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IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 4, NO. 4, JULY 2014
Correlation of Built-In Potential and I–V Crossover
in Thin-Film Solar Cells
James E. Moore, Member, IEEE, Sourabh Dongaonkar, Member, IEEE, Raghu Vamsi
Krishna Chavali, Student Member, IEEE, Muhammad Ashraful Alam, Fellow, IEEE,
and Mark S. Lundstrom, Fellow, IEEE
Abstract—Thin-film solar cells often show a crossover between
the illuminated and dark I–V characteristics. Several device specific reasons for crossover exist and have been discussed extensively.
In this paper, we show that a low contact-to-contact built-in potential can produce a voltage-dependent photocurrent that leads to
I–V crossover at a voltage that is almost exactly the device built-in
potential. This mechanism can produce crossover in the absence
of carrier trapping or recombination. It can be a contributing factor to crossover, but when an anomalously low contact-to-contact
built-in potential exists, it can be the dominant factor. Using numerical simulations, we examine a variety of model solar cell structures with low contact-to-contact built-in potential and show a
strong correlation of the crossover and built-in potential voltages.
These simulations also suggest that a plot of the illuminated minus
dark current may help identify when a low Vbi is limiting device
performance.
Index Terms—Built-in potential, device modeling, injection current, photogenerated current, superposition.
I. INTRODUCTION
HE superposition principle commonly used to describe the
current in crystalline solar cells [1] defines the total current
measured under illumination as
T
Jlight (V, G) = Jinj (V ) + Jgen (G)
(1a)
where Jinj is an injection current, which is typically assumed
to be equal to the measured dark current, and Jgen is the photogenerated current, which is typically assumed to be voltage
independent and equal to the measured short-circuit current.
Electrical measurements of thin-film solar cells, however, often show a crossover point between the illuminated and dark
I–V characteristics, which cannot occur according to (1a). At
the crossover voltage Vx , the dark and illuminated currents are
equal (see Fig. 1).
Although I–V crossover is not always as dramatic as illustrated in Fig. 1, it can, in some cases, lower open-circuit voltages
and fill factors [2]–[5]. Moreover, such a crossover complicates
Manuscript received October 11, 2013; revised March 4, 2014; accepted
March 18, 2014. Date of publication May 6, 2014; date of current version June
18, 2014. This work was supported by the SRC Energy Research Initiative (ERI)
and the SunShot Program funded by the U.S. Department of Energy.
The authors are with the School of Electrical and Computer Engineering,
Purdue University, West Lafayette, IN 47907 USA (e-mail: moore32@purdue.
edu; sourdon@gmail.com; c.raghuvamsi@gmail.com; lundstro@purdue.edu;
alam@purdue.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JPHOTOV.2014.2316364
Fig. 1. Experimental data showing I–V crossover in a CdTe solar cell with a
Schottky back contact [4], an a-Si solar cell and a CZTSSe solar cell similar
to those reported on in [26]. Solid lines show dark I–V and dashed lines show
illuminated I–V.
the reciprocity relation between absorption and emission in a solar cell [6]; therefore, it is important to understand the physical
origin of the crossover phenomenon.
Several different mechanisms can cause J–V crossover and
these have been extensively discussed in the literature (e.g.,
[2]–[5]). The key message of this paper is that a low overall
device built-in voltage Vbi,d can produce a voltage-dependent
photocurrent and cause I–V crossover. Here, Vbi,d refers to the
contact-to-contact total built-in potential difference of the solar
cell, while Vbi,j refers to the built-in potential of the individual
junction within the solar cell. For example, the presence of a
Schottky contact does not change Vbi,j , but it lowers Vbi,d . Surprisingly, we will show that Vbi,d related crossover can occur in
many different device structures, even in the absence of traps
and bulk recombination. It provides a simple, physically transparent explanation for crossover due to a Schottky back contact
and for one of the factors that are contributing to cross-over in
p-i-n cells. It can be the dominant factor for cells with anomalously low Vbi,d . This paper highlights the need to consider low
Vbi,d as a possible cause of crossover in solar cells.
II. BACKGROUND AND APPROACH
Physical mechanisms that can cause superposition failure
have been discussed extensively in the literature [2]–[6], but
most of these studies focus on a specific technology. Fig. 1
shows I–V crossover in three different types of solar cells. In
this paper, we will describe a physical mechanism, low contactto-contact built-in potential that can produce I–V crossover and
will discuss how this mechanism can contribute to the observed
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MOORE et al.: CORRELATION OF BUILT-IN POTENTIAL AND I–V CROSSOVER IN THIN-FILM SOLAR CELLS
1139
I–V crossover for the three different types of cells illustrated
in Fig. 1. This includes ink-based CZTSSe, for which recent
work shows an experimentally measured crossover voltage that
is correlated with Vbi,d and independent of light wavelength
and intensity [7]. Understanding these measurements is one key
motivation for the study presented in this paper.
Explanations for I–V crossover due to voltage-dependent current collection have been given previously [2], [3], focusing
mainly on devices with short diffusion lengths. Similarly, I–V
crossover is observed in a-Si solar cells and has been found
to be directly related to the built-in voltage of the p-i-n junction [5]. General equations describing the mechanism have been
given [8], [9], but have only recently been noticed and applied to
solar cells [10]. The presence of a Schottky barrier back contact
produces two diodes in series [11]. Crossover in the presence
of a Schottky barrier has been analyzed in terms of this twodiode model [4], [12]. Several papers have also noted that the
photoexcited deep trap levels in the window layer can change
the electrostatics of the device with illumination that lead to
crossover in Cu2 S and CIGS [13]–[16]. It has been shown that
this effect can be identified by a change in the I–V characteristic
under long- or short-wavelength light [13], [14].
To examine the physics of I–V crossover, (1a) must be generalized to
with low contact-to-contact Vbi,d demonstrate that the same effect can occur in a wide variety of solar cell structures—when
the contact-to-contact Vbi,d is low. In Section V, we discuss the
validity of the assumptions made in the previous sections, and
we conclude in Section VI that Vbi,d related crossover should
be considered as a possible cause when diagnosing crossover in
solar cells.
Jlight (V, G) = Jinj (V, G) + Jgen (V, G) .
Consider first an n-i-p homostructure as shown in Figs. 2(a)
and (b). For this structure, I–V crossover occurs because of a
voltage-dependent photocurrent, which can be readily understood in terms of carrier partitioning between two contacts.
As shown in Fig. 2(a), under short-circuit conditions, most
photogenerated carriers in the i-layer are swept out to the correct
contacts by the built-in electric field. However, at the crossover
voltage Vx , the bands are nearly flat, and there is an equal
probability that photogenerated carriers will diffuse to either
of the two contacts [see Fig. 2(b)]. This equal partitioning of
the photogenerated carriers between the two contacts causes the
photogenerated current to be zero at the built-in potential where
crossover occurs. For even larger biases, the electric field and
the photocurrent change direction. Electrons are now collected
by the p+ contact and holes by the n+ contact. These effects
are well documented experimentally [7], [10].
Next, we examine an N-i-p heterostructure with a wide
bandgap N+ layer, which creates a blocking barrier in the valence band [see Fig. 2(c)]. Solar cells such as CIGS and CdTe
often have a wide bandgap window layer at the front contact
similar to this structure [20], and therefore, this model can help
us understand crossover in these types of cells.
In this model cell, carrier partitioning between the two contacts occurs for electrons, but the holes can only exit from the
p+ contact [see Fig. 2(c)]. The photocurrent will be voltage
dependent, but it cannot reach zero unless all electrons and
holes exit from the p+ contact [see Fig. 2(d)]. However, as
Fig. 2(d) also shows, photogenerated holes will accumulate at
the heterojunction under large forward bias. The internal electric fields will, therefore, be different under illumination and
in the dark, and therefore, the diode injection current becomes
generation dependent. For this structure, the crossover voltage
(1b)
We find it useful to plot
ΔJ = Jlight − Jdark = {Jinj (V, G) − Jdark } + Jgen (V, G)
(1c)
which shows that I–V crossover can occur in two ways. First,
the diode injection current at a given forward bias may change
under illumination. In this case, the first term in brackets in (1c)
is dominant. Second, the photogenerated current may have a
strong voltage dependence and go to zero at V = Vx . In this
case, the second term of (1c) is dominant. Both terms may play
a role in crossover; depending on which one is dominant, the
plot of (1c) will display different characteristic shapes. Using a
specially modified version of the numerical simulation program,
ADEPT [17], [18], we will perform the current decomposition
of (1b) exactly and examine the relative contributions of the two
terms for various solar cell structures. Additional simulations
were also done using SCAPS-1D [19].
The objectives of this paper are to demonstrate a close connection between the crossover and device built-in voltages for
a variety of solar cells and to discuss the physics of this close
correlation. This study provides a simple, unifying description
of effects that have been considered to be distinct. We examine
several different model structures with detailed, numerical simulations. We begin in Section III by analyzing a simple n-i-p
structure for which I–V crossover can be simply explained by
a voltage-dependent photocurrent. We then examine the effect
of a wide bandgap n layer with a hole blocking heterojunction
(because this type of structure is common [20]). This example
shows how the shape of the ΔJ characteristic is related to the
internal physics of the cell. In Section IV, we examine other solar cell structures and show that simulations of model structures
III. N-I-P MODEL STRUCTURES
In this section, we consider n-i-p homo- and heterostructures,
as it allows us to develop an analytic model that illustrates the
basic physics of crossover associated with the built-in potential. For simplicity, we initially assume no trap states within
the bandgap, no bulk recombination, and uniform photogeneration (these nonidealities will be addressed in Section V). We
will initially assume that the injection current term in (1c) is
negligible, although we will later show that it is important in
the heterojunction case. Although other effects such as short
diffusion lengths will most likely affect superposition failure in
actual solar cells, we show that Vbi related crossover can occur,
even in this ideal case.
A. Physics of I–V Crossover
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IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 4, NO. 4, JULY 2014
Fig. 2. Numerically simulated energy bands for (a) n-i-p homojunction at short circuit, (b) n-i-p homojunction at crossover, (c) n-i-p heterojunction with valence
band offset at short circuit, and (d) n-i-p heterojunction with valence band offset at crossover.
will be determined by both a bias-dependent photocurrent and
a generation-dependent injection current.
The qualitative physics of these two model cells can be described quantitatively by analytical analysis and by full numerical simulation.
B. Analytic Treatment of n-i-p Structures
Analytic expressions for the two terms in (1b) have been derived for an insulator with two contacts in [9] and shown to
apply to an a-Si n-i-p solar cell in [10]. These expressions are
summarized in Appendix A. The resulting expressions are found
to be in agreement with those presented in [8]; however, separation of injection and photocurrent provide additional insight
into the physical mechanisms behind I–V crossover. At voltages
significantly below Vbi,d , these equations show that the device
behaves according to (1a). As the voltage approaches Vbi,d , the
generation current goes to zero for the reason summarized in
Figs. 2(a) and (b). In this case, Vx and Vbi,d are therefore identical. The C–V measurements, which can normally be used to
estimate Vbi,j do not work for n-i-p cells, and therefore, this
observation offers a novel alternative for measuring n-i-p solar
cells such as a-Si. For V > Vbi,d , the Jinj equation reduces to
the resistor current equation Jinj = (nqμn + pqμp )E, and Jgen
changes direction, as was explained in the previous section.
Fig. 3. Comparison of analytical (symbols) and numerically simulated photocurrent (lines) for n-i-p homojunction and heterojunction. The difference in
ΔJ is due to additional band bending induced by photogenerated charge accumulation, which is not accounted for in the analytic expression.
The symbols plotted in Fig. 3 (crosses) show the analytical
result for the voltage-dependent generation current of the n-i-p
cell. The generation current goes to zero at Vbi,d , and crossover
occurs. If the n and p regions were highly doped, Vbi would
be large, and Jinj Jgen at the crossover point. Open-circuit
conditions would occur well below Vbi,d , where the generation
current is approximately constant, and superposition would apply. When Vbi,d is small, however, crossover occurs at relatively
low injection current, i.e., near Voc .
MOORE et al.: CORRELATION OF BUILT-IN POTENTIAL AND I–V CROSSOVER IN THIN-FILM SOLAR CELLS
Analytic expressions can be similarly derived for the structure
of Fig. 2(c) with a hole blocking heterojunction (the equations
are presented in Appendix A). The injection current is the same
as for the n-i-p homojunction case, except the term for holes vanishes, since the hole current is blocked by the valence band barrier (ignoring bulk recombination). In this case, however, Jgen
is not symmetric about Vbi,d but is asymptotic to zero at large
forward bias (see Fig. 3 circles). This occurs because at large forward bias, all electrons and holes leave from the p+ contact, and
therefore, the generation current approaches zero. Because Jgen
never reaches zero, crossover should not occur, but this analysis
does not account for the change in electric field due to hole accumulation, as indicated in Fig. 2(d). The added positive charge
from this accumulation acts as an additional forward bias on the
junction and allows crossover to occur. An analytical discussion
of this accumulation effect is also presented in Appendix A.
To understand why the current partitioning limited crossover
is determined by the contact-to-contact potential Vbi,d (rather
than the junction potential Vbj,j ), consider, for example, a heterojunction formed by two intrinsic semiconductors with an
energy barrier of Δχ at the interface due to different material
electron affinities, similar to the HOMO/LUMO levels of an
organic photovoltaic cell. Such a device will have no Vbi,j contribution from doping, yet the energy barrier provides a means of
partitioning carriers between contacts [21]. Crossover may still
occur, therefore, when the device is biased above Vbi,d = Δχ.
C. Numerical Simulation of n-i-p Structures
The analytic calculations that are discussed in Section III-B
support the qualitative picture of crossover presented in
Section III-A and can be confirmed by full numerical simulations. The decomposition of (1b) can be done exactly using
numerical simulation. All simulation parameters, including mobility, lifetime, layer thickness, bandgap, and doping concentration can be found in Appendix B. First, the illuminated I–V
characteristic Jlight (V, G) is computed by solving the semiconductor equations numerically. At each bias, we then turn OFF
the illumination and compute the injection current Jinj (V, G)
by using the self-consistent electrostatic potential that was computed under illumination [22], [23]. The photogenerated current
Jgen (V, G) is then computed from (1b). The numerical results
can then be compared with the analytical results presented as
symbols in Fig. 3.
It should be noted that the efficiencies of the simulated structures with low Vbi,d tend to be below 12%. Although this would
be considered quite low performance for a mature solar cell
technology, it would not be unreasonable to find these efficiencies in new technologies still under development. We, therefore,
note that the theory presented in this paper would be useful to
characterize experimental solar cells but would not necessarily
be relevant to mature technologies such as c-Si.
As shown in Fig. 3, the analytic results for Jgen (symbols) and the numerical results (solid lines) compare well.
In addition, shown in Fig. 3 is the numerically computed
ΔJ = Jlight − Jdark . For the n-i-p homostructure, Jgen (dotted line) and ΔJ (solid line) are identical up to the crossover
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voltage. The crossover is, therefore, almost entirely due to the
voltage-dependent photogenerated current. The crossover voltage is exactly equal to the built-in potential, as expected. For
the case with the hole blocking heterojunction, charging effects
make {Jinj (V, G) − Jdark } > 0 in (1c). As a result, crossover
does occur, due to both terms in (1c). In this case Vx is close to,
but not exactly equal to, Vbi,d .
We have shown by physical arguments, with analytic calculations, and by numerical simulations, that I–V crossover in n-i-p
cells is directly related to the built-in voltage. Surprisingly, as
we will show next, this can also be the case in very different
types of solar cell structures.
IV. N-P MODEL STRUCTURES
In the previous section, it was shown that the voltage at which
I–V crossover occurs is equal to Vbi,d for an ideal n-i-p homostructure and at a voltage close to Vbi,d for the case with a
heterojunction that blocks holes. In this section, we will look at
numerical simulations of the following three additional device
structures: 1) a symmetrically doped n-p junction [see Figs. 4(a)
and (b)], 2) an n-p junction with a Schottky back contact similar to that reported for CdTe [4] (see Figs. 4(c) and (d)], and
3) an N-p heterojunction similar to that found in CIGS or CZTS.
The structures we examine here are designed to have built-in
potentials well below the bandgap so that crossover can be observed under normal operating conditions at moderate forward
bias. Specific simulation parameters for all devices are listed in
Appendix B.
Because these structures do not have a constant electric field
like the n-i-p structures, analytic expressions are more difficult to
obtain, and therefore, we will examine the relationship between
Vbi,d and Vx , as obtained by simulation.
A. Homojunction Model Structures
The first structure is a symmetric n-p junction with modest
doping on both sides of the junction [see Fig. 4(a)]. We vary the
doping in the n- and p-layers from 1014 to 1017 cm−3 and plot
Vx versus Vbi,d . For these modest doping densities, the builtin potential is low; therefore, crossover occurs near Voc . The
behavior of the n-p structure near Vx is similar to that of the
n-i-p structure, as expected [see Fig. 4(b)]. Simulations show
that the crossover voltage Vx is equal to Vbi,d , just as it was for
the n-i-p structure (see Fig. 5 diamonds).
For the second structure, we leave the doping of the n and p
regions constant at 1016 cm−3 and vary the work function of
the back contact, changing the built-in Schottky barrier height
between 0 and 0.3 eV. Simulation results again show that I–V
crossover occurs very near the contact-to-contact Vbi,d , which
is given by
Vbi,d = Vbi,j − Vbi,sb
(2)
where Vbi,j is the built-in voltage of the p-n junction, and Vbi,sb
is the built-in voltage of the Schottky junction, which is given
by the difference in work functions of the p-layer and the metal
contact, as shown in Fig. 4(c).
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IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 4, NO. 4, JULY 2014
Fig. 4. Energy band diagrams at short circuit plotted using SCAPS-1D [18] for (a) symmetric n-p junction (n and p doping 101 6 ), (b) n-p junction biased at V b i ,
(c) n-p junction with a Schottky back contact (V b i , s b = 0.3 eV), and (d) device with the Schottky back contact biased at V b i .
Fig. 5. Numerical simulation results in SCAPS-1D for crossover voltage plotted versus built in voltage for the three homojunction device structures discussed.
All are closely related to V b i , but the n-p and sb results may be shifted slightly
above or below due to small changes in band bending under illumination.
is symmetrical [see Fig. 4(d)]. The photogenerated carriers in
the bulk partition equally between the front and back contacts
and Jgen = 0. Even though the bands are not perfectly flat, the
symmetry in band bending leads to I–V crossover just as we
saw for the simple n-p device. Again, we see that Vx is equal
to Vbi,d (see Fig. 5 triangles). This approach provides a simple
explanation for the enhanced electron current described in [13].
Fig. 5 shows a strong correlation of Vbi,d with Vx for a wide
voltage range, indicating that the current partitioning effect is
present in all devices. In practice, however, we would only
expect to see this effect in devices with Vbi,d well below the
bandgap (1.1 eV for this simulation). This is simply due to the
fact that when a large forward bias is applied to the device, the
injection current would be far too large to measure properly.
This type of crossover would, therefore, probably not be visible
in measurements of real devices above Vbi,d ≈ 2/3Eg .
B. CIGS-Like Model Structure
As explained in [4] and [11], the Schottky junction changes
from forward to reverse bias as the operating point moves from
the fourth to the third quadrant, but it is remarkably easy to
explain the crossover voltage. At V = Vbi,d , the band bending
Finally, we examine an N-p heterojunction with a valence
band barrier blocking the N contact. This structure is similar
to that found in thin-film solar cells like CIGS [15]. It is similar to the N-i-p heterostructure in Fig. 2(c) but without the
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1143
Fig. 6. Band diagram for N-p heterojunction plotted using SCAPS-1D (a) at short circuit and (b) biased at V b i , d . The window layer was modeled with parameters
similar to CdS, and the absorber was modeled with parameters similar to CZTS.
ilar CdTe structure with a p-side doping of 1014 . If the n-side
doping is small, less band bending will occur on the p-side and
there will be a smaller accumulated charge caused by photogenerated holes, so the device will require a higher voltage to
achieve crossover. In either case, we see that a cell of this type
can experience I–V crossover, even in the absence of traps or
bulk recombination if the device simply has a low overall Vbi,d .
Because low Vbi,d is associated with a low open-circuit voltage,
the type of crossover discussed in each of the cases so far is an
indicator of lower efficiency.
V. DISCUSSION
A. Effects of Traps at a Heterojunction Interface
Fig. 7. Simulation results using SCAPS-1D for a CIGS like heterojunction.
The dotted line is a reference showing the built-in voltage equal to the crossover
voltage. Each set of data points shows the trend for a given n-layer doping, and
the p-type base doping is varied from 101 3 to 10 1 6 in each case.
intrinsic layer and, therefore, behaves in a similar manner. The
model structure uses an n-type doping that results in a low Vbi,d
determined by the front contact work function and the p-type
absorber doping. By lowering the Vbi,d in this manner, the Voc
is lowered and crossover occurs near Voc . Recent experiments
show that this mechanism could be the cause of crossover in
CZTSSe cells [7]. The simulated band structure under short circuit is shown in Fig. 6(a) and at crossover in Fig. 6(b). The
structure contains a small conduction band offset at the n-p
interface, but in this case, its effect on the crossover voltage is
negligible. In Section V, we will discuss a structure with a larger,
more significant conduction band offset.
For this simulation, we vary the n- and p-side doping individually. Fig. 7 shows a crossover voltage strongly correlated with
Vbi,d , but Vx may be slightly above or below Vbi,d , depending on
which side of the junction is more heavily doped. If the n-side is
highly doped, most of the band bending will occur on the p-side,
leading to increased photogenerated charge accumulation and a
crossover voltage lower than Vbi,d , as observed in [2] for a sim-
In the previous section, we showed how accumulated charge
at a heterojunction interface can affect the generation-dependent
injection current of the device and cause crossover, even in the
absence of traps. Several papers have discussed I–V crossover
caused by traps in the widow layer or interface [13], [14], [16],
[23], [24]. In this section, we will show that interface traps
populated by photoexcitation can have a similar effect to mobile
charge accumulation at a heterojunction.
We first consider the effect of traps in a heterojunction without
any conduction band offset, such as that shown in Fig. 6(a).
Like the accumulation charge discussed previously, the trapped
charge is a sheet of charge at the interface, which increases the
forward bias of the junction under illumination. Adding acceptor
traps at the interface increases the band bending, thus increasing
the injection current (see Fig. 8). Donor traps have the opposite
effect, suppressing the injection current. Larger trap densities
will have a larger effect on the injection current.
There is, however, one important difference in behavior between the two types of charge. As shown in Fig. 3, the accumulation charge only increases the injection current above
Vbi,j , because this is the regime where accumulation occurs.
The interface traps, on the other hand, influence the current at
all voltages, because a fraction of the traps are charged even
below Vbi,j (see Fig. 8).
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Fig. 8. Simulated results for different interface trap distributions. As the
trapped charge at the interface increases, the electron injection current increases
under illumination.
IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 4, NO. 4, JULY 2014
Fig. 10. Plot showing change in crossover voltage with carrier lifetime for
three different structures: n-i-p, n-p, and n-p with 0.2-eV Schottky built-in
voltage. The dotted lines show the built in potential of each device.
behavior in the light J–V curve, with a second bend appearing in
high forward bias [25]. The offset will affect crossover insofar
as it lowers Vbi,d . There may also be some inversion charge due
to the band offset, which can slightly increase Vx , even pushing
it slightly above Vbi,d in the case with no traps (see Fig. 9, blue
squares).
B. Evaluation of Assumptions
Fig. 9. Simulated results for a CIGS structure with a V b i of 1.0 V and varying
deep acceptor trap density in the CdS layer. The modeled CdS/CIGS band offset
is 0.4 eV. Inset: light (solid line) and dark (dashed line) I–V curves for the case
with no traps.
In [13], Gloeckler et al. discuss a CIGS solar cell where
crossover is caused by deep acceptor traps in the CdS. The
structure of interest has a relatively large Vbi,d of 1.0 V. This
is fundamentally different from the previous structures, where
crossover was predominantly caused by a voltage-dependent
photocurrent.
We simulate a CIGS structure like that discussed in [13] with a
conduction band offset of 0.4 eV at the CdS/CIGS interface and
plot ΔJ versus V – Vbi,d in Fig. 9. For structures with traps in
CdS, we find that Vx is significantly lower than Vbi,d . Therefore,
Vx is uncorrelated with Voc ; therefore, in this case, crossover is
not necessarily an indicator of lower efficiency, since Voc is not
limited by Vbi,d , as seen in the previous case.
As the acceptor trap density in the CdS layer increases, the
crossover is pushed farther below the trap free case. We also do
not see symmetry in the plot of ΔJ because the photocurrent is
not limited to any specific value by current partitioning.
This structure also contains a significant conduction band
offset, which we have included in our model to better match
simulated results previously reported in the literature [15]. This
offset results in a two-diode effect, which can lead to unexpected
Up to this point, all simulations have assumed no bulk recombination and uniform optical generation. In this section, we will
test the robustness of our model by simulating structures with
finite recombination lifetimes and different generation profiles.
Bulk recombination gives the photogenerated carriers an additional means of removal from the device. Fig. 10 shows the
results for crossover voltage as a function of carrier lifetime for
three device structures. Carriers that recombine in the bulk of the
device reduce the collected photocurrent but do not change the
current partitioning. For reasonably long lifetimes, we find that
the crossover voltage remains invariant. At very low lifetimes,
we find that the crossover voltage increases only slightly for the
n-i-p and n-p structures and decreases slightly for the Schottky
barrier structure. However, even with a short lifetime of 1 ns the
crossover voltage remains within about 50 mV of the built-in
potential. In this case, the diffusion lengths will affect the low
bias properties such as fill factor, as discussed in [8]; however,
at high forward bias, current partitioning remains the dominant
effect, and therefore, Vx is not significantly changed.
Fig. 11 shows the I–V characteristic for different generation
profiles. Simulations of solar cells illuminated from either the
n- or p-side give identical I–V curves, since for our model,
we assume that electrons and holes have identical transport
properties. The overall I–V is different for uniform generation
and illumination from one side. However, when the device is
illuminated from the n-side, electrons are able to exit from the n
contact more easily, but holes also have an easier time escaping
from the n contact. The crossover voltage, therefore, remains
the same, even if the generation is nonuniform.
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1145
When taking experimental measurements, it is also important
to account for series resistance. The models that are discussed
in this paper presumed that series-resistance (Rs ), is negligible;
although, there are finite series resistances in all real devices.
This resistance will increase the measured voltage at current I
by I ∗ Rs . The measured crossover voltage must, therefore, be
corrected in analysis by Vx = Vx,m eas – Ix Rs , where Ix is the
measured current at the crossover voltage [2]. However, caution
must be taken when correcting for series resistance above V =
Vbi since, as mentioned in Section III, the exponential portion
of the diode current disappears in this region, and the current
becomes dominated by the bulk resistance.
Fig. 11. Plot showing the effect of different generation profiles on the I–V
characteristics of an n-i-p structure. The I–V is slightly different in each case,
but the crossover point remains tied to the built-in potential, regardless of the
generation profile.
C. Experimental Interpretation of Crossover
In this section, we will discuss possible measurement techniques that can be used on real devices to conclusively determine
whether crossover is due to a low Vbi,d or some other cause such
as light-induced charge trapping. We have already shown how
plotting ΔJ can be useful to determine the cause of crossover.
For Vbi,d limited crossover in a homojunction, the ΔJ curve
will appear symmetric, as shown in Fig. 3, which is not the case
for other causes of crossover.
The case of a heterojunction is slightly more complicated.
If the doping of both sides of the junction is known, then the
crossover can be compared with a simulation such as that shown
in Fig. 7. The crossover voltage can also be affected by traps
in the window layer. As demonstrated in [13] and [14], this
effect can be identified using wavelength-dependent light bias
measurements. If the crossover voltage remains constant under
low-wavelength illumination (i.e., with photon energy below
the bandgap of the window layer), then photoexcitation in the
window layer can be effectively ruled out as the primary cause
of crossover, as demonstrated in [7].
Comparison of the crossover voltage with Vbj,j measured
from the intercept of 1/C 2 versus voltage can be useful to
characterize crossover. If the two values match closely, then it
can be concluded that the crossover is limited by the built-in
voltage of the junction Vbj,j . Unfortunately, as noted in [2], the
C–V of thin-film devices is often difficult to interpret and may
not give the correct intercept for Vbi,j , and therefore, care must
be taken in this measurement.
If the 1/C2 intercept is above the voltage predicted by
crossover, a Schottky barrier at one of the contacts may be
responsible. As noted previously, the crossover is dependent
on the contact-to-contact Vbi,d , while C–V is dependent only
on Vbi,j . The presence of a Schottky barrier can also be confirmed by other methods as well, such as temperature-dependent
I–V [12]. The Vbi,d limited crossover voltage should also have
the same temperature dependence as Vbi,d and will extrapolate
to Eg at T = 0 K if limited by the junction or Eg – φsb if a
Schottky barrier is present [7].
VI. CONCLUSION
Determining the cause of I–V crossover in a solar cell can be
helpful in understanding performance limitations. As discussed
extensively in the literature, many different effects can cause
crossover. All causes of crossover can be classified into two
categories. Crossover may be caused by a voltage-dependent
photocurrent, or by a generation-dependent diode injection current caused by a change in band bending under illumination.
Both mechanisms often play a role, and the overall behavior of
the solar cell depends on which term in (1c) is dominant.
Previous studies have attributed voltage-dependent collection
in thin-film solar cells to short diffusion lengths, but as shown in
this paper, voltage-dependent collection can occur even in ideal
devices with long diffusion lengths, if the contact-to-contact
built-in potential is low. This effect can occur in a wide variety
of cell structures. For example, we showed that I–V crossover
caused by a Schottky back contact can be simply explained by
the reduced contact-to-contact Vbi . We also showed that a simple plot of ΔJ = Jlight – Jdark can be useful in identifying
the specific cause of crossover if series resistance is properly
accounted for. These conclusions were supported with physical
arguments, analytical calculations, and with detailed numerical
simulation. In light of these findings, we believe that the existence of a low contact-to-contact built-in potential should be
considered as a possible cause for I–V crossover.
APPENDIX A
ANALYTIC TREATMENT OF N-I-P STRUCTURES
Assuming the electric field is constant throughout the absorber layer, the injection and photogenerated currents for the
structure of Fig. 2(a) are as follows:
Jinj = q
n2
n2
μn i + μp i
ND
NA
Jgen = qGL coth
(V − Vbi )
eq (V −V b i )/k T − 1
q (V − Vbi )
2kT
−
2kT
q(V − Vbi )
eq V /k T − 1
(A1)
(A2)
where G is the generation rate, and L is the length of the channel.
We see from this equation that the photogenerated current has
some voltage dependence. The full derivation of these equations
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IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 4, NO. 4, JULY 2014
can be found in [9], where they were originally derived for an
m-i-m structure. They are shown to apply to a n-i-p solar cell
in [10].
An analytic expression can be similarly derived for the structure in Fig. 2(c) with a hole blocking heterojunction by slightly
modifying the derivation of (2). Setting the boundary condition
for the hole current at the n-contact to Jp = 0, the equations for
the injection and photogenerated currents become
(V − Vbi )
n2
eq V /k T − 1 (A3)
Jinj = q μn i
q
(V
−V
)/k
T
bi
ND e
−1
1
kT
Jgen = qGL q (V −V )/k T
−
. (A4)
bi
q(V − Vbi )
e
−1
The injection current in (A3) is the same as for the homojunction case, except the term for holes vanishes, since the hole
current is blocked by the valence band barrier.
The extra injection current is explained by a change in the
charge distribution under illumination. At high forward bias,
the electric field will try to push the holes toward the n-contact
but holes cannot enter the n-contact due to the large valence
band barrier. The density of photogenerated holes at the blocking contact builds up because they cannot escape through this
contact [see Fig. 2(d)]. This accumulation of holes acts as an
additional forward bias on the junction, increasing the electron
injection current. The additional injection current under illumination increases ΔJ above zero, as shown in Fig. 3. The analytic
expression does not account for this Poisson effect, which is why
it differs from the numerical simulation in this case.
It would be difficult to derive a full analytic expression for
this accumulation driven current; however, a rough approximation can be made by treating the accumulation charge as a sheet
charge at the n-p interface. This sheet charge simply adds a constant component to the electric field which can then be rewritten
as
E = E0 + E Qs (V, G)
2Ks ε0
APPENDIX B
PARAMETERS USED IN NUMERICAL MODELING
Material parameters for CIGS [15]
Material parameters for CdS [15]
n-i-p homostructure
(A5)
where E0 is the constant electric field in the intrinsic layer of
the p-i-n junction given by (V − Vbi )/d, and E is the additional
electric field due to the accumulation or trapped charge. The
equation for this additional electric field is
E =
As previously mentioned, this equation is only an approximation of the accumulation current due to the estimation of
the accumulated holes purely as a surface charge. In reality, the
excess holes under illumination may have a more complicated
distribution. We, therefore, introduce a correction factor α to
account for this difference.
We can evaluate the validity of this expression using our results from numerical simulation. The simulation gives us the
hole concentration at the interface. We find that the relationship
between current and charge accumulation is in close agreement
with our prediction from (A7) for α = 0.8. This equation provides a simple way to understand the full numerical simulation
shown in Fig. 3.
N-i-p heterostructure
(A6)
where Qs is the surface charge per unit area, and Ks is the
dielectric constant of the intrinsic layer. Note that Qs is also
a function of both V and G. Additionally, trap states at the
interface may also provide an additional source of surface charge
as discussed previously. Because the electric field of a sheet
of charge is uniform with position, the equation for our total
electric field remains position independent. At high forward
bias, the extra current caused by photogenerated accumulation
or trapped charge can now be written as
Qs, light − Qs, dark
.
(A7)
Jacc = αND qμn
2Ks ε0
NP structure
NP structure with Schottky Barrier
Work function of back contact: 5.1559 eV (back contact of
all other structures assumed to be ohmic).
MOORE et al.: CORRELATION OF BUILT-IN POTENTIAL AND I–V CROSSOVER IN THIN-FILM SOLAR CELLS
ACKNOWLEDGMENT
The authors acknowledge the work of Prof. R. Agrawal’s
group, particularly his students C. Hages and N. Carter, whose
experiments in CZTSSe devices were the primary motivation
for this work, as well as Prof. J. Sites, for his helpful comments
and suggestions on this work.
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[13] M. Gloeckler, C. R. Jenkins, and J. R. Sites, “Explanation of light/dark
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James E. Moore (M’11) received the B.S. degree in
engineering physics from Taylor University, Upland,
IN, USA, in 2009 and the M.S. degree in electrical
and computer engineering from Purdue University,
West Lafayette, IN, in 2011, where he is currently
working toward the Ph.D. degree in electrical and
computer engineering.
His research interests include numerical and compact modeling of semiconductor devices, as well as
experimental work involving electrical and optical
characterization.
Sourabh Dongaonkar (M’13) received the B.Tech.
degree in electrical engineering from the Indian Institute of Technology Kanpur, Kanpur, India, in 2007
and the Ph.D. degree from the School of Electrical
and Computer Engineering, Purdue University, West
Lafayette, IN, USA, in 2013.
He is currently a Compact Device Modeling Engineer with Intel Corporation, Hillsboro, OR, USA.
From 2007 to 2008, he was with the Global Market
Center, Deutsche Bank, Mumbai, India, as a Quantitative Analyst. His research interests include the
analysis and design for variability and reliability in solar cells, modeling of
novel semiconductor devices, and solid-state storage technologies.
Dr. Dongaonkar received the 2008 Ross Fellowship from Purdue University Graduate School, as well as the Best Student Presentation and the Best
Poster Awards at the 37th and 38th IEEE Photovoltaic Specialists Conference,
respectively.
Raghu Vamsi Krishna Chavali (S’13) received the
B.E (Hons.) degree in electrical and electronics engineering from the Birla Institute of Technology and
Science, Pilani, India, in 2009 and the M.S degree
in electrical and computer engineering from Purdue
University, West Lafayette, IN, USA, in 2011, where
since 2011, he has been working toward the Ph.D.
degree with the School of Electrical and Computer
Engineering.
His research interests include modeling, simulation, and characterization of semiconductor devices.
1148
Muhammad Ashraful Alam (M’96–SM’01–F’06)
received the B.S.E.E. degree from the Bangladesh
University of Engineering and Technology, Dhaka,
Bangladesh, in 1988; the M.S. degree from Clarkson University, Potsdam, NY, USA, in 1991; and the
Ph.D. degree from Purdue University, Lafayette, IN,
USA, in 1994, all in electrical engineering.
He is currently a Professor of electrical and computer engineering with the School of Electrical Engineering and Computer Science, Purdue University,
where his research and teaching focus on physics,
simulation, characterization, and technology of classical and novel semiconductor devices. From 1995 to 2001, he was with Bell Laboratories, Lucent
Technologies, Murray Hill, NJ, USA, as a member of Technical Staff with
the Silicon ULSI Research Department. From 2001 to 2003, he was a Distinguished Member of Technical Staff and the Technical Manager of the IC
Reliability Group, Agere Systems, Murray Hill. In 2004, he joined Purdue University. He has contributed to more than 150 papers in international journals and
has presented many invited and contributed talks at international conferences.
His current research interests include stochastic transport theory of oxide reliability, transport in nanonet thin-film transistors, nanobio sensors, and solar cells.
Dr. Alam received the IEEE Kiyo Tomiyasu Award for his contributions to
device technology for communication systems. He is a Fellow of the American Physical Society and the American Association for the Advancement of
Science.
IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 4, NO. 4, JULY 2014
Mark S. Lundstrom (S’72–M’74–SM’80–F’94) received the B.E.E. and M.S.E.E. degrees from the
University of Minnesota, Minneapolis, MN, USA,
in 1973 and 1974 and the Ph.D. degree from Purdue
University, West Lafayette, IN, USA, in 1980.
He is currently the Don and a Carol Scifres
Distinguished Professor of Electrical and Computer
Engineering, Purdue University, West Lafayette,
IN, USA. His current research interests include
the physics of small electronic devices, particularly nanoscale transistors, on carrier transport in
semiconductor devices and devices for energy conversion, storage, and
conservation.
Dr. Lundstrom is a Fellow of the American Physical Society and the American Association for the Advancement of Science, as well as a Member of the
U.S. National Academy of Engineering.