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A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells

02 Jul 2007-Journal of Applied Physics (American Institute of Physics)-Vol. 102, Iss: 1, pp 013102

AbstractOne of the authors K.R.C. acknowledges the support of an Australian Research Council fellowship, and also the support of the Centre of Excellence for Advanced Silicon Photovoltaics and Photonics, supported by the Australian Research Council.

Summary (3 min read)

A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells

  • Furthermore, the authors show that for thin film solar cells with front surface gratings and flat rear reflectors, modal analysis can be used to predict the optimum parameters for maximum light trapping.
  • Subsequently, a series of studies on similar structures has been carried out,4–7 since these structures allow strong light diffraction to be achieved with a relatively simple fabrication process.
  • It has been shown recently15,16 that the modal method17 can be used as the basis for a phenomenological interpretation that allows a deeper understanding of the behavior of highly efficient transmission gratings, where two diffraction orders propagate, and other cases including the subwave- length limit, where only one diffraction order propagates.
  • The haze parameter is the diffuse transmittance of a textured thin film measured in air, whereas the diffuse transmittance as used in this paper is the light that is transmitted diffusely into the silicon.

II. METHOD

  • A diffraction grating couples incident light into a number of diffracted orders or modes.
  • Modal analysis divides the fields propagating in each region into different types of modes: the incident mode, modes propagating within the grating, and reflected and transmitted diffracted modes.
  • Coupling and interference between the various modes determine the diffraction efficiency of the grating.
  • This is a good approximation for thin silicon gratings in the long wavelength region where light trapping is important.

III. RESULTS AND DISCUSSION

  • The authors first investigate how the effects of varying the period and height of silicon and TiO2 gratings can be understood with modal analysis.
  • The authors then look at the relationship between diffuse transmittance and light trapping for thin silicon solar cells with front surface gratings and flat rear reflectors, and finally they apply their understanding to thin film solar cells with ZnO/Si gratings.

A. TE case

  • The peaks in the first diffracted order occur at odd integer multiples of h = 2 n2 eff − n0 eff .
  • The authors can also use the graphs of the electric fields of the grating modes to obtain a clearer picture of the meaning of the overlap integral.
  • Before the onset of the fourth grating mode near 720 nm, there is the onset of the third grating mode around 790 nm, but this has no effect on the diffuse transmittance since it does not couple to normally incident light.
  • Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp value for technological reasons, there are still sets of parameters where the diffuse transmittance is high.

B. TM case

  • The authors can see in Fig. 6 a that interference between the zeroth and second grating modes gives good agreement with the diffuse transmittance peaks for =700–1100 nm.
  • After the onset of the fourth grating mode at =700 nm, interference between the zeroth and fourth grating modes is the dominant effect.
  • This is due to a low overlap between the second grating mode and the zeroth diffracted order, as can be seen from Table III.
  • Interference between the zeroth and fourth grating modes, and between the zeroth and sixth grating modes, dominates in different wavelength regions.
  • Table III shows that in each case, the overlaps with the zeroth diffracted order are strongest for the zeroth grating mode and the highest propagating grating mode.

C. Criteria for high diffuse transmittance

  • The design criteria for a diffraction grating for a solar cell are that diffuse transmittance should be high and that this should occur over a large wavelength range.
  • In order to choose the optimum grating period, the authors note that the zeroth diffraction order tends to couple most strongly with the grating modes when there are few grating modes present.
  • Thus to get high diffuse transmittance over a large wavelength range, neff for the second or fourth, etc. grating mode should be equal to na near the longest wavelength for which light trapping is required, i.e., about 1100 nm for Si.
  • By knowing the periodicity of h, the authors can also estimate the height range over which they will get high diffuse transmittance.

D. Predicting optimum parameters for gratings with different refractive indices

  • The authors can use modal analysis to make predictions of the optimum choice of parameters for gratings without the need to perform RCWA.
  • Often modal analysis is all that is needed to predict the range where the optimum values will be found.
  • As for the silicon grating, the authors expect that the optimal light diffusion will occur when n2 eff=1 occurs near =1100 nm.
  • The results of RCWA for this grating are plotted in Fig. 7, along with the maxima in diffuse transmittance expected from Eqs. 6 and 7 .
  • This leads to an overall optimum height of 327±70 nm, which is in very good agreement with the optimum height as calculated by RCWA of 300±5 nm.

E. Diffuse transmittance and light trapping

  • For thin film structures with gratings that have sufficiently low periods, all of the light that is diffusely transmitted into the silicon lies outside the escape cone.
  • Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp results; this is because the optimum value for the second peak is more sensitive to wavelength than the value for the first peak, so the overall position of the second peak depends on the relative contribution of each wavelength to Jsc.
  • The variation of diffuse transmittance with grating period at =950 nm, together with Jsc for the thin film structure calculated over the wavelength range of 650–1100 nm, is shown in Fig. 8 c .
  • In some cases the secondary maxima are strongly dependent on wavelength.
  • For the device structures studied here with periods near the optimum value, diffuse transmittance provides a good measure of light trapping.

F. Behavior of ZnO/Si gratings

  • With light incident from the ZnO.the authors.
  • Grating structures formed from these materials have been the subject of recent experimental and numerical studies, because the structure can be fabricated relatively simply by deposition of silicon on a ZnO grating/thin layer structure which in turn lies on a glass superstrate.
  • For L=750 nm, the optimum height predicted by modal analysis at =950 nm, averaged over TE and TM illuminations is 383 nm.
  • This is supported by the fact that near the optimum parameters for the front surface grating, single grating and dual grating structures had similar predicted Jsc values under long wavelength illumination.

IV. CONCLUSIONS

  • The authors have shown that simplified modal analysis can be used to understand the diffuse transmittance behavior of onedimensional 1D lamellar gratings for solar cells, and that it can be used to predict the optimum values for 1D gratings, without the need to undertake rigorous numerical modeling.
  • For thin film solar cells with a front surface grating and flat rear reflector, this allows us to predict the optimum period and height for maximum light trapping.
  • The approach used here also provides the basis for understanding more complicated diffractive structures that may be even more effective for light trapping and light extraction, such as pillar-type gratings.

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A conceptual model of the diffuse transmittance of lamellar diffraction
gratings on solar cells
K. R. Catchpole
a
School of Photovoltaic and Renewable Energy Engineering, University of New South Wales,
Sydney NSW 2052 Australia
Received 21 May 2006; accepted 2 April 2007; published online 2 July 2007
Diffraction gratings are effective ways of increasing the light absorption of solar cells and the light
extraction of light-emitting diodes. In this paper, we show that simplified modal analysis can be used
as a conceptual model for understanding the behavior of the diffuse transmittance of lamellar
diffraction gratings on infinite substrates. We use simplified modal analysis to predict the optimum
values of period and height for the gratings, and achieve excellent agreement with rigorous coupled
wave analysis. Furthermore, we show that for thin film solar cells with front surface gratings and flat
rear reflectors, modal analysis can be used to predict the optimum parameters for maximum light
trapping. © 2007 American Institute of Physics. DOI: 10.1063/1.2737628
I. INTRODUCTION
Light trapping is becoming increasingly important in so-
lar cells as the devices become thinner. For cells with thick-
nesses that can range from a few hundred nanometers for
amorphous silicon to a few microns for microcrystalline sili-
con, conventional types of light trapping which have feature
sizes around 10
m are not suitable.
Periodic and quasiperiodic diffraction grating structures
are attractive alternative methods for applying light trapping
to thin solar cell structures. Morf et al. introduced the use of
grating structures for solar cells, studying, in particular,
blazed gratings and subwavelength antireflection gratings,
and showed that very high absorptances can be obtained for
silicon cell thicknesses of only a few microns.
1,2
Thin film
a-Si solar cells on rectangular ZnO gratings were experimen-
tally investigated by Eisele et al.
3
Subsequently, a series of
studies on similar structures has been carried out,
47
since
these structures allow strong light diffraction to be achieved
with a relatively simple fabrication process. In these studies,
the diffraction gratings were fabricated using interference li-
thography. Other fabrication methods include molding
8
for
periodic structures and nanosphere lithography for quasiperi-
odic structures.
9,10
Quasiperiodic diffraction gratings formed
by nanosphere lithography have also been used to increase
the light extraction from light-emitting diodes.
11
Diffraction grating structures can be modeled numeri-
cally by a number of methods, such as rigourous coupled
wave analysis RCWA.
1214
However, there are a large num-
ber of free parameters in designing a diffraction grating, and
numerical modeling alone does not offer much insight as to
why a particular combination of parameters should be opti-
mal.
It has been shown recently
15,16
that the modal method
17
can be used as the basis for a phenomenological interpreta-
tion that allows a deeper understanding of the behavior of
highly efficient transmission gratings, where two diffraction
orders propagate, and other cases including the subwave-
length limit, where only one diffraction order propagates. In
this paper, we call this approach simplified modal analysis,
and we apply the interpretation to somewhat more compli-
cated cases, where at least three diffraction orders propagate.
We show that nevertheless this simplified modal analysis can
be used to provide a conceptual understanding of the diffuse
transmittance of rectangular gratings suitable for solar cells.
We show that the model can also be used to predict the
optimum parameters for a light trapping front surface grating
on a thin film solar cell with a flat rear reflector, and we
confirm the results using RCWA. The approach used in this
paper also provides the basis for understanding more com-
plicated diffractive structures such as pillar-type gratings,
18
which can be even more effective for light trapping and light
extraction.
It is important to distinguish the diffuse transmittance
used in this work from the haze parameter that has been used
to characterize textured transparent conductive oxide layers
designed for application in thin film
cSi:H and a-Si solar
cells. The haze parameter is the diffuse transmittance of a
textured thin film measured in air, whereas the diffuse trans-
mittance as used in this paper is the light that is transmitted
diffusely into the silicon. While the diffuse transmittance into
the silicon is not easily accessible experimentally, it is useful
for understanding the optimal values of grating parameters. It
has been pointed out recently that the haze is not a sufficient
parameter to characterize the light-trapping properties of a
surface texture; the angular distribution of the light contrib-
uting to the haze is also important.
4,19
For a general surface
texture, the diffuse transmittance into the silicon is also not a
sufficient parameter to characterize light trapping because it
does not contain information about the angular distribution
of the light, and, in particular, the fraction of light outside the
escape cone. However, for certain device structures all of the
light that is diffusely transmitted lies outside the escape cone,
and for such structures there is a close correlation of the
maximum possible short-circuit current J
sc
with the diffuse
transmittance, as described in Sec. III E.
a
Electronic mail: k.catchpole@amolf.nl
JOURNAL OF APPLIED PHYSICS 102, 013102 2007
0021-8979/2007/1021/013102/8/$23.00 © 2007 American Institute of Physics102, 013102-1
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II. METHOD
A diffraction grating couples incident light into a number
of diffracted orders or modes. For a transmission grating on
a substrate with refractive index n
s
, the propagation angles of
the diffracted orders are given by
sin
p
=
p
n
s
L
, p = 0,1,2, ... , 1
for normal incidence, where L is the period of the grating.
While the angles into which light is coupled are easily cal-
culated, to find the optimum grating parameters that maxi-
mize the diffuse transmittance would generally require a
complicated numerical method.
The essence of simplified modal analysis is summarized
below; more details can be found in the relevant
papers.
15,16,20
The theory applies to rectangular lamellar
gratings of the type shown in Fig. 1. Modal analysis divides
the fields propagating in each region into different types of
modes: the incident mode, modes propagating within the
grating, and reflected and transmitted diffracted modes. Cou-
pling and interference between the various modes determine
the diffraction efficiency of the grating. For the grating
shown in Fig. 1 the grating modes can be found analytically.
Each mode is characterized by an effective refractive index
n
eff
which is a solution of the equation
20
fn
eff
= cosk
xr
L
r
cosk
xg
L
g
1
2
k
xr
k
xg
+
1
k
xg
k
xr
sink
xr
L
r
sink
xg
L
g
= cosk
x
L, 2
where k
0
is the wave vector of the incident wave, k
x
=k
0
sin
in
= 2
/n
a
sin
in
is the x component of the
wave vector of the incident wave, and
k
xi
= k
0
n
i
2
n
eff
2
1/2
, i = r,g 3
are the x components of the wave vectors in the ridges and
grooves of the grating, respectively.
is equal to 1 for
transverse-electric TE polarization, n
r
2
/n
g
2
for transverse-
magnetic TM polarization, and L =L
r
+L
g
. Because of the
symmetry of the problem, the field in the grating region can
be separated into an x-dependent part ux and a z-dependent
part
z. For TE polarization, for example, only a y compo-
nent of the electric field is present, which can be written as
E
y
x,z = ux
v
z. 4
The field distributions associated with each mode are given
by Sheng et al.
20
Here, we note that for normal incidence, the
x-dependent parts of the grating modes are periodic, with the
same period as the grating. One of the factors affecting the
efficiency of excitation of the mth grating mode by the inci-
dent mode E
y
in
, i.e., the fraction of power that is transferred
from the incident mode to the mth grating mode is the over-
lap integral of the two modes,
21
E
y
in
x,0,u
m
x兲典 =
E
y
in
x,0u
m
xdx
2
E
y
in
x,0兲兩
2
dx
u
m
x兲兩
2
dx
. 5
The other factors affecting the excitation of the grating
modes by the incident mode, and the excitation of diffracted
modes by the grating modes, are Fresnel-like reflection and
transmission coefficients.
15,17
These coefficients give low re-
flectance and high transmittance when the y components of
the wave vectors of the two modes are similar. This is known
as impedance matching. Thus for the air/grating interface,
the efficiency of excitation of a given grating mode increases
as n
eff
approaches n
a
cos
in
=n
a
for normally incident light.
For the grating/substrate interface, the efficiency of excita-
tion of diffracted mode p by a given grating mode increases
as n
eff
approaches n
s
cos
p
.
Absorption is taken into account
20
in Eq. 2. However,
in the calculations presented in this paper, we have neglected
absorption when calculating the effective refractive indices
for the various grating modes. This is a good approximation
for thin silicon gratings in the long wavelength region where
light trapping is important. At short wavelengths, where sili-
con is strongly absorbing, it would be important to take ab-
sorption into account as the grating modes will be absorbed
differently depending on their field distribution.
III. RESULTS AND DISCUSSION
In this section, we first investigate how the effects of
varying the period and height of silicon and TiO
2
gratings
can be understood with modal analysis. We then look at the
relationship between diffuse transmittance and light trapping
for thin silicon solar cells with front surface gratings and flat
rear reflectors, and finally we apply our understanding to thin
film solar cells with ZnO/ Si gratings.
A. TE case
We consider the case of TE illumination first i.e., elec-
tric field parallel to the grooves of the grating.ForaSi
grating with L =650 nm at = 1000 nm n
r
=3.58, there are
three propagating grating modes with effective refractive in-
dices n
0
eff
=3.37, n
1
eff
=2.70, and n
2
eff
=1.40. There are also
three propagating diffraction orders in the substrate. When
we calculate the diffraction efficiency i.e., the fraction of
light that is transmitted into each order as a function of
height using RCWA for this case, we see an almost periodic
FIG. 1. Parameters for the grating under consideration.
013102-2 K. R. Catchpole J. Appl. Phys. 102, 013102 2007
Downloaded 28 Mar 2010 to 150.203.243.53. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

oscillation of the zeroth and first diffracted orders, with much
less power going into the second diffracted order Fig. 2.
We can get a more complete picture of the grating be-
havior with contour plots of the diffuse transmittance as a
function of wavelength and grating height. The diffuse trans-
mittance is the fraction of light that is transmitted into the
silicon into orders higher than the zeroth order, i.e., it is the
total transmittance minus the zeroth order transmittance. In
this section and in Secs. III B–III D, we consider the diffuse
transmittance for gratings on an infinite Si substrate; in Secs.
III E and III F, we discuss how this is related to light trap-
ping in thin film silicon cells.
A contour plot for L =650 nm calculated with RCWA is
shown in Fig. 3. The major feature of the plot is a series of
peaks in the diffuse transmittance at periodic values of h
with the peak values of h decreasing with increasing wave-
length. This oscillation can be explained as being due to a
phase difference accumulated during propagation of the dif-
ferent grating modes through the grating. When the phase
shift is zero essentially all the light coupled into the grating
is directed into the zeroth diffracted order, while when the
phase shift is
most of the light is coupled into the first
diffracted order. The peaks in the first diffracted order occur
at odd integer multiples of
h =
2n
2
eff
n
0
eff
. 6
The positions of the peaks as calculated with Eq. 6 are
shown as solid black lines in Fig. 3. It can be seen that there
is an excellent agreement with the results of RCWA.
To determine which modes contribute to the interfer-
ence, we calculate the overlap integrals between the grating
modes and the diffracted orders Table I. For normal inci-
dence the overlap integral between the first grating mode and
the zeroth diffracted order is zero because the first grating
mode is an odd function with respect to the vertical axis of
symmetry of the grating. Thus for normally incident light
there is no coupling to the first grating mode.
The x dependence of the electric field of the grating
modes for L =650 nm and =1000 nm is plotted in Fig. 4a.
We can see that for the zeroth grating mode most of the
energy propagates in the Si while for the second grating
mode the energy is more evenly distributed between the air
and the Si. In fact, from the intensity in each region, pro-
portional to the square of the electric field, we find that 97%
of the energy is in the Si for the zeroth mode, compared with
48% for the second mode. This is the reason that the effec-
tive refractive index of the zeroth mode is high 3.37 while
the effective refractive index of the second mode is relatively
low 1.40. We can also use the graphs of the electric fields
of the grating modes to obtain a clearer picture of the mean-
ing of the overlap integral. In Fig. 4b the integrand in the
numerator of Eq. 5 is plotted shaded area, along with the
x-dependent parts of the fields for the zeroth grating mode
and first diffracted order for which the value of the overlap
integral is 0.2. The figure shows how the value of the over-
lap integral is increased where the fields due to both modes
are high. The total value of the overlap integral is the square
of the shaded area divided by the intensities of the modes
taking part.
We now turn our attention to the long wavelength region
of the contour plot in Fig. 3. The overlap integrals for L
=650 nm with = 1300 nm are given in Table II. For
=1300 nm there are only two propagating grating modes.
The coupling between the first diffracted order and the zeroth
grating mode is weaker than at =1000 nm, but the main
difference between the two cases is that at =1300 nm there
is no second grating mode available to couple to the zeroth
diffracted order. As at =1000 nm, the first grating mode
does not couple with normally incident light.
FIG. 2. The diffraction efficiency vs height h for the transmitted diffracted
orders for a Si grating on an infinite Si substrate with L =650 nm at L
=1000 nm n
r
=3.58.
FIG. 3. Diffuse transmittance for a rectangular Si grating on an infinite Si
substrate with L =650 nm calculated with RCWA. The lines show the
maxima predicted with modal analysis, including contributions from inter-
ference between the zeroth and second grating modes solid lines, Fabry-
Pérot interference of the zeroth grating mode with itself dash-dot lines, and
interference between zeroth and fourth grating modes dashed lines.
TABLE I. Overlap integrals between propagating diffracted orders p and
propagating grating modes m for L = 650 nm and =1000 nm.
n
eff
p=−2 p =−1 p=0 p=1 p=2
m=0 3.373 0.005 0.200 0.589 0.200 0.005
m=1 2.700 0.108 0.389 0 0.389 0.108
m=2 1.396 0.154 0.173 0.299 0.173 0.154
013102-3 K. R. Catchpole J. Appl. Phys. 102, 013102 2007
Downloaded 28 Mar 2010 to 150.203.243.53. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

Thus interference between different grating modes does
not occur for =1300 nm. Nevertheless we can see from
Fig. 3 that there is a variation of diffuse transmittance with
height for = 1300 nm. The source of this variation is Fabry-
Pérot interference of the zeroth grating mode with itself. The
Fabry-Pérot interference has maxima in the diffuse transmit-
tance at odd integer multiples of
h =
2n
0
eff
. 7
The maxima in diffuse transmittance according to Eq. 7 are
also plotted in Fig. 3 dash-dot lines. From the RCWA re-
sults we find that the Fabry-Pérot resonance effect becomes
weaker as the grating height increases, so only the first reso-
nance is plotted for the full wavelength range of
8001300 nm. The second and third resonances are plotted
near 1300 nm only. We can see that the first maximum in
diffuse transmittance as calculated by RCWA in the region
where there are three propagating grating modes
=790 1250 nm is due to a combination of interference be-
tween the zeroth and second grating modes and Fabry-Pérot
interference of the zeroth grating mode with itself. This leads
to less variation in the optimum height with wavelength than
would be the case if Fabry-Pérot resonance did not play a
role.
As the wavelength decreases for a given grating period,
the number of propagating grating modes increases. For ex-
ample, at = 700 nm, there are five propagating grating
modes. Of these, the zeroth, second, and fourth can couple to
normally incident light. The fourth grating mode has a stron-
ger overlap with the zeroth diffracted order than the second
grating mode, so interference between the zeroth and fourth
grating modes dominates the height dependence. The
maxima in diffuse transmittance due to interference between
the zeroth and fourth grating modes are plotted in Fig. 3 for
in the range of 600700 nm. Before the onset of the fourth
grating mode near 720 nm, there is the onset of the third
grating mode around 790 nm, but this has no effect on the
diffuse transmittance since it does not couple to normally
incident light.
The effect of a larger period is similar to the effect of a
decreased wavelength in that the number of grating modes
increases. For L =1000 nm in the range = 8301000 nm,
there are five grating modes. The first and third grating
modes do not couple to normally incident light, and the sec-
ond grating mode has a relatively small overlap with the
zeroth diffraction order of 0.09, so the height dependence is
due to interference between the zeroth and fourth grating
modes which have overlap integrals 0.50 and 0.37, respec-
tively, as shown in Fig. 5 solid lines. Also shown in Fig. 5
dotted lines is the height dependence due to the interfer-
ence of the zeroth and sixth grating modes for the wave-
length range of 650 750 nm, where there are seven propa-
gating grating modes.
From the above examples, we can see the importance of
choosing the optimum height for a given grating period. If
the grating period has to be chosen larger than the optimum
FIG. 4. a.Thex dependence of the electric field of the three grating modes
for L = 650 nm and = 1000 nm. The gray regions are the Si and the white is
air. b A schematic visualization of the meaning of the overlap integral,
calculated for the overlap between the zeroth grating mode and first dif-
fracted order for L = 650 nm and = 1000 nm. The solid line shows the x
dependence of the zeroth grating mode while the dashed line is the x depen-
dence of the first diffracted order.
TABLE II. Overlap integrals for L=650 nm with =1300 nm.
n
eff
p=−2 p=−1 p=0 p=1 p=2
m=0 3.202 0.002 0.162 0.672 0.162 0.002
m=1 2.103 0.056 0.444 0 0.444 0.056
FIG. 5. Diffuse transmittance for a rectangular grating with infinite Si sub-
strate with L = 1000 nm. The contour plot shows the results calculated with
RCWA. The lines show the maxima predicted with modal analysis, due to
interference between the zeroth and fourth grating modes for
=8301000 nm solid lines and interference between the zeroth and sixth
grating modes for = 650750 nm dashed lines.
013102-4 K. R. Catchpole J. Appl. Phys. 102, 013102 2007
Downloaded 28 Mar 2010 to 150.203.243.53. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

value for technological reasons, there are still sets of param-
eters where the diffuse transmittance is high. Modal analysis
can be used to predict the optimum height for a given grating
period, without the necessity of doing a full calculation using
RCWA. The overlap integrals tell us which modes are im-
portant and the effective indices then allow us to calculate
where the peaks in diffuse transmittance are.
B. TM case
For the TM case the diffuse transmittance can also be
predicted by the interference of two grating modes, as shown
in Figs. 6a and 6b and also in the next section in Fig.
7b. We can see in Fig. 6a that interference between the
zeroth and second grating modes gives good agreement with
the diffuse transmittance peaks for = 700 1100 nm. After
the onset of the fourth grating mode at = 700 nm, interfer-
ence between the zeroth and fourth grating modes is the
dominant effect. This is due to a low overlap between the
second grating mode and the zeroth diffracted order, as can
be seen from Table III. The trend is similar for L = 1000 nm,
as shown in Fig. 6b. In this case, interference between the
zeroth and fourth grating modes, and between the zeroth and
sixth grating modes, dominates in different wavelength re-
gions. Table III shows that in each case, the overlaps with the
zeroth diffracted order are strongest for the zeroth grating
mode and the highest propagating grating mode.
C. Criteria for high diffuse transmittance
The design criteria for a diffraction grating for a solar
cell are that diffuse transmittance should be high and that this
should occur over a large wavelength range. The diffusely
transmitted light should also lie outside the escape cone; this
is discussed further in Sec. III E. From Eq. 6 we see that in
order to get a high diffuse transmittance over a large wave-
length range a grating height corresponding to the first peak
in the diffuse transmittance i.e., phase difference of
rather
than 3
or 5
, etc. should be chosen. This fixes the opti-
mum height for a given grating period, for TE or for TM
illumination. In order to choose the optimum grating period,
we note that the zeroth diffraction order tends to couple most
strongly with the grating modes when there are few grating
modes present. Since the first grating mode does not couple
FIG. 6. Peaks in diffuse transmittance for a L = 650 nm and b L
=1000 nm for the TM case for a rectangular Si grating on an infinite Si
substrate. The dashed lines show the interference of the zeroth and second
grating modes, the solid lines show the interference of the zeroth and fourth
grating modes and the dash-dot lines show the interference of the zeroth and
sixth grating modes. Calculation of the modal overlap allows us to predict
the dominant modes in each case.
FIG. 7. Diffuse transmittance for a TiO
2
n
s
=2.6 grating on a silicon sub-
strate with L = 900 nm, for a TE and b TM illuminations. The plots also
show the maxima in diffuse transmittance expected due to inteference be-
tween the zeroth and second grating modes dashed lines, between the
zeroth and fourth grating modes dash-dot lines, and Fabry-Pérot interfer-
ence of the zeroth grating mode for the TE case solid line.
TABLE III. Overlap integrals between zeroth diffracted orders and propa-
gating grating modes m for TM incidence for L = 650 nm and L
=1000 nm, with =600 nm.
L=650 nm L=1000 nm
n
eff
Overlap n
eff
Overlap
m=0 3.832 0.413 3.894 0.411
m=2 2.839 0.051 3.510 0.046
m=4 1.014 0.340 2.588 0.020
m=6 ¯¯1.031 0.297
013102-5 K. R. Catchpole J. Appl. Phys. 102, 013102 2007
Downloaded 28 Mar 2010 to 150.203.243.53. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

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Journal ArticleDOI
Abstract: Nanophotonic light trapping for solar cells is an exciting field that has seen exponential growth in the last few years. There has been a growing appreciation for solar energy as a major solution to the world’s energy problems, and the need to reduce materials costs by the use of thinner solar cells. At the same time, we have the newly developed ability to fabricate controlled structures on the nanoscale quickly and cheaply, and the computational power to optimize the structures and extract physical insights. In this paper, we review the theory of nanophotonic light trapping, with experimental examples given where possible. We focus particularly on periodic structures, since this is where physical understanding is most developed, and where theory and experiment can be most directly compared. We also provide a discussion on the parasitic losses and electrical effects that need to be considered when designing nanophotonic solar cells.

273 citations


Journal ArticleDOI
Abstract: The authors acknowledge the A. R. C. and NOW for research conducted at the FOM as a part of the Joint Solar Programme for financial support.

218 citations


Journal ArticleDOI
TL;DR: The main physical effects of patterning are focused on, namely a reduction of reflection losses, diffraction of light in air or inside the cell, and coupling of incident radiation into quasi-guided optical modes of the structure, which is characteristic of photonic light-trapping.
Abstract: We theoretically investigate the light-trapping properties of one- and two-dimensional periodic patterns etched on the front surface of c-Si and a-Si thin film solar cells with a silver back reflector and an anti-reflection coating. For each active material and configuration, absorbance A and short-circuit current density Jsc are calculated by means of rigorous coupled wave analysis (RCWA), for different active materials thicknesses in the range of interest of thin film solar cells and in a wide range of geometrical parameters. The results are then compared with Lambertian limits to light-trapping for the case of zero absorption and for the general case of finite absorption in the active material. With a proper optimization, patterns can give substantial absorption enhancement, especially for 2D patterns and for thinner cells. The effects of the photonic patterns on light harvesting are investigated from the optical spectra of the optimized configurations. We focus on the main physical effects of patterning, namely a reduction of reflection losses (better impedance matching conditions), diffraction of light in air or inside the cell, and coupling of incident radiation into quasi-guided optical modes of the structure, which is characteristic of photonic light-trapping.

164 citations


Journal ArticleDOI
Abstract: The optics of microcrystalline silicon thin-film solar cells with integrated light trapping structures was investigated. Periodic grating couplers were integrated in microcrystalline silicon thin-film solar cells and the influence of the grating dimensions on the short circuit current and the quantum efficiency was investigated by the numerical simulation of Maxwell’s equations utilizing the finite difference time domain algorithm. The grating structure leads to scattering and higher order diffraction resulting in an increased absorption of the incident light in the silicon thin-film solar cell. The influence of the grating period and the grating height on the short circuit current and the quantum efficiency was investigated. Enhanced quantum efficiencies are observed for the red and infrared parts of the optical spectrum. Optimal dimensions of the grating coupler were obtained.

129 citations


Journal ArticleDOI
Abstract: Dielectric gratings are a promising method of achieving light trapping for thin crystalline silicon solar cells. In this paper, we systematically examine the potential performance of thin silicon solar cells with either silicon (Si) or titanium dioxide (TiO2) gratings using numerical simulations. The square pyramid structure with silicon nitride coating provides the best light trapping among all the symmetric structures investigated, with 89% of the expected short circuit current density of the Lambertian case. For structures where the grating is at the rear of the cell, we show that the light trapping provided by the square pyramid and the checkerboard structure is almost identical. Introducing asymmetry into the grating structures can further improve their light trapping properties. An optimized Si skewed pyramid grating on the front surface of the solar cell results in a maximum short circuit current density, Jsc, of 33.4 mA cm−2, which is 91% of the Jsc expected from an ideal Lambertian scatterer. An optimized Si skewed pyramid grating on the rear performs as well as a rear Lambertian scatterer and an optimized TiO2 grating on the rear results in 84% of the Jsc expected from an optimized Si grating. The results show that submicron symmetric and skewed pyramids of Si or TiO2 are a highly effective way of achieving light trapping in thin film solar cells. TiO2 structures would have the additional advantage of not increasing recombination within the cell.

83 citations


References
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Frequently Asked Questions (1)
Q1. What are the contributions in "A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells" ?

In this paper, the authors show that simplified modal analysis can be used as a conceptual model for understanding the behavior of the diffuse transmittance of lamellar diffraction gratings on infinite substrates. Furthermore, the authors show that for thin film solar cells with front surface gratings and flat rear reflectors, modal analysis can be used to predict the optimum parameters for maximum light trapping.