NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

The Use of 2nd and 3rd Level 

Correlation Analysis for 

Studying Degradation in 

Polycrystalline Thin-Film Solar 

Cells 

D.S. Albin and J.A. del Cueto

National Renewable Energy Laboratory

S.H. Demtsu

Primestar Solar

S. Bansal

GE Global Research Center

Presented at the 35th IEEE Photovoltaic Specialists Conference 

(PVSC '10)

Honolulu, Hawaii

June 20-25, 2010

Conference Paper

NREL/CP-5200-48393

March 2011
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1 

THE USE OF 2ND AND 3RD LEVEL CORRELATION ANALYSIS FOR STUDYING 

DEGRADATION IN POLYCRYSTALLINE THIN-FILM SOLAR CELLS

D. S. Albin1

, J. A. del Cueto1, S. H. Demtsu2, and S. Bansal3

1

National Renewable Energy Laboratory (NREL), Golden, CO USA 2

Primestar Solar, Arvada, CO USA 3

GE Global Research Center, Niskayuna, NY USA

ABSTRACT

The correlation of stress-induced changes in the 

performance of laboratory-made CdTe solar cells with 

various 2nd and 3rd level metrics is discussed. The overall 

behavior of aggregated data showing how cell efficiency 

changes as a function of open-circuit voltage (Voc), shortcircuit current density (Jsc), and fill factor (FF) is explained 

using a two-diode, PSpice model in which degradation is 

simulated by systematically changing model parameters. 

FF shows the highest correlation with performance during 

stress, and is subsequently shown to be most affected by 

shunt resistance, recombination and in some cases 

voltage-dependent collection. Large decreases in Jsc as 

well as increasing rates of Voc degradation are related to 

voltage-dependent collection effects and catastrophic 

shunting respectively. Large decreases in Voc in the 

absence of catastrophic shunting are attributed to 

increased recombination. The relevance of capacitancederived data correlated with both Voc and FF is discussed. 

INTRODUCTION

Polycrystalline CdS/CdTe thin-film solar cells have 

demonstrated small-area, laboratory efficiencies of 16.5% 

[1]. The highest published efficiency for CdTe modules is 

11.1% [2]. In addition to considerable research 

addressing efficiency, recent work has focused on 

understanding the durability of these thin-film 

semiconductor devices. The effects of back contact and 

processes on cell durability have been discussed for 

example in [3-5]. More recently, the detrimental effects of 

localized shunts [6], the general effects of film micrononuniformities [7], and the effect of cell-fabrication 

processing through formal design-of-experiments 

methodologies [8] have been reported.

The basic structure of a thin-film CdS/CdTe solar cell is 

that of a glass superstrate design in which light passes 

through a conducting/insulating (buffer) oxide film layer

stack deposited on glass. Tin oxide (SnO2) is the most 

common transparent conducting oxide (TCO), though 

there is considerable interest in the stannate compounds 

(Cd2SnO4 and SnZnOx) for their superior optical properties 

[1]. 

Once transmitted through the glass/TCO/buffer 

superstrate, light is absorbed in the n-CdS/p-CdTe 

heterojunction structure, which provides the field 

necessary for charge separation. A back contact structure 

completes the cell. A schematic of this basic design is 

shown in Fig. 1.

Figure 1 Basic CdS/CdTe solar cell design. 

To ascertain durability, cells were fabricated and then 

exposed to 1-sun illumination under open-circuit, Voc, bias 

and acceleration temperatures of 60 – 120 oC for times 

exceeding 1000 hours [9]. Two dominant degradation 

mechanisms were identified in the temperature range 

studied. As shown in Fig. 2(a), from 60-80 oC, an 

activation energy of 2.94 eV was measured and was 

attributed to S-outdiffusion from the CdS layer into the 

CdTe based on a reported value of 2.8 eV for bulk 

diffusion of S in CdTe [10]. This assertion was also 

supported by the observation of Kirkendall voids in the 

CdS layer. In the temperature range 100-120 oC, an 

activation energy of 0.63 eV was determined, in good 

agreement with the reported value of 0.67 eV for Cu 

diffusion in CdTe [11].

Figure 2 Degradation activation energies (a) and how 

the linear correlation (R2

) of ∆η% changes with Voc, 

Jsc, and FF as a function of stress temperature (b).

(a) (b)
2 

Since cell efficiency, η%, is determined by the equation:

inc

oc JV sc FF

φ η

×× % =

(where Φinc, the incident power density, is typically 

normalized to a solar value of 100 mW/cm2), a comparison 

of the linear correlation coefficients of ∆η% versus 

changes in Voc, short-circuit current density, Jsc, and fill 

factor, FF, during stress testing as a function of stress 

temperature was performed. This analysis is shown in 

Fig. 2(b). The moderate correlation of η% with Jsc seen at 

lower stress temperatures is due to reduced optical 

attenuation associated with S-outdiffusion from the CdS. 

The most important variable affecting η% at all 

temperatures, approaching an ideal value at 120 oC, was 

FF. 

Similar trends are observed when looking at aggregate 

cell stress data. Figure 3 is a plot of the percent change in 

efficiency (del Eff) vs. changes in Voc, Jsc, and FF (del Voc, 

del Jsc, and del FF, respectively) obtained in a set of 2052 

measurements performed on CdTe devices stress tested 

at NREL with no regard to subtle changes in processing or 

stress conditions.

Figure 3 del Eff vs. del Voc, Jsc, and FF. 

The strong correlation of decreasing η% with both Voc and 

FF is obvious, as is the general lack of correlation with Jsc. 

These figures also show possible annealing effects 

resulting in improved performance through increasing FF 

as well as the general observation that Jsc often increases 

during stress testing. 

In this paper, through a combination of modelling and the 

examination of past data, we hope to provide a better 

understanding of why these correlations are observed 

between performance and changes in Voc, Jsc, and FF.

J-V Curve Modeling 

PSpice modeling provides an easy method by which to 

understand basic solar cell behavior. In Fig. 4, a parallel 

combination of forward-biased diodes is used to 

independently simulate recombination currents in the 

quasi neutral (JQNR) and space charge (JSCR). The back 

contact behavior is represented by the parallel 

combination of a reverse-biased diode (Jb) and shunt 

conductance (1/Rb) that was previously used to model

performance characteristics observed in the first quadrant 

[12]. The theoretical basis for the two-diode model as a 

general expression for the current produced in a solar cell 

can be found in Ref. [13].

Figure 4 Two-diode solar cell discrete circuit model. 

When the barrier height, Eb, associated with the back 

contact is low (Eb < 0.5 eV, Jb >100 mA/cm2

), the back 

contact impact on 4th quadrant behaviour, is minimized 

[12] such that the current-density, J, and voltage, V, of the 

solar cell is represented by:

(2)

where JSCR, JQNR, and Jph are the recombination currents 

and photo generated current respectively, while Rs and Rsh

represent series and parallel (shunt) resistance losses. 

JSCR and JQNR are further represented by the following: 

(3)

(4)

where J01 and J02 are further dependent upon minority 

carrier transport properties.

(a)

(b)

(c)

ph

sh

s

SCR QNR J

R

V JR JJJ −













 − ++=





 



 = − −

1

)(

01

kT

Vq JR

QNR

s

eJJ





 



 = − −

1 2

)(

02

kT

Vq JR

SCR

s

eJJ
3 

In the fourth quadrant, maximum power output is achieved 

by maximizing the term, Jph, and minimizing the first three 

terms in (2), often referred to collectively as the “forward” 

current. Each of the forward current terms contributes to 

decreased cell performance. It should be noted that the 

recombination currents, JQNR, and JSCR are themselves 

dependent on resistive effects as shown in (3) and (4).

Using the J-V curve measured for a 14.4% cell fabricated 

at NREL, the percent that each parameter contributes to 

the forward current (and thus loss) in the power quadrant 

was determined and is shown in Fig. 5. This calculation 

used a fit-determined value of 3 Ω⋅cm2 for Rs which is a 

reasonable upper value observed during stress testing of 

these cells [9].

Figure 5 Forward current contributions to loss. 

As seen in this figure, recombination occurs mostly in the 

space charge near Voc where recombination in the quasi 

neutral region, i.e., between the depletion width and back 

contact, begins to dominate. Note that resistive 

contributions for both effectively go to zero at Voc where J 

= 0 in equations (3) and (4). The voltage, V, at which JQNR

exceeds JSCR, is given by the relation:





 



 = 01

02 ln 2

J

J

q

kT V

(5) 

In high efficiency cells (smaller J02) with higher Voc values, 

the J01 term has a strong effect on Voc. It is primarily for 

this reason that a two-diode model is preferred over onediode models in which an “average diode factor (typically 

between 1 and 2) is usually determined. In general, if Voc

is less than 800 mV, the importance of J01 is reduced.

RESULTS AND DISCUSSION

Cell degradation during stress can be modeled by 

changing the discrete elements of Fig. 4. Increased 

recombination (J01, J02), resistive losses (increasing Rs, 

decreasing Rsh), and degraded backcontacts (decreasing 

Jb due to an increase in Eb, increased Rb) all cause 

performance to decrease. These effects on both the J-V 

characteristics of cells, and performance correlations 

similar to those shown in Fig. 3 were determined using 

PSpice as a modeling tool. For the sake of brevity, only 

important results explaining the behavior shown in Fig. 3 

will be presented. An initial “baseline” cell with J01=3e-16

A/cm2

, J02=1e-09 A/cm2, Rsh = 200 KΩ⋅cm2, and Rs = 3 

Ω⋅cm2 was used as the point from which degradation in 

Voc, Jsc, FF, and η% were determined. 

Back contact degradation

A commonly observed degradation characteristic of CdTe 

solar cells is the formation of “roll-over” in the 1st quadrant. 

As mentioned previously, this effect can be modeled 

through introducing a contact diode leakage current, Jb

which is inversely related to the back contact barrier height 

Eb [12]. Graphically, Jb can be estimated as the current at 

which the J-V curve “rolls over” in the 1st quadrant. For 

our simulations, a “baseline” value of Eb = 0.56 eV (which 

equates to Jb~10 mA/cm2

) and a value of Rb = 0. When 

Rb=0, the barrier is effectively absent in the 1st quadrant 

however it’s appearance with degradation (as well as with 

low-temperature J-V measurements) confirms it’s 

presence. Simulations were performed with Eb increasing 

to 0.68 eV (Jb~0.1 mA/cm2

) and Rb increasing as high as 

100 Ω⋅cm2

. 

The correlation of del Eff vs. del Voc, del Jsc, and del FF 

assuming a nominal setting of Rb equal to 5 Ω⋅cm2 is 

shown in Figure 6.

Figure 6 Efficiency change with contact barrier 

increase. 

Figure 6 (and subsequent figures) has been scaled to 

allow comparison with Fig. 3. As seen in Fig. 6, a change 

in the back contact diode behavior only affects efficiency

through FF. As the contact degrades, it affects neither Voc

nor Jsc. The increase in Eb shown in Fig. 5 is a worse-
4 

case estimate as 1st quadrant behavior typically does not 

show such roll-over. The decrease in η% shown in Fig. 6 

is thus not large enough to explain the η% vs. FF behavior 

shown in Fig. 3(c). 

Resistive effects (Rsh, Rs) 

Decreasing shunt resistance, Rsh, is a strong root cause 

for the behavior shown in Fig. 3(c). Rsh begins to affect FF 

as it approaches ~ 1000 Ω⋅cm2

, and Voc and Jsc at a level 

closer to 100 Ω⋅cm2

. The correlation of del Eff vs. del Voc, 

del Jsc, and del FF as Rsh decreases is shown in Figure 7.

Figure 7 Efficiency change with decreasing Rsh. 

Values of Rsh as low as 50-100 Ω⋅cm2 can be observed 

during cell stress testing and can account for the large 

performance drops observed in Fig. 3. In cases of 

extreme shunting, Jsc can degrade by as much as 10%. 

The sharp increase in Voc degradation occurring in 

extreme shunting (what might be referred to as 

catastrophic degradation) is also reflected in Fig. 3(a) 

where the rate of Voc drop begins to increase greatly at a 

decreased performance level of ~40%.

A similar plot of the effect of series resistance, Rs, is 

shown in Fig. 8. 

Figure 8 Efficiency change with increasing Rs. 

Similar to Rsh, Rs can have a strong effect on FF, but 

differs in that it has no effect on Voc, and only a very slight 

impact on Jsc at very high degradation levels. In practice, 

Rs is not observed to increase much from baseline levels 

and thus Rsh is believed to be more important in 

determining FF during stress testing.

Recombination

The effect of back contact and resistive degradation 

shown in Figs. 6, 7, and 8 do not adequately explain 

features observed in Fig. 3(a) and 3(b). Of these, only 

very large catastrophic decreases in Rsh (Fig. 7) can 

explain some of the large changes in Voc and Jsc shown in 

Fig. 3. However, such changes are frequently observed in 

cells that do not exhibit “catastrophic” shunting.

Increased recombination was modeled by systematically 

increasing J02 from its baseline value of 1e-09 to 1e-06

A/cm2

. The corresponding correlation plot is shown in Fig.

9.

Figure 9 Efficiency change with increasing J02. 

An order magnitude increase in recombination current can 

account for most of the observed decrease in Voc shown in 

Fig. 3(a) as well as a portion of the decrease in FF seen in 

Fig. 3(c). The fact that Voc degrades during stress testing 

cells is a clear indicator of the importance in mitigating 

recombination in order to achieve stable cells.

As mentioned previously, significant decreases in Jsc have 

been observed where catastrophic shunting is not 

observed. Yet, the modeling performed with PSpice 

cannot explain this. The reason for this is due to the 

inability of PSpice to model a well-known effect in 

polycrystalline thin film cells known as voltage-dependent 

collection [14].

In PSpice simulations, this additional loss mechanism can 

be captured by properly adjusting the resistive model 

parameters. At V=0, the slope of the J-V curve can be 

used to accurately determine Rsh. The 1st quadrant 

behavior can then be used to adjust Rs in the absence of 

roll-over. If the latter is present, Rs can still be determined 

by fitting the curve between the maximum power point, 

Pmax and Voc. In doing so, voltage-dependent collection is 
5 

revealed as the difference between the model fit, and the 

actual J-V curve. Figure 8 is an example of this procedure 

applied to a cell where after 722 h of stress, η% had 

decreased 33% with a corresponding decrease in Jsc of 

4.8%. In this example, neither Rsh (fixed by dV/dJ at V=0, 

or Rs (fixed by dV/dJ in the 1st quadrant) could adequately 

model the drop in Jsc. A comparison between the best 

model fit and the actual J-V curve is shown in Fig. 10. 

Figure 10 Voltage-dependent collection

The difference observed between the model-determined fit 

and the actual J-V curve after 722 h of stress (T = 125 oC, 

Voc bias, 1-sun illumination) is believed to reveal an 

additional voltage-dependent loss mechanism (not present 

at t=0) beyond recombination as represented by equations 

(3) and (4). No iteration of the discrete element 

parameters of the model shown in Fig. 4 can replicate the 

actual J-V curve. For the cell shown in Fig. 10, the 

additional voltage-dependent degradation introduced into 

the cell by stress testing corresponds to approximately a 

0.96% and 3.78% increase in Jsc and FF loss, respectively.

3rd-Level Metric Correlations

The previous approach to understanding degradation 

requires little more than measuring and modeling details 

associated with J-V curves. In some measure, the 

incorporation of recombination terms provides a slight 

foray into more complex, higher-order parameters often 

extracted by sophisticated metrology tools and techniques. 

Unfortunately, the latter tools are often expensive and not 

readily available to those requiring many measurements in 

order to reach statistically viable conclusions. The 

techniques themselves often suffer from being too timeconsuming or are entirely destructive or requiring ambient 

conditions that might introduce degradation not 

characteristic of field degradation in the cell. In order to 

increase our ability to probe for changes occurring in cells 

and modules during stress testing, we have recently 

developed and reported on the use of bi-directional 

voltage scans during capacitance measurements to both 

understand degradation [15] as well as to study and 

improve upon module stabilization procedures [16]. 

In Ref. [15], the relative durability of CdS/CdTe cells grown 

on either a bi-layer SnO2 or cadmium/zinc stannate 

(CTO/ZTO) superstrate was evaluated. Cells grown on 

the latter typically exhibit greater Voc and FF degradation 

than cells grown on the more common SnO2-coated 

substrate. Using the two-direction, voltage scan method, 

capacitance-voltage (C-V) hysteresis in cells was 

determined as a function of stress time. Hysteresis is 

arbitrarily defined as the difference in Wd at V = 0 between 

reverse and forward direction scans, i.e., Wd,rev – Wd,fwd . 

Fig. 11 summarizes the variation of hysteresis with stress 

time as well as the correlation of both Voc and FF with 

hysteresis determined in this study.

Figure 11 Variation of C-V hysteresis with stress time 

(a) for cells based on TCO-type and subsequent 

correlations of (b) Voc and (c) FF with hysteresis.

The increased hysteresis shown in Fig. 11(a) was 

paralleled by a decrease in performance during stress. 

The correlation of hysteresis with both Voc and FF (shown 

in Fig. 8(b) and 8(c), respectively) is also apparent. The 

correlation coefficient, R2

, of Voc with hysteresis for 

0.8

0.7

0.6

0.5

0.4

0.3

Wd,rev - Wd,fwd (um)

0 20 40 60 80 100

Stress Time, hrs

 CTO/ZTO #1

 CTO/ZTO #2

SnO2 #1

SnO2 #2

(a)

0.78

0.79

0.8

0.81

0.82

0.83

Voc (volts)

0.2 0.4 0.6 0.8

Wd_rev - Wd_fwd

(b)

60

65

70

FF %

0.2 0.4 0.6 0.8

Wd_rev - Wd_fwd

(c)
6 

CTO/ZTO cells #1 and #2, and SnO2 cells #1 and #2, were 

0.98, 0.46, 0.75, and 0.82, respectively. The same values 

for FF were 0.99, 0.58, 0.63, and 0.87, respectively. 

Similar analysis performed without considering hysteresis, 

i.e., based on either Wd,fwd or Wd,rev separately were not 

nearly as high and tended to show non-monotonic 

behavior.

As discussed in Ref.[16], finished thin-film modules 

(including those based on Cu(In,Ga)Se2) exhibit various 

degrees of hysteresis. The trends shown in Fig. 11 are in 

some cases replicated by the observations made on 

modules undergoing various tests to determine 

procedures to achieve stabilization prior to benchmarking 

performance. Figure 12 shows for example, the C-V 

hysteresis curves associated with two different completed 

CdTe modules after various conditioning exercises. More 

detail on the latter procedures can be found in Ref.[17].

Figure 12 C-V hysteresis observed in finished 

production CdTe modules after various conditioning 

tests.

The greater hysteresis, and movement of the C-V 

determined Wd observed in module #X1 correlated with a 

performance degradation of ~8.3%. For module #X7, 

module performance decreased ~ 1.0% during test. 

ACKNOWLEDGEMENT

The authors wish to thank our various industrial partners 

through whom we work collectively to understand and 

improve the fundamental durability of CdTe cells. We also 

acknowledge the support of the U.S. Department of 

Energy under Contract No. DOE-AC36-08G028308.

REFERENCES

[1] X. Wu, “High efficiency polycrystalline CdTe thin-film 

solar cells,” Solar Energy 77, 2004, pp. 803-814.

[2] Publically available at www.firstsolar.com 

[3] K. Dobson, I. Visoly-Fisher, G. Hodes, and D. Cahen, 

“Stabilizing CdTe/CdS Solar Cells with Cu-containing 

Contacts to p-CdTe,” Advanced Materials, 13 (9), 2001, 

pp. 1495-1499.

[4] C. Corwine, A. Pudov, M. Gloeckler, S. Demtsu, and J. 

Sites, “Copper inclusion and migration from the back 

contact in CdTe solar cells, “Sol. Energy Mater. Sol. 

Cells, 82, 2004, pp. 481- 489.

[5] D. Albin, D. Levi, S. Asher, A. Balcioglu, “Precontact 

surface chemistry effects on CdS/CdTe solar cell 

performance and stability,” Proc. 28th IEEE 

Photovoltaics Specialists Conference, New York, 2000, 

pp. 583-586.

[6] T. McMahon, T.J. Berniard, and D.S. Albin, “Nonlinear 

shunt paths in thin-film CdTe solar cells,” J. Appl. Phys., 

97, 2005, pg. 054503.

[7] V.G. Karpov, A.D. Compaan, and D. Shvydka, “Effects 

of nonuniformity in thin-film photovoltaics,” Appl. Phys. 

Lett., 80 (22), 2005, pp. 4256-4258.

[8] D.S. Albin, S.H. Demtsu, and T.J. McMahon, “Film 

thickness and chemical processing effects on the 

stability of cadmium telluride solar cells,” Thin Solid 

Films, 515, 2006, pp. 2659-2668.

[9] D.S. Albin, “Accelerated Stress Testing and Diagnostic 

Analysis of Degradation in CdTe Solar Cells,” in 

Reliability of Photovoltaic Cells, Modules, Components, 

and Systems, edited by Neelkanth G. Dhere, 

Proceedings of SPIE Vol. 7048 (SPIE, Bellingham, WA, 

2008) 70480N.

[10] B. McCandless, M. Engelmann, and R. Birkmire, 

“Interdiffusion of CdS/CdTe thin films: Modeling x-ray 

diffraction line profiles,” J. Appl. Phys. 89(2), 2001, pp. 

988-994.

[11] H. Woodbury and M. Aven, “Some Diffusion and 

Solubility Measurements of Cu in CdTe,” J. Appl. Phys. 

39(12), 1968, pp. 5485-5488.

[12] Demtsu, S., Sites, J., "Effect of back-contact barrier 

on thin-film CdTe solar cells," Thin Solid Films 510, 

2006, pp. 320-324.

[13] Luque, A., Hegedus, S., eds., [Handbook of 

Photovoltaic Science and Engineering], John Wiley & 

Sons, in particular Gray, J., Chapter 3 "The Physics of 

the Solar Cell,", (2003).

[14] S.S. Hegedus and W.N. Shafarman, “Thin-Film Solar 

Cells: Device Measurements and Analysis,” Prog. 

Photovolt: Res. Appl. 2004; 12:155-176.

[15] D.S. Albin, R.G. Dhere, S.C. Glynn, et al., 

“Degradation and capacitance-voltage hysteresis in 

CdTe Devices,” Proceedings of SPIE Vol. 7412, 74120I 

(2009). 

[16] del Cueto, J.A., Deline, C.A., Albin, D.S., et al., 

“Striving for a standard protocol for preconditioning or 

stabilization of polycrystalline thin film photovoltaic 

modules,” Proceedings of SPIE Vol. 7412, 741204 

(2009).

Vb (V/cell)

Wd (microns)

-0.6 -0.4 -0.2 0.0 0.2 0.4

baseline

Precon 36 kWh/m2 out

dark thermal anneal

1st 24 h bias dark expo 216 h bias dark expo

CdTe X1: Fwd Biased Dark Expo

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Vb (V/cell)

Wd (microns)

-0.6 -0.4 -0.2 0.0 0.2 0.4

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.6

baseline

Precon 36 kWh/m2 out

dark thermal anneal

1st 24 h bias dark expo 216 h bias dark expo

CdTe X7: Fwd Biased Dark Expo
7 

[17] del Cueto, J.A., Deline, C.S., Rummel, S.R., and 

Anderberg, A., “Progress Toward a Stabilization and 

Preconditioning Protocol for Polycrystalline Thin-Film 

Photovoltaic Modules”, [ibid].
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14. ABSTRACT (Maximum 200 Words)

The correlation of stress-induced changes in the performance of laboratory-made CdTe solar cells with various 2nd 

and 3rd level metrics is discussed. The overall behavior of aggregated data showing how cell efficiency changes as 

a function of open-circuit voltage (Voc), short-circuit current density (Jsc), and fill factor (FF) is explained using a twodiode, PSpice model in which degradation is simulated by systematically changing model parameters. FF shows the 

highest correlation with performance during stress, and is subsequently shown to be most affected by shunt 

resistance, recombination and in some cases voltage-dependent collection. Large decreases in Jsc as well as 

increasing rates of Voc degradation are related to voltage-dependent collection effects and catastrophic shunting 

respectively. Large decreases in Voc in the absence of catastrophic shunting are attributed to increased 

recombination. The relevance of capacitance-derived data correlated with both Voc and FF is discussed.

15. SUBJECT TERMS

CdTe solar cell; degradation; stress

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