Modeling and Quantifying Optimal Dynamics of Extraction of Charge Carriers in the Operation of Perovskite Solar Cells

In this paper, a mathematical model and the relevant computer code are developed to quantify the extraction probability rate of charge carriers (EPRCC) in a perovskite solar cell of the structure: Glass/ITO/PEDOT: PSS/CH3NH3PbI3/PC60BM/Al to investigate the influence of interfaces and grain boundaries. It is found that, without passivation, the probability of an electron generated near the anode reaches to the cathode is only 35%, while by passivating the interfaces and grain boundaries, this probability increases to about 60% at maximum power point condition. Likewise, without passivation, the probability of a hole generated near the electron transport layer‐active layer interface reaches to the hole transport layer is only 15%, while by passivating the interfaces and grain boundaries, this probability increases to about 45% at maximum power point condition. The same calculation has been done at the short‐circuit current condition, and it is found that at the maximum power point condition, passivation works better for increasing the EPRCC than at the short circuit current condition. The authors have also investigated the influence of grain boundary sizes on the EPRCC, and the results show that the EPRCC is almost grain boundary size independent.


Introduction
Solar energy is clean and nearly inexhaustible. Solar cells are used to convert solar energy directly into electricity; thereby, these devices can be regarded as an alternative to meet the www.advmatinterfaces.de recombine before reaching the corresponding transport layer, [17] leading to lower PCE. Hence, a shorter path from HTL or ETL for charge transport can minimize the recombination of charge carriers and reduce the energetic losses. [18] It is very well known that the passivation of interfaces is an effective strategy to improve the efficiency of PSCs. [19] However, no work has been done to model and quantify the influence of passivating the interfaces and grain boundaries on the extraction probability rate of charge carriers (EPRCC) in PSCs. In this paper, a mathematical model and the relevant computer codes are developed to quantify EPRCC in a perovskite solar cell of the structure: Glass/ITO/PEDOT: PSS/CH3NH3PbI3/PC60BM/ Al. In our model, different rates of the tail state recombination are considered at grain boundaries, interfaces and other parts of the active layer to simulate the EPRCC at different positions x in the active layer. The concept of introducing the extraction probability rate is based on the fact that when electrons and holes are generated within the active layer, they move toward the cathode and anode, respectively, and some of these electrons and holes may be recombined radiatively or nonradiatively on their way. In the model, we have also incorporated the influence of the size of grain boundaries and passivation at the interfaces and grain boundaries on the EPRCC.
In addition to the capability of our model in simulating EPRCC, it is more comprehensive and rigorous than other mathematical models [19c,20] presented in the literature review. In our model, the heat transfer equations are solved by iteration to determine the operating temperature of a PSC. As such, it can be regarded as presenting the operation of PSCs more accurately than the previous conventional models, [19c,20] where the operating temperature is assumed to be a constant. Here, the rate of tail state recombination, which converts the charge carrier's energy to thermal energy, changes at each iteration; hence, the operating temperature of solar cells also varies accordingly. Therefore, considering a constant operating temperature misrepresents the operation of an ordinary solar cell where the operating temperature always rises due to the tail state recombination, particularly at higher voltages where tail state recombination is high.
The outcomes of this mathematical model may be expected to quantify and understand the influence of different factors on the EPRCC. To the best of the authors knowledge, this is the first attempt to introduce such a model with the concept of probability in simulating the EPRCC in PSCs.

Experimental Section
To simulate the EPRCC, first, the authors applied the finite difference method by dividing the active layer of a PSC with the structure of Glass/ITO/ PEDOT:PSS/CH 3 NH 3 PbI 3 /PC 60 BM/Al into N (1, 2, 3…, N) meshes as shown in Figure 1. Then, the authors had applied the Optical Transfer Matrix Method (OTMM) to calculate the charge carrier-generation rate at each position in the active layer. The details of OTMM were presented in the earlier work. [21] In the model, Poisson's and drift-diffusion equations were solved by incorporating the heat-transfer equations to improve the accuracy as described in detail in the earlier works. [22] The authors assumed that all the electrons that were generated in each mesh within the active layer move toward the cathode (right) and all holes were moved toward the anode (left) due to the work function difference in the electrodes. Furthermore, considering the operation of a PSC under the steadystate condition, the number of charge carriers that were generated and recombine was proportional to the corresponding generation rates. The authors denoted the rate of generation and recombination of electrons in the i th mesh by G i , and R i , respectively, then the probability of recombination, P Re−m of an electron within the m th mesh could be written as: www.advmatinterfaces.de total number of charge carriers recombined within meshes from i = 1 to m-1. Thus, the denominator of Eq. (1) gave the total charge carriers present in the m th mesh. Then, the probability, P Ge−m , that this electron did not recombine in the m th mesh could be given by: The electron surviving in the m th mesh would move to the next mesh and then might reach the cathode by surviving in each mesh (Figure 2). The authors had derived the probability P e−m of an electron located at mesh number m reached the cathode as: where P Gh−m was the probability analogous to Eq.
(2) that a hole did not recombine in the m th mesh, and it could be given as: In PSCs, the interfaces between the active layer, ETL (electron transport layer, which was PC60BM) and the active layer and HTL (hole transport layer, which was PEDOT:PSS) were found to have more defects than other parts within the active layer and act as trapping centers leading to nonradiative recombination. [22b,23] Likewise, the presence of grain boundaries (GBs) in the active layer also enhanced contribution in the nonradiative recombination. Therefore, in this mathematical model, the nonradiative recombination of charge carriers at the interfaces and grain boundaries was incorporated separately in the drift-diffusion equations to improve the accuracy of the  results. In addition, such incorporation could help the authors simulate the influence of the size of grain boundaries and that of the passivation at the interfaces and grain boundaries on the EPRCC. A summary of the methodology was represented in the flow chart given in Figure 4. For testing the validity of the numerical simulation, the authors had calculated the J − V charactristics of a PSC of the structure: Glass/ITO/PEDOT:PSS/ CH 3 NH 3 PbI 3 /PC 60 BM/Al from the simulation and compared with the experimental results as shown in Figure 5. [22b,24] As it could be seen in Figure 5, the simulation results agreed very well with the experimental results and hence validated the accuracy of the mathematical model.

Results and Discussions
We have developed a mathematical model to simulate the probability that an electron or hole photogenerated in a PSC can reach its respective electrode without recombination. Here, we discuss the influence of different factors such as the passivation at interfaces and grain boundaries and size of grain boundaries on the thus simulated EPRCC at maximum power point (P max ) and short-circuit current (J sc ) conditions.

Under Maximum Power Point Condition
Using Eqs. (1) -(3), we have calculated the probability P e−m (Eq. (3)) that electrons that are located at the position x in the active layer can reach the cathode without recombination. These calculations are carried out assuming that the tail state recombination rate at the interfaces, N ti , and at the grain boundaries, N tgb are equal and a PSC operates at the maximum power point P max at a voltage of V = 0.76 V. Then, P e−m is calculated for two different tail state recombination rates N ti = N tgb = 10 18 m −3 (eV) −1 and N ti = N tgb = 10 15 m −3 (eV) −1 , and plotted as a function of x as shown in Figure 6. It may be noted that P e−m shown in Figure 6 for x < 10 nm is smaller for the higher recombination rate than that for the smaller recombination rate. This is apparently correct because if the recombination rate is higher, the probability will be lower for an electron to reach the cathode if located far away from the cathode. According to Figure 6, the probability that electrons located near the interface of HTL and active layer at x ≈ 0 reach the cathode is relatively small, whereas if the electrons are located closer to the cathode, say at x = 100 nm, then P e−m increases to more than 90%. Such a variation makes sense because electrons closer to the cathode have a greater probability of reaching it. It may be noted here that the uncertainty is directly derived from the relevant figures to the nearest whole number, i.e., instead of writing 15.1%, it is approximated as 15%. Thus, all the percentage numbers have an uncertainty of 1%.
It may be desired to discuss also the effect of passivation of interfaces and grain boundaries here. According to Figure 6, the probability P e−m is about 35% at x = 0 and increases to about 70% at x = 10 nm for N ti = N tgb = 10 18 m −3 (eV) −1 , whereas it is about 60% at x = 0 and reaches to around 70% at x = 10 nm for N ti = 10 15 m −3 (eV) −1 . This is the region where the effect of passivation can be regarded to be more significant than for any other x > 10 nm, and it gets saturated almost to the same value beyond about 50 nm. In other words, by passivating the interfaces and grain boundaries by three orders of magnitude, the probability that electrons generated at the HTL-active layer interface reach the cathode increases by 25% at the maximum power point condition.
The probability P h−m that holes located at position x from the interface of HTL and the active layer reaches the anode is plotted in Figure 7 as a function of x. Analogous to the case of electrons, these calculations are carried out using Eqs. (4) and (5) for the same two different tail state recombination rates at the interfaces and grain boundaries as N ti = N tgb = 10 18 m −3 (eV) −1 and N ti = N tgb = 10 15 m −3 (eV) −1 . As shown in Figure 7, the probability P h−m is minimum when holes are located farthest from the interface of HTL and the active layer and closest to the active layer-ETL interface and maximum when holes are located nearest to the HTL-active layer interface.

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Like in the case of electrons, this behavior is expected to be correct because holes generated far away from the anode will have less probability of reaching the anode.
According to Figure 7, P h−m increases from about 15% to 60% when holes move toward the anode from 200 to 190 nm for N ti = N tgb = 10 18 m −3 (eV) −1 and from about 45% to 60% for N ti = N tgb = 10 15 m −3 (eV) −1 . That means P h−m increases by about 30% when the interfaces and grain boundaries are passivated and N ti and N tgb reduce to 10 15 m −3 (eV) −1 . In other words, by passivating the interfaces, the probability that holes generated closest to the active layer-ETL interface reach the anode increases by 30% at the maximum power point condition. However, as shown in Figure 7, passivation of interfaces may not influence P h−m significantly when holes are located <190 nm away from the anode, because then the P h−m gets saturated nearly to the same value for both with and without passivation.

Under Short-Circuit Current Condition (V = 0)
We have also simulated P e−m and P h−m at the short-circuit current (J sc ) condition (V = 0), and the results are plotted as a function of x in Figures 8 and 9, respectively, for the two tail state recombination rates at the interfaces and grain boundaries as N ti = N tgb = 10 18 m −3 (eV) −1 and N ti = N tgb = 10 15 m −3 (eV) −1 . As shown in Figure 8, P e−m first increases from about 88% at x = 0 to about 99% at x ≈ 10 nm for N ti = N tgb = 10 18 m −3 (eV) −1 and then it gets saturated to 99% for x > 10 nm. Likewise, under the passivated condition for N ti = N tgb = 10 15 m −3 (eV) −1 , P e−m increases first from about 97% at x = 0 to 99% at x = 10 nm, and then gets saturated at 99% for x > 10 nm. Here again, the effect of passivation is only significant when the electrons are generated close to the anode, and then P e−m increases by 11%. However, this increase is less than that obtained at the P max condition and shown in Figure 6; or in other words, the influence of passivation on the EPRCC increases by increasing the voltage. This may be attributed to the fact that at the J sc condition (V = 0), a smaller number of electrons and holes go through recombination.
Likewise, as shown in Figure 9, P h−m increases from about 86% at x = 200 nm to more than 99% at x = 190 nm for N ti = N tgb = 10 18 m −3 (eV) −1 and about more than 96% to more than 99% for N ti = N tgb = 10 15 m −3 (eV) −1 . After that, for x < 190 nm, both P h−m get saturated nearly to the same value. In this case also, the effect of passivation is only significant when holes are generated close to the cathode, and then  www.advmatinterfaces.de P h−m increases by 10%, and this increase is also less than that obtained at the P max condition because of the same reason as explained above.

Influence of Size of Grain Boundaries
We have also investigated the influence of grain boundary (GB) sizes on the P e−m and P h−m by calculating these for three different sizes of 50, 100, and 200 nm of grain boundaries. P e−m and P h−m thus simulated are plotted as a function of the position x in Figures 10 and 11, respectively. As it is clear from Figures 10 and 11, the size of GB has no significant influence on P e−m and P h−m ; almost independent of the size of GB.

Conclusions
In this paper, a mathematical model and the relevant computer code are developed to quantify the EPRCC in perovskite solar cell with the typical structure of Glass/ITO/PEDOT:PSS/ CH 3 NH 3 PbI 3 /PC 60 BM/Al to investigate the influence of interfaces and grain boundaries. To the best of the authors knowledge, this is the first attempt to introduce such a model with the concept of probability in simulating the EPRCC in PSCs.
The results have shown that the probability that electrons reach from the very left-hand side to the cathode is only 35%, while by passivating the interfaces and grain boundaries, this probability increases to about 60% at maximum power point condition. The results have shown that within 10 nm from the HTL-active layer, the effect of passivation can be regarded to be more significant than for any other x > 10 nm, and it gets saturated almost to the same value beyond about 50 nm. Likewise, for holes to reach from the very right side of the active layer to the anode is only 15%, while by passivation of the interfaces and grain boundaries, the probability increases to 45% at the maximum power point. Analogs to electrons, passivation of interfaces may not influence P h−m significantly when holes are located <190 nm away from the anode because then the P h−m gets saturated nearly to the same value for both with and without passivation. The same calculation has been done at the short-circuit current condition, and the results show that the influence of passivation on the EPRCC is more significant at the maximum power point condition than that of short-circuit current condition. In other words, the influence of passivation on the EPRCC increases by increasing the voltage. Also, the results show that the EPRCC is almost grain boundary size independent. As this model can be extended to any perovskite solar cell, the information in this paper can help the experimentalists to predict and assess the influence of passivation of Figure 9. P n-m is the probability that holes at position x to reach the anode without recombination at J sc condition. Figure 10. P e-m for three different size of grain boundaries (GB = 200 nm, GB = 100 nm, and GB = 50 nm). As it is shown, P e−m is GB independent (the curves are overlapped).