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Solar Energy 122 (2015) 76–86

Multilayer antireflection coating design for GaAs0.69P0.31/Sidual-junction solar cells

Sueda Saylan a,⇑, Timothy Milakovich b, Sabina Abdul Hadi a, Ammar Nayfeh a,Eugene A. Fitzgerald b, Marcus S. Dahlem a

aMasdar Institute of Science and Technology, Abu Dhabi, UAEbMassachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 4 March 2015; received in revised form 23 June 2015; accepted 25 July 2015

Communicated by: Associate Editor Arturo Morales-Acevedo

Abstract

In this paper, we systematically investigate the use of multilayer antireflection coatings (ARC) on GaAs0.69P0.31/Si dual-junction solarcells for achieving broadband reflection suppression. We incorporate dispersion, absorption, and both coherent and incoherent interfer-ence, by modeling the solar cell as a multilayer medium of thin films and a substrate, using the transfer matrix method. The analyticalmodel is verified through reflectance measurements on a GaAs(1�x)Px/GaP stack. Our results show that optimized double- andtriple-layer ARCs can minimize reflectance to below 5% within the spectral range of �400–945 nm, with the latter maintaining thisperformance over a broader spectrum of 390–1000 nm, in comparison to 25–45% reflectance for the bare solar cell.� 2015 Elsevier Ltd. All rights reserved.

Keywords: Antireflection coating; Multi-junction solar cell; Transfer matrix method; GaAs(1�x)Px/silicon tandem cell

1. Introduction

Multi-junction solar cells (MJSCs) address the problemof energy loss due to thermalization and lack of absorp-tion. By combining materials with different band gaps ina single cell, a higher fraction of the available solar energycan be converted into electrical energy. Despite theirimportance in achieving ultimate solar conversion efficien-cies, MJSCs have been mainly used for space and highlyconcentrated terrestrial applications. The challenge ofextending their use to one-sun or low-concentrationterrestrial applications requires generating cost-effective,

http://dx.doi.org/10.1016/j.solener.2015.07.049

0038-092X/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: P.O. Box 54224, Abu Dhabi, UAE.Tel.: +971 2 810 9104.

E-mail address: ssaylan@masdar.ac.ae (S. Saylan).

high-efficiency multi-junction solar cells (InternationalEnergy Agency, 2014).

In recent years, numerous studies have been conductedaiming at optimizing the solar cell structures of two ormore junctions, using different band gap materials. Amongthe materials that have attracted tremendous interest arethe III–V compounds, due to several advantages such asband gap tunability and high absorption, as well as fabri-cation advances that allow these layers to be grown on Sithrough the use of various graded buffers (Andre et al.,2005; Carlin et al., 2000; Geisz et al., 2006; Grassmanet al., 2009, 2010; Lueck et al., 2006). This monolithicintegration of III–V materials on Si platforms combinesboth the techniques and the tools from the Si manufactur-ing technology with the rich selection of different band gapmaterials of III–V compounds.

S. Saylan et al. / Solar Energy 122 (2015) 76–86 77

As shown by Kurtz et al., a combination of band gaps of1.1 eV for the bottom cell and 1.7–1.8 eV for the top cellleads to the highest theoretical efficiency (up to 37% underone-sun AM1.5G) for a dual-junction, series-connected cell(Kurtz et al., 1990), and this arrangement can be obtainedby stacking a GaAs(1�x)Px cell with a Si cell, connected bya tunnel junction.

In order to minimize energy losses due to reflection,antireflection coatings (ARCs) can be employed on topof the solar cells. Most of the studies on ARCs focuson single-junction solar cells (Aziz et al., 2011;Gangopadhyay et al., 2004; Zhao and Green, 1991).However, the design of effective antireflection coatingsfor multi-junction solar cells is indispensable to approachthe high theoretical limits associated with those cells.Compared to the design of ARCs for single-junctioncells, the main challenge with MJSC ARCs is relatedto the broader reflection suppression requirement, sincethe combination of the different subcells will absorb ina broader range of the solar spectrum. This additionalrequirement can be satisfied with multilayer ARCdesigns.

The literature on the study of antireflection coatingsfor III–V multi-junction solar cells can be grouped intotwo main categories – the number of junctions formingthe solar cell, and the method adopted during the analy-sis. The Rouard method can be used to optimize theparameters associated with the design of multilayer antire-flection coatings (Homier et al., 2012), while some of thereported studies use heuristic search methods, such asgenetic algorithms (Schubert et al., 2008; Yan et al.,2013) to perform the same task. The transfer matrixmethod (TMM) (Conrad et al., 2014; Zhao and Green,1991; Lu and Gang, 2007) is also effective for calculatingthe reflection spectra, and can easily take into accountboth the coherent and incoherent light behavior. All theseresearch efforts, except for (Conrad et al., 2014), focus ontriple-junction solar cells, where the top cell is based onGaInP.

The main focus of our work is to implement an analyt-ical model to calculate and minimize reflection of a tandemsolar cell through the use of multilayer antireflection coat-ings. We employ an approach based on the theory ofquarter-wave dielectric layers and the transfer matrixmethod to determine the materials and the thicknesses ofthe layers forming the multilayer ARCs. The TMMapproach offers the advantage of having complete controlof the physical parameters of the ARC stack. As an advan-tage over the heuristic approaches, the analytical modelallows one to assess the reflection suppression performanceof the ARC under thickness deviations subject to the toler-ance windows of the fabrication processes. Using this ana-lytical model, we address the energy-related losses due toreflection (and hence reduced absorption of photons withenergies above the band gap of the absorber layers), in aGaAs0.69P0.31/Si dual-junction solar cell. Our approachcan be easily generalized to any solar cell structure with

various numbers of junctions and materials, assuming thatthe optical constants of all the constituent layers areknown. It can also be adapted to any number of ARClayers.

The number of junctions forming the solar cell determi-nes the cell design and the respective materials to be usedfor maximizing the efficiency. These materials also deter-mine the reflection response of the cell and the ARC designthat is required to achieve optimum antireflection. Thematerials that are traditionally used in the literature to sup-press reflection in solar cells set the basis for our materialselection. That list of materials includes TiO2, SiO2, SiNx,MgF2, ZnS, ZnO, Al2O3, Ta2O5, SiC, AlON, and HfO2.

With this material selection, we focused our work on thedesign of double-layer ARC (DLARC) and triple-layerARC (TLARC) stacks. For the DLARCs, we used SiO2/TiO2 and SiO2/SiC, and for the TLARCs we used MgF2/HfO2/SiC and MgF2/HfO2/TiO2. TiO2 has been used inthis type of applications (Aiken, 2000; Jin et al., 2003;Lien et al., 2006). In addition, the use of SiC as antireflec-tion coatings in Si-based solar cells (Klyui et al., 2002; Shenet al., 2012) and as dielectric gratings in thin-film solar cells(Esteban et al., 2010), due to its high refractive index andlow extinction coefficient, has also been reported in the lit-erature. Furthermore, SiC is known for its good passiva-tion properties (Lee et al., 2012) and hence has beenwidely used as a window layer material in thin-film solarcells (Heidt et al., 2011; Kim et al., 2012; Yu et al.,2001). Thus, the proposed coating of MgF2/HfO2/SiC isexpected to adequately meet both the passivation andantireflection requirements.

Our results show that the reflectance for the optimizedtriple-layer coating (using MgF2/HfO2/SiC) is less than10% over the entire visible and near-infrared (NIR) spec-trum, as opposed to 25% to 45% reflectance calculatedfor the bare GaAs0.69P0.31/Si solar cell. The optimizedtriple-layer design shows a significant improvement in thenear-ultraviolet (NUV) band, over the optimized double-layer design. The solar spectrum weighted reflectance(SWR) over the entire spectrum of 300–1100 nm variesbetween 2.86% and 4.37% for the investigated triple- anddouble-layer designs, respectively, compared to 32.22%for the bare GaAs0.69P0.31/Si solar cell.

2. Materials and methods

In this paper, we implement the transfer matrix methodto study the effect of antireflection coatings on the suppres-sion of reflection at the top surface of a GaAs0.69P0.31/Sidual-junction, series-connected solar cell, which is treatedas a multilayered medium in the model. TMM provides asystematic procedure for finding the transmittance andreflectance of a stack of dielectric layers of prescribedthicknesses and refractive indices. These input parametersare varied within a certain range, in order to find thecombination that leads to the lowest reflection. Thealgorithm incorporates dispersion and absorption in each

78 S. Saylan et al. / Solar Energy 122 (2015) 76–86

layer by using the complex refractive indices of eachmaterial.

A complete model should consider all the interfacesand layers involved in the solar cell (illustrated inFig. 1); however, we exploit two different simplified mod-els of the actual solar cell structure, that serve two differ-ent purposes. The first model (see Fig. 2(a)) has beenused for comparing the calculated reflectance with themeasured reflectance at the bare GaAs0.69P0.31-air inter-face. The structure used in the experiments had an addi-tional �25 nm-thick GaP layer grown on top as a caplayer. We provide a comparison of the calculated andmeasured reflectance spectra for this structure in Sec-tion 4. The second model (see Fig. 2(b)) is used forobtaining a response as close as possible to that of theactual cell. Therefore, it incorporates both of the absorb-ing materials (GaAs0.69P0.31 and Si), while simplifying thesystem by eliminating unknown parameters such as theoptical constants of the intermediate layers. The backcontact is included, in consistency with the actual solarcell structure, in order to generalize the model such thatit can be used for modeling any substrate material andthickness. However, in our calculations it is not relevantwhether the back interface is metal or air, since the lightof interest is almost totally absorbed by the 650 lm Sisubstrate. These two models form the basis for our anal-ysis, which can be easily extended to represent any actualsolar cell design, given the optical constants of the con-stituent layers are known.

The optimum refractive indices for the layers composingthe ARCs, and thus the ARC materials, are determinedbased on the principles of quarter-wave films (Macleod,1986). For a double-layer ARC, the optimum refractiveindices are found by

n1 ¼ffiffiffiffiffiffiffiffiffiffiffiffin20 � ns3

qð1Þ

and

Back contact

Front contact

window layer

subcell

Back surfacefield (BSF)

Buffer andcap layers

Si subcellTunnel junction

In(1-x)GaxP

GaAs(1-x)Px

Fig. 1. Illustration of a GaAs(1�x)Px/Si series-connected dual-junction celldesign.

n2 ¼ffiffiffiffiffiffiffiffiffiffiffiffin0 � n2s3

q; ð2Þ

where n0 and ns are the refractive indices of the two layerssurrounding the ARC, which in this case are air at the topand GaAs0.69P0.31 at the bottom, respectively. n1 and n2correspond to the two materials forming the ARC, fromtop to bottom. Similarly, for a triple-layer ARC, the fol-lowing relationships hold for the refractive indices of theconstituent layers:

n1 ¼ffiffiffiffiffiffiffiffiffiffiffiffin30 � ns4

qð3Þ

n2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffin0 � nsp ð4Þ

n3 ¼ffiffiffiffiffiffiffiffiffiffiffiffin0 � n3s4

q; ð5Þ

where n1, n2 and n3 are the optimum refractive indices forthe top, middle, and bottom layers of the TLAR coating,respectively. Using n0 as air and ns as GaAs0.69P0.31, theseequations return the indices of refraction summarized inTables 1 and 2, for the DLARC and TLARC structures,respectively.

Based on these values, the most suitable materials forthe different layers are the following: SiO2 (1st layer) andSiC or TiO2 (2nd layer) for the DLARC, and MgF2 (1stlayer), HfO2 (2nd layer) and TiO2 or SiC (3rd layer) forthe TLARC. The refractive indices (n) and extinctioncoefficients (k) for all the relevant materials are plottedin Figs. 3 and 4, respectively. All the optical constantsare taken from SOPRA (SOPRA), except for TiO2 (Devore,1951; Vos and Krusemeyer, 1977), GaP (Milakovich et al.,unpublished results), and GaAs0.69P0.31 (Milakovich et al.,unpublished results). In all calculations, we assume normalincidence.

3. Quantitative analysis of multilayer interference

In general, each layer of the multilayer stack will gener-ate interference fringes, assuming a coherent behavior.However, when the thickness of any of the layers in themultilayer medium is larger than several wavelengths, asis the case for the GaP and Si substrates in the model pre-sented in Fig. 2, these fringes are usually not resolvedexperimentally (Harbecke, 1986). Furthermore, in thicklayers, the phase relation between the front and backreflected waves is random. It is therefore required thatthe analytical model incorporates both coherent and inco-herent interference effects between the light reflected at thedifferent interfaces involved. In our model, we consider the2 lm thick GaAs0.69P0.31 layer to be coherent.

Table 1Optimum refractive indices for the different layers of the DLARC.

Wavelength (nm) GaAs0.69P0.31 Top layer Bottom layer

400 4.74 1.68 2.82500 3.90 1.57 2.48600 3.65 1.54 2.37800 3.38 1.50 2.25

Table 2Optimum refractive indices for the different layers of the TLARC.

Wavelength (nm) Top layer Middle layer Bottom layer

400 1.48 2.18 3.21500 1.40 1.97 2.77600 1.38 1.91 2.64800 1.36 1.84 2.49

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.110−6

10−5

10−4

10−310−210

−110

0

Wavelength (µm)

Extin

ctio

n co

effic

ient

SiC GaP GaAsTiO2

GaAs0.69P0.31

GaAs0.69P0.31 (extrapol.)

Fig. 4. Extinction coefficients of the materials used for the ARCoptimizations. The values corresponding to GaP and GaAs are alsoplotted for reference.

1

2

3

4

5

Wavelength (µm)

Ref

ract

ive

inde

x

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

MgF2 SiO2 HfO2

SiC GaPTiO2

GaAs GaAs0.69P0.31

Fig. 3. Refractive indices of the materials used for the ARC optimizations.The values corresponding to GaP and GaAs are also plotted for reference.

S. Saylan et al. / Solar Energy 122 (2015) 76–86 79

For coherent layers, i.e. for all the layers except for thesubstrates, the matrix method followed by Chuang (2009)has been used to obtain the reflection coefficient for normalincidence light. Following this notation, the amplitudes ofthe incident and reflected waves in layer 0 of the multilay-ered medium, illustrated in Fig. 5 are related to the ampli-tude in the transmitted region by

E0

rE0

� �¼ B01B12B23 . . .BNðNþ1Þ

tE0

0

� �¼ b11 b12

b21 b22

� �tE0

0

� �:

ð6ÞHere, we assume that there is no incident wave from the +x

direction. The reflection coefficient of the multilayeredmedium is therefore

r ¼ b21b11

: ð7Þ

The backward-propagation matrices, Bl(l+1) in (6) aresimply formed by the product of two matrices: interfacetransmission matrix Tl(l+1), followed by a uniform layerpropagation matrix Pl+1, in medium l + 1, where Tl(l+1)

and Pl+1 are defined as

T lðlþ1Þ ¼ 1

2

1þ P lðlþ1Þ 1� Plðlþ1Þ1� P lðlþ1Þ 1þ Plðlþ1Þ

� �ð8Þ

Plþ1¼e�ikðlþ1Þxhðlþ1Þ 0

0 eikðlþ1Þxhðlþ1Þ

� �: ð9Þ

Back contact

Si substrate

Si thin film

filmthin GaAs(1-x)Px

(a)

Fig. 2. Physical models used in the analyses: (a) two-layer stack composed of acalculated vs. measured reflectance; (b) four-layer stack comprising both GaAs0reflector, closely mimicking the actual cell.

The P lðlþ1Þ term in (8) is a manifestation of the impe-dance mismatch between the two consecutive media, corre-sponding to an electromagnetic wave traveling from one

GaP substrate

filmthin GaAs(1-x)Px

(b)

GaAs0.69P0.31 film (2 lm) and a GaP substrate (400 lm), for comparing the

.69P0.31 (2 lm) and Si (1 lm) films on a Si substrate (650 lm), and the back

Fig. 5. Illustration of the multilayered medium considered in the transfermatrix method. Redrawn and adapted from (Chuang, 2009).

Fig. 6. Illustration of the multilayered medium considered in the transfermatrix method, incorporating the optically thick substrate.

80 S. Saylan et al. / Solar Energy 122 (2015) 76–86

medium to another medium with different characteristicimpedances, and it is defined as

P lðlþ1Þ ¼ llkðlþ1Þxllþ1klx

¼ glglþ1

; ð10Þ

where li and gi are, respectively, the permeability and char-acteristic impedance of each media, and kix is the wave vec-tor component in the direction in which the medium isinhomogeneous (x-direction in our figure).

In the solar cell model, any wave reflected at the bottominterface of the optically thick substrate interferes incoher-ently with the waves reflected at the remaining coherentinterfaces. Here, we present the mathematical formalismused to incorporate the optically thick substrate into oursolar cell reflectance calculations. The reflected power fromthe stack of consecutive coherent layers (1 to N in Fig. 5)can be represented by a single term R0(N+1), where

R0ðNþ1Þ ¼ jrj2, with r given by (7).We now assume that we introduce a thick substrate after

layer N, as illustrated in Fig. 6. The reflection and thetransmission coefficients at the substrate/(N + 1) interfaceare given by

rsðNþ1Þ ¼ ~ns � nNþ1

~ns þ nNþ1

; tsðNþ1Þ ¼ 2~ns~ns þ nNþ1

; ð11Þ

where the refractive index of the substrate material ~ns iscomplex and nNþ1 is 1 in case of air. In the case of a metal backcontact, rs(N+1) and ts(N+1) can be approximated to �1 and0, respectively, representing the behavior of a perfect metal.

The incoherence between the waves reflected from thesubstrate/(N + 1) interface and the remaining coherent lay-ers imposes the addition of intensities rather than fieldamplitudes, as shown in the literature (Katsidis andSiapkas, 2002). The intensities of the incident and reflectedwaves of a multilayered medium with an optically thicksubstrate, such as the one shown in Fig. 6, are related tothe transmitted intensity by

jE0j2R0ðNþ1ÞjE0j2

" #¼ M0sM sM sðNþ1Þ

T 0ðNþ1ÞjE0j20

" #

¼ m11 m12

m21 m22

� �T 0ðNþ1ÞjE0j2

0

" # ; ð12Þ

where M0s, Ms, and Ms(N+1) are defined as

M0s ¼ 1

jt0sj21 �jrs0j2jr0sj2 jt0s � ts0j2 � jr0s � rs0j2

" #ð13Þ

M s ¼ j expð�iksx � hsÞj2 0

0 j expðiksx � hsÞj2" #

ð14Þ

M sðNþ1Þ ¼ 1

jtsðNþ1Þj21 jrsðNþ1Þj2

jrsðNþ1Þj2 1

" #: ð15Þ

Here, ksx = (ns + ijs)(2p/k)cos(hi) is the component ofthe wave vector in x-direction in the substrate, and nsand js are the real and imaginary parts of the refractiveindex of the substrate material, respectively. At normalincidence, hi = 0�, ksx = ks = (ns + ijs)(2p/k).

r0s used in (13) is defined in the same way as the reflec-tion coefficient defined in (7), and rs0, t0s, and ts0 can also bedefined using the elements of the coherent transfer matrix,similar to (7), through the following expressions:

rs0 ¼ � b12b11

ð16Þ

t0s ¼ 1

b11ð17Þ

ts0 ¼ b11 � b22 � b12 � b21b11

: ð18Þ

The fraction of power reflected from the entire stack, tak-ing into account the coherent layers and the incoherentsubstrate, is given by the reflectance R0(N+1) as

R0ðNþ1Þ ¼ m21

m11

; ð19Þ

where m11 and m21 can be obtained from the product ofM0sM sM sðNþ1Þ as follows:

m21 ¼ jr0sj2jt0sj2:jtsðNþ1Þj2

j expð�iksx � hsÞj2 þ jrsðNþ1Þj2jt0sj2:jtsðNþ1Þj2

�ðjt0s � ts0j2 � jr0s � rs0j2Þj expðiksx � hsÞj2 ð20Þ

m11 ¼ 1

jt0sj2:jtsðNþ1Þj2j expð�iksx � hsÞj2

þ jrsðNþ1Þj2jt0sj2:jtsðNþ1Þj2

jrs0j2j expðiksx � hsÞj2 ð21Þ

S. Saylan et al. / Solar Energy 122 (2015) 76–86 81

Using (20) and (21) in (19), and dividing both the nomina-tor and denominator by exp(2jshs)(2p/k), we obtain:

R0ðNþ1Þ ¼ jr0sj2 þjrsðNþ1Þj2:jt0s � ts0j2: expð�4jshs 2pk Þ1� jrs0j2:jrsðNþ1Þj2: expð�4jshs 2pk Þ

: ð22Þ

4. Results and discussion

We first discuss the experimental and theoretical resultsobtained for a stack of GaAs(1�x)Px (2 lm-thick) and GaP(substrate), with a 25 nm-thick GaP cap layer on top, usingthe model described in the previous section. Fig. 7 is a com-parison of the reflectance calculated and measured for thisstructure. Both computed and measured curves match well,where the small differences between them can be explainedby possible differences between the actual and experimentaloptical constants that are used for the analysis. The reflec-tance measurement was performed using a Solar Cell Spec-tral Response QE/IPCE Measurement System (QEX7, PVMeasurements, Inc., Boulder, Colorado USA). We havefurther validated the model against experimental resultsfrom the literature that focus on fabrication of differentARC designs (Bouhafs et al., 1998; Liu et al., 2014, 2013).

We now use the same model to optimize the thicknessesof the ARC layers, by minimizing the reflectance spectrafor each layer combination. We start with a double-layerARC, and then extend the analysis to a triple-layer ARC.

4.1. Optimization of double-layer ARCs

As mentioned before, for the DLARC, SiO2 as the 1stlayer and SiC or TiO2 as the 2nd layer are the best suitablematerials for mitigating the reflection. We start by studyingthe effect of a SiC layer on top of the stack illustrated inFig. 2(a), assuming an overcladding of SiO2. The resultsare mapped in Fig. 8, which shows the reflectance for dif-ferent thicknesses of SiC.

As seen from the figure, the reflectance exhibits a peri-odic pattern as a function of the thickness of the SiC layer.At certain thickness values, the reflectance is minimized(dark blue zones in the colormap). This indicates thatcontinuously increasing the SiC thickness does not provide

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.2

0.3

0.4

0.5

Wavelength (μm)

Ref

lect

ance

TMM − bare surfaceExpt. − bare surface

Fig. 7. Reflectance calculated and measured at the top surface of theGaAs0.69P0.31(2-lm-thick)/GaP (substrate) stack, with a 25 nm-thick GaPcap layer.

a better performance (decreased reflectance). In fact, due tothe absorption of SiC below 450 nm, it is desirable to use alayer as thin as possible. One can therefore effectively selectthe optimum SiC thickness using the reflectance–thicknessmap, where the selected thickness minimizes the reflectanceat a spectral band as wide as possible, while keeping thelayer as thin as possible. From the figure, we determine thatthe optimum thickness of the SiC layer is around 45 nm.For this thickness, the reflectance is minimized within awide wavelength range above �400 nm, where theAM1.5G solar spectrum shows a steep increase and reachesits maximum.

By using this selected thickness for the bottom layer ofthe DLARC, we generate a second map incorporatingthe top layer, forming the complete ARC structure (whichis comprised of SiC and SiO2). The incident medium is nowassumed to be air. This map determines the optimum thick-ness for the top layer, for the previously fixed thickness ofthe bottom layer. Fig. 9 shows the reflectance calculatedwith the SiO2 layer on top of the SiC layer, which has anoptimized thickness of 45 nm.

It is observed from Fig. 9 that the optimum thickness forthe SiO2 layer is about 95 nm, around which the reflectanceis decreased for the wavelengths in the range of370–850 nm, compared to other thicknesses. Similarly, weoptimize the thicknesses of a second set of materials:SiO2/TiO2. The optimum thicknesses found for this mate-rial combination are 45 nm and 90 nm, for the TiO2 andSiO2, respectively. Having material options for similarARC performances may prove to be useful in the fabrica-tion of these antireflection coatings. Fig. 10 plots the reflec-tance spectrum for the optimized DLARC sets [set 1: SiO2

(95 nm)/SiC (45 nm) and set 2: SiO2 (90 nm)/TiO2

(45 nm)], as well as the curve for the bare GaAsP/GaPstack, obtained from the TMM model.

As seen from Fig. 10, in the spectral range between �700and 1100 nm, the reflectance is still significantly high,despite the integration of the ARC. It is important tounderstand the source of the high reflectance observed atthis portion of the spectrum. Fig. 11 shows that thewavelengths above �700 nm are not absorbed by either

Wavelength (µm)

SiC

thic

knes

s (nm

)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

40

80

120

160

200

0.05

0.1

0.15

0.2

0.25

Fig. 8. Reflectance calculated from the multilayer stack of SiO2/SiC/GaAs0.69P0.31/GaP, where SiO2 is the incident material and the thicknessof the SiC layer is varied.

Wavelength (µm)

SiO 2 th

ickn

ess (

nm)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

40

80

120

160

200

0.1

0.2

0.3

Fig. 9. Reflectance calculated from the stack of SiO2/SiC (45 nm)/GaAs0.69P0.31/GaP, where the thickness of the SiO2 layer is varied.

Inte

nsity

per

pas

s

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.250.5

0.751

GaAsP − 2 µm

Wavelength (µm)

Wavelength (µm)

Inte

nsity

per

pas

s

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5

00.250.

0.751

GaP − 400 µm

(a)

(b)

Fig. 11. Normalized intensity for GaAs0.69P0.31 and GaP, after one passthrough the respective layer, with thicknesses of 2 lm and 400 lm,respectively.

82 S. Saylan et al. / Solar Energy 122 (2015) 76–86

the GaAs0.69P0.31 or the GaP layers. Therefore, thiswavelength range is reflected at the bottom GaP/air inter-face, back to the top surface. In the final cell shown inFig. 1, this will not be the case, since the bottom Si cell willabsorb the light at the longer wavelengths.

We now calculate the reflectance from the structure inFig. 2(b) covered with the optimized double-layer antireflec-tion coatings, illustrated in Fig. 12. As seen from the figure,the proposed DLARC sets reduce the reflectance below15% over the visible and NIR wavelengths, as opposed toabove 25% for the bare surface. The strongest antireflectioneffect is observed in the visible band, where the AM1.5Gsolar radiation is strongest and mostly absorbed by thetop subcell. About 15% of the incident light is absorbedby the 45 nm SiC layer at a 400 nm wavelength, with theabsorption gradually dropping to zero at around 450 nm.At 400 nm, the reflectance calculated at the bare surfaceof the structure in Fig. 2(b) is around 45%. Although someof the incoming sunlight is absorbed by the SiC layer, thisoccurs at wavelengths where the solar radiation is weakand the reflectance from the bare surface is the highest, asshown in Fig. 12. Under these conditions, the DLARC set1, based on SiC, still allows for about 82% of the incominglight to be transmitted into the underlying subcells, even at400 nm. The parasitic losses associated with DLARC set 2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

Bare surfaceDLARC (set 1) DLARC (set 2)

Ref

lect

ance

Wavelength (µm)

Fig. 10. Reflectance calculated with the optimized DLARC sets [set 1:SiO2/SiC and set 2: SiO2/TiO2], integrated at the top surface of theGaAs0.69P0.31/GaP stack, along with the reflectance modeled for the samestructure without the ARC. The optimized thicknesses for the twoDLARC combinations are given in Table 3.

are even less, since a 45 nm-thick TiO2 layer absorbs lessthan a 45 nm-thick SiC layer, with a negligible absorption(in TiO2) above wavelengths of 400 nm.

4.2. Optimization of triple-layer ARCs

We now focus on the antireflection performance oftriple-layer coatings to be used with the GaAs0.69P0.31/Sidual-junction solar cells, and we present the results fortwo sets of material systems: MgF2/HfO2/SiC and MgF2/HfO2/TiO2. Our analysis is similar to the one describedfor the DLARC.

We first start with the optimization of the bottom SiClayer thickness, assuming an overcladding of HfO2.Fig. 13 maps the reflectance for the stack illustrated inFig. 2(a), where the thickness of the SiC layer is varied.The optimized thickness for the SiC film is estimated tobe around 40 nm from the figure. Similarly, Fig. 14 mapsthe reflectance for the same structure with a HfO2/SiC

0.1

0.2

0.3

0.4

0.5

0

5x1018

4x1018

3x1018

2x1018

1x1018

00.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Ref

lect

ance

Wavelength (μm)

Phot

on fl

ux d

ensi

ty (m

- ²s-1nm

-1)

DLARC (set 1) DLARC (set 2)Bare surfaceAM1.5G

Fig. 12. Reflectance calculated with two different sets of DLARC andwithout ARC, using the structure in Fig. 2(b). The optimized thicknessesfor the different ARC layers are given in Table 3, and the solid linerepresents the Solar Spectral Irradiance AM1.5G (Reference SolarSpectral Irradiance Air Mass AM1.5).

Wavelength (µm)

HfO

2 thic

knes

s (nm

)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

40

80

120

160

200

0.05

0.1

0.15

0.2

0.25

Fig. 14. Reflectance calculated from the stack of MgF2/HfO2/SiC(40 nm)/GaAs0.69P0.31/GaP multilayer structure, where MgF2 is theincident material, and the thickness of the HfO2 layer is varied.

Wavelength (µm)

MgF

2 thic

knes

s (nm

)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

40

80

120

160

200

0.05

0.1

0.15

0.2

0.25

Fig. 15. Reflectance calculated from the stack of MgF2/HfO2 (45 nm)/SiC(40 nm)/GaAs0.69P0.31/GaP multilayer structure, where the thickness ofthe MgF2 layer is varied.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

Bare surfaceDLARC (set 1) TLARC (set 1)

Ref

lect

ance

Wavelength (µm)

Fig. 16. Reflectance calculated with the triple-layer ARC set 1 (MgF /

S. Saylan et al. / Solar Energy 122 (2015) 76–86 83

(40 nm) stack, where the thickness of the top layer is var-ied, and the incident material is assumed to be MgF2.Based on the figure, the optimum thickness for the HfO2

film is estimated to be around 45 nm, minimizing the reflec-tance substantially in a broad spectral range, especially inthe visible. The thickness of the MgF2 layer has also beendetermined using the same approach, and the respectivecolormap is shown in Fig. 15. This figure returns an opti-mized MgF2 thickness of about 75 nm. The performanceof the optimized TLARC is plotted in Fig. 16, along withthe curves for the optimized DLARC (set 1) and bareGaAsP/GaP stack, obtained from the TMM model.

The performance of the TLARC is now shown on top ofthe stack illustrated in Fig. 2(b), in order to evaluate theexpected performance of the final GaAs0.69P0.31/Sidouble-junction solar cell. The results are plotted inFig. 17, showing that the proposed triple-layer ARC (set1) can reduce the reflectance below 10% over the entirevisible and NIR spectrum, when compared to over 25%reflectance for the bare GaAs0.69P0.31/Si solar cell. TheTLARC has an improved performance in the NUV bandcompared to the DLARC.

The same approach is applied to the second set of mate-rials selected for the TLARC: MgF2/HfO2/TiO2. Theresults for this option are also presented in Fig. 17, usingthe optimized film thicknesses of 70 nm (MgF2), 40 nm(HfO2), and 35 nm (TiO2). As shown in the figure, incorpo-rating this design reduces the reflectance to values below�10% over the entire visible and NIR spectrum, similarlyto the TLARC set 1. It is worth mentioning that all thedesigns presented in Fig. 17 reduce the reflectance to valuesbelow 5% in the wavelength range from �390 nm to�940 nm. The optimized thicknesses and the solar spec-trum weighted reflectance over the entire range of 300–1100 nm for the different ARC combinations are summa-rized in Table 3.

4.3. Current output analysis using a proposed triple-layer

ARC

The optimal operation of series-connectedmulti-junctionsolar cells requires the subcells to be current-matched.

Wavelength (µm)

SiC

thic

knes

s (nm

)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

40

80

120

160

200

0.05

0.1

0.15

0.2

Fig. 13. Reflectance calculated from the stack of HfO2/SiC/GaAs0.69P0.31/GaP multilayer structure, where HfO2 is the incident material, and thethickness of the SiC layer is varied.

2

HfO2/SiC) and the double-layer ARC set 1 (SiO2/SiC), integrated at thetop surface of the structure in Fig. 2(a), along with the reflectance modeledfor the same structure without the ARC. The optimized thicknesses aregiven in Table 3.

In order to investigate this fulfilment, we calculate theshort-circuit current density for each subcell, using one ofthe proposed triple-layer ARC combinations (set 1), basedon the well-known short-circuit photocurrent equation:

J sc ¼ qZ

ð1� RðEÞÞF ðEÞð1� e�aðEÞdÞdE ð23Þ

where R(E) is the spectral reflectance calculated by themodel, F(E) the photon flux density of the AM1.5G solar

0

0.1

0.2

0.3

0.4

0.5

Wavelength (µm)

Ref

lect

ance

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

DLARC (set 1) TLARC (set 1)TLARC (set 2) Bare surface

Fig. 17. Reflectance calculated with two different triple-layer ARCdesigns; set 1 (MgF2/HfO2/SiC) and set 2 (MgF2/HfO2/TiO2), and thedouble-layer design set 1 (SiO2/SiC), integrated at the top surface of thestructure in Fig. 2(b), along with the reflectance modeled for the samestructure without the ARC. This emulates the response of the final solarcell from Fig. 1. The optimized thicknesses for the different ARCcombinations are given in Table 3.

15

16

17

18

19

20

21

0.75 1 1.25 1.5 1.75 2 2.25

Top cellBottom cell

GaAs0.69P0.31 thickness (μm)

Cur

rent

den

sity

(mA

/cm

2 )

Fig. 18. Short-circuit current density calculated for different GaAs0.69P0.31

thicknesses, using a TLARC (set1).

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1. 10

0.1

0.2

Ref

lect

ance

t = 1 μm

t = 1.25 μmt = 1.5 μm

2 = t μm

Wavelength (μm)

Fig. 19. Reflectance calculated for different GaAs0.69P0.31 thicknesses,using a TLARC (set1).

84 S. Saylan et al. / Solar Energy 122 (2015) 76–86

spectrum, aðEÞ the spectral absorption coefficient, and d

the absorber layer thickness of the subcell. We assume per-fect collection of the photogenerated carriers, and zerocontact grid area. For the light absorbed in the bottomSi cell, we assume that the optical path length is doubleddue to the back contact. In order to eliminate the effectof the back contact reflectivity on the short-circuit current,the back contact is assumed to be a perfectly reflecting mir-ror. The band gap of the GaAs0.69P0.31 is estimated to beabout 1.8 eV (Pitt and Stewart, 1975). Fig. 18 shows thetop and bottom subcell current outputs for differentGaAsP thicknesses, and Fig. 19 plots the respective reflec-tance curves for these same thicknesses, calculated for atandem cell covered with the designed TLARC (set 1). Asseen from Fig. 19, as the thickness of the GaAsP layerchanges, the reflectance remains unchanged for wave-lengths below the band gap of the top cell. For longerwavelengths, the locations of the interference fringeschange; however, the spectral average of the fringesremains the same.

As seen from Fig. 19, the proposed triple-layer ARCsuccessfully minimizes the reflectance, regardless of theGaAs0.69P0.31 thickness. Furthermore, the tandem cellshort-circuit current density is maximized (at about18 mA/cm2) when the GaAs0.69P0.31 thickness is set toabout 1.45 lm (seen in Fig. 18).

Table 3Optimized ARC designs.

Label Material Thickness (nm) SWR (%)

DLARC (set 1) SiO2/SiC 95/45 3.98DLARC (set 2) SiO2/TiO2 90/45 4.37TLARC (set 1) MgF2/HfO2/SiC 75/45/40 2.86TLARC (set 2) MgF2/HfO2/TiO2 70/40/35 3.45Reference – – 32.22

5. Conclusion

In this paper, we study the performance of optimizeddouble-layer and triple-layer antireflection coatings forGaAs0.69P0.31/Si double-junction solar cells. A TMMmodel has been implemented in order to calculate the opti-cal response of multilayer stacks with thick substrates. Thebest result is obtained for a triple-layer ARC, formed by aMgF2/HfO2/SiC stack, where the reflectance from aGaAs0.69P0.31/Si solar cell can be suppressed to valuesbelow 5% over the spectrum of 390 nm to 1000 nm, asopposed to values between 25 and 45% for the uncoatedcell. Alternatively, a similar performance up to wavelengthsaround 940 nm can be achieved by replacing the SiC filmby TiO2. A reflectance below 5% is also achievable withthe proposed DLARC designs in the wavelength range of400–945 nm. The TLARC improves the performance inthe NUV band when compared to the DLARC. Usingthe best performing TLARC design (set 1), the top/bottomcell current-matching is obtained when the top cell thick-ness is about 1.45 lm, leading to a tandem cell short-circuit current density of about 18 mA/cm2. This studycan be easily extended to design ARCs for different cells,

S. Saylan et al. / Solar Energy 122 (2015) 76–86 85

or with a higher number of ARC layers. It can also be usedto incorporate the effects of the window layer and the inter-mediate layers, including the SiGe graded buffer layersgrown between the Si and the GaAs(1�x)Px subcells.

Acknowledgement

This work was supported by Masdar Institute ofScience and Technology, under the MI-MIT flagship grant12RAMD2.

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