Optical and recombination losses in thin-film Cu(In,Ga)Se2 solar cells

L.A. Kosyachenko a,n, X. Mathew b, P.D. Paulson c, V.Ya. Lytvynenko a, O.L. Maslyanchuk a

a ChernivtsiNationalUniversity, Kotsyubinsky Str. 2, Chernivtsi 58012, Ukraineb Instituto de Energias Renovables, Universidad Nacional Autonoma de Mexico, Temixco, Morelos 62580, Méxicoc Nuvosun Inc., 1565 Barber Ln, Milpitas, CA 95035, USA

a r t i c l e i n f o

Article history:Received 16 May 2014Received in revised form26 June 2014Accepted 11 July 2014

Keywords:CuIn(Ga,In)Se2 solar cellsOptical lossesRecombination losses

a b s t r a c t

Optical and recombination losses in thin-film solar cells based on CuInxGa1-xSe2 with the bandgaps1.14–1.16 and 1.36–1.38 eV have been evaluated. Parameters used for the analysis and calculations wereverified by comparing the measured quantum efficiency spectra with the results of calculations. Opticallosses due to reflections from the interfaces and absorption in the ZnO and CdS layers are found usingthe optical constants of materials taking into account anti-reflection coating on the ZnO surface. It isshown that for typical parameters of solar cells studied the optical losses amount to 16–18%. Losses dueto the recombination of charge carriers at the front and back surfaces of the absorbing layer and in thespace charge region (SCR) are calculated based on their dependences on the carrier lifetime, theconcentration of uncompensated acceptors (determined the width of the SCR) and other parameters ofthe absorber. Total recombination losses in solar cells with the bandgaps of CuIn(Ga,In)Se2 1.14 and1.36 eV are equal to 7.0% and 4.5%, respectively. It is also shown that for solar cell with the bandgap ofabsorber 1.14 eV, an incomplete charge collection is caused also due to non-optimal width of the SCR. Animprovement of charge collection in this cell can be achieved by varying the concentration ofuncompensated acceptors in the absorber. However, it is impossible to achieve such improvement forsolar cells with the bandgap of 1.36 eV due to shorter carrier lifetime in the material.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

For a long time, solar cells based on CuInxGa1�xSe2 (CIGS),similar to CdTe-based devices, keep a stable position in thin-filmphotovoltaics as an alternative to solar modules based on mono-and poly-silicon wafers. Mass production of cost-effective CIGS-based modules has been achieved by many companies worldwide.The efficiency of such modules is in the range of 12–15%, but forsmall area laboratory cells, the efficiency achieved 20.3% in 2010,which was a record level among thin-film solar cells [1]. In early2014, Solar Frontier has achieved 20.8% energy conversion effi-ciency for small area CIS cells and shortly Zentrum für Sonnene-nergie- und Wasserstoff-Forschung Baden-Württemberg (ZSW)improved the cell efficiency to 20.9% [2,3]. It should be noted thatthe CIGS devices have shown excellent long-term stability andhigh radiation resistance, can be made lightweight and flexiblethat are desirable for space, portable and building integratedapplications [4]. Since the efficiency of 12–15% in modules is abouthalf of the theoretical limit (28–30%), works to improve the CIGS

module efficiency are extremely relevant both from scientific andeconomic points of view.

The processes of photoelectric conversion in CIGS and CdTesolar cells are largely similar. In both cases, the thin-film p-nheterostructure is the key element in determining the perfor-mance of the device. As in CdTe solar cell, a thin layer of n-CdSserves as “window”, through which radiation penetrates into theCIGS absorber. The difference between these devices lies in theirsuperstrate (CdTe) and substrate (CIGS) configuration. In super-strate configuration, the sunlight enters the absorber through theglass plate and transparent conductive layer (TCO, usually SnO2:F)while only through the TCO layer (usually ZnO:Al) in substrateconfiguration. These design features are not of fundamentalimportance from the point of view of the physical processes takingplace, but only require different device fabrication technologies. Itfollows that the physical models developed for the interpretationof the CdTe solar cell properties can be applied with somemodifications to the CIGS devices [5,6].

In this paper we consider the optical and recombination lossesin CIGS devices, which are important causes of low solar-to-electric energy conversion and quantum efficiency in solar cells.We discuss the quantitative determination of the losses andidentify possible pathways to reduce it. Calculations of the opticallosses are carried out based on the optical constants of the

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Solar Energy Materials & Solar Cells

http://dx.doi.org/10.1016/j.solmat.2014.07.0190927-0248/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ380 372 244 22; fax: þ380 372 551 809.E-mail address: l.a.kosyachenko@gmail.com (L.A. Kosyachenko).

Solar Energy Materials & Solar Cells 130 (2014) 291–302

materials used, i.e., the refractive indices and extinction coeffi-cients. Recombination losses are determined on the basis of thecontinuity equation taking into account the drift and diffusioncomponents of the photocurrent.

The analysis of the physical process discussed in this paper canbe useful from a practical point of view since undoubted successesin the development of efficient CuInxGa1-xSe2 solar cells have beenachieved mainly empirically [4].

2. Main parameters and optical constants of the materialsused

We present analysis on CIGS solar cells with two absorberlayer: one with bandgap Eg¼1.14–1.16 eV (Ga/(InþGa)E0.3)which is the material used in the mass production of highefficiency solar modules; the second is the wider bandgapEg¼1.36–1.38 eV (Ga/(InþGa)E0.6–0.7) which is close to thetheoretically optimal value for maximum efficiency 28–30%.

Fig. 1 shows the schematic cross-section of a typical CIGS solarcell, where the notations of the optical constants ni and κi and thereflection coefficients Rij at the interfaces used in the calculations

are indicated. Photoelectric conversion in this solar cell occurs inthe CIGS absorber with a thickness of 2 μm, while the CdS filmwith a thickness 30–40 nm is a window layer for the absorber.A 500-nm-thick ZnO:Al is the transparent conducting oxide (TCO)and application of an undoped high-resistivity 50-nm-thickZnO layer between TCO and CdS is very common in substrateconfiguration devices. In high-efficiency devices, an antireflection�100-nm-thick MgF2 layer is also deposited onto the front surfaceof ZnO.

In the calculations of quantum efficiency of CdS/CIGS cell, oneneeds to know the optical transmission of the ZnO/CdS structureT(λ), which is determined by reflections from the interfaces air/ZnO, ZnO/CdS, CdS/CIGS and absorption in the ZnO and CdS layers.

For determining the optical transmission T(λ), it is necessary toknow the refraction indices ni and extinction coefficients κi of ZnO,CdS and CIGS. According to the Fresnel equations, when the light isat near-normal incidence, the reflection coefficients (reflectances)from three interfaces R12, R23 and R34 can be calculated as

Rij ¼nn

i �nn

j

��� ���2nn

i þnn

j

��� ���2 ¼ðni�njÞ2þðκi�κjÞ2ðniþnjÞ2þðκiþκjÞ2

: ð1Þ

where nn

i and nn

j are the refractive indices of the materials, whichaccount for their conductivity containing imaginary parts and arewritten as nn

i ¼ni–iκi and nn

j ¼nj–iκj.Fig. 2 shows the spectral dependences of n and κ for ZnO taken

from [4,7], for CdS from [8] and CIGS from [9] (in some cases theextinction coefficient κ was determined as αλ/4π, where α is theabsorption coefficient). It is worth to note the fact that for CIGS,the extinction coefficient in the photon energy range hvoEg has avalue of 0.04–0.05, which corresponds to the absorption coeffi-cient α¼4πκ/λ¼(4–5)�103 cm–1 [9]. With such values of α, afairly high quantum efficiency should be obtained for the solar cell,but, in fact, this is not observed. At λ4λg¼hс/Eg, the quantumefficiency decreases quite rapidly with wavelength up to zerowithin a range about 150 nm above λg. This can be explained bythe presence of so-called ‘tails’ of the density of states in thebandgap of the semiconductor with strong doping or/and dis-ordered crystal structure. In this case, the electron wave functionsand force fields of impurity atoms overlap, whereby the discreteimpurity levels are broadened and transformed into an impurityband. At a certain critical concentration, the impurity band joinswith the conduction (valence) band, i.e., these tails of the densityof states appear. The absorption coefficient in this range depends

ZnO:Al (n2, κ2)

CdS (n3, κ3)

Cu(In,Ga)Se2 (n4, κ4)

R12

R23

R34

R a d i a t i o n

Metal

Metal grid Air (n1, κ1)

Substrate(glass, polyimide, metal foil)

High-resistive ZnO

Anti-reflection coating

Fig. 1. Cross-section of a typical CIGS solar cell.

n(λ

)

200 400 600 800 1.5

2.0

2.5

3.0

3.5

λ (nm)

CdS

ZnO

Cu(In,Ga)Se2Eg= 1.16 eV

Eg= 1.38 eV

κ(λ

)

1000 1200 1400 200 400 600 800 1000 1200 1400λ (nm)

Cu(In,Ga)Se2

CdS

ZnO

1.0

0.1

10–2

10–3

10–4

10

Eg= 1.16 eV

Eg= 1.38 eV

Fig. 2. Wavelength dependences of the refractive indices (a) and extinction coefficients (b) of ZnO, CdS, CuIn0.69Ga0.31Se2 and CuIn0.34Ga0.66Se2.

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302292

exponentially on the photon energy α(hv)pexp(hv/Eo), where Eois the spectral independent value. This empirical dependence iscalled the Urbach rule (Eo can be proportional to kT at sufficientlyhigh T).

In order to make the interband transition and to take part inthe formation of the photocurrent, the electron must get theenergy, which is equal to or greater than Eg. In the case of the tailabsorption at hvoEg, this occurs due to the fact that the part ofenergy, equal to hv, electron gets from photon, while its deficit iscovered by phonons (the so-called multiphonon transitions). TheUrbach rule reflects the probability of the multiphonon process,which decreases with reducing hv, reproducing the spectral curveof photocurrent but with a stronger dependence on the photonenergy. It is important that an electron having absorbed photonwith energy hvoEg takes part in the photocurrent formation justas in the absorption of a photon with energy hvZEg. Based on theabove comments, in our calculations we use the κ(λ) curves ofCIGS corrected in the long-wavelength region, as shown in Fig. 2bby broken lines. Anticipating things note that part of the spectrumhvoEg makes a relatively small contribution to the short-circuitcurrent in solar cells: 0.4 mA/cm2 (1.1%) and 0.8 mA/cm2 (2.9%) forthe absorber bandgap 1.16 and 1.38 eV, respectively.

In the spectral range hvZEg, the absorption coefficient for CIGSfollows the law for a direct-gap semiconductor:

α¼ αoðhv�EgÞ1=2

hv: ð2Þ

The above correction of the spectral curve in the long wave-length region does not exclude the possibility of determining thebandgap of the samples by using Eq. (2) and the measured valuesof α as illustrated in Fig. 3.

As seen in Fig. 3, for the two materials distinct straight-lineportions intercepting the bandgaps on the photon energy axis areobserved. From the intercept the bandgap values are estimated as1.16 and 1.38 eV for CuIn0.69Ga0.31Se2 and CuIn0.34Ga0.66Se2,respectively. These bandgap values correspond to the wavelengths1068 and 898 nm as shown by arrows in Fig. 2b.

3. Optical losses caused by reflection and absorption in ZnOand CdS layers

The reflection coefficients for the interfaces air/ZnO (R12),ZnO/CdS (R23) and CdS/CIGS (R34) can be calculated using Eq. (1),whereas absorption in the layers ZnO and CdS can be presented asexp(–α2d2) and exp(–α3d3), where α2 and α3, d2 and d3 are theabsorption coefficients and thicknesses of the ZnO and CdS layers,respectively. Then the transmission T(λ) of these layers can bewritten as

TðλÞ ¼ Tgrð1�R12Þexpð�α2d2Þð1�R23Þexpð�α3d3Þð1�R34Þ: ð3ÞThe factor Tgr in Eq. (3) takes into account the shading effect

due to the lateral current-collecting metal grid contacts depositedonto the front surface of ZnO. In practice, the dimensions of thegrid are reduced such that it shades only about 4–5% of the frontarea of the solar cell. With minimum error the Tgr value can be setequal to 0.96 [4].

3.1. Effect of antireflection coating

Eq. (3) does not take into account the effect of the antireflectioncoating, which significantly reduces the reflectance of the frontsurface R12. Magnesium fluoride, MgF2, is commonly used as theantireflection coating for ZnO because MgF2 has the refractiveindex narc close to the theoretical optimal value (n2)1/2, where n2 isthe refractive index of ZnO. This is confirmed by the data shown in

0.8 1.0 1.2 1.4 1.6

1

0

2

3

4

hv (eV)

(α. h

v)2 (

107 c

m–2

)

x = 0.66

x = 0.31

CuIn1–xGaxSe2

Fig. 3. Comparison of dependence of the absorption coefficient α of CuIn1–xGaxSe2on the photon energy hv with Eq. (2).

200 400 600 8000

2

3

1000 1200

1 na

(n2)1/2

n2

n a, n

2, (n

2)1/

2

200 400 600 800 0

0.2

0.4

0.6

0.8

1.0

T( λ

)

1

2

1200 (nm) λ (nm) λ

Fig. 4. (a) Comparison of the spectral dependences of the refraction indices for ZnO (n2), MgF2 (narc) and (n2)1/2¼narc and (b) the optical transmission of the ZnO/CdS layerswith and without anti-reflection coating over ZnO (curve 1 and curve 2, respectively).

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302 293

Fig. 4a, where a comparison of the dependences narc, n2 and (n2)1/2

on the wavelength are presented.The expression for the reflection coefficient from the material

with an antireflection coating has the form [10]

Rarc ¼r2f þr2bþ2rf rb cos ð2βÞ1þr2f r

2bþ2rf rb cos ð2βÞ; ð4Þ

where rf¼(narc�n1)/(narcþn1) and rb¼(n2�narc)/(n2þnarc) are theamplitude values of reflectivity (Fresnel coefficients) from the frontand back surfaces of the antireflection material and

β¼ 2πλnarcdarc: ð5Þ

Fig. 5 shows the reflectance spectrum of the ZnO surface with a100-nm-thick MgF2 coating and without antireflection coating.

The position of the reflectance minimum of the curve 2 isdependent on the thickness of antireflection coating darc. As seen,for darc¼100 nm the minimum is at the wavelength �550 nmcorresponding approximately to the maximum of the solar radia-tion under the AM1.5 conditions.

A comparison of the two curves in Fig. 5 shows a significantdecrease in reflection in the range 500–800 nm due to theantireflection coating. Sharp increase in reflectance at λo500 nmdoes not produce a significant effect in solar cell performancesince at short wavelength region the intensity of solar radiationdecreases rapidly. In addition, the region below 500 nm is domi-nated by absorption in the CdS buffer layer and the TCO (seeFig. 4b). However, an increase in reflectance at λ4500 nm canhave negative effect on solar cell performance.

Calculation shows that the antireflection coating increases theshort-circuit current by 9.1–9.2% for both types of solar cells.

It should be noted that the reflection of ZnO can also be reducedby texturing the surface, as a result of chemical treatment of thesurface or even its interaction with the ambient atmosphere [4].

3.2. Reflection and Absorptive losses

Quantitative characterization of the influence of optical losseson the solar cell performance can be obtained by calculating theshort-circuit current density Jsc. We will calculate Jsc for thespectral distribution of AM1.5 total solar radiation using the TableISO 9845-1:1992 [11]. If Φ is the spectral radiation power densityand hν is the photon energy, then the spectral density of the

incident photon flux is Φ/hν; hence we can write Jsc as

Jsc ¼ q∑iTðλiÞ

ΦiðλiÞhvi

Δλi; ð6Þ

where q is the electron charge, Δλi is the interval between theneighboring values of λi in the ISO table. The summation in Eq. (6)should be made over the spectral range from λ¼300 nm (atλo300 nm the terrestrial solar radiation is negligible) toλ¼λg¼hc/Eg¼1068 nm for CIGS solar cell with the bandgap ofthe absorber Eg¼1.16 eV and λ¼898 nm for Eg¼1.38 eV. Analyzingthe optical losses, we assume that the quantum efficiency of thesolar cell η¼1, so Eq. (6) does not contain η.

Taking into account antireflective coating, the reflection coeffi-cient R12 in Eq. (3) should be replaced by the coefficient Rarc, whichis determined by Eq. (4):

TðλÞ ¼ Tgrð1�RarcÞexpð�α2d2Þð1�R23Þexpð�α3d3Þð1�R34Þ: ð7ÞThe calculations show that in the case of 100% quantum

efficiency and 100% transmission through the ZnO and CdS layersthe short-circuit current density Jsc of the CIGS solar cell with theabsorber bandgap 1.16 eV would be equal to 42.0 mA/cm2. How-ever, for real T(λ) after taking into account reflection from theinterfaces and absorption in the ZnO and CdS layers (d¼500 and40 nm, respectively) without and with antireflection coating, theJsc value becomes equal to 30.6 and 33.7 mA/cm2, respectively. Itfollows that the antireflection coating increases Jsc by 7.4%. In thecase of the CIGS solar cell with absorber bandgap Eg¼1.38 eV for100% quantum efficiency and 100% transmission, Jsc would beequal to 33.0 mA/cm2. This Jsc value decreases to 24.8 and 27.4 mA/cm2, respectively without and with antireflection coating, i.e., theantireflection coating increases Jsc by 7.9%.

From a practical standpoint, it is important to assess the varioustypes of losses due to reflections at the interfaces and absorptionin the ZnO and CdS layers. Reflection loss at the interface can bedetermined using Eq. (7) as the difference between the flux ofphotons that approached the interface and passed through it.Absorptive losses are determined from the difference between thephoton fluxes that entered the layer and that reached theother side.

Decrease in short-circuit current due to reflection from thefront surface of ZnO with an antireflection coating can be obtainedfrom Eq. (6) in which transmission T(λ) is taken from Eq. (7).Calculations show that for CuIn0.69Ga0.31Se2 and CuIn0.34Ga0.66Se2solar cells, these losses for the 100 nm thickness of the MgF2 layerare equal to 1.9% and 1.4%, respectively.

200 400 600 8000

5

10

15

1000 1100 1200

20

λ (nm)

R (%

)

2

1

Fig. 5. Reflectivity of the ZnO surface without and with MgF2 antireflection coating(curve 1 and curve 2, respectively).

0

1

2

3

4

ΔJ s

c /J s

c (%

)

60 80 100 120

darc

CuIn0.34Ga0.66Se2

CuIn0.69Ga0.31Se2

160140

Fig. 6. Dependences of short-circuit current losses on the thickness of antireflec-tion coating applied on the ZnO surface for CuIn0.69Ga0.31Se2 and CuIn0.34Ga0.66Se2solar cells.

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302294

Fig. 5 shows that the zero reflectance of ZnO with an antire-flection coating corresponds to the wavelength of 570 nm for a100 nm thick layer of MgF2. It is interesting to see how the short-circuit current density Jsc (which takes into account the spectraldistribution of solar radiation) depends on the thickness of thelayer darc. The calculated dependences of decrease in Jsc on darc areshown in Fig. 6 (for the two cells, the curves are different becausethey cover different spectral regions).

As seen in Fig. 6, the loss is minimal when the thickness of theMgF2 film is in the range of 105–115 nm that is close to thethickness commonly used for MgF2 film but the losses becomesignificant when the MgF2 thickness is below or above the 105–115 nm range. Note that the reflection losses reach 9.2–9.3% inthe absence of antireflection coating while with coating it is only1.4–1.9%.

The reflection loss at the ZnO/CdS interface calculated usingEqs. (6) and (7) is equal to 0.9% and 1.0% respectively forCuIn0.69Ga0.31Se2 and CuIn0.33Ga0.66Se2 solar cells, whereas forthe CdS/CIGS interface the loss is 1.3% and 1.1%, respectively. Suchlow reflection losses at the interfaces are explained by therelatively small difference between the optical constants ofthe optically contacting materials. This is illustrated in Fig. 7 bythe calculation results of the reflectivity spectra for ZnO, CdS andCuIn0.34Ga0.66Se2 in the air and in optical contact with each other.The results show that the reflection coefficients at the interfaces

air/MgF2/ZnO, ZnO/CdS and CdS/CIGS are much smaller than thoseat the interfaces air/ZnO, air/CdS and air/CIGS (by 10–17 times (!)at λ¼700 nm). The sharp increase in reflectance at λo500 nm forthe air/MgF2 interface does not produce a significant effect asdiscussed in Section 3.1.

Absorption losses in the 300 nm thick ZnO and 40 nm thick CdSlayers are larger as compared to reflection losses and equalrespectively to 2.2% and 5.6% for CuIn0.69Ga0.31Se2 solar cell and1.9% and 7.9% for CuIn0.34Ga0.66Se2 solar cell. As previously noted,the difference in losses for the two types of samples is due to thedifference in their bandgaps.

Fig. 8 shows the dependence of optical losses (expressed as adecrease in short-circuit current) caused by the absorption in ZnOand CdS layers on their thicknesses. As seen, for dCdS¼40 nm, theZnO layer with thickness in the range of 100–500 nm causes a lossof 1–3%, and for thinner layers of 100–200 nm the loss reduces to�1%. The loss originating from the absorption in CdS layer ishigher even at its lowest thickness of 20–30 nm (�5%), and isclose to 20% for a 150 nm thick film. It seems that the thinning ofthe CdS and ZnO layers may lead to a decrease in the optical lossesno more than 4–5%.

The calculated values of optical losses along with the corre-sponding decrease in Jsc (given in brackets) are summarized inTable 1.

The calculated data of optical losses are largely in agreementwith the results reported in literature, but differ appreciably insome cases. For example, according to our calculations the

200 400 600 8000

10

20

30

40

λ (nm)

R(%

)

ZnO/air

CIGS/air

CdS/air

air/MgF2 /ZnO

ZnO/CdSCdS/CIGS

1000 1200

Fig. 7. Reflectivity of ZnO, CdS, MgF2 and CuIn0.34Ga0.66Se2 for optical contact withair and other material.

dZnO (nm)

ΔJ s

c /J

sc (%

)

dCdS = 40 nm

0 200 400 600 0

1

2

3

4

5

6

dCdS (nm)

ΔJ s

c /J

sc (%

)

dZnO = 300 nm

0 50800 1000 100 1500

5

10

15

20

Fig. 8. Dependences of the losses caused by absorption in the CdS and ZnO layers on their thicknesses.

Table 1Optical and the corresponding Jsc losses in CIGS solar cells.

Origin of optical losses Losses in solar cell

CuIn0.69Ga0.31Se2 CuIn0.34Ga0.66Se2

Reflection lossesAir/ZnO interface 1.9% (0.8 mA/cm2) 1.4% (0.5 mA/cm2)ZnO/CdS interface 0.9% (0.4 mA/cm2) 1.0% (0.3 mA/cm2)CdS/CIGS interface 1.3% (0.5 mA/cm2) 1.1% (0.4 mA/cm2)Shading from grid 4.0% (1.7 mA/cm2) 4.0% (1.3 mA/cm2)Total reflection losses 8.1% (3.4 mA/cm2) 7.5% (2.5 mA/cm2)

Absorption lossesAbsorption in ZnO 2.9% (1.2 mA/cm2) 1.9% (0.6 mA/cm2)Absorption in CdS 6.3% (2.6 mA/cm2) 7.9% (2.6 mA/cm2)Insufficient absorptivity 0.2% (0.1 mA/cm2) 0.4% (0.1 mA/cm2)Total absorption losses 9.4% (3.9 mA/cm2) 10.2% (3.4 mA/cm2)Total optical losses 17.5% (7.3 mA/cm2) 17.7% (5.8 mA/cm2)

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302 295

reflection losses in CuIn0.69Ga0.31Se2 solar cell (without shadingfrom grid) are equal to 4.1 mA/cm2 are 3.8 mA/cm2 as indicated in[4], i.e., are close to each other. However, the absorption losses inthe CdS film for CuIn0.69Ga0.31Se2 and CuIn0.34Ga0.66Se2 solar cellsin our calculations and in [4] are equal to 2.6 and 0.8 mA/cm2; thelosses due to insufficient absorptivity of the CIGS absorber alsodiffer considerably being equal to 0.1 and 1.9 mA/cm2, respectively.

3.3. Effect of multiple reflection in Zno and CdS layers

In the above calculations, the multiple reflections and inter-ference effects in the ZnO and CdS layers were not included,although they occur like those in anti-reflection coating. On theother hand, periodic changes (oscillations) in the quantum effi-ciency spectra of the CIGS solar cells (evidence of interferenceeffects) in most cases are not observed. To explain this, we willcalculate the optical transmission of the ZnO/CdS layered structuretaking into account multiple reflections, for which one can use theformula for the double-layer antireflection coatings [12]:

where r1, r2, r3 in our case are the amplitude reflection coeffi-cients:

r21 ¼ðn2�1Þ2þðκ2Þ2ðn2þ1Þ2þðκ2Þ2

; ð9Þ

r22 ¼ðn2�n3Þ2þðκ2�κ3Þ2ðn2þn3Þ2þðκ2þκ3Þ2

; ð10Þ

r23 ¼ðn3�n4Þ2þðκ3�κ4Þ2ðn3þn4Þ2þðκ3þκ4Þ2

; ð11Þ

n2, n3, n4 are, as before, the refraction indices of ZnO, CdS and CIGS,respectively,

β1 ¼2πλn2dZnO; ð12Þ

β2 ¼2πλn3dCdS: ð13Þ

Taking into account the presence of the grid contact on thefront surface of the ZnO layer and using Eq. (8), one can write theexpression for the transmission of the ZnO/CdS stack:

TðλÞ ¼ Tgrð1�RdlÞ: ð14ÞCurve 1 in Fig. 9 shows the calculation results from Eq. (14)

using Eq. (8) for CuIn0.69Ga0.31Se2 solar cell, Tgr¼0.95,dZnO¼300 nm, dCdS¼40 nm. The oscillations of the transmissionare clearly seen whose amplitude is relatively small. The latter isexplained by a low reflectance at the interfaces due to the smalldifference of the refractive indices of the contacting layers. Notethat the amplitude of oscillations increases by more than twotimes when the transmission is calculated without antireflectionlayer on the ZnO surface.

Eq. (8) describes the oscillations in the reflection spectra due tointerference effects in the ZnO and CdS layers under consideration.However this formula does not take into account absorption inthese layers. Evidently the oscillations should remain for theabsorbed layers with small κ, but for higher κ they cannot beobserved at all. We can take into account the absorption of lightfor one passage through the ZnO and CdS layers by entering in theright side of Eq. (14) the exponential factors appearing in Eqs. (3)and (7):

TðλÞ ¼ Tgrð1�RdlÞexpð�α2d2Þexpð�α3d3Þ: ð15ÞThe results of calculations obtained from Eq. (15) are shown in

Fig. 9 as curve 2. In this case, the oscillation amplitude in thespectral range λ4500 nm (where the absorption in ZnO and CdSis weak) has remained almost unchanged, but in the range of theinterband transitions in CdS at λo500 nm, the oscillation ampli-tude decreased significantly.

Fig. 9 also shows the transmission curve 3 calculated from Eq.(3), i.e., without considering the multiple reflections in the ZnOand CdS layers. As can be seen, when taking into account themultiple reflections, only minor periodic deviations occur aboveand below the curve 3. Obviously, when calculating the short-circuit current density of a solar cell, which is the result of thesummation (integration) over the entire spectral range, ignoringmultiple reflections cannot cause appreciable errors. Calculation ofthe short-circuit current density using Eqs. (7) and (15) for T(λ)gives results differing only by 0.9% and 0.5% for CuIn0.69Ga0.31Se2and CuIn0.34Ga0.66Se2 solar cells, respectively.

Note that Eq. (8) is obtained for flat, perfect and strictly parallelinterfaces air/ZnO, ZnO/CdS and CdS/CIGS. These conditions can-not be met completely in practical cases and hence the oscillationsin the transmission spectra can be less or not observed at all. Incontrast, if the interfaces are of good quality, the periodicalvariations in the transmission and photoresponse spectra of thedevices are clearly pronounced [13].

3.4. Absorptive capacity of the absorber

Calculating the short-circuit current in the previous sections;we assumed 100% efficiency of photoelectric conversion in theCIGS layer. In fact, there are losses related to the incompleteabsorption in this layer. Considering this effect, the exact values ofintegrated absorptive capacity of the CIGS layer must be estimatedtaking into consideration the spectral distribution of solar radia-tion and the absorption coefficient of the material rather than to

400 600 8000

0.2

0.4

0.6

0.8

1.0

1000 1200λ (nm)

T(λ )

1 2 3

Fig. 9. Transmission of the ZnO/CdS layers calculated taking into account themulti-reflections without considering absorption (line 1) the same but taking intoaccount absorption in the ZnO and CdS layers (line 2) and without the multi-reflections (line 3).

Rdl ¼r21þr22þr23þr21r

22r

23þ2r1r2ð1þr23Þ cos ð2β1Þþ2r2r3ð1þr21Þ cos ð2β2Þþ2r1r3 cos ð2β1þ2β2Þ

1þr21r22þr22r

23þr21r

23þ2r1r2ð1þr23Þ cos ð2β1Þþ2r2r3ð1þr21Þ cos ð2β2Þþ2r1r3 cos ð2β1þ2β2Þ

; ð8Þ

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302296

determine the absorptive capacity for one wavelength in the rangehv4Eg as usually estimated.

It should be noted that while considering the dependence ofthe absorptivity of solar radiation power AΦ (rather than photonflux Ahv) on the thickness of the semiconductor, one does not takeinto account that an electron–hole pair arises independently of thephoton energy hv absorbed in the range from Eg to 4.1 eV (thelatter corresponds to a wavelength of 300 nm). For this reason, thenumber of electron–hole pairs, and hence the short-circuit currentgenerated in the solar cell, is not proportional to the power of solarradiation. Therefore, the dependences of Ahv and AΦ on thethickness of the semiconductor are different, but, of course, theabsorption of all photons and all energy of radiation takes place atthe same thickness of the semiconductor.

The value of integrated absorptive capacity of the semiconduc-tor layer depends not only on the spectral distribution of solarradiation but also on the optical transmission of the ZnO and CdSlayers that modifies the solar radiation spectrum. Thus, theintegrated absorptivity is equal to the ratio of the number ofphotons absorbed in the CIGS layer to the number of photonspenetrated through the ZnO and CdS layers [14]:

AhvðdÞ ¼∑iTðλÞðΦi=hviÞ½1�expð�αidÞ�Δλi

∑iTðλÞðΦi=hviÞΔλi; ð16Þ

where the summation should be made, as earlier, over the spectralrange from λ¼300 nm to λ¼1068 or 898 nm for CuIn0.69Ga0.31Se2and CuIn0.34Ga0.66Se2 solar cells, respectively.

The calculation using Eq. (16) for a layer thickness of 2 μm givesvirtually total photon absorption (99.6–99.8%) for both materialsas is often indicated in the literature. But, for CuIn0.69Ga0.31Se2 andCuIn0.34Ga0.66Se2 layers with a thickness of 1 μm the photonabsorptivity is equal to 98.3% and 97.7% respectively, i.e., opticallosses become noticeable. Note that, in CdTe, also a direct-gapsemiconductor, the photon absorptivity of 99.6–99.8% takes placeat a thickness of about 30 μm [14].

It should also be borne in mind that there is a metal electrodeat the back surface of the CIGS layer. As a result, radiation in thelow absorption coefficient region is reflected from the back sur-face. In the event of 100% reflectance from the back surface, theabsorptivity is the same as for the double thickness of the absorberlayer, i.e., the Ahv(d) value in such a solar cell can be found bydoubling the thickness d in Eq. (16). This means that 99.6–99.8%photon absorption in CIGS takes place at a thickness of 1 μmrather than 2 μm.

4. Recombination losses

Electrons created as a result of photon absorption in the spacecharge region (SCR) of CdS/CIGS p–n heterostructure move underthe influence of the junction built-in voltage to the front metalgrid contact through the CdS and ZnO layers, while photogener-ated holes move to the neutral part of the CIGS layer and further tothe Mo back contact. However, some of the photogeneratedcarriers in the SCR recombine before reaching the contacts and,therefore, do not participate in the formation of the photocurrent.Recombination may also occur on the front surface of the absorbercontacting with the CdS film. In the formation of the photocurrent,electrons (minority carriers), which are generated outside the SCR,also take part reaching the SCR as a result of diffusion. This processcompetes with the recombination of electrons with the majoritycarriers (holes). Finally, recombination of electrons can occur onthe back surface of the absorber, i.e., at the CIGS/metal interface.

For a given thickness of the absorber, recombination lossesdepend primarily on the carrier lifetime and the width of the SCR,which is determined by the concentration of uncompensated

acceptors Na�Nd. Therefore, investigating the influence of thematerial parameters on the recombination losses, we will calculatethe Jsc dependence on Na�Nd for several values of carrier lifetimes.Consideration of the statistics of electrons and holes in a none-quilibrium state leads to the conclusion that their lifetimes in theSCR, τno and τpo, are practically equal to the lifetime of minoritycarriers in heavily doped p-and n-type materials, respectively [15].In the CIGS solar cell, the absorber is a fairly heavily dopedsemiconductor (free hole concentration is about 1015–1017 cm�3

[4]), therefore one can assume that the lifetimes of electrons in theneutral part and the SCR can be close to each other. Also assumethat hole lifetime in the SCR can be comparable in order ofmagnitude with the electron lifetime.

We will judge about recombination losses according to thevalue of short-circuit current density Jsc using the formula:

Jsc ¼ q∑i

ΦiðλÞhvi

TðλÞηintðλÞΔλi: ð17Þ

where T(λ) is, as earlier, the optical transmission of the ZnO andCdS layers:

TðλÞ ¼ Tgrð1�RarcÞexpð�α2d2Þð1�R23Þexpð�α3d3Þð1�R34Þ ð18Þ

and ηint(λ) is the internal quantum efficiency of photoelectricconversion.

It is generally accepted to consider the CdS/CIGS solar cell as anabrupt asymmetric p–n heterostructure, in which the space-charge region (depletion layer) is practically located in the p-CIGS and the photoelectric conversion takes place almost in thislayer (see, for example, [4] and references therein). The built-inpotential and field distribution in abrupt asymmetric p–n hetero-junction is the same as in Schottky diode; therefore, furtherconsideration of the processes in CdS/CIGS solar cells can be madebased on the concepts developed for Schottky diodes.

The internal quantum efficiency of a CdS/CIGS solar cell con-tained in Eq. (17) includes the drift and diffusion components,which are obliged to photogeneration of electron-hole pairs in theSCR and in the neutral part of the diode structure, respectively.

Considering the drift component, it is necessary to take intoaccount losses due to recombination at the CdS/CIGS interface, i.e.,at the front surface of the absorber layer. The solution of thecontinuity equation is cumbersome in general case, but withrespect to the used model it can be reduced with sufficientaccuracy to a simpler expression [16]:

ηdrift ¼1þðSf=DpÞ½αþð2=WÞðφo�qVÞ=kT ��1

1þðSf=DpÞ½ð2=WÞðφo�qVÞ=kT ��1 �expð�αWÞ; ð19Þ

where Sf is the velocity of recombination at the front surface of theCIGS layer; V is the voltage; φbi is the barrier height (the bandbending) at the contact of CIGS related to the diffusion potentialVbi by the equality φbi¼qVbi; Dp is the diffusion coefficient of holesrelated to their mobility μp by the Einstein relation qDp/kT¼μp;and W is the width of the SCR.

In a CIGS always containing both acceptor and donor impu-rities, the width of the SCR is determined by the expression [17]:

W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2εεoðφbi�qVÞq2ðNa�NdÞ

s; ð20Þ

where ε is the relative permittivity of CIGS (ε¼13.6), εo is thepermittivity of vacuum, Na�Nd is the concentration of uncompen-sated acceptors (rather than the free hole density).

The diffusion component of the quantum efficiency ηdif is alsofound from the continuity equation. Taking into account recombi-nation at the back surface of the CIGS layer and the boundary

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302 297

condition Δn¼0 at x¼W, it can be written as [17]

where Dn and Ln is the electron diffusion coefficient and diffusionlength; Sb is the velocity of recombination at the rear surface of theCIGS layer; and d is its thickness.

The internal quantum efficiency ηint is the sum of drift anddiffusion components:

ηint ¼ ηdriftþηdif ; ð22Þ

whereas the external efficiency can be written in the form

η¼ TðλÞðηdriftþηdif Þ; ð23Þ

where T(λ) is the optical transmission of the ZnO and CdS layersdetermined by Eq. (18).

Fig. 10 shows a comparison of the measured quantum effi-ciency spectra of CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 solar cellstaken from [4] with the results of calculations using Eqs. (17)–(23).Note that the data on the extinction coefficients (and hence theabsorption coefficient α) of CIGS used in the following calculationswere measured for Eg¼1.16 eV (not 1.14 eV) and Eg¼1.38 eV (not1.36 eV), respectively, that explains the slight blue shift (10–20 nm) in the calculated long-wavelength edge of the spectra inFig. 10.

In the calculations, the parameters of the absorber layer werevaried within the limits reported in the literature. It should benoted that for polycrystalline CIGS, there is a large spread inmobility values of electrons and holes. At room temperature, thevalues of the hole mobility are most often indicated in the rangefrom 1–5 to 30–50 cm2/(V s) and 1–100 cm2/(V s) for the electronmobility [4,18–20]. Unlike this, it was found in [21] that the

electron and hole mobilities in Cu(In,Ga)Se2 are much lower than1 cm2/(V s). As mentioned previously, such low mobilities refer to

the charge transport in the impurity sub-band in the bandgapjoined with the conduction (valence) band due to high doping or/and disorder in the crystal lattice. Electrons and holes arising as aresult of photon absorption and involving in the photocurrentformation are moving in the conduction and valence band, wheretheir mobilities are much higher.

The lifetime of minority carriers (electrons in this case),determining their diffusion length Ln¼(τnDn)1/2 has also a sig-nificant impact on the efficiency of solar cells. As reported back in1996 [22], the lifetimes of electrons in thin film CuInSe2 are in therange of tens of picoseconds to a few nanoseconds, which wassubsequently confirmed [18,23]. Lifetime of minority carriers affectthe diffusion component of photocurrent and therefore revealsitself in the long-wavelength part of spectrum at λ4600–700 nm.

Yet another important parameter determining the quantumefficiency spectra of solar cells is the concentration of uncompen-sated acceptors Na�Nd in the absorber, which according to Eq.(20) determines the width of the SCR W. At high Na�Nd, the widthof the SCR amounts to a small portion of the absorber thickness.With decreasing Na�Nd (extension of the SCR), the efficiency ofcell increases since more and more part of the radiation isabsorbed in the SCR, where the photogenerated electrons andholes move in the opposite direction (due to the built-in electricfield) and can reach the boundaries of the SCR without recombi-nation. However, an increase of the quantum efficiency due toexpansion of the SCR occurs only to a certain extent, because withincreasingW, the electric field in the SCR is reduced and, therefore,recombination at the front surface of the absorber layer and in theSCR is intensified. As a result, the growth of the quantumefficiency becomes slower and even starts to reduce due to theexpansion of the SCR.

Finally, the recombination at the front surface of the CIGS layeraffects the efficiency in a short-wavelength range and stronger in awider SCR (smaller Na�Nd). Our calculations in Section 4.3 showthat recombination at the rear surface of the CuIn0.76Ga0.24Se2 andCuIn0.39Ga0.61Se2 absorber layers for their thickness 2 μm manifestitself very weakly and only for long lifetimes of electrons.

It follows that the main parameters of the absorber layer affectthe quantum efficiency spectrum of the solar cell differently [24].This facilitates the choice of the parameter values for betteragreement between the calculated and experimental data. Theresults of such a matching of the parameters are summarized inTable 2.

Notice that in the investigated solar cells, the SCR width(0.4–0.6 μm) covers a small part of the thickness of the absorberlayer and the recombination velocity Sf is equal to 104–105 cm/s, i.e., is low as compared to the case of thin-film CdS/CdTe hetero-structure. This is explained by the known features of CdS/CIGSheterojunction. Back in the late 1980s, it was shown that the CIGSsolar cells are insensitive to defects at the junction caused by alattice mismatch or impurities at the CdS/CIGS interface. In fact,

Table 2Parameters of CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2 solar cells giving better match between measured and calculated data.

Solar cell Eg (eV) φbi (eV) μp (cm2/V s) μn (cm2/V s) τn (ns) Na�Nd (cm�3) Sf (cm/s) dCdS (nm)

CuIn0.76Ga0.24Se2 1.14 0.75 30 20 20 9�1015 5�104 40CuIn0.39Ga0.61Se2 1.36 0.95 30 20 2 5�1015 2�105 30

200 400 600 8000

0.2

0.4

0.6

0.8

1.0

1000 1200

λ (nm)

η(λ)

Eg =1.14 eV

Eg =1.36 eV

Fig. 10. Comparison of the measured (circles) and calculated (solid lines) quantumefficiency spectra of CuIn0.76Ga0.24Se2 (Eg¼1.14 eV) and CuIn0.39Ga0.61Se2(Eg¼1.36 eV) solar cells.

ηdif ¼αLn

α2L2n�1expð�αWÞ � αLn�

ðSbLn=DnÞ½ coshððd�WÞ=LnÞ�expð�αðd�WÞÞ�þ sinhððd�WÞ=LnÞþαLnexpð�αðd�WÞÞðSbLn=DnÞsinh ½ðd�WÞ=Ln�þ cosh ½ðd�WÞ=Ln�

� �; ð21Þ

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302298

(i) the lattice mismatch is very small for CdS and CuInSe2 (�1%)and weakly increases with the Ga content, (ii) the deposition ofCdS on the treated and cleaned surface of CIGS is characterized bypseudo-epitaxial growth, (iii) the intermixing of the heterojunc-tion constituents is observed even at a relatively low-temperaturetechnological processes [25,26].

A comparison of measured and calculated results presented inFig. 10 shows that the theoretical model describes in detail thespectral distribution of the quantum efficiency of CIGS solar cellsthat is important for further analysis of recombination losses. Butthe question arises, how it may be possible in polycrystallinematerial with its inhomogeneity and recombination at the grainboundaries. A possible explanation for the applicability of themodel in question to efficient solar cells based on polycrystallineCIGS is that during the growth of the absorber layer and post-grown processing, the material recrystallization, grain growth andtheir coalescence occur. No less important is also the fact that thestructure in the form of ordered column oriented perpendicular tothe electrodes is created in the CIGS layer (see reviews [4,27] andreferences therein). One can assume that in a layer of columnarstructure, collection of photogenerated charge occurs withoutcrossing the grain boundaries. In addition, the scattering andrecombination on the lateral surfaces of the columns also haveno significant effect due to the strong electric field in the barrierregion. Indeed, the width of the SCR in the studied solar cell isabout 0.5 μm, and the electric field at a barrier height φbi of about1 eV is higher than 104 V/cm. Under such conditions, the driftlength of charge carriers with the mobility of 20–30 cm2/(V s)and lifetimes 10–9 s is several microns which is significantlygreater than the width of SCR and makes recombinationimprobable.

Outside the SCR, where the diffusion component of photocur-rent is formed, the electric field does not affect, but due to the highabsorption capacity of CIGS the vast majority of solar radiation isabsorbed in the SCR. Calculation by Eq. (17) shows that for theparameters listed in Table 2, the contribution of the diffusioncomponent in the short circuit current in CuIn0.39Ga0.61Se2 solarcell is only �4%. In CuIn0.76Ga0.24Se2 solar cell with longer electronlifetime, the contribution of the diffusion component is somewhatgreater (�8%), but far inferior to the drift component. Thissignificantly weakens the role of recombination at the grainboundaries of CIGS solar cells.

4.1. Recombination losses at the front surface

To determine the recombination losses, we will calculate theshort circuit current density by varying the parameters of whichthe losses are dependent: the recombination velocity on theabsorber surface, the concentration of uncompensated acceptors,the carrier lifetimes, the thickness of the absorber. Short-circuitcurrent density will be calculated by Eq. (17).1

First consider how the short-circuit current density Jsc variesdepending on the concentration of uncompensated acceptors fordifferent recombination velocities at the front surface of the CIGSabsorber. According to theory, the recombination velocities Sf doesnot depend on the carrier lifetime in this case (see Eq. (19)).

Fig. 11 shows the dependences of Jsc on Na�Nd in the range5�1014–1018 cm–3 and Sf in the range 104–107 cm/s (as well asSf¼0) for CuIn0.39Ga0.61Se2 solar cell. The lower range of Na�Nd islimited to 5�1014 cm–3, because at this values the width of theSCR becomes greater than the thickness of the absorber layer

d¼2 μm. As seen in Fig. 11a, if the concentration of uncompen-sated acceptors exceeds 1017 cm–3, surface recombination does notreveals itself, but at lowering the Na�Nd, the decrease in Jsc becomesappreciable. At Sf¼104 cm/s, recombination practically does notreduce the short-circuit current and slightly lowers it at the realvalues of Na�Nd¼5�1015 cm–3 and Sf¼2�105 cm/s for this solarcell. Our calculations show that for CuIn0.39Ga0.61Se2 solar cell atNa�Nd¼5�1015 cm–3 surface recombination reduces Jsc by 1.9% andby 0.2% for CuIn0.76Ga0.24Se2 at Na�Nd¼9�1015 cm–3. Small recom-bination losses are explained by the strong electric field in the SCRdue to rather high concentration of uncompensated acceptors.However lowering the Na�Nd significantly increases the recombina-tion losses especially at high velocities of surface recombination.

4.2. Recombination losses in the SCR

Recombination of photogenerated charge carriers in the SCR canbe taken into account, using the well-known Hecht equation [28]:

ηHðxÞ ¼λnW

1�exp � xλn

� �� þλpW

1�exp �W�xλp

� �� ; ð24Þ

where x is the coordinate, where an electron–hole pair arises (x ismeasured from the CdS/CIGS interface), λn and λp are the driftlengths of electrons and holes in the SCR:

λn ¼ μnFτno; ð25Þ

λp ¼ μpFτpo; ð26Þ

F is the electric field strength; τno and τpo are the lifetimes ofelectrons and holes in the SCR, respectively, which we will accept tobe equal to τn (assuming that the CIGS absorber is sufficiently highlydoped).

In CdS/CIGS heterostructure, the electric field is not uniform,but the problem of the nonuniformity is simplified, since the fieldstrength F decreases in a Schottky diode linearly with the xcoordinate. In this case, F in expressions (25) and (26) can bereplaced by the average values of F in the sections (0,x) and (x,W)for electrons and holes, respectively, i.e., F(0,x)¼[(φo�qV)/qW](2�x/W) and F(x,W)¼[(φo�qV)/qW](1�x/W) [16].

Evidently, charge collection efficiency in the dx interval of theSCR is ηHα � exp(–αx)dx and then the quantum efficiency at thewavelength λi is determined by expression:

ηHðλiÞ ¼Z W

0ηHðxÞαiexpð�αixÞdx: ð27Þ

Fig. 12 shows the short circuit current density Jsc as a functionof the concentration of uncompensated acceptors Na�Nd atdifferent carrier lifetimes calculated by taking into account therecombination losses in the SCR given by the expression:

Jsc ¼ q∑i

ΦiðλiÞhvi

TðλiÞηHðλiÞΔλi: ð28Þ

As can be seen in Fig. 12a, when Na� Nd43�1016 cm–3

(Wo0.2–0.3 μm), the curves calculated for different lifetimes ofcharge carriers τno¼τpo¼τn coincide, but as the SCR widthincreases, the curves diverge and become more pronounced whenτn is less. The reduction of Jsc due to recombination can be foundby subtracting the current at a given τn from the value of currentjosc when recombination does not occur. The latter is possible for asufficiently large lifetime of charge carriers. A simple calculationshows that when τn4500–600 ns, the current becomes indepen-dent of τn, therefore the values of josc at different Na�Nd werefound for τn¼1000 ns (the curve josc vs. Na�Nd is shown inFig. 12a).

Fig. 12b shows the reduction in Jsc due to recombination in theSCR and its dependence on the Na�Nd and τn. As seen, when Na

1 Generally speaking, the recombination losses should be calculated in theoperating mode of solar cell (at V≠0), when the photocurrent is somewhat lower,but its relative changes vary slightly.

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302 299

�Ndo1016 cm–3 (W41 μm), the recombination losses increaserapidly up to 60% with decreasing Na – Nd and τn. The datapresented in this figure allow determining the recombinationlosses for any values of Na�Nd and τn. For the real carrier lifetimeof 2 ns and Na�Nd¼5�1015 cm–3, which are characteristics ofCuIn0.39Ga0.61Se2 solar cell, the recombination losses amount to2.1%, whereas for CuIn0.76Ga0.24Se2 solar cell the losses is only 0.1%because of higher values τn¼20 ns and Na�Nd¼9�1015 cm–3.

4.3. Recombination losses at the back surface

Information about the effect of recombination at the backsurface of the solar cell on the photocurrent can be obtained bystudying the dependence of Jsc on the thickness of the absorber. Ifd is large, the effect of recombination is imperceptible, but whenthe rear surface is close to the SCR by a distance comparable to thediffusion length of electrons, the role of recombination increases,which leads to a decrease in the diffusion component of the shortcircuit current. Subtracting currents calculated for the recombina-tion velocity Sb¼107 cm/s and Sb¼0, one can obtain the change of

Jsc due to recombination at the back surface. Obviously, for longerminority-carrier lifetime, recombination at the back surface of theabsorber is intensified and manifests itself strongly for largerabsorber thickness.

Fig. 13a shows the dependences of Jsc on the thickness of theCuIn0.39Ga0.61Se2 absorber layer d calculated for Sb¼107 cm/s andSb¼0 and at different lifetimes of electrons.

As seen in Fig. 13a, when the CuIn0.39Ga0.61Se2 layer becomesthinner, the short-circuit current for Sb¼0 first slightly grows andthen rapidly decreases (dashed lines). The observed currentgrowth can be explained by the fact that the diffusion componentof the photocurrent is determined by the derivative of the excessconcentration of photogenerated electrons Δn/dx, which may belarger than that in the absence of recombination at the surface, butwhen approaching the cross section x¼W, the diffusion compo-nent eventually decreases due to reduction inΔn value andΔn¼0when x¼W [17]. This also explains why the dependence of ΔJsc/Jscon d, shown in Fig. 13b is described by a curve with a maximum.

As can be seen from Fig. 13b, a decrease in the short-circuitcurrent in CIGS solar cells with lifetimes of electrons 20 and 2 ns

15

20

25

30

1014 1015 10171016

J sc (

mA

/cm

2 )

Na – Nd (cm–3)

τn = 0.5 ns

2 ns

10 ns

5 ns

20 ns

τn = 1000 ns

1014 1015 10171016

ΔJ s

c /J

sc (%

)

Na – Nd (cm–3)

0

20

40

60τn = 0.5 ns

2 ns

10 ns

5 ns

20 ns

Fig. 12. (a) Short-circuit current density Jsc as a function of the concentration of uncompensated acceptors Na�Nd for different carrier lifetimes τn in the CIGS absorber.(b) Reduction of the short-circuit current density expressed in percentage.

20

22

24

26

28

30

1014 1015 1016 1017 1018

Sf = 107 cm/sJ sc (

mA

/cm

2 )

Na – Nd (cm–3)

Sf = 0

106 cm/s

105 cm/s

104 cm/s

0

5

10

15

1014 1015 101016 1017 18

Sf = 107 cm/s

ΔJ s

c /Δ

J sc (

%)

Na – Nd (cm–3)

106 cm/s

105 cm/s

104 cm/s

Fig. 11. (a) Short-circuit current density Jsc as a function of the concentration of uncompensated acceptors Na�Nd for different recombination velocity Sf at the front surfaceof the CIGS absorber. (b) Reduction in the short-circuit current density, expressed as in percentage.

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302300

does not exceed 0.8% for CuIn0.76Ga0.24Se2 and o0.1% for CuIn0.39-Ga0.61Se2 cell. It seems that with such low losses, the creation of aheavily doped layer adjacent to the back contact as it is proposedin CdS/CdTe solar cells [6] or to form a bandgap gradient outsidethe SCR of the CIGS absorber [4] to reduce the negative impact ofrecombination at the rear surface of the absorber seems appar-ently unreasonable. In addition, a sufficiently small fraction ofcarriers, taking part in the photocurrent formation (4–8%), falls onthe neutral part of the studied CIGS solar cells that also should beborne in mind.

Of course, one has to bear in mind that the recombination ofphotogenerated carriers occurs not only on the back surface of theabsorber, but also in whole its neutral part (outside the SCR).Simple calculations show that such recombination losses are 6.0%for CuIn0.76Ga0.24Se2 cell and 2.9% for CuIn0.39Ga0.61Se2 cell.

The results obtained in Sections 4.1–4.3 show that the totalrecombination losses in CuIn0.76Ga0.24Se2 and CuIn0.39Ga0.61Se2solar cells amount to 7.0% and 4.1%, respectively. On the otherhand, the charge collection efficiency of photogenerated carrierscan be determined using Eq. (17) as the ratio of short-circuitcurrent for a real internal quantum efficiency ηint(λ) to that forηint(λ)¼1. Such a calculation shows that the charge collectionefficiencies are 96.7% and 96.0% for the CuIn0.76Ga0.24Se2 andCuIn0.39Ga0.61Se2 solar cells, respectively. This means that anincomplete charge collection is caused due to the non-optimalparameters of the devices.

Charge collection can be improved by extending the SCR due tothe fact that a larger part of the radiation will be absorbed in thelayer where the built-in electric field of the p–n heterojunctionexists. As was already mentioned, the SCR width W according toEq. (20) is determined by the concentration of uncompensatedacceptors Na�Nd in the material (but not the free carrier density).In fact, the charge collection in CuIn0.76Ga0.24Se2 solar cellenhances remarkably from 96.7% to 98.7% by reducing Na�Nd

from 9�1015 to 1.8�1015 cm–3. Such an improvement in chargecollection cannot be achieved for solar cells with bandgap of theabsorber 1.36 eV due to shorter carrier lifetime which increasesconsiderably the recombination losses when the SCR is widened.There is also a negative impact of the above mentioned expansionof the SCR since it leads to an increase in the forward recombina-tion current in CIGS solar cell, which is proportional to W.Analyzing the electrical characteristics of the solar cell, it is notdifficult to show that the decrease in Na�Nd from 9�1015 to

1.8�1015 cm–3 causes lowering the open circuit voltage Voc byabout 0.04 V. This effect can be weakened by increasing the built-in voltage Vbi, which depends on the position of the Fermi level inthe neutral part of the absorber. One can increase the value of Vbi

by increasing the hole concentration.

5. Conclusions

In a CIGS solar cell the optical losses caused by reflections fromthe interfaces and absorption in the ZnO and CdS layers have beencalculated using the optical constants of the materials. Anti-reflection coating reduces the reflection loss at the front surfaceof solar cells from 9.2% to 1.9% for a CIGS absorber with bandgap1.16 eV and from 9.3% to 1.4% for CIGS bandgap 1.38 eV. The totalreflection losses for typical parameters of these solar cells are equalto 4.1% and 3.5%, respectively (excluding shading from grid). Thelosses caused by absorption in the ZnO and CdS layers amount to1.9–2.2% and 5.6–7.7% for these solar cells, respectively. The lossesdue to insufficient absorptivity of the CIGS layer of 2 μm thick are0.2% and 0.4% and increase to 1.7% and 2.3% for a thickness of 1 μmfor above bandgaps, respectively. The total optical losses with theantireflection coating amount to 16.0% and 17.5% respectively forsolar cells with absorber bandgaps 1.16 and 1.38 eV, respectively.When calculating the photoelectric characteristics of a solar cell,ignoring the multiple reflections and interference effects in theZnO and CdS layers cannot cause remarkable errors.

The recombination losses in the studied solar cells have beendetermined by comparing the measured quantum efficiencyspectra with the calculation results. This approach allowed us todetermine quite accurately the main parameters of the devices:lifetimes of charge carriers, concentration of uncompensatedacceptors in the absorber, recombination velocity at the frontand back surfaces, the thickness of the CdS film. It has been shownthat recombination at the front surface of the absorber reduces theshort-circuit current by �1.9% and �0.2% respectively in solarcells with the CIGS bandgap 1.14 and 1.36 eV. Recombination in thespace-charge region causes a reduction of the short-circuit currentdensity by 1.0% in CIGS solar cells with Eg¼1.36 eV due to shortlifetime of charge carriers (�2 ns). At the same time, in solar cellswith Eg¼1.14 eV and the carrier lifetime larger by an order ofmagnitude (�20 ns), the losses caused by recombination in theSCR is less than 0.1%. At the back surface of solar cells,

0 1 2 3 26

27

28

29

d (μm)

J sc (

mA

/cm

2 )

τn = 1 ns

τn = 20 ns

τn = 5 ns

0 1 2 3 4 5 4 50

0.5

1.0

1.5

d (μm)

ΔJ s

c/Jsc

(%)

τn = 20 ns

τn = 10 ns

τn = 5 ns

τn = 2 ns

τn = 1 ns

Fig. 13. (a) Dependences of the short-circuit current density on thickness of the CuIn0.39Ga0.61Se2 layer calculated for different electron lifetimes with and without taking intoaccount recombination at the back surface of the absorber (solid and dashed lines, respectively). (b) Decrease in short-circuit current density due to recombination at theback surface calculated in percent for different electron lifetimes.

L.A. Kosyachenko et al. / Solar Energy Materials & Solar Cells 130 (2014) 291–302 301

recombination causes a decrease in the short-circuit current by0.8% for CIGS solar cells with the bandgap of the absorber 1.14 eVand practically zero when Eg¼1.36 eV. Total recombination losses inthe studied CIGS solar cells with bandgap of absorber 1.14 and1.36 eV are equal to 7.0% and 4.1%, respectively.

The efficiencies of charge collection calculated for CuIn0.76-Ga0.24Se2 (Eg¼1.14 eV) and CuIn0.39Ga0.61Se2 (Eg¼1.36 eV) solarcells are 96.7% and 96.0%, respectively. For solar cells with theabsorber bandgap of 1.14 eV, an incomplete charge collection iscaused by non-optimal parameters of the device. In this solar cell,the charge collection enhances remarkably from 96.7% to 98.7% ifone reduces the concentration of uncompensated acceptors from9�1015 to 1.8�1015 cm–3. Similar improvement of charge collec-tion cannot be achieved for solar cell with bandgap of the absorber1.4 eV due to shorter carrier lifetime that can increase therecombination losses when the SCR becomes wider. This may beone of the possible explanations for the fact that the efficiency ofCuIn(Ga,In)Se2 solar cells is reduced when bandgap of the absor-ber increases.

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