Solar Energy Materials & Solar Cells 98 (2012) 455–464

Contents lists available at SciVerse ScienceDirect

Solar Energy Materials & Solar Cells

0927-02

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/solmat

Theoretical enhancement of solar cell efficiency by the application of an ideal‘down-shifting’ thin film

C.P. Thomas a,n, A.B. Wedding a, S.O. Martin b

a School of Electrical and Information Engineering, University of South Australia, Mawson Lakes 5095, Australiab Sustainable Energy Centre, University of South Australia, Mawson Lakes 5095, Australia

a r t i c l e i n f o

Article history:

Received 24 January 2011

Received in revised form

10 November 2011

Accepted 15 November 2011Available online 16 December 2011

Keywords:

Detailed balance

Down-shifting

Efficiency enhancement

Luminescence

Optical thin films

Silicon solar cells

48/$ - see front matter Crown Copyright & 2

016/j.solmat.2011.11.027

esponding author.

ail address: Christopher.Thomas@unisa.edu.au

a b s t r a c t

Poor ultraviolet (UV) quantum conversion efficiency contributes to a reduction in the efficiency of

silicon based photovoltaic cells. In the UV, the main loss mechanism is through surface recombination

of photo-generated carriers due to the shallow absorption depth of high energy photons. One method

for greater utilisation of the UV region is by down-shifting UV photons to lower energies where the

quantum efficiency of silicon is higher. This work determines the potential enhancement in efficiency

that can be obtained by a luminescent down-shifting layer applied to silicon based solar cells. The

efficiency is determined through detailed balance arguments. The maximum calculated efficiency

enhancement due to an ideal down shifting process is 0.6% absolute using the AM1.5G standard spectra.

Applying a similar analysis to a multicrystalline silicon solar cell results in an efficiency enhancement

due to the down-shifting process of 0.17% absolute.

Crown Copyright & 2011 Published by Elsevier B.V. All rights reserved.

1. Introduction

One way of overcoming the spectral losses in conventionalsingle junction solar cells is through the manipulation of theincident spectrum before absorption by the solar cell. The subjectof this work focuses on the poor UV response of silicon basedphotovoltaic devices. The shallow absorption depth of UV photonsincreases the probability that the generated carriers will recom-bine via surface defect states, thereby reducing the UV response.The refractive index of silicon is comparatively higher in the UV/blue region, which further reduces the UV response due to anincreased reflection of UV photons. The high dispersivity of therefractive index in the UV region also introduces complications inoptimising anti-reflection (AR) coatings. A method of increasingthe efficiency of solar cells with poor UV response, which was firstproposed by Hovel et al. [1] is through the down-shifting of UVphotons to longer wavelengths where silicon has greater quan-tum efficiency.

A common method of increasing the UV quantum efficiency ofsilicon based photovoltaic devices is via thermally grown oxideand PECVD SiN layers to passivate surface defects in order toreduce the front surface recombination velocity [2]. However, toprovide further enhancements on the efficiency to cost ratio of

011 Published by Elsevier B.V. All

(C.P. Thomas).

silicon based photovoltaic devices, a suitable method, which addslittle to the cost and complexity of such devices is required. Anapproach that has attracted interest is via the application of anorganic luminescent down-shifting (LDS) layer on the front face ofa photovoltaic device [3,4]. Such organic dyes or pigments are lowcost and add little to device complexity and production energyconsumption; for example, it has been estimated that the addi-tion of an organic dye material in an encapsulation layer for a1 m2 module will add an additional cost of less than US$1b [3].

The focus of this work is to determine the potential enhance-ment in efficiency that can be obtained from an ideal down-shifting thin film. The approach taken has involved an adaption ofthe Shockley–Queisser detailed balance limit [5], with furtherincorporations of the work presented by Trupke et al. and Badescuet al. [6,7]. Several models have been published, which extend thedetailed balance method of Shockley and Queisser beyond the30.9% efficiency limit [8–10]. The first theoretical model based ondetailed balance to implement multiple carrier generationthrough the down-conversion of photons however, was describedby Trupke et al. [6]. It has been noted that this model involved acombination of down-conversion and down-shifting where themodelled solar cell had zero quantum efficiency and was com-pletely transparent to photons with energies greater than twicethe bandgap of the material [7].

In this work, the model developed by Trupke et al. [6] isadapted so that only down-shifting is considered. A furthermodification that is implemented in this work is to consider the

rights reserved.

Incident LightLuminescent Down Shifter

C.P. Thomas et al. / Solar Energy Materials & Solar Cells 98 (2012) 455–464456

internal quantum efficiency of silicon solar cells corresponding toboth mono-crystalline and multi-crystalline silicon solar cells (asoutlined in Sections 5 and 6). The ideal case of a top hat response isconsidered first and is discussed in Section 2. The results obtainedfrom this model and presented here include an analysis of theenhancement in efficiency of a down-shifter under the AM1.5standard spectra as opposed to the theoretical black body spec-trum at 6000 K.

Solar Cell

S0→ S1 absorption S1→ S0 emissionλ→

Fluorescence Intensity

ExtinctionCoefficient

S0

S1

S0

S1

Fig. 1. (a) Schematic representation of the solar cell and LDS layer. (b) Potential

energy level diagram of the LDS layer [12].

2. Description of model

There are several important characteristics that a LDS layershould contain in order to be effective. Obviously, the luminescentquantum efficiency (LQE), defined as the ratio of emitted photonsto the number absorbed by a luminescent material needs to behigh and ideally greater than or equal to unity. Less than unity LQEwill result in an additional loss mechanism. This effect is com-pounded further if the absorption spectrum of the down-shifteroverlaps with the spectral response of the cell. The absorptioncoefficient and film thickness should enable 100% absorption ofshort-wavelength light and be completely transparent to allwavelengths within the responsivity range of the cell. The LDSlayer should also show a strong red shift to minimise any overlapbetween the absorption and emission spectra. Reabsorption ofluminescent light by the LDS layer will provide a further lossmechanism if the LQE is less than unity. As the luminescent light isemitted isotropically, there are spectral losses that result fromluminescence escaping through the front surface of the device. Thisis again further compounded if the LDS layer’s absorption spectrumoverlaps with the spectral response of the cell. Optimally, theemission spectrum should be narrow and occur at the peak of theIQE of the cell; this however is not considered initially as it isassumed that the cell has unity quantum efficiency within itsresponsivity range. The LDS layer also requires a suitable refractiveindex to minimise device reflections and maximise the forwardcoupling of luminescent light into the cell. This establishes theconditions required for an idealised LDS layer, which provides thebasis of the model analysed in this work. For a more detaileddiscussion of these characteristics, the review of Klampaftis et al.[3] is recommended.

The initial device configuration, which is analysed hereconsists of a conventional single junction silicon solar cell witha physically deposited solid film of an LDS material on the frontsurface (Fig. 1(a)). It is assumed that the LDS layer and the solarcell surfaces are spectrally flat; however for a solid film of anorganic LDS layer, this assumption may be unrealistic [11]. TheLDS layer is modelled to have complete absorption of all incidentlight with wavelength shorter than 400 nm and is otherwisetransparent. It is assumed that the LDS layer has unit LQE andexhibits luminescence at wavelengths longer than 400 nm toensure that there is no overlap of its absorption and emissionspectra. It is also assumed that the LDS layer is a perfectinsulator.

The LDS layer modelled here is considered as a crystalline thinfilm of an organic dye molecule. The following discussion pre-sents a qualitative description of the luminescence processes oforganic species in the gas and crystal phases. The spectralproperties of such materials introduce a restriction on thecharacteristics of an ideal LDS layer. It was mentioned previouslythat an ideal LDS layer should contain an emission spectrum withits peak located at the maximum of the IQE of the cell. Thisideality is unrealistic for an organic LDS layer and is later relaxedwhere it is assumed that the red shift is just enough to considerthe effects of any reabsorption processes as negligible. The idealcase is presented and discussed first.

The electronic structure of isolated organic molecules consistsof energetic states comprised of electronic, vibrational and rota-tional components. Electronic transitions correspond to excitationenergies in the order of ultra-violet and visible photons. Vibra-tional and rotational transition energies are in the near infra-redand far infra-red, respectively. Electronic states are coupled to amanifold of vibrational states as well as other electronic states.Upon optical excitation of an electron from the ground to anexcited electronic state, the nuclear configuration of the moleculeis no longer in a steady equilibrium allowing the electron to relaxnon-radiatively to lower vibrational states within the vibrationalmanifold. Non-radiative vibronic relaxation occurs much fasterthan radiative relaxation to the ground state meaning thatradiative relaxation to the ground state generally occurs from alower energetic state. The energy lost through vibrational relaxa-tion results in an emitted photon of less energy than theexcitation photon. The frequency shift of the transitions asso-ciated with the 0-0 vibrational energy levels (lowest vibrationalenergy levels of the electronic states) of the ground and excitedelectronic states is termed the Stoke’s shift. The excited electronicstate is also coupled to other electronic manifolds of different spinmultiplicity. Radiative relaxation from these states (triplet states)is termed phosphorescence whereas the previous process

100 200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1

Wavelength (nm)

Rel

ativ

e A

bsor

ptio

n an

d E

mis

sion

Spe

ctra

Fig. 2. Generic luminescent spectra of the modelled LDS layer.

C.P. Thomas et al. / Solar Energy Materials & Solar Cells 98 (2012) 455–464 457

(transitions from singlet states) is known as fluorescence.Fig. 1(b) shows a qualitative depiction of a simplified potentialenergy level diagram showing the fluorescence process from thefirst excited singlet electronic state.

The spectral properties of molecular aggregates and crystalsdiffer in general to the spectral structure observed in isolatedmolecules, or rather molecules in the gaseous phase. The collec-tive interaction between exited molecules and other (not neces-sarily excited) molecules is termed an exciton. This interactionprovides additional and or alternate relaxation pathways for anexcited molecule in the crystal compared to the isolated case. Theexcitation energy can be lost through thermalisation or redis-tribution via inter-molecular vibrations in the crystal.

In the crystal phase, the interaction energy of neighbouringmolecules affects their equilibrium energy distribution comparedto the gas phase. In this paper, the spectral properties of the LDSlayer are idealised where it is assumed that the molecule displaysstrong absorption of UV light and shows a corresponding lumi-nescence distribution with a Stoke’s shift that enables thereabsorption properties to be neglected (Fig. 2). The emissionspectrum of Fig. 2 is used in the analysis of Sections 5 and 6. Theexact mode of how this absorption and emission spectra may beachieved in an LDS layer is beyond the scope of this paper. For anelementary treatment of the optical properties of organic mole-cular crystals, Refs. [12–15] are recommended.

Initially the solar cell is modelled to have a top hat response,which is unity between 400 nm and approximately 1100 nm(corresponding to the bandgap of silicon as 1.12 eV) and is zeroat all other wavelengths.

This is a simplistic approach, which provides an upper mostlimit to the potential enhancement that can be provided from aplanar LDS layer. The top hat response is later replaced by a morerealistic IQE of a typical silicon solar cell.

3. Theory

The theoretical determination of the upper limiting efficiencyof a single junction solar cell as a function of bandgap energy wasfirst carried out by Shockley and Queisser [5] and was termed asthe ‘detailed balance limit of efficiency’. The theorem of detailedbalance states that when a system is in statistical equilibrium, thefrequency of a transition in one direction is matched by thereverse process [16]. The approach used to find the detailed

balance limit of efficiency (to model the current–voltage (I–V)characteristic of a solar cell) therefore assumes a state of statis-tical equilibrium by setting the sum of all current generationprocesses to zero. The electrical current, which is delivered to aload is therefore equated to the difference of the rate of photo-generation of carriers and the rate of radiative and radiationlesscarrier recombination. The maximum power delivered to amatched load is then determined by the condition d(IV)/d(V)¼0.The efficiency is then found as the ratio of the power delivered tothe load and the incident spectral power. The detailed balancelimit of efficiency was found to be 30% at a bandgap of 1.12 eV foran idealised cell where only radiative carrier recombination wasconsidered [5].

The system modelled here has followed the detailed balancecriterion in order to find the limiting efficiency of an idealisedmodel of a down-shifter applied to the front surface of a siliconbased cell. The LDS layer is modelled by considering its opticalproperties as discussed in Section 2 as an augmentation of thesolar spectrum that is absorbed by the solar cell. Ignoring thisaugmentation for the moment, the absorbed and emitted photoncurrent density of the cell is modelled as a generalisation of theStefan–Boltzmann law utilising Kirchoff’s law as shown in Eq. (1)

NðA ,Eu,Eg ,m,TÞ ¼ A2

c2h3

Z Eu

Eg

E2

exp E�mkT

� ��1

dE ð1Þ

The lower energy limit of the integral is non-zero correspond-ing to the bandgap of the material and the upper energy limit (Eu)is reduced from infinity corresponding to a short wavelength cut-off for the top hat response. In Eq. (1), N is the photon currentdensity,A is the etendue, E the spectral energy and m is thechemical potential of the emitted light corresponding to thedifference of the quasi Fermi energy levels of a two band systemof a conventional cell [17,18]. The solar disc is initially modelledas a blackbody radiator with a surface temperature of Ts¼6000 K(the AM1.5 standard spectra is also used and is discussed in alater section). The Earth’s atmosphere and solar cell are modelledas a blackbody and a non-thermal emitter, respectively, withchemical potential m both at an ambient temperature ofTa¼300 K. The etendue (A) describes the propagation of lightthrough an optical system and depends on the refractive index n

of the medium, the angle y of propagation to the normal of anelement of surface area dA and the solid angle dO as [19]

A ¼ZZ

n2 cosðyÞdOdA ð2Þ

Taking the surface area A to be unity, the etendue can beexplicitly stated as

A ¼ pn2 sin2yh ð3Þ

In Eq. (3), yh is the half angle of the emitted and absorbedradiation. In the first stage of this model, the refractive indices ofthe solar cell and down-shifting material are both set to unity(non-unity refractive index is also considered and shown below).For unconcentrated solar irradiation, the view factor of the sun isfound by setting yh in Eq. (3) equal to the half angle subtended bythe sun, which is 0.2661. This produces a sun view factor ofAs¼6.85�10�5. The view factor produced by the emission of ablackbody into a hemisphere, or into a material of the samerefractive index as modelled here is determined using yh¼p/2 inEq. (3), which results in Ah¼p. The radiation absorbed by the cellfrom the environment is described as the view factor from ahemisphere minus the view factor formed by the sun. The totalcurrent, I, generated by the solar cell is calculated as the productof the total photon flux N

0

(Eq. (1)) (corresponding to the sum ofphoto-generation and radiative recombination), the quantum

C.P. Thomas et al. / Solar Energy Materials & Solar Cells 98 (2012) 455–464458

efficiency Z and the electronic charge q as

I¼ qN0Z ð4Þ

The quantum efficiency of the solar cell is initially modelled asa top hat and is hence set to unity within its selectivity range andzero elsewhere as follows:

Z¼1, Eg rErEu

0, EoEg or E4Eu

(ð5Þ

The total current generated in the solar cell has contributionsfrom the solar disc, the environment, the luminescent emission(radiative recombination) of the solar cell and the photo lumines-cence of the down-shifting material

I1ðmÞ ¼ qNðA s,Eu,Eg ,0,TsÞþNðAh,Eu,Eg ,0,TaÞ

�NðAh,Eu,Eg ,m,TaÞþ12 NLDS

!ð6Þ

The first two terms in Eq. (6) describe the photo-generatedcurrent due to the absorption of the photon flux from the solardisc and the atmosphere. The third term of Eq. (6) describes theluminescent emission corresponding to the radiative recombina-tion in the solar cell. The factor of 1/2 before the fourth term inEq. (6) takes into account that the photo luminescence from theLDS layer occurs isotropically where half of the photons are lostthrough the front surface of the device. It has been assumed thatthere is a perfectly reflecting surface at the rear of the solar cellmeaning that radiative recombination losses occur through thefront surface only.

The optimum efficiency of the cell is determined by finding themaximum power point of the corresponding I–V curve wherethe chemical potential of the cell can be altered by varying thevoltage across the cell via the relation m¼qV. The efficiency isdetermined as the ratio of the power generated by the cell at itsmaximum power point divided by the total incident power fromthe solar disc.

4. Results

The results of the model described above are shown in Fig. 3.The upper most line is the detailed balance limit of Shockley andQueisser [5] with a maximum efficiency of 30.9% at 1.3 eV. The

0.5 1 1.5 212

16

20

24

28

32

Energy Gap (eV)

Effi

cien

cy (%

)

Fig. 3. Calculated cell efficiency as a function of energy gap. The fine line

represents the detailed balance limit of Shockley and Queisser and the thick line

shows the same system with a shortwave cut-off. The broken line represents the

efficiency of the model including the LDS layer.

lowest line represents the same system but with a shortwaveenergy cut-off at 400 nm resulting in a maximum efficiency of27.5%. The middle line corresponds to the efficiency of the overalldevice including the LDS layer, which shows an enhancement ofapproximately 1.5% over the adapted detailed balance limit at1.12 eV. The loss that has occurred in the LDS coated model tothat of the conventional detailed balance limit is due to the factthat half of the luminescence from the layer is emitted away fromthe cell to the atmosphere.

It is important to note that in Fig. 3, the theoretical black bodyspectrum has been used to evaluate the result. This over-esti-mates the potential enhancement in efficiency that can beobtained via a UV down-shifting scheme by terrestrial solar cellsas a significant proportion of UV light is absorbed in the atmo-sphere. To provide a more meaningful result, the ASTM G 173-03standard reference tables for terrestrial irradiation have beenused in the manner described above in place of the theoreticalblack body spectrum. This standard includes AM1.5 referencetables for direct normal and hemispherical radiation on a 371tilted surface. The results obtained using the AM1.5 standardspectrum are shown in Fig. 4. The upper most curve indicates thedetailed balance limit with the standard spectra and is compar-able with the work presented in Tiedje et al. [20], which has (as isthe case here) a limiting efficiency of approximately 32.9% at thebandgap of silicon (1.12 eV). The lowest curve as before indicatesthe efficiency of the same system but with a shortwave cut-off at400 nm. The middle line shows that the efficiency enhancementthat can be achieved utilising a UV LDS layer under an AM1.5standard spectrum is now approximately 0.6%.

It is clear from the above results that the potential enhance-ment that an LDS layer can provide can be essentially doubled ifan effective light trapping scheme is constructed to ensureminimal luminescent light is lost through the front surface ofthe cell. The refractive index of the LDS layer therefore plays acrucial role in ensuring there is both a good optical couplingbetween the cell and the LDS layer to maximise forward trans-mission of luminescent light into the cell and also minimisedevice reflections.

In the above model, the refractive index of the solar cell andLDS layer are set to unity for simplicity. This is not realistic, and asshown in Sections 4.1 and 6, the refractive index of the LDS layerhas a significant impact on the down shifting process.

0.5 1 1.5 218

22

26

30

34

Energy Gap (eV)

Effi

cien

cy (%

)

Fig. 4. Calculated cell efficiency as a function of energy gap using AM1.5 standard

spectral irradiance. The line marking scheme is the same as for Fig. 3.

0.5 1 1.5 214

18

22

26

30

Energy Gap (eV)

Effi

cien

cy (%

)

Fig. 5. Cell efficiency of the solar cell and LDS layer with non-unity refractive

index. The fine line corresponds to the efficiency of the system with a LDS layer.

The broken line corresponds to the same system with the down shifting process

removed where the LDS layer is providing enhanced optical coupling alone. The

thick line corresponds to an uncoated cell.

C.P. Thomas et al. / Solar Energy Materials & Solar Cells 98 (2012) 455–464 459

4.1. Refractive index

The refractive indices of the LDS layer and the cell material willaffect the overall efficiency of the solar cell via device reflections.Non unity refractive indices were considered in the work presentedin Trupke et al. [6] where the importance of a high value of therefractive index of the conversion layer was demonstrated. Thiseffect is amplified in the down-conversion case where (in the caseof unity refractive indices) isotropic emission of down-convertedlight effectively negates the down-conversion process. Also, as thedown-converter modelled by Trupke et al. [6] had a responsivity,which overlaps with that of a silicon solar cell, half of the photonsin the energy range between the bandgap and twice the bandgap ofthe cell that would be otherwise absorbed by the cell are lost byisotropic re-radiation back to the atmosphere. This process resultsin an overall efficiency loss and clearly demonstrates the need forlight trapping. The overall device efficiency from the model ofTrupke et al. [6] was optimised by a conversion layer whoserefractive index approached infinity. An important factor thatwas neglected and pointed out in the work by Badescu et al. [7]is that increased device reflections would result from increasingvalues of the conversion layers refractive index. An increasingrefractive index is still desirable however to ensure that most if notall of the luminescent light from the conversion (or LDS layer inthis case) is transmitted into the cell. It may be possible toovercome the increased device reflections that would ensue froma large refractive index using a LDS layer, which exhibits a highdegree of surface texturisation. The simplest case of a spectrally flatLDS layer is evaluated and presented below.

To model the transmission of light quanta through interfacesof differing dielectric materials, the relations shown in Eqs. (7)and (8) are used as discussed in Badescu et al. [7] and Stern [21].These relations are derived from Fresnel’s formulae for thetransmission of perpendicular and parallel polarised light denotedby s and p, respectively. In this case, terrestrial and isotropicblackbody radiation are considered as unpolarised light withtransmission values given by the average of Eqs. (7) and (8)(modelled to have equal quantities of s and p polarised light)integrated over the solid angle dO¼ sinydydf (which is projectedto the normal of the interface, hence the cosy term), then dividedby the total incident power as shown in Eq. (9)

t12,sðy,nijÞ ¼ 4cosyðn2ij�sin2yÞ1=2

½cosyþðn2ij�sin2yÞ1=2

��2 ð7Þ

t12,pðy,nijÞ ¼ 4n2ij cosyðn2

ij�sin2yÞ1=2½n2

ij cosyþðn2ij�sin2yÞ1=2

��2 ð8Þ

T12ðnijÞ ¼

R ymax

0

Rf

12 ½t12,sðy,nijÞþt12,pðy,nijÞ�SðoÞcosysinydydfR ymax

0

RfSðoÞcosysinydydf

ð9Þ

In Eqs. (7)–(9), y is the angle of incident radiation to thenormal of the interface, nij is the ratio (j/i) of the refractive indicesof two dielectric media denoted as i and j, S is the energy fluxdensity and f is the azimuthal angle. The upper limit of theintegral in Eq. (9) is the maximum half angle of the cone ofradiation incident on the dielectric media. Two sets of constraintsare applied to the upper limit of the integral of Eq. (9), whichdepend on the direction radiation is crossing an interface of twodielectric materials. If transmission is occurring to a dielectricwith a larger or equal refractive index (nijZ1), then ymax is p/2corresponding to a hemispherical radiation source such as theatmosphere. However if the radiation source is the sun, then ymax

is the half angle subtended by the sun, which is 0.2661 or4.64�10�3 rad. The second constraint is applied when radiationis transmitted to a medium of smaller or equal refractive index(nijr1). In this case the upper limit in Eq. (9) is satisfied by therelation sin2y¼ n2

ij. For angles that do not satisfy this constraint,

no radiation is transmitted. For the system modelled here, thetransmission of radiation through two interfaces of three dielec-tric media is required; and as shown in Badescu et al. [7] isdescribed as follows:

t12ðy,nijÞ ¼t12,sðy,nijÞþt12,pðy,nijÞ

2ð10Þ

t23ðy2,njkÞ ¼t23,sðy2,njkÞþt23,pðy2,njkÞ

2ð11Þ

t123ðy,nij,njkÞ ¼t23ðy2,njkÞt12ðy,nijÞ

t12ðy,nijÞþt23ðy2,njkÞ�t12ðy,nijÞt23ðy2,njkÞð12Þ

T123ðnij,njkÞ ¼

R ymax

0

Rft123ðy,nij,njkÞSðoÞcosysinydydfR ymax

0

RfSðoÞcosysinydydf

ð13Þ

The subscript k is introduced to denote the refractive index ofdielectric 3. By applying Snell’s law at the interface of dielectrics1 and 2, y2 can be determined. Eqs. (7)–(13) are used here as amultiplication factor for the etendue of Eq. (6).

As an ideal LDS layer is modelled here, the optimum refractiveindex for a planar coating for a silicon solar cell is chosen to be therefractive index of the LDS layer and serve as the starting point ofthis investigation. The refractive index is therefore initially set tobe equal to the square root of the refractive index of silicon [22].The refractive index data used for silicon was obtained fromGreen and Keevers [23].

The results of applying the transmission relations to Eq. (6)with the refractive index data of silicon and the LDS layer areshown in Fig. 5. As seen by a comparison of Figs. 4 and 5, asignificant reduction in the efficiency of an uncoated solar cell(lowest line of Fig. 5) occurs due to reflections caused bymismatched refractive indices between the solar cell and theatmosphere. The middle line in Fig. 5 shows the efficiency of thecell where the LDS layer is working as a transparent planarcoating alone. The upper most line indicates the further potentialenhancement over the planar coating (approximately 0.6% at1.12 eV) by utilising the down-shifting process.

Though it may be expected that the down-shifting processmay be improved in this case due to internal reflections of

C.P. Thomas et al. / Solar Energy Materials & Solar Cells 98 (2012) 455–464460

luminescent light at the LDS layer–air interface; extra losses alsooccur via reflections of luminescent light at the LDS layer–cellinterface. An effective light trapping scheme is necessary to utilisethe lost luminescent light.

300 500 700 900 11000

20

40

60

80

100

Wavelength (nm)

Inte

rnal

Qua

ntum

Effi

cien

cy (%

)

Se = 5.8x104 cm/s

Se = 2x105 cm/s

Se = 3x106 cm/s

Fig. 6. IQE of the solar cell obtained via Eqs. (14)–(17), corresponding to three

different front surface recombination velocities.

104 105 106 107-.1

0

.1

.2

.3

.4

.5

Front Surface Recombination Velocity (cm/s)

Effi

cien

cy (%

)

Fig. 7. Efficiency enhancement provided by the LDS layer is plotted as a function

of the front surface recombination velocity of the cell using Eqs. (14)–(17) for

the IQE.

5. Internal quantum efficiency of silicon solar cells

In the preceding analysis, it has been assumed that the IQE ofsilicon is either unity or zero as described by Eq. (5). This analysishas been performed to provide an ideal limiting scenario of theefficiency enhancement that can be provided by utilising a LDSlayer. Relaxing this assumption provides further loss mechanismswhere the non-zero and non-unity IQE of silicon solar cells in theregions defined by Eq. (5) reduce the performance of the solar cellwith a LDS layer from the results presented above. To investigatethe effect of the non-ideal IQE of silicon based solar cells on theLDS process, the solution of the continuity equations of aconventional solar cell is used to obtain the IQE as shown bySze [24] and is shown as follows:

IQEemitter ¼aLe

ðaLeÞ2�1

ððSeLe=DeÞþaLeÞ�e�aWe ððSeLe=DeÞcoshðWe=LeÞþ sinhðWe=LeÞÞ

ðSeLe=DeÞsinhðWe=LeÞþ coshðWe=LeÞ

�aLee�aWe

24

35

ð14Þ

IQEbase ¼aLb

ðaLbÞ2�1

e�aðWe þWscr Þ

�

aLb

�SbLbDbfcoshðWb=LbÞ�e�aWb g

þsinhðWb=LbÞþaLbe�aWb

ðSbLb=DbÞsinhðWb=LbÞþcoshðWb=LbÞ

26664

37775ð15Þ

IQEscr ¼ e�aWe ð1�e�aWscr Þ ð16Þ

IQE¼ IQEemitterþ IQEbaseþ IQEscr ð17Þ

In Eqs. (14)–(17), IQEemitter, IQEbase and IQEscr correspond tothe internal quantum efficiencies of the emitter, base and spacecharge regions of a conventional single junction solar cell,respectively. The solar cell therefore consists of an emitterregion of thickness We with a minority carrier diffusion coeffi-cient De and a diffusion length Le; a base region of thickness Wb

with a minority carrier diffusion coefficient Db with diffusionlength Lb and a space charge region of thickness Wscr. In Eqs.(14)–(17) a is the absorption coefficient of the solar cell. Theabsorption coefficient data was obtained from Green andKeevers [23]. The IQE described by Eqs. (14)–(17) correspondto a solar cell with an assumed constant doping concentrationand zero recombination in the space charge region. Typicalvalues of the parameters outlined above are used, which aredisplayed in Table 1.

Three IQEs determined by Eqs. (14)–(17) using the parametersoutlined in Table 1 for different emitter surface recombinationvelocities are displayed in Fig. 6. There are other processes, whichwill limit the UV conversion efficiency in a single junction solarcell such as Auger recombination in the emitter region and

Table 1Parameters used in the evaluation of Eqs. (14)–(17).

Solar cell region Thickness (mm) Diffusion co

(cm2/s)

Emitter We¼0.5 De¼5

Space charge region Wscr¼0.5

Base Wbase¼300 Db¼30

absorption of UV light in material over layers, which are ignoredhere. As can be seen from Fig. 6, a front surface recombinationvelocity of 5.8�104 cm/s results in an IQE that is greater than 50%in the UV region.

As described above, applying an LDS layer to a cell whose UVquantum conversion efficiency is greater than 50% will likelylead to efficiency losses due to the isotropic emission ofluminescent light in the LDS process. To provide a qualitative

efficient Diffusion

length (mm)

SRV (cm/s)

Le¼15 Se¼5.8�104-3�106

Lb¼100 Sb¼1�107

C.P. Thomas et al. / Solar Energy Materials & Solar Cells 98 (2012) 455–464 461

analysis of the effect of a non-ideal IQE on the LDS process, theIQE of Eqs. (14)–(17) has been applied to the model outlined inSections 2–4 where the efficiency is calculated as a function ofthe front surface recombination velocity. The results are dis-played in Fig. 7.

As can be seen from Fig. 7, the LDS process begins to becomeparasitic when (in this case) the front surface recombinationvelocity approaches 104 cm/s. Commercially produced passivatedsingle crystal silicon solar cells have been shown to achievesurface recombination velocities less than 100 cm/s [25]. Theapplication of a LDS layer to mono-crystalline silicon solar cellswill likely lead to efficiency losses. A poor blue response may stillbe observed in mono-crystalline solar cells due to the absorptionof UV light in encapsulate materials such as EVA. For cells with agood blue response, alternative encapsulate materials with highUV transparency may achieve greater efficiency gains thanthrough the application of a LDS material [26].

6. Multicrystalline cells

As shown and discussed above, commercially produced singlecrystal silicon solar cells that incorporate surface passivation arenot suitable candidates for the inclusion of a LDS layer. There isgreater potential for the LDS process in solar cells fabricated withmaterials that show poor UV properties such as cells constructedfrom multicrystalline silicon. Fig. 8 shows an IQE of a multi-crystalline silicon solar cell which has been obtained via anapproximated extrapolation of IQE’s of PECVD SiN mc-Si solarcells from the literature [32,33]. In order for a LDS layer to bebeneficial, its application should have minimal effect on any ofthe processing steps involved in the fabrication of the solar cell. Itis also important that the design considerations of the solar cell’sfabrication steps are independent of an applied LDS layer. If this isnot the case, an applied LDS layer may have a parasitic effect onthe cell’s efficiency and any cost involved in altering process stepsmay result in a decreased desirability for the introduction of LDSlayers to the solar cell manufacturing process. In this section, theapplicability of an LDS layer to multicrystalline solar cells isevaluated and is based only on optical considerations.

The use of PECVD SiN technology has been incorporated in thefabrication of multicrystalline solar cells both for its surfacepassivation and anti-reflection properties. Further advantagesare that the refractive index of SiN layers can be tuned in the

300 450 600 750 900 1050 12000

20

40

60

80

100

Wavelength (nm)

Inte

rnal

Qua

ntum

Effi

cien

cy (%

)

Fig. 8. Internal quantum efficiency of a multicrystalline silicon solar cell.

range of 1.8–2.2/2.3 [27,28] by alterations to the stoichiometry ofthe gas mix in the PECVD process. The diffusion of hydrogen intothe cell from the SiN layer formation also leads to bulk defectpassivation. For best optical coupling in an encapsulated cell witha SiN anti-reflection coating (ARC), the refractive index of the ARCshould be closer to 2.3. At the higher range of the refractive indexof the SiN layer, its extinction coefficient also becomes significantand increased UV absorption is observed. For this reason, therefractive index of the SiN layer is chosen to be approximately2.0 to optimise the trade-off between enhanced optical couplingand UV absorption.

The optimum thickness for the SiN layer corresponds to itsoperation as an effective quarter wave plate corresponding to awavelength range of 580–600 nm. This translates to an optimumthickness range of 70–80 nm for the SiN ARC [29]. The applicationof an additional LDS layer will change the optical coupling of thesolar cell to the atmosphere. Also, as the LDS layer effectivelyalters the incident spectrum, the spectral distribution of theemission from the LDS layer may introduce further implicationson the thickness and refractive index of the SiN ARC.

In what follows, two scenarios are modelled to evaluate theeffects of an applied LDS layer to a multicrystalline solar cell witha PECVD SiN coating. The aim is to determine a suitable operatingrange of the optical characteristics of the LDS layer (namely therefractive index) and the potential enhancement in efficiency thatmay be obtained.

To investigate the effect of applying a LDS layer to a cell asdescribed above, a method of determining the optical transmis-sion and reflection properties of a multilayer system is requiredthat also includes the interference effects of the ARC. Theapproach used here is through use of the Abeles relations whichcan be applied to a stack of stratified isotropic optical films orlayers [30]. This is a matrix transfer method where each layer inthe stack can be described by a characteristic matrix given by

Mj zj

� �¼

cosðk0njwjcosyjÞ �ði=pjÞsinðk0njwjcosyjÞ

�ipjsinðk0njwjcosyjÞ cosðk0njwjcosyjÞ

!ð18Þ

where for TE waves

pj ¼ nj cosyj ð19Þ

For TM waves pj is replaced by

qj ¼1

njcosyj ð20Þ

In the above, k0 is the wavenumber of the electromagneticwave in vacuum, while nj and wj are the refractive index andthickness of the jth layer, respectively. The propagation angle ofthe jth layer is denoted by yj. The characteristic matrix of a seriesstack of optical films can be obtained from the product of thecharacteristic matrices for each layer, i.e.

M ¼M1ðw1ÞM2ðw2Þ:::MjðwjÞ ð21Þ

Another series of matrices is used to describe the amplitudesof the electric Ex and magnetic Hy field vectors throughout thestack of optical films and for the TE case (where the yz plane is theplane of incidence) is denoted as

Q j ¼9Exj99Hyj9

����������¼

UjðzÞ

VjðzÞ

���������� ð22Þ

The product of the characteristic matrix and the matrix in (22)describes the propagation of the electromagnetic wave throughthe multilayered stack

Q 0 ¼MQ j ð23Þ

C.P. Thomas et al. / Solar Energy Materials & Solar Cells 98 (2012) 455–464462

Here, Q 0 describes the electromagnetic wave amplitudes beforeincidence on the first interface of the multilayered stack. Thetransmission and reflection coefficients of the stack can bedetermined via

r¼ðm011þm012pjÞp1�ðm

021þm022pjÞ

ðm011þm012pjÞp1þðm021þm022pjÞ

ð24Þ

t¼2p1

ðm011þm012pjÞp1þðm021þm022pjÞ

ð25Þ

In the above m0 ij correspond to the matrix elements of thecharacteristic matrix of the stack. The transmissivity and reflec-tivity of the stack can be found via

R¼ rj j2 ð26Þ

and

T ¼pj

p1tj j2 ð27Þ

For the case of a TM wave, the substitution of (20) in (18), (24)and (25) is required. With the appropriate manipulation of theabove, the transmission and reflection of light through any film ofthe stack can be found. For a thorough description and derivationof this formalism, consult Born and Wolf [30].

The matrix transfer method as discussed above has been usedto model the efficiency enhancement of the LDS process in mc-Sisolar cells in an optical five layer stack. The five layer stackcorresponds to a scenario consisting of the solar cell, a SiN ARC,the LDS layer, an encapsulation/glass layer and the atmosphere. Inthe scenario, it is assumed that any additional interfaces such asdue to glass laminates, which generally encase photovoltaicmodules do not provide an additional optical interface wherereflection can occur. A diagram of the configuration is provided inFig. 9. The results of this scenario are shown in Fig. 10.

The efficiency has been calculated as a function of the refractiveindex of the LDS medium and the thickness of the ARC coating tofind the optimum configuration. The ARC thickness has been variedin order to see what impact the altered incident spectrum onto the

Solar CellARC

LDS LayerEncapsulant

Fig. 9. Diagrammatic representation of the 5 layer optical stack.

50 88.3136.2

184.0231.9270.2

1.11.54

2.032.51

3.0018

20

22

24

Thickness of ARC (nm)Refractive Indexof LDS Layer

Effi

cien

cy (%

)

Fig. 10. Results of the LDS process applied to a mc-Si solar cell with a SiN ARC

coating and an encapsulation layer.

ARC has on determining its optimum thickness. The efficiencycalculations have been determined for the refractive index of theARC at values of 2.0 and 2.3. This investigation was carried out todetermine what optical gains can be achieved by a higher refractiveindex ARC where UV absorption in the ARC may no longer need tobe considered due to the LDS process. The thicknesses of all layersexcept for the ARC coating have been made thick enough so thatinterference effects are negligible. The refractive index of theencapsulant has been set to 1.48 roughly corresponding to therefractive index of EVA. The same analysis as in Section 4 is appliedto calculate the solar cell efficiency where the unit response of thecell is replaced by the IQE of a mc-Si as shown in Fig. 8 and thetransmission values are replaced by those calculated by the matrixtransfer method outlined above.

The maximum calculated efficiency for the case where therefractive index of the ARC is set to 2.0 is found to be 23.13% witha refractive index of approximately 1.4 for the LDS layer and athickness of approximately 78 nm for the ARC layer. The enhance-ment over a cell without the LDS layer is found to be 0.19% wherethe enhancement due to the downshifting process alone is foundto be 0.16%. The additional 0.03% optical gain may be due tochanges of the optical path length of light in the ARC caused bythe added LDS layer. When the refractive index of the LDS layermatches the encapsulation layer (as may be the case if theluminescent species was added or dissolved into the encapsulantduring its formation), the gain in efficiency is also found to be0.16%; and obviously there is no optical gain or loss.

For the case where the refractive index of the ARC layer is setto 2.3, the optimum calculated efficiency is found to be 23.66% ata refractive index of 1.54 for the LDS layer and a thickness ofapproximately 69 nm for the ARC layer. The gain due to downshifting alone in this case is found to be 0.17%. A more significantimprovement is made in this case where the refractive index ofthe ARC is set to 2.3 achieving an efficiency gain of 0.53% over theprevious case. A down shifting scheme may allow the use of a SiNARC coating with a higher refractive index value where UVabsorption in the ARC is sidestepped by the down shiftingprocess. The potential enhancement in efficiency of the opticalgain due to a SiN ARC coating with a refractive index of 2.3 insteadof 2.0 is greater than the down shifting process in this scenario.Again, for the case when the refractive index of the LDS layermatches the encapsulation layer, the efficiency gain due to thedown shifting process is found to be 0.16%.

If the LDS layer is deposited directly onto the passivated celland underneath the encapsulation layer, there is a tolerance rangeof the refractive index of the LDS layer that provides an efficiencygain. Outside of this range, the optical losses lead to a parasiticeffect on the efficiency of the cell. This is shown in Fig. 11 wherethe efficiency is calculated at the optimum thickness of the ARClayer and as a function of the refractive index of the LDS layer.This has been performed for both a cell that has an ARC coatingwith a refractive index of 2.0 and 2.3. For the scenario that modelsthe cell with a refractive index of 2.3, the gain is calculated over acell with an ARC layer that has a refractive index of 2.0.

For the case when the refractive index of the ARC layer is 2.0, therefractive index of the LDS layer needs to be between 1.205 and 1.555for a gain to be achieved. The range increases to lie between 1.2 and1.975 if the ARC coating is fabricated to have a refractive index of 2.3.The majority of the gain observed in this scenario is due to increasedoptical coupling provided by the higher refractive index of the ARCand not the down shifting process. A down shifting layer howevermay allow for the use of a higher refractive index ARC. The downshifting process in both scenarios has not shown any significantimpact on the optimum thickness of the ARC. For the case where therefractive index of the ARC is 2.0, the optimum thickness is 78 nm,which lies within the 70–80 nm range for a typical coating.

-.4

-.3

-.2

-.1

0

.1

.2

.3

Refractive Index of LDS Layer

Effi

cien

cy (%

)

1.1 1.3 1.5 1.7 1.9 2.1-.4

-.2

0

.2

.4

.6

.8

Refractive Index of LDS Layer

1.1 1.3 1.5 1.7 1.9 2.1

Effi

cien

cy (%

)

Fig. 11. Potential efficiency gain of a cell with a LDS layer over a conventional cell,

for (a) a mc-Si cell with a SiN ARC with refractive index of 2.0 and (b) a mc-Si cell

with a SiN ARC with refractive index of 2.3.

C.P. Thomas et al. / Solar Energy Materials & Solar Cells 98 (2012) 455–464 463

The efficiency calculations in these scenarios are based on aparticular IQE, which is shown in Fig. 8. Greater gains in solar cellefficiency due to the LDS process will obviously be observed withpoorer quality cells exhibiting a short wave response that is worsethan that of the IQE analysed here. It is also important to note thatthe refractive index of all layers except the solar cell have beenmodelled to be independent of wavelength. Ignoring the disper-sivity and inequality of the refractive index of the laminate andencapsulation layers may introduce some inaccuracies in theresults presented here. It is also assumed that no UV absorptionoccurs in the overlayers of the cell (apart from the LDS layer),which would compete with the LDS process in real systemsinfluencing the potential gain in efficiency that can be achieved.

7. Conclusion

A theoretical limit of efficiency has been calculated andpresented for the use of an ideal front surface LDS layer on singlejunction silicon based solar cells. The maximum efficiency gain of

the down-shifting process with a LDS layer has been found to be1.5% with a theoretical blackbody spectrum and 0.6% using AM1.5standard spectra. These results are comparable to the approx-imate 1% efficiency gains stated in Marchionna et al. [31] andMcIntosh et al. [32]. The calculations have assumed unit refractiveindex values and ideal spectral responses of the solar cell anddown shifter. Removing these idealisations has resulted in effi-ciency gains due to the downshifting process that are significantlyless than the ideal case. The IQE and refractive indices of theoptical layers of the solar cell has a significant impact on theperformance of the down-shifting process. This is shown wherethe maximum efficiency gain of the down-shifting process wasfound to be 0.17% using an IQE of a multicrystalline cell andintroducing optical layers to resemble the ARC and encapsulationlayers of a photovoltaic module. This result shows a closeagreement with the experimental results of Klampaftis andRichards [33].

Only radiative recombination has been considered in the bulkof the cell in this work; non radiative processes such as Auger andShockley–Read–Hall recombination may further reduce theenhancement that a UV down-shifter can supply. Other processessuch as photon recycling and free carrier absorption have alsobeen ignored. Photon recycling and free carrier absorption pre-dominantly affect the long wavelength response of a silicon solarcell and may not affect the results presents here. Spectrally flatsurfaces at all interfaces have been assumed. Incorporating a solidfilm of an LDS material may mean that this assumption isunrealistic leading to variations from the results presented herefor such a configuration.

The model used here incorporates idealisations of luminescentmaterials, which approximate the fluorescence characteristics oforganic materials. Organic dye pigments can exhibit near unityLQE and have absorption and emission spectra suitable for use asa down shifting material with silicon based cells [34]. Researchinterest in such materials as down-shifters is usually demon-strated via their use as a dopant in encapsulation layers forluminescent solar concentrators (LSC) and flat plate modules[3,4,32–34]. Another configuration has also been considered herewhere the LDS layer is modelled as constituting an optical layer ofits own. The results here support the case where the LDS materialis embedded in the encapsulation layer, which results in aconfiguration close to the optimum from an optical perspective.This also removes the issue of needing a film of a LDS material toexhibit a suitable refractive index.

References

[1] H.J. Hovel, R.T. Hodgson, J.M. Woodall, The effect of fluorescent wavelengthshifting on solar cell spectral response, Solar Energy Materials 2 (1979)19–29.

[2] A. Cuevas, P.A. Basore, G. Giroult-Matlakowski, C. Dubois, Surface recombina-tion velocity of highly doped n-type silicon, Journal of Applied Physics 80(1996) 3370–3375.

[3] E. Klampaftis, D. Ross, K. Mcintosh, B. Richards, Enhancing the performance ofsolar cells via luminescent down-shifting of the incident spectrum: a review,Solar Energy Materials and Solar Cells 93 (2009) 1182–1194.

[4] W.G.J.H.M. van Sark, K.W.J. Barnham, L.H. Slooff, A.J. Chatten, A. Buchtemann,A. Meyer, S.J. McCormack, R. Koole, D.J. Farrell, R. Bose, E.E. Bende,A.R. Burgers, T. Budel, J. Quilitz, M. Kennedy, T. Meyer, C. De Mello Donega,A. Meijerink, D. Vanmaekelbergh, Luminescent solar concentrators—a reviewof recent results, Optics Express 16 (2008) 21773–21792.

[5] W. Shockley, H. Queisser, Detailed balance limit of efficiency of p–n junctionsolar cells, Journal of Applied Physics 32 (1961) 510–519.

[6] T. Trupke, M.A. Green, P. Wurfel, Improving solar cell efficiencies by down-conversion of high-energy photons, Journal of Applied Physics 92 (2002)1668–1674.

[7] V. Badescu, A. De Vos, A.M. Badescu, A. Szymanska, Improved model for solarcells with down-conversion and down-shifting of high-energy photons,Journal of Physics D: Applied Physics 40 (2007) 341–352.

[8] A. De Vos, Detailed balance of the efficiency of tandem solar cells, Journal ofPhysics D: Applied Physics 13 (1980) 839–846.

C.P. Thomas et al. / Solar Energy Materials & Solar Cells 98 (2012) 455–464464

[9] R. Brendel, J.H. Werner, H.J. Queisser, Thermodynamic efficiency limits forsemiconductor solar cells with carrier multiplication, Solar Energy Materialsand Solar Cells 41/42 (1996) 419–425.

[10] A. Luque, A. Marti, Increasing the efficiency of ideal solar cells by the photoninduced transitions at intermediate levels, Physical Review Letters 78 (1997)5014–5017.

[11] S.J. Keough, T.L. Hanley, A.B. Wedding, J.S. Quinton, Grazing incidence x-raystudies of ultra-thin lumogen films, Surface Science 601 (2007) 5744–5749.

[12] M. Pope, C.E. Swenberg, Electronic Processes in Organic Crystals and Poly-mers, 2nd Ed, Oxford University Press, 1999 (1328 pp).

[13] A.S. Davydov, Theory of Molecular Excitons, Plenum Press, 1971 (313pp).[14] A.S. Davydov, The theory of molecular excitons, Soviet PhysicsUspekhi 7

(1964) 145–178.[15] R.M. Hochstrasser, The luminescence of organic molecular crystals, Reviews

of Modern Physics 34 (1962) 531–550.[16] R.C. Tolman, The Principles of Statistical Mechanics, Dover Publications, 1979

521–523.[17] V.F. Chel’tzov, Interaction of a two-band semiconductor with radiation field,

Solid State Communications 9 (1971) 1343–1346.[18] P. Wurfel, The chemical potential of radiation, Journal of Physics C: Solid

State Physics 15 (1982) 3967–3985.[19] T. Markvart, The thermodynamics of optical etendue, Journal of Optics A:

Pure and Applied Optics 10 (2008) 015008 7pp.[20] T. Tiedje, E. Yablonovitch, G. Cody, B. Brooks, Limiting efficiency of silicon

solar cells, IEEE Transactions on Electronic Devices ED-31 (1984) 711–716.[21] F. Stern, Transmission of isotropic radiation across an interface between two

dielectrics, Applied Optics 3 (1964) 111–114.[22] A. Vasicek, Optics of Thin Films, North-Holland Publishing Company, 1960 65–68.[23] M. Green, M. Keevers, Optical properties of intrinsic silicon at 300 K, Progress

in Photovoltaics 3 (3) (1995) 189–192.

[24] S. Sze, Physics of Semiconductor Devices, 2nd ed., John Wiley and Sons, 1981790–816.

[25] G. Aberle, Surface passivation of crystalline silicon solar cells: a review,Progress in Photovoltaics: Research and Applications 8 (2000) 473–487.

[26] K. McIntosh, J.N. Cotsell, A.W. Norris, N.E. Powell, B.M. Ketola, An opticalcomparison of silicone and EVA encapsulants under various spectra, in:Proceedings of the 35th Photovoltaic Specialists Conference, p. 269.

[27] A. Rohatgi, Z. Chen, P. Sana, J. Crotty, J. Salami, High efficiency multicrystal-line silicon solar cells, Solar Energy Materials and Solar Cells 34 (1994)227–236.

[28] F. Duerinckx, J. Szlufcik, Defect passivation of industrial multictrystallinesolar cells based on PECVD silicon nitride, Solar Energy Materials and SolarCells 72 (2002) 231–246.

[29] A. Elamrani, I. Menous, L. Mahiou, R. Tadjine, A. Touadand, A. Lergoum,Silicon nitride film for solar cells, Renewable Energy 33 (2008) 2289–2293.

[30] M. Born, E. Wolf, Principles of Optics, 4th ed., Pergamon Press, 1970 51–70.[31] S. Marchionna, F. Meinardi, M. Acciarri, S. Binetti, A. Papagni, S. Pizzini,

V. Malatesta, R. Tubino, Photovoltaic quantum efficiency enhancement bylight trapping harvesting of organo-lanthanide complexes, Journal of Lumi-nescence 118 (2006) 325–329.

[32] K.R. McIntosh, G. Lau, J.N. Cotsell, K. Hanton, D.L. Batzner, Increase in externalquantum efficiency of encapsulated silicon solar cells from a luminescentdown-shifting layer, Progress in Photovoltaics 17 (2009) 191–197.

[33] E. Klampaftis, B.S. Richards, Improvement in multi-crystalline silicon solarcell efficiency via addition of luminescent material to EVA encapsulationlayer, Progress in Photovoltaics 19 (2011) 345–351.

[34] O. Moudam, B.C. Rowan, M. Alamiry, P. Richardson, B.S. Richards, A.C. Jones,N. Robertson, Europium complexes with high total photoluminescencequantum yields in solution and in PMMA, Chemical Communications(2009) 6649–6651.