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Optimizing two-level hierarchical particles for
thin-film solar cells
Shiwei Zhou,1,* Xiaodong Hunang,1 Qing Li,2 and Yi Min Xie1,*
1Centre for Innovative Structures and Materials, School of Civil, Environmental and Chemical Engineering, RMIT
University, GPO Box 2476, Melbourne 3001, Australia
2School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006,
Australia
*shiwei.zhou, mike.xie@rmit.edu.au
Abstract: For the thin-film solar cells embedded with nanostructures at
their rear dielectric layer, the shape and location of the nanostructures are
crucial for higher conversion efficiency. A novel two-level hierarchical
nanostructure (a sphere evenly covered with half truncated smaller spheres)
can facilitate stronger intensity and wider scattering angles due to the
coexistence of the merits of the nanospheres in two scales. We show in this
article that the evolutionary algorithm allows for obtaining the optimal
parameters of this two-scale nanostructure in terms of the maximization of
the short circuit current density. In comparison with the thin-film solar cells
with convex and flat metal back, whose parameters are optimized singly,
the short circuit current density is improved by 7.48% and 10.23%,
respectively. The exploration of such a two-level hierarchical nanostructure
within an optimization framework signifies a new domain of study and
allows to better identify the role of sophisticated shape in light trapping in
the absorbing film, which is believed to be the main reason for the
enhancement of short circuit current density.
©2013 Optical Society of America
OCIS codes: (350.6050) Solar energy; (310.6628) Subwavelength structures, nanostructures.
References and links
1. A. Shah, P. Torres, R. Tscharner, N. Wyrsch, and H. Keppner, “Photovoltaic technology: the case for thin-film
solar cells,” Science 285(5428), 692–698 (1999).
2. B. Rech and H. Wagner, “Potential of amorphous silicon for solar cells,” Appl. Phys., A Mater. Sci. Process.
69(2), 155–167 (1999).
3. P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and
A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt. 14(2), 024002 (2012).
4. P. Campbell and M. A. Green, “The limiting efficiency of silicon solar-cells under concentrated sunlight,” IEEE
Trans. Electron. Dev. 33(2), 234–239 (1986).
5. S. S. Lo, C. C. Chen, F. Garwe, and T. Pertch, “Broad-band anti-reflection coupler for a:Si thin-film solar cell,”
J. Phys. D Appl. Phys. 40(3), 754–758 (2007).
6. C. Eisele, C. E. Nebel, and M. Stutzmann, “Periodic light coupler gratings in amorphous thin film solar cells,” J.
Appl. Phys. 89(12), 7722–7726 (2001).
7. R. Dewan, M. Marinkovic, R. Noriega, S. Phadke, A. Salleo, and D. Knipp, “Light trapping in thin-film silicon
solar cells with submicron surface texture,” Opt. Express 17(25), 23058–23065 (2009).
8. S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells,” J. Appl.
Phys. 101(9), 093105 (2007).
9. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213
(2010).
10. V. E. Ferry, M. A. Verschuuren, H. B. T. Li, E. Verhagen, R. J. Walters, R. E. I. Schropp, H. A. Atwater, and A.
Polman, “Light trapping in ultrathin plasmonic solar cells,” Opt. Express 18(S2 Suppl 2), A237–A245 (2010).
11. S. H. Lim, W. Mar, P. Matheu, D. Derkacs, and E. T. Yu, “Photocurrent spectroscopy of optical absorption
enhancement via scattering from surface plasmon polaritons in gold nanoparticles,” J. Appl. Phys. 101, 104309
(2007).
12. F. J. Beck, A. Polman, and K. R. Catchpole, “Tunable light trapping for solar cells using localized surface
plasmons,” J. Appl. Phys. 105(11), 114310 (2009).
13. M. Gu, Z. Ouyang, B. H. Jia, N. Stokes, X. Chen, N. Fahim, X. P. Li, M. J. Ventura, and Z. R. Shi,
“Nanoplasmonics: a frontier of photovoltaic solar cells,” Nanophotonics 1(3-4), 235–248 (2012).
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Received 11 Sep 2012; revised 11 Jan 2013; accepted 18 Feb 2013; published 26 Feb 2013
(C) 2013 OSA
11 March 2013 / Vol. 21, No. S2 / OPTICS EXPRESS A285
14. K. R. Catchpole and A. Polman, “Design principles for particle plasmon enhanced solar cells,” Appl. Phys. Lett.
93(19), 191113 (2008).
15. X. Chen, B. H. Jia, J. K. Saha, B. Y. Cai, N. Stokes, Q. Qiao, Y. Q. Wang, Z. R. Shi, and M. Gu, “Broadband
enhancement in thin-film amorphous silicon solar cells enabled by nucleated silver nanoparticles,” Nano Lett.
12(5), 2187–2192 (2012).
16. M. P. Bendsøe and O. Sigmund, Topology Optimisation: Theory, Methods, and Applications (Springer, 2003).
17. O. Sigmund and S. Torquato, “Composites with extremal thermal expansion coefficients,” Appl. Phys. Lett.
69(21), 3203–3205 (1996).
18. Y. M. Xie and G. P. Steven, “A simple evolutionary procedure for structural optimization,” Comput. Struc.
49(5), 885–896 (1993).
19. S. W. Zhou and Q. Li, “Design of three-dimensional periodic metamaterials for electromagnetic properties,”
IEEE Trans. Microw. Theory Tech. 58, 910–916 (2010).
20. S. W. Zhou, W. Li, Y. H. Chen, G. Y. Sun, and Q. Li, “Topology optimization for negative permeability
metamaterials using level-set algorithm,” Acta Mater. 59(7), 2624–2636 (2011).
21. X. Huang, Y. M. Xie, B. Jia, Q. Li, and S. W. Zhou, “Evolutionary topology optimization of periodic composites
for extremal magnetic permeability and electrical permittivity,” Struct. Multidisc. Optim. 46(3), 385–398 (2012).
22. R. Storn and K. Price, “Differential Evolution – A simple and efficient heuristic for global optimization over
continuous spaces,” J. Glob. Optim. 11(4), 341–359 (1997).
23. A. H. Hielscher, A. D. Klose, and J. Beuthan, “Evolution strategies for optical tomographic characterization of
homogeneous media,” Opt. Express 7(13), 507–518 (2000).
24. M. Mitchell, An Introduction to Genetic Algorithms (MIT Press, 1996).
25. P. Y. Chen, C. H. Chen, H. Wang, J. H. Tsai, and W. X. Ni, “Synthesis design of artificial magnetic
metamaterials using a genetic algorithm,” Opt. Express 16(17), 12806–12818 (2008).
26. L. F. Shen, Z. Ye, and S. L. He, “Design of two-dimensional photonic crystals with large absolute band gaps
using a genetic algorithm,” Phys. Rev. B 68(3), 035109 (2003).
27. M. d’Avezac, J. W. Luo, T. Chanier, and A. Zunger, “Genetic-algorithm discovery of a direct-gap and optically
allowed superstructure from indirect-gap Si and Ge semiconductors,” Phys. Rev. Lett. 108(2), 027401 (2012).
28. N. Yamada and T. Nishikawa, “Evolutionary algorithm for optimization of nonimaging Fresnel lens geometry,”
Opt. Express 18(S2 Suppl 2), A126–A132 (2010).
29. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic
media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966).
30. http://www.lumerical.com.
31. M. Yang, Z. P. Fu, F. Lin, and X. Zhu, “Incident angle dependence of absorption enhancement in plasmonic
solar cells,” Opt. Express 19(S4 Suppl 4), A763–A771 (2011).
32. E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou, “Minimal discrete energy on the sphere,” Math. Res. Lett. 1,
647–662 (1994).
33. R. Dewan and D. Knipp, “Light trapping in thin-film silicon solar cells with integrated diffraction grating,” J.
Appl. Phys. 106(7), 074901 (2009).
1. Introduction
Even though the material consumption of conventional bulk crystalline silicon solar cells has
come down considerably over many years of development, it still accounts for up to 40% of
the total cost. Thus thin-film solar cells [1, 2], which reduce the thickness of active layer (the
absorbent semiconductor film) from hundreds of micrometers to several micrometers or even
to nanometers, become promising photovoltaic devices capable of generating electrical power
at competitive cost. Moreover, thin-film solar cells have additional merits with reduced bulk
recombination and photo-degradation as well as better carrier collection [3] and high open-
circuit voltage [4]. As a tradeoff, however, the energy conversion efficiency becomes lower
due to the light path in silicon film of absorption is shortened and the transmission losses
increase. Furthermore, the absorption of thin-film solar cells seems to be weaker in the
frequency range near the semiconductor band gap.
To overcome the abovementioned weaknesses, it needs to optically increase the light path
within the absorbent semiconductor film so that the physically-shortened active layer could be
compensated reasonably. Since conventional light trapping methods used for bulk solar cells
like anti-reflection coatings [5, 6] and grated structures (e.g. pyramid-like and triangular
texture) [7] are not applicable to thin-film solar cells, the localized surface plasmons (LSPs)
induced by the metallic structures have been employed as they offer an alternative mechanism
for trapping more light in the thin semiconductor layers [3, 8–10]. When metallic structures
are scaled into sub-wavelength of incident light, their interactions with the sunlight become
dramatically strong at visible range and thus produce high absorption in semiconductor even
though it is as thin as hundreds of nanometers [8, 10, 11]. The presence of metallic
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nanoparticles can result in unwanted surface recombination and therefore lower photocurrent
and open-circuit voltage. But the side-effects can be depressed as the recombination is
isolated by the dielectric layer between the nanoparticles and the semiconductor.
In the original study on the role of LSPs in solar cells, irregularly-shaped particles was
randomly distributed on the front surface and it is found that the absorption is significantly
enhanced with respect to the flat surface [8]. Nevertheless, the abovementioned enhancement
can be partially counteracted by the Fano resonance effect, i.e. the light is not absorbed but
reflected if the frequency is out of the resonance range [12]. A remedy to this unwanted effect
is building the nanostructures in the rear of semiconductor. It is observed that this kind of
nanostructure disposition keeps the advantages of LSPs for absorption enhancement with blue
and green lights while allows the long wavelength light (red) to be coupled into the Si film
[12]. Thus the absorption is expected to be enhanced within the sunlight spectrum
corresponding to the band gap absorption in a-Si. However, the rear-located nanostructures
could induce significant Ohmic loss around them and the absorption is enhanced only within
a narrow band and fluctuates near the resonant frequency. Such disadvantages can be partially
overcome by optimizing the particle size and shape so that the electric intensity is relatively
lower around the metallic particles and the absorption peak occurs at demanded frequency
region. However, as conventional light trapping techniques, the disadvantages cannot be
taken away completely in the existing structure and require further investigations [13].
The roles of LSP in the solar cells depend on the size, shape, location and coverage
density of the nanostructures. Therefore these geometric parameters should be systemically
investigated and whenever possible, optimally determined. For front-located irregular
nanoparticles, the studies showed that small nanoparticles result in maximal overall
enhancement while large ones help to improve light emission [8]. Subsequent work revealed
that the shape of particle is crucial to the plasmon-enhanced solar cells and the cylindrical and
hemispherical particles have better performance than spherical particles [14]. Recently, Chen
et al. [15] proposed a novel two-level hierarchical nanostructure that combines the virtues of
larger and smaller spheres into a single structure and therefore realizes enhancement of
broadband absorption. From previous studies [12, 15], the short circuit current density (Jsc)
and the conversion efficiency could be enhanced if the radius of large spheres and the
coverage density of the nanostructure are properly selected.
The determination of optimal parameters in the conventional work is largely based on a
trial-and-error exercise, in which interactions of different parameters are barely considered.
To a certain extent, such an approach could help accelerate the design process for a range of
solar cells, but attaining the optimized parameters using the trial-and-error approach could be
difficult as the number of permutations can be astronomic even for a small number of
parameters. For this reason, a more methodical optimization technique is needed for the
efficient design of solar cells. Nevertheless, little systematic study has been conducted in this
regard. Recently, great efforts have been directed towards the optimization of physical
properties by rearranging its microstructures using topology optimization techniques [16–21].
Thin-film solar cells could be considered as a class of new composites comprised by
periodically repeated Represented Volume Elements (RVE). For the devised nanostructures
in the present study, only the location and a few geometric parameters of a given shape are
considered as the design variables.
Differential evolutionary algorithm [22] offers a viable approach to size optimization for a
parameter-based and sophisticated cost function, in which the gradient information may be
difficult to be obtained. Unlike traditional gradient-based algorithms that can be susceptible to
being trapped at local minima, malfunction in cases of insensitive gradient, and slack-off if
the initial guess is far from the optimum [23], the evolutionary algorithm proves to be
versatile, powerful and efficient for optical design problems [23]. Compared with other
gradient-free algorithms like genetic algorithm [24] that is based upon natural selection and
survival-of-the-fittest, the evolutionary algorithm usually exhibits faster convergence as
shown by extensive numerical tests [23]. It is noted that the prevalence of using genetic
algorithms in the design of optical devices has been seen in the metamaterial design for
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negative refractive index [25], the photonic crystal with large absolute band gaps [26], and the
superstructure with different electronic and optical gaps [27]. However, the application of
more powerful algorithms, such as the differential evolutionary algorithm, to improve the
solar cells has not been reported even though its recent applications in solar devices, such as
nonimaging Fresnel lens [28], seem promising.
2. Statement of the problem
As shown in the schematic (Fig. 1(a)), a typical thin-film solar cell consists of four layers: a
80 nm SnO2:F coated glass which is used as a transparent conductive oxide on the front;
followed by a 350 nm hydrogenated amorphous silicon (a-Si:H) layer; subsequently a layer of
Al-doped ZnO (ZnO:Al) thin-film embedded with periodically-repeated Ag particles.
Noticeably, the LSP resonant frequency red-shifts when the surrounding medium of metallic
particle yields large refractive index. But it cannot be the design variable as we are concerned
with the size and shape of the nanostructures in this study. Finally, a 120 nm Ag layer is used
as the back reflector. It is noted that the back surface is not flat but distributed with bulging
dots (Figs. 1(b) and (c)), where the distance between the bottommost points of ZnO:Al and
Ag back layers maintains the same value (120 nm) as the planar part, otherwise it could lead
to electricity leakage. Such a design also allows to consistently compare the roles of
nanospheres in different sizes [15]. For the same reason, the thickness of ZnO:Al layer is not
given in Fig. 1(a) and it should be determined in terms of the size of nanostructure. The
location of larger and small spheres is illustrated by the cross section of this solar sell in Fig.
1(b) with an inset (red dashed region) to zoom in inner geometrical characterizations. Figure
1(c) gives a 3D view of this thin-film solar cell.
a-Si:H
Ag
SnO
2
:F
ZnO:Al
80 nm
350 nm
120 nm
(a)
?
r
1
d
1
d
2
p
(b)
(c)
Large Ag
particle
Small Ag
particle
Fig. 1. Schematic of the thin-film solar cell: (a) front view; (b) cross sectional view; (c) 3D
view.
Since the nanostructures are periodically-repeated and the incident light is normal to the
front surface, it is sufficient to have the computational region limited to the RVE with
periodic boundary conditions on its bilateral surfaces (Fig. 2(a)). The open boundaries on the
input and output sides are truncated by the perfectly matched boundary layers (PMLs), whose
distances to the top and bottom of solar cell are 200 nm and 100 nm, respectively. The
electromagnetic field within the RVE is governed by the well-known Maxwell’s equations
and solved by the finite difference time domain (FDTD) algorithm [29]. In the present study,
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we utilize the same FDTD software from Lumerical [30] as used by Chen et al. [15]. By
taking the advantage of the symmetry and anti-symmetry at x and y directions, the
computational region can be reduced to a quarter of the base cell without destroying its
periodicity. As a result, the computing time is reduced significantly. Once the Maxwell’s
system is solved, the short circuit current density, electric intensity, absorption as well as
other relevant factors can be computed to any pre-defined level of accuracy.
PML layer
Incident
light plane
Periodic
boundaries
PML layer
(a) (b)
Fig. 2. Schematic of a hierarchical silver particle in a thin-film solar cell: (a) the representative
volume element; (b) two-level nanoshperes.
Numerical experiments illustrate some parameters are unlikely to be influenced by others
(e.g. the thickness of Si Film invariantly converges to its upper bound), therefore we herein
only consider more dependent factors such as the location and geometrical parameters of the
abovementioned two-scale nanostructure (Fig. 2(b)). Because of the central points of small
spheres are evenly distributed on the large sphere, this novel structure exhibits the combined
virtues of a large sphere, stronger scattering intensity, and small spheres, wider scattering
angles. It is believed that the stronger intensity induced by large sphere contributes to the
improvement of the scattering/absorption cross-section ratio and wider scattering angles
owing to small spheres help the light take a prolonged path when it is scattered into the
absorbing Si layer. As a result, the conversion efficiency of the solar cell is improved
considerably [15]. Moreover, the combination of large and small spheres effectively prevents
significant Ohmic loss as well as generate a larger scattering rate because the Ohmic loss is
proportional to the volume of metal object (υ) whilst scattering rate is scaled with υ [2, 9].
The radius of the large sphere and its periodical distribution length (denoted as p in Fig.
1(b)) were studied singly by Chen et al. [15], and their optimal values were determined to be
100 nm and 792.67 nm (corresponding 10% optimal nanostructure coverage) as a result of
investigating their roles. In their pioneering study, the radius of small sphere was assumed to
be 1/5 of the large one and the location of the nanostructure in the ZnO:Al layer is fixed with
d1 = 20 nm and d2 = 80 nm (Fig. 1(b)). It should be pointed out that, however, these factors
interact with each other and create different effects on solar cells. Thus they should be
designed simultaneously so that their roles can be balanced. In addition to the size and shape
of nanoparticles, the incident angle of light is another important factor affecting the
performance of thin-film solar cell as the light polarization changes the LSP characterizations
(e.g. the shift of resonance frequency). Recent studies in [31] suggested that the optimal
incident angle should be confined to a well-defined range of 0° to 35° for a thin-film solar cell
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whose Ag nanospheres are located on the top of a-Si substrate directly. However, for the thin-
film solar cell considered herein that consists of nanostructures in the rear of semiconductor,
relevant research is deficient. Thus we assume the solar cell is placed exactly perpendicular to
the solar rays in this paper.
In the spherical coordinate, the azimuthal angle θk and polar angle φk of the kth small
sphere on the large sphere can be determined using the following simple formulae [32].
()( )
()
arccos 1 2 1 1 1 .
kkN kN
θ
=−+−− ≤≤
(1)
()( )
()
()
2
13.6 1 1 2 1 1 .
kk NkN
ϕϕ
−
=+ −−+− − (2)
where N is the number of evenly distributed small spheres. The radius of small sphere is
calculated as 21
1.44RRN=. The periodic length p, radius R1, location parameters d1 and
d2, and number of small spheres form a hyperspace for the optimization problem.
The optimization of the nanostructures is aimed at searching for the stationary point in
that hyperspace spanned by the design variables so that a figure of merit is achieved. Either a
larger short circuit current density or a higher open circuit voltage can enhance the conversion
efficiency. A larger Jsc commonly means a higher absorption and better light trap. In such a
configuration, we define Jsc as the cost function (figure of merit) to be maximized. If all light
absorbed into the a-Si:H layer is collected, Jsc is formulated as
( ) () ()
sc 1.5
/.
AM
J
ehcQEI d
λλλλ
= (3)
where e denotes the charge on an electron, h Plank’s constant, λ wavelength and c the speed
of light in vacuum. The quantum efficiency
() () ()
abs in
QE P P
λλλ
= is the ratio of the
power of the absorbed light
()
abs
P
λ
to that of the incident light
()
in
P
λ
within the Si film.
IAM1.5 stands for the relevant part of the solar spectral irradiance. In the simulation, 300
equally spaced frequency points are considered in the solar spectrum between λ = 300nm and
λ = 840nm.
The constraints on the design variables should be considered in the optimization to reflect
the real physical environment. Table 1 gives the lower and upper bounds for these parameters.
It is emphasized that it generally takes longer time for the evolution algorithm to find the
optimal values if the parameters are located on the bounds. Thus it is necessary to set wide
ranges for these parameters, otherwise feasible solutions would be hard to achieve.
Table 1. Specifications of two-scale nanostructure
Variable Name Symbol Lower bound Upper boun
d
The radius of large sphere (nm)
r
150 200
The periodicity of the RVE (nm)
p
300 1000
The distance from the top of nanosphere to Si film
(nm) d1 1 50
The distance from the bottom of nanosphere to Ag
layer (nm) d2 20 140
The number of small spheres
N
32 128
The evolution algorithm starts from a set of parent vectors, which are randomly selected
in the given parameter space. For each parent vector, a fitness (figure of merit) is obtained by
solving Maxwell’s equations. Following this, the offspring are produced by a certain mutation
rule that is defined as the summation of the weighted difference between a pair of parent
vectors and the third one. It is required that the offspring should be mended by a cross-over
process to improve their diversity. In this step the weighing factor and cross-over probability
are set to be 0.85 and 1, respectively, for following examples. The mutation and cross-over
processes are repeated until the best performance is obtained.
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3. Results and discussion
Two examples are aimed at demonstrating the capabilities of the proposed evolution
algorithm for the two-level hierarchical nanosphere design. We abbreviate Example 1 with a
bulging back (Fig. 1(b)) to ‘convex’, as opposed to ‘flat’ with planar back Example 2. In
terms of the maximization of Jsc, the best parameters (Table 2) are found within 55
generations. In comparison with the data obtained from the parameters reported in [15], it is
remarkable that Jsc has increased from 17.78 (mA/cm2) to 19.11 (mA/cm2), around 7.48%
enhancement. The equivalent coverage of the nanostructure in terms of r1 and p approximates
to 3.77%, much less than the previously assumed optimum 10% [9, 15]. Interestingly, the
distance between the top of nanostructure and Si film (d1 = 48.81 nm) is of a moderate value,
indicating a consequence of the compromise between the destructive interference effect and
the near-field coupling effect [14]. The former prefers a larger d1 to avoid the incidence being
offset by the reflected fields, whilst the later wants a smaller d1 to radiate a strong near-field
into the Si film.
Table 2. The parameters and the performance of the convex solar cells
r1
(nm)
p
(nm)
d1
(nm)
d2
(nm) N Jsc
(mA /cm2)
Singly-optimized 100 792.67 20 80 81 17.78
Optimized 50.08 646.58 48.81 135.95 95 19.11
A near-field optical picture is required to explain the enhancement of short circuit
currency density. Figure 3 shows a side-by-side comparison of the near-field electric intensity
(|E(x,y)|2 in the x-y plane and |E(x,z)|2 in the x-z plane) distribution of the separately-
optimized and optimized nanostructures at a time step where the maximal |E(x,y)|2 is found in
the x-y plane. It is emphasized that both planes should cover the center of the large sphere (if
there is one) for a realistic comparison. We can observe that the field intensity is evidently
boosted after the optimization both in vertical (Fig. 3(b)) and horizontal (Fig. 3(e)) directions.
It is noted the electric intensity in the absorbing Si film also increases considerably, but it is
not reflected in Fig. 3 unless plotted in logarithmic scale as its magnitude is relatively smaller
than that in the near-field around nanoparticles. The gradient direction (the white arrows in
Fig. 3(b), only the left part is arrowed because of the symmetry) of the electric field intensity
in vertical symmetric plane illustrates that the nanoparticles help scatter and diffract light at
wider angles before impinging upon the absorbing Si film.
Apart from easy fabrication, it is noted that changing the convex back to flat back for
solar cells can generate a larger short circuit current through the proposed optimization
process. Table 3 summarizes the parameters for four types of flat solar cells and their optical
performance. The first has no rear-located nanostructure and can be used as a reference for
quantifying the influence of the nanostructure on Jsc. The second consists of only large sphere
and its Jsc (18.38 mA/cm2) is a small fraction less than the third one (18.45 mA/cm2)
embedded with separately-optimized two-level spheres. These two cases indicate the
significance of optimization without which the two-level hierarchical nanostructure might
have little effect (or even negative influence). The last highlights the role of the optimization
as Jsc is increased dramatically to 20.15 (mA/cm2), indicating a 10.23% enhancement with
respect to the first generation. If consider the worst case (Jsc = 17.19 mA/cm2) in the first
generation that has tested 100 configurations, the Jsc is increased by 17.22%. Note that it takes
65 generations for Jsc to reach convergence (Fig. 4(a)), which is slightly longer than that in
Example 1. The nanostructure coverage for this example is 11.09%, which is close to the 10%
optimal value assumed in the previous study [15]. It is interesting to note that d1 = 1.2 nm is
very small, indicating that the flat solar cell more relies on the near-field to improve Jsc than
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the convex solar cell. However, d1 should not be equal to zero, as a dielectric gap between the
nanoparticles and the Si film is essential to maintain electrical isolation so that additional
surface recombination owing to the presence of the metal is prohibited [14].
(a) (b) (c)
(d) (e) (f)
|E| 2
x
z
x
y
|E| 2
Fig. 3. The vertical/horizontal cross sections of the electric intensity: (a) and (d) around the
separately-optimized nanospheres for convex solar cell; (b) and (e) around the optimized
nanospheres for convex solar cell; (c) and (f) around the optimized nanospheres for flat solar
cell.
Table 3. The parameters and the performance of the flat solar cells
r
1
(nm)
p
(nm)
d1
(nm)
d2
(nm) N
J
sc
(mA /cm2)
J
sc enhanc-
ement (%)
no sphere - - - - - 18.28 -
Singly-optimized
(large sphere only) 100 792.67 20 80 - 18.38 0.55
Singly-optimized
(two-level spheres) 100 792.67 20 80 81 18.45 0.93
Optimized (two-
level spheres) 61.02 459.29 1.20 39.28 103 20.15 10.23
To compare the convergence of different parameters, they are normalized with respect to
their maximal values in the optimization. As shown in Fig. 4(b), the radius of the large sphere
r1 converges almost straightaway, which indicates that it is the most significant factor to the
nanostructure according to the evolutionary algorithm. Another key factor to the solar cell is
the thickness of Si film which converges to its upper bound within a few generations in the
other examples that are not shown in the paper. The parameters that have less impact on the
enhancement of performance, such as d1, take more generations to converge. It is not
surprising to observe that the cost function does not fluctuate significantly even though d1
drops at the late stage. The number of small spheres is another important factor as they
exhibit the strongest electric intensity as shown in Figs. 3(c) and 3(f). These two figures also
illustrate that the flat solar cell has stronger electric intensity than the convex one. Thus it can
be regarded as one of the explanations for obtaining larger Jsc because the light path in Si film
is improved accordingly by a stronger resonance.
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Received 11 Sep 2012; revised 11 Jan 2013; accepted 18 Feb 2013; published 26 Feb 2013
(C) 2013 OSA
11 March 2013 / Vol. 21, No. S2 / OPTICS EXPRESS A292
Fig. 4. (a) The convergence history of the optimization process: (a) the short circuit current
density; (b) five design parameters.
The visualization of the relative absorption per unit volume (A(x,z) =
log10[Im(ε)|E(x,z)|2)] helps reveal the mechanism of the enhancement of Jsc. The snapshots in
Fig. 5 present the profiles of the relative absorption for three types of solar cells with the same
arrangement as in Fig. 2. For the flat solar cell without nanostructure, Fig. 5(a) exhibits
parallel absorption pattern in the Si film as light is reflected from the back metal in an inverse
direction of the incident wave, indicating that the light travels in the Si film via the shortest
path. The reflection from the convex back Ag layer, together with the scattering and
diffraction caused by the nanostructure in Fig. 5(b), results in a ripple-like absorption in the Si
film. Thus there exists a region (on the upper middle region of the Si layer) where absorption
is rather weak (blue color) for the convex solar cell. For the flat solar cell with the rear-
located optimized nanostructure, since the gap between the nanostructure to Si layer is
relative small, the near-field of the surface plasmon resonance is capable of coupling into the
lower part of Si film (red region in Fig. 5(c)) and therefore the effective absorption is
improved. The absorption distribution in Fig. 5(c) clearly exhibits better absorption than the
one in Fig. 5(b).
Fig. 5. The relative absorption distribution per unit volume: (a) flat solar cell without
nanostructure; (b) convex solar cell with rear-located optimized nanostructure; (c) flat solar
cell with rear-located optimized nanostructure.
Figure 6 depicts Jsc in the sunlight spectrum for the abovementioned three solar cells. For
the flat solar cell embedded with optimized nanostructure, the Jsc (red curve) almost matches
that of the flat solar cell without nanostructure (black curve) at short wavelength. This is
because the light is sufficiently absorbed within the front Si film before it reaches the
nanostructure at this wavelength range. While in a longer wavelength region, the flat solar
cell without nanostructures exhibits small Jsc owing to a lower absorption coefficient of Si in
this spectrum [33]. However, when the light at a longer wavelength encounters nanostructure,
plasmonic resonance is induced on the metal surface and thus the light is scattered and
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Received 11 Sep 2012; revised 11 Jan 2013; accepted 18 Feb 2013; published 26 Feb 2013
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11 March 2013 / Vol. 21, No. S2 / OPTICS EXPRESS A293
diffracted into the Si film at wider angles and a longer light path is achieved. This is the
reason why both solar cells embedded with nanostructures show large Jsc than the flat solar
cell between 650 and 840 nm over spectrum. An explanation for the difference between blue
and red curves in Fig. 6 is that the propagating mode, one of the modes of the surface
plasmon polaritons at back metal, is influenced by the inner concave metal surface. Therefore
more energy is absorbed in the metal film due to the Ohmic loss. However, further
investigation into the difference is needed in the future studies.
Fig. 6. The short circuit current density in the sunlight spectrum for three solar cells.
4. Conclusion
This work has investigated the benefits of large sphere and small spheres in thin-film solar
cells, which yield a stronger intensity and wider scattering angles, can coexist in a single two-
level hierarchical nanostructure consisting of evenly distributed small spheres on the surface
of a large sphere. The short circuit current density of the thin-film solar cells can be improved
considerably if the location and key parameters of those nanostructures embedded in the real
dielectric layer are optimized by the evolutionary algorithm. We have found the coverage of
the nanostructure is only 3.77% which is much smaller than the optimum 10% suggested by
other researchers. Of five parameters in the design, the size of the large sphere is most
significant, whilst the distance between the nanostructure and absorbing Si film is less
important. The findings from the present study can be indicative to further enhancement of
the conversion efficiency of thin-film solar cells by seeking the optimal values of their key
factors.
Acknowledgment
This work was supported by an Australian Research Council Discovery Early Career
Researcher Award (project number DE120102906) and an Australian Research Council
Discovery Project grant (DP110104698). The authors gratefully acknowledge Dr. Baohua Jia
and Prof. Min Gu, from Centre for Micro-Photonics, Faculty of Engineering and Industrial
Sciences, Swinburne University of Technology, for helpful discussions.
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Received 11 Sep 2012; revised 11 Jan 2013; accepted 18 Feb 2013; published 26 Feb 2013
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11 March 2013 / Vol. 21, No. S2 / OPTICS EXPRESS A294
