Predictive Modeling of Ion Migration Induced Degradation in

Nov 3, 2017 - Surface Chemistry of All Inorganic Halide Perovskite Nanocrystals: Passivation Mechanism and Stability. Dandan Yang , Xiaoming Li , Haib...
0 downloads 0 Views 1MB Size
Download PDF
www.acsnano.org

Predictive Modeling of Ion Migration Induced Degradation in Perovskite Solar Cells Vikas Nandal* and Pradeep R. Nair* Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India S Supporting Information *

ABSTRACT: With excellent efficiencies being reported from multiple laboratories across the world, device stability and the degradation mechanisms have emerged as the key aspects that could determine the future prospects of perovskite solar cells. However, the related experimental efforts remain scattered due to the lack of any unifying theoretical framework. In this context, here we provide a comprehensive analysis of ion migration effects in perovskite solar cells. Specifically, we show that (a) the effect of ionic charges is almost indistinguishable from that of dopant ions, (b) ion migration could lead to simultaneous improvement in Voc and degradation in Jscan observation which is beyond the realm of mere parametric variation in carrier mobility and lifetime, (c) champion devices are more resilient toward the ill effects of ion migration; (d) we propose characterization schemes to determine both magnitude and polarity of ionic species, and finally, (e) we illustrate that ion migration could be differentiated from ion redistribution based on the distinct trends in performance degradation. Our results, supported by detailed numerical simulations and direct comparison with experimental data, are of broad interest and provide a much needed predictive capability toward the research on performance degradation mechanisms in perovskite solar cells. KEYWORDS: organic−inorganic halide perovskite, stability, hysteresis, efficiency, open-circuit voltage, fill factor

O

yet to settle on the specifics of ion species that migrate and its origin. Theoretical23−25 and experimental22,26,27 studies indicate that I− and MA+ could be the migrating species. However, the dominant mode of ion migration (i.e., through bulk crystal vs grain boundaries) is still an unresolved topic. Moreover, recent reports indicate many surprising trends on perovskite degradation, which include an improvement in Voc with a simultaneous degradation in Jsc.28,29 These trends indicate that the degradation phenomena in such solar cells are more complex and require detailed modeling. In this article, we provide a detailed mapping of ion migration induced performance degradation of perovskite solar cells (schematic of the cell is shown in Figure 1a). In particular, we show that ion migration affects the device electrostatics in a nontrivial manner, which could explain some of the puzzling phenomena described earlier. Our results indicate that ion migration can, surprisingly, lead to Voc improvement, whereas, in general, a degradation in device performance is usually anticipated. Further, we propose schemes to extract both the density of ionic charge and its polarity from device current−

rganic−inorganic hybrid perovskites have emerged as a promising candidate for photovoltaics due to their excellent optoelectronic properties such as large diffusion length,1−3 large absorption coefficient,4 and flexibility in band gap tuning.5−8 Perovskite-based solar cells already report power conversion efficiencies of 22.1%9 and show excellent improvement compared to CdTe, CIGS, and c-silicon solar cells.10 Low-temperature solution processing, low-cost raw material, and relative lack of defects are some of the other appealing attributes of this emerging class of devices.11−13 Given the impressive gains in power conversion efficiency, it is natural that long time stability becomes the main research focus of the community. In general, degradation in solar cells is mainly manifested by a decrease in the performance metrics like Jsc, Voc, fill factor (FF), and hence the efficiency. Perovskite solar cells are no different, and a degradation in the abovementioned parameters is routinely reported in the literature. Further, there has been significant research to explore effects like hysteresis,14−16 giant switchable photovoltaic effect,17 giant dielectric constant,18 enhanced transistor performance at low temperature,19 photoinduced phase separation,20 self-poling,21 and electric field driven reversible conversion between PbI2 and MAPbI3.22 Many of these studies indicate that ion migration could be one of the dominant phenomena, while the debate is © 2017 American Chemical Society

Received: September 4, 2017 Accepted: November 3, 2017 Published: November 3, 2017 11505

DOI: 10.1021/acsnano.7b06294 ACS Nano 2017, 11, 11505−11512

Article

Cite This: ACS Nano 2017, 11, 11505-11512

Article

ACS Nano

Based on the above arguments, we now develop an analytical model to predict the effect of ion migration on solar cell performance. For the ease of analysis, we first develop a model for uniform distribution of ions in the sample. Later, through detailed analysis and numerical simulations, we consider nonuniform ion distributions, as well (and also address whether ion migration could be differentiated from ion redistribution). Accordingly, here, we consider a simple perovskite solar cell (P−I−N configuration) as the model system. The Voc of a solar cell, by definition, is the separation between the quasi-Fermi levels at open-circuit conditions. Rather, we have Voc =

(2)

where kT is thermal energy and charge carrier densities (n, p) are assumed to be spatially uniform in the active layer (ni is the intrinsic carrier concentration). At Voc, detailed balance indicates that the rate of generation of carriers (G) should be the same as the rate of recombination (R) and is given as

Figure 1. (a) Schematic of the model system with planar PIN device architecture. (b) Energy band diagram of the device with and without bulk ion charge density of ±NI = 1017 cm−3. Depending on the polarity of the ions, the original PIN configuration changes to PPN or PNN scheme.

G=R = (np − n i2)/(τn(p + p1 ) + τp(n + n1)) + B(np − n i2)

voltage characteristics. Indeed, our results indicate that Voc values of champion devices are less likely to be affected by ion migration, and the effect is also critically influenced by the contact layer properties, as well. The article is arranged as follows: We first provide a theoretical analysis on the influence of ions on device electrostatics and then its effects on perovskite solar cell performance. The model predictions are then validated through detailed numerical simulations and direct comparison with available experimental data. Theoretical Approach. Migration of ionic species is a welldocumented topic in the field of inorganic semiconductor devices. For example, the field effect transistor device reliability vastly improved due to the detailed understanding of Na+ ion diffusion in gate insulators.30,31 As such, both diffusion and drift (due to an electric field) can contribute to the migration of ions. To understand the effect of ionic charges on solar cell performance, the ionic charge density needs to be accounted in Poisson’s equation as follows ∇·ε∇ϕ = −q(p − n + Nd − Na ± NI)

kT ⎛ np ⎞ ln⎜ ⎟ q ⎝ n i2 ⎠

+ (A np + A pn)(np − n i2)

(3)

where the first term on the right-hand side denotes the trap assisted Shockley−Read−Hall (SRH) recombination, the second term denotes the radiative recombination, and the third term denotes the Auger recombination. Here, τn and τp are electron and hole SRH carrier life time, whereas, n1 and p1 are the associated SRH recombination parameters which depend on the trap energy level.32 Further, B and (An, Ap) are the radiative and Auger recombination coefficients, respectively. The photogenerated excess minority carrier density (Δn or Δp) can be expressed in terms of the effective lifetime (τeff) due to the above recombination mechanisms as Δn ∼ Gτeff. Specifically, eq 3 indicates that the effective lifetime for carriers in the intrinsic active region (i.e., for PIN structure with n ∼ Δn = Δp ∼ p, τn = τp = τ, and An = Ap = A) is given as 1 1 = + Bn + An2 τeff, p 2τ

(1)

where ϕ is the potential, ε is electrical permittivity of the medium, q is electronic charge, p and n are hole and electron densities, and Na, Nd, and NI are acceptor, donor, and bulk ionic charge densities, respectively. Here, the carrier densities, n and p, depend on the electrostatic potential and the quasi-Fermi levelsnot per se on the doping level (however, under charge neutrality conditions, the majority carrier density will be the same as effective doping levels). Hence, it is evident from the above equation that ionic charges are indistinguishable from that of ionized dopantseven though they do not provide any mobile charge carriers on their own. In such a scenario, mobile carriers, if present, will be sourced from elsewhere, for example, other regions or the contacts. In fact, the origin or source of such mobile carriers is not predicted by eq 1. Accordingly, our detailed simulations show that the effect of ionic charge is almost indistinguishable from ionized dopants even for carrier selective P−I−N device configurations. Hence, depending on the polarity of ionic species, an original P−I−N structure changes to either P−N−N or P−P−N due to ion migration (see Figure 1b).

(4)

However, eq 1 indicates that ion migration changes the effective doping density and hence changes the majority carrier density in the active layer. Hence, for negatively charged ions of density NI, we have p = NI, and the effective lifetime for minority carriers (i.e., electrons) is given as 1 1 = + BNI + ANI2 τeff, d τ

(5)

Accordingly, the Voc for pristine devices (i.e., PIN structure) is given as Voc, p =

2kT ⎛ Gτeff, p ⎞ log⎜ ⎟ q ⎝ ni ⎠

(6)

whereas that for degraded devices (with n ≈ Δn, and p = NI) is given as Voc, d = 11506

kT ⎛ GNIτeff, d ⎞ ⎟ log⎜ 2 q ⎝ ni ⎠

(7) DOI: 10.1021/acsnano.7b06294 ACS Nano 2017, 11, 11505−11512

Article

ACS Nano

Efficiency: The above discussion indicates that if ion migration is the dominant phenomenon, then there could be an improvement in Voc and a decrease in Jsc. Accordingly, the efficiency could show interesting trends. For example, if Jsc decrease dominates over Voc improvement, then the efficiency of degraded device will decrease.

Equations 4−7 predict the Voc of the device before and after ion migration. As such, these equations are quite general and are applicable even in the presence of significant interface recombination by modifying the bulk recombination as 1/τ = 1/τB + S/l, where τB is the bulk carrier lifetime and S is the carrier recombination velocity due to traps at the electron transport layer (ETL)/perovskite and hole transport layer (HTL)/perovskite interface, and l is the thickness of the perovskite layer (we note that this analysis could be extended to recombination at grain boundaries, as well). These equations are based on the assumption that the dominant effect of ion migration is in the electrostatics of the device. However, ion migration could create more traps in the active layer. Although this effect is not explicitly discussed in this paper, eqs 4−7 predict the Voc in such a scenario with appropriate τ. Further, eq 7 is valid for both polarities of ions, irrespective of the specific assumptions made in the derivation. The analytical model described above makes some interesting predictions which are listed below. Voc Improvement: The theoretical limit for open-circuit voltage for perovskite-based solar cells is 1.17 V (for an assumed band gap of 1.55 eV). This limit is due to radiative and Auger recombination processes.33 However, the reported Voc values in literature are significantly lower than the theoretical limits, which indicates that trap assisted recombination (could be bulk or interface) is the dominant mechanism.11,28,34 Under such conditions, we have τeff,p = 2τ and τeff,d = τ. Accordingly, comparison of eq 6 with eq 7 indicates that the Voc of degraded devices can be better than that of the pristine devices if NI > 4Gτ. Further, the Voc is expected to be increase with NI as the effective lifetime is still governed by trap assisted SRH recombination (and hence is insensitive to NI; see eq 5). This Voc improvement will reduce for larger values of NI due to increased radiative and Auger recombination (as per eq 5). It is interesting to note that in the absence of any increase of ionic charges during degradation, the Voc can improve only if the carrier lifetime increases, which might not be the scenario in most cases. Jsc Degradation: The presence of ionic charges can affect the electrostatics of the device quite significantly as shown in Figure 1b. Accordingly, the dominant carrier collection phenomena can change from drift- to diffusion-based transport. However, this need not result in any appreciable change in Jsc as carrier collection length ld (ld = Dτ , where D is diffusion coefficient) could still be much larger than the device thickness. For very large NI, the effective lifetime decreases due to increased radiative and Auger recombination (see eq 5), and this could lead to a reduction in collection efficiency and hence the Jsc. In addition, the Jsc can further decrease due to the unfavorable band bending at the perovskite/transport layer interface. Note that in the absence of any ion migration effects, the Jsc can decrease if parameters like carrier mobility or carrier lifetime decrease. However, a decrease in τ will result in a simultaneous decrease in Voc, whereas a change in mobility will not affect the Voc at all. Hence, a simultaneous observation of Jsc decrease and Voc improvement could be a signature of ion migration effects. FF Variation: The trends for FF, which is a measure of charge collection efficiency near Voc conditions (to be precise, at maximum power point condition), are rather difficult to predict compared to that of Jsc and Voc. In general, the FF is shown to follow the Voc trends for inorganic solar cells,35 and we expect similar results here, as well.

RESULTS AND DISCUSSION To test the model predictions, we performed detailed numerical simulations. In addition, we also compare our results with recent experimental trends in the literature. For the numerical simulations, we consider a perovskite-based P−I−N solar cell. The device schematic and the energy level of various materials is provided in Figure 1a and Figure SS1 (Supporting Information), respectively. Uniform photogeneration is assumed in the active layer (with thickness l = 300 nm and G = 4.75 × 1021 cm−3 s−1, which corresponds to Jsc = 22.8 mA/ cm2). The recombination mechanisms in the perovskite layer were already mentioned in eq 3. ETL and HTL are n- and ptype-doped, respectively, which facilitates in the extraction of corresponding charge carriers. The details of simulation methodology are provided in the Methods section. Figure 2 shows the variation of Voc, Jsc, FF, and efficiency as a function of ionic charge density. To explore the effects of ionic

Figure 2. Effect of bulk ionic charge density NI on performance matrices: (a) Voc, (b) Jsc, (c) fill factor, and (d) efficiency of perovskite solar cells. The red curve denotes a device with large τ and, hence, a large diffusion length, whereas the blue curve denotes a device with small diffusion length. The results indicate that simultaneous improvement in Voc and degradation in Jsc is a unique feature of ion migration effects.

charge in detail, here we consider two cases: (a) device with diffusion length more than the perovskite thickness (assumed τ = 3 μs, which corresponds to ld = 1.24 μm) and (b) device with diffusion length less than the active layer thickness (assumed τ = 10 ns, which corresponds to ld = 72 nm). For both cases, in addition to the above-mentioned SRH recombination process, we explicitly consider radiative (B = 3 × 10−11cm3/s) and Auger (An = Ap = 10−29cm6/s)33 processes, as well. The results shown here are for positive polarity of the ions; however, the trends are the same for negative ions, as well (as shown by our numerical simulations; however, there could be significant differences if the ETL/HTL has dissimilar properties like doping density, dielectric constant, etc.). 11507

DOI: 10.1021/acsnano.7b06294 ACS Nano 2017, 11, 11505−11512

Article

ACS Nano Part (a) of Figure 2 shows that the Voc of the degraded device indeed improves compared to that of the pristine device for low τ, whereas there is no Voc improvement for devices with large τ. These surprising results are very much in accordance with the model predictions. The minimum ionic charge density at which Voc shows an improvement is given by NI = 4Gτ. For τ = 10 ns, this correspond to NI ≈ 1015 cm−3, and the Voc indeed shows an improvement in this case. For τ = 3 μs, the ionic charge density required for any improvement in Voc is on the order of 1017 cm−3. However, for such large ionic densities, the effective lifetime decreases due to increased radiative and Auger recombination, and hence, the Voc decreases. Specifically, eq 5 indicates that for NI = 1017 cm−3 the radiative lifetime is given as (BNI)−1 = 0.3 μs, which is much lower than the assumed SRH lifetime of τ = 3 μs and hence lowers the Voc. The results in Figure SS2 of Supporting Information indicate that the Voc improvement is expected to be significant for devices with low τ. Champion devices (with excellent Voc or large τ), quite surprisingly, are least affected by the electrostatics of ion migration. Hence, it is imperative that the active layer material quality should be excellent for ion migration tolerant performance. The effect of ionic charge on Jsc is shown in part (b) of Figure 2. For devices with PIN configuration, the charge collection efficiency η is given as η = lc(1−e−l/lc)/l, where lc is the drift collection length (lc = μτeffE, where E is the built-in electric field and μ is the charge carrier mobility) and l is the active layer thickness.36 The above equation predicts that the collection efficiency is 100% for τ = 3 μs, whereas it reduces to 88% for τ = 10 ns. Note that the simulation results in Figure 2b (i.e., the ratio of Jsc at low NI) show the same trends as the above theoretical predictions. Figure 2b also shows that the Jsc is not affected for low values of NI as the effective lifetime remains a constant (see eq 5). However, as NI increases, the electrostatics of the device is significantly affected and the PIN scheme changes to PPN, thus changing the dominant charge collection mechanism from drift to diffusion. For very large NI, the effective lifetime decreases due to the increase in radiative and Auger recombination process, and hence the Jsc degrades significantly. Note that these results are in accordance with the model predictions. The fill factor variation due to ion migration is provided in part (c) of Figure 2. Here, we find that the FF, more or less, follows the trends in Voc. Specifically, we find that the FF is not much affected for devices with large τ, whereas for small τ, the FF shows an increase for large ionic densities. As NI increases, the fraction of depletion width shared by active layer (Wac) decreases. This improves the fill factor as the voltage dependence on charge collection efficiency decreases. Putting all together, we find that the efficiency of devices degrades due to ion migration (Figure 2d). It is evident that the ion density required for a performance degradation in champion devices is more than that of low-performance devices. This implies that the device lifetime or the stability is also expected to be better for champion devices (provided ion migration is the dominant degradation mechanism). We now compare our simulation results with experiments reported in the literature. Recently, Voc improvement and Jsc reduction during degradation were reported by Guerrero et al.28 (see inset of Figure 3). Here, the Jsc reduces by 40%, while the Voc improves by 150 mV during device degradation. These experimental trends are consistent with our model predictions and detailed numerical simulations. Interestingly, our numerical

Figure 3. Comparison of numerical simulation of ion migration effects with experimental data from ref 28 (see inset; open square symbols represent pristine device, and the open triangles represent the degraded device). The experimentally observed Voc improvement and Jsc degradation is well anticipated by numerical simulations (with ±NI = 1017 cm−3).

simulations indicate that the observed effects could be due to the presence of ionic density of NI ≈ 1017 cm−3 inside the perovskite layer (under the assumption that the transport layers are similarly doped; see Figure SS3 of Supporting Information for discussion on the effects of transport layer doping). In addition, our results also indicate that observed experimental trends could not be explained through simple parametric variations in μ and τ alone (shown in Figure SS4, Supporting Information). For example, it would require a significant improvement in τ (a factor of around 20 to account for 150 mV improvement in Voc as per eq 6) and dramatic reduction in μ (a factor of 10−2 to account for the 40% decrease in Jsc) to explain the observed experimental characteristics in the absence of any ion migration effects. While not impossible, such a simultaneous change in mobility and lifetime leads to a significant reduction in fill factorwhich is contradictory to experimental observations (see Figure SS5, Supporting Information). These results indicate that ion migration could be the dominant phenomena that contribute to the device degradation. Beyond the performance degradation of perovskite solar cells, there is a significant interest among the photovoltaic community to identify the ionic species that contributes to ion migration effects. As mentioned before, both I− and MA+ ions are among the likely species.22−27 In this regard, here we discuss the possibility of finding the polarity and the magnitude (or density) of migrated ionic species in the active layer. Back Extraction of Ionic Charge Density. Ionic charge density can be extracted from the Voc improvement (ΔVoc = Voc,d − Voc,p) observed in the degraded device. With trap assisted SRH as the dominant recombination mechanism, eqs 6 and 7 indicate that ionic charge density can be back extracted as ⎛ qΔVoc qVoc, p ⎞ NI = 2n i exp⎜ + ⎟ 2kT ⎠ ⎝ kT

(8)

Figure 4 shows a comparison of the back extracted ionic density using the above equation. Here, the x-axis denotes the ion density assumed in simulations, and the y-axis denotes the back extracted ion density using eq 8 (i.e., obtained from the simulated current voltage characteristics under illumination). The results indicate that eq 8 predicts the ionic density very well. However, back extraction from Voc improvement is not very accurate at higher input NI because ΔVoc is limited by Auger recombination. To further validate our model, we 11508

DOI: 10.1021/acsnano.7b06294 ACS Nano 2017, 11, 11505−11512

Article

ACS Nano

Figure 4. Comparison of back extracted ionic charge density from CV measurements under dark conditions (using eq 9, circle symbols) and Voc improvement (using eq 8, square symbols). Here, the x-axis denotes the assumed NI for dark CV or illuminated IV numerical simulations, and the y-axis denotes the back extracted NI.

Figure 5. Proposed characterization scheme to identify the polarity of ions in the active layer. Insets show the EB diagrams of different cases considered: Case A (NI = 0), Case B (NI = −1017 cm−3), and Case C (NI = +1017 cm−3). Localized photogeneration is assumed near the ETL/perovskite interface, and the Jsc is normalized to that of 100% collection efficiency. The results show that the Jsc for Case C is significantly lower than that of Case A, whereas the Jsc for Case B is either better than or the same as that of Case A. This contrasting nature of Jsc is due to the ion polarity dependent band bending in the active layer and helps to identify the polarity of ions in the perovskite layer.

employed the well-known dark capacitance voltage (CV) characteristic technique to extract ion charge density. Doping density for the P+N (or PN+) diode at the edge of the lightly doped depletion region can be determined by using Mott− Schottky (MS) analysis, and hence the back extracted ion density is given by 2 NI = − ⎛ d( 12 ) ⎞ ϵq⎜⎜ dVC ⎟⎟ ⎝ ⎠ (9)

negative ions in perovskite. Case C: positive ions in perovskite. Case A has uniform electric field in perovskite, and the charge collection mechanism is due to drift. Case B has significant band bending near the ETL/perovskite interface. For the localized generation near this interface, the collection efficiency is higher as the electric field is much larger than that in Case A. However, for Case C, the band bending is near the HTL/ perovskite interface, and hence the dominant collection mechanism for photocarriers generated near the ETL/perovskite interface is diffusion. For all the above scenarios, the collection length (lc or ld) in Case B is larger than the collection length in Case A, which is in turn larger than the collection length in Case C. As a result, the collection efficiency for Case B saturates to 100% at much lower τ than for Case A because of the higher electric field ETL/perovskite interface. Even for devices with large τ, the collection efficiency for Case C is not 100% as the radiative recombination becomes the dominant phenomena (see Figure SS6 of Supporting Information for details). This contrasting behavior in collection efficiency is due to different band bending near the ETL/perovskite interface and thus enables us to characterize polarity of ions. Therefore, EQE measurements of pristine and degraded devices can be used as a tool to identify the polarity of ions present inside the active layer. Note that being a technique that relies only on device terminal characteristics, the proposed method is very well suited for fully packaged solar cells, as well. Ion Migration vs Ion Redistribution. In the preceding sections, we considered uniform distribution of ionic charge density to provide a comprehensive analysis of degradation of perovskite solar cells. In addition, we have introduced characterization schemes to quantify ionic charge density and polarity of ionic species that are responsible for degradation of perovskite solar cells (PSCs). However, as a general case, the distribution of ions inside perovskite could be spatially nonuniform in contrast to the assumption of uniform distribution used in this paper. In addition, the built-in electric field could result in a concentration gradient of ion density with possible migration of positive (negative) ions toward HTL/AL

Here, C is capacitance and V is applied DC bias voltage. We find that back extracted NI from MS analysis follows a similar trend and is in close agreement with input NI. Therefore, Voc improvement from a pristine to a degraded device and dark CV characteristics of the degraded device can be used to quantify the ion species present inside the perovskite layer. In addition, eq 8 provides a simple alternative to CV analysis for back extraction of ionic charge densities. Polarity of Ion Species. The previous analysis indicates that CV and Voc improvement can be used to back extract the density of ionic species in the active layer. However, both these methods are not suited to identify the polarity of ions, and this requires a different characterization scheme. We note that the polarity of ion species dictates the electrostatics of the active layer (as evident from Figure 1b). Accordingly, depending on the polarity of the ionic species, we find that the band bending in the active layer could be near the ETL or HTL. This distinct nature of band bending helps us to identify the polarity of ions. Depending on band bending, collection efficiency of localized photogenerated electron−hole pairs near the perovskite/ETL interface could be different. Therefore, we propose that quantum efficiency measurements using localized carrier generation (using photons of appropriate wavelength) can characterize the polarity of ions. Figure 5 compares the collection efficiencies (or normalized current density at short-circuit conditions) for various scenarios with light being incident from the ETL side as τ (i.e., only the SRH lifetime) is varied, while the radiative and Auger recombination rates are the same as that in Figure 2. The localized generation of photocarriers is accounted through a spatially uniform carrier generation over a region with a thickness of 60 nm near the ETL/perovskite interface. Here, we consider three scenarios. Case A: Intrinsic perovskite. Case B: 11509

DOI: 10.1021/acsnano.7b06294 ACS Nano 2017, 11, 11505−11512

Article

ACS Nano

region of active region being under the influence of ion migration, the results in Figure 6c indicate that we observe a simultaneous improvement in Voc, a reduction in Jsc, and no apparent change in FF. We note that these broad features are also observed in recent experiments29 (also see Figure 3), thus supporting the proposed model. In contrast, Figure 6d shows the effect of ion redistribution on performance metrics of PSCs. Here, we find that the Voc improvement is independent of lw and is similar to the uniform ion distribution. However, the trends for fill factor and Jsc for such ion redistribution are distinctly different from that of ion migration and hence can be used to quantify the relative contribution of ion migration vs ion redistribution toward the change in performance parameters.

(ETL/AL) interface. Hence, we consider two possible scenarios to further test the validity and broad appeal of our theoretical model. Case 1: ion migration, in which a wave of ions migrates through the active layer (see Figure 6a). Here, an ion density of

CONCLUSIONS To summarize, here we provided a comprehensive theoretical analysissupported by detailed numerical simulations and experimental observationson the electrostatic effects of ion migration in perovskite solar cells. Our predictive modeling provides physical insights into unique features observed in performance (Voc improvement and Jsc degradation in comparison to a pristine device) of a degraded device. We find that ion migration might be the most probable physical mechanism that leads to such experimentally observed puzzling results. We indicate that champion devices are more tolerant toward ion migration effects. In addition, we have also proposed characterization schemes to probe the polarity and density of the migrating ionic species and help to distinguish between ion migration and ion redistribution. Indeed, our results and the solution methodology are of broad interest and help the community to understand the ion migration effects in isolation and thus explore the relative contribution of other mechanisms, if any, in greater detail.

Figure 6. Effect of spatially nonuniform ion distribution on perovskite solar cells for (a) Case 1, ion migration with NI = 1017 cm−3 with varying width lw, and (b) Case 2, ion redistribution with NI × lw = constant. Parts (c) and (d) illustrate the effect of positive ions on the performance metrics for Case 1 and Case 2, respectively, as a function of width lw. Performance metrics are normalized with respect to NI = 0 or the PIN device. Dotted lines correspond to the theoretical predictions based on uniform distribution of positive ion density with NI = 1017 cm−3 across the perovskite layer.

METHODS The Poisson and continuity equations32 for electrons and holes are self-consistently solved37 to explore the effect of ionic charge on device performance. The effects of ions are explicitly considered in the electrostatics, as mentioned in eq 1. The simulation methodology is well calibrated with experiments, as previously reported,33,38 and the parameters used for this study are provided in Table S1 of the Supporting Information.

NI = 1017 cm−3 is assumed to be present over a region of width lw. As lw increases, significant extent of the active layer will be under the influence of ion migration). Case 2: ion redistribution, in which the total number of ions remains the same but only the spatial distribution changes (see Figure 6b). Here, the total number of ions (NI × lw) is assumed to be a constant with varying width lw and hence NI reduces with lw. For both cases, we vary lw over a broad range to study the effects of nonuniform ion distribution on device performance. Figure 6c,d shows the effect of ion migration and ion redistribution on the normalized performance metrics such as Voc, Jsc, and fill factor. The results indicate that simultaneous improvement in Voc and degradation in Jsc is a unique feature of ion migrationan observation which is consistent with the proposed model. In addition, for scenarios in which a significant region of the perovskite active region is under the influence of ion migration (see Figure 6c), the effects of such nonuniform ion distribution are clearly anticipated by the proposed model (see dotted lines in Figure 6c), which indicates that the analytical model, although simple, is quite useful to predict broad trends of even nonuniform ion distribution in the active region. Further, under such a scenario of a significant

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b06294. Energy level alignment diagram along with different parameters used for simulations; comparison between model predictions and simulations for Voc; effects of transport layer doping density on ion migration; and effect of different parametrs (μ, τ, and NI) on device performance (PDF)

AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Vikas Nandal: 0000-0002-7356-3926 11510

DOI: 10.1021/acsnano.7b06294 ACS Nano 2017, 11, 11505−11512

Article

ACS Nano Notes

(13) Stranks, S. D.; Snaith, H. J. Metal-Halide Perovskites for Photovoltaic and Light-Emitting Devices. Nat. Nanotechnol. 2015, 10, 391−402. (14) Jeon, N. J.; Noh, J. H.; Kim, Y. C.; Yang, W. S.; Ryu, S.; Seok, S. Il. Solvent Engineering for High-Performance Inorganic-Organic Hybrid Perovskite Solar Cells. Nat. Mater. 2014, 13, 897−903. (15) Unger, E. L.; Hoke, E. T.; Bailie, C. D.; Nguyen, W. H.; Bowring, a. R.; Heumüller, T.; Christoforo, M. G.; McGehee, M. D. Hysteresis and Transient Behavior in Current−voltage Measurements of Hybrid-Perovskite Absorber Solar Cells. Energy Environ. Sci. 2014, 7, 3690−3698. (16) Snaith, H. J.; Abate, A.; Ball, J. M.; Eperon, G. E.; Leijtens, T.; Noel, N. K.; Stranks, S. D.; Wang, J. T.-W.; Wojciechowski, K.; Zhang, W. Anomalous Hysteresis in Perovskite Solar Cells. J. Phys. Chem. Lett. 2014, 5, 1511−1515. (17) Xiao, Z.; Yuan, Y.; Shao, Y.; Wang, Q.; Dong, Q.; Bi, C.; Sharma, P.; Gruverman, A.; Huang, J. Giant Switchable Photovoltaic Effect in Organometal Trihalide Perovskite Devices. Nat. Mater. 2014, 14, 193−198. (18) Juarez-Perez, E. J.; Sanchez, R. S.; Badia, L.; Garcia-Belmonte, G.; Kang, Y. S.; Mora-Sero, I.; Bisquert, J. Photoinduced Giant Dielectric Constant in Lead Halide Perovskite Solar Cells. J. Phys. Chem. Lett. 2014, 5, 2390−2394. (19) Chin, X. Y.; Cortecchia, D.; Yin, J.; Bruno, A.; Soci, C. Lead Iodide Perovskite Light-Emitting Field-Effect Transistor. Nat. Commun. 2015, 6, 7383. (20) Hoke, E. T.; Slotcavage, D. J.; Dohner, E. R.; Bowring, A. R.; Karunadasa, H. I.; McGehee, M. D. Reversible Photo-Induced Trap Formation in Mixed-Halide Hybrid Perovskites for Photovoltaics. Chem. Sci. 2015, 6, 613−617. (21) Deng, Y.; Xiao, Z.; Huang, J. Light-Induced Self-Poling Effect on Organometal Trihalide Perovskite Solar Cells for Increased Device Efficiency and Stability. Adv. Energy Mater. 2015, 5, 1500721. (22) Yuan, Y.; Wang, Q.; Shao, Y.; Lu, H.; Li, T.; Gruverman, A.; Huang, J. Electric-Field-Driven Reversible Conversion Between Methylammonium Lead Triiodide Perovskites and Lead Iodide at Elevated Temperatures. Adv. Energy Mater. 2016, 6, 1501803. (23) Eames, C.; Frost, J. M.; Barnes, P. R. F.; O’Regan, B. C.; Walsh, A.; Islam, M. S. Ionic Transport in Hybrid Lead Iodide Perovskite Solar Cells. Nat. Commun. 2015, 6, 7497. (24) Azpiroz, J. M.; Mosconi, E.; Bisquert, J.; De Angelis, F. Defect Migration in Methylammonium Lead Iodide and Its Role in Perovskite Solar Cell Operation. Energy Environ. Sci. 2015, 8, 2118−2127. (25) Haruyama, J.; Sodeyama, K.; Han, L.; Tateyama, Y. FirstPrinciples Study of Ion Diffusion in Perovskite Solar Cell Sensitizers. J. Am. Chem. Soc. 2015, 137, 10048−10051. (26) Yuan, Y.; Chae, J.; Shao, Y.; Wang, Q.; Xiao, Z.; Centrone, A.; Huang, J. Photovoltaic Switching Mechanism in Lateral Structure Hybrid Perovskite Solar Cells. Adv. Energy Mater. 2015, 5, 1500615. (27) Yang, T.-Y.; Gregori, G.; Pellet, N.; Grätzel, M.; Maier, J. The Significance of Ion Conduction in a Hybrid Organic-Inorganic LeadIodide-Based Perovskite Photosensitizer. Angew. Chem., Int. Ed. 2015, 54, 7905−7910. (28) Guerrero, A.; You, J.; Aranda, C.; Kang, Y. S.; Garcia-Belmonte, G.; Zhou, H.; Bisquert, J.; Yang, Y. Interfacial Degradation of Planar Lead Halide Perovskite Solar Cells. ACS Nano 2016, 10, 218−224. (29) Joshi, P. H.; Zhang, L.; Hossain, I. M.; Abbas, H. A.; Kottokkaran, R.; Nehra, S. P.; Dhaka, M.; Noack, M.; Dalal, V. L. The Physics of Photon Induced Degradation of Perovskite Solar Cells. AIP Adv. 2016, 6, 115114. (30) Snow, E. H.; Grove, A. S.; Deal, B. E.; Sah, C. T. Ion Transport Phenomena in Insulating Films. J. Appl. Phys. 1965, 36, 1664−1673. (31) Kuhn, M.; Silversmith, D. J. Ionic Contamination and Transport of Mobile Ions in MOS Structures. J. Electrochem. Soc. 1971, 118, 966− 970. (32) Sze, S. M. Physics of Semiconductor Devices; John Wiley & Sons: Hoboken, NJ, 1981.

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This article is based upon work supported under the U.S.− India Partnership to Advance Clean Energy-Research (PACER) for the Solar Energy Research Institute for India and the United States (SERIIUS), funded jointly by the U.S. Department of Energy (Office of Science, Office of Basic Energy Sciences, and Energy Efficiency and Renewable Energy, Solar Energy Technology Program, under Subcontract DE-AC3608GO28308 to the National Renewable Energy Laboratory, Golden, Colorado), and the Government of India, through the Department of Science and Technology under Subcontract IUSSTF/JCERDC-SERIIUS/2012 dated 22nd November 2012. The authors also acknowledge the Center of Excellence in Nanoelectronics (CEN) and National Center for Photovoltaic Research and Education (NCPRE), IIT Bombay, for computational facilities. REFERENCES (1) Stranks, S. D.; Eperon, G. E.; Grancini, G.; Menelaou, C.; Alcocer, M. J. P.; Leijtens, T.; Herz, L. M.; Petrozza, A.; Snaith, H. J. Electron-Hole Diffusion Lengths Exceeding 1 Micrometer in an Organometal Trihalide Perovskite Absorber. Science 2013, 342, 341− 344. (2) Xing, G.; Mathews, N.; Sun, S.; Lim, S. S.; Lam, Y. M.; Grätzel, M.; Mhaisalkar, S.; Sum, T. C. Long-Range Balanced Electron- and Hole-Transport Lengths in Organic-Inorganic CH3NH3PbI3. Science 2013, 342, 344−347. (3) Edri, E.; Kirmayer, S.; Henning, A.; Mukhopadhyay, S.; Gartsman, K.; Rosenwaks, Y.; Hodes, G.; Cahen, D. Why Lead Methylammonium Tri-Iodide Perovskite-Based Solar Cells Require a Mesoporous Electron Transporting Scaffold (but Not Necessarily a Hole Conductor). Nano Lett. 2014, 14, 1000−1004. (4) De Wolf, S.; Holovsky, J.; Moon, S.-J.; Löper, P.; Niesen, B.; Ledinsky, M.; Haug, F.-J.; Yum, J.-H.; Ballif, C. Organometallic Halide Perovskites: Sharp Optical Absorption Edge and Its Relation to Photovoltaic Performance. J. Phys. Chem. Lett. 2014, 5, 1035−1039. (5) Filip, M. R.; Eperon, G. E.; Snaith, H. J.; Giustino, F. Steric Engineering of Metal-Halide Perovskites with Tunable Optical Band Gaps. Nat. Commun. 2014, 5, 5757. (6) Kumawat, N. K.; Dey, A.; Kumar, A.; Gopinathan, S. P.; Narasimhan, K. L.; Kabra, D. Band Gap Tuning of CH3NH3Pb(Br1− xClx)3 Hybrid Perovskite for Blue Electroluminescence. ACS Appl. Mater. Interfaces 2015, 7, 13119−13124. (7) Kumawat, N. K.; Dey, A.; Narasimhan, K. L.; Kabra, D. Near Infrared to Visible Electroluminescent Diodes Based on Organometallic Halide Perovskites: Structural and Optical Investigation. ACS Photonics 2015, 2, 349−354. (8) Kulkarni, S. A.; Baikie, T.; Boix, P. P.; Yantara, N.; Mathews, N.; Mhaisalkar, S. Band-Gap Tuning of Lead Halide Perovskites Using a Sequential Deposition Process. J. Mater. Chem. A 2014, 2, 9221−9225. (9) Yang, W. S.; Park, B.-W.; Jung, E. H.; Jeon, N. J.; Kim, Y. C.; Lee, D. U.; Shin, S. S.; Seo, J.; Kim, E. K.; Noh, J. H.; Seok, S. I. Iodide Management in Formamidinium-Lead-Halide−based Perovskite Layers for Efficient Solar Cells. Science 2017, 356, 1376−1379. (10) Green, M. a; Bein, T. Photovoltaics: Perovskite Cells Charge Forward. Nat. Mater. 2015, 14, 559−561. (11) You, J.; Hong, Z.; Yang, Y.; Chen, Q.; Cai, M.; Song, T.-B.; Chen, C.-C.; Lu, S.; Liu, Y.; Zhou, H.; Yang, Y. Low-Temperature Solution-Processed Perovskite Solar Cells with High Efficiency and Flexibility. ACS Nano 2014, 8, 1674−1680. (12) Snaith, H. J. Perovskites: The Emergence of a New Era for LowCost, High-Efficiency Solar Cells. J. Phys. Chem. Lett. 2013, 4, 3623− 3630. 11511

DOI: 10.1021/acsnano.7b06294 ACS Nano 2017, 11, 11505−11512

Article

ACS Nano (33) Agarwal, S.; Nair, P. R. Device Engineering of Perovskite Solar Cells to Achieve near Ideal Efficiency. Appl. Phys. Lett. 2015, 107, 123901. (34) Nie, W.; Tsai, H.; Asadpour, R.; Blancon, J.-C.; Neukirch, A. J.; Gupta, G.; Crochet, J. J.; Chhowalla, M.; Tretiak, S.; Alam, M. A.; Wang, H. L.; Mohite, A. D. High-Efficiency Solution-Processed Perovskite Solar Cells with Millimeter-Scale Grains. Science 2015, 347, 522−525. (35) Green, M. A. Solar Cell Fill Factors: General Graph and Empirical Expressions. Solid-State Electron. 1981, 24, 788−789. (36) Crandall, R. S. Modeling of Thin Film Solar Cells: Uniform Field Approximation. J. Appl. Phys. 1983, 54, 7176. (37) Synopsys, Sentaurus Device Simulation Tool, 2011. (38) Agarwal, S.; Seetharaman, M.; Kumawat, N. K.; Subbiah, A. S.; Sarkar, S. K.; Kabra, D.; Namboothiry, M. A. G.; Nair, P. R. On the Uniqueness of Ideality Factor and Voltage Exponent of PerovskiteBased Solar Cells. J. Phys. Chem. Lett. 2014, 5, 4115−4121.

11512

DOI: 10.1021/acsnano.7b06294 ACS Nano 2017, 11, 11505−11512