Designing periodic arrays of metal nanoparticles for light-trapping
applications in solar cells
S. Mokkapati,
1,a兲
F. J. Beck,
1
A. Polman,
2
and K. R. Catchpole
1
1
Center for Sustainable Energy Systems, College of Engineering and Computer Science, The Australian
National University, Canberra 0200, Australia
2
Center for Nanophotonics, FOM Institute AMOLF, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
共Received 30 April 2009; accepted 17 July 2009; published online 7 August 2009兲
We present criteria for optimizing the light-trapping efficiency of periodic arrays of metal
nanoparticles for Si solar cell applications. The scattering cross section of the nanoparticles and the
diffraction efficiency of the grating should be maximized in the long wavelength range. The grating
pitch should be chosen to allow higher order diffraction modes for long wavelengths while
maintaining the highest possible fill factor. These conditions place strong constraints on the optimal
parameters 共particle size of ⬃200 nm and pitch of ⬃400 nm兲 for periodic arrays of metal
nanoparticles, in contrast to dielectric gratings, where a relatively wide range of periods and feature
sizes can be used for efficient light trapping. © 2009 American Institute of Physics.
关DOI: 10.1063/1.3200948兴
There is currently a lot of interest in silicon 共Si兲 solar
cells with thin active regions that minimize material costs
while maintaining high efficiencies. Thin-film solar cells re-
quire efficient light trapping 共LT兲 in order to increase the
effective absorption length in the cell. LT is especially im-
portant at long wavelengths 共close to the bandgap兲 in Si solar
cells since Si is a poor absorber due to its indirect bandgap.
Surface texturing, with feature sizes of a few microns, is
routinely used for LT in wafer based Si solar cells
1
but can-
not be used in the case of thin active regions, which may
themselves be of the order of a few microns thick. Light
scattering by the excitation of localized surface plasmons in
metal nanoparticles has been used to demonstrate effective
LT for photovoltaic applications.
2–4
Both random arrays
3
and
quasiperiodic arrays
2
of metal nanoparticles have demon-
strated significant absorption enhancement in solar cells.
While there has been a systematic study on the requirements
for efficient LT using random arrays,
5,6
a range of additional
parameters is introduced for periodic arrays of metal nano-
particles that need to be optimized. In this article, we study
the effect of particle dimensions, the grating pitch, and fill
factor and introduce essential criteria for optimizing the LT
efficiency of periodic arrays of silver 共Ag兲 nanoparticles. We
demonstrate that for efficient LT, the optimal geometry of the
metal nanoparticle array is relatively narrowly defined due to
the limited range of particle sizes that result in efficient light
scattering into the substrate in the long wavelength range.
This is in contrast to dielectric gratings, where a relatively
wide range of periods and feature sizes can be used for effi-
cient LT.
7,8
Numerical simulations to estimate the LT efficiency of
periodic arrays of nanoparticles were performed using
LUMERICAL finite dimension time domain software. Semi-
infinite Si substrates with periodic arrays of Ag nanoparticles
on the surface are simulated in this study. Light is incident on
the nanoparticle array from the Si side 共Fig. 1兲. Only struc-
tures with Ag arrays on the rear have been simulated for this
study, as Ag arrays on the front of the solar cell would result
in lower external quantum efficiency compared to a cell
without the nanoparticles at wavelengths below the plasmon
resonance wavelength of the nanoparticles.
9,10
A 20 nm thick
SiO
2
layer, typical of surface passivation layers used in Si
solar cells, is also included between the Si substrate and the
nanoparticles. An ordered array of Ag nanoparticles with a
square base was simulated using square unit cell and periodic
boundary conditions for the simulation volume. The particle
height was 150 nm. A plane wave source under normal inci-
dence was used. The grating structure had equal periods in
both x- and y-directions, and light was incident in z direction.
For an isolated particle, the scattering cross section pro-
vides a useful measure of interaction of light with the par-
ticle. The scattering cross section of the nanoparticle peaks at
its plasmon resonance wavelength. However, for an array of
particles, the scattering cross section is not defined. We
therefore use absorption calculations to determine the plas-
mon resonance wavelength of the Ag nanoparticles in the
periodic array since to first order, the absorption in the nano-
particles also peaks at the plasmon resonance wavelength.
Absorption in Ag nanoparticles was determined from the dif-
ference in input and output power flows into a closed box
surrounding the particle. Figure 2共a兲 shows the fraction of
incident power absorbed in the Ag nanoparticle array for
varying particle dimensions. The array pitch was fixed at 400
nm. As can be seen, by increasing the particle size from 50 to
150 nm, the plasmon resonance wavelength redshifts from
⬃505 to ⬃770 nm, while the absorption strength increases.
The broadening of the absorption spectra with an increase in
particle size is due to radiative damping and dynamic depo-
a兲
Electronic mail: sudha.mokkapati@anu.edu.au. FIG. 1. 共Color online兲 Schematic of the structures simulated for this study.
APPLIED PHYSICS LETTERS 95, 053115 共2009兲
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larization. The shift in the resonance wavelength is compa-
rable to the resonance redshift for single particles 共i.e., with-
out periodic boundary conditions兲 with increasing size. The
absorption in the nanoparticle array scales with the particle
volume and the surface coverage. For particles with lateral
dimensions of 200 and 300 nm, the plasmon resonance
wavelength is beyond 1000 nm. For particle lateral dimen-
sions of 300 nm, an absorption peak at ⬃580 nm is ob-
served, which coincides with the absorption peak in the par-
ticle with base dimensions of 100 nm. It is well known that
as the metal nanoparticle size increases, its plasmon reso-
nance wavelength corresponding to the dipole oscillations
redshifts and higher order charge oscillations appear at
shorter wavelengths.
11
Based on this, we associate this peak
with higher order charge oscillations in the 300 nm Ag nano-
particles.
After interaction with the nanoparticle array, a certain
fraction of incident light is transmitted into air 共I
esc
in Fig. 1兲,
and the rest is diffracted back into Si. The light diffracted
back into Si is coupled into various allowed diffraction
modes. The light coupled into the zeroth diffraction order,
I
oth
in Fig. 1 共propagating along the direction of surface nor-
mal兲, is not trapped inside the substrate, as it lies within the
escape cone of Si. Only light coupled into higher order dif-
fraction modes, I
hod
in Fig. 1, which propagate outside the
escape cone, is trapped inside Si. Figure 2共b兲 shows the light
coupled into higher order diffraction modes in Si 关f
hod
=I
hod
/ 共I
hod
+I
oth
兲兴 after interaction with the nanoparticle ar-
ray as a fraction of total light coupled back into Si. As the
particle size increases from 50 to 150 nm, the wavelength
range in which a large fraction of light is coupled into higher
order diffraction modes redshifts. The largest f
hod
共⬃96%兲
is obtained for a particle size of 150 nm. For particle dimen-
sions of 200 nm, the wavelength range over which the maxi-
mum fraction of light is coupled into higher order diffraction
modes lies beyond 1000 nm. Clearly, a high fraction of light
is coupled into higher order diffraction modes over a broad
spectral range around the dipole resonance wavelengths 关Fig.
2共a兲兴. So for efficient LT at wavelengths close to the bandgap
of Si, the particle dimensions should be chosen such that the
plasmon resonance wavelength corresponding to the dipole
charge oscillations in the particles lies in this wavelength
range.
Figure 2共c兲 shows the fraction of incident power re-
flected into Si, coupled into all diffraction orders, after inter-
action with the metal nanoparticle grating 关f
ref
=共I
hod
+I
0th
兲/ I
in
in Fig. 1兴. In the long wavelength region, particles
with dimensions of 200 and 300 nm have a higher f
ref
than
particles with smaller dimensions. Figure 2共d兲 shows the net
LT efficiency
LT
共=f
hod
f
ref
兲 for the arrays. It is clear that at
wavelengths above 850 nm,
LT
is lower for arrays of par-
ticles with base dimensions of 150 nm than for particles with
base dimensions of 200 and 300 nm, in spite of a higher
fraction of scattered light going into higher order grating
modes 关Fig. 2共b兲兴. This is due to the smaller scattering cross
section for the smaller particles. The data in Figs. 2共c兲 and
2共d兲 show oscillations that are ascribed to interference be-
tween light coupled into different diffraction modes.
The results in Fig. 2 indicate that for an ordered array of
particles, a large fraction of light coupled into higher order
diffraction modes 共f
hod
兲 in the substrate by itself does not
imply a high LT efficiency. The LT efficiency is determined
by both the particle dimensions 共which determine the plas-
mon resonance wavelength for an individual particle and
hence f
hod
兲 and the grating parameters 共that determine f
ref
兲.
For efficient LT, it is essential to tune the grating parameters
so that a high fraction of incident light is retained in Si at
wavelengths around the particle plasmon resonance wave-
length.
We now examine the effect of pitch and fill factor on the
LT efficiency of an ordered array of Ag particles. For LT
applications using periodic arrays of metal nanoparticles, at
least one higher order diffraction mode should exist in Si,
with propagation angle of ⬎16° with respect to the surface
normal. Using the one-dimensional grating equation given
by sin
p
= p/ nL, where p is the order of the diffracted
mode, is the wavelength of incident light, n is the refrac-
tive index of the substrate, L is the pitch of the grating, and
p
is the associated angle of propagation, the requirement of
higher order propagating diffraction modes places a con-
straint on the minimum value of grating pitch that can be
employed. In order for one higher order diffraction mode to
exist at wavelength of 1000 nm in Si, the grating pitch
should be at least 300 nm.
The net LT efficiency,
LT
, as defined above, as a func-
tion of wavelength, for various grating pitches and a fixed
particle size of 200 nm is depicted in Fig. 3. The curve for
the 400 nm pitch is identical to that in Fig. 2共d兲. For a grating
pitch of 250 nm, long wavelength light is not trapped as no
propagating higher order diffraction modes exist in Si. As the
pitch is increased to 300 nm, first order diffraction modes for
longer wavelengths are allowed in Si and the light is trapped.
The LT efficiency of the grating is further increased by in-
FIG. 2. 共Color online兲共a兲 Absorption in Ag nanoparticles, abs
Ag
. 共b兲 Frac-
tion of light coupled into higher order diffraction modes in Si, f
hod
. 共c兲
Fraction of incident source power reflected back into Si, f
ref
. 共d兲 The LT
efficiency,
LT
, of a periodic array of Ag nanoparticles as a function of
incident wavelength for different particle sizes. The particles have a square
base with a height of 150 nm. The lateral dimensions are indicated in the
legend. The pitch of the grating is 400 nm.
053115-2 Mokkapati et al. Appl. Phys. Lett. 95, 053115 共2009兲
Downloaded 09 Aug 2009 to 150.203.45.94. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp
creasing the pitch to 400 nm. This is because the propagation
angles for the diffraction modes decrease as the pitch of the
grating is increased. At a wavelength of 1000 nm, the propa-
gation angle for the first order diffraction modes in Si re-
duces from ⬃64° to ⬃42.5° by increasing the pitch from 300
to 400 nm, resulting in an increase in
LT
by a factor of four.
For diffraction orders with large propagation angles, there is
poor impedance matching between the incident light, the al-
lowed grating modes, and the diffraction modes,
7
reducing
the fraction of light coupled into these modes. As the propa-
gation angle reduces 共with increasing pitch兲, light is effi-
ciently coupled into these diffraction modes. Once the propa-
gation angles for higher order diffraction modes are
relatively small, a further increase in the pitch of the grating
results in reduced LT efficiency because of reduced fill factor
共fraction of the unit cell area occupied by the nanoparticle兲.
The scattering cross sections of isolated metal nanoparticles
on a substrate, normalized to the particles geometrical cross
section, are generally in the range from two to six;
5
from
this, we expect that a fill factor of at least ⬃20% would be
required for efficient light scattering. The effect of reducing
fill factor is evident from Fig. 3 by comparing the LT effi-
ciency for pitches of 400 nm 共corresponding to a fill factor of
25%兲 and 600 nm 共fill factor of 11%兲. These results indicate
that for efficient LT applications, the pitch of the grating
should be chosen such that one higher order diffraction mode
exists in Si with reasonably small propagating angles while
maintaining a sufficiently high fill factor.
In summary, for efficient LT using periodic arrays of
metal nanoparticles, the particle dimensions and its dielectric
environment should be chosen such that the dipole oscilla-
tion resonance lies in the wavelength range of interest. It is
also necessary to tune the grating parameters so that high
diffraction efficiency is obtained in a wavelength range close
to the dipole oscillation resonance of the individual particles,
ensuring that most of the incident power is retained in Si.
The pitch of the grating should be chosen to allow at least
one diffraction mode propagating outside the escape cone in
Si for long wavelength light. This imposes a lower limit of
⬃400 nm on the grating pitch. In addition, a minimum fill
factor of ⬃20% should be maintained for efficient interac-
tion between the incident light and the nanoparticle array.
Taken together, these conditions place strong constraints on
the optimal particle size and grating parameters for a peri-
odic array, and based on these arguments, we conclude that
arrays of particles with particle dimensions of ⬃200 nm and
a pitch of ⬃400 nm are ideal for LT applications for Si solar
cells.
The authors acknowledge the A. R. C. and NOW 共for
research conducted at the FOM as a part of the Joint Solar
Programme兲 for financial support. We acknowledge SARA
computing and networking services, supported by NWO.
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FIG. 3. 共Color online兲 LT efficiency 共
LT
兲 of a periodic array of Ag nano-
particles as a function of wavelength for different pitches 共indicated in the
legend兲. The particles have a square base with a side of 200 nm and a height
of 150 nm.
053115-3 Mokkapati et al. Appl. Phys. Lett. 95, 053115 共2009兲
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