Light trapping using diffraction gratings is a promising approach to increasing absorption in solar  cells. In this paper, the computationally calculated absorption enhancement expected from a diffraction grating  consisting of a triangular array of cylindrical wells is presented. Angle-extended polychromatic illumination is  considered, and special attention is paid to absorption of sub-bandgap photons in an intermediate band solar cell.  Results are compared to the absorption enhancement expected from an ideal Lambertian (randomizing) scatterer,  which is considered as a baseline. It is found that for cells which absorb very weakly, the diffraction grating  provides absorption enhancement above that of the ideal Lambertian scatterer over a wide wavelength range. For  cells which absorb more strongly, the grating underperforms the ideal Lambertian scatterer over almost all  wavelengths. Finally, the grating period, well height and well radius are optimised.  Keywords: Light Trapping, Diffraction Grating, Intermediate Band Solar Cel

Luque López, Antonio

Martí Vega, Antonio

Mellor Null, Alexander Virgil

Tobías Galicia, Ignacio

English

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Light trapping properties of cylindrical well diffraction gratings in solar cells: Computational calculations

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LIGHT TRAPPING PROPERTIES OF CYLINDRICAL WELL DIFFRACTION GRATINGS IN SOLAR 
CELLS: COMPUTATIONAL CALCULATIONS 
 
 
Mellor A. Tobías I. Martí A. Luque A. 
Instituto de Energía Solar, Universidad Politécnica de Madrid, Madrid 28040, Spain 
 
 
ABSTRACT: Light trapping using diffraction gratings is a promising approach to increasing absorption in solar 
cells. In this paper, the computationally calculated absorption enhancement expected from a diffraction grating 
consisting of a triangular array of cylindrical wells is presented. Angle-extended polychromatic illumination is 
considered, and special attention is paid to absorption of sub-bandgap photons in an intermediate band solar cell. 
Results are compared to the absorption enhancement expected from an ideal Lambertian (randomizing) scatterer, 
which is considered as a baseline. It is found that for cells which absorb very weakly, the diffraction grating 
provides absorption enhancement above that of the ideal Lambertian scatterer over a wide wavelength range. For 
cells which absorb more strongly, the grating underperforms the ideal Lambertian scatterer over almost all 
wavelengths. Finally, the grating period, well height and well radius are optimised. 
Keywords: Light Trapping, Diffraction Grating, Intermediate Band Solar Cell 
 
 
1 INTRODUCTION 
 
This paper presents computational calculations of the 
absorption enhancement that can be expected from a 
proposed diffraction grating when incorporated into a 
solar cell. 
The use of optical structures to increase the path 
length of light inside a solar cell, known as light trapping, 
can increase absorption and hence efficiency. Amongst 
other methods, this can be achieved by placing a 
diffraction grating on one of the cell surfaces, which 
diffracts the incident light into oblique directions with 
long optical path lengths, some of which are internally 
confined[1]. This technique has normally been applied to 
thin film silicon devices, which absorb band-edge 
photons very weakly[2]. It has been shown that a near-
field analysis is necessary to accurately describe 
absorption enhancement in such a system[3]. 
The advent of the Intermediate Band Solar Cell 
(IBSC) provides renewed motivation for the development 
of light trapping structures. The IBSC is capable of 
generating photocurrent from sub-bandgap photons due 
to the existence of a metallic intermediate band (IB) 
between the valence band and the conduction band 
(CB)[4]. A prominent means of creating an IB is to use 
the confined ground states in quantum dots[5]. So called 
Quantum Dot IBSCs (QD-IBSCs) have been shown to 
display IB behaviour, but suffer from extremely weak 
absorption of sub-bandgap photons, particularly those 
relating to the IB – CB transition. Effective light trapping 
is therefore required. QD-IBSCs are fabricated on wafer 
substrates whose thickness is on the order of a few 
hundred microns. This presents the possibility of 
employing a diffractive scheme in which the active layer 
is in the far-field of the diffraction grating, facilitating a 
simpler analysis. 
In this paper the computationally calculated 
absorption enhancements expected from diffraction 
gratings consisting of a triangular array of cylindrical 
wells are presented. Analysis is made for angle-extended 
(conical) polychromatic illumination. Particular attention 
is paid to the wavelength range corresponding to the IB – 
CB transition in an IBSC, which is taken to be 1.19μm ≤ 
λ0 ≤  3.35μm. The grating period and well radius and 
depth are optimised. 
2 SIMULATION TECHNIQUE 
 
The simulated cell structure is shown in Figure 1. 
The diffraction grating is located at the front face of the 
solar cell. An assumed perfect reflector is located at the 
rear surface. The absorbing ‘active’ region of the cell is 
modelled as a uniform layer of width w and absorption 
coefficient α, which is wavelength independent. The 
simulated grating profile consists of a triangular array of 
cylindrical wells that are etched directly into the front 
face of the GaAs solar cell substrate. The grating profile 
is shown in Figure 2. Incident light is diffracted into a 
discrete set of diffracted orders; the transmitted orders 
traverse the cell, are reflected at the rear surface and are 
re-incident on the grating (Figure 1). It has been shown 
that these orders are re-diffracted into a set of orders 
whose trajectories are identical to the originally 
diffracted set[6], which in turn repeat the process. The 
brightness of each diffracted order in the steady state is 
then the solution to a matrix equation, where the matrix 
elements are the complex scattering efficiencies of the 
diffraction grating. The scattering matrix elements are 
calculated using commercial software package Gd-Calc®: 
an electromagnetic simulation program based on rigorous 
coupled-wave (RCW) theory[7]. The cell is assumed to 
be illuminated by a cone of light of half angle 0.5°. 
Simulations are performed for a sample of incidence 
directions within the illumination cone. This produces a 
full description of the angular dependence of the internal 
brightness, from which the absorption is calculated. A 
rigorous description of this formalism is described in[8]. 
It is implicitly assumed that different diffracted orders 
lose coherence between diffraction from and incidence 
on the grating. Hence this analysis is only valid for cells 
which are thick compared to the wavelength of the light. 
 
25th European Photovoltaic Solar Energy Conference and Exhibition /
5th World Conference on Photovoltaic Energy Conversion, 6-10 September 2010, Valencia, Spain
647
 
 
Figure 1. The simulated cell structure. A diffraction 
grating is located at the front surface and a perfect 
reflector located at the rear. Red and blue arrows 
represent light incident on and diffracted from the grating 
respectively. 
 
 
 
Figure 2. The simulated grating profile. The grating 
consists of a triangular array (period d) of cylindrical 
wells of radius r and height h. 
 
The RCW method is applied to arbitrary grating 
profiles using a staircase approximation: the grating is 
described by a finite stack of lamellar strata. The 
scattering matrix elements are computed using a 
truncated Fourier series expansion. The number of strata 
in the grating profile description and the number of 
orders retained in the truncation of the Fourier series 
have been chosen through two-parameter convergence 
testing. 
 
 
3 RESULTS 
 
The fraction of incident power absorbed (absorption) 
has been calculated as a function of vacuum wavelength 
λ0 normalised to units of the grating period d. Results are 
shown in Figure 3 for cylindrical wells of radius r=0.35d 
and height h=0.2d. Results for cells with active layer 
absorbances of αw=1x10-3, αw=0.01 and αw=0.1 are 
displayed as solid curves in red, blue and green 
respectively. The dashed lines show the absorptions for 
the same cells with the grating replaced by an ideal 
Lambertian (randomizing) scatterer, calculated using the 
formula derived in [9]. Lambertian scattering is a 
conventional method of light trapping in solar cells, and 
is considered as a baseline. 
The red low-absorbance curve is characterised by 
well defined peaks. These occur at wavelengths at which 
certain diffraction orders are at very oblique angles 
leading to high optical path lengths within the cell and 
hence high absorption. Oblique orders dominate 
behaviour for cells with low absorbances because only 
light rays which travel long distances within the cell are 
appreciably absorbed. 
 
 
 
Figure 3. Fraction of incident power absorbed as a 
function of vacuum wavelength λ0, normalised to units of 
the grating period d, for the proposed structure with 
r=0.35d and h=0.3d. Results for cells with active layer 
absorbances of αw=1x10-3, αw=0.01 and αw=0.1 are 
displayed as solid curves in red, blue and green 
respectively. Dashed lines show Lambertian baseline for 
each absorbance. 
 
On increasing the cell absorbance (blue and green 
curves), other factors such as in-coupling of the incident 
light become more important causing peaks to become 
less defined. This is accompanied with a relative 
decrease in absorption compared to the Lambertian 
scatterer; for absorbances of αw=1x10-3, and αw=0.01, 
absorptions higher than the Lambertian baseline are 
predicted over a wide wavelength range, whereas for 
αw=0.1 there is only marginal improvement over a 
narrow wavelength range. 
For each absorbance, the drop in absorption for 
wavelengths below λ0/d≈0.85  results from the 
introduction of multiple diffracted orders on the side of 
the grating that is external to the solar cell, which leads 
to greater out-coupling of confined light. 
The grating period is a design parameter and can be 
chosen so as to maximise the absorption over the 
wavelength range of interest. In the case of r=0.35d, 
h=0.3d and αw=0.1, the optimum period is found to be 
d=2.25μm. The average absorption over the wavelength 
range of 1.19μm ≤ λ0 ≤ 3.35μm is then A=0.68. The grey 
area in Figure 3 shows the section of the curve which has 
been integrated to yield this result. 
Performing the above procedure for a range of well 
radii and heights, these parameters have been optimized. 
Figure 4 shows a contour plot of the average absorption 
over the above mentioned wavelength range as a function 
of well radius and well height, for a cell with an active 
layer absorbance of αw=0.1. For each combination, d has 
been chosen to maximise the average absorption. There 
is a clear maximum at r=0.35d, h=0.3d. The optimum 
radius, height, period and the resulting mean absorption 
for the IB-CB range are therefore those presented in the 
previous paragraph. 
25th European Photovoltaic Solar Energy Conference and Exhibition /
5th World Conference on Photovoltaic Energy Conversion, 6-10 September 2010, Valencia, Spain
648
 
 
Figure 4. Average absorption over the wavelength range 
corresponding to the IB-CB transition in an IBSC as a 
function of well height and radius, both normalised to 
units of the grating period d. Both the vertical axis and 
colour-scale show the average absorption. 
 
The optimisation procedure has been performed for 
absorbances of αw=1x10-3, αw=0.01 and αw=0.1. Table I 
shows the optimised parameters and resulting average 
absorption enhancement over the IB-CB wavelength 
range relative to the Lambertian baseline. The optimised 
diffraction gratings perform better than the ideal 
Lambertian scatterer for very weakly absorbing cells but 
worse for cells which absorb light more strongly. 
 
 
4 CONCLUSIONS 
 
The light trapping properties of diffraction gratings 
consisting of a triangular array of cylindrical wells have 
been simulated computationally, and the absorption 
calculated as a function of wavelength for different cell 
absorbances. For cells with absorbances of αw=1x10-3 
and αw=0.01, absorptions higher than the Lambertian 
baseline are predicted over a wide wavelength range. For 
cells with higher absorbances, the absorption achieved by 
the diffraction gratings is shown to be lower than for the 
ideal Lambertian scatterer over almost all wavelengths, 
with only a marginal improvement made over a narrow 
wavelength range. 
The grating period, well radius and well height have 
been optimised, seeking to maximise the average 
absorption over the wavelength range 1.19μm ≤  λ0 ≤ 
3.35μm, which corresponds to the IB-CB transition in an 
IBSC. The optimum values are d=2.25μm, r=0.79μm and 
h=0.68μm respectively for all studied cell absorbances. 
The average absorption for these optimised parameters is 
higher than the Lambertian baseline for αw=1x10-3 and 
αw=0.01, but lower for αw=0.1. 
 
 
ACKNOWLEDGEMENTS 
 
This work is supported by the European Commission 
project IBPOWER  (Contract 211640), the NUMANCIA 
(S2009/ENE/1477)  project funded by the Comunidad de 
Madrid, and the Nanogeffes project, which is part of  the 
Spanish National Program. Alexander Mellor gratefully 
acknowledges the Comunidad de Madrid for financial 
support through the scholarship Personal Investigador de 
Apoyo. 
The simulations used in this work employ 
commercial software package Gd-Calc® as a subroutine 
to calculate grating efficiencies. 
 
 
REFERENCES 
 
[1] Heine C and Morf RH. Submicrometer Gratings for 
Solar-Energy Applications. Applied Optics 1995; 34 
(14): 2476-2482 
[2] Green MA. Silicon Solar Cells: Advanced Principles 
and Practice. Centre for Photovoltaic Devices and 
Systems: Sydney, 1995; pp.32-39. 
[3] Stiebig H, Senoussaoui N, Zahren C, Haase C and 
Muller J. Silicon thin-film solar cells with 
rectangular-shaped grating couplers. Progress in 
Photovoltaics 2006; 14 (1): 13-24. 
[4] Luque A and Marti A. Increasing the efficiency of 
ideal solar cells by photon induced transitions at 
intermediate levels. Physical Review Letters 1997; 78 
(26): 5014-5017. 
[5] Marti A, Cuadra L and Luque A. Quantum dot 
intermediate band solar cell. Conference Record of 
the Twenty-Eighth Ieee Photovoltaic Specialists 
Conference - 2000 2000: 940-943. 
[6] Tobias I, Luque A and Marti A. Light intensity 
enhancement by diffracting structures in solar cells. 
Journal of Applied Physics 2008; 104 (3). 
[7] Neviere M and Popov E. Light Propagation in 
Periodic Media: Differential Theory and Design 
(Optical Engineering). CRC: 2002. 
[8] Mellor A, Tobías I, Martí A, Mendes MJ and Luque 
A. Upper Limits to Absorption Enhancement in Solar 
Cells Using Diffraction Gratings. Submitted to 
Progress in Photovoltaics 2010. 
[9] Green MA. Lambertian light trapping in textured 
solar cells and light-emitting diodes: Analytical 
solutions. Progress in Photovoltaics 2002; 10 (4): 
235-241. 
 
 
Cell absorbance 
 
Grating period 
(μm) 
Well radius 
(μm) 
Well height 
(μm) 
Absorption 
 
Absorption enhancement relative 
to Lambertian baseline 
0.001 2.25 0.79 0.68 0.06 +47% 
0.01 2.25 0.79 0.68 0.33 +9% 
0.1 2.25 0.79 0.68 0.68 -17% 
 
Table I. Optimised grating parameters and resulting absorption and absorption enhancement relative to Lambertian 
baseline 
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5th World Conference on Photovoltaic Energy Conversion, 6-10 September 2010, Valencia, Spain
649


Light trapping properties of cylindrical well diffraction gratings in solar cells: Computational calculations

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