Accelerating the Assessment of Hysteresis in Perovskite Solar Cells

Halide perovskite materials have reached important milestones in the photovoltaic field, positioning them as realistic alternatives to conventional solar cells. However, unavoidable kinetic phenomena have represented a major concern for reliable steady-state performance assessment from standard current–voltage measurements. In particular, the dynamic hysteresis of current–voltage curves requires relatively long stabilization to achieve a credible figure for the power conversion efficiency. Hysteresis is caused by complex current transient phenomena that become active during staircase voltammetry. Here, we address the root of this problem. We pinpoint the dynamic characteristics behind the slow transient responses to strategically predict a minimum time delay and thus estimate the power conversion efficiency under steady-state conditions. Circuit-element analysis and impedance spectroscopy confirm our predictions based on an advanced transient study. Our results fundamentally explore the possibility of reducing data time acquisition in a reliable performance assessment, providing disruptive solutions and perspectives toward systematic production of photovoltaic perovskites.

T o evaluate device performance in perovskite photo- voltaics, it is important to understand the memory properties, based on internal ionic−electronic effects, of this semiconductor material. 1,2−6 By interface engineering strategies 7,8 and adequate measurement protocols, 9−12 it has been possible to minimize the "hysteria around current-voltage hysteresis" in the research field.
Nevertheless, hysteresis mechanisms persist even in recent perovskite devices with high performance metrics. 13In order to circumvent the hysteresis problem, 14,15 alternative techniques for assessing the performance of photovoltaic perovskites have been employed in the process of independent power conversion efficiency certification by cell calibration laboratories such as NREL or Newport (e.g., quasi steady-state current−voltage measurements and protocols based on stabilized power outputs near the maximum power point, MPP). 13,16However, these standardized but time-consuming practices can lead, in certain scenarios, to potential intrinsic device instabilities due to the use of ad hoc slow scan rates.Thus, the stability of the perovskites under the efficiency testing protocols based on current−voltage scanning represents a significant challenge for pushing this technology toward commercialization.In each current−voltage curve, the current is sampled in a discrete linear sense via staircase voltammetry.It follows that a reliable analysis of transient phenomenology in a stepwise voltage scanning, as indicator of charge-carrier dynamics in the transition between consecutive current recordings, can provide control over internal state variables and the respective relaxation effects that result in complex memory properties. 5,17,18n this Letter, we describe the multiple transient responses exhibited by perovskite solar cells along a current−voltage curve.As nonlinearity and memory effects play a dominant role in "hysteretical" current−voltage characteristics, we aim to determine a minimum delay time, in terms of scan rate, to carry out current−voltage measurements under steady-state conditions.Recently, hysteresis effects have been classified into two major types, capacitive and inductive, according to the dominant response of impedance spectroscopy. 19We use circuit theory as inspiration that describes the extraordinarily complex physical processes that turn out to be ubiquitous in halide perovskites, requiring eventual transformations of electrical elements along the voltage changes.Throughout our study, the multiple versions of transient responses exhibited in a halide perovskite that commonly undergo transformation from capacitive to inductive properties in a single current− voltage curve are introduced in accordance with the experimental observations reported in the literature.Analysis of our representative photovoltaic perovskites shows that the proposed numerical approximations are satisfied by the experimental data.Our procedure, easily implementable as an industrial-level automatic routine that monitors the realtime current during the current−voltage protocol, could continuously adapt the measurement conditions as the perovskite solar cells show important changes in the dynamic transient responses regarding device operation in the context of different degradation pathways. 20he representative structure of the photovoltaic perovskites in this study is fluorine-doped tin oxide (FTO)/compact titanium dioxide (c-TiO 2 , 30 nm)/mesoporous titanium dioxide (m-TiO 2 , 125 nm)/perovskite (350 nm)/2,2′,7,7′-Tetrakis[N,N-di(4-methoxyphenyl)amino]-9,9′-spiro-bifluorene (spiro-OMeTAD, 150 nm)/Au (100 nm), with a nominal a c t i v e l a y e r c o m p o s i t i o n o f Cs 0.05 Rb 0.05 MA 0.15 FA 0.75 Pb 1.05 (Br 0.05 I 0.95 ) 3 .All fabrication procedures and electrical measurements are described in the Supporting Information.We obtained device performance metrics of around 20% derived from current−voltage measurements in the forward and reverse scan directions, which are within the typical values reported for this formulation. 21evertheless, as shown in Figure 1a, the estimate of the real efficiency is not trivial since considerable hysteresis mechanisms emerge, both normal and inverted.At this point, it is necessary to put the transition between discrete photocurrent values under a magnifying glass, thus requiring an "intelligent transient analysis" to accurately capture steady-state current− voltage responses.This approach is, however, perceived as highly complicated because it requires certain mathematical and engineering expertise from the perspective of nonlinear electrical circuits. 12Our representative metal halide perovskite, based on a mesoporous n-i-p configuration, shows different types of transient responses in separate voltage domains, as also reflected by literature results. 22,23From this closer look at the stepwise current−voltage measurements, we observe, after an initial faithful reproduction of the step transition, voltagedependent gradual transient decays in the region of regular hysteresis for low applied voltages (see Figure 1b,c).In contrast, at high voltages, final abrupt rises in current dynamics are found in Figure 1d,e leading to the inverted hysteresis of the perovskite device.All these transient effects clearly underlie the physical mechanisms associated with electrical relaxation processes in the sense of transport, accumulation, and recombination of ionic and electronic charges. 24,25Therefore, the design of transient analysis protocols in current−voltage measurements to follow or, at least, to use as complementary information is essential to adhere to standard procedures and thus ensure that the reported perovskite solar cells have been obtained under steady-state conditions.
Based on the ionic−electronic properties of the perovskite semiconductors commonly manifested throughout a current− voltage curve in perovskite solar cells, we formulated a familiar dynamical model 26−28 outlined primarily from the following expression: where the total current flowing through the solar cell j is divided into four distinct pathways: (i) a displacement current that charges the geometrical capacitance C g of the perovskite material; conduction channels in which the current may be extracted from the contacts (ii) instantaneously via recombination processes J rec or (iii) slowly with an ion-modulated current j d ; and (iv) an interfacial current in the sense of a corresponding charge Q s .For the recombination current, note that we consider, throughout the Letter, J rec = J rec0 e qV/n rec kT , where q is the electron charge, n rec is an ideality factor, k is Boltzmann's constant, and T is the absolute temperature.This gives rise to the conductance g rec = dJ rec /dV.Nevertheless, the current j d cannot follow the external voltage V rapidly, reacting slowly to achieve the steady-state conditions. 27,29On the other hand, the surface charge Q s , a function of an internal voltage V s and thus introducing an interfacial capacitance C s = dQ s /dV s , also shows a retarded dynamic due to ionic charges. 26,30In this way, the variables V s and j d generate the famous memory effects in perovskites with characteristic behaviors described, respectively, as that control the voltage-dependent conductivity in the perovskite materials depending on the ionic kinetics.In eq 2, the slow variable, for convenience, is the surface polarization voltage V s , governed by the characteristic time τ s , which introduces a transitory ionic conductance, g ion = 1/τ s (dQ s /dV s ) = C s /τ s . 26This relaxation process is commonly visible in the low-voltage region.On the other hand, the dynamical behavior of the current j d in eq 3, dominant at high voltages, models the process that establishes an additional conductivity channel causing the memory effect by a slow recovery of relaxation time τ d in response to the external stimulus V. 26,30 In effect, metal halide perovskite devices commonly undergo a transformation in the dynamics of the slow current components, both related to ionic-controlled recombination processes via surface polarization effects. 12,27The kinetic properties at long time scales are modulated by slight variations of the characteristic time constants (τ s and τ d ) that determine the dominant relaxation response (eq 2 or 3) at different voltage regions.−34 Thus, the additional conductivity pathway in equilibrium can be also modeled from g elect = dJ elect /dV.Two types of electronic conductances (g rec and g elect ) indeed emerge in the model with similar nature but exhibiting different physics beyond the time scales behind them.The first represents the immediate recombination of photogenerated charges.The second is related to an ionic reorganization in the device that enhances the electronic current in terms of charge collection or trapping.Importantly, this delayed photocurrent mode forms the prominent feature of the chemical inductor, As mentioned in previous works, 35,36 this inductive process is a general feature present in many processes and materials, caused by a delay effect on the electronic phenomena associated with slow ionic mechanisms and related to the negative capacitance.The idea behind the general term "chemical inductor" is that this electrical element is not based on electromagnetic effects but arises from mixed ionic− electronic conduction in complex materials.Therefore, the model has a total of two memory variables that interchange the governability of the response at long time scales in the current−voltage curves of perovskite solar cells due to the nonlinear ionic−electronic dynamic complexity of this material.However, the slow kinetic properties in perovskites, based on ionic-controlled surface recombination processes, 37−39 are commonly regulated by a single relaxation time due to the characteristic memory-based coupling of these materials whose phenomenological consequence is that τ s ∼ τ d . 22,29,40In essence, this identification means that the slow current and surface polarization are both caused by a single variable, as reported experimentally. 27A similar framework analysis has been successfully used to explain the dynamic hysteresis 26 and linear impedance patterns 29,30 of perovskite solar cells.Here, we will try to explain the experimental transient dynamics observed during consecutive recordings in current−voltage measurements with the aim of establishing determined conditions to reach steady-state values and eliminate hysteresis.According to our physical model, we consider the electrical representation of a perovskite solar cell, as shown in Figure 2a.Inspired by the multiple versions of current transient responses to stepwise voltage scanning observed in Figure 1, this advanced nonlinear electrical circuit models the voltagedependent behavior in the time-domain along the current− voltage curve of halide perovskites, transforming dominant hysteretic dynamics from capacitive to inductive in nature as the bias point increases.This equivalent circuit, in addition to qualitatively explaining transient behavior in terms of conductances, helps to articulate a consistent interpretation of the complex physics behind this ionic−electronic conductor 29,30 in comparison with other electrical models reported in the literature. 27,31hen one changes the applied potential in stepwise current−voltage measurements, the new equilibrium point totally modifies the transient response 12,17 as, in effect, illustrated in Figure 1b−e.To analyze the temporal evolution at a given voltage bias under small-signal conditions, we obtain important insights into the transient dynamics from the linearized equations of our three-dimensional physical model (eqs 1−3) at a steady-state point.For a step change of applied potential Δv in the context of stepwise current−voltage measurements, the resulting small-amplitude current response Δj(t) in the time-domain can be determined, in terms of operational Δj ss and memory Δj mem (t) contributions, from the following model equations: obtained as the natural solution, via numerical integration methods or the Laplace transform technique, of the previous set of the differential equations (refer to eqs 1−3) in the linear version.The corresponding theoretical analysis is described in detail in the Supporting Information.
Although it is irrelevant in the destabilization process (hysteresis) of the current−voltage curves, we first describe the additional term in eq 5 as an ultrafast process corresponding to the charge of the geometrical capacitance, by a characteristic time of τ v = R s C g . 41,42This capacitive current can be indeed found in all the transient responses of Figure 1, in the form of initial sharp jump discontinuities followed by ultrafast decays, due to the series resistance R s and the charging of the constant capacitance C g whose origin is purely bulk dielectric in nature.Note that we consider throughout the Letter, for simplicity, that the parasitic series resistance effects are negligible (R s → 0) in comparison to the conductance states of the perovskite over the voltage range swept in the measurement under study.
Our model, with a clear physical meaning, shows that there are two recombination pathways in halide perovskites as reported in the literature: direct g rec and delayed by ionic reorganization g elect , 28 evidenced in dc conditions (refer to eq 6).The memory function Δj mem (t), on the other hand, results in an exponential function with a prefactor consisting of a combination of conductances with an ionic and electronic nature (g ion and g elect , respectively) and the characteristic relaxation time τ kin that generalizes the electrical charge coupling in halide perovskites.In effect, eq 7 provides a theoretical visualization of the interplay between perovskites' ionic and electronic responses in the appearance of current− voltage hysteresis. 43Following the rule of transient dynamics in Figures 1b−e, we obtain g ion > g elect at low voltages (gradual decays, V → 0) and g ion < g elect in the high-voltage region (abrupt rises in current at long time scales, V → V oc ).To summarize, Figure 2b shows the graphical representation of simulated currents in the time domain varying the value of the applied voltage.Representative transient responses, with a dynamical behavior similar to those of Figure 1, have been labeled to reveal relevant memory-based currents in relation to our theory: Δv(g rec + g ion ) and Δvg dc for t ≪ τ kin and t ≫ τ kin , respectively, and the difference between both terms that results in the interesting value Δv(g ion − g elect ).It is clearly seen that this last parameter decreases as the bias voltage increases, going from positive to negative values and thus inverting the transient dynamics.Importantly, the results obtained from numerical simulations displayed a good qualitative agreement with experimental data, giving a step beyond current−voltage curves and impedance spectra 25−30 in modeling the rich phenomenology of perovskites by using this family of mathematical models.
Theoretically speaking, the necessary time delay for steadystate device operation, Δj(t) → Δj ss , and thus, to eliminate the memory traces (Δj mem (Δt ss ) → 0) that give rise to the current−voltage hysteresis is that for which Δj ss ≫ Δj mem (Δt ss ). 44We consider the 5% criterion, a typical tolerance band used in control engineering, yielding In eq 8 appears the classical result 3τ kin described in basic control engineering courses, 45 but also an additional term originated by the ionic−electronic coupling of perovskite solar cells that modulates the conventional suitable time delay and makes emerge the discrepancy between forward and reverse scans when one simply selects the classical value.Thus, although the speed of transient dynamics depends fundamentally on the value of the time constant τ kin , the settling time Δt ss associated with steady-state conditions is also a function of a ratio of conductances.The final step is the selection of the most restrictive voltagedependent Δt ss in the large amplitude voltage scanning due to nonlinear characteristics of the perovskite devices.In effect, our physical model needs to be supplemented with the voltage dependence of the conductances.Assuming that the different conductances have approximately the same ideality factor (as in the numerical simulations of Figure 2b), 12,33,46 we observe that Δt ss exhibits a voltage-independent behavior because τ kin commonly remains constant along the voltage sweep (see above).Finally, the integration of the required time delay therefore gives the optimum scan rate to develop steady-state current−voltage measurements in perovskites solar cells: stabilizing all the transient responses with multiple versions, due to their different origins, that constitute the current− voltage curves in halide perovskites, with the best overall compromise between non-time-consuming experiments, in-trinsic device stability, and reliable estimation of the real device performance.
The time constant τ kin of each time transient is plotted in Figure 3 as a function of voltage.We observe, in effect, that the applied voltage has not a considerable effect on the characteristic relaxation times, as remarked by Garci ́a-Belmonte and co-workers. 46Furthermore, we point out that the relaxation time remains at approximately 75 ms even though a transition in the slow dominant mechanism occurs, from capacitive to inductive.This result justifies the identification of relaxation times presented in eq 4 (if the times are physically very different, a jump of τ kin would be observed in Figure 3).Now, the corresponding time delays Δt ss are calculated via the model parameters extracted from the fitting of experimental time transients and are also presented in Figure 3.As we expected, we found an approximately voltageindependent value in the range of 2 s.This optimum time delay correlates to a value of scan rate of 5 mV/s considering a 10-mV of voltage step size from eq 9.
To finalize from the general landscape of the experimental requirements criteria to stabilize the current−voltage curves in perovskite solar cells, we show a batch of curves at different voltage sweep rates in Figure 4. We represent the observed hysteresis traces, transforming the dominant behavior from normal (Figure 4a−c) to inverted (Figure 4d,e) as the scan rate decreases.The current−voltage curve is stabilized when the time transients reach equilibrium along the stepwise scanning at the hold time values Δt ss in decreasing sweep rates determined from transient analysis, as illustrated in Figure 4f.In effect, if one uses larger values for the steady-state time selected without any basis, one will also obtain stabilized current−voltage curves.However, this scenario would accelerate degradation pathways in the perovskite solar cells, derived from a long-term evaluation of devices, 47 e.g., the measurement time of our current−voltage curve with a scan rate of 5 mV/s is 9 min, being 43 min if the sweep speed is 1 mV/s.The reproducibility of our methodology was checked and assured by conducting the experimental measurements in 12 samples with the same device configuration.Furthermore, we also present experimental evidence that confirms the validity of the control strategy of hysteresis presented here in planar p-i-n photovoltaic perovskites in the Supporting Information.
Taking into account the convenient equivalent circuit interpretation of the model, we can safely assume that impedance measurements are an alternative method to extract the dynamic operating parameters, with the aid of the interpretative potential of this technique. 31,48,49Therefore, such measurements were performed, although we will represent the experimental data in the form of admittance responses to better display the correlation of the conductances and time constants with the current−voltage experiments.
Figure 5 shows the evolution of the characteristic admittance patterns of perovskite solar cells, along the different points of the current−voltage curve, by representing the measured points in complex plane plots with the implicit frequency clockwise increasing.The spectral pictures, at low voltages, show two capacitive semicircles that provide information on the recombination kinetics in the bulk and the contacts.−52 For the interpretation of impedance spectroscopy of our experimental results, we develop again our three-dimensional model (eqs 1−3) into the smallperturbation equations; we subsequently take the Laplace transform in terms of the variable s = jω where j 1 = (do not confuse with current density), and we finally calculate the admittance Y(jω) = j ̂/V ̂that gives the result where the "hysteretical relaxation processes" are controlled by the kinetic time constant τ kin (pole of the transfer function, see eq 4).Note that the circumflex accent indicates small perturbation quantities.The admittance in eq 10 becomes that of the equivalent circuit of Figure 2a, with the addition of the series resistance R s .Further details on how to obtain the admittance function and the equivalent impedance plots of Figure 5 can be found in the Supporting Information.Beyond this visual inspection of the spectral features, we carried out a quantitative analysis to identify the value of the electrical parameters that determine the value of the steady-state time (see eqs 8 and 9).In effect, the procedure for the analysis of the admittance spectra is the same as that of impedance plots, requiring a fitting of the experimental results with an equivalent model (here, the electrical circuit of Figure 2a).However, the limiting behaviors of the admittance now are expressed in terms of conductances, and even more importantly, the characteristic frequencies are obtained as dY j /dω = 0 (ω HF and ω LF ), exhibiting different values for the relaxation times of the impedance response.Graphically, time constants correspond to the inverse of the characteristic frequencies found from the maximum magnitudes of the imaginary part of the admittance in the resulting arcs (τ v = 1/ω HF and τ kin = 1/ω LF ), mathematically correlated with those of the current transient responses. 53From mathematical operations, we obtain again constant values of Δt ss ≈ 2 s and s c ≈ 5 mV/s to reach equilibrium in all the transient responses along the stepwise current−voltage measurements.
−56 One of the most important drawbacks of this type of measurement is the time consumption of the order of ten minutes (equally distributed to extract the kinetic parameters and apply the MPP tracking), 57 leading to a continuous demand of optimized protocols that circumvent the recurrent problem of time-variance in metal halide perovskites due to reversible and irreversible degradation during the performance assessment. 58,59The essential step here is to achieve a significant reduction in the data acquisition time with reliable precision in the cell efficiency measurement in comparison to the classical protocol commonly used by experimentalists, maximizing in addition the information about the electrical behavior of photovoltaic devices and retaining the advantages of classical current−voltage measurements.
To provide a guideline to experimentalists, we explain the measurement protocol proposed that, in summary, is as  outlined in Figure 6: (i) It starts with an introductory fast current−voltage curve to explore the level of hysteresis.Thereafter, (ii) one has to carry out small perturbation choroamperometric experiments at low and high voltages to determine if there exists a transformation in the slow transient dynamics.An alternative approach to small-amplitude timedomain measurements consists of implementing the equivalent technique of impedance spectroscopy to obtain the value of the corresponding conductances and time constants that determine the value of Δt ss and s c .Note that impedance measurements have been already correlated with the hysteresis level in current−voltage curves. 12,41,58Then, (iii) one must extract the optimal scan rate from the fitting of experimental time-domain (admittance) responses to eqs 5−7 (eq 10) in order to find hysteresis-free current−voltage curves with the aid of eqs 8 and 9. Similar values were obtained from both responses (eq 4).Once the stabilization of the transient responses has been guaranteed, (iv) the current−voltage measurement under the voltage sweep velocity determined in (iii) can be carried out in both forward and reverse directions (for verification purposes) to finally determine the power conversion efficiency from light to electricity of the perovskite solar cell.In effect, there is a significant reduction in the measurement time when using this optimal protocol, decreasing by at least half�a few seconds in (i), in the order of microseconds in (ii), and a few hundred seconds (always less than 300 s) in (iv) applying a single sweep direction�to obtain the electrical responses.
Standard testing protocols are of particular interest for assessing the solar energy conversion of photovoltaic perovskites in an industrialization phase due to the intense debate that remains around current−voltage hysteresis (normal and inverted) at the research level since the first reports of these electronic devices.In this context, we propose here a comprehensive exploration of stepwise time transient effects rooted in hysteresis mechanisms.By an exhaustive analysis of the discrete nature of the classical current−voltage curve in perovskite solar cells, we unveiled the intricate nature of slow transient responses (capacitive or inductive) that introduce memory traces under non-steady-state conditions.Through an analytical methodology that monitors the real-time current during current−voltage measurements, we delineated a suitable time delay to stabilize all the transient responses throughout the nonlinear curve without reducing the operational lifetime of perovskite devices due to unnecessary long-time experiments due to the use of ad hoc slow scan rates.Our results predict experimentally the dynamic behavior of memory modes in perovskite semiconductors, enabling rapid and massive characterization of high-performance cells by automated methods.
Theoretical analysis of the model in time-and frequency-domain; impedance spectra fitting results; application of the hysteresis control strategy in inverted perovskite devices; experimental section (PDF) ■

Figure 1 .
Figure 1.(a) Representative current−voltage curve of our quadruple-cation perovskite solar cell obtained using a step-size of 10 mV and a voltage scan rate of 50 mV/s.Halide perovskite exhibits normal and inverted hysteresis derived by the voltagedependent transient dynamics of the photocurrent responses, consisting of decays at low voltages, (b) 0 V and (c) 1 V, and negative spike components with abrupt final rises in current for the high-voltage domain, (c) 1.2 V and (c) 1.3 V.

Figure 3 .
Figure 3.Time constants and optimum time delay extracted from the transient responses throughout the voltage range.In both characteristic times, the values remain independent of voltage at around 75 ms and 2 s for τ kin and Δt ss , respectively.Note that Δt ss is far from the classical value of 3τ kin .

Figure 4 .
Figure 4. Catalogue of measured current−voltage responses for our perovskite solar cell as a function of the scan rate: (a) 500, (b) 250, (c) 100, (d) 25, (e) 10, and (f) 5 mV/s.Hysteresis mechanisms are, in effect, eliminated around the value of time delay and scan rate estimated here.

Figure 5 .
Figure 5. Admittance spectra patterns measured at different bias voltages to help model and interpret mathematically equivalent steady-state photocurrent transient responses.Y r and Y j are the real and imaginary parts of the admittance function, respectively.

Figure 6 .
Figure 6.Fast experimental methodology to determine the stationary performance of perovskite solar cells with hysteresis effects.The experimental data correspond to an inverted or p-i-n perovskite device, described in detail in the Supporting Information.