Practical Efficiency Limit of Methylammonium Lead Iodide Perovskite

College of Science and Engineering, Hamad Bin Khalifa University, Doha, Qatar. ‡Qatar Environment and Energy Research Institute, Hamad Bin Khalifa U...
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Cite This: J. Phys. Chem. Lett. 2018, 9, 426−434

Practical Efficiency Limit of Methylammonium Lead Iodide Perovskite (CH3NH3PbI3) Solar Cells

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ecently, hybrid perovskite solar cells (PSCs) have emerged in photovoltaic (PV) research activities as an active competitor. As a result, its power conversion efficiency (PCE) has shown an unprecedented benchmark increasing from 3.8% in 20091 to 22.1% in just 7 years.2 The key feature that has enabled such a PV family to reach high performance within a short period of time is the conjunction of the best of both conventional inorganic and organic solar cells by utilizing the efficient operation principles from the first one while utilizing relatively simple fabrication techniques from the second one, thereby providing a possible pathway toward economical solar energy alternatives.3−5 Basically, the high efficiency achieved by a PSC has been attributed to its convenient optoelectronic properties such as the suitable band gap,6−11 high absorption coefficient over the solar spectrum,5,12 low exciton binding energy,13,14 and efficient charge transport characteristics.15−19 Interestingly, a very wide range of device designs and fabrication methods has resulted in efficient cells. Besides illustrating how rich the field is, this indicates more importantly that there is still a lot of room for optimization of the device. Despite the considerable progress toward high PCE, which is one of the key performance indicators for solar cells,20 detailed analysis is needed to achieve a better understanding of the actual factors limiting the performance of PSCs and to approach the thermodynamic efficiency limits of around 30− 33% for the corresponding band gap of 1.2−1.6 eV.21 More specifically, this necessitates further investigations about the nature of the recombination mechanisms at work and the impact on charge transfer dynamics and PCEs in the actual PSC devices.13,22,23 Previous studies have confirmed that recombination kinetics has served as the basis for technological advancement for other semiconductors such as silicon.24 The recombination losses arise from band-to-band radiative recombination (UB) and nonradiative defect-assisted recombination known as Shockley−Read−Hall (USRH)25 under a low injection level in solar cells. Radiative recombination is a process based on the reciprocity relation for photon emission and absorption, whereas nonradiative trap-assisted recombination is due to the defects in the material.22 In PSCs, the effect of nonradiative recombination is dominant and has a stronger influence, when compared to radiative recombination; thereby, it is the mechanism that practically limits the efficiency of PSCs.13,25−29 Remarkably, the reported rates of defect-assisted recombination for CH3NH3PBI3 vary within a very wide range13 due to the large variety of processes used for perovskite film growth and the complexity of measuring trap-assisted recombination rates. Depending on the material quality and process conditions employed for perovskite absorber growth, the reported trap densities are between 10 8 and 10 16 cm−3.22,30−36 Figure 1 illustrates the variation of the band-toband recombination and product of the trap density and capture cross section for perovskite solar absorbers reported in the literature. The radiative recombination is found to vary by 2 orders of magnitude, whereas material quality (quantified by © 2018 American Chemical Society

Figure 1. Range of (a) the band-to-band radiative recombination constant and (b) product of the trap density and cross section recombination reported in the literature for CH3NH3PBI3. For trapassisted recombination, a product of the trap density (NT) and capture cross section (σ) is employed as a measure of SRH recombination. The reported rates of radiative and nonradiative trap-assisted recombination have a broad range due to the large variety of fabrication practices used for the perovskite film and the complexity of measuring trap-assisted recombination rates.

the product of the trap density (NT) and capture cross section (σ)) has a huge range from ∼101 to 10−4 cm−1. The extraction or recombination depends collectively on multiple properties such as recombination kinetics and charge dynamics and device design.37 Therefore, a comprehensive understanding of the complex relationship between these different factors is needed to estimate the practical limits of PSCs and identify the achievable value. Performance wise, despite the vast effort and huge set of developed PSCs, the best reported short-circuit currents Jsc (24.2 mA/cm2 2) are considerably smaller than the expected Published: January 18, 2018 426

DOI: 10.1021/acs.jpclett.7b03343 J. Phys. Chem. Lett. 2018, 9, 426−434

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The Journal of Physical Chemistry Letters

the extreme reported values of both UB and USRH in the literature. Unlike most of the previous works, a comprehensive device optimization model is coupled with actual experimental measurements. This gives a better estimation of the practical efficiency limits of PSCs in an extensive range of absorber thicknesses and recombination rates. By employing a multilayered device simulation and optimization method and reproducing the typical experimental conditions of PSCs, it was found that for a given material quality an optimal thickness that maximizes the conversion efficiency can be much more than the usually employed 0.3−0.4 μm range.43,51,52 By increasing the quality of growth and device fabrication, the recombination rates can be reduced, in principle to their fundamental intrinsic values. In such a case, the calculations demonstrate that the maximum possible thickness should be around 1.3 μm for low-recombination regimes, which can boost Jsc to 28.78 mA/cm2 with an efficiency of 29.53%, thereby approaching the detailed balance limit of 31%.39 Remarkably, Jsc is bordering on the upper bound of 29 mA/cm2 for an energy gap of 1.5 eV.38 This is consistent with other solar cell technologies as well.53−55 Moreover, it is shown that there exists a wide range of the absorber thickness to approach PCE limits with minor compromise of ∼0.5%. By considering collectively the effects of radiative and nonradiative recombination on the performance of existing absorber qualities, the presented approach can provide a guide to design optimal device geometry for efficient CH3NH3PBI3-based PSCs. As aforementioned, charge carrier recombination and its effects on the PCE are vital in predicting the upper limits for PSCs. The performance depends on both UB and USRH recombination losses. To assess the effect of the existing PV absorber qualities, different scenarios for recombination losses are studied. Figure 2 presents the resulting optimal Jsc, Voc, and η for the four cases. The numerical values beside the optimized thickness are summarized in Table 1. The maximum obtained value for η is 29.53%, with a corresponding Jsc of 28.78 mA/cm2 and Voc of

theoretical values of a semiconducting absorber of 1.5 eV. Conceptually, 29 mA/cm2 should be achievable, and this should result in a better PCE (η), if the open-circuit voltage (Voc) and fill factor (FF) are within the theoretical limit ranges.38 However, the increase of Jsc needs either to have better light management or to increase the thickness of the perovskite layer.39 The second is mainly limited by materials’ quality, and hence, the maximum obtained value of Jsc is not fundamental. This is due to the reduced qualities of both the perovskite layer and the interfaces with other layers in the device. The first one limits the possible thickness of the perovskite layer and, hence, the photogenerated carriers and Jsc, while the second affects the carrier’s extraction and results in further reduction in Jsc. Optimum thicknesses for device layers will ensure that charge collection is efficient by balancing both absorption and recombination as a whole in the device. There has been some focus on the practical limitations of Jsc as a function of geometry and morphology.40−43 Correa-Baena et al.42 analyzed the effect of the perovskite capping layer on the generated light current and optical properties. It was found that an absorber layer of 0.5 μm with high-quality crystal size was sufficient to provide champion cell efficiency of 20.8%. Yuan et al.44 showed that a pinhole-free absorb layer of 0.8 μm was possible with 16.8% PCE using solvent engineering. Although initial reports have suggested the optimized device thickness in the range of 0.3−0.4 μm,4,43 this should increase as the crystalline quality of the perovskite has been improving ever since.45 The first hybrid perovskite electrochemical cell showed a Jsc of 11 mA/cm2,1 which has evolved rapidly by employing all-solid device architectures to reach the maximum of 24.2 mA/cm2 in 2017.2 The question of “How thin is too thin for efficient PSCs?” is primarily driven by the fact that the levelized cost of energy is determined by the material cost and the efficiency of the solar cell, which are both dependent on absorber thickness.46 Thus, it is timely to investigate computationally the practical efficiency limit of PSCs utilizing the experimental report values of the diverse quality of CH3NH3PbI3. So far, the PCE limit for PSCs has been estimated generally by two methods. In the first, PCE analysis has been carried out for external radiative limits,47,48 while, on the other one, the detailed balance approach with alterations for nonradiative recombination and light trapping is investigated.39,49,50 Sha et al. have shown that the detailed balance limit for CH3NH3PBI3 perovskite is 31%.39 However, to have better practical estimators, it is important to consider nonradiative recombination rates as well, which is the dominant recombination effect in PSCs. The detailed balance method is not well suited from the perspective of charge transport and extraction.49 Contrarily, the recently developed full space optimization method37 performs an in-depth analysis for PSC devices by incorporating the practical device architecture and charge dynamics such as carrier recombination, mobility, and geometry optimization.37 This can provide better assessment for the limits achievable by a multilayered device. Hence, by considering both radiative and nonradiative recombination pathways for extreme cases (minimum and maximum reported recombination rates) reported in the literature for CH3NH3PbI3, realistic limits for PSCs can be estimated. This shall provide a guideline to experimentalists for improving multilayered perovskite device performance. In this work, quantification of charge extraction and recombination kinetics in PSCs is carried out by considering

Figure 2. Performance limits for Jsc and Voc with the corresponding η estimated for the four studied cases. An η limit of 29.53% with the corresponding 28.78 mA/cm2 and 1.151 V is expected when minimum reported USRH(NT = 7.60 × 108 cm−3 and σ = 2.93 × 10−13 cm2) and B of 7.4 × 10−11 cm3/s are considered. The detailed balance limit for a CH3NH3PBI3-based perovskite absorber is η = 31%,39 with Jsc of 29 mA/cm2 for a band gap material of 1.5 eV.38 427

DOI: 10.1021/acs.jpclett.7b03343 J. Phys. Chem. Lett. 2018, 9, 426−434

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The Journal of Physical Chemistry Letters

Table 1. Cell Characteristics from Device Optimization for Different Values of the Recombination Constants B and NT × σ thickness [μm] Case

B, Nt × σ

spiro- OMeTAD

perovskite

TiO2

Voc [V]

Jsc [mA/cm2]

FF [%]

η [%]

1 2 3 4

min, min min, max max, min max, max

0.012 0.053 0.012 0.053

1.32 0.81 1.27 0.81

0.04 0.04 0.04 0.04

1.151 0.994 1.086 0.993

28.78 27.64 28.73 27.63

88.51 73.17 86.86 73.24

29.53 20.11 27.12 20.11

Table 2. Considered Cases for Maximum and Minimum Reported Band-to-Band and Trap-Assisted Recombination band-to-band recombination (UB) Case 1 2 3 4

B [cm3/s] 7.4 7.4 1.10 1.10

× × × ×

−1160

10 10−1160 10−958 10−958

trap-assisted SRH recombination (USRH) trap density NT [cm−3] 7.60 1.30 7.60 1.30

× × × ×

861

10 101530 10861 101530

cross section σ [cm2] 2.93 2.50 2.93 2.50

1.15 V. Considering the reported single-crystal properties, it is assumed to be the practical limit where it is obtained with minimum reported USRH and B. The results are promising considering that the reported detailed balance limit for a CH3NH3PBI3-based perovskite absorber is 31%.39 Likewise, Jsc is approaching the upper limit of 29 mA/cm2 for a band gap material of 1.5 eV.38 The second best performance is for Case 3, with a maximum reported B and minimum USRH, which has a maximum η of 27.12%, where Jsc is found to be 28.73 along with a Voc of 1.086. On the other hand, both Cases 2 and Case 4 (associated with maximum USRH) result in relatively lower performance. This indicates that USRH is more critical than UB in PSCs. As for the optimized device design (thicknesses in this case), Table 1 highlights the optimized values with the associated performance parameters. Many conclusions can be derived from the analysis of these results. A few important observations can be pointed out: Effect of UB: At a minimum USRH, the change of B from 1.10 × 10−9 to 7.4 × 10−11 cm3/s enhances the relative η by 8% from Case 3 to Case 1. The main enhancement comes from the increase of Voc with a relative enhancement of 5.64%. However, when USRH is at its peak, UB becomes secondary and we notice only a small difference between Case 2 and Case 4. Effect of USRH: A significant effect on the cell performance is observed between the lowest USRH (single-crystal perovskite) and its maximum case (for the mesoscopic structure). Between the two cases, the relative difference of η is about 47% when B is the minimum. This is much larger than what was observed when we considered the extreme cases of band-to-band recombination. This is due to the wider range of variation of USRH as noted in Table 2. The 47% improvement in η is related to the enhanced quality of the absorber as we move from mesoscopic deep level defect material to single crystals. Additional simulations were performed to assess the effect of trap level on the performance of the PSC in Case 1. By varying ET from the conduction band (shallow level) to midgap (deep level), the absolute difference in η was 0.1%. This is expected as Case 1 has low trap density (∼108), and to notice any difference in shallow level, higher trap density is required. On the other hand, we observe 8% enhancement in η as the bandto-band recombination constant B changes from its highest to its minimum value. Effect on the optimum absorber thickness: The optimal absorber thickness is a trade-off between absorption and recombination. Nonetheless, the estimated optimal thickness of

× × × ×

−1361

10 10−1530 10−1361 10−1530

NT × σ [cm−1]

ET − Ei [eV]

2.23 × 10−4 3.25 2.23 × 10−4 3.25

Eg/2 0.1330 Eg/2 0.1330

Figure 3. Calculated EQE for the optimized cells of the four considered cases. The main discrepancy is due to the wider fluctuation of USRH.

CH3NH3PBI3 is noticeably larger than the commonly reported values. At minimum USRH cases (Case 1 and Case 3), the maximum η are obtained for thicknesses ranging between 1.27 and 1.32 μm. However, in the high USRH regime, there is a negligible change in optimal thickness of 0.81 μm for Case 2 and Case 4 due to the dominance of USRH in this limit. Experimentally, thicker perovskite films of 0.8 μm for reducing the pinholes associated with thin films have already been reported with a champion efficiency of 16.8%.56 Effect on hole transport material (HTM) and electron transport material (ETM) thickness: For spiro-OMeTAD as a HTM with low charge carrier mobility, i.e., ∼10−4 cm2/(V s),4 the optimized values of 0.013, 0.053, 0.012, and 0.053 μm for Cases 1−4, respectively, are estimated to optimally collect the charges. Likewise, for TiO2, the optimization yields a thickness value of lower bound 0.04 μm for low carrier mobility of 10−1 cm2/(V s), as specified in Table 1. This is in agreement with the reported cells. For example, Choi et al.57 showed that with an average thickness of 0.04 μm a maximum η can be observed experimentally. A further increase in TiO2 thickness resulted in higher series resistance, thereby increasing losses. Figure 3 presents the calculated external quantum efficiency (EQE) plot for the optimized cells of the four considered cases. 428

DOI: 10.1021/acs.jpclett.7b03343 J. Phys. Chem. Lett. 2018, 9, 426−434

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The Journal of Physical Chemistry Letters

Figure 4. Key performance parameters (a) η, (b) Voc, (c) FF, (d) Jsc, and (e) Jo vs the perovskite absorber thickness for the optimized PSCs in the four considered cases. (f) Legends for the curves. Yellow and blue highlighted regions show arbitrary thickness ranges of 0.5−0.8 and 0.8−1.4 μm, respectively. The losses on Voc and FF are related to dependence of the leakage current Jo of the junction. Cases 1 and 3 have an optimized thickness range of around 0.8−1.4 μm, whereas Cases 2 and 4’s optimal thickness range is around 0.5−0.8 μm. The relative percentage losses in η are found to be 2.53 and 1.81% for Cases 1 and 3 when the thickness is reduced from 1.4 to 0.8 μm, respectively. Cases 2 and 4 result in η reductions of 2.71 and 2.72% in the range of 0.5−0.8 μm.

4a−d. Although an optimum device thickness results from the competition between absorption and recombination and results in a single value of thickness, it is clear that there is wide ranges of tabs with small losses from the optimal values of all four considered cases. The yellow and blue highlighted regions in the plots show thickness ranges of 0.5−0.8 and 0.8−1.4 μm, respectively. The efficiency losses are found to be mainly governed by Voc and FF. Moreover, the recombination losses on Voc are related to dependence of the leakage current Jo, which is a strong function of absorber thickness. On the basis of the analysis in the different recombination and thickness regimes, these PSCs were found to limited by, in descending order of importance, Voc, FF, and Jsc. Cases 1 and 3 have an optimized thickness range of around 0.8−1.4 μm, whereas Cases 2 and 4 have optimal thickness ranges of around 0.5−0.8 μm for nearly the same maximum efficiency. Under low recombination rates

This is to compare the effect of recombination on the photogenerated current under AM1.5g illumination. EQE, defined as the fraction of extracted free charge carriers collected from incident photons, is another measure of the collective effect of absorption and charge extraction on the cell performance. Overall, for all cases, the results show that under the visible spectrum (400−700 nm) PSCs showed good performance. Clearly, the variation of B over a range of 2 orders of magnitude does not affect the EQE if USRH takes its minimum. This can be observed in Case 1 and Case 3 with nearly 100% EQE until it plummets down to the band gap limit. On the other hand, Case 2 and Case 4 show, on average, 96% EQE within the range of 400−550 nm; then it starts reducing at longer wavelengths, indicating recombination losses near the back-contact side of the cell. The absorber thickness effects on key performance parameters Jsc, Voc, FF, and η of PSCs are shown in Figure 429

DOI: 10.1021/acs.jpclett.7b03343 J. Phys. Chem. Lett. 2018, 9, 426−434

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The Journal of Physical Chemistry Letters (Case 1), η is found to slightly get reduced from 29.51 to 28.76% in the thickness range from 1.4 to 0.8 μm, as shown in Figure 4a. Conversion efficiency dies rapidly for further thickness reduction. Similarly, in Case 3, the absolute change in η is found to be 0.49% in the absorber thickness range of 0.8−1.4 μm. Interestingly, a thickness between 0.4 and 0.5 μm should be enough to increase efficiency from the currently reported a maximum of 22.1%3 to about 25% with high-quality perovskites. For highUSRH, both Case 2 and Case 4, showed an absolute η reduction of 0.54% from an optimum 20.11% by reducing the thickness from 0.8 to 0.5 μm. Again, it is important to re-emphasize that η as a function of absorber thickness has a wide plateau allowing easier realization. Voc and its dependence on thickness are presented in Figure 4b. As Voc = (kT/q) ln (Jsc/Jo + 1) and from Figure 4d (Jsc), the effect on Voc is found to be more significant than those on Jsc. The efficiency losses are mainly due to Voc and FF. The losses on Voc are related to dependence of the leakage current (Jo) of the junction and the thickness of the absorber. Figure 4e shows Jo of different cases studied as a function of absorber thickness. It is clear that the Jo grows faster with thickness than Jsc. This results in a net reduction of Voc with thickness. Therefore, Voc was found to be reduced as a function of absorber thickness due to an increase in recombination losses. The relative changes of 1.17 and 1.25% for Case 1 and Case 3 were observed for Voc reduction by increasing the thickness from 0.8 to 1.4 μm. Cases 2 and 4 showed a similar negative correlation with thickness, with relative percentage reductions of 2.65 and 2.71% from 0.8 to 0.5 μm. Figure 4c highlights the impact on fill factor with different recombination scenarios. Because FF is a strong function of Voc and Jo, a similar trend as that for Voc was observed, which can be observed by the exponential increase of Jo with respect to the thickness, as highlighted in Figure 4e. The FF for Case 1, with minimum B and USRH, was decreased from 88.9 to 88.6% from 0.8 to 1.4 μm, with a relative percentage reduction of 0.36%. Case 3, with maximum UB and minimum B, showed a relative percentage reduction of 1.01% from 87.56% at 0.8 μm to 86.67% at 1.4 μm. Case 2 and Case 4 working, with maximum B, showed a relatively higher reduction rate with 2.34 and 2.39% when changed from 0.5 to 0.8 μm. Figure 4d shows the effect of absorber thickness on the photogenerated current. Optimizing the absorber thickness ensures that the charges are efficiently generated and collected. However, there exists a broad maximum of absorber thickness that shows that maximum achievable Jsc is similar after a particular thickness. Jsc was found to saturate beyond 1.2 μm even in the worst-case scenario of highest SRH. This is because, considering the used values, the diffusion length in the worse scenario remains slightly higher than 2 μm. In the range of 0.8− 1.4 μm, the simulated Jsc changes by no more than 1.15 mA/ cm2 (absolute) for Case 1 and Case 3. Likewise, for Cases 2 and 4 with maximum SRH, the calculated Jsc was reduced by an absolute 2.05 mA/cm2 when the thickness was reduced from 0.8 to 0.5 μm. Therefore, understanding of ideal thickness to improve Voc, FF and Jsc can assist in reaching performance limits. Moreover, by increasing the quality of the films and device, the recombination losses can be diminished to intrinsic values. To further understand the charge carrier dynamics of extreme recombination cases, a simulation was performed with a constant η of 20% with fixed thicknesses of TiO2 and spiro-OMeTAD at optimum values, as highlighted in Figure 5. The aim was to analyze the current reported values of

Figure 5. Optimum perovskite absorber thickness to achieve an η of 20%. Fixed thicknesses of TiO2 and spiro-OMeTAD at optimum values were considered. The required perovskite thickness was 0.19 μm for Case 1, 0.60 μm for Case 2, 0.22 μm for Case 3, and 0.61 μm for Case 4.

thicknesses and their associated efficiencies. It shows that even at a thickness of 0.19 μm with a high-quality material, such as Case 1, an η of 20% can be achieved. By increasing the radiative recombination rate to 1.10 × 10−9 cm3/s58 as in Case 3, a thickness of 0.22 μm was required to achieve an η of 20%. Cases 2 and 4 with more deep level defects showed absorber thicknesses of 0.60 and 0.61 μm for achieving an η of 20%. Higher extrinsic SRH recombination showed more losses in bulk, thereby increasing the needed absorber thickness and limiting the practical values for efficiencies due to the optimization of absorption and recombination. Practical limits for PSCs were explored using device simulation and an optimization method. For realistic conditions, extreme values for the band-to-band recombination and trap-assisted SRH recombination reported in the literature were used. This provides an understanding of efficiency limits of PSCs in an extensive range of absorber thicknesses and recombination rates. The maximum PCE limit of 29.53% with corresponding Jsc = 28.78 mA/cm2 and Voc = 1.15 V was observed with a minimum reported SRH, and band-to-band recombination was simulated for PSC devices. The results were promising considering that the reported detailed balance limit for the CH3NH3PBI3-based perovskite absorber is 31%.39 Likewise, Jsc was found to approach the upper limit of 29 mA/ cm2 for a band gap material of 1.5 eV.38 The primary cause for the variation is that in our methodology, experimental recombination losses are considered along with the driftdiffusion approach for the complete device. On the contrary, the detailed balance approach does not take into account the aspects of charge carrier recombination and nonradiative recombination, which are extremely vital for optimization between absorption and recombination and its impact on absorber thickness. By introducing the maximum reported extrinsic SRH recombination, the PCE was reduced to 20.1%. Overall, trap-assisted recombination was found to be the main degradation factor for the optoelectronic performance of PSCs. We demonstrated that by increasing the growth quality and reducing recombination the thickness of the absorber can be increased up to ∼1300 nm to approach the performance limits. Moreover, it was also shown that there exists a range of optimal 430

DOI: 10.1021/acs.jpclett.7b03343 J. Phys. Chem. Lett. 2018, 9, 426−434

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The Journal of Physical Chemistry Letters absorber thicknesses that can be employed to obtain a nearpractical limit with an absolute percentage PCE reduction of ∼0.5%. Therefore, an understanding of ideal thickness to improve key parameters can assist in reaching limits. With the presented combination of the theoretical framework and experimentally obtained measurements, an accurate analysis for practical efficiency limits achievable by existing perovskite absorber qualities can be estimated.

Case 1: [(Bmin ), (NT,min , σmin)]at shallowest level (ET ≈ Ec or Ev )

Case 2: [(Bmin ), (NT,max , σmax)]

Case 3: [(Bmax ), (NT,min , σmin)] Case 4: [(Bmax ), (NT,max , σmax )]at deepest level (ET ≈ Ei)

■

where the subscripts min and max show the minimum and maximum reported values for respective recombination losses as listed in Table 2. The full recombination rates for UB and USRH are shown by eqs 1 and 2.

COMPUTATIONAL METHODOLOGY Using multiproperty device optimization and the extreme reported values of both radiative band-to-band and nonradiative SRH recombination in CH3NH3PBI3, the practical limits for PSCs are estimated. Full device performance simulation and multiproperty optimization have been performed using a sequentially coupled interface between MATLAB and the Solar Cell Capacitance Simulator (SCAPS).37,59 The simulation code calculates the performance of given cell designs, while the device design is optimized in MATLAB software. Figure 6a shows the general device structure for planar PSCs consisting of contacts, TiO2 as an ETM, CH3NH3PBI3

UB(t ) = B(n(t )p(t ) − n i 2)

(1)

and USRH(t ) = NTσvth

n(t )p(t ) − n i 2

( E kT− E )

n(t ) + p(t ) + 2n i cosh

i

T

(2)

where ni is the intrinsic carrier density while n(t) and p(t) are their densities after excitation, vth is the carrier thermal velocity (assumed 107 cm/s at room temperature), ET is the trap energy level and Ei is the midgap, kB is Boltzmann’s constant, and T is the absolute temperature in Kelvin. The maximum reported B is 1.10 × 10−9 cm3/s,58 while the minimum is 7.4 × 10−11 cm3/ s,60 as shown in Table 2. USRH is of fundamental importance in determining the limits of the solar cell device as it dominates the recombination and hence limits the absorber layer thickness. Therefore, from a device perspective, an optimum absorber thickness for maximum efficiency would be less for a defective material occupied with a high density of deep traps than a material with a low density of shallow traps. The parameters NT, σ, and ET are employed as performance metrics to check the limits of defect-assisted recombination rates reported as shown in Table 2. For the maximum reported USRH losses, results from Deep Level Transient Spectroscopy (DLTS) were considered.30 The maximum reported value for NT is 1.30 × 1015 cm−3, with σ as 2.50 × 10−15 cm2 and a deep defect level at ET = 0.62 eV below the conduction band (ET − Ei = 0.13).30 It can be noted that the value of NT × σ varies over a 4 orders of magnitude range (3.25−2.23 × 10−4), while the values suggested for the constant B vary only over 2 orders of magnitude. This is mainly due to the improved material quality when a single-crystal perovskite is considered with a density of traps as low as 7.60 × 108 cm−3 and σ = 2.93 × 10−13 cm2.61 The variation in band-to-band radiative recombination for the same material, apart from experimental uncertainty, may arise from a change in crystallite size and dielectric function because of the complex compounds resulting from atomic diffusion from a mesoporous metal oxide and other layers to the perovskite.16 Note that all considered values are related to a deep defect level close to the midgap. Therefore, one can expect lower recombination rates in the case of shallow traps. For the absorbing CH3NH3PbI3 properties, the used values are obtained from the literature, where a band gap (Eg) of 1.5 eV62 and electron affinity (χ) of 3.9 eV63 are considered for the simulations. The relative dielectric permittivity (ε) of 10 4, conduction band density of states of 3.9 × 1018 cm−3, and valence band density of states of 2.75 × 1018 cm−34 are used. For mobilities, 20 cm2/V·s is assumed for both electron and holes (μn and μp).64 This can be justified by the weaker dependence of the mobilities upon point defect concentration,

Figure 6. (a) Device structure and energy band diagram of a CH3NH3PBI3 PSC consisting of TiO2 as an electron transport material and spiro-OMeTAD as a hole transport material. (b) Radiative recombination pathway for perovskite absorber. Here, B denotes the band-to-band recombination constant. (c) Trap-assisted SRH recombination for perovskite with key parameters of trap energy level (ET), trap density (NT), and capture cross section area (σ).

perovskite absorber, and spiro-OMeTAD as a HTM. To explore the device improvement range and predict efficiency limits, four extreme scenarios under 1 sun conditions are considered by considering the maximum and minimum reported UB (quantified by the band-to-band recombination constant B) and USRH (quantified by NT, σ, and trap level (ET)) recombination, as highlighted in Figure 6b,c, respectively. Under low injection conditions, the recombination rate USRH associated with deep traps (ET close to Ei) is higher than that related to shallow traps (ET close to Ec or Ev). The subscripts i, c, and v are for intrinsic, conduction band, and valence band, respectively. The difference depends on the ratio of the intrinsic density to the effective doping concentration and the defect level. Therefore, the considered cases are 431

DOI: 10.1021/acs.jpclett.7b03343 J. Phys. Chem. Lett. 2018, 9, 426−434

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The Journal of Physical Chemistry Letters

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while the lifetime can vary over many orders of magnitude. Finally, ideal ohmic front and back contacts with no resistance are considered. The absorption spectrum is extracted from ref 4. All of the simulations are conducted under standard conditions corresponding to an AM1.5g solar spectrum and a temperature of 300 K. Details of TiO2 and spiro-OMeTAD properties employed in optimization have been extracted also from the same ref 4 and references therein. The optimized parameters are the thicknesses of the three layers in the complete perovskite device. Thicknesses of the ETM, perovskite, and HTM are varied within very wide ranges to maximize the efficiency based on the descriptor and objective function formulated by eq 3 ηmax = max η t ETM , tabs , t HTM

(3)

where η(t) = JscVocFF/Pin is the PCE and tETM, tabs, and tHTM are the thicknesses of the ETM, absorber, and HTM layers, respectively. The optimization is constrained by lower and upper bounds. The incident power (Pin) of AM1.5g is 100.3 mW/cm2. The multiproperty optimization is performed using gradient-based optimization for faster computational convergence.37,65 Numerical convergence is set at a stopping function condition of 1 × 10−6 for tolerance in function evaluation, whereas initial guesses are provided for each parameter within a physically acceptable range. As a result, the optimal thicknesses for all of the layers are obtained for different recombination pathways.

Ahmer A. B. Baloch*,† M. I. Hossain‡ N. Tabet†,‡ F. H. Alharbi†,‡ †

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College of Science and Engineering, Hamad Bin Khalifa University, Doha, Qatar ‡ Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Doha, Qatar

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been funded by the Qatar National Research Fund under a Qatar National Priority Project, NPRP-6-931-2382. We also acknowledge the support of the Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Qatar Foundation.

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REFERENCES

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