A conceptual model of the diffuse transmittance of lamellar diffraction
gratings on solar cells
K. R. Catchpole
a兲
School of Photovoltaic and Renewable Energy Engineering, University of New South Wales,
Sydney NSW 2052 Australia
共Received 21 May 2006; accepted 2 April 2007; published online 2 July 2007兲
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. 关DOI: 10.1063/1.2737628兴
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,
4–7
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 共RCWA兲.
12–14
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
c–Si: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兲
Electronic mail: k.catchpole@amolf.nl
JOURNAL OF APPLIED PHYSICS 102, 013102 共2007兲
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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, ... , 共1兲
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 共lamellar兲
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
f共n
eff
兲 = cos共k
xr
L
r
兲cos共k
xg
L
g
兲 −
1
2
冉
k
xr
k
xg
+
1
k
xg
k
xr
冊
⫻sin共k
xr
L
r
兲sin共k
xg
L
g
兲 = cos共k
x
L兲, 共2兲
where k
0
is the wave vector of the incident wave, k
x
=k
0
sin共
in
兲= 共2
/兲n
a
sin共
in
兲 is the x component of the
wave vector of the incident wave, and
k
xi
= k
0
关n
i
2
− 共n
eff
兲
2
兴
1/2
, i = r,g 共3兲
are the x components of the wave vectors in the ridges and
grooves of the grating, respectively.
is equal to 1 for
transverse-electric 共TE兲 polarization, n
r
2
/n
g
2
for transverse-
magnetic 共TM兲 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 u共x兲 and a z-dependent
part
共z兲. For TE polarization, for example, only a y compo-
nent of the electric field is present, which can be written as
E
y
共x,z兲 = u共x兲
v
共z兲. 共4兲
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
, 共i.e., the fraction of power that is transferred
from the incident mode to the mth grating mode兲 is the over-
lap integral of the two modes,
21
具E
y
in
共x,0兲,u
m
共x兲典 =
冏
冕
E
y
in
共x,0兲u
m
共x兲dx
冏
2
冕
兩E
y
in
共x,0兲兩
2
dx
冕
兩u
m
共x兲兩
2
dx
. 共5兲
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. 共2兲. 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. 共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.兲
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 共i.e., elec-
tric field parallel to the grooves of the grating兲.ForaSi
grating with L =650 nm at = 1000 nm 共n
r
=3.58兲, 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 共i.e., the fraction of
light that is transmitted into each order兲 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 共2007兲
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oscillation of the zeroth and first diffracted orders, with much
less power going into the second diffracted order 共Fig. 2兲.
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 =
2兩n
2
eff
− n
0
eff
兩
. 共6兲
The positions of the peaks as calculated with Eq. 共6兲 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 共Table I兲. 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. 4共a兲.
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. 共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.兲 This is the reason that the effec-
tive refractive index of the zeroth mode is high 共3.37兲 while
the effective refractive index of the second mode is relatively
low 共1.40兲. 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. 4共b兲 the integrand in the
numerator of Eq. 共5兲 is plotted 共shaded area兲, along with the
x-dependent parts of the fields for the zeroth grating mode
and first diffracted order 共for which the value of the overlap
integral is 0.2兲. 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 共n
r
=3.58兲.
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 共solid lines兲, Fabry-
Pérot interference of the zeroth grating mode with itself 共dash-dot lines兲, and
interference between zeroth and fourth grating modes 共dashed lines兲.
TABLE I. Overlap integrals between propagating diffracted orders 共p兲 and
propagating grating modes 共m兲 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 共2007兲
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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
. 共7兲
The maxima in diffuse transmittance according to Eq. 共7兲 are
also plotted in Fig. 3 共dash-dot lines兲. 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
800–1300 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 共
=790– 1250 nm兲 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 600–700 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 = 830–1000 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 共which have overlap integrals 0.50 and 0.37, respec-
tively兲, as shown in Fig. 5 共solid lines兲. Also shown in Fig. 5
共dotted lines兲 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. 共a兲.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. 共b兲 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
=830–1000 nm 共solid lines兲 and interference between the zeroth and sixth
grating modes for = 650–750 nm 共dashed lines兲.
013102-4 K. R. Catchpole J. Appl. Phys. 102, 013102 共2007兲
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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. 6共a兲 and 6共b兲 and also in the next section in Fig.
7共b兲. We can see in Fig. 6共a兲 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. 6共b兲. 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. 共The diffusely
transmitted light should also lie outside the escape cone; this
is discussed further in Sec. III E.兲 From Eq. 共6兲 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 共i.e., phase difference of
rather
than 3
or 5
, etc.兲 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 共a兲 L = 650 nm and 共b兲 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
共n
s
=2.6兲 grating on a silicon sub-
strate with L = 900 nm, for 共a兲 TE and 共b兲 TM illuminations. The plots also
show the maxima in diffuse transmittance expected due to inteference be-
tween the zeroth and second grating modes 共dashed lines兲, between the
zeroth and fourth grating modes 共dash-dot lines兲, and Fabry-Pérot interfer-
ence of the zeroth grating mode for the TE case 共solid line兲.
TABLE III. Overlap integrals between zeroth diffracted orders and propa-
gating grating modes 共 m兲 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 共2007兲
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