Modeling the degradation and recovery of perovskite solar cells

Abstract

Degradation and recovery in the device parameters of perovskite-based solar cells are modeled for the devices aged by exposing to air humidity/moisture. Several devices are considered with the same CH₃NH₃PbI_3−xCl_x absorber layer but with different windows, hole transport layers and metallic back contacts. We proposed several models to fit the experimental data, reported in the literature, on the degradation of fill factor, efficiency and short-circuit current density. The kinetics proposed for the modeling of degradation/recovery in the fill factor are based on variation in the fast and slow metastable defect states in the respective layer. The degradation/recovery of the short-circuit current density is investigated by proposing a new model which accounts for the density of metastable defects before/after aging. Finally, the degradation of the device efficiency is modeled by fitting the data with proper functions mostly following the Gaussian bi-exponential shape. It is shown that instability of the device parameters might be due to variation in metastable defects in any component of a perovskite solar cell. However, the fill factor, or consequently the series resistance, is the main degradation source in perovskite solar cells.

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1. Introduction

Perovskite-based solar cells with >20% efficiency, obtained over the last 5 years, still suffer from high degradation rates or poor stability. The main instability source of such organic–inorganic devices is moisture and exposure to air humidity. Several research groups have been systematically studying this aspect in the last three years, which is briefly discussed here. In 2013, Prof. Snaith et al. identified that the light-induced desorption of surface-adsorbed oxygen causes a critical instability in mesoporous TiO₂-sensitized solar cells.¹ They proposed a rather stable TiO₂-free device whose fill factor, (FF), decreased by about 30% within 200 hours aging (under one sun illumination) and then became almost stable over the next 1000 hours of aging. The short-circuit current density, J_sc, and open-circuit voltage, V_oc, of this device showed less variation. Several reasons were suggested for the FF degradation and efficiency, η, of such devices e.g. oxygen desorption at the vacancy sites of TiO₂ and changes imposed at the interface of Spiro-OMETAD/Au at the back region of the cell. In 2014, the research group of Prof. Gratzel reported a novel electron-rich molecule called H101 as a replacement with a Spiro-OMETAD layer.² The V_oc of this device decreased slightly, whereas, the J_sc and FF were the main parameters which were affected by heating at 70 °C for 7 days. This degradation behavior was primarily attributed to a moisture induced mechanism in a very hygroscopic perovskite layer. In 2015, Chen et al. inserted a thin layer of CuI instead of PEDOT:PSS as the hole transport layer, HTL, and used Ag instead of Al as the back contact.³ This device resulted in a 13.28% efficiency and improved the air stability of the cell as compared to devices employing PEDOT:PSS as HTL. An efficiency drop of Δη ≈ −10% was reported after 14 days storage in air compared to Δη ≈ −73% for PEDOT:PSS-based devices. This degradation behavior was again attributed mainly to FF degradation and a gradual decrease of J_sc. In 2016, Li et al. reported an 11.76% efficiency with a AgAl back contacted perovskite cell and PEDOT:PSS as the front contact.⁴ Again, the efficiency degraded mainly due to a loss in fill factor, ΔFF, after 360 hours aging under a relative humidity of 10% in air. The V_oc of this design didn’t change too much and the J_sc improved or dropped slightly with aging. A similar degradation trend was observed for regular perovskite devices with TiO₂ and Spiro-OMETAD partners under continuous illumination.⁵ XPS measurements of such cells revealed that broken interface binding of the hole transport layer and Au metallic contact is the degradation source while the TiO₂/perovskite interface doesn’t have an effective role in device instability. Therefore, Δη was mainly driven by ΔJ_sc and ΔFF. The latter was also responsible for the cell recovery. Some other literature has also reported a similar degradation and recovery trend for perovskite devices with different designs and aged under similar conditions.^6–8 Nevertheless, in all the above publications, the device instability was mostly driven by the variation in FF and to some extent by J_sc. Therefore, we base our modeling on the given data in four different references (ref. 3–6) on experimentally realized perovskite solar cells with similar active layers but different windows (CuI & PEDOT:PSS³), different HTLs (PEDOT:PSS & Spiro:OMETAD⁴), and back contacts (AgAl & Ag & Al^5,6). The aging methods of these devices were similar as they were all tested for air stability and moisture effect for hours (0 < Δt < 350 hours). Our modeling approach, for the degradation/recovery of FF, is based on a kinetic model provided by Myong and Lim in 2006.⁹ This model was primarily proposed to fit the degradation/recovery trend of FF in a-Si:H solar cells aged by light and temperature. This model accounts for the concentration of fast and slow metastable defect states in the i-region of p–i–n structures. Since perovskite solar cells are also a sandwich of pin layers we apply their model to explain the FF variation by humidity.¹⁰ In addition, the gradual change in J_sc of perovskite cells by air exposure is modeled by developing a new approach based on variation of drift length and defect density. Finally, the efficiency variation over aging time is modeled using several new fitting functions and the fitting coefficients are correlated to device properties such as defect density and recombination lifetime. The readers are advised that this model does not directly take into account the contact effect on the stability since that makes the modeling quite complicated.¹¹ However, we simply assume that the defect density was increased over time under stress conditions. The research group of Prof. Garcia-Belmonte has been actively investigating the stability of perovskite solar cells considering the effects of contacts.^12,13

2. Theory

The loss in device parameters, Δη, ΔFF, and ΔJ_sc, was calculated for the aging period Δt, after which the respective parameter did not degrade or became stable or started to recover. We first address the degradation/recovery of FF as this parameter has been shown to be responsible for the degradation of efficiency of the considered devices (1, 2, 3). It has been reported that perovskite materials have a long carrier diffusion length, L_D = 10 μm, due to planar diffusion of indirect excitons.¹⁴ Perovskite-based solar cells are also considered as p–i–n structures as was experimentally investigated in Fig. 5 of ref. 15. Routinely, for a uniform electric field, E = (V_bi − V_a)/d, in the i-region, the carrier collection is given by L_D = μτE. Note that, the electric field, E, is the total of built-in voltage, V_bi, and applied voltage V_a over the perovskite thickness, d. In this case, the ambipolar recombination lifetime, τ, allows the carrier to drift to the respective electrodes with a drift mobility, μ. Recently, the transport mean free path, l₀, in perovskite devices was connected to L_D by Gevorkian et al.¹⁴ as,

L_D = l₀(pρ/ℏ)³ where l₀ = N^−1/3 (1)

and p, ρ, and N are the exciton momentum (∼2.7 × 10⁻²⁵ kg m s⁻¹), indirect exciton radius (∼5 nm), and bulk concentration of defects, respectively. Due to the similarity in device physics of a-Si:H and perovskite thin film solar cells, e.g. p–i–n structure and drastic change of FF, we can use the Nguyen approach to relate FF to L_D for perovskite devices here,where N is the bulk defect density, and C_0,1 = 0.39 and A = 0.3 are constants. Interestingly, for perovskite devices |A| = 1/3 ≈ 0.33 which is the power of n in eqn (1). Connecting the model to the experimental data, bringing the parameter time into the model (FF(t)) and in order to get rid of the coefficients which vary from one device to another, we get the ratio (normalized) value of the fill factor as,where FF_d is the degraded fill factor after aging by air moisture, FF_i is the initial fill factor before aging, and N₀ and N_d are the defect density in the perovskite layer before and after aging, respectively. Note that, this modeling is not limited to only the defect increment in the perovskite layer, but it is also applied to the defect increment in the TiO₂ or HTL layers too. In all ref. 1–8, the degradation of the full device was attributed to defect increments in either sides of the perovskite interfaces and HTL thickness. However, the proposed model for FF(t) is flexible for replacing the perovskite layer with TiO₂ or HTL. It just needs assuming that the FF of the full device changes by N increments in a respective layer. A first-principal study by Yin et al.¹⁶ has shown that CH₃NH₃PbI₃ and methylammonium lead iodide (MAPbI₃) perovskites have unusual defect physics which is mostly associated with the Pb ions. Although an in-depth understanding of the defects in perovskites is still lacking, a photoluminescence, PL, study by Wen et al.¹⁷ showed that both slow and fast metastable defects are present in these materials. This PL decay was fitted to a bi-exponential function including both slow and fast trapping/detrapping lifetimes and they concluded that both slow, N_s, and fast, N_f, metastabilities are present in a perovskite layer with comparable weights and probabilities. Therefore, here we also ascribe the possibility of the presence of both slow and fast metastabilities in the i-region of the perovskite cell under study and get,

N = N₀ + N_f + N_s & N_d = N₀ + N_df + N_ds (4)

where the indices df and ds stand for fast and slow defects in a degraded cell. Recently, we developed a time-dependent model for defect increment under stress conditions and showed that the density of fast and slow defects is obtained as¹⁸where τ_s and τ_f are time constants for the fast and slow metastable defects to return to the stable state, respectively. By replacing the above equations in (3), the following kinetic model is obtained,¹⁹where n is given by,and α and β are defined as the ratio of photo-created fast and slow metastable defect densities, respectively, and are given by,

α = N_df/N₀ and β = N_ds/N₀ = 1/n − 1 − α. (8)

β is obtained if we know α and n. n is obtained if we know the time constants. Therefore, eqn (6) is a useful formula as it allows one to fit the model with data simply by finding the proper values of the three independent parameters: α, τ_f, and τ_s. When the cell is under air exposure (which is the case here), the metastable defects are created by the reaction between the perovskite and oxygen. Oxygen is abundant in the ambient atmosphere and will also influence the quality and stability of CH₃NH₃PbI₃ in a way that perovskite tends to hydrolyze and degrade as follows,²⁰

CH₃NH₃PbI₃(s) → PbI₂(s) + CH₃NH₃I(aq) (9)

CH₃NH₃I(aq) → CH₃NH₂(aq) + HI(aq) (10)

4HI(aq) + O₂(g) → 2I₂(s) + 2H₂O(l) (11)

2HI(aq) → H₂(g) + I₂(s) (12)

where O₂ and I₂ are Oxygen and Iodide molecules. The consumption of HI, according to the third and last reactions, leads to the degradation of the whole device. It’s been also addressed by Tress et al. that the reactions of perovskite with water (humidity) cause decomposition of the perovskite material and a change in the color from black to yellow which is an indication of lead iodide, PbI₂, formation.¹⁵

3. Modeling results and discussion

3.1. Modeling the degradation/recovery of FF

Fig. 1 provides the modeling results for the degradation/recovery of FF as a function of aging time (hours) using eqn (6). All the Devices 1, 2, and 3 fabricated by different research groups, were exposed to air with relative humidity. This similar aging test allowed comparison of the modeling results. In addition, we selected three different perovskite solar cells with very similar structures but different in windows, HTLs or back contact layers. This allows the reader to see the modeling applied to a variety of designs of perovskite devices.¹⁵ As shown in Fig. 1a–c, we separated the recovery and degradation trends of every device and modeled the variation of FF accordingly. For this, we recorded the initial and degraded values of the fill factor, FF_i and FF_d, for every degradation/recovery feature within the respective aging time of every device, and then fitted the data with a model using eqn (6) and finding the proper values of the fitting parameters given in Tables 1–3. The fitting parameters are τ_f, τ_s, and n for every device, respectively. The data of Fig. 1a were taken from ref. 3 and show both recovery and degradation trends when CuI is used as the window layer. Regardless of the actual initial value, FF increased when exposed to air for 80 hours. According to Table 1 it improved from 95% to 109%. The model fits the data when we get τ_f < τ_s (30 vs. 300 ns). Exposing the device to humidity for longer degraded the FF. This occurs from the initial value of 107–95% and τ_s is one order of magnitude smaller than the recovery case. This means that humidity is creating metastable defects with higher density which cause faster recombination (smaller lifetime). This assignment is verified by comparing the n values of 0.99 to 0.7 for the recovery and degradation trends, respectively. Smaller n means higher N_d according to eqn (7) and a higher N_d causes degradation due to a lower FF or increased series resistance. We considered the effect of stress on FF and series resistance of solar cells in our previous publication.¹⁸ The increase in parameter α (from 0.53 to 0.8) is also consistent with decreased n as both represent defect increment with time. Note that, in ref. 3 the aged devices were not encapsulated and thus the influence of moisture is more probable. We also verified the variation in defect density by modeling the J_sc(t) of every device in the last section of this manuscript. Obviously, CuI contacted cells showed a smaller degradation rate to the one with PEDOT:PSS. The lower stability of the PEDOT:PSS contacted device may be because of the acidic and hygroscopic nature of PEDOT:PSS as well as the moisture-sensitive perovskite layer.³ No recovery trend was observed for the PEDOT:PSS contacted cell. The modeling coefficients are given in Table 1. Despite the fitting parameters being within a similar range to those of the CuI-based device, a smaller initial FF_i and degraded FF_d led to a more drastic slope for PEDOT:PSS. Fig. 1b shows a rather different trend than the previous case. In this case, the researchers investigated the aging effect on the stability of perovskite devices with different back contact materials. Several different metallic contacts were applied, AgAl, Al, and Ag.⁴ The FF values of the cells with Al and Ag electrodes quickly decreased to 70% of FF_i while the AgAl-contacted cells retained 90% of their FF_i values after 24 hours aging. The degradation pathways were attributed to chemical degradation of the perovskite layer, interfacial materials, electrodes and the interfaces according to the authors of ref. 4. The normalized FF values were fitted by eqn (6) with the fitting coefficients given in Table 2. τ_f was taken as 1 ns for all three devices and τ_s was taken in the same order of magnitude. The α values (=N_d/N₀) followed the degradation intensity where the Al-contacted cell got the biggest α = 0.85. The reason for such extreme degradation was attributed to the fact that CH₃NH₃PbI₃ films can be easily decomposed into HI under exposure to moisture and high temperature which can strongly corrode Al.⁴ Note that the V_oc of these devices retained it’s stability over aging time indicating that the reduction in FF and/or J_sc is the cause.

Fig. 1 Modeling of FF degradation/recovery for three perovskite devices with: (a) PEDOT:PSS and CuI window layers exposed to air without encapsulation, (b) AgAl, Ag and Al back contacts exposed to 10% relative humidity, (c) PEDOT:PSS and Spiro-OMETAD hole transport layers exposed to 40% relative humidity. Insets are the schematic of every device. The experimental data are shown as filled symbols by triangles, squares, stars and circles. These data were taken from ref. 3 (for (a)), ref. 4 (for (b)), and ref. 5 and 6 (for (c)), respectively.

Table 1 Fitting parameters used in Fig. 1(a)

Device #	Deg./rec.	FFi	FFd	τf	τs	α	n
Device 1 with CuI	Recovery	0.95	1.09	30	300	0.53	0.99
Device 2 with CuI	Degradation	1.07	0.95	20	90	0.8	0.7
Device 3 with PEDOT	Degradation	0.98	0.45	30	100	0.67	0.7

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Table 2 Fitting parameters used in Fig. 2(b)

Device #	Deg./rec.	FFi	FFd	τf	τs	α	n
Device 2 with AgAl	Degradation	0.974	0.538	1	70	0.65	0.73
Device 2 with Al	Degradation	0.958	0.308	1	20	0.85	0.88
Device 2 with Al	Recovery	0.359	0.317	40	100	0.58	0.91
Device 2 with Ag	Degradation	0.952	0.523	1	20	0.65	0.75
							0.7
Device 2 with Ag	Recovery	0.523	0.556	0.7	70	0.56	0.90

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Table 3 Fitting parameters used in Fig. 1(c)

Device #	Deg./rec.	FFi	FFd	τf	τs	α	n
Device 3 with PEDOT	Recovery	0.94	0.99	4	100	0.66	0.6
Device 3 with PEDOT	Degradation	0.968	0.452	15	90	0.75	0.85
Device 3 with Spiro	Degradation	0.958	0.308	1	70	0.6	0.72
Device 3 with Spiro	Recovery	0.540	0.583	30	110	0.4	0.88

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After over 100 hours of aging of Device 2, the FF was recovered except for the AgAl-contacted cell which became stable at around 53%. The α and n fitting parameters have respectively smaller and bigger values for the recovering features than the degrading ones. This means smaller metastable defects are created by continuous aging over 100 hours. Thus our modeling showed that a higher defect density might be the cause of degradation in these devices. This statement was considered by our research group elsewhere to model metallic ion diffusion into the absorber layer of thin film solar cells which consequently increased the series resistance or reduced FF.²¹ Alternatively, Fig. 1c shows that the hole transporting layer also has an effect on the stability of perovskite devices. Two devices with the same design but different HTLs, PEDOT:PSS and Spiro-OMETAD, were examined in ref. 5 and 6 and the reported data are used in this work. Device 3 with a PEDOT:PSS HTL was recovered at the beginning of the exposure to humidity. This might be compared to the slightly stable feature of Device 1 with PEDOT:PSS as the window layer in Fig. 1a. Both devices with PEDOT:PSS show a slightly recovered FF of about 10% < ΔFF < 20%. However, it has a significant effect on FF degradation with a drop in the fill factor in the range of 60% < ΔFF < 70%. It is also interesting to note that the FF of Device 3 with PEDOT:PSS shows a reverse trend to Device 3 with Spiro-OMETAD after 100 hours aging. In ref. 5 and 6 several other materials were also proposed as HTLs in perovskite devices. However, we stick to considering the most known materials as HTLs in the literature. The FF of Device 3 with PEDOT:PSS degrades when exposed to humidity and does not become stable even after 250 hours. In contrast, the FF of Device 3 with Spiro-OMETAD recovers for a longer aging time (>100 hours). Also the values of the fitting parameters are consistent with data where α is 0.66 for recovery and 0.75 for degradation as shown in Table 3. Again our modeling, which deals with defect density variation and it’s role in stability, predicts that an increase in metastable defects causes degradation of PEDOT:PSS, and the saturation/reduction of these defects causes recovery of Sprio-OMETAD. This model, however, is not able to predict the effect of corrosive chemical dopants (acetonitrile, TBP, etc.) or additives on the Spiro-OMETAD layer which are essential to process the surface of such materials.⁵ It’s interesting to note that to model the recovery trend, we inserted τ_s > τ_f (100 ns vs.4 ns for PEDOT:PSS or 110 ns vs. 30 ns for Spiro-OMETAD). A longer lifetime causes a lower recombination rate and allows the electric field to collect the carriers more efficiently.²² A longer lifetime and lower recombination rate are both due to a lower metastable defect density in the HTL.

A general look at the modeling results in Tables 1–3 shows that the back contact material has the most effective role in the stability of perovskite layers. This doesn’t mean that the other components, e.g. the HTL and window layer, are not effective, but a careful selection of the metallic top layer will critically improve or drastically damage the cell stability. This is understood by a steep slope of degradation in Fig. 1b which is steeper than the other ones. Oppositely, the recovery rate is smaller for Device 2. We should not forget to analyze the relation between the FF and series resistance, R_s of the materials. A lower FF represents a bigger R_s.⁶ Therefore, Device 1 and 3 have a bigger R_s than Device 2 or they are found rather defective absorber layers. Shunting of such a device is also more probable. Junsangsri et al. introduced a model based on practical data by considering the single diode model where the FF decreases with increasing R_s.²³ The resistivity or conductivity of aged perovskite solar cells was investigated by transient spectroscopy.²⁴ Their estimation of τ is around 10 nanoseconds (Fig. 4 of ref. 24) which is well in the range of our modeling fitting values (20–100 ns). The practical outcome of this analysis is that a thinner metallic back contact is more desirable for perovskite devices. Also, a moisture-protective layer such as C or graphite between the HTL and metallic contact may reduce the migration of metallic ions into the HTL or perovskite layer and finally enhances the device stability.^25,26 It may be useful to investigate the report of Bao et al., which reports that after 20 days of exposing the perovskite device to air humidity, the color of the perovskite changes to yellow as a result of dissociated grains or increased grain boundaries in the absorber layer.²⁷ Our modeling can only be applied to such a degradation source if one considers that the metastable defects may be created at the boundaries or mobile ions migrate towards the interfaces via these grain pathways. This is still possible as we included several parameters in the baseline of our modeling, e.g. τ_s, F, which can be assigned to several defects at different levels of bandgap, and parameter K which may be related to other device parameters that have not been considered directly in the model. Looking at the aging test results of ref. 3–6 it’s obvious that the FF variation is significant in all devices, the open-circuit voltage is almost unchanged and J_sc changes slightly less than the FF.

3.2. Modeling the degradation/recovery of efficiency

Fig. 2a–c show the modeling results of the degradation/recovery trends of efficiency data versus aging time of Device 1, 2 and 3, respectively. We used the cftool of MATLAB to model the data taken from ref. 3–6. Except for a few cases most of the data were accurately fitted to a Gaussian distribution and only two of them fitted to a single or double term exponential function. The idea of such a fitting was taken from ref. 28 where the authors fitted the degradation/recovery trend of an a-Si:H based solar cell with some complicated exponential functions. We nevertheless found better and rather straight fitting functions for our data for perovskite devices. Except Device 1 with Ag and Device 3 with Spiro-OMETAD, the other data were fitted with a Gaussian function as,where a_1,2, b_1,2, and c_1,2 are fitting parameters with the values given in Table 4. The double exponential function is in agreement with our previous assumption of having slow and fast metastable defects responsible for the degradation/recovery behavior of perovskite solar cells. It is also in agreement with the double exponential function used to fit the PL data of perovskite devices in ref. 17. Different than the other devices, Device 1 with Ag and Device 3 with Spiro-OMETAD are fitted with,

η_Ag(t) = a₁ exp(b₁t) + a₂ exp(b₂t) (14)

η_Spiro(t) = a₁ exp(b₁t). (15)

Fig. 2 Modeling the degradation/recovery of efficiency vs. aging time of (a) Device 1 exposed to air without encapsulation, (b) Device 2 exposed to 10% moisture, and (c) Device 3 exposed to 40% humidity. Most of the data were fitted with a Gaussian double exponential distribution. All data were normalized to a max. value to facilitate the comparison of their stability trend. The experimental data are shown as triangles, squares, stars and circles (filled or unfilled symbols). These data were taken from ref. 3 (for (a)), ref. 4 (for (b)), and ref. 5 and 6 (for (c)), respectively.

Table 4 Fitting coefficients for η vs. aging time of Fig. 2

Device #	a1	b1	c1	a2	b2	c2
Device 1 with CuI	0.1	32.65	69	1.03	160	496
Device 1 with PEDOT	0.24	115	78	1.78	1.61	281
Device 2 with AgAl	0.91	34.24	153	0.48	320	284
Device 2 with Ag	0.56	0.06	0.52	0	0	0
Device 2 with Al	1.5	75.23	46	0.16	139	267
Device 3 with PEDOT	0.87	8.20	50	4.94	765	132
Device 4 with Spiro	1.04	0.56	0	0	0	0

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The degradation trend of η is similar to that of the FF where Device 2 has the steepest degradation rate and slightly recovers after 50 hours. Device 1 shows a recovery in efficiency for the first 100 hours of aging while Device 3 with PEDOT:PSS shows a continuous degradation with a near constant trend for the first 50 hours of aging. In Table 4, parameters b₁ and b₂ resemble the slow or fast recombination lifetime parameters as we discussed for the FF. Interestingly, they are in the same order of magnitude as the ones for the FF trends; 10–10² ns. On the other hand, a₁ or a₂ can represent the initial efficiency or saturation efficiency as was discussed in ref. 28. Again a great correlation is obtained with the normalized values of efficiency where 0 < a_1,2 < 1. Overall, we found a great correlation between the degradation/recovery of η and FF. Thus it can be said that η is correlated to R_s too. It is known that bigger grains reduce R_s since many transport barriers are surpassed by a lower number of grain boundaries.²⁹ A detailed temperature-dependent PL study has indicated that the higher efficiency of large grain perovskites originates from the larger radius and lower binding energy of the donor–acceptor-pair in larger grains (≥1 μm), which in turn causes faster separation and collection of the pairs therein. It was also suggested by Dong et al., and Nie et al., that the diffusion length and mobility of a carrier can be improved significantly in large grains even over 3 μm.^30,31 This is in line with the discussion in the previous section where a longer diffusion length was related to a higher FF. Again we see the consistency between the modeling results between η and FF. According to aging test data given in ref. 3–6 a part of the degradation/recovery in efficiency arises from a change in J_sc which is considered in following section.

3.3. Modeling of degradation/recovery of J_sc

According to Ohshima et al. J_sc degrades for shorter diffusion or drift lengths, L_diff or L_drift, for minority carriers.³² It is known that an increased defect density reduces the diffusion/drift length via J_sc ∝ L ∝ τ ∝ 1/N. In ultrathin perovskite devices, i.e. d_perovskite ≤500 nm, we can assume that only the drift component can be very effective in a pin structure perovskite solar cell. Then we can relate J_sc to L_drift through the ratio of initial and degraded/recovered current density as,where J_sc,i and J_sc,d refer to the initial and degraded short-circuit current density after aging, respectively. Furthermore, the relationship between N_d and N₀ is exponential as we calculated in ref. 25,

N_d(t) = N₀ exp(−γt) (17)

where γ is the defect annealing rate and t is current injection time (in this case exposure to moisture). Therefore, eqn (16) leads to,

The above equation describes well the degradation/recovery of J_sc over aging time. For N₀ < N_d the ratio of and thus J_sc(t) degrades. Oppositely, when for N₀ > N_d, the ratio of and J_sc(t) increases over time or recovers. In the case of the data we considered in this paper, ref. 3–6, all J_sc trends show degradation which means an increment of metastable defect density by a moisture effect. The reduction in J_sc of Device 1 with PEDOT:PSS after about 350 hours was only 10%.³ This shows a primary degradation which is then recovered after about 100 hours aging. Oppositely, the J_sc of Device 1 with CuI goes degrades from the beginning and after 350 hours aging, gradually decreases to reach 70% × J_sc,i. N₀ and N_d are then obtained in the order of 10¹⁵ cm⁻³ and 10¹⁷ cm³ using eqn (18). These values are, clearly, smaller in the case of Device 1 with PEDOT:PSS but with the difference that at the beginning of aging we get N₀ = 10¹⁵ cm⁻³ > N_d = 10¹⁴ cm⁻³. When J_sc drastically reduces with aging, N_d becomes much greater and may even go beyond 10²⁰ cm⁻³. This is what we see in Device 2 with Al and Device 3 with Spiro-OMETAD. The other devices show incomparable −ΔJ_sc and −ΔFF e.g. within 10–40%. This confirms that most of degradation arises by variation in the FF. Recovery occurs when the electric field is strong such that it compensates decreased L_drift (because of increment in metastable defects). It is worth noting that among all the devices considered here, only Device 2 with AgAl shows recovery in J_sc which is actually ΔJ_sc = +10%; an increase from the beginning of aging. We calculated N₀ = 10¹⁷ cm⁻³ and N₀ = 10¹⁶ cm⁻³ for this case. Finally, we found no remarkable change in V_oc of the considered devices or the variation in V_oc was very random.

4. Conclusion

We theoretically investigated the degradation/recovery of fill factor, efficiency and short-circuit current density of perovskite solar cells aged at air-humidity exposure. The experimental data were taken from ref. 3–6. Three devices were considered with the same absorber layer but different windows, HTLs and back contacts all aged in the same way (exposing to air humidity). Fill Factor was modeled by considering the variation in metastable defect density and its role in the degradation/recovery of the cell. We started by connecting FF to carrier drift length, L_D, which is in turn related to the metastabilities of both slow/fast types. This allowed us to track the variation of FF over time, FF(t), and fit the data with data for every recovery/degradation trend. Alternatively, the efficiency was fitted with proper functions present in a MATLAB-fitting tool. The efficiency variation trend mostly fit with bi-exponential functions indicating that both slow/fast metastable defects are responsible for changes in efficiency over time. Finally, short-circuit current density was modeled by a new proposed equation based on defect creation/annihilation over the aging period. The modeling results shed light on the underlying physics behind the instability of perovskite-based solar cells. It is concluded that fill factor is the responsible parameter in the degradation/recovery of perovskite solar cells. Fill factor as a measure of series resistance degrades by defect increment and recovers by defect annihilation under aging tests. The metallic contacts are crucial for a stable device and the efficiency stability of every cell is determined by the stability of fill factor rather than short-circuit current density or open-circuit voltage. Our modeling shows that a single degradation/recovery model cannot be applied to the whole device instability behavior which actually makes the modeling of such phenomena rather complicated.

Acknowledgements

We are grateful for the Senior Research Fellowship by the Iran Nanotechnology Initiative Council, 2016; Grant No. 104423.

References

1. Y. J. Jeon, S. Lee, R. Kang and J. E. Kim, et al., Sci. Rep., 2014, 4, 6953.
2. H. Li, K. Fu, A. Hagfeldt and M. Gratzel, et al., Angew. Chem., Int. Ed., 2014, 53, 4085–4088.
3. W. Chen, L. Deng and S. Dai, et al., J. Mater. Chem. A, 2015, 3, 19353–19359.
4. Z. Jiang, X. Chen and X. Lin, et al., Sol. Energy Mater. Sol. Cells, 2016, 146, 35–43.
5. D. Wei, T. Wang and J. Ji, et al., J. Mater. Chem. A, 2016, 4, 1991–1998.
6. J. You, L. Meng, T. Song and T. Guo, et al., Nat. Nanotechnol., 2016, 11, 75–82.
7. H. Li, K. Fu, P. P. Boix and L. H. Wong, et al., ChemSusChem, 2014, 7, 3420–3425.
8. A. Dubey, N. Adhikari and S. Venkatesan, et al., Sol. Energy Mater. Sol. Cells, 2016, 145, 193–199.
9. S. Y. Myonga and K. S. Lim, Appl. Phys. Lett., 2006, 88, 243510.
10. T. Minemoto and M. Murata, J. Appl. Phys., 2014, 116, 054505.
11. A. Guerrero, J. You, C. Aranda, Y. S. Kang, G. G. Belmonte, H. Zhou, J. Bisquert and Y. Yang, ACS Nano, 2016, 10, 218–224.
12. J. Carrillo, A. Guerrero, S. Rahimnejad, O. Almora, I. Zarazua, E. Mas-Marza, J. Bisquert and G. Garcia-Belmonte, Adv. Energy Mater., 2016, 2, 1502246.
13. B. Chen, M. Yang, X. Zheng, C. Wu, W. Li, Y. Yan, J. Bisquert, G. Garcia-Belmonte, K. Zhu and Sh. Priya, J. Phys. Chem. Lett., 2015, 6, 4693–4700.
14. Zh. Gevorkian, V. Gasparian and Y. Lozovik, Appl. Phys. Lett., 2016, 108, 051109.
15. W. Tress, N. Marinova, T. Moehl, S. M. Zakeeruddin, M. Nazeeruddin and M. Gratzel, Energy Environ. Sci., 2015, 8, 995.
16. W. J. Yin, T. Shi and Y. Yan, Appl. Phys. Lett., 2014, 104, 063903.
17. X. Wen, Y. Feng, Sh. Huang and F. Huang, et al., J. Mater. Chem. C, 2016, 4, 793.
18. N. E. Gorji, U. Reggiani and L. Sandrolini, IEEE Trans. Device Mater. Reliab., 2015, 15, 198–205.
19. S. Y. Myong, Adv. OptoElectron., 2007, 2007, 30569.
20. S. Yuan, Z. Qiu, H. Zhang, H. Gong, Y. Hao and B. Cao, Appl. Phys. Lett., 2016, 108, 033904.
21. M. Houshmand, M. H. Zandi and N. E. Gorji, JOM, 2015, 67:9, 2062–2070.
22. S. Schuller, P. schilinsky, J. Hauch and C. J. Brabec, Appl. Phys. Lett., 2004, 79, 37–40.
23. Ph. Junsangsri and F. Lombardi, 25th International Symposium on Defect and Fault Tolerance in VLSI Systems, 1550-5774/10, 2010, pp. 240–249.
24. T. Leijtens, G. E. Eperon, S. Pathak, A. Abate, M. M. Lee and H. J. Snaith, Nat. Commun., 2013, 4:2885, 1–8.
25. M. Aldosari, L. Grigoriu, H. Sohrabpoor and N. E. Gorji, Mod. Phys. Lett. B, 2016, 30:5, 1650044.
26. M. Houshmand, H. Esmaili, M. H. Zandi and N. E. Gorji, Phys. E, 2013, 53, 130–136.
27. X. Bao, Y. Wang and Q. Zhu, et al., J. Power Sources, 2015, 297, 53–58.
28. K. Takahisa, T. Kojima, K. Nakamura, T. Koyanagi and T. Yanagisawa, Sol. Energy Mater. Sol. Cells, 1997, 49, 179–186.
29. J. Chen, T. Shi, X. Li, B. Zhou, H. Cao and Y. Wang, Appl. Phys. Lett., 2016, 108, 053302.
30. Q. Dong, Y. Fang and Y. Shao, et al., Science, 2015, 347:6225, 967–970.
31. W. Nie, H. Tsai and R. Asadpour, Science, 2015, 347:6221, 522–525.
32. T. Ohshima and S. Sato, et al., Sol. Energy Mater. Sol. Cells, 2013, 108, 263–268.

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