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Extraction of solar cell series resistance without presumed current–voltage

functional form

Ming-Kun Lee

a

, Jen-Chun Wang

a

, Sheng-Fu Horng

a,



, Hsin-Fei Meng

b

a

Department of Electrical Engineering, National Tsing Hua University, Hsinchu 300, Taiwan, Republic of China bInstitute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan, Republic of China

a r t i c l e

i n f o

Article history: Received 30 June 2009 Received in revised form 22 November 2009 Accepted 24 November 2009 Available online 29 December 2009 Keywords:

Solar cell Circuit model

Series resistance extraction

a b s t r a c t

A new method which does not require presumed current–voltage functional form is proposed for the determination of the series resistance, the shunt resistance, the photocurrent, and the intrinsic current– voltage characteristics of solar cells. This method was applied to analyze a bulk heterojunction organic solar cell. It was found that the extracted intrinsic current–voltage characteristic clearly exhibits a linear hopping current component and a quadratic space-charge limited current component. Furthermore, the reconstructed dark current–voltage curve is found to differ significantly from the measured dark current–voltage curve, revealing the importance of electric field in the operation of bulk heterojunction organic solar cells.

&2009 Elsevier B.V. All rights reserved.

1. Introduction

Solar cells are promising devices for clean electric generation and have attracted intensive research. Like all other electrical power generators, solar cells possess internal series resistance which affects significantly their power conversion efficiency (PCE). More-over, the simulation and design of solar cell systems also require an accurate knowledge of the series resistance and other related device parameters to describe their nonlinear electrical behavior. Extract-ing the series resistance as well as other device parameters for solar cells is therefore of vital importance.

Over the years, various methods have been proposed for extracting the series resistance and related device parameters of solar cells[1–13]. These methods either involve current–voltage (I–V) measurements with different illumination levels[1–3,8], or apply curve fitting method to some presumed functional relation-ship[5–7,9–12], or employ integration procedures based on the computation of the area under the I–V curves[4], or use linear regression[13].

However, all these previously proposed methods are based on the assumption that the intrinsic I–V relationship of the solar cell follows a specific functional form, which is usually taken to be one or combination of the Shockley-type single exponential I–V characteristic with ideality factor. While the exponential I–V assumption may produce convenient equivalent-circuit model for use in conventional simulation tools, its validity, and hence its

usefulness in understanding the underlying physics, is generally not guaranteed. This is especially the case for non p–n junction type devices such as organic solar cell (OSC) or dye-sensitized solar cell. For example, one would expect polynomial type intrinsic I–V characteristics for OSC if the charge transport is dominantly space-charge-limited (SCL). It is therefore advanta-geous to be able to extract the series resistance and device parameters without presumed I–V functional form. Such a series resistance extraction method without presumed I–V functional form is proposed in this paper. We found that with certain physically plausible assumptions such a scheme will lead to unique determination of all the device parameters as well as the intrinsic I–V characteristics. This method was applied to OSC and first-order hopping current and second-order SCL current compo-nents were observed in the intrinsic I–V characteristics.

2. Theory

The solar cell is characterized using the equivalent circuit model as shown inFig. 1and the relation between the measured current Imand the measured voltage Vmis given by

Im¼ VD Rsh þf ðVDÞIph ð1Þ Vm¼VDþ VD Rsh þf ðVDÞIph   Rs ð2Þ

where VD, f(VD), Iph, Rsand Rsh, are the voltage across the diode, the

intrinsic I–V characteristics of the diode, the photocurrent, the Contents lists available atScienceDirect

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

Solar Energy Materials & Solar Cells

0927-0248/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.solmat.2009.11.026



Corresponding author. Tel.: + 886 35915873. E-mail address: sfhorng@itri.org.tw (S.-F. Horng).

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series resistance and the shunt resistance, respectively. Note from Fig. 1that since both f ðVDÞand Rshare to be determined and are in

parallel connection, any combination of f ðVDÞ and Rsh that

preserves the value VD=Rshþf ðVDÞ will leave the measured I–V

characteristics, Imand Vm, unchanged. One therefore needs more

assumptions to allow for unique determination of these device parameters. The following for assumptions are taken in our fitting scheme:

1. Rs, Rsh, Iphremain constant during the measurements;

2. f ð0Þ ¼ 0, that is, the power generation is all attributed to Iph;

3. f(VD)-f0as VD5 0; that is, the leakage current is attributed

to Rsh;

4. f(VD) is a monotonic function of VD;

5. f(VD) is nonlinear for VD40;

6. Iphchanges monotonically with illumination level.

From assumption 4 and (2), Vmis also a monotonic function of VD,

and Vm50 when VD50. From assumption 3 for Vm50, one can

eliminate VDfrom both (1) and (2) and obtain

Im¼ 1 ðRshþRsÞ Vm Rsh ðRshþRsÞ  ðf0þIphÞ ð3Þ

The total resistance Rt¼RshþRs can therefore be obtained

from the slope dIm=dVmat Vm50.

From (1) and (2) one can obtain

VD¼VmRsIm ð4Þ f ðVDÞ ¼ Rt ðRtRsÞ Im Vm ðRtRsÞ þIph ð5Þ

Considering also assumption 2, one also has

Iph¼ ImðVm@VD¼0Þ ð6Þ

It is clear from (3), (4) and (5) that since Rtcan be extracted

from the measured I–V characteristics, once Rsis determined, all

other device parameters, namely Rsh, Iph and the intrinsic I–V

characteristics f ðVDÞcan all be extracted. We now need a scheme

to determine Rs.

In order to devise a scheme for the unique determination of Rs,

we construct the following test quantities for arbitrary R:

VDR¼VmR  Im ð7Þ IphR¼ ImðVm@VDR¼0Þ ð8Þ fR¼ Rt ðRtRÞ Im Vm ðRtRÞ þIphR ð9Þ

Denote as VD0the voltage across the diode when VDR= 0. It is

clear that if R= Rs, then VDR, fR and IphR reduce to VD, f and Iph,

respectively, and VD0 =0.

For RaRs, as derived in appendix, we have the following three

equalities: f ðVD0Þ ¼ ðRtRÞ Rsh ðRRsÞ VD0þIph ð10Þ fR¼ Rsh ðRtRÞ  ðf ðVDÞf ðVD0ÞÞ ð11Þ fR¼ Rsh ðRtRÞ  ðRsRÞ VDRþ 1 ðRRsÞ  ðVDVD0Þ ð12Þ

It is obvious from (10) that VD0depends on both Iphand R.

For a given f ðVDÞ, one can expand it into Taylor series around

VD0, f ðVDÞ ¼f ðVD0Þ þf0ðVD0Þ  ðVDVD0Þ þ 1 2!f 00ðV D0Þ  ðVDVD0Þ2þ    ð13Þ Combining (11)–(13), we have Rsh ðRtRÞðRsRÞ VDRþ 1 ðRRsÞ ðVDVD0Þ ¼ Rsh ðRtRÞ f0ðV D0ÞðVDVD0Þ þ 1 2!f 00ðV D0ÞðVDVD0Þ2þ      ð14Þ

For a given f ðVDÞ, one can solve ðVDVD0Þ from (14) and

substitute the result into (12) to obtain the functional relationship between fR and VDR. Since, from 5, f ðVDÞis nonlinear in VD, the

coefficients in the expansion (13) cannot be all constant and must depend on VD0. Therefore, ðVDVD0Þ and hence fR must also

depend on VD0.

From the aforementioned discussion, we know that unless R ¼ Rs, at which VD0vanishes identically, fRðVDRÞmust depend on

VD0and hence on Iphand illumination level (6). At a specific R, we

may therefore use the root-mean-square error (RMSE) between fRðVDRÞat different illumination levels to determine if R coincides

with Rs. Alternatively one may use the extracted fRðVDRÞfrom the

I–V measurement at one illumination level to reconstruct the measured I–V at another illumination and calculate the RMSE. It is noteworthy that to avoid the temperature difference due to different illumination levels, which might lead to significant extraction error [12,14], it is suggested that close illumination levels are used. In practice close illumination levels also ensure the constancy of Rs, Rsh required in assumption 1 and ascertain

that identical Rt from both I–V characteristics can be obtained.

Since Rsand Rshare sensitive to cell temperature and illumination

level, the constancy of Rt is an important indicator that

assumption 1 is met. Note also that our extraction algorithm does not require the precise ratio in the illumination levels.

Our Rsextracting scheme is summarized as follows:

1. measure I–V characteristics Im1ðVmÞ and Im2ðVmÞ at two

different illumination levels;

2. extract Rtfrom both Im1ðVmÞand Im2ðVmÞaccording to (3);

3. for a given R construct VDR1, fR1and IphR1according to (7)–(9)

from Im1ðVmÞ;

4. extract IphR2according to (7) and (8) from Im2ðVmÞ;

5. reconstruct Ireconstruct2ðVmÞfrom VDR1, fR1and IphR2;

6. calculate the RMSE between Im2ðVmÞand Ireconstruct2ðVmÞ;

7. repeat 3–5 for another R and search for minimum of RMSE. Although too straightforward to be included in this paper, we have checked the validity of this fitting scheme (steps 3–7) with numerically generated data. It was found that the discrepancy in the fitting result was set by the error in the estimated Rt. It is

therefore advisable to estimate Rtat sufficiently negative bias.

Vm -+ f(vD) Rs Rsh Im -+ VD Iph

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Since our Rs extracting scheme does not require specific I–V

functional form, it can be applied to various kinds of solar cells as long as the assumptions are met. In particular it is interesting to apply this method to non p–n junction type photovoltaic devices to see if new insights may be obtained. In this paper a bulk heterojunction (BHJ) OSC was employed.

Organic photovoltaic devices have attracted much attention due to their promising properties such as mechanical flexibility, light weight and environmental benignity. In addition, because of their low-temperature and solution-based processability, they can be fabricated on flexible substrate, which can be adapted to high throughput continuous roll-to-roll manufacturing and leads to greatly reduced production cost. Due to strong Coulomb interaction in organic materials, photo-excited charged carriers quickly form excitons, with which charge separation by electric field typically used in p–n junction type solar cells is ineffective. Donor-acceptor type-II heterojunctions such as BHJ were demon-strated to be effective for exciton dissociation. With various structure, material and process advancements, the PCE of OSCs has been significantly improved and a record high PCE of 7.6% was recently reported [15]. All-solution roll-to-roll manufacturing processes for OSCs have also been designed and studied[16,17].

3. Experiment

In this study the BHJ OSC is made of a blend of poly (3-hexylthiophene) (P3HT) and 6,6-phenyl-C61-butyric acid methyl ester (PCBM) in a 1:1 weight ratio. The blend is sandwiched in between an indium tin oxide (ITO) coated-glass (sheet resistance 10

O

=&)/poly(3,4 ethylenedioxythiophene): poly(styrenesulfonate) (PEDOT:PSS), and an evaporated calcium (50 nm)/silver (80 nm) top electrode. The thickness of the active layer is 220 nm and the area of the OSC is 4 mm2. The fabrication procedure is standard

and follows that in Ref.[18]. After fabrication the I–V characteristics was measured with a calibrated solar simulator at AM1.5G (100 mW=cm2). To obtain the I–V curve at a different light intensity,

a microscope cover glass was placed on top of the OSC, leading to a light attenuation of about 8%. The RSME is calculated for the range from VD¼0 to open-circuit voltage (Voc) in the measured I–V

characteristic with less light illumination.

4. Results and discussion

Fig. 2shows the measured Im1ðVm) and Im2ðVm) as well as dark

I–V. The comparison of Im2ðVm) with the reconstructed I–V, as

described in step 5, in the extraction scheme, is shown inFig. 3, showing very good agreement between the measurement and the fitting curve. The fitting result is summarized inTable 1.

Fig. 4(a) and (b) show the extracted intrinsic device I–V characteristics in log–log and semi-log scale, respectively. A linear and a quadratic component can be clearly resolved in Fig. 4(a). The linear component is tentatively attributed to the hopping conduction, which dominates the carrier transport in disordered organic materials and can be linearized at low field [19]. The quadratic component is attributed to the SCL current transport frequently observed in organic materials. For comparison, the dark I–V, along with lines with the required slope, was also plotted in the insets ofFig. 4(a) and (b). It was found that the linear and quadratic current components can be resolved only after the removal of series and shunt resistance. It is also interesting to see from Fig. 4(b) that there is a 120 mV/dec exponential component in the extract intrinsic I–V, which may result from trap-assisted photo-carrier recombination. However,

the origin of this exponential component remains unclear and further investigation is required.

Fig. 5 shows the comparison of the measured and the reconstructed dark I–V characteristics. The latter was calculated as in step 5 yet with photocurrent set to zero. It was found that dramatic difference exists between these two I–V curves. It is well known that the dominant charge transport mechanisms in disordered organic materials are either hopping or SCL transport and depend strongly on the field distribution within the device. Since the carrier concentration, and therefore the field distribution, in the device changes significantly with light illumination, the reconstructed I–V characteristics from device parameters extracted from illuminated device will be different from the measured dark I–V. This result also corroborates the

-0.5

-0.2 0.0 0.2 0.4 0.6 0.8

0.0 0.5 1.0

Measured Current (mA)

Measured Voltage (V) Im1

Im2 Id

Fig. 2. Measured I–V characteristics at two illumination levels (Im1, Im2) and dark I–V (Id) of the OSC.

-0.4 -0.2 0.0 0.2 0.4 0.6 -0.2 0.0 0.2 0.4 Current (m A) Voltage (V) Im2 Im2f

Fig. 3. The comparison of the measured illuminated I–V (Im2) and the reconstructed I–V (Im2f) as described in step 5 of the extraction scheme.

Table 1

The fitting parameters including Rs, Rsh, Jph1and Jph2for organic solar cell. RsðOcm2Þ RshðOcm2Þ Jph1ðmA=cm2Þ Jph2ðmA=cm2Þ Organic solar cell 5.24 3060 11.3 10.5

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needs for close illumination levels previously proposed in our extraction scheme.

For comparison we have also applied our Rsextraction scheme

to a multi-crystal Si solar cell as well as an amorphous Si solar cell. These two solar cells were obtained from the Photovoltaics Technology Center at Industrial Technology Research Institute, Taiwan. The size for the multi-crystal (amorphous) Si solar cell is 2:13 ð1:68Þ cm2. The fitting results are summarized inTable 2.

Fig. 6(a) and (b) show the comparison of the measured and the reconstructed dark I–V characteristics for multi-crystal and amorphous Si solar cell, respectively. FromFig. 6, it was found that, while very good agreement between the measured and the reconstructed dark I–V characteristics was obtained for the multi-crystal solar cell, there is dramatic difference in the case of amorphous Si solar cell. These results are not surprising since they simply reflect the difference between diffusion controlled carrier transport in bulk Si solar cell and drift current controlled, and hence electric field dependent, carrier transport in amorphous Si solar cell. Though disorder related traps in amorphous Si may also contribute additionally to the difference between the measured and the reconstructed dark I–V characteristics, these results are consistent with our previous argument of electric field effect on dark I–V in OSC.

Fig. 7(a) and (b) show the extracted intrinsic device I–V characteristics plotted in semi-log scale for the multi-crystal and

log–log scale for amorphous Si solar cell, respectively. From Fig. 7(a) multi-crystal Si solar cell exhibits mostly single exponential I–V characteristic with ideality factor 2 as expected. On the other hand, it is clear from Fig. 7(b) that there is a quadratic current component at low voltage for amorphous Si

0.01 0.1 1

Diode Current (mA)

Diode Voltage (V) 120mv/dec 1E-6 0.2 0.4 0.6 1E-5 1E-4 1E-3 0.01 0.1 slope=1

Dark Current (mA)

Voltage (V) 1E-6 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.5 0.5 0.5 0.6 0.6 1E-5 1E-4 1E-3 0.01 0.1 120mv/decade

Dark Current (mA)

Voltage (V) 1E-3 0.2 0.4 0.6 0.01 0.1 slope=1 slope=2

Diode Current (mA)

Diode Voltage (V)

Fig. 4. The extracted intrinsic I–V in (a) log–log scale, showing clearly a linear and a quadratic current component; (b) in semi-log scale; note that an exponential component with a slope of 120 mV/dec was clear seen. The measured dark I–V was also plotted in the insets for comparison.

0.0 -0.2 0.0 0.2 0.4 0.6 0.1 0.2 Current (mA) Voltage (V) Id Idf

Fig. 5. The comparison of the measured dark I–V (Id) and the reconstructed dark I–V (Idf) as described in step 5 of the extraction scheme, albeit with Iphset to zero.

Table 2

The fitting parameters including Rs, Rsh, Jph1and Jph2for silicon solar cell. RsðOcm2Þ RshðOcm2Þ Jph1ðmA=cm2Þ Jph2ðmA=cm2Þ Multi-crystal silicon 3.5 489 40.7 38.8 Amorphous silicon 14.7 2619 12.6 11.8 0 -0.2 0.0 0.2 0.4 0.6 10 20 30 40 Voltage (V) Current (mA) Id Idf 0 2 4 6 Current (mA ) Id Idf -0.2 0.0 0.2 0.4 0.6 0.8 Voltage (V)

Fig. 6. The comparison of the measured dark I–V ðIdÞand the reconstructed dark I– V (Idf) for (a) multi-crystal and (b) amorphous Si solar cell, respectively.

0.1

0.3 0.4 0.5 0.6 0.2 0.4 0.6 0.8 1

1 10 100

Diode Current (mA)

Diode Voltage (V) 120mv/dec 0.1 1 10 100

Diode Current (mA)

Diode Voltage (V)

slope : 2

Fig. 7. The extracted intrinsic device I–V characteristics plotted in (a) semi-log scale for the multi-crystal and (b) log–log scale for amorphous Si solar cell, respectively.

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solar cell. This quadratic current component may also result from SCL transport; however, further investigation is required to clarify its nature.

5. Conclusion

In summary, a series resistance extraction scheme which requires no presumed I–V functional form was proposed. With I–V measurement at two illumination levels, all device-level para-meters, including series resistance, shunt resistance, photocurrent and intrinsic I–V characteristics can be determined. This method was applied to analyze a BHJ OSC. We found that, after the removal of the series resistance, a linear hopping current component as well as a quadratic SCL current component can be clearly identified in the extracted intrinsic I–V characteristics. Furthermore, the reconstructed dark I–V is found to differ from the measured dark I–V, revealing the important role of electric field in BHJ OSC.

Acknowledgment

This work is supported by the National Science Council of Taiwan under Grant no. 2120-M-007-004 and NSC97-2628-M-009-016.

Appendix

Combine (1), (8) and (10), we get IphR¼

VD0

RsR

ð15Þ Substitute (1), (2) into (7), we have

VDR¼

ðRtRÞ

Rsh

VDþ ðRsRÞ  f ðVDÞðRsRÞ  Iph ð16Þ

Therefore, one gets f ðVDÞ ¼ VDR ðRsRÞ  ðRtRÞ Rsh ðRsRÞ VDþIph ð17Þ and VD¼ Rsh ðRtRÞ VDR Rsh ðRsRÞ ðRtRÞ f ðVDÞ þ Rsh ðRsRÞ ðRtRÞ Iph ð18Þ

Note also from (4) and (7) Im¼

VDRVD

RsR ð19Þ

Solve Vmfrom (4) and substitute the result as well as (15), (18)

and (19) into (9), we have fR¼ Rsh ðRtRÞ f ðVDÞ Rsh ðRtRÞ Iphþ 1 ðRsRÞ VD0 ð20Þ From (10) Iph¼f ðVD0Þ ðRtRÞ Rsh ðRRsÞ VD0 ð21Þ

Substituting (21) into (20) leads to (11) and substituting (10) and (17) into (11) leads to (12).

References

[1] M. Wolf, H. Rauschenbach, Series resistance effects on solar cells measure-ments, Adv. Energy Convers. 3 (1963) 455–479.

[2] R.J. Handy, Theoretical analysis of the series resistance of a solar cell, Solid-State Electron. 10 (1967) 765–775.

[3] K. Rajkanan, J. Shewchun, A better approach to the evaluation of the series resistance of solar cells, Solid-State Electron. 22 (1979) 193–197.

[4] G.L. Araujo, E. Sanchez, A new method for experimental determination of the series resistance of a solar cell, IEEE Trans. Electron. Dev. 29 (1982) 1511–1513.

[5] T. Easwarakhanthan, J. Bottin, I. Bouhouch, C. Boutrit, Nonlinear minimization algorithm for determining the solar cell parameters with microcomputers, Int. J. Sol. Energy 4 (1986) 1–12.

[6] M. Chegaar, Z. Ouennouchi, A. Hoffmann, A new method for evaluating illuminated solar cell parameters, Solid-State Electron. 45 (2001) 293–296. [7] A. Kaminski, J.J. Marchand, A. Laugier, I–V methods to extract junction

parameters with special emphasis on low series resistance, Solid-State Electron. 43 (1999) 741–745.

[8] E. Radziemska, Dark I-U-T measurements of single crystalline silicon solar cells, Energy Convers. Manage. 46 (2005) 1485–1494.

[9] A. Jain, A. Kapoor, A new approach to study organic solar cell using Lambert W-function, Sol. Energy Mater. Sol. Cells 86 (2005) 197–205.

[10] M. Murayama, T. Mori, Equivalent circuit analysis of dye-sensitized solar cell by using one-diode model: effect of carboxylic acid treatment of TiO2 electrode, Jpn. J. Appl. Phys., Part 1 45 (2006) 542–545.

[11] A. Ortiz-Conde, F.J. Garcia Sanchez, J. Muci, New methods to extract model parameters of solar cells from the explicit analytic solutions of their illuminated I–V characteristics, Sol. Energy Mater. Sol. Cells 90 (2006) 352–361.

[12] K.I. Ishibashi, Y. Kimura, M. Niwano, An extensively valid and stable method for derivation of all parameters of a solar cell from a single current–voltage characteristics, J. Appl. Phys. 103 (2008) 094507.

[13] M. Chegaar, N. Nehaoua, A. Bouhemadou, Organic and inorganic solar cells parameters evaluation from single I–V plot, Energy Convers. Manage. 49 (2008) 1376–1379.

[14] P. Mialhe, A. Khoury, J.P. Charles, Device-related phenomena a review of techniques to determine the series resistance of solar cells, Phys. Status Solidi A 83 (1984) 403–409.

[15] G. Li, et al., in: Exhibition in Solar Power International Conference 2009, Anaheim, California, October 27–29, 2009.

[16] F.C. Krebs, S.A. Gevorgyan, J. Alstrup, A roll-to-roll process to flexible polymer solar cells, manufacture and operational stability studies: models studies, J. Mater. Chem. 19 (2009) 5442–5451.

[17] F.C. Krebs, Fabrication and processing of polymer solar cells: a review of printing and coating techniques, Sol. Energy Mater. Sol. Cells 93 (2009) 394–412.

[18] G. Li, Y. Yao, H. Yang, V. Shrotriya, G. Yang, Y. Yang, Solvent annealing effect in polymer solar cells based on (Poly3 -hexylthiophene) and methanofullerenes, Adv. Funct. Mater. 17 (2007) 1636–1644.

[19] V.I. Arkhipov, P. Heremans, E.V. Emelianova, G.J. Adriaenssens, J. Bassler, Weak-field carrier hopping in disordered organic semiconductors: the effects of deep traps and partly filled density-of-states distribution, J. Phys. Condens. Matter. 14 (2002) 9899–9911.

數據

Fig. 1. The circuit model for solar cells.
Fig. 2 shows the measured I m1 ðV m ) and I m2 ðV m ) as well as dark

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