Avdelningen för energi och byggnadsdesignInstitutionen för arkitektur och byggd miljöLunds tekniska högskolaLunds universitet, 2008Rapport EBD-R--08/19
Modelling and Optimization of CIGS Solar Cell Modules
Joar Johansson
Modelling and Optimization of
CIGS Solar Cell Modules
Joar Johansson
Master's Thesis
December 2007
Supervisors: Uwe Zimmermann
Marika Edo
Examiner: Björn Karlsson
Abstract
The thin-lm solar cell module based on Cu(In,Ga)Se2 (CIGS) is a technologywith great potential. Two reasons for this are low material consumption and arelatively high eciency. Low-concentrating systems, which use a large reec-tor to focus light onto a small area of solar cells, are another technology withpotential. Research on both these solar energy technologies is carried out atuniversities and companies in Sweden.
In this work a computer model of a CIGS solar cell module is built and simu-lations are performed to optimize the design, i.e. to nd the optimal relationbetween the width of the cells and the sheet resistance of the transparent frontcontact, at both standard and higher irradiances. The numerical model is basedon the one-diode model and takes into account electrical, optical and geometri-cal parameters. The model is implemented in COMSOL MultiphysicsTM, whichsolves the problem using the nite element method. The optimization is per-formed in MATLAB R©.
Parameters and equations are adjusted to t measured data and the model isveried by data from manufactured CIGS solar cell mini-modules with dierentcell widths. The accuracy of the model is shown to be within±3 % for the outputparameters open-circuit voltage, short-circuit current, ll factor and eciency.The results from simulations show that a module optimized for an irradianceof 1000 W/m2 has the highest eciency for a cell width of 3 mm and a frontcontact sheet resistance of 20 Ω/. A module optimized for low concentration,in this case an irradiance of 8000 W/m2, has a cell width of 2 mm and a frontcontact sheet resistance of 10 Ω/.
Contents
1 Introduction 5
2 Theory 7
2.1 CIGS Solar Cell Modules . . . . . . . . . . . . . . . . . . . . . . 72.2 Solar Cell Output Parameters . . . . . . . . . . . . . . . . . . . . 82.3 Diode Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Diode Saturation Current . . . . . . . . . . . . . . . . . . 92.3.2 Current Density . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 One-Diode Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Series and Shunt Resistances . . . . . . . . . . . . . . . . . . . . 11
2.5.1 Resistivity and Conductivity . . . . . . . . . . . . . . . . 112.5.2 Sheet Resistance . . . . . . . . . . . . . . . . . . . . . . . 112.5.3 Contact Resistance . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Modelling 13
3.1 CIGS Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Transparent Front Contact . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Light Transmission . . . . . . . . . . . . . . . . . . . . . . 143.3 Back Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Interconnect Structure . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Simulations, Results and Discussions 21
4.1 Parameters Adjustment . . . . . . . . . . . . . . . . . . . . . . . 214.2 Experimental Verication . . . . . . . . . . . . . . . . . . . . . . 234.3 Cell Width Optimization . . . . . . . . . . . . . . . . . . . . . . . 244.4 ZnO:Al Sheet Resistance Optimization . . . . . . . . . . . . . . . 264.5 Irradiance Optimization . . . . . . . . . . . . . . . . . . . . . . . 27
5 Conclusions 29
Acknowledgements 31
References 33
5
1 Introduction
A solar cell converts radiant energy into electrical energy. This conversion,which occurs in some semiconductors, is called the photovoltaic eect and wasrst observed in 1839 by Becquerel. Solar cells manufactured from wafers ofcrystalline or polycrystalline silicon are today the dominant technology in thecommercial market. Solar cells based on a thin lm of the semiconductor are an-other technology with great potential. One such thin-lm technology is based ona compound of the elements copper, indium, gallium and selenium, abbreviatedCu(In,Ga)Se2 or CIGS. Two advantages of this thin-lm technology are thelow material consumption and the high eciency that has been demonstrated,which both make it economically competitive. Research on CIGS thin-lm so-lar cells is carried out at both Uppsala University and at the company SolibroResearch AB.
Low-concentrating systems, which use a large reector to focus light onto asmall area of solar cells, are another technology with potential. The idea is toreplace solar cells with cheaper light concentrating devices. Research on suchsystems is carried out at both the Faculty of Engineering at Lund Universityand at the company Arontis Solar Concentrator AB.
This study will combine these two research areas. The purpose is to build a nu-merical model of the CIGS solar cell module which takes into account electrical,optical and geometrical parameters. The model is then used for optimizing thedesign of the module at dierent light intensities.
First a model of the CIGS solar cell module is derived. The model, which isbased on the one-diode model, is implemented in COMSOL MultiphysicsTM
and the optimization script is implemented in MATLAB R©. Some parametersand equations in the model are adjusted to t measured data and the model isveried by using data from manufactured CIGS solar cell mini-modules, shownin Figure 1. Finally, simulations are performed to optimize the performance ofthe CIGS solar cell module with respect to eciency.
Figure 1. CIGS solar cell mini-modules, with an aperture area of 80 cm2 and cellwidths of 3 mm, 5 mm, 7 mm and 9 mm, manufactured at Uppsala University.
7
2 Theory
2.1 CIGS Solar Cell Modules
A CIGS solar cell is built up of a substrate of soda-lime glass, a back contact ofmolybdenum (Mo), a light absorbing layer consisting of copper indium galliumdiselenide (Cu(In,Ga)Se2), a buer layer of cadmium sulphide (CdS) or zincoxy sulphide (Zn(O,S)), a thin layer of high resistive zinc oxide (ZnO) and atransparent front contact of aluminium doped zinc oxide (ZnO:Al). Figure 2shows the cross section of a CIGS solar cell with typical thicknesses of thedierent layers indicated.
Figure 2. Transmission electron micrograph of the layers in a CIGS solar cell.
In a CIGS solar cell module several cells are connected in series. The seriesconnection is made in an interconnect structure consisting of the three scribescalled P1, P2 and P3. Figure 3 shows a series connection of two cells with thecell width, the active cell width, the interconnect structure, the three scribesand the dierent layers indicated.
Active cellInterconnectstructure
ZnO:Al
Cu(In,Ga)Se2
MoGlass substrate
ZnOBuffer
P1 P2 P3
Cell width
Figure 3. Series connection of two CIGS solar cells. The cell width, the active cellwidth, the interconnect structure, the three scribes (P1, P2 and P3) and the dierentlayers are indicated. The sketch is not to scale.
CIGS solar cell modules are produced by the research team at Uppsala Univer-sity and the process involves several dierent steps. First the Mo is sputteredonto the soda-lime glass. The Mo layer is patterned using a laser and in thisway P1 is created. The Cu(In,Ga)Se2 layer is deposited using co-evaporation.
8 2 THEORY
If CdS is used it is prepared by chemical bath deposition (CBD), but if insteadthe cadmium-free Zn(O,S) is used, it is deposited by atomic layer deposition(ALD). The ZnO layer is sputtered, and then the layers are mechanically scribedand P2 is created. The ZnO:Al layer is sputtered from a ceramic target. Theisolation scribe P3 is then created mechanically. The nal steps include attach-ment of electrical wires and encapsulation of the solar cell module using a coverglass, usually with EVA as the laminate material.
2.2 Solar Cell Output Parameters
IV measurements are performed to characterize solar cells. Figure 4 showsa typical IV curve with some of the following solar cell output parametersindicated [2]:
• Short-circuit current, Isc, is the maximum current at zero voltage. Theshort-circuit current density, Jsc, is often used (see Section 2.3.2).
• Open-circuit voltage, Voc, is the maximum voltage at zero current.
• Maximum power point, Pmp, is the maximum power output at optimaloperating condition, i.e. Pmp = VmpImp.
• Fill factor, FF , is a measure of how square the IV curve is. It is denedas
FF =VmpImpVocIsc
=PmpVocIsc
(1)
• Eciency, η, is the energy-conversion eciency. It is given by [1]
η =PmpPin
=VocIscFF
Pin(2)
where Pin is the total power of the incident light.
• Peak power or peak watts, Wp, is the power output at the maximumpower point under standard test conditions (STC), i.e. Wp = Pmp at STC.The IV measurement is often performed under STC, which requires anirradiance of 1000 W/m2, a temperature of 25 C and the standard globalAM 1.5 spectrum. Modules are usually rated in terms of peak watts.
QE measurements are another method for characterization of solar cells:
• Quantum eciency, QE, is the number of generated electron-hole pairs perincident photon in the solar cell. When measured with an external circuitthis quantity is also referred to as the external quantum eciency, EQE. Itis often measured for wavelengths, λ, in the range from 300 nm to 1300 nm.The short-circuit current density can be calculated from the measurementas
Jsc =∫ ∞
0
EQE(λ) Φ(λ) dλ (3)
where Φ is the photon ux at the AM 1.5 spectrum.
2.3 Diode Equation 9
Vmp Voc
Isc
Imp
Typical curveDecreasing series resistance, Rs
Increasing shunt resistance, Rsh
Voltage
Cur
rent
Figure 4. Typical IV curve showing short-circuit current, Isc, open-circuit voltage,Voc, and the maximum power point, Vmp and Imp. The eects of decreasing seriesresistance, Rs, and increasing shunt resistance, Rsh, are also shown.
2.3 Diode Equation
A CIGS solar cell is essentially a heterojunction diode. A heterojunction isformed by joining two dierent semiconductor materials, such as p-type CIGSand n-type buer/ZnO. The current through a diode, ID, is generally describedby the diode equation [2]
ID = I0(eqV/AkT − 1) (4)
where V is the applied voltage, I0 is the diode saturation current (see Sec-tion 2.3.1), q is the elementary charge, A is the ideality factor, k is the Boltz-mann constant and T is the temperature. For an illuminated solar cell the diodeequation becomes [2]
I = ID − IL = I0(eqV/AkT − 1)− IL (5)
where IL is the light-generated current. By denition the IV curve is oftenplotted in the rst quadrant, and represented by [2]
I = IL − I0(eqV/AkT − 1) (6)
2.3.1 Diode Saturation Current
The diode saturation current, I0, for a CIGS solar cell is given by [3]
I0 = I00e−Φb/AkT (7)
10 2 THEORY
where I00 is a prefactor, Φb is the barrier height and A is the ideality factor.These quantities depend on the details of each recombination mechanism [4].It is possible to have recombination in the bulk absorber, in the space chargeregion of the Cu(In,Ga)Se2 or at the Cu(In,Ga)Se2/buer interface [3, 4].
If the barrier height is equal to the band gap of Cu(In,Ga)Se2, i.e. Φb = Eg,this implies [3]
I0 = I00e−Eg/AkT (8)
2.3.2 Current Density
Current density, J , is dened as electric current per cross-sectional area and theunit is A/m2, but normally the unit mA/cm2 is used. The diode equation canbe expressed using the current density
J = J0(eqV/AkT − 1)− JL (9)
withJ0 = J00e
−Eg/AkT (10)
Note that the unit for the Boltzmann constant is J/K for Eq. 9 and eV/K forEq. 10.
2.4 One-Diode Model
A solar cell can be modelled as an electric circuit in the so-called one-diodemodel. The one-diode model consists of a current generator in parallel with adiode and a shunt resistance, Rsh, which are all connected in series with a seriesresistance, Rs. The equivalent circuit is shown in Figure 5. An equation for theone-diode model is derived using the equivalent circuit and the diode equation.
IL
I
Rsh
ID Ish
Rs
V’ V
Figure 5. Equivalent circuit of a solar cell according to the one-diode model.
Following Kirchho's current law
I = ID + Ish − IL (11)
Kirchho's voltage law and Ohm's law give
V ′ = V − IRs (12)
The diode equation (Eq. 4) and Eq. 12 give
ID = I0(eqV′/AkT − 1) = I0(eq(V−IRs)/AkT − 1) (13)
2.5 Series and Shunt Resistances 11
Ohm's law and Eq. 12
Ish =V ′
Rsh=V − IRsRsh
(14)
Finally, Eq. 13 and Eq. 14 in Eq. 11 give an equation for the one-diode model [2]
I = I0(eq(V−IRs)/AkT − 1) +V − IRsRsh
− IL (15)
2.5 Series and Shunt Resistances
The series resistance, Rs, in a CIGS module is caused for example by the bulkresistance of the semiconductor material [2], the resistance of the transparentfront contact and the contact resistance between the front and back contact [6].The eect of series resistance is shown in Figure 4. The slope of the curve atthe open-circuit voltage, Voc, changes as the series resistance changes.
The shunt resistance, Rsh, in a CIGS module is caused by for example partialshorting of cells near edges [2] and connection between two adjacent cells throughthe CIGS in the P1 scribe [6]. The eect of shunt resistance is shown in Figure 4,and it is clear that the shunt resistance changes the slope of the curve at theshort-circuit current, Isc.
2.5.1 Resistivity and Conductivity
The electrical resistance, R, of a material is given by
R = ρl
A(16)
where ρ is the electrical resistivity, A is the cross-sectional area and l is thelength. The unit of resistance is Ω which implies that the unit of resistivity isΩm. The electrical conductivity, σ, of a material is the reciprocal of electricalresistivity
σ =1ρ
(17)
The unit of conductivity is S/m.
2.5.2 Sheet Resistance
Sheet resistance, R, is a measure of electrical resistance of thin lms that havea uniform thickness, d, and is dened as [2]
R =ρ
d=
1dσ
(18)
Sheet resistance is normally expressed as Ω/. Sheet resistance is measuredexperimentally using a four-point probe and the thickness can be measuredusing a prolometer.
12 2 THEORY
2.5.3 Contact Resistance
Contact resistance, Rc, is a measure of electrical resistance between two con-tacting surfaces and is given by
Rc = RAc (19)
where R is the resistance and Ac is the area of contact. The unit of contactresistance is normally Ωcm2.
2.6 Transmittance
The transmittance is the fraction of incident light at a specied wavelength thatpasses through a sample
T (λ) =IoutIin
(20)
where Iin is the intensity of the incident light and Iout is the intensity of thelight coming out of the sample. The transmittance can also be expressed usingthe exponential Beer-Lambert law
T (λ) = e−α(λ)d (21)
where α(λ) is the attenuation coecient and d is the path length.
13
3 Modelling
The model of a CIGS solar cell module has to take into account electrical, opticaland geometrical parameters [6]. Light is absorbed in the solar cell and a currentis generated following the diode equation. There is an absorption of light in thetransparent front contact which leads to optical losses. Major electrical seriesand shunt resistances of a CIGS solar cell module are indicated in Figure 6. Thelateral current in the transparent front contact (ZnO:Al) and the back contact(Mo) is not uniform, and hence these layers constitute distributed series resis-tances. The distributed series resistances are RZnO:Al,1, RZnO:Al,2, · · · , RZnO:Al,n
for the ZnO:Al and RMo,1, RMo,2, · · · , RMo,n for the Mo, as indicated in Figure 6.At the interconnect structure the ZnO:Al carries the full current and is mod-elled as a discrete series resistance, Rd. There is an additional resistance, Rc,at the ZnO:Al/Mo contact. The CIGS layer provides a shunting path betweenthe front contact and the back contact, indicated by Rsh,1, Rsh,2, · · · , Rsh,n inFigure 6. The interconnect structure leads to losses in active area. Altogetherthis leads to a set of nonlinear partial dierential equations (PDEs) which canbe solved numerically.
wcell
ZnO:Al
CIGS
Mo
Glass substrate
Buffer/ZnO
JL Rsh,1
J00A
RMo,1
RZnO:Al,1
Rc
RdJ1
Eg
TV1
Rsh,2
RMo,2
RZnO:Al,2J2
V2
Rsh,n
RMo,n
RZnO:Al,nJn
Vn
. . .
. . . VcellJ
wRcwRd
d
dCIGS
dMo
wic
w
Figure 6. Equivalent electrical circuit of a CIGS solar cell. The one-diode model,shunt resistances and series resistances are shown. Voltages, currents and dimensionsare dened. After Ref. [14] and [6].
3.1 CIGS Absorber
A heterojunction is formed at the interface between the p-type Cu(In,Ga)Se2
and the n-type buer/ZnO layer. The p-type material has a high concentrationof holes and a low concentration of electrons, hence holes ow readily throughthe p-type material [1]. The opposite is true for the n-type material. Excesselectron-hole pairs are generated by the light that is absorbed in the CIGSlayer. The assymmetrical properties of the heterojunction encourage a ow ofgenerated holes to the back contact and a ow of generated electrons to then-type material. Therefore, an illuminated heterojunction which is electricallyshorted will cause a net current ow. In the model, the current is generated in
14 3 MODELLING
each lateral point of the active cell, according to the diode equation (Eq. 9) andEq. 10 as
Ji = J0eqVi/AkT − JL (22)
withJ0 = J00e
−Eg/AkT (23)
where i = 1, 2, · · · , n are the lateral points.
The light-generated current, JL, depends on the transmission of the incidentlight through the transparent front contact as
JL = TZnO:AlJL,in (24)
where TZnO:Al is a function of the transmittance of the incident light throughthe ZnO:Al (see Section 3.2.1) and JL,in is the incident light-generated current.The incident light-generated current has a typical value about 350 A/m2 at STC.
In high-performance CIGS solar cells, the Cu(In,Ga)Se2 band gap, Eg, is be-tween 1.1 eV and 1.2 eV [3]. The ideality factor, A, depends on the dominantrecombination mechanism. It has been demonstrated that the ideality factor is1 < A < 2 for CIGS solar cells [3]. The prefactor, J00, has a typical value ofabout 1010 A/m2 and the temperature, T , is 298 K at STC. The thickness of theCIGS layer is between 1.5µm and 2.0µm [5]. In the model, the conductivity ofCIGS determines the total shunt resistance, Rsh, of the module.
The buer layer is a prerequisite to form the heterojunction. The thickness ofthe buer layer is about 0.05µm [5]. The ZnO layer increases the performanceof the solar cell. The thickness of the ZnO is about 0.1µm [5]. The propertiesof these layers are included in the modelled CIGS layer.
3.2 Transparent Front Contact
The following relationship holds for the sheet resistance, conductivity and thick-ness of the ZnO:Al layer (cf. Eq. 18)
R,ZnO:Al =1
dσZnO:Al(25)
Experimental data of the sheet resistance and the thickness of the ZnO:Al layercan be seen in Table 1. In the table the calculated value of the conductivity isalso shown. It is clear that the conductivity increases as the thickness increases.
3.2.1 Light Transmission
The optical transmission of the incident light through the transparent frontcontact decreases sharply when the ZnO:Al layer becomes thicker and as a resulthas less sheet resistance. A model that describes the transmittance, TZnO:Al, asa function of the sheet resistance, R,ZnO:Al, is [6]
TZnO:Al = T1 −(
R1
R,ZnO:Al
)m1
(26)
3.3 Back Contact 15
where T1, R1 and m1 are constants adjusted to t experimental data. For thesake of simplicity the transmission of light is assumed to be independent of thewavelength of the incident light.
Experiments have been performed to determine the transmission through theZnO:Al layer as a function of its sheet resistance at Solarex Corporation [7],at Uppsala University [8] and in the present study. The tting parameters areT1 = 0.96, R1 = 3.3 Ω/ and m1 = 3.3 [6] for the experimental data fromSolarex Corporation, and T1 = 1, R1 = 0.3707 Ω/ and m1 = 0.8226 [8] for theexperimental data from Uppsala University. Eq. 26 and the tting parametersfor the experimental data from Solarex Corporation and Uppsala Universitygive the following equations for the transmittance
TZnO:Al = 0.96−(
3.3R,ZnO:Al
)3.3
(27)
TZnO:Al = 1−(
0.3707R,ZnO:Al
)0.8226
(28)
An experiment was carried out to characterize the ZnO:Al from another sputtermachine, MRC II, which is used when CIGS mini-modules are manufactured atUppsala University. Together with monitor samples on plain glass substrates,solar cells with an area of 0.5 cm2 were prepared with an increasing thicknessof the ZnO:Al layer. The short-circuit current density, Jsc, for each solar cellwas calculated following Eq. 3, see Table 1. The thickness of the ZnO:Al, d,was measured using a prolometer, the sheet resistance of the layer, R,ZnO:Al,was measured with a four-point probe and as a result the conductivity of theZnO:Al, σZnO:Al, could be calculated, see Table 1. The short-circuit currentdensity decreases as the thickness of the ZnO:Al layer increases due to thehigher optical absorption of light in a thicker layer. Using the exponential Beer-Lambert law (Eq. 21) and Eq. 25, a curve, which describes the transmission ofthe incident light through the ZnO:Al layer, was tted to the experimental data
TZnO:Al = e−αd = e−α/(R,ZnO:AlσZnO:Al) = e−1.095/R,ZnO:Al (29)
where the tting parameters are α = 1.04·105 m−1 and σZnO:Al = 0.95·105 S/m.
Figure 7 shows experimental data of 1 − TZnO:Al as a function of the sheetresistance, R,ZnO:Al. The analytical approximations, i.e. Eq. 27, Eq. 28 andEq. 29, are also shown in the gure.
3.3 Back Contact
The thickness of the Mo layer is typically 0.5µm and the sheet resistance isabout 0.65 Ω/ [12]. The conductivity of Mo, σMo, can thus be calculated,using Eq. 18, to 3.1 · 106 S/m.
3.4 Interconnect Structure
The interconnect structure aects the performance of the solar cell module neg-atively due to area loss, discrete resistance and contact resistance. The width
16 3 MODELLING
Table 1. Experimental data. The short-circuit current density is measured on CIGSsolar cells using quantum eciency measurement. The ZnO:Al sheet resistance andthickness are measured on glass substrates using a four-point probe and a prolometerrespectively. The conductivity is calculated.
Sub. no. Jsc Sub. no. R,ZnO:Al d σZnO:Al
CIGS [mA/cm2] glass [Ω/] [µm] [S/m]3074-3838-04 33.6 4141-2 28 0.56 0.64 · 105
3080-3838-12 31.6 4141-1 11 1.1 0.83 · 105
3073-3838-05 30.3 4147-2 7.2 1.5 0.93 · 105
3076-3838-04 29.3 4141-5 5.3 1.9 0.99 · 105
3078-3838-05 26.9 4142-1 3.4 2.7 1.1 · 105
3075-3838-12 24.4 4142-2 2.3 3.3 1.3 · 105
0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
0.2
0.25
0.3
0.35Experimental data 1, Solarex CorporationExperimental data 2, Uppsala UniversityExperimental data 3, Uppsala UniversityAnalytical approximation 1Analytical approximation 2Analytical approximation 3
R,ZnO:Al [Ω/]
1−
TZnO
:Al[-]
Figure 7. 1 − TZnO:Al of the ZnO:Al layer as a function of its sheet resistance,R,ZnO:Al. Experimental data 1 were taken from Solarex Corporation [7] and data 2were taken from Uppsala University [8]. Experimental data 3 were produced duringthe present study. The approximations, i.e. Eq. 27, Eq. 28 and Eq. 29, for each set ofexperimental data are also shown [6, 8].
3.5 Model Implementation 17
of the interconnect structure, wic, is typically about 300µm to 400µm, see Fig-ure 6 [11]. This part of the solar cell is inactive, i.e no current is generatedthere. The active area loss is simply wic divided by the cell width, w. In theinterconnect structure the full current is carried by the ZnO:Al. The widthof this part of the ZnO:Al, wRd, is typically 160µm [11]. Both the thicknessand the conductivity of this part of the transparent front contact have the samevalue as the rest of the ZnO:Al.
Contact resistance, Rc, is found at the interconnect between ZnO:Al and Mo.The contact resistance should be lower than 0.01 Ωcm2 and preferably lowerthan 0.002 Ωcm2 according to Ref. [6], and it is about 0.0008 Ωcm2 accordingto Ref. [8]. An ongoing study at Uppsala University shows that the contactresistance, which is still a comparatively unknown parameter, is anything from0.003 Ωcm2 to very small [13]. The width of the contact, wRc, is about 50µm.
3.5 Model Implementation
In broad outline the modelling is done using COMSOL MultiphysicsTM and theoptimization is performed using MATLAB R©. COMSOL Multiphysics (earlierFEMLAB) is a commercial software package for modelling and solving problemsbased on partial dierential equations (PDEs). The PDEs are solved usingthe nite element method (FEM). FEM is a computer-based general methodfor numerically solving PDEs. The package contains a number of so-calledapplication modes. An application mode consists of a predened template anda user interface already set up with equations and variables for a specic area ofphysics. The software has a graphical user interface but it is also possible to doscript programming in the MATLAB language. To do this, the model has to beexported to the MATLAB language as an M-le. Then COMSOL Multiphysicshas to be connected to MATLAB, which makes it possible to modify and runthe model in MATLAB [9].
The way COMSOL Multiphysics is used in the modelling and solving processis here described in brief. The model is implemented using the ConductiveMedia DC Application Mode. A 2D geometry model of one single cell andthe interconnect structure is created. When the geometry is completed theboundary conditions are set. The electric-potential boundary condition, V =Vcell, and the ground boundary condition, V = 0, specify the voltages at two ofthe outer boundaries [10]. The electric-insulation boundary condition, n · J = 0,species that no current ows across any of the other outer boundaries. Thecontinuity boundary condition, n · (J1 − J2) = 0, species that the normalcomponents of the electric current are continuous across the interior boundary.The conductivity of the materials, constants, and expressions for the generatedcurrent, Eq. 22, Eq. 23 and Eq. 24, are all set. An equation that describes thetransmission of the incident light, e.g. Eq. 28, is also set.
The software then automatically applies a mesh to the geometry. The mesh-generation process can be controlled through a set of control parameters. Apredened mesh size is chosen. Because of the large dierence in dimensionbetween the width and the height of the cell, the geometry has to be scaledbefore meshing. This geometrical scaling is performed by the software. Next
18 3 MODELLING
comes the solution stage for which the software includes a set of numericalsolvers. The Nonlinear Parametric solver is chosen and solver parameters areset. COMSOL Multiphysics internally compiles a set of PDEs representing theentire model and solves the problem. The model is solved for the potential, Vcell,in the range of 0 V to 0.7 V and each voltage will give an outgoing current density,J . Figure 8 shows the potential distribution of the layers at Vcell = 0.1 V. Themodel is exported as an M-le.
The optimization script is written in MATLAB. The optimization is carried outby simply varying the cell width, w, and the thickness of the ZnO:Al, d. Foreach such conguration the model M-le is called and from the solution, whichis a J-V curve, it is possible to calculate the solar cell eciency, η. The optimaldesign is the one which has the maximum eciency. Since a CIGS module has axed area and a xed number of cells, the current, I, and the voltage, V , for themodule can be calculated. A function is written which calculates other outputparameters, such as Isc, Voc and FF .
Scripts and functions have been created with which it is possible to describe thebehaviour of a CIGS module. It is also possible to nd the optimal relationshipbetween the cell width and the thickness of the ZnO:Al layer. From now onthese scripts and functions are just called the model. The model has severalinput parameters. The parameters are summarized in Table 2.
Figure 8. A snapshot of COMSOL Multiphysics showing the potential distributionof the layers in the model at Vcell = 0.1 V.
3.5 Model Implementation 19
Table 2. Input parameters for the model. The parameters are Optimized, Given,Measured, Adjusted, an Equation or a Constant.
Name Parameter Unit TypeNumber of cells n - O or GModule width wmodule cm GModule length lmodule cm GCell width w mm O or GActive cell width wcell mm O or GInterconnect width wic µm MInterconnect ZnO:Al width wRd µm MThickness ZnO:Al d µm O or MThickness CIGS dCIGS µm MThickness Mo dMo µm MContact resistance Rc Ωcm2 A or MContact width wRc µm MConductivity ZnO:Al σZnO:Al S/m A or MSheet resistance ZnO:Al R,ZnO:Al Ω/ MConductivity CIGS σCIGS S/m AConductivity Mo σMo S/m MPrefactor J00 A/m2 ABand gap Eg eV A or MIdeality factor A - A or GTemperature T K GIrradiance Pin W/m2 GIncident light-generated current JL,in A/m2 AElementary charge q C CBoltzmann constant k eV/K CDiode saturation current J0 A/m2 ELight-generated current JL A/m2 ETransmittance ZnO:Al TZnO:Al - E, MDiode equation Ji A/m2 E
21
4 Simulations, Results and Discussions
4.1 Parameters Adjustment
The parameters in the model are set to t measured data of an actual mini-module with a cell width of 5 mm and a ZnO:Al sheet resistance of 32 Ω/.This value of the sheet resistance is too high, and not the optimal one for a cellwidth of 5 mm. The parameters are set so that the measured IV curve andsimulated IV curve match each other. Some input parameters are given, someare measured and some have to be adjusted. To get the correct short-circuitcurrent, Jsc, the incident light-generated current, JL,in, has to be adjusted. Thecorrect open-circuit voltage, Voc, in the model can be achieved by adjusting theprefactor, J00, the band gap, Eg, and/or the ideality factor, A. The idealityfactor also has an inuence on the shape of the curve.
The series resistance depends on the thickness, d, on the conductivity, σZnO:Al, ofthe ZnO:Al layer, and on the contact resistance, Rc. Both the sheet resistance,R,ZnO:Al, and the thickness, d, of the ZnO:Al layer are measured, which impliesthat the conductivity, σZnO:Al, is also known. So, the contact resistance, Rc, isadjusted to t the given series resistance. In the same way the conductivity ofthe CIGS layer, σCIGS, is adjusted to t the given shunt resistance. An equationfor the transmission of light through the ZnO:Al layer, TZnO:Al, has also tobe chosen. Figure 9 shows the IV curves of both an actual measured mini-module and a simulated module for which the parameters have been adjustedto correspond to the actual module. The parameters used in the simulation canbe seen in Table 3.
0 2 4 6 8 10 120
0.03
0.06
0.09
0.12
0.15
0.18
Measured, cell width 5 mmSimulated, cell width 5 mm
Voltage [V]
Cur
rent
[A]
Figure 9. Simulated module with a cell width of 5 mm, which has been adjusted tocorrespond to an actual module (sub. no. 3709).
22 4 SIMULATIONS, RESULTS AND DISCUSSIONS
Table 3. Input parameters and their values during a specic simulation. Materialparameters were adjusted to t an actual module (sub. no. 3709).
Parameter Unit Valuen - 16wmodule cm 8.00lmodule cm 9.70w mm 5.00wcell mm 4.70wic µm 300wRd µm 160d µm 0.7dCIGS µm 2dMo µm 0.5Rc Ωcm2 0.005wRc µm 50σZnO:Al S/m 4.5·104
R,ZnO:Al Ω/ 31.7σCIGS S/m 5·10−5
σMo S/m 3.10·106
J00 A/m2 1.0·1010
Eg eV 1.2A - 1.245T K 298Pin W/m2 1000JL,in A/m2 350q C 1.6·10−19
k eV/K 8.62·10−5
J0 A/m2 Eq. 23JL A/m2 Eq. 24TZnO:Al - Eq. 28Ji A/m2 Eq. 22
4.2 Experimental Verication 23
4.2 Experimental Verication
Mini-modules with cell widths of 3 mm, 7 mm and 9 mm are simulated andthe performances of these modules are compared with measurement data frommanufactured modules, see Table 4. This will give an idea of how well the modelpredicts the performance of modules with dierent cell widths. All parametersexcept the number of cells, n, module width, wmodule, cell width, w, and activecell width, wcell, are kept constant.
Table 5 shows measured and simulated data of mini-modules with cell widths of3 mm, 7 mm and 9 mm. The corresponding IV curves can be seen in Figure 10.The conclusion is that the measured and simulated performances of the CIGSsolar cell modules agree well. The accuracy of the model is within ±3 % for theoutput parameters Voc, Isc, FF and η. It should be noted that the low ecien-cies at wider cell widths is a consequence of the relatively high sheet resistanceof the ZnO:Al layer. The sheet resistance of the manufactured modules wasmeasured at 32 Ω/.
Table 4. Measured data of CIGS solar cell mini-modules from the same batch but withdierent cell widths and ZnO:Al sheet resistances. The solar cell output parametersare measured under STC. The sheet resistance is measured using a four-point probe.
Sub. w n R,ZnO:Al Area Area Voc/cell Jsc FF ηno. loss
[mm] [-] [Ω/] [cm2] [%] [V] [mA/cm2] [%] [%]3810 3 27 32 78.6 10 0.660 30.4 72.2 14.53812 3 27 32 78.6 10 0.657 30.3 72.1 14.43713† 3 27 32 78.6 10 0.662 31.9 71.4 15.13709 5 16 32 77.6 6 0.648 31.7 64.1 13.23808 5 16 27 77.6 6 0.648 31.8 64.2 13.23809 5 16 27 77.6 6 0.640 31.5 65.4 13.23716 5 16 20 77.6 6 0.642 30.5 69.5 13.63712 5 16 20 77.6 6 0.645 30.7 68.8 13.63813 7 12 32 81.5 4 0.645 32.3 53.9 11.23814 7 12 27 81.5 4 0.634 32.3 58.3 11.93714 7 12 27 81.5 4 0.638 32.0 60.3 12.33711 9 9 32 78.6 3 0.658 33.0 40.6 8.823815 9 9 27 78.6 3 0.649 32.7 46.6 9.893715 9 9 27 78.6 3 0.640 32.4 47.9 9.93†This module was prepared with an anti-reection coating.
Table 5. Measured and simulated output parameters of CIGS solar cell mini-moduleswith dierent cell widths. ZnO:Al sheet resistance is equal to 32 Ω/.
Sub. no. w Voc Isc FF η[mm] [V] [A] [%] [%]
3810 3 17.8 0.0885 72.2 14.5Simulated 3 17.5 0.0890 72.6 14.43709 5 10.4 0.154 64.1 13.2Simulated 5 10.4 0.154 63.9 13.23813 7 7.74 0.219 53.9 11.2Simulated 7 7.78 0.218 52.0 10.93711 9 5.92 0.288 40.6 8.82Simulated 9 5.84 0.281 41.3 8.63
24 4 SIMULATIONS, RESULTS AND DISCUSSIONS
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
Measured, cell width 3 mm, 5 mm, 7 mm and 9 mmSimulated, cell width 3 mm, 5 mm, 7 mm and 9 mm
Voltage [V]
Cur
rent
[A]
Figure 10. Measured modules (sub. no. 3810, 3709, 3813 and 3711) and simulatedmodules with cell widths of 3 mm, 5 mm, 7 mm and 9 mm. The material propertieswere adjusted to match the actual module with a cell width of 5 mm (sub. no. 3709)and then kept constant for the other cell widths.
4.3 Cell Width Optimization
The model is now used to nd the optimal cell width, i.e. the cell width whichgives the highest module eciency at STC. The thickness of the ZnO:Al layer, d,is optimized for each investigated cell width. The same values of the parametersare used as in Table 3, except for the conductivity of the ZnO:Al, σZnO:Al,which is increased from 4.5 · 104 S/m to 1.0 · 105 S/m. The higher value of theconductivity is taken from the experimental data in Table 1. For each thickness,d, the corresponding sheet resistance, R,ZnO:Al, can be calculated. The numberof cells, n, module width, wmodule, and active cell width, wcell, all vary withthe chosen cell width. The simulations are performed for the three equations(Eq. 27, 28 and 29) describing the transmission of light through the ZnO:Allayer, TZnO:Al.
Figure 11 shows the eciency, η, of modules with dierent cell widths. WithTZnO:Al set equal to Eq. 28 the highest module eciency, 14.5 %, is found for cellwidths between 3 mm and 4 mm. The optimal sheet resistance of the ZnO:Allayer for a cell width of 3 mm is 20.0 Ω/, and for a cell width of 4 mm theoptimal sheet resistance is 14.5 Ω/. A module using a cell width of 5 mm willhave a maximum eciency of 14.3 % with a sheet resistance of 10.5 Ω/. Inthese simulations the interconnect width, wic, has been set to 300µm, whichimplies an active area loss of 10 % for a cell width of 3 mm and 6 % for a cellwidth of 5 mm. The higher active area loss for a cell width of 3 mm is morethan compensated for by the lower optical losses caused by smaller absorption
4.3 Cell Width Optimization 25
of light in the thinner ZnO:Al layer as compared to wider cell widths. For acell width of 2 mm the active area loss in this case is 15 %, which is too largeto be compensated for by the lower optical losses. The highest eciency for amodule with a cell width of 2 mm is 14.1 % with an optimal sheet resistance of31.1 Ω/. It is clear that it is more important to decrease the active area lossesby minimizing the interconnect width for CIGS modules with narrower cells.
This can be compared with simulations performed in Ref. [5] which gave anoptimal cell width of 5 mm at a 10 Ω/ sheet resistance of the ZnO:Al. TZnO:Al
for these simulations was also described by Eq. 28. Some other input parameterswere not the same, e.g. the maximum eciency was 12.2 % and the interconnectwidth was set to 400µm.
The short-circuit current, Isc, and the open-circuit voltage, Voc, are two otheroutput parameters which change when the cell width of the module is modied,see Table 4. The change of these two output parameters is merely a consequenceof the dierent number of cells.
If TZnO:Al is set equal to Eq. 27, the maximum eciency, 14.6 %, is found fora cell width of 3.5 mm. In this case, the sheet resistance of the ZnO:Al layeris 13.7 Ω/. For TZnO:Al set equal to Eq. 29, the highest eciency, 14.3 %, isreached using a cell width of 3 mm and a sheet resistance of 24.4 Ω/. Theresults of the performed simulations show that any of the three equations de-scribing the transmission of light through the ZnO:Al layer will give about thesame optimal cell width.
0 1 2 3 4 5 6 7 8 9 100.115
0.12
0.125
0.13
0.135
0.14
0.145
0.15
TZnO:Al set equal to Eq. 27TZnO:Al set equal to Eq. 28TZnO:Al set equal to Eq. 29
Cell width [mm]
Effi
cien
cy[-]
Figure 11. Simulations showing the module eciency at dierent cell widths. Threedierent approximations describing the transmission of light through the ZnO:Al layerare used. The optimal cell width is found between 3.0 mm and 3.5 mm.
26 4 SIMULATIONS, RESULTS AND DISCUSSIONS
4.4 ZnO:Al Sheet Resistance Optimization
Simulations are performed to investigate how the sheet resistance of the ZnO:Alinuences the eciency of the CIGS solar cell module. For each investigatedcell width the thickness of the ZnO:Al layer, d, is varied. Since d is varied, thesheet resistance, R,ZnO:Al, also changes. The same values of the parameters areused as in Table 3, except for the conductivity of the ZnO:Al, σZnO:Al, which isset to 1.0·105 S/m. The number of cells, n, module width, wmodule, and activecell width , wcell, all vary with the chosen cell width.
Figure 12 shows the module eciency, η, at dierent values of the ZnO:Alsheet resistance, R,ZnO:Al. The simulations were performed for modules withcell widths of 3 mm, 5 mm, 7 mm and 9 mm. Measured data from manufacturedmodules are also plotted in the gure, cf. Table 4. The rst conclusion is thatthe maximum eciency increases with decreasing cell width for the cell widthsinvestigated. If the shape of the four curves are compared it is clearly seen thatthe curve becomes atter around its maximum value with decreasing cell width.So, the narrower cell width the module has, the less sensitive it is to variations ofthe ZnO:Al sheet resistance, see Table 6. For instance, if the cell width is 3 mmthe eciency is still within 95 % of its maximum value if R,ZnO:Al is in therange 660 Ω/. For a cell with of 9 mm the corresponding range is 311 Ω/,which implies a much smaller process window.
0 5 10 15 20 25 30 35 400.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Simulated, cell width 3 mmSimulated, cell width 5 mmSimulated, cell width 7 mmSimulated, cell width 9 mmMeasured, cell width 3 mmMeasured, cell width 5 mmMeasured, cell width 7 mmMeasured, cell width 9 mm
R,ZnO:Al [Ω/]
Effi
cien
cy[-]
Figure 12. Simulations showing the module eciency at dierent values of theZnO:Al sheet resistance for modules with cell widths of 3 mm, 5 mm, 7 mm and 9 mm.Measured data from Table 4 are also plotted.
It is also interesting to look at the consumption of ZnO:Al during productionof CIGS solar cell modules. Modules with cell widths of 3 mm, 5 mm, 7 mmand 9 mm have an optimal R,ZnO:Al of 20.0 Ω/, 10.5 Ω/, 7.5 Ω/, 5.5 Ω/
4.5 Irradiance Optimization 27
respectively. Since the sheet resistance is inversely proportional to the ZnO:Althickness, the consumption of ZnO:Al is about twice as high for a cell widthof 5 mm, three times higher for a cell width of 7 mm and four times higher fora cell width of 9 mm compared to using a cell width of 3 mm. Except for thesmaller material consumption, a thinner layer of ZnO:Al also allows for a fastersputter process. On the other hand, the smaller the cell width of the modulethe more time it will need in the scribing machine since there are more cells tobe scribed. It is clear that other factors than just the eciency (or peak watts)have to be taken into account when optimizing a CIGS solar cell module forlarge-scale production.
Table 6. Simulations carried out to optimize the CIGS solar cell modules with respectto eciency.
w [mm] 3 5 7 9R,ZnO:Al at ηmax [Ω/] 20.0 10.5 7.5 5.5R,ZnO:Al at η > 0.95 ηmax [Ω/] 660 425 415 311Active area loss [%] 10 6 4 3ηmax [%] 14.5 14.3 13.7 13.1
4.5 Irradiance Optimization
Simulations are carried out to investigate the performance of the CIGS solarcell module at other irradiances than 1000 W/m2. The optimal relationshipbetween cell width and ZnO:Al sheet resistance is simulated for irradiances of8 kW/m2 and 20 kW/m2. The main parameter varied during these simulationsis the irradiance, Pin. The incident light-generated current, JL,in, will changewith the same factor as Pin. The parameter σZnO:Al is set to 1.0 · 105 S/m. Theparameters n, wmodule and wcell all vary with the chosen cell width. The otherparameters can be found in Table 3. Note that the temperature, T , is constantat 298 K throughout the simulations.
For an irradiance of 8 kW/m2 the optimal cell width is 2 mm with a ZnO:Alsheet resistance of 9.9 Ω/. This module, optimized for 8 kW/m2, is called the8X-module. The so-called 20X-module has an optimal cell width of 1.5 mm witha ZnO:Al sheet resistance of 8.0 Ω/. In Figure 13 the eciency is plotted asa function of the irradiance. The 1X-module with a cell width of 5 mm has itsmaximal eciency, 14.3 %, not far from 1 kW/m2. For the 1X-module with acell width of 3 mm the maximal value, 14.7 %, is shifted and occurs at about1.5 kW/m2. The eciency decreases fast at higher irradiances for both thesemodules. The 8X-module has its maximum eciency, 14.6 %, at 4 kW/m2, andthe 20X-module has its maximum eciency, 14.0 %, at 7 kW/m2. The 1X-module has the highest overall eciency, and it is not possible to reach equallyhigh eciency for the 8X- and 20X-modules because of the bad transmissionof light through the ZnO:Al. The eciency decreases much more slowly athigher irradiances for both 8X- and 20X-module compared to the 1X-module.This quality makes both the 8X- and 20X-module suitable for low-concentratingsystems, such as the Swedish developed systems MaReCo from the companyVattenfall Utveckling AB and Solar8 from the company Arontis Solar Concen-trator AB. The results are summarized in Table 7.
28 4 SIMULATIONS, RESULTS AND DISCUSSIONS
0 2 4 6 8 10 12 14 16 18 200.1
0.11
0.12
0.13
0.14
0.15
1X-module, cell width 5 mm1X-module, cell width 3 mm8X-module, cell width 2 mm20X-module, cell width 1.5 mm
Effi
cien
cy[-]
Irradiance [kW/m2]
Figure 13. Simulations showing the module eciency as a function of irradiance forCIGS modules optimized for 1 kW/m2, 8 kW/m2and 20 kW/m2.
Table 7. Simulated eciencies at dierent irradiances for optimized CIGS modules.
Module w R,ZnO:Al η [%] at Pin [kW/m2][mm] [Ω/] 0.5 1 2 8 20
1X 5 10.5 13.7 14.3 14.0 9.13 5.281X 3 20.0 13.7 14.5 14.7 11.4 7.098X 2 9.90 12.7 13.7 14.3 14.3 12.220X 1.5 8.00 11.8 12.8 13.5 14.0 13.2
29
5 Conclusions
A numerical model of a CIGS solar cell module which is based on the one-diodemodel and takes into account electrical, optical and geometrical parameters wasderived. The model consists of a set of partial dierential equations which cansuccessfully be solved using the nite element method. The software packageCOMSOLMultiphysics, in which the model is implemented, utilizes this methodin the solving process. An optimization script was written in the MATLABlanguage.
A crucial part of the modelling was to describe correctly the optical and electricalbehaviour of the transparent front contact. For this reason an experiment wascarried out to characterize the ZnO:Al from the MRC II sputter, resultingin an analytical expression. A batch of CIGS mini-modules with an aperturearea of 80 cm2, cell widths of 3 mm, 5 mm, 7 mm and 9 mm and dierent sheetresistances of the ZnO:Al layer were manufactured.
An eciency as high as 15.1 % was measured under standard test conditionsfor one of the modules with a cell width of 3 mm. Another module, with a cellwidth of 5 mm and a ZnO:Al sheet resistance of 32 Ω/, was used as a referencefor adjusting parameters in the model. This adjustment was performed bysimply varying some of the input parameters in the model until the simulatedand measured IV curves matched each other. An experimental verication ofthe model was performed by comparing the measured and simulated outputparameters of modules with cell widths of 3 mm, 7 mm and 9 mm, all with thesame ZnO:Al sheet resistance. The accuracy of the model was shown to bewithin ±3 % for the output parameters Voc, Isc, FF and η.
Simulations were performed to optimize the performance of the CIGS solar cellmodule with respect to eciency. A module optimized for an irradiance of1000 W/m2 has a cell width of 3 mm and a ZnO:Al sheet resistance of 20 Ω/.The simulations showed that the higher active area loss for a cell width of 3 mmis more than compensated for by the lower optical losses caused by higher trans-mission of light through the thinner transparent front contact as compared towider cell widths. The simulations also showed that the narrower cell width themodule has, the less sensitive it is to variations of the ZnO:Al sheet resistance.Simulations were performed to nd the optimal design of CIGS modules forlow-concentrating systems such as MaReCo and Solar8. One such module, op-timized for an irradiance of 8000 W/m2, has a cell width of 2 mm and a ZnO:Alsheet resistance of 10 Ω/.
31
Acknowledgements
This project, which completes my studies in Engineering Physics, was carriedout at Ångström Solar Center, Uppsala University. It was nanced by theSwedish SolEl 03-07 programme. First of all I would like to thank my super-visors Uwe Zimmermann and Marika Edo for their help and guidance. I amparticularly grateful to Uwe for fruitful dicussions about the modelling and forhelp when performing the experiments and manufacturing modules. I wouldalso like to thank the other members of the research team for a pleasant timein their company and for their support, especially Marta Ruth and Per-OskarWestin for their assistance when manufacturing the modules. I am also in-debted to Joakim Byström at Arontis Solar Concentrator AB for suggestingthis project to Marika. Finally, I would like to thank Christer and Karen-Mariefor the proofreading.
REFERENCES 33
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ISSN 1651-8128ISBN 978-91-85147-27-4
