An extended model for upconversion in solar cells

An extended model for upconversion in solar cellsViorel Badescu Citation J Appl Phys 104 113120 (2008) doi 10106313040692 View online httpdxdoiorg10106313040692 View Table of Contents httpjapaiporgresource1JAPIAUv104i11 Published by the AIP Publishing LLC Additional information on J Appl PhysJournal Homepage httpjapaiporg Journal Information httpjapaiporgaboutabout_the_journal Top downloads httpjapaiporgfeaturesmost_downloaded Information for Authors httpjapaiporgauthors

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An extended model for upconversion in solar cellsViorel Badescua

Candida Oancea Institute Polytechnic University of Bucharest Spl Independentei 313Bucharest 060042 Romania

Received 25 July 2008 accepted 24 October 2008 published online 15 December 2008

Here we analyze the system proposed by Trupke et al J Appl Phys 92 4117 2002 to increasesolar cell efficiency The system consists in adding to the cell a so-called upconverter which is adevice able to convert the low-energy subband-gap incident solar photons into photons of higherenergy The model takes account of i the nonradiative recombination in both solar cell andconverter and ii the refractive index of both cell and converter Two configurations are studied celland rear converter C-RC and front converter and cell The main conclusions are as follows 1When nonradiative recombination is neglected for both cell and converter the energy conversionefficiency of a C-RC system slightly exceeds the efficiency of a solar cell operating alone under 1sun illumination 2 When similar realistic values for the radiative recombination efficiency areconsidered for both cell and converter the energy conversion efficiency of a C-RC system is lowerthan the efficiency of a solar cell operating alone under 1 sun illumination 3 Adding a rearupconverter to the solar cell is beneficial in the case of present-day quality solar cells underconcentrated solar radiation 4 At small values of the cell refractive index roughly less than 2 theconversion efficiency does not depend on the converter refractive index 5 At higher values of thecell refractive index the conversion efficiency decreases by increasing the converter refractiveindex 6 The energy conversion efficiency does not increase by adding a front upconverter to thecell whatever the values of the radiative recombination efficiency and solar radiation concentrationratio are copy 2008 American Institute of Physics DOI 10106313040692

I INTRODUCTION

Many techniques have been proposed to increase the ef-ficiency of solar cells They are based on different processesand properties such as photon recycling12 impactionization3ndash5 angle dependent selectivity6 down-conversionwhich is sometimes referred to as ldquoquantum cutting viadown-conversionrdquo7ndash12 and down-shifting13ndash17 Combina-tions of down-shifting and down-conversion processes werealso studied see eg Refs 18ndash20

Another technique studied in the past years converts theincident low-energy subband-gap solar photons into pho-tons of energy higher than solar cell band gap Thus thenumber of high-energy photons absorbed in the cell in-creases This ldquoupconversionrdquo technique has a number ofadvantages27 no modification of the active layer is neededas is the case with the impurity photovoltaic effect21 and theintermediate band solar cell22 and it does not require com-plicated structures such as that used for multiple-junctionsolar cells23 or quantum well solar cells24 Gibart et al25

reported the application of upconversion on a GaAs solar cellwith a vitroceramic codoped with trivalent erbium and triva-lent ytterbium at an energy of 1391 eV under excitation of1 W 0039 cm2 Shalav et al26 reported the application to abifacial silicon solar cell with a rear upconverter consistingof NaYF4 doped with trivalent erbium for monochromaticlight of 1523 nm under excitation of 24 W cm2

For the application to silicon solar cells erbium dopedmaterials are most promising for upconversion27 Alsobarium chloride doped with trivalent erbium BaCl2 Er3+ isproposed to be an efficient upconverter due to advantageousphononic properties28 Upconversion processes have alsobeen demonstrated in organic materials and in transitionmetals2930 A good review of the properties of available up-converters with focus on silicon solar cells has been made inRef 27

As the experiments so far have been carried out withlaser illumination little is known about the behavior of thesystem under sun illumination31 The efficiency of upconver-sion increases with increasing intensity Thus solar radiationconcentration is needed to achieve conditions comparable tothe mentioned laser illumination A 500 sun concentration isexpected to achieve conversion efficiencies comparable withthose obtained under laser excitation32

Theoretical studies predict that the maximum conversionefficiency for an ideal upconverter on the rear side of a cellunder nonconcentrated sunlight increases as compared to thecase of the solar cell operating alone A maximum conver-sion efficiency of 374 is predicted in Ref 32 while for asilicon solar cell the estimated upper limit is 40233 Theefficiency enhancement due to an upconverter is expected toincrease up to about 55 at 100 sun concentration27 Com-paring these theoretical values with values accessible bypresent-day technology may give a broader perspective forthe upconversion technique Todayrsquos most efficient solar celltechnology is based on multijunctions made of III-V com-pound semiconductors The best material combination is alattice matched solar cell that consists of three active junc-

aAuthor to whom correspondence should be addressedTel 40214029428 FAX 40214104251 Electronic mailbadescuthetatermopubro

JOURNAL OF APPLIED PHYSICS 104 113120 2008

0021-897920081041111312010$2300 copy 2008 American Institute of Physics104 113120-1


tions of Ga049In051P Ga099In001As and Ge The highestefficiencies under the AM0 space solar spectrum are about3053435 For concentration ratios between 200 and 600the highest efficiencies are in the range of 35ndash393436ndash39

Note however that a triple-junction solar cell is a compli-cated device consisting of up to 30 individual semiconductorlayers including barrier layers and tunnel junctions that haveto be optimized for reaching ultimate performance40 Thismeans that for similar conversion efficiency the upconver-sion option might be preferred in terms of cost over III-Vbased triple junctions

Although there has been much work on down-conversion photoluminescence and upconversion the appli-cation to solar cells has formed a comparatively small branchof this research to date27 An interesting model was proposedin Trupke et al41 for the upconversion technique These au-thors have taken account of the fact that the radiation fluxescoming from a medium depend on the refractive index ofthat medium However these authors did not include in theiranalysis all the effects that normally a different from unityrefractive index has Mainly the fact that the radiation flux ischanged when passing through interfaces of different refrac-tive indices was not considered Here we propose a moreinvolved model for the upconversion technique which prop-erly takes into account the effects of the radiation transferthrough interfaces Our analysis refers to cells and upcon-verters of different refractive indices This extends thework41 where cells and upconverters of equal refractive in-dices were considered Three other improvements are alsoperformed here First our model includes nonradiative re-combination which was neglected in the previous approachSecond the effect of using antireflection AR layers is takeninto account Third the only configuration considered in Ref41 is solar cellndashrear upconverter C-RC Here the configu-ration front upconverterndashsolar cell FC-C is analyzed tooThe main question we want answered is the following doesadding an upconverter to the solar cell really improve thesolar energy conversion efficiency

II MODEL OF CELL AND UPCONVERTER SYSTEM

The system studied in Ref 41 consists of four plan-parallel layers a solar cell an upconverter an insulator be-tween these two components and a reflector This systemrequires a bifacial solar cell The electrical insulator betweenthe solar cell and the upconverter makes the coupling be-tween these two components purely radiative The reason touse the reflector is that only one face of this system allowsradiative losses to the ambient A short description follows

The upconverter is made of a dielectric whose energyband gap contains an intermediary energy level E1 Thus theupconverter is a three-level system that may be used to con-vert two lower energy photons into one higher energy pho-ton The absorption of the low-energy photons in the va-lence band VB and intermediate level IL leads to twoelectronic transitions A one-step recombination of an elec-tron in the conduction band CB and a hole in the VB isaccompanied by emission of one higher energy photon Moredetails may be found in Ref 41

The photon number flux density emitted by a semicon-ductor material with the refractive index n1 at temperature T1

is obtained by integration of the spectral photon number fluxdensity

NElEun1T11B1 =n1

2B1

433c02

El

Eu

2

exp minus 1

kT1 minus 1minus1

d 1

Here the relation E= between frequency and energy Eis used Also El and Eu denote the lower and upper energiesfor the transition involved and 1 is the chemical potentialof the emitted radiation The photon number flux densitycoming from a medium with refractive index n1 and tem-perature T1 that is absorbed in medium 2 after passing asingle interface is given by

NElEun1T11B1rarr2 =n1

2B1rarr2

433c02

El

Eu

2

exp minus 1

kT1 minus 1minus1

d

2

In the case when the incident photon flux passes through twointerfaces the absorbed photon number flux density is givenby

NElEun1T11B1rarr2rarr3 =n1

2B1rarr2rarr3

433c02

El

Eu

2

exp minus 1

kT1 minus 1minus1

d

3

The factors B1 B1rarr2 and B1rarr2rarr3 in Eqs 1ndash3 may beevaluated by using the procedure presented in Appendix A ofRef 19 for the case of axisymmetrical sources of radiationsee also Tables I and II in the present work Equations1ndash3 assume that the cell absorptance is unity between thelower and upper energy thresholds and zero outside this en-ergy interval

The output power of a solar cell is evaluated as a productof the voltage V across the cell and the electric current Ithrough the cell In the academic case considered in Ref 41one assumes that all transitions are radiative and that absorp-tion of a photon generates a single electron-hole pair whilerecombination of a carrier pair emits a single photon Thenthe current I is given by the difference between the absorbedphoton number flux and the emitted photon number fluxtimes the electron electric charge q In practice not all theabsorbed photons generate electron-hole pairs and not allcarrier recombinations are radiative A way of writing the I-Vcharacteristic of a realistic solar cell is19

113120-2 Viorel Badescu J Appl Phys 104 113120 2008


I

q= fabsNabs minus f recNemV 4

where Nabs and NemV are absorbed and emitted photon fluxdensities respectively whereas fabs and f rec are absorptionand recombination factors respectively which are given bysee Table 1 of Ref 19

fabs = abs 5

f rec = 1 minus absrecr

rec1 minus r 13 6

where the brackets denote an average value over the cellvolume while abs rec and r are the efficiency of carriergeneration by photon absorption the efficiency of radiativerecombination and the efficiency of photon recycling re-spectively for definitions see Table 1 of Ref 19

Two geometrical configurations may be considered Tak-ing the solar radiation direction as a reference the first con-figuration C-RC is solar cell thin insulator rear upcon-verter and reflector while the second configuration FC-Cis front upconverter thin insulator solar cell and reflec-tor Both configurations are treated below

A Configuration solar cell and rear upconverter

The energy diagram of the C-RC system is shown in Fig1 where sunlight is incident from the left side The energyband model used in Ref 41 is adopted here Fig 1a Threetypes of transitions occur inside the converter a band-to-

band transition associated with electron-hole pair recombi-nation and two intermediate transitions between the bottomedge of the CB and the IL and between the IL and the topedge of the VB These two intermediate transitions are asso-ciated with electron-hole pair generation These three typesof transitions may be seen as three independent two-bandsystems with individual electrochemical potentials and thewhole upconverter may be represented in an equivalent cir-cuit by three fictitious cells connected in series Fig 1bCell C2 corresponds to the band-to-band transition withelectrochemical potential difference C2 Devices C3 andC4 which correspond to the two intermediate transitionscan be modeled as cells with electrochemical potential dif-ferences C3 and C4 respectively Finally C1 in Fig 1brepresents the real solar cell with electrochemical potentialdifference C1

Cell C2 of band-gap Eu=Eg can emit and absorb pho-tons in the energy range Eu Eu+E2 Cells C3 and C4 emitand absorb low-energy photons in well defined energy inter-

TABLE I Factors entering Eqs 7ndash10 of the present work and Eqs 3ndash6 of Ref 41 in the case of a C-RCsystem This table refers only to geometrical factors and not to energy levels Thus here we treat cells C3 andC4 as identical they both have the same refractive index In the case of two materials with equal refractiveindices the eacutetendue adopted in Ref 41 for the radiation emitted from one material into the other is n2

FactorNo of crossed

interfacesTrupke et al

Ref 41Presentwork

F1srarr2 1 Bs B1srarr2=20maxs1rarr21sin 1 cos 1d1

F1ararr2 1 minusBs B1ararr2=2021rarr21sin 1 cos 1d1

F3rarr2 1 n2 B3rarr2=2023rarr21sin 1 cos 1d1

F2rarr1 1 B2rarr1=2022rarr11sin 1 cos 1d1


F1srarr3 2 Bs B1srarr2rarr3=20maxs1rarr2rarr31sin 1 cos 1d1

F3rarr1 2 B3rarr2rarr1=2023rarr2rarr11sin 1 cos 1d1

F1ararr3 2 minusBs B1ararr2rarr3=2021rarr2rarr31sin 1 cos 1d1

TABLE II FC-C system Factors entering Eqs 11ndash14

FactorNo of crossed

interfaces Present work

F1srarr3 2 B1srarr2rarr3=20maxs1rarr2rarr31sin 1 cos 1d1

F1ararr3 2 B1ararr2rarr3=2021rarr2rarr31sin 1 cos 1d1




F1srarr2 1 B1srarr2=20maxs1rarr21sin 1 cos 1d1

F1ararr2 1 B1ararr2=2021rarr21sin 1 cos 1d1


FIG 1 C-RC system The cell medium 2 has the refractive index n2 andthe rear upconverter medium 3 has the refractive index n3 The solar celland the upconverter are electronically isolated from each other Thesubband-gap photons transmitted by the solar cell are partially upconvertedinto high-energy photons which are subsequently absorbed in the solar cellA reflector is located behind the upconverter a The energy band diagramAn IL is located inside the converterrsquos band gap at energy E1 above the VBedge The widths of the valence and CBs are limited to ascertain photonselectivity Dotted arrows symbolize the two-step generation of charge car-rier pairs via the IL The dashed arrow symbolizes the radiative recombina-tion of a charge carrier pair via a band-to-band transition Eg energy bandgap ECB and EVB the bottom CB energy level and the top VB energy levelrespectively in the solar cell b The equivalent electric circuit The ficti-tious cells C2 C3 and C4 represent the band-to-band transitions and thetwo intermediate transitions respectively C1 represents the real solar cell



vals corresponding to transitions through the intermediarylevel This way a photon of energy E1Eu hits an electronin the VB that climbs on the intermediary level A secondphoton of energy E2Eu transfers its energy to it and theelectron reach the CB After the recombination of this elec-tron a photon of energy EEg may be emitted and absorbedby solar cell C1 Cells C2ndashC4 are connected in series

The model allows another energy transfer mechanismtoo An incident photon can hit an electron from the VB Ifthe photon energy has a well determined value E2 that elec-tron can reach the intermediary level A second incident pho-ton of energy Eg may produce the electron transition on alevel higher than the minimum of the CB

All the photon ldquoselection rulesrdquo used in Ref 41 areadopted here i Cell C1 emits and absorbs photons in theenergy interval Eg ii Cell C2 emits and absorbs pho-tons in the energy interval Eg Eg+E2 iii Cell C3 emitsand absorbs photons in the energy interval E1 E2 iv CellC4 emits and absorbs photons in the energy interval E2 Eg

A few details follow The incident photons coming fromthe Sun with energies in the interval Eg are absorbed bycell C1 generating an electron-hole pair After a time inter-val those carriers will recombine and a photon with energyin the same interval Eg will be emitted The emittedphoton is either absorbed by cell C2 or is lost in the ambientThe incident photons with energy in the interval E2 Eg passfreely through cell C1 and reach the converter There theyhit the IL electrons which are jumping to the CB After atime interval these electrons recombine through band-to-band transitions producing a photon with energy in the rangeEg Eg+E2 The incident photons with energy in the inter-val E1 E2 pass freely through cell C1 They are absorbed inthe converterrsquos VB and an electron there is transferred to theIL In this case the energies should obey the following rela-tion E1E2Eg

Three types of contributions to the electric current areconsidered when the I-V characteristics of the four cells arecalculated First there is the contribution of the photonscoming from the Sun which is described as a source sub-tending a cone of half-angle maxs=4610minus3 rad Secondthere is the contribution coming from the ambient which isseen as a hemispherical radiation source maxa= 2Third there is the contribution of the photons emitted by thenearby cells seen as hemispherical radiation sources Thesecontributions are used according to the photon selection ruleslisted above

The configuration of Fig 1a shows that the ambient ismedium 1 refractive index n1 solar cell C1 is medium 2refractive index n2 and the rear upconverter consisting ofcells C2ndashC4 is medium 3 refractive index n3 The thininsulator is neglected here The I-V characteristics of cellsC1ndashC4 are

IC1

q= fabsC1NEgn1Ts0F1srarr2

+ NEgn1Ta0F1ararr2 + NEgEg

+ E2n3TcC2F3rarr2

minus f recC1NEgn2TcC1F2rarr1 + NEgEg

+ E2n2TcC1F2rarr3 7

IC2

q= fabsC2NEgE2 + Egn2TcC1F2rarr3

minus f recC2NEgE2 + Egn3TcC2F3rarr2 8

IC3

q= fabsC3NE1E2n1Ts0F1srarr3

+ NE1E2n1Ta0F1ararr3

minus f recC3NE1E2n3TcC3F3rarr1 9

IC4

q= fabsC4NE2Egn1Ts0F1srarr3

+ NE2Egn1Ta0F1ararr3

minus f recC4NE2Egn3TcC4F3rarr1 10

where Ts Ta and Tc are the temperatures of the three sourcesof radiation ie Sun ambient and the four cells assumedin thermal equilibrium respectively The factors F enteringEqs 7ndash10 are shown in Table I

The current IC1 generated by cell C1 Eq 7 is theresult of photons absorbed from the Sun first term and fromthe ambient second term and of photons emitted by the cellto the ambient fourth term and to the converter fifth termThe third term corresponds to photons emitted by the con-verter and absorbed by cell C1 The current IC2 generated bycell C2 Eq 8 is the result of absorbed photons comingfrom cell C1 first term and emitted photons by cell C2 lastterm The currents generated by cells C3 and C4 Eqs 9and 10 respectively are the result of absorbed photonscoming from the Sun and the ambient first and secondterms respectively and photons emitted by those cells lastterm

There is a difference between Eqs 3ndash6 of Ref 41 andthe present Eqs 7ndash10 which consists of the factors FThese factors are shown in Table I for both models Onereminds us that the factors F in the model by Trupke et al41

do not take account of the interface transmittance

B Configuration front upconverter and solar cell

The energy diagram and the equivalent circuit of theFC-C system are shown in Fig 2 Sunlight is incident fromthe left side The ambient is medium 1 refractive index n1the front upconverter consisting of cells C2ndashC4 is medium2 refractive index n2 and solar cell C1 is medium 3 re-fractive index n3 Generally the photon selection rules usedin the case of the C-RC Sec II A apply here too

The I-V characteristics of the FC-C system are



IC1

q= fabsC1NEg + E2n1Ts0F1srarr3 + NEg

+ E2n1Ta0F1ararr3 + NEgEg

+ E2n2TcC2F2rarr3 minus f recC1NEgEg

+ E2n2TcC1F3rarr2 + NEg

+ E2n2TcC1F3rarr1 11

IC2

q= fabsC2NEgEg + E2n1Ts0F1srarr2 + NEgEg



+ E2n2TcC2F2rarr1 + NEgEg


IC3




IC4




The factors F entering Eqs 11ndash14 are shown in Table IIThe current IC1 generated by cell C1 Eq 11 is the resultof photons absorbed from the Sun first term and from theambient second term The third term gives the photon num-ber flux emitted by cell C2 and absorbed by cell C1 whilethe last two terms are associated with the photons emitted byC1 The current IC2 generated by cell C2 Eq 12 is theresult of absorbed photons coming from the Sun first termand from the ambient second term The third term corre-sponds to photons emitted by cell C1 and absorbed by cell

C2 while the last two terms are the photon number fluxdensities emitted by cell C2 The currents generated by cellsC3 and C4 Eqs 13 and 14 respectively are the result ofabsorbed photons coming from the Sun and the ambientfirst and second terms respectively and photons emitted bythose cells last term

C Maximum efficiency evaluation

The voltages VC1 VC2 VC3 and VC4 across the cells Cii=12 3 4 are computed respectively by

qVC1 = C1 15a

qVC2 = C2 15b

qVC3 = C3 15c

qVC4 = C4 15d

Cells C2ndashC4 are series connected see Figs 1b and 2bTherefore the same electric current flows through thesecells

minus IC2 = IC3 = IC4 16

The minus sign shows that the current has different direc-tions through cells C2 and C3 The sum of the voltages VC3

and VC4 across cells C3 and C4 equals the voltage VC2 acrosscell C2 In other words

C3 + C4 = C2 17

The solar cell output power is evaluated as a productbetween the voltage VC1 across the solar cell and the electriccurrent IC1 through the cell The solar energy conversionefficiency of the system cell-upconverter is given by theratio between the power provided per unit surface of solarcell C1 making part of the C-RC or FC-C system and thesolar irradiance incident on the interface between the envi-ronment and the C-RC or FC-C system

=IC1VC1

13s 18

where 13s is solar irradiance given by

13s = BsTs4 19

The following procedure is adopted to evaluate themaximum energy conversion efficiency max The number ofunknowns is 9 ie the electric currents ICii=12 3 4 theenergy level E1 and the four chemical potentials C1 C2C3 and C4 The number of equations is 7 ie the fourI-V characteristics and Eqs 16 and 17 Therefore thereare two degrees of freedom say E1 and C1 The efficiency given by Eq 18 is maximized as a function of E1 andC1 Once the optimum value of C1 is found the aboveseven equations are used to obtain the electric currentsICii=12 3 4 and the chemical potentials C2 C3 andC4 Finally the voltages VC1 VC2 VC3 and VC4 are com-puted by using Eqs 15andash15d respectively

FIG 2 FC-C system The upconverter medium 2 has the refractive indexn2 and the solar cell medium 3 has the refractive index n3 a The energyband diagram For details see the legend of Fig 1a b The equivalentelectric circuit For details see the legend of Fig 1b



D Radiation concentration

One denotes by s the solid angle subtended by the Sunwhen viewed from Earth Note that20

s = 21 minus cos maxs 20

When concentrated radiation is considered the cell sees thesun under the enlarged solid angle C which depends on theconcentration ratio C The half-angle maxC of the cone sub-tended by the enlarged source of radiation may be obtainedfrom

C = 21 minus cos maxC 21

Also note that20

C =C4 minus Cs4 minus s

22

Once the value of the concentration ratio C is given fromEqs 21 and 22 one may easily find C and maxC re-spectively

III RESULTS

The input data adopted here are as follows Ts

=5760 K for the sun temperature and Ta=Tc=300 K for theambient and the cell and converter temperatures respec-tively The environment is air the refractive index is equal to1

Comparisons between the conversion efficiency of a so-lar cell operating alone and the efficiency of the C-RC andFC-C systems are reported below Two different cases areconsidered In the first case there is no AR layer on the solarcell or on the C-RC or FC-C system This is the casetreated in41 However here the reflectance factors are com-puted by using Fresnel-like relationships for details see19while in41 the reflection at interfaces was neglected In thesecond case AR coatings are provided and a reflectance R=002 is assumed for the solar radiation incident on the in-terface between the solar cell and the ambient or on the in-terface between the C-RC or FC-C system and the ambient

Two different solar cells are considered next The firstcell say Cgh is a good quality cell with high radiativerecombination efficiency reccell=01 This may be associ-ated for instance with the radiative recombination efficien-cies of more than 10 for the photoluminescence from well-passivated high quality float-zone silicon reported in Ref 42These values are orders of magnitude higher than the resultspreviously published Second there is a poor quality siliconcell Cp with a common value of the radiative recombina-tion efficiency reccell=10minus4 The absorption and recombi-nation factors for the two solar cells are determined as in aand b below

a The cell Cgh is calibrated at the refractive index ncell

=36 to provide at one sun illumination C=1 the ef-ficiency of 1449 of the standard baseline n-p-p+mc-Si cell studied in Refs 14 and 15 The cell absorp-tion efficiency resulting from calibration is abscell

=074 For details see Ref 19b An a-SiH cell with 655 efficiency simulated in Ref

16 is used to calibrate the Cp cell This poor qualityhydrogenated amorphous silicon solar cell is character-ized by low absorption efficiency abscell=04 for theCp cell For details see Ref 19

A high value is adopted for the absorption efficiency ofthe upconverter absconv1 This is consistent with recentstudies on the performance of solar cells coated by spectralconverters with quantum dots QDs1416 Indeed QDs ab-sorb all the light of a wavelength smaller than the absorptionmaximum16

Different high values of the radiative recombination ef-ficiency recconv in the upconverter are used below They areconsistent with the high quantum efficiency QE value 08adopted in Refs 14ndash16 for QD materials and with the quan-tum yield of unity assumed in Ref 13 for the fluorescentconcentrators placed on the top of solar cells Note that ex-perimental results for sulforhodamine 101 molecularprobes in ethyl alcohol show luminescence QE close to9043

In the first set of calculations both the converter and thesolar cell have the same refraction index Also the fourldquocellsrdquo have the same absorption factors These are the as-sumptions adopted in Ref 41 In addition here one assumesthat the four cells have the same recombination factorsTherefore

fabs fabsC1 = fabsC2 = fabsC34 23

f rec f recC1 = f recC2 = f recC34 24

Calculations are performed for unconcentrated solar ra-diation C=1 Results are shown in Fig 3 The case whenthe refractive indices of cell and upconverter equal each

FIG 3 Dependence of the maximum solar energy conversion efficiency oncell band-gap Eg for a solar cell operating alone a C-RC configuration anda FC-C configuration Devices with and without AR coatings were consid-ered The following input values were used unconcentrated solar radiationC=1 and similar refractive index and absorption factors for both solar celland converter n2=n3=36 and fabsC1= fabsC2= fabsC3= fabsC4=1 a Idealradiative recombination factors f recC1= f recC2= f recC3= f recC4=1 b verygood recombination factors f recC1= f recC2= f recC3= f recC4=10 and c com-mon present-day technology values for the recombination factors f recC1

= f recC2= f recC3= f recC4=10 000



other has been considered there Note that Trupke et al41 didnot consider the FC-C system This configuration is analyzedhere too taking into account that it may improve the perfor-mance of a solar cell operating alone when down-conversionof high-energy photons is considered1920 Using AR coatingsgenerally increases the solar energy conversion efficiency ofcells operating alone as well as of C-RC and FC-C systemsFig 3

The ideal case studied in Ref 41 abs=1 and rec=1 isconsidered first By using Eqs 5 and 6 one derives fabs

=1 and f rec=1 The conversion efficiency of a solar cell withfabs= f rec=1 operating alone shows a maximum for a band-gap energy at around 13 eV Fig 3a This applies in thecase of cells with or without AR coatings For reader conve-nience one reminds that Si and GaAs have band-gap ener-gies of Eg=11 and 14 eV respectively their refractive in-dex is about 34 The efficiency of cells with AR coatings isabout 10 higher than the efficiency of cells without ARcoatings and is slightly smaller than the ShockleyndashQueisserupper limit1841 for unconcentrated radiation Fig 3a Add-ing a rear upconverter to the cell is beneficial in the case ofhigher band gaps but the increase in the conversion effi-ciency is small This applies to C-RC systems with or with-out AR coatings Fig 3a The conversion efficiency of aC-RC system with AR coating is very close to the efficiencyof the C-RC system studied in Ref 41 where the effects ofthe refractive index were partially taken into account Addinga front upconverter to the solar cell diminishes the perfor-mance of the cell Fig 3a

In practical devices the energy conversion efficiency issmaller than estimated in Fig 3a where nonradiative re-combination was neglected This assumption is relaxed inFigs 3b and 3c The conversion efficiency of solar cellswith rec=01 f rec=10 is obviously smaller than that in thecase of the ideal cells with f rec=1 compare Figs 3b and3a Adding a rear or a front upconverter to the cell is notbeneficial in this case Fig 3b These features remain inthe case of present-day quality solar cells with rec

=00001 f rec=10 000 Fig 3c FC-C systems show gen-erally lower performance than C-RC systems Decreasing theradiative recombination efficiency or in other words in-creasing the recombination factor moves the optimum bandgap toward higher values compare Figs 3andash3c

Our previous results19 on the down-conversion techniqueshow that when converter materials with recombination effi-ciency as low as that of the solar cell are used the achievedconversion efficiency is lower than that of the cell workingalone This is consistent with conclusions drawn by research-ers studying ldquodown-shiftingrdquo layers with both low refractiveindex131417 and high refractive index15

The QE of the upconverter in the experimental study byShalav et al26 was calculated to be 16731 In practicaldevices the converter must have higher radiative recombina-tion efficiency than the solar cell Note that materials withrather low nonradiative recombination efficiency are alreadyavailable For example the silica nanocrystals used in Ref17 have a radiative recombination efficiency of 80 Alsocore-shell highly luminescent QDs which could easily be

dispersed in a low refractive index transparent polymer layerhave been produced with a luminescent efficiency of 8444

In the second set of calculations one uses more realisticentry values for the absorption and recombination factorsThe three cells C2ndashC4 used to describe the converter areconsidered similar from the viewpoint of the absorption andrecombination factors but different from the solar cell C1Therefore the following notation is used

fabscell fabsC1 25

fabsconv fabsC2 = fabsC34 26

f reccell f recC1 27

f recconv f recC2 = f recC34 28

However both the cell and the converter are still treated asldquoidealrdquo in the sense that their properties are constant over thewhole spectrum interval where absorption occurs Also pho-ton recycling is neglected r=0 Equations 5 and 6 yieldfabscell=abscell fabsconv=absconv and f reccell=1 reccellf recconv=1 recconv respectively Calculations are performedfor unconcentrated solar radiation C=1 Results are shownin Fig 4 for C-RC and FC-C systems based on Cgh and Cpsolar cells respectively Upconverters with high and idealradiative recombination efficiencies have been consideredrecconv=08 and 1 respectively in other words f recconv

=125 and 1 respectively Both C-RC and FC-C systemsshow lower conversion efficiency than the solar cells operat-ing alone Figs 4andash4d Upconverter systems based onCp cells perform worse than similar systems based on Cgh

FIG 4 Dependence of the maximum solar energy conversion efficiency oncell band-gap Eg for two different quality solar cells ie a high quality cellCgh fabsC1=074 and f recC1=10 and a standard low quality cell CpfabsC1=04 and f recC1=10 000 The graphs show the cells operating aloneand included into a C-RC configuration or a FC-C configuration respec-tively Devices with and without AR coatings respectively were consideredThe following input values were used unconcentrated solar radiation C=1 and the same refractive index for both solar cell and converter n2

=n3=36 The converter absorption factors are always the same fabsC2

= fabsC3= fabsC4=1 a Cgh and converter with f recC2= f recC3= f recC4=125b Cp and converter with f recC2= f recC3= f recC4=125 c Cgh and converterwith f recC2= f recC3= f recC4=1 and d Cp and converter with f recC2= f recC3

= f recC4=1



cells as expected compare Figs 4b and 4d on one handwith Figs 4a and 4c on the other hand Increasing theradiative recombination efficiency of the upconverter or inother words decreasing its recombination factor increasesthe conversion efficiency in the case of both C-RC and FC-Csystems compare Figs 4c and 4d on one hand withFigs 4a and 4b on the other hand

Trupke et al41 considered only the case when the celland upconverter refractive indices equal each other In prac-tice the two quantities may be different Indeed rather lowvalues of the refractive index in the range of 15ndash2 areappropriate for transparent polymer layers and organic mate-rials such as polymethyl methacrylate used fordown-conversion19 Different values of the refractive indexfor solar cell and converter are considered in the next calcu-lations which are performed for a C-RC system based onCgh and Cp solar cells In all cases the band-gap energy isEg=11 eV similar to silicon and the C-RC system is pro-vided with AR coating Unconcentrated solar radiation isconsidered C=1 Results are shown in Fig 5 Note that theconversion efficiency of a Cgh cell operating alone increasesfrom 01978 for n2=1 to a maximum of 02006 for n2=24and then decreases to 02013 for n2=4 Also the efficiencyof a Cp cell operating alone increases from 00777 for n2

=1 to a maximum of 00808 for n2=32 and then decreasesto 00796 for n2=4 Upconverters with high and ideal radia-tive recombination efficiencies have been consideredrecconv=08 and 1 respectively in other words f recconv

=125 and 1 respectively The C-RC system based on theCgh cell has lower conversion efficiency than the efficiency

of the Cgh cell operating alone This applies in the case ofC-RC systems based on Cp solar cells too At small valuesof the cell refractive index n2 roughly less than 2 the con-version efficiency does not depend on the converter refrac-tive index n3 Figs 5andash5d However at higher values ofn2 the conversion efficiency decreases by increasing n3

Figs 5andash5d This observation differs from the qualita-tive conclusion of Ref 41 where the reflectance at the inter-face has been neglected

The influence of the concentration ratio is studied for aCgh cell provided with AR coating operating alone or insidea C-RC system Upconverters with high and ideal radiativerecombination efficiencies have been considered recconv

=08 and 1 respectively in other words f recconv=125 and 1respectively Results are shown in Fig 6 The conversionefficiency of both the cell operating along and the C-RCsystem increases by increasing the concentration ratio Ccompare Figs 6andash6c Adding a rear upconverter to thecell is always beneficial This applies in the case of convert-ers with an ideal recombination factor f recconv=1 and in thecase of the present-day technology recconv=08 or in otherwords f recconv=125 The increase in conversion efficiencyby adding the converter increases at higher values of theconcentration ratio C compare Figs 6andash6c

IV CONCLUSIONS

A system of converting the low-energy incident photonsinto photons of higher energy than the solar cell band gapwas proposed in Ref 41 Here we reanalyzed that system byusing a more elaborated model Two configurations are stud-ied C-RC and FC-C Computations are performed to coverthe energy band gaps of usual semiconductors ie less than2 eV

FIG 5 Dependence of the maximum solar energy conversion efficiency onthe cell and converter refractive index n2 and n3 respectively for a C-RCsystem based on two different quality solar cells ie a high quality cellCgh fabsC1=074 and f recC1=10 and a standard low quality cell CpfabsC1=04 and f recC1=10 000 Devices with AR coatings were consideredThe following input values were used unconcentrated solar radiation C=1 and cell band-gap Eg=11 eV The converter absorption factors arealways the same fabsC2= fabsC3= fabsC4=1 a Cgh inside a C-RC systemwith f recC2= f recC3= f recC4=125 b Cp inside a C-RC system with f recC2

= f recC3= f recC4=125 c Cgh inside a C-RC system with f recC2= f recC3

= f recC4=1 and d Cp inside a C-RC system with f recC2= f recC3= f recC4=1

FIG 6 Dependence of the maximum solar energy conversion efficiency oncell band-gap Eg for a high quality solar cell Cgh fabsC1=074 and f recC1

=10 operating alone and inside a C-RC configuration respectively Deviceswith AR coatings were considered The following input values were usedsimilar refractive index and absorption factors for both solar cell and con-verter n2=n3=36 and fabsC1= fabsC2= fabsC3= fabsC4=1 Two values of theconverter recombination factor were used f recconvf recC1= f recC2= f recC3

= f recC4=125 and 1 Concentration ratios a C=100 b C=500 and cC=1000



The results we obtained are generally different fromthose presented in Ref 41 The difference is due to modelimprovement as well as to the more involved analysis per-formed here Modeling has been improved by taking accountof the transmittance effects at interfaces in a more detailedway than how it was treated in Ref 41 Also we considereda more realistic solar cell model which includes nonradiativerecombination this process was neglected in Ref 41 Ouranalysis has taken into account the effect of antireflectionlayers and we considered cells and upconverters of differentrefractive indices These aspects extend the work41 wherecells and upconverters of equal refractive indices and withoutantireflection layers were considered

The main conclusions of this work are as follows

1 The reflectance at the interfaces of the C-RC and FC-Csystem has significant influence on the solar energy con-version efficiency

2 When maximum ideal radiative recombination effi-ciency is considered for both cell and converter the con-version efficiency slightly increases in the case of aC-RC system under unconcentrated solar radiation ascompared to the efficiency of a solar cell operatingalone These results confirm previous findings by Trupkeet al41

3 When similar but realistic values for the radiative re-combination efficiency are considered for both cell andconverter the conversion efficiency decreases in thecase of a C-RC system under unconcentrated solar ra-diation as compared to the efficiency of a solar cell op-erating alone

4 Adding a rear upconverter to the solar cell is beneficialin the case of present-day quality solar cells operatingunder concentrated solar radiation A few comments areuseful The so-called ldquospectral concentrationrdquo techniquehas been proposed recently31 to increase the perfor-mance of C-RC systems operating under unconcentratedsolar radiation The starting point was the observationthat rare earth based upconverters have a weak and nar-row absorption range This leaves most of the solarspectra with energies below the band gap unused andtherefore limits the ability of this particular upconver-sion materials to reduce the subband-gap losses31 Toavoid this inconvenience Struumlmpel et al45 proposed tocombine the upconverter with a fluorescent materialThe fluorescent material absorbs photons with wave-lengths between the solar cell band gap and the absorp-tion range of the upconverter and emits in the narrowabsorption range of the upconverting material The up-converter then converts these photons to photons withenergies above the band gap of the cell This process hasbeen called spectral concentration31 because the photondensity in the upconverter absorption range is increasedBut possible fluorescent materials absorb in the emissionrange of the upconverter too To avoid unwanted ab-sorption Goldschmidt et al31 separated the upconverterand the fluorescent material by a selectively reflectivephotonic structure The geometric concentration in-creases by incorporation of the fluorescent material in a

fluorescent concentrator and this increases the upcon-version process efficiency

5 At small values of the cell refractive index n2 roughlyless than 2 the conversion efficiency does not dependon the converter refractive index n3 in a C-RC systemAt higher values of n2 the conversion efficiency de-creases by increasing n3

6 The energy conversion efficiency does not increase byadding a front upconverter to the cell whatever the val-ues of the radiative recombination efficiency and con-centration ratio are

ACKNOWLEDGMENTS

The author thanks the referees for useful comments andsuggestions and Dr Claudia Struumlmpel University of Kon-stanz and Dr Jan Christoph Goldschmidt and Dr FrankDimroth Fraunhofer ISE Freiburg for providing researcharticles describing recent experimental results This paper isdedicated to the memory of Eugenia Badescu the authorrsquosmother

1V Badescu and P T Landsberg Semicond Sci Technol 8 1267 19932V Badescu and P T Landsberg Semicond Sci Technol 12 1491 19973P T Landsberg H Nussbaumer and G Willeke J Appl Phys 74 14511993

4V Badescu P T Landsberg A De Vos and B Desoete J Appl Phys 892482 2001

5P T Landsberg and V Badescu J Phys D 35 1236 20026V Badescu J Phys D 38 2166 20057B S Richards Sol Energy Mater Sol Cells 90 1189 20068A Meijerink R Wegh P Veerger and T Vlugt Opt Mater AmsterdamNeth 28 575 2006

9N Kodama and Y Watanabe Appl Phys Lett 84 4141 200410R T Wegh E V D van Loef and A Meijerink J Lumin 90 111

200011R T Wegh H Donker E V D van Loef K D Oskam and A

Meijerink J Lumin 87ndash89 1017 200012K D Oskam R T Wegh H Donker E V D van Loef and A

Meijerink J Alloys Compd 300ndash301 421 200013T Markvart J Appl Phys 99 026101 200614W G J H M van Sark Appl Phys Lett 87 151117 200515W G J H M van Sark A Meijerink R E I Schropp J A M van

Roosmalen and E H Lysen Sol Energy Mater Sol Cells 87 395 200516W G J H M van Sark A Meijerink R E I Schropp J A M van

Roosmalen and E H Lysen Semiconductors 38 962 200417V Švrček A Slaoui and J-C Muller Thin Solid Films 451ndash452 384

200418T Trupke M A Green and P Wurfel J Appl Phys 92 1668 200219V Badescu A De Vos A M Badescu and A Szymanska J Phys D 40

341 200720V Badescu and A De Vos J Appl Phys 102 073102 200721M Wolf Proc IRE 48 1246 196022A Luque and A Marti Phys Rev Lett 78 5014 199723E Jackson Transactions of the Conference on Use of Solar Energy Tus-

con Arizona 1955 unpublished Vol 5 p 12224K Barnham and G Duggan J Appl Phys 67 3490 199025P Gibart F Auzel J-C Guillaume and K Zahraman 13th EPVSEC

Nice France 1995 unpublished p 8526A Shalav B Richards T Trupke R Corkish K Kramer H Gudel and

M Green Proceedings of the Third World Conference on PhotovoltaicEnergy Conversion 11ndash18 May 2003 unpublished Vol 1 pp 248ndash250

27C Struumlmpel M J McCann G Beaucarne V Arkhipov A Slaoui VSvrcek C del Cantildeizo and I Tobias Sol Energy Mater Sol Cells 91 2382007

28J Ohwaki and Y Wang Jpn J Appl Phys Part 2 33 L334 199429C Struumlmpel M J McCann and G Hahn 23rd European Photovoltaic

Solar Energy Conference and Exhibition Valencia Spain 1ndash5 September



2008 unpublished Paper No 1DV21730C Struumlmpel M J McCann C del Cantildeizo I Tobias and P Fath Pro-

ceedings of the 20th European Solar Energy Conference and ExhibitionBarcelona Spain 6ndash10 June 2005 unpublished pp 43ndash46

31J C Goldschmidt P Loumlper S Fischer S Janz M Peters S W GlunzG Willeke E Lifshitz K Kraumlmer and D Biner IUMRS-ICEM 2008Sydney Australia 28 Julyndash1 August 2008 unpublished Paper No G5-S14

32A W Bett F Dimroth S W Glunz A Mohr G Siefer and G WillekeProceedings of the 19th European Photovoltaic Solar Energy ConferenceParis 7ndash11 June 2004 unpublished pp 2088ndash2491

33T Trupke A Shalav B S Richards P Wurfel and M A Green SolEnergy Mater Sol Cells 90 3327 2006

34R R King D C Law C M Fetzer R A Sherif K M Edmondson SKurtz G S Kinsey H L Cotal D D Krut J H Ermer and N HKaram Proceedings of the 20th European Solar Energy Conference andExhibition Barcelona Spain 6ndash10 June 2005 unpublished pp 118ndash121

35M Yamaguchi T Takamoto T Agui M Kaneiwa K Nishimura YYagi T Sasaki N J Ekins-Daukes H S Lee N Kojima and Y OhshitaProceedings of the 19th European Photovoltaic Solar Energy ConferenceParis 7ndash11 June 2004 unpublished pp 3610ndash3613

36M Yamaguchi T Takamoto T Agui M Kaneiwa K Araki M Kondo

H Uozumi M Hiramatsu Y Miyazaki T Egami and N J YoshishigeKemmoku Proceedings of the 19th European Photovoltaic Solar EnergyConference Paris 7ndash11 June 2004 unpublished pp 2014ndash2017

37Sharp Corporation Proceedings of the Seventh PVTEC Technology De-brief Session 2005 unpublished pp 18ndash22

38F Dimroth G Letay M Hein M Meusel S van Riesen and U Schu-bert Proceedings of the 31st IEEE PVSC Orlando FL 2005 unpub-lished pp 525ndash529

39A W Bett G Siefer C Baur S v Riesen G Peharz H Lerchenmuumlllerand F Dimroth Proceedings of the 20th European Solar Energy Confer-ence and Exhibition Barcelona Spain 6ndash10 June 2005 unpublished pp114ndash117

40F Dimroth Phys Status Solidi C 3 373 200641T Trupke M A Green and P Wuumlrfel J Appl Phys 92 4117 200242T Trupke J Zhao A Wang R Corkish and M A Green Appl Phys

Lett 82 2996 200343W G J H M van Sark P L T M Frederix A A Bol H C Gerritsen

and A Meijerink ChemPhysChem 3 871 200244A P Alivisatos MRS Bull 23 18 199845C Struumlmpel M McCann C del Cantildeizo I Tobias and P Fath Proceed-

ings of the 20th European Photovoltaic Solar Energy Barcelona Spain6ndash10 June 2005 pp 43ndash46



An extended model for upconversion in solar cellsViorel Badescua

Candida Oancea Institute Polytechnic University of Bucharest Spl Independentei 313Bucharest 060042 Romania

Received 25 July 2008 accepted 24 October 2008 published online 15 December 2008

Here we analyze the system proposed by Trupke et al J Appl Phys 92 4117 2002 to increasesolar cell efficiency The system consists in adding to the cell a so-called upconverter which is adevice able to convert the low-energy subband-gap incident solar photons into photons of higherenergy The model takes account of i the nonradiative recombination in both solar cell andconverter and ii the refractive index of both cell and converter Two configurations are studied celland rear converter C-RC and front converter and cell The main conclusions are as follows 1When nonradiative recombination is neglected for both cell and converter the energy conversionefficiency of a C-RC system slightly exceeds the efficiency of a solar cell operating alone under 1sun illumination 2 When similar realistic values for the radiative recombination efficiency areconsidered for both cell and converter the energy conversion efficiency of a C-RC system is lowerthan the efficiency of a solar cell operating alone under 1 sun illumination 3 Adding a rearupconverter to the solar cell is beneficial in the case of present-day quality solar cells underconcentrated solar radiation 4 At small values of the cell refractive index roughly less than 2 theconversion efficiency does not depend on the converter refractive index 5 At higher values of thecell refractive index the conversion efficiency decreases by increasing the converter refractiveindex 6 The energy conversion efficiency does not increase by adding a front upconverter to thecell whatever the values of the radiative recombination efficiency and solar radiation concentrationratio are copy 2008 American Institute of Physics DOI 10106313040692

I INTRODUCTION

Many techniques have been proposed to increase the ef-ficiency of solar cells They are based on different processesand properties such as photon recycling12 impactionization3ndash5 angle dependent selectivity6 down-conversionwhich is sometimes referred to as ldquoquantum cutting viadown-conversionrdquo7ndash12 and down-shifting13ndash17 Combina-tions of down-shifting and down-conversion processes werealso studied see eg Refs 18ndash20

Another technique studied in the past years converts theincident low-energy subband-gap solar photons into pho-tons of energy higher than solar cell band gap Thus thenumber of high-energy photons absorbed in the cell in-creases This ldquoupconversionrdquo technique has a number ofadvantages27 no modification of the active layer is neededas is the case with the impurity photovoltaic effect21 and theintermediate band solar cell22 and it does not require com-plicated structures such as that used for multiple-junctionsolar cells23 or quantum well solar cells24 Gibart et al25

reported the application of upconversion on a GaAs solar cellwith a vitroceramic codoped with trivalent erbium and triva-lent ytterbium at an energy of 1391 eV under excitation of1 W 0039 cm2 Shalav et al26 reported the application to abifacial silicon solar cell with a rear upconverter consistingof NaYF4 doped with trivalent erbium for monochromaticlight of 1523 nm under excitation of 24 W cm2

For the application to silicon solar cells erbium dopedmaterials are most promising for upconversion27 Alsobarium chloride doped with trivalent erbium BaCl2 Er3+ isproposed to be an efficient upconverter due to advantageousphononic properties28 Upconversion processes have alsobeen demonstrated in organic materials and in transitionmetals2930 A good review of the properties of available up-converters with focus on silicon solar cells has been made inRef 27

As the experiments so far have been carried out withlaser illumination little is known about the behavior of thesystem under sun illumination31 The efficiency of upconver-sion increases with increasing intensity Thus solar radiationconcentration is needed to achieve conditions comparable tothe mentioned laser illumination A 500 sun concentration isexpected to achieve conversion efficiencies comparable withthose obtained under laser excitation32

Theoretical studies predict that the maximum conversionefficiency for an ideal upconverter on the rear side of a cellunder nonconcentrated sunlight increases as compared to thecase of the solar cell operating alone A maximum conver-sion efficiency of 374 is predicted in Ref 32 while for asilicon solar cell the estimated upper limit is 40233 Theefficiency enhancement due to an upconverter is expected toincrease up to about 55 at 100 sun concentration27 Com-paring these theoretical values with values accessible bypresent-day technology may give a broader perspective forthe upconversion technique Todayrsquos most efficient solar celltechnology is based on multijunctions made of III-V com-pound semiconductors The best material combination is alattice matched solar cell that consists of three active junc-

aAuthor to whom correspondence should be addressedTel 40214029428 FAX 40214104251 Electronic mailbadescuthetatermopubro

JOURNAL OF APPLIED PHYSICS 104 113120 2008

0021-897920081041111312010$2300 copy 2008 American Institute of Physics104 113120-1










NElEun1T11B1 =n1

2B1

433c02

El

Eu

2

exp minus 1

kT1 minus 1minus1

d 1



2B1rarr2

433c02

El

Eu

2

exp minus 1

kT1 minus 1minus1

d

2



2B1rarr2rarr3

433c02

El

Eu

2

exp minus 1

kT1 minus 1minus1

d

3





I



fabs = abs 5


rec1 minus r 13 6








FactorNo of crossed


Ref 41Presentwork










FactorNo of crossed



















IC1



+ E2n3TcC2F3rarr2


+ E2n2TcC1F2rarr3 7

IC2



IC3




IC4













IC1






IC2






IC3




IC4








qVC1 = C1 15a

qVC2 = C2 15b

qVC3 = C3 15c

qVC4 = C4 15d





C3 + C4 = C2 17


=IC1VC1

13s 18


13s = BsTs4 19










Also note that20


22


III RESULTS




























fabscell fabsC1 25









= f recC4=1











IV CONCLUSIONS



















ACKNOWLEDGMENTS















































NElEun1T11B1 =n1

2B1

433c02

El

Eu

2

exp minus 1

kT1 minus 1minus1

d 1



2B1rarr2

433c02

El

Eu

2

exp minus 1

kT1 minus 1minus1

d

2



2B1rarr2rarr3

433c02

El

Eu

2

exp minus 1

kT1 minus 1minus1

d

3





I



fabs = abs 5


rec1 minus r 13 6








FactorNo of crossed


Ref 41Presentwork










FactorNo of crossed



















IC1



+ E2n3TcC2F3rarr2


+ E2n2TcC1F2rarr3 7

IC2



IC3




IC4













IC1






IC2






IC3




IC4








qVC1 = C1 15a

qVC2 = C2 15b

qVC3 = C3 15c

qVC4 = C4 15d





C3 + C4 = C2 17


=IC1VC1

13s 18


13s = BsTs4 19










Also note that20


22


III RESULTS




























fabscell fabsC1 25









= f recC4=1











IV CONCLUSIONS



















ACKNOWLEDGMENTS







































I



fabs = abs 5


rec1 minus r 13 6








FactorNo of crossed


Ref 41Presentwork










FactorNo of crossed



















IC1



+ E2n3TcC2F3rarr2


+ E2n2TcC1F2rarr3 7

IC2



IC3




IC4













IC1






IC2






IC3




IC4








qVC1 = C1 15a

qVC2 = C2 15b

qVC3 = C3 15c

qVC4 = C4 15d





C3 + C4 = C2 17


=IC1VC1

13s 18


13s = BsTs4 19










Also note that20


22


III RESULTS




























fabscell fabsC1 25









= f recC4=1











IV CONCLUSIONS



















ACKNOWLEDGMENTS













































IC1



+ E2n3TcC2F3rarr2


+ E2n2TcC1F2rarr3 7

IC2



IC3




IC4













IC1






IC2






IC3




IC4








qVC1 = C1 15a

qVC2 = C2 15b

qVC3 = C3 15c

qVC4 = C4 15d





C3 + C4 = C2 17


=IC1VC1

13s 18


13s = BsTs4 19










Also note that20


22


III RESULTS




























fabscell fabsC1 25









= f recC4=1











IV CONCLUSIONS



















ACKNOWLEDGMENTS







































IC1






IC2






IC3




IC4








qVC1 = C1 15a

qVC2 = C2 15b

qVC3 = C3 15c

qVC4 = C4 15d





C3 + C4 = C2 17


=IC1VC1

13s 18


13s = BsTs4 19










Also note that20


22


III RESULTS




























fabscell fabsC1 25









= f recC4=1











IV CONCLUSIONS



















ACKNOWLEDGMENTS












































Also note that20


22


III RESULTS




























fabscell fabsC1 25









= f recC4=1











IV CONCLUSIONS



















ACKNOWLEDGMENTS
















































fabscell fabsC1 25









= f recC4=1











IV CONCLUSIONS



















ACKNOWLEDGMENTS















































IV CONCLUSIONS



















ACKNOWLEDGMENTS
















































ACKNOWLEDGMENTS

























































Date post:	12-Dec-2016
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An extended model for upconversion in solar cells

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