on the reference diode and sample, we also performed 
the EQE measurement with a NIST traceable calibrated photodiode with known EQE in the sample position. The laser beam 
width was about 1 mm, which is smaller than the area of the solar cells and the calibrated diode, enabling area-independent 
measurements. Moreover, on the length scale of nanospheres, the laser beam approximates a plane wave. 
 
10 Electrical parameters 
The electrical parameters for the device physics simulation (Fig. 2b) were taken from Schropp and Zeman [6], with 
the exceptions that all the a-Si:H layers have an assumed bandgap of 1.78 eV, electron affinity of 4 eV, a relative 
permittivity of 11.9, and the peak dangling bond concentration in the intrinsic region was set to 5 x 1017 cm-3. In addition, we 
added a distributed series resistance of 15 Ω cm2 to reproduce the slope of the current-voltage curve near open circuit. We 
also found it necessary to scale down the generation rate calculated by FDTD by a multiplicative factor of 0.9 to match the 
experimental IV curve, this increase from the factor of 0.8 used in the analysis of the optical simulations is justified by 
noting that the device physics simulation accounts for imperfect carrier collection. 
References 
[1] E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic Press, 1998). 
[2] J. Grandidier, M. G. Deceglie, D. M. Callahan, and H. A. Atwater, J. Photon. Energy 2, 024502 (2012). 
[3] J. Grandidier, D. M. Callahan, J. N. Munday, and H. A. Atwater, Adv. Mater. 23, 1272-1276 (2011). 
[4] B. Marion, B. Kroposki, K. Emery, J. del Cueto, D. Myers, and C. Osterwald, Validation of a photovoltaic module energy ratings 
procedure at NREL, Technical report NREL/TP-520-26909 (1999). 
[5] S. J. Orfanidis, Electromagnetic Waves and Antennas http://www.ece.rutgers.edu/~orfanidi/ewa/, May 2012. 
[6] R. E. I. Schropp and M. Zeman, Amorphous and Microcrystalline Silicon Solar Cells: Modeling, Materials, and Device Technology: 
(Kluwer Academic, Norwell MA, 1998). 
 


Solar cell efficiency enhancement via light trapping in printable resonant dielectric nanosphere arrays

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Solar cell efficiency enhancement via light trapping in printable resonant dielectric nanosphere arrays

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