Peizhe Liao^{1}, Xiaojuan Zhao^{1}, Guolong Li^{2}, Yan Shen^{1}, Mingkui Wang^{1,}* 
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NanoMicro Lett. (2018) 10: 5 

First Online: 11 September 2017 (Article) 

DOI:10.1007/s408200170159z 

*Corresponding author. Email: mingkui.wang@mail.hust.edu.cn 
Abstract
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1 Introduction
The photoelectric effect converts solar energy into electricity, which is one of promising way to solve the global energy crisis and environmental pollution. Due to the excellent light absorption and carrier transportation characteristics, perovskite type semiconductors with a general ABX_{3} formula have attracted intensive interest in recent years [1]. The power conversion efficiency (PCE) of perovskite solar cell (PSC) has a rapid growth from 3.8% in 2009 to 22.1% in 2016 [2, 3]. Despite this, it is important to understand the carrier transport mechanism of PSCs, while it is a good way to fit currentvoltage (JV) curves. Hence the JV curves for silicon solar cells and thinfilm solar cells have been fitted to analyze the working mechanism and performance of solar cells [46]. Considering the absence of specific equivalent circuit and fitting formula for PIN model, an ideal single PN junction circuit has been built to simulate the JV characteristic of various PSCs [7]. By fitting JV curves under light and in dark, three parameters including series resistance (R_{s}), diode ideal factor (m), and reverse saturation current (J_{0}) can be obtained. Compared to the reverse saturation current of conventional semiconductor diodes (such as CdTe, GIGS), the parameter J_{0} of PSC is relatively low, which explains its smaller bandgapvoltage loss (~0.4 eV) [8]. Therefore, it is helpful to improve the efficiency of PSC by understanding the two parameters of R_{s} and J_{0} [9]. Furthermore, the parameter m has been utilized as an indication of the heterojunction solar cell [7].
To date, planar structured PSCs have been developing rapidly due to their various advantages of simple device structure, lowtemperature processable fabrication, and so on [10]. Nevertheless, in planar PSCs, the value of m obtained by single PN junction modeling does not fill in the theoretical expectation, indicating that the heterojunction property in planar PSCs further discussion [11, 12]. Generally, for a single heterojunction model, the ideal factor approaches to 1 when the carrier diffusion in the neutral region of semiconductors dominates the diode current through a PN junction. On the other hand, the ideal factor approaches to 2 when the diode current is dominated by carrier indirect recombination in depleted spacecharge region [9]. Theoretically, the smaller value of m reflects the less carrier recombination induced by the interface defect state. In most cases, both diffusion and composite currents exist simultaneously and therefore the parameter of m is in the range of 12. Interestingly, we note that, as shown in Table 1, most of the calculation results are larger than 2. Hence a single PN junction model (Fig. 1a) is not suitable to planar heterojunction PSCs [13, 14].
Table 1 The calculated ideal factor (m) on planar heterojunction PSCs based on single PN junction model reported in literatures
In this work, we present a new equivalent circuit to investigate the heterojunction property for planar PSCs (in light and dark). Based on the new double heterojunction circuit we found that smaller value of m reflects better PN junction quality in PSCs. Moreover, carrier recombination and transportation characteristics can be further explored by fitting JV curve in dark with the new model for describing these important processes in efficient PSC devices.
Fig. 1 a Single PN junction model for PSCs, and b new double PN junction model improved for planar perovskite solar cells with J_{L} (the light induced current), J_{D} (the dark current or the forward current of PN junction diode under the sunlight), R_{s} (the series resistance), R_{sh} (shunt resistance, a fictional parameter to represent the size of leakage current), J (output current of the cell), and V (voltage flowing through the external load). c Planar heterojunction perovskite solar cells with TiO_{2}/CH_{3}NH_{3}PbI_{3x}Cl_{x}/SprioOMeTAD/Au device architecture 
2 Theoretical Background
Firstly, the rectification characteristic of heterojunction solar cell can be typically described by the Shockley diode equation (Eq. 1) [14].
where J_{D} is the dark current, V is the applied voltage, J_{0} is the reverse saturation current density, q is the elementary charge, m is the ideal factor of a heterojunction, K is the Boltzmann constant, T is the absolute temperature. Under the ideal condition of sunlight, photocurrent can be added into Eq. 1:
where J_{ph} is the photocurrent. In fact, output current (J) in Eq. 2 is limited by internal resistance and leakage current in PSCs. Figure 1a presents ideal circuit model with a single PN junction, from which the JV curve (in light) of heterojunction PSC can be further described with Eq. 3 [15],
where R_{s} and R_{sh} are the series and shunt resistance, respectively. Under the circumstances, R_{s}, R_{sh}, J_{0}, and m can be numerically obtained by simulation the JV curves (both in light and dark) of PSCs with Eq. 3. The R_{s} reflects the internal resistance and R_{sh} is a fiction parameter to represent the leakage current. The value of J_{0} is directly related to the recombination rate, indicating the thermal emission rate of electrons from the valence band to the conduction band in light absorption layer [16], which also has an impact on the opencircuit voltage. Nevertheless, comparing with R_{s} and J_{0}, m, correlating with ShockleyReadHall recombination [6], is a rarely discussed parameter in PSCs when fitting JV curves with using single PN junction model [1720].
Fig. 2 Energy band diagram of different PN junction photovoltaic devices a A PN junction solar cell. b A PIN solar cell with homogenous builtin electric field c CH_{3}NH_{3}PbI_{3x}Cl_{x} perovskitebased cell with inhomogenous builtin electric field. 
Compared to traditional PIN structure solar cells (‘aSi: H’like), inhomogeneous builtin field of PSCs results in different band structure (Fig. 2) [21, 22]. When the perovskite light absorption layer is sandwiched between n and ptype charge selective contacts (Fig. 1c), two active junctions immediately form at the ntype electron transport layer (ETL) and the ptype hole transport layer (HTL) sides [22]. Therefore, we suggest two PN junctions in series for explaining planar heterojunction PSCs [2327], rather than a single PN junction. {Edri, 2014 #103} Equation 4 is applied according to {Edri, 2014 #103}the equivalent circuit of double PN junction as shown in Fig.1b:
According to the characteristics of the series circuit, the current through the double PN junction should be identical (Eq. 5).
where m_{1}, V_{1}, J_{01}, m_{2}, V_{2}, J_{02} are diode ideality factor, voltage and reverse saturation current of ETL/perovskite and perovskite/HTL two PN junctions, respectively.
In this study, perovskite absorption layer acts as intrinsic semiconductor, which is fully depleted with highly doped P/N selective layers to form versatile PIN photovoltaics [28]. Considering the condition of J_{01}∝P_{n}D_{p}/L_{p}, J_{02}∝N_{p}D_{n}/L_{n} and a similar carrier density for electrons and holes [29, 30], the value of D_{p}/L_{p} can be approximately equal to D_{n}/L_{n }[31]. Therefore, the difference of the calculated J_{01} and J_{02} values lies in the same magnitude in this case according to the derivation process (Eq. 6).
Then Eq. 3 can be further revised as Eq. 7:
According to Eqs. 47, Eq. 8 and Eq. 9 can be inferred:
Equation 8 describes the JV curve of planar PSCs under illumination. Since m in Eq. 3 includes the contribution from the double junctions, the sum of m_{1} and m_{2} can be in the range of 2 to 4. In short, the calculation results of m (~24) in Table 1 confirm the suitability of the proposed double heterojunction equivalent circuit for the planar PSC devices, with which carrier transportation (including direct recombination) and recombination (ShockleyReadHall recombination) processes can be preciously described.
3 Results and Discussion
In order to elucidate the effect of m on planar PSCs, we fabricated CH_{3}NH_{3}PbI_{3}based planar PSC devices with structure of ITO/TiO_{2}/CH_{3}NH_{3}PbI_{3x}Cl_{x}/SpiroOMeTAD/Au using twostep deposition method [23], in which methylammonium chlorine (MACl) were added to increase the perovskite films quality. Figure 3a shows the typical JV characteristics of these devices under simulated sunlight at 100 mW cm^{2} (AM 1.5G). Table 2 shows the photovoltaic parameters of devices with perovskite layers by varying MACl concentration.
Table 2 The photovoltaic performance of perovskite solar cells fabricated from doping varied MACl concentration
The addition of Cl^{} in the perovskite film significantly improved the efficiency of planar PSCs. The maximum performance was obtained for doping the appropriate amount of MACl (MAI:MACl=50:5), which resulted in highly efficient devices exhibiting short circuit current (J_{sc}) of 20.35 mA cm^{2}, open circuit voltage (V_{oc}) of 1.05 V, fill factor (FF) of 72.73%, corresponding to PCE of 15.6%. The keys to improve the performance of CH_{3}NH_{3}PbI_{3x}Cl_{x} devices were better crystallinity of perovskite film, pin holefree coverage of the perovskite films and fewer interface defect states [32, 33].
Fig. 3 a Currentvoltage curve for planar perovskite solar cells using TiO_{2}/CH_{3}NH_{3}PbI_{3x}Cl_{x}/SpiroOMeTAD/Au architecture, the perovskite films prepared by mixing with different concentrations of Cl ions. The measurements are carried out under 100 mW cm^{2}. b Plots of dV/dJ vs (symbols) and the linear fitted curve (solid lines), c Plots of ln(J_{sc}JV/R_{sh}) vs V+JR_{s} (symbols) and the linear fitted curve (solid lines). 
We further fitted these JV curves of devices in Fig. 3a to investigate the effect of doping Cl on perovskite layers by analyzing the parameters of R_{sh}, R_{s}, m_{1}+m_{2}, and J_{0}. R_{sh }can be calculated from the inverse of the slope of the JV curves at 0 V [34]. The other three parameters (R_{s}, m_{1}+m_{2}, J_{0}) can be obtained through deduction of Eq. 8 as shown in the following,
R_{s} can be obtained by calculating the intercept of fitting curve of vs. (Eq. 10) in Fig. 3b. Similarly, J_{0 }can be obtained by fitting the curve of ln(J_{sc}JV/R_{sh}) vs. (V+JR_{s}) (Eq. 11) in Fig. 3c. The value of (m_{1}+m_{2}) can be simultaneously inferred from the slope of fitting curves (Eq. 11) in Fig. 3c. The calculation results of four parameters (R_{sh}, R_{s}, m_{1}+m_{2}, J_{0}) are shown in Table 3.
Table 3 R_{sh}, R_{s}, m_{1}+m_{2}, and J_{0 }derived from Fig. 3b, c
As shown in Table 3, the shunt resistance R_{sh} dramatically increases from 840 to 5050, 4280, and 3400 Ω for addition of Cl ions (MAI:MACl from 50:0 to 50:7.5) in perovskite films. A larger R_{sh} indicates less leakage current. This could be related to less pinholes for perovskite layers with Cl^{} than that without additives [32, 35]. It is significant that the ideal factor (m_{1}+m_{2}) drastically decreases from 3.27 to 2.66 after adding Cl^{}, then keeps similar value of 2.66. The value of ideal factor (m_{1}+m_{2}) is 3.27, 3.07, 2.66, and 2.64 for corresponding devices respectively, which further conforms the feasibility of the double PN junction model. Smaller value of m_{1}+m_{2} indicates better PN junction quality [6]. The reverse saturation current (J_{0}) of corresponding devices was estimated to be 1.69×10^{4}, 8.23×10^{5}, 6.89×10^{6}, and 4.10×10^{6 }mA cm^{2}. The smaller J_{0} is a sign of substantially suppression of the thermal emission rate of electrons from the VB to the CB [16], resulting in higher output voltage. This is verified with the V_{oc} (being 1.05 V) of device with the suitable addition of Cl ions (MAI:MACl = 50:5) in perovskite layer, which is higher than devices without Cl. In short, the calculated R_{sh}, m_{1}+m_{2} further indicates that larger shortcircuit current (20.35 mA cm^{2}) for the device with Cl can be attributed to less carrier recombination and loss in ETL/perovskite and perovskite/HTL interfaces.
Fig. 4 Plots of dark current of planar PSCs as well as fitting curve by using Eq. 13. a The device structure is shown in Fig. 1c, the perovskite films prepared by mixing with different concentrations of Cl ions. b The device structure is PEDOT:PSS/CH_{3}NH_{3}PbI_{3x}Cl_{x}/PCBM/Hole blocking layer/Al, with the original data coming from the Ref. [8]. The inset region A, B, C, D is mainly determined by shunt current, recombination current in diode space charge region, diffusion current, diode diffusion current limited by series resistance, respectively. 
The dark current was fitted with Eq. 3 (single PN junction model), however, very limited information can be obtained except of three parameters (m, J_{0}, R_{s}) [7]. Furthermore, in the exponential coordinates, the dark current curve slope is different with the voltage increases, reflecting different physical processes. This physical process cannot be reflected by Eq. 3 (single PN junction model). In fact, as shown in Fig. 4a, the region A, B, C of dark current is related to shunt current, recombination current and diffusion current, respectively. At last, above the builtin potential at about 1.2 V in the region D, the effect of the recombination current is negligible, the curve is determined only by the diffusion current, limited by the series resistance (R_{s}) of the cell [14]. In order to more accurate quantitative analysis of the dark JV characteristic in region A, B, C in the perovskite solar cells regardless of region D, Eq. 8 can be futherly deduced in dark (not consider J_{L}, R_{s}):
In region A of Fig. 4a, the dark current is mainly affected by the shunt current under the small applied bias voltage. With the bias voltage increase, recombination current is much larger than diffusion current in dark JV characteristics of planar PSCs, as shown in region B of Fig. 4a. The slope of region B is less than slope of region C, the steep increment of the current results from a diffusiondominated current [14]. The dark currentvoltage characteristic is in a single exponential relationship in region B and region C, respectively.
In order to quantitative calculate diffusion current and recombination current respectively, the Eq. 12 is further rewritten by taking into account the heterojunction diffusion model [29, 30]:
In Eq. 13, the first term is the shunt current corresponding to region A in Fig. 4a. The second term is the recombination current (ShockleyReadHall recombination), m_{1r}=2. The third term is the diffusion current (including carrier directly recombination), m_{1d}=1 [14, 36, 37]. According to Eq. 9, Eq. 13 can be furtherly derived:
In the Eq. 14, m_{r}=m_{1r}+m_{2r}=4, m_{d}=m_{1d}+m_{2d}=2. The dark current in Fig. 4a can be fitted by Eq. 14. The calculation results of three parameters (R_{sh}, J_{r}, J_{d}) are shown in Table 4.
Table 4 R_{sh}, J_{r}, J_{d }values derived from fitting dark current in Fig. 4a by using Eq. 14
We fit the dark current of inverted planar heterojunction PSCs in other literatures in order to further verify the formula based on double PN junction equivalent circuit (Fig. 1b) [8].
Table 5 R_{sh}, J_{r}, J_{d }derived from fitting dark current of inverted planar PSCs in Fig. 4b
As shown in Table 5, the parameter R_{sh} of inverted planar PSCs with different hole blocking layers are 1, 5, and 35 MΩ cm^{2}. The recombination current (J_{r}) of corresponding devices are 3×10^{6}, 1.5×10^{6}, and 5×10^{7 }mA cm^{2}. Both of them indicate that hole blocking layer blocks hole injection into the diode, effectively reducing the shunt current and recombination current. Compared to the BCP, the device with PFN shows better holeblocking property. The same conclusion obtained by comparing the dark current at 100 mV as discussed in literatures. Meanwhile, the devices without holeblocking layer showed larger dark current under reverse bias, mainly due to the larger hole injection into the diode [8]. Moreover, the PFN enhanced electron injection and extraction, which can be verified by drastically increased diffusion current J_{d} (from 8×10^{10 }to 1×10^{9 }mA cm^{2}). This conclusion confirms the speculation in the literature: PFN improves the electron injection and extraction in PSC devices [8]. Therefore, the new model proposed in this study can be universal and effective to analyze carrier recombination and transportation.
4 Conclusion
In conclusion, we built up a double PN junction equivalent circuit to fit JV curves of PIN planar structure heterojunction PSCs. The new method focuses on the relationship between the diode ideal factor and the carrier recombination from the interface defects. By varying Cl^{} content in the CH_{3}NH_{3}PbI_{3} perovskite film, we found that the value of m drastically diminished (decreased) with the perovskite film quality improvement. In order to quantitatively analyze the correlation mechanism of dark current under different bias voltages, a new equation based on the double PN junction equivalent circuit has been proposed to analyze the dark currentvoltage curve. Consequently, carrier recombination and loss reduction could be reflected in R_{sh} and J_{r}. The carrier transmission could be reflected on the parameter J_{d}. Based on the double PN junction equivalent circuit, the JV curve in light and in dark could be fitted respectively, helping us analyze the working mechanism and improve the efficiency of planar PSCs.
Acknowledgment
Financial support from the 973 Program of China (No. 2014CB643506 and 2013CB922104), the China Scholarship Council (No. 201506165038), the Natural Science Foundation of China (No. 21673091), the Natural Science Foundation of Hubei Province (No. ZRZ2015000203), Technology Creative Project of Excellent Middle & Young Team of Hubei Province (No. T201511), and the Wuhan National High Magnetic Field Center (2015KF18) is acknowledged.
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Citation Information
Peizhe Liao, Xiaojuan Zhao, Guolong Li, Yan Shen, Mingkui Wang, A New Method for Fitting CurrentVoltage Curves of Planar Heterojunction Perovskite Solar Cells. NanoMicro Lett.(2018) 10: 5. http://dx.doi.org/10.1007/s408200170159z
History
Received: 22 July 2017 / Accepted: 11 September 2017