Analytical design of antire¯ection coatings for siliconphotovoltaic devices

P. Nubile

Instituto de Pesquisas Espaciais, LaboratoÂrio Associado de Sensores e Materiais, Caixa Postal 515, CEP 12201-001 SaÄo Jose dos Campos, Brazil

Received 6 March 1998; accepted 25 September 1998

Abstract

Re¯ectivity and maximum short-circuit current for multilayer optical coating structures have been calculated analytically. This approach

has been used to optimize antire¯ection coatings for silicon solar cells. Triple coating structures are indicated instead of the conventional two

layer ones, especially when the goal is to minimize re¯ectivity in the UV range. The same approach was used to calculate the maximum

short-circuit current for grooved surface solar cells. The results for ¯at and grooved surface solar cells were compared. This work shows that

texturized cells need only one coating layer to achieve the same photocurrent obtained for ¯at surface cells with three layer coatings. q 1999

Elsevier Science S.A. All rights reserved.

Keywords: Silicon solar cells; Antire¯ection coatings; Texturized solar cells

1. Introduction

Antire¯ection coatings (AR) have become a key feature

for solar cell design [1,2]. Among the existing methods to

produce such coatings, the most used are evaporation in

high vacuum, reactive sputtering and electron beam

evaporation. All these methods are capable of producing

®lms of uniform thickness and good optical properties.

The main problem for the design of antire¯ective coatings

is to calculate the values for the refraction index and layer

thickness to achieve the best performance along the desired

spectrum.

The choice of the number of coatings depends on the

range of optical spectra to which the device will be

submitted. Low cost terrestrial solar cells can work with a

single coating, since the emphasis is not in the device ef®-

ciency but in its cost. Otherwise, if the objective is improv-

ing the device performance, double and triple coatings are

necessary. The wider is the spectral range to be covered, the

higher is the number of coatings necessary to keep the

re¯ectivity below a critical value.

Double antire¯ective coatings have been used commer-

cially in most space use and high-ef®ciency devices [3].

This two-layer system replaced the single-layer one with

very good results in the visible and near-IR range [4,5].

Otherwise, double-layer systems fail when the re¯ectivity

has to be kept below 10% in the visible region as well as in

the near-UV.

This work discusses the utilization of a three-layer system

for ¯at and grooved surface space use solar cells, especially

when the idea is to improve the AR performance in the UV

region without degrading its performance in the visible and

IR regions.

2. Mathematical approach

The design of antire¯ection coatings consists in calculat-

ing and minimizing the re¯ectivity as a function of the

wavelength in the range of interest. Fig. 1 shows the studied

system with all involved parameters. The refraction index,

nj, and thickness, dj characterizes each coating layer The

absorbing material, the solar cell bulk in our case, is char-

acterized by its refraction index ns, supposing that its thick-

ness is in®nite regarding optical considerations.

The N-layer problem can be analyzed by a matrix

approach [6]. Each layer is represented by a 2 £ 2 matrix

Mj, which has the form:

Mj �cosuj

2isinuj

nj

2injsinuj cosuj

0B@1CA �1�

where u j is the optical phase shift, given by:

Thin Solid Films 342 (1999) 257±261

0040-6090/99/$ - see front matter q 1999 Elsevier Science S.A. All rights reserved.

PII: S0040-6090(98)01446-1

E-mail address: nubile@las.inpe.br (P. Nubile)

uj �2pnjdj

lcosa �2�

where l is the light beam wavelength and a is the incident

angle.

The matrix equation to solve the three antire¯ection coat-

ing problem is given by:

a

b

" #�

n0 21

n0 1

" #*M1*M2*M3*

1

ns

" #�3�

for normal incidence (a � 0). The re¯ection coef®cient is

given by:

R � a

b

���� ����2 �4�

The complex numbers a and b can be written as:

a � a1 1 ia2 �5�and

b � b1 1 ib2 �6�and the re¯ection coef®cient can be calculated by:

R � aa*

bb*� a2

1 1 a22

b21 1 b2

2

�7�

The terms present in Eq. (5) are expressed as below:

a1 � 1 2 A12B21 2 A12C21 2 B12C21

2 ns 1 2 A21B12 2 A21C12 2 B21C12

ÿ � �8�

a2 � A12B21C21 2 A21 2 B21 2 C21

1 ns A21 1 B12 1 C12 2 A12B21C12

ÿ � �9�

b1 � 1 2 A12B21 2 A12C21 2 B12C21

1 ns 1 2 A21B12 2 A21C12 2 B21C12

ÿ � �10�

b2 � 2A12B21C21 1 A21 1 B21 1 C21

1 ns A21 1 B12 1 C12 2 A12B21C12

ÿ � �11�

where A12 � n1tan�u1�, A21 � �1=n1�tan�u1�,B12 � n2tan�u2�, B21 � �1=n2�tan�u2�, C12 � n3tan�u3� and

C21 � �1=n3�tan�u3�.The analytical solution for the re¯ectivity, represented by

Eq. (7), with its coef®cients de®ned in Eqs. (8)-(11) has

been used to study triple-layer systems with the possibility

of minimizing or maximizing the re¯ectivity in special

regions of the absorbing spectrum.

3. Criteria for antire¯ection coating design

Although minimizing the re¯ectivity is highly desirable,

it is not the best criterion to achieve the AR coating para-

meters for which solar cells have the best performance. The

spectral characteristics of the incident light have to be taken

into account. From the observation of the data of the photon

¯ux for AM0, AM1 and AM1.5G conditions, it is clear that

the AR coating re¯ectivity for solar cells working under

different illumination conditions has to be minimized at

different wavelengths. At AM0 condition, it must be mini-

mized for 500 nm. Otherwise, if the incident light is close to

the AM1.5G condition, the re¯ectivity has to be minimized

for 700 nm.

Zhao and Green [3] pointed out that the maximum theo-

retical photogenerated current density is the best criterion

for AR coating design, that is, the short-circuit current

density generated by a solar cell with internal quantum ef®-

ciency of 100% in the whole absorbing spectrum. The maxi-

mum photogenerated current density is given by:

Jmaxph d1;2;3; n1;2;3

ÿ � � Zlmax

lmin

qf l� � 1 2 R l; d1;2;3; n1;2;3

ÿ �ÿ �dl

�12�where f (l ) is the photon ¯ux, lmin is the minimum radia-

tion wavelength, de®ned by quantum ef®ciency and re¯ec-

tivity measurements, and lmax is the maximum radiation

wavelength given by:

lmax � hc

Eg

�13�

where h is the Plank constant, c is the light velocity and Eg is

the energy gap.

More accurate results can be achieved if one considers the

actual internal quantum ef®ciency. In this case, Eq. (11) is

P. Nubile / Thin Solid Films 342 (1999) 257±261258

Fig. 1. Diagram for a triple-layer antire¯ection coating.

written as:

Jrealph d1;2;3; n1;2;3

ÿ � � Zlmax

lmin

qf l� �hq 1 2 R l; d1;2;3; n1;2;3

ÿ �ÿ �dl

�14�

Calculations using Eqs. (12) and (14) show that the differ-

ences in the obtained AR coating parameters in these two

cases are less than 1% for high ef®ciency silicon solar cells.

This error is comparable to the uncertainty in the determi-

nation of the layer thickness during the deposition process.

The number of adjusting parameters is quite high, but

some simpli®cations can be done in order to help achieve

one of these two conditions:

2Jmaxph

2p� 0;

2Jrealph

2p� 0 �15�

where p is the adjusting parameter. Normally, ns, n1 and d1

are known, since these quantities refer to the substrate and

passivation layer.

4. Antire¯ection coatings for ¯at and grooved siliconsolar cells

The approach developed in this work has been used to

calculate the deposition parameters for antire¯ection coat-

ings in optoelectronic devices, as solar cells and photode-

tectors. Beside this classical application, this theory is

suitable to calculate re¯ective ®lms, as indicated in some

types of band pass ®lters.

A classical problem for photovoltaic devices is the design

of antire¯ection coatings for silicon solar cells passivated

with an SiO2 layer. This layer appears when the device is

thermally treated during the emitter formation or, after it,

during the annealing procedure. The presence of the SiO2

layer decreases the recombination velocity of minority

carriers at the emitter surface. Normally, a double antire-

¯ection coating is deposited over the SiO2/Si-bulk structure.

This system has, in fact, three optical layers. Fig. 2 shows

calculated re¯ectivity for a double antire¯ection coating

formed by a top layer of MgF2 of 120 nm and a bottom

layer of ZnS of 56 nm. The SiO2 layer thickness is one of

the variable parameter in this 3D plot. This plot shows that

the re¯ectivity does not change signi®cantly with the SiO2

layer thickness in the range 400±1000 nm. But, if one

considers the re¯ectivity in the far-IR above 1000 nm, and

near-UV, below 400 nm, the re¯ectivity can change drama-

tically with the SiO2 layer thickness. Conventional space

solar cells use the value of around 10 nm for this parameter

[5,6]. This value is relatively dif®cult to change because the

silicon oxide is a native layer formed during the annealing

procedure after the emitter diffusion or implantation.

Figs. 3 and 4 show the plots of Jph(max) as a function of

the three layer thicknesses for both AM0 and AM1 condi-

tions. In these calculations the start parameters were set at

the following values: d�SiO2� � 10 nm, d�ZnS� � 56 nm

and d�MgF2� � 132 nm. These ®gures show the relatively

small in¯uence of the SiO2 layer thickness in the maximum

current collection, as qualitatively pointed out by other

authors [3,7]. In this work we also ®nd that the solar cell

performance is weakly sensitive to the MgF2 layer thickness

in the range 50±150 nm. There is no gain in the current

collection for thicker or thinner layers. Instead, if one

considers the ZnS layer, the maximum current can be

strongly affected by its thickness. The maximum current

can vary 5% between the best thickness value (around

50 nm) to the worse one (around 90 nm).

This method can also consider the case of microgrooved

cell surfaces. Fig. 5 shows a two-dimensional sketch of a

texturized cell. In this case there is no perpendicular inci-

dence of incoming light and there are two re¯ections at

different incidence angles, u and f . The parameters R1

P. Nubile / Thin Solid Films 342 (1999) 257±261 259

Fig. 2. Calculated re¯ectivity of a double-layer MgF2/ZnS antire¯ection

coating deposited over an SiO2/Si-bulk structure.

Fig. 3. Calculated maximum short-circuit current for monocrystalline sili-

con solar cells as a function of antire¯ection layer thickness at AM0 condi-

tion.

and R2 of Fig. 5 are given by: R1 � R�l; u� and

R2 � R�l; u�R�l;f�.In the case where u � f � p=4, the simplest approach to

calculate the re¯ectivity of grooved surfaces,

R2 � �R�l;p=4��2. Using this value of re¯ectivity in Eq.

(12), the maximum short-circuit current in this texturized

solar cell is shown in Fig. 6. The total current density was

calculated for AM0 and AM1 conditions. Results for non-

texturized cells were plotted for comparison.

The results shown in Fig. 6 show that, for solar cells with

microgrooved surfaces, the current is practically constant in

relation to the MgF2 thickness. In both AM0 and AM1

conditions, the maximum photogenerated current does not

depend on the layer thickness. In fact, one can achieve the

best current feature with a grooved texturized surface and

only one antire¯ection layer.

Fig. 7 shows the same results for the ZnS layer. Textur-

ization makes the maximum current photogenerated almost

invariant with respect to the ZnS layer thickness. The same

results for the maximum current can be achieved using a

texturized surface and one single ZnS layer.

This work is not considering the effects of texturization in

the surface recombination velocity. Perhaps, it is a factor of

decreasing the internal quantum ef®ciency. But it is clear

that, with convenient passivation methods, texturization is a

powerful tool for improving the carrier collection and simul-

taneously simplifying the optical coating deposition

process.

5. Conclusions

The development of an analytical method to calculate

antire¯ection coatings for solar cells is an important step

in solar cell design. Applying this method to known struc-

tures, the following conclusions can be drawn.

The maximum photogenerated current of triple antire¯ec-

tion coatings in ¯at surface solar cells for both AM0 and

AM1 conditions depends strongly on the ZnS layer thick-

ness and considerably less on the SiO2 and MgF2 layers.

Multilayer coatings are not necessary for grooved

surfaces. Calculations show that the short-circuit current

obtained with only one ZnS or MgF2 coating is practically

P. Nubile / Thin Solid Films 342 (1999) 257±261260

Fig. 4. Calculated maximum short-circuit current for monocrystalline sili-

con solar cells as a function of antire¯ection layer thickness at AM1 condi-

tion.

Fig. 5. Schematic diagram of a microgrooved solar cell.

Fig. 6. Calculated maximum short-circuit current for monocrystalline ¯at

and grooved silicon solar cells as a function of MgF2 layer thickness. For

the AM0 condition: (A) three layers with texturization, (B) three layers

planar and (C) single layer planar. For AM1 condition: (D) three layers

with texturization, (E) three layers planar and (F) single layer planar.

Fig. 7. Calculated maximum short-circuit current for monocrystalline ¯at

and grooved silicon solar cells as a function of ZnS layer thickness. For the

AM0 condition: (A) three layers with texturization, (B) three layers planar

and (C) single layer planar. For AM1 condition: (D) three layers with

texturization, (E) three layers planar and (F) single layer planar.

the same in comparison with results for triple coating struc-

tures deposited over these surfaces.

Acknowledgements

This work was partially supported by FAPESP, with

contract number 96/1958-5.

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P. Nubile / Thin Solid Films 342 (1999) 257±261 261