Supporting Information
On the Uniqueness of Ideality Factor and Voltage Exponent of
Perovskite Based Solar Cells
Sumanshu Agarwal1,*, Madhu Seetharaman2, Naresh K. Kumawat3, Anand S. Subbiah1, Shaibal
K. Sarkar1, Dinesh Kabra3, Manoj A. G. Namboothiry2, and Pradeep R. Nair4,*
1
Department of Energy Science and Engineering, IIT Bombay, Powai, Mumbai, India,
2
School of Physics, IISER Thiruvananthapuram, Kerala, India,
3
Department of Physics, IIT Bombay, Powai, Mumbai, India,
4
Department of Electrical Engineering, IIT Bombay, Powai, Mumbai, India,
E-mail: sumanshu@iitb.ac.in , prnair@ee.iitb.ac.in
Correspondence should be addressed to:
*
Sumanshu Agarwal, Department of Energy Science and Engineering, IIT Bombay, Powai,
Mumbai-400076, email: sumanshu@iitb.ac.in
*
Prof. Pradeep R. Nair, Department of Electrical Engineering, IIT Bombay, Powai, Mumbai400076, email: prnair@ee.iitb.ac.in
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Section I: Device fabrication details:
Cell C: To test the validity of our model predictions, we fabricated perovskite based solar
cells with device structure of ITO/PEDOT:PSS/CH3NH3PbIxCl1-x/PCBM/Ag in inert
conditions in Department of Physics at IIT Bombay. These cells showed reasonably good
solar cell performance with typical PCE of ~ 6.3%. Perovskites films were prepared via
solution process (by spin coating using 2000rpm for 45sec). After deposition films were
annealed at 90⁰ C for 90 minutes in inert conditions with moisture level of < 0.1 ppm and
typical thickness of these films were in the range ~ 330 nm.
Cell D: This cell was made in IISER Thiruvananthapuram, India. This device has P3HT as
HTL and TiO2 as ETL. TiO2 was deposited on FTO by spin coating TiO2 solution, made in
1-butanol, at 2000 rpm for 60 s. A solution of perovskite (CH3NH3PbI3-xClx) was prepared by
dissolving methylammonium iodide and lead chloride in anhydrous DMF. This solution was
spin deposited over dense TiO2 layer and annealed at 100oC for 90 min. The hole
transporting layer of Poly(3-hexylthiophene-2,5-diyl) (P3HT) was prepared by spin coating
15 mg/ml solution in chlorobenzene at 1500 rpm for 2 mins. The substrates were loaded into
a vacuum thermal evaporator and 120 nm of silver was thermal evaporated as counter
electrode at a pressure of 6x10-6 mbar.
Cell E: The TiO2 sol gel was prepared by adding 1 g of di-ethanol amine with 2.7 ml of
Titanium iso-propoxide in 71 ml of ethanol. After subjecting this solution to heavy stirring,
70 µl of deionized water was added to the precursor solution and aged for a few hours. This
solution was then spun on FTO substrates at 4000 rpm for 25 s and then the substrates were
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annealed at 500o C for an hour. This procedure is carried out 5 times repeatedly to get a
uniform thin film of compact TiO2 (~100 nm) on FTO. For thermal co-evaporation of
CH3NH3PbI3-xClx, two separate sources Methyl ammonium iodide and Lead Chloride were
evaporated simultaneously at the rate of 0.95±0.15 A/s and 8.5±1.5A/s respectively for a
time period of 24 min. The films were then annealed at 90o C for 60 min to achieve
perovskite absorber of approximately 400 nm thickness. Spiro-OMeTAD precursor solution
was prepared by adding 80 mg of Spiro-OMeTAD in 1 ml of chloro-benzene with 34µl of
170 mg Li-TFSI/1 ml Acetonitrile and 14 µl of tert-butyl pyridine. This solution was spun at
2000 rpm for 30 s on TiO2/Perovskite structure followed by thermal evaporation of the metal
Ag contacts using a metal mask (aperture area of ~ 0.07 cm2) to complete the device
structure.
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Section II: Extraction of Ideality factor from the dark I-V characteristics of
experimental devices
Supplementary Figure 1 |. Ideality factor extraction for experimental cells. Semilog plot
of J-V characteristics with corresponding exponential fit (solid line) for experimental cells are
shown here. Extracted ideality factor is also given in respective plot. Lower plot in each panel
shows ideality factor extraction using formula
.
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Section III: Terminal currents for simulated device
Supplementary Figure 2 | Terminal currents for the simulation results shown in
Fig. 3 of main text. The solid line indicates the total current through the device,
which is the summation of electron and hole components. Square symbols indicate the
electron component at FC while open triangles represent the hole component at the
BC. As the electron component at FC equals the total current, it is evident that there is
negligible over the barrier transport of holes from perovskite to the ETL. Similarly,
the hole component at BC equals the total current indicating negligible electron
transport from perovskite to the HTL. The total current is given by the recombination
of carriers in the perovskite region.
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Section IV: Theoretical analysis for ideality factor and voltage exponents
From the E-B diagram and the simulation results provided earlier (see Fig. 1 and 3 of main
document, and Figure 2 of this document), it is clear that the dark current is due to the
recombination of charge carriers inside the perovskite region. The rate of recombination due
to trap limited SRH mechanism is given by eq. 1
np  ni2
R
 n ( p  p1 )   p (n  n1 )
J
 Rdx
(1)
(2)
Lp
where, L p is the thickness of perovskite layer.
From the classical theory of PN and PIN diodes 1, it is well known that SRH recombination
at various locations leads to different ideality factors. For example, eq. (1) indicates that
maximum recombination will occur at point where n  p (for  n   p ). Accordingly, a PN
junction diode exhibits an ideality factor of 2 during the low bias regime due to the
recombination inside the depletion region. Similar arguments holds good for the
recombination inside the I layer of a PIN diode, again resulting in an ideality factor of 2.
However, SRH recombination can also lead to different ideality factors, other than 2. For
example, in a classical long base diode, once the carriers cross the depletion region, they
become minority carriers in the corresponding region. This minority carrier recombination
also follows eq. (1). However, this case leads to an ideality factor of 1, as opposed to the
recombination inside the depletion region. Hence the ideality factor is an important
parameter that provides key insights regarding not just the type of recombination, but also
about the spatial location that contributes significantly to the dark current.
We now explain/interpret the range of ideality factors and voltage exponents observed in our
simulations. Note that for all the cases considered in Fig. 4 of main text, the expected ideality
factors and high bias voltage exponents can be explained using a simple analogy with either
PN or PIN diode, with one crucial difference. For the perovskite cell, over the barrier
transport is negligible, while such over the barrier transport plays a crucial role for traditional
diodes. Hence, the operation of a perovskite solar cell can be conveniently understood in
terms of the appropriate recombination current of traditional diodes.
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Case I:
ETL: undoped
Perovskite & HTL: undoped
Perovskite dielectric constant: low
Large dielectric constant of ETL (TiO2) causes negligible potential drop in ETL and hence all
the potential drop takes place in perovskite and HTL layer (see the EB diagram, Fig. 1b).
Therefore maximum recombination will occur where n  p . If n0 ( x) and p0 ( x) were the
electron and hole concentration at point x under equilibrium conditions, then these
concentrations at any applied bias Va can be expressed as
n( x)  n0 ( x) exp(Va / Vt )
(3)
p( x)  p0 ( x) exp((1   )Va / Vt )
(4)
Here  is the fraction of applied voltage dropped till point x from one contact and Vt is the
thermal voltage. Solving eq. 3 and 4 for n( x)  p( x) , we get
1

Vt
ln(n0 ( x) / p0 ( x))
Va
2
Putting value of  back in eq. 3 we get
n( x)  n0 ( x) exp(
Va 1 n0 ( x)
 ln
)
2Vt 2 p0 ( x)
(5)
Under such conditions, it is evident from eqs. (1), (2), and (5), that the ideality factor will be
2. This case corresponds to Fig. 3 of main text. As the HTL is either intrinsic or low doped,
the high bias regime could exhibit space charge effects.
Now we will provide detailed analysis for all the cases discussed in the figure 4 of main text
on the uniqueness of observed features. We first consider the role of ETL doping vs. ETL
dielectric constant.
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Case II: Effect of ETL doping vs. ETL dielectric constant
The E-B diagrams shown below indicate that the effect of ETL dielectric constant and doping
on the EB diagrams. For both cases, we assume that the ETL dielectric constant is large (
r  173 ) The figure on the left indicates the E-B diagram when the ETL is undoped, while
the figure on the right indicates the case when the ETL is heavily doped. It is evident that
both E-B diagrams are almost similar (due to the large dielectric constant of ETL), regardless
of the doping level of ETL. In both the cases observed potential drop across ETL will be
negligible and the device characteristics will be controlled by the drop across perovskite and
the HTL. As a result the IV characteristics show similar trends for ideality factors and
voltage exponents.
Supplementary Figure 3 | Comparison of equilibrium energy band diagrams for ETL
is undoped (left), and ETL is heavily doped (right).
Now we provide detailed analysis for all the cases considered in Fig. 4 of main text. Here we
assume that ETL is heavily doped and its dielectric constant is large.
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Case III: When dielectric constant of perovskite is low (comparable to HTL)
III.A: perovskite and HTL are intrinsic or low doped
Low doping ( 11014 cm3 ) is similar to the case of intrinsic layer. The analysis done for Case
I holds here as well. Hence the expected ideality factor is 2, along with a high bias voltage
exponent of 2.
III.B: perovskite is intrinsic or low doped with heavily doped HTL
As already mentioned, low doping is almost similar to intrinsic material. Therefore, the
device will acts as a P-I-N junction diode, which is well known to give an ideality factor of 2.
As both the contacting layers are heavily doped, the device shows ohmic behavior with a
voltage exponent of 1.
III.C: perovskite heavily doped with HTL intrinsic or low doped
Heavy doping in the perovskite with intrinsic or low doped HTL causes almost all the
potential to drop inside HTL only. In this case the holes injected into the perovskite will be a
minority carrier, and hence the scenario is very similar to the bulk recombination of injected
minority carriers in a PN diode. As the applied potential drops entirely across the HTL (due
to its low doping), this leads to an ideality factor 1. This low doped HTL will lead to a high
bias voltage exponent of 2, due to space charge effects.
III. D: Heavily doped perovskite and HTL
High doping of both perovskite and HTL make it as PN junction and potential drop across
depletion region will be distributed equally between the perovskite and HTL. Maximum
recombination in this case again occur at the perovskite and HTL interface, but only half of
the total potential is dropped inside the HTL. Hence an ideality factor 2 is expected. As all
layers are highly doped, high bias regime exhibits a voltage exponent of 1.
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Case IV: When dielectric constant of perovskite is large
IV. A: undoped or low doped HTL
Supplementary figure 5 compares the E-B diagram as a function of perovskite dielectric
constants. The left figure shows the case for large dielectric constant while the right figure
shows the E-B diagram when perovskite dielectric constant is low. It is evident that the large
dielectric constant of perovskite (or heavy doping) as compared to HTL will cause the entire
potential to drop across HTL this will give rise to ideality factor 1. This is similar to case
III.C.
Supplementary Figure 4 | Comparison of equilibrium energy band diagrams as a
function of perovskite dielectric constant. The left figure corresponds to large
dielectric constant while the right figure corresponds to low dielectric constant.
IV.B: Heavily doped HTL with undoped or low doped perovskite
With HTL being heavily doped, the structure resembles a PIN diode. It is expected that under
such conditions the recombination inside the perovskite would lead to an ideality factor of 2
and high bias voltage exponent of 1.
IV.C: Heavy doping in HTL and perovskite
This case is almost similar to abrupt PN junction. The depletion region width will be much
more in HTL than that in perovskite (note that the depletion region width in respective
material at junction of two materials varies as
2
/
1
, where
1
the material for which we want to have the depletion width and
is the dielectric constant for
2
is the dielectric constant
for other material). As a result the entire applied bias will be dropped in HTL. As a result
injected holes will act as a minority carrier in perovskite which leads to an ideality factor of
1. As all the layers are heavily doped, the high bias regime is expected to be ohmic with a
voltage exponent of 1.
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Section V: Ideality factor and voltage exponents from simulation results.
Supplementary Figure 5 | Extraction of ideality factor and voltage exponent. Part
(a) shows semilog plot of J vs V for simulation result for undoped HTL and
perovskite. Solid line in the same plot shows an exponential fit giving ideality factor
2. Part (b) shows the ideality factor extracted from the derivative of the J-V
characteristics. A voltage exponent fit, giving exponent 2, for intermediate bias range
is shown in part c, while similar analysis for very high bias range (part d) gives
asymptotic exponent equal to 1.8, a value very close to the exponent obtained from
intermediate bias regime.
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Section VI: Mobility extraction using Mott Gurney relation:
Mobility was extracted from the slope of J vs V curve using Mott-Gurney relation2 which
is given below
J
Here  is the mobility and
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0
 V2
8 L3
is the dielectric constant of the material. L is the thickness of
the layer causing space charge limited current. This equation says that
J vs V curve will
be linear in SCLC regime with slope of the curve being mobility dependent. Supplementary
Figure 6 shows the linear regime of
J vs V and the corresponding slopes.
Supplementary Figure 6 | Extraction of mobility using Mott-Gurney relation.
Part (a) shows
vs
for simulation result with HTL thickness being 450 nm. Part
(b) shows the
vs V for Cell C (experimental results). Current density units are
mA/cm2. For mobility extraction HTL thickness was used and its dielectric constant
was taken as 10 for all the cells (although the dielectric constant varies with HTL
material. However this approximation is not expected to change the results/trends
significantly).
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References:
(1)
Sze, S. M.; Ng, K. K. Physics of Semiconductor Devices; Wiley-Interscience: Hoboken,
NJ, 2007.
(2)
Mott, N. F.; Gurney, R. W. Electronic Processes in Ionic Crystals; Clarendon Press, 1948;
p. 275.
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