Discerning recombination mechanisms and ideality factors through impedance analysis of high-efficiency perovskite solar cells

Abstract The ubiquitous hysteresis in the current-voltage characteristic of perovskite solar cells (PSCs) interferes in a proper determination of the diode ideality factor ( n ), a key parameter commonly adopted to analyze recombination mechanisms. An alternative way of determining n is by measuring the voltage variation of the ac resistances in conditions of negligible steady-state dc currents. A reliable analysis of n based on the determination of the resistive response, free of hysteretic influences, reveals two separated voltage exponential dependences using different perovskite absorbers (3D perovskites layer based on CH3NH3PbI3 or mixed Cs0.1FA0.74MA0.13PbI2.48Br0.39) and a variety of interlayers (2D perovskite thin capping). The dominant resistive element always exhibits an exponential dependence with factor n ≈ 2 , irrespective of the type of perovskite and capping layers. In addition, a non-negligible resistive mechanism occurs at low-frequencies (with voltage-independent time constant ~ 1 s) which is related to the kinetic properties of the outer interfaces, with varying ideality factor ( n = 2 for CH3NH3PbI3, and n = 1.5 for Cs0.1FA0.74MA0.13PbI2.48Br0.39). Our work identifies common features in the carrier recombination mechanisms among different types of high-efficiency PSCs, and simultaneously signals particularities on specific architectures, mostly in the carrier dynamics at outer interfaces.


Abstract
The ubiquitous hysteresis in the current-voltage characteristic of perovskite solar cells (PSCs) interferes in a proper determination of the diode ideality factor ( n ), a key parameter commonly adopted to analyze recombination mechanisms. An alternative way of determining n is by measuring the voltage variation of the ac resistances in conditions of negligible steady-state dc currents. A reliable analysis of n based on the determination of the resistive response, free of hysteretic influences, reveals two separated voltage exponential dependences using different perovskite absorbers (3D perovskites layer based on CH3NH3PbI3 or mixed Cs0.1FA0.74MA0.13PbI2.48Br0.39) and a variety of interlayers (2D perovskite thin capping). The dominant resistive element always exhibits an exponential dependence with factor 2  n , irrespective of the type of perovskite and capping layers. In addition, a non-negligible resistive mechanism occurs at low-frequencies (with voltage-independent time constant ~1 s) which is related to the kinetic properties of the outer interfaces, with varying ideality factor ( 2  n for CH3NH3PbI3, and 5 . 1  n for Cs0.1FA0.74MA0.13PbI2.48Br0.39). Our work identifies common features in the carrier recombination mechanisms among different types of high-efficiency PSCs, and simultaneously signals particularities on specific architectures, mostly in the carrier dynamics at outer interfaces.

Introduction
Over the last few years perovskite solar cells (PSCs) gained great attention in the field of photovoltaics with an unprecedented increase in power conversion efficiency (PCE) for facile solution processing [1][2][3][4]. Currently, the main challenge in order to reach commercialization in the perovskite solar research lies in fabricating high efficient perovskite-based devices exhibiting good long-term stability under real operation conditions. In order to fulfill these ambitious goals, investigations on materials science are mainly focused on three directions: (i) development of new stable light harvesting perovskites alternatives to CH3NH3PbI3 (MAPI), (ii) engineering selective charge extracting materials and (iii) optimization of fabrication processes/materials quality.
However, despite progressive improvements are systematically reported, there is still a lack of comprehension on important physical aspects ruling the device performance [11]. Several standard characterization protocols and tools used in photovoltaics present difficulties when applied to PSCs because of the occurrence of anomalous behaviors which prevent typical interpretations. The ubiquitous hysteresis in the current densityvoltage ( J V  ) curve has forced the definition of good practices in order to properly Ideality factor ( n ) has been traditionally adopted as an indicating parameter of the carrier recombination processes dominating the solar cell operation. However, its analysis from J V  curves has resulted inconclusive [15] for PSCs because of the mentioned masking contributions of hysteretic currents. Apart from open-circuit dependence on light intensity and temperature, and electroluminescence analyses [15], an alternative way of determining the diode ideality factor is by measuring the voltage variation of alternating current (ac) resistances in conditions of negligible steady-state direct currents (dc). Differential resistances are no more than current derivatives with respect to voltage perturbations in such a way that recombination losses and ideality factors distinctively appear by analyzing resistances as well. This is feasible by performing impedance measurements in open-circuit conditions under varying illumination intensities following a well-established protocol.
Several works have drawn upon the relative contribution of bulk and outer interface carrier recombination routes on the overall performance of PSCs [15][16][17][18][19]. It has been specifically suggested that a transition occurs from bulk-to interface-dominated recombination when pristine solar cells experience ionic contact polarization upon cycling [16]. For less performing or degraded PSCs, interfaces seem to play a determining role on carrier recombination [20].
The ideality factors defined by Eqs. (3) and (4) may accord in the ranges where all the considered approximations are valid, if electronic reciprocity is obeyed [28,29].
Interestingly, the definition of n as from Eq. (3) is directly proportional to the recombination resistance per unit area, which also can be obtained from Eq. (2) with two contributions in parallel (assuming voltage-independent photo-generation), namely sh R and rec R as The resulting total dc resistance is presented in Eq. (S6). It is evident that by substituting Eqs. (1) and (2) in (3), a constant n is expected. In any case, the evaluation of n can be accessed from the measurement of the resistance. In the case of deviation from a single where the integration limits may ensure the above specified.
Note that despite the equations for   n V can be used for the particular estimation of n at a given V , our approach here will be more interested in getting n from the exponential trends of the resistance. However the determination of such exponential in acetonitrile (10mg/ml) onto the mesoporous layer, followed by a sintering step at 500 °C for 10 min to decompose the Li-salt as previously described [34].
The 3D MAPI perovskite layers were fabricated by a single step spin-coating procedure reported by Seok et al. [35] ( Fig. 1a-d).  Films were then annealed at 100°C for 60 min. For forming additional 2D perovskite film on top of this perovskite film, and between TiO2 and CFMPIB, (Fig. 1f,g) substrates were treated with a PEAI isopropanol solution. 100 μL of PEAI solution (15 mg/ml) were spin-coated on the TiO2 substrate or the as-prepared perovskite films at 4000 rpm, and annealed at 100°C for 5 min.

Characterization
All electrical characterizations were carried out at room temperature and atmosphere (humidity bellow 10%). For the impedance analysis and the J-V curve bias sweep monitoring an SP-200 BioLogic potentiostat was used. For standard PCE evaluation a AAA solar simulator from Newport equipped with a 1000 W Xenon lamp was employed while a dc regulated Illuminator from Oriel Instruments was used for tuning oc V as the light intensity was varied.

Results
In this work we analyzed PSCs with the structure FTO/TiO2  Fig. 1a,e, respectively). In addition, MAPI-based variants were measured (Fig. 1b-d) in which 2D perovskite thin capping was included with AVA2PbI4 between TiO2 and MAPI, and/or Bl2PbI4 between MAPI and spiro-OMeTAD. Similarly, CFMPIB variants included PEA2PbI4 as a thin coating at both interfaces (Fig. 1f, g). With these cells we conducted J V  measurements in dark and under illumination, as well as IS measurements at open-circuit with different light intensities, as previously described. The data of IS were fitted to equivalent circuit models. Based on the resulting resistive and capacitive behaviors, differences in the recombination features between the 3D perovskite type and the presence of 2D capping at the interfaces were found. conditions [36,37]. Specifically, a significant loss in fill factor (FF) of around 70% of the PCE is observed for 2D/MAPI devices. On the other hand, the decrease of PCE in mix-perovskite samples and MAPI samples without 2D layer did not exceed 10% of initial values, in agreement with our previous study [38]. These data are summarized in Table S3 where corresponding performance parameters: oc V , sc J , FF and PCE can be found, as well as the PCE ratio between fresh and stressed samples. On the MAPI cells these results seems to contrast with our previous findings [39] where the stability was remarkably improved when including 2D capping layers.

Jcap effects and n values from dark curves
It is important to highlight that in photovoltaic responses shown in Fig. S1 are only J V  curves by sweeping the bias in the direction from open-circuit to short-circuit.
The discussion on the efficiencies on the sweep direction and corresponding stability can be found in recent previous works [39]. Nevertheless, in this sense the basic measurement of dark J V  curves at both scan directions is included [21,40,41].
Dark J V  hysteretic curves are illustrated in Fig. 2 where typical capacitive square patterns are observed, pointing to the presence of an approximately constant capacitance C in reverse and low forward bias regimens [21]. Accordingly, one can take cap J s C   , being s the voltage scan rate, and the estimated capacitances result in the order of 2 μF cm   , which match the low-frequency capacitance values extracted from dark impedance, as discussed in the next section. Slight capacitive differences appeared between MAPI-based samples (Fig. 2a), while no evident cap J dissimilarities were detected for the CFMPIB-based samples (Fig. 2b) [42]. Differences between MAPI samples and similarities between CFMPIB cells are also evidenced towards the region of high injection currents. Here, the ideality factor can be extracted from the exponential response following Eq. (2). MAPI-based cells without interlayers (or comprising spiro-OMeTAD/Bl2PbI4 interlayer) in Fig. 2 (a), and CFMPIB-based cells in Fig. 2b exhibit similar ideality factors, around 2. However, the two structures comprising TiO2/AVA2PbI4 in MAPI-based cells (Fig. 2a) showed ideality factors even larger than 3. These two later cases also included the presence of small inverted hysteresis between 0.5 V and 1.0 V, which suggest a complex interplay of non-capacitive effects and/or changes in the capacitance itself. The differences in exponential laws are better illustrated in Fig. S2c,d where the dc resistance was obtained following the current derivative on the average dc currents of Fig. 2a Also by using the definition of Eq. (3), ( ) n V can be obtained as displayed in Fig. S2e,f.
Following the relationship between recombination mechanisms summarized in Table   S2 and their corresponding ideality factors extracted from J V  curves, we might understand the behavior of 3D-MAPI, 2D/spiro-MAPI and CFMPIB-based samples ( 2 n  ) as indicative of large generation-recombination processes in the depletion region involving both minority and majority carriers. This agrees with the nearly voltageindependent capacitance extracted at reverse and low forward bias (Fig. S2a,b), suggesting a fully depleted quasi-intrinsic perovskite layer forming a p-i-n heterojunction. Actually, it is known that p-i-n diodes operate in the high-injection condition ( 2 n  ) and hence recombination is within the i-region [43]. One would then expect that the classical model of application for these devices is that of a p-i-n diode with dispersion in the ideality factor possibly due to non-capacitive effects including series resistances. The latter effects could be behind behaviors for those samples including the interphase TiO2/AVA2PbI4, i.e. 2D/TiO2-MAPI and 2D/3D/2D-MAPI, where n is significantly larger than 2. Contrastingly, the exponential slopes were not so scattered in mixed perovskite based samples with 2D capping. This would be a second hint pointing to the predominance of bulk recombination in CFMPIB-based samples in comparison to the larger importance of interfaces in MAPI cells.

Impedance spectroscopy analysis
IS analysis was carried out in each of the above analyzed samples. Complete spectra with respective modeling are presented in Fig. S3,S4 for MAPI-and CFMPIB-based samples, respectively. The employed equivalent circuits are those of Fig. 3a,b and illustrative capacitance and impedance spectra are displayed in Fig. 3c,d for different illumination intensities in open-circuit conditions. Fig. 3a where s L and s R are series inductive and resistive elements (related to wires and connections), respectively.

For MAPI cells the used equivalent circuit is drawn in
The core of the circuit is composed by two couples of resistive-capacitive (RC) elements in matryoshka configuration. At frequencies higher than 100 Hz, the capacitance spectra describe a plateau around a value Hf C , and for almost all illumination intensities a clear arc of diameter Hf R is observed in the impedance plots. However, as frequency is decreased below 100 Hz, capacitance is significantly enhanced towards a second plateau around Lf C in the Bode plots and a second arc Lf R is evident in the impedance plot.
The equivalent circuit in Fig. 3a, is probably the most reported model for describing IS spectra in PSCs because of its simplicity and physical meaning [20,44,45]. In general Hf C and Hf R have been identified with dielectric and recombination properties, while the origin of Lf C and Lf R , although still are under debate, has been associated with the slow mechanisms of accumulation capacitance that also produce the hysteretic currents in the J V  curves [11,41,46]. It is already known that recombination properties of the TiO2/absorber interfaces largely depend on specific preparation routes and processing conditions. For instance, the use of ammonia [47] or engineering of hierarchical microstructures [48] alters the TiO2 surface states and concomitant electronic structure with evident changes in interface recombination.
In contrast, for CFMPIB-based cells the model of Fig. 3b for the analysis of IS spectra results more featured. Two main sets of elements have been highlighted in red.
First, affecting the lowest frequencies, an inductive element Lf L with series resistance L R are shunting Lf C and Lf R . These elements can produce a decrease of capacitance in the Bode plot or a distortion from a semicircle in impedance plot (see Fig. 3d  The main resistive and capacitive parameters (namely Lf C , Lf R , Hf C and Hf R ) determined from fits are presented in Fig. 4. Corresponding general behaviors in PSCs have been earlier reported in the literature [20,51] and here we present an illustrative phenomenological modeling (solid lines in Fig. 4) based on the recombination resistance in Eq. (5) in parallel with sh R , and for the capacitances following the empirical form where  , 0 C and 1 C serve as fitting parameters. Specifically,  is an exponential At low frequency, we notice that Lf C exhibits the exponential trend of Eq. (7)  C that points to the occurrence of electrode polarization caused by ionic double layers, as introduced in previous papers [41] that also agrees with the capacitive currents reported in Fig. 2.
Regarding high-frequency capacitive responses, Hf C behavior in Fig. 4b,d is nearly constant at forward bias around the geometric capacitance proportionally to the expected thicknesses of the perovskite layers, as earlier commented on Fig. S2a,b. Only a slight increase is found towards larger forward biases that hardly fit an exponential law. Such minor Hf C augment could be originated by thermal effects after prolonged illumination [52], and it will be carefully studied in future works.
High-frequency resistance Hf R is understood in terms of Eq. (5) as previously discussed.
At low forward bias (i.e. 0.1 V) the ac resistances in Fig. 4a,c seems to be in the same order that dc resistances obtained from dark J V  curves in Fig. S2c,d, and associated to shunts. However, above 0.6 V an exponential decrease was found with different trends among the samples. For the CFMPIB-based samples n was found to be between Comparing exponential trends (extracted from IS analysis) with the results from dark J V  curves, CFMPIB samples also deliver lower n values than MAPI cells.
However, overestimation of n does not occurs here provided that IS analysis excludes s R effects in the Hf R evaluation. Once more, as summarized in Table S2,

Discussion
The results shown in previous sections confirm that CFMPIB samples exhibit lesser n values than MAPI cells. Our IS approach, by measuring in open-circuit conditions under varying light intensity, seems to provide more reliable n values than those inferred from J V  curves in which the capacitive currents certainly complicate the analysis. Although resistances determined by IS can be easily interpreted in terms of known recombination mechanisms accounting for the operation of inorganic photovoltaic technologies, perovskite solar cells present a distinctive capacitive feature in the low-frequency wing of the spectra (Fig. 4b,e). A huge increase of capacitance that grows with illumination and charge injection up to -2 mF cm  (under 1 sun light or 1.0 V) has been measured and connected to the hysteretic features and surface/interface phenomena [20,21,[40][41][42]51]. This is indeed a prominent issue that also correlates with the resistive behavior which exhibits the same exponential trends with voltage (Fig. 4).
In On the other hand, capacitance increments following an exponential law with oc V has been typically understood in terms of chemical capacitances C  , which are known to account for bulk charging of density of states. For carrier occupancy of conduction and valence band states following the Boltzmann statistics, one can calculate the bulk chemical capacitance (BCC) per unit area [30,54] and also the edge chemical capacitance (ECC) confined at the interfaces (see Eqs. (S8)-(S15) and corresponding explanatory comments). Assuming a bulk origin for the low-frequency capacitance, and noticing that resistive and capacitive responses are highly correlated (exhibiting in Fig.   4 Fig. S8) [11,57]. Obviously, further theoretical analyses are still needed here for clarifications, but note that whatever the mechanism behind Lf C is, it is strongly constricted by a slow relaxation time constant (subsequently commented), possibly connected to the ionic kinetics. This, for instance, prevent the probably chemical capacitance to affect Hf C , in excess of g C , at frequencies typically above 1 kHz [52].
Anyway, regardless of the chemical or dielectric electronic nature of Lf C , its correlation with ionic processes seems to be a fact, as tackled in several studies also referring the J V  curve hysteresis phenomena [11]. These ionic mechanisms could be behind the large photo-generated Lf C , so they should be considered in subsequent theoretical approaches. Accordingly, the assumptions and approximations made in order to match models and experiments are illustrated in Fig. S8 and the above mentioned models are compared in Table S4. In any case our approaches entail the importance of interfaces as the locus explaining the operating mechanisms in PSCs [51,58,59].
An alternative way of approaching recombination features is by examining the response times from IS measurements. The corresponding high-frequency characteristic time calculated from the circuit elements as Hf is presented in Fig. 5

 
, approximately constant within the order of seconds (see Fig. 5), is responsible for the IS patterns at low frequencies. For higher illumination and consequently larger ph J , the process remains almost unaltered in time scale (see Fig. 5), and since more current flows ), as evidenced from exponential slopes in Fig. 4b,d and Fig. S6. Response times of the order of seconds were previously identified by us using IS with less performing PSCs [20,53]. Similar trends are now verified using high-efficiency perovskite solar cells. As discussed previously [20], a constant (illumination-intensity independent) response time is a strong indication that electronic mechanisms originating capacitance and resistance should be inherently coupled. But again the time constant corresponding to the low-frequency mechanism cannot be considered a true carrier lifetime because the chemical nature of the capacitive process is not assured. Nevertheless, here we also highlight the possible connection between the ideality factor, from resistances and photocurrents, and the exponential coefficient  of Lf C . In case of pure chemical capacitance the exact match n   is expected [43], while in our case a general trend n   Fig. 5b, independently of the presence of thin film 2D capping layers.

Concluding Remarks
Our impedance analysis of different 3D perovskite absorbers (CH3NH3PbI3 or mixed Cs0.1FA0.74MA0.13PbI2.48Br0.39) and 2D perovskite interlayers (AVA2PbI4, Bl2PbI4, and PEA2PbI4) allow identifying common patterns and dissimilarities. We can say that (i) the dominant resistive elements (high-frequency) behave for both set of samples following an exponential dependence on voltage with 2  n , slightly lower in the case of CFMPIB-based cells. The simpler interpretation is connecting this resistance to the recombination mechanism within the absorber bulk. However, it cannot be completely discarded that outer interfaces actively participate in the carrier recombination determining Hf R values.
Concerning the low-frequency behavior, it is demonstrated that (ii) exponential indexes vary differently as a function of the absorber type: again 2  n for MAPI-based