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

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