Metal halide perovskites have brought about a disruptive shift in the field of third-generation photovoltaics. Their potential as remarkably efficient solar cell absorbers was first demonstrated in the beginning of the 2010s. However, right from their inception, persistent challenges have impeded the smooth adoption of this technology in the industry. These challenges encompass issues such as the lack of reproducibility in fabrication, limited mid- and long-term stability, and concerns over toxicity. Despite achieving record efficiencies that have outperformed even well-established technologies, such as polycrystalline silicon, these hurdles have hindered the seamless transition of this technology into industrial applications. In this Perspective, we discuss which of these challenges are rooted in the unique dual nature of metal halide perovskites, which simultaneously function as electronic and ionic semiconductors. This duality results in the intermingling of processes occurring at vastly different timescales, still complicating both their comprehensive investigation and the development of robust and dependable devices. Our discussion here undertakes a critical analysis of the field, addressing the current status of knowledge for devices based on halide perovskites in view of electronic and ionic conduction, the underlying models, and the challenges encountered when these devices are optoelectronically characterized. We place a distinct emphasis on the positive contributions that this area of research has not only made to the advancement of photovoltaics but also to the broader progress of solid-state physics and photoelectrochemistry.

In the year 2009, the group of Miyasaka reported1 the first photovoltaic application of a novel material2 capable of efficiently capturing light within the visible range of the solar spectrum. The material had the crystalline structure of “classical” perovskites3,4 with stoichiometry ABX3, but including organic cations in the A lattice position and halides in the X positions. Metal halide perovskites (MHPs) used for photovoltaics and other optoelectronic applications typically have lead in the B position. Their crystalline structure is very characteristic involving corner-sharing BX6 octahedra with A-cations occupying the spaces in between the octahedra. The crystallization of metal halide perovskites occurs in cubic, tetragonal, or orthorhombic phases depending on the prevailing temperature conditions.5 The range of structural stability is contingent upon the relative sizes of the constituent ions, as determined by the Goldschmidt tolerance factor.6 

While, in Miyasaka’s research, MHPs functioned as a “sensitizer” for a high-bandgap semiconductor, similar to the approach of a conventional dye-sensitized solar cell, numerous research groups discovered that these materials possessed excellent properties for making solar cells. These perovskites demonstrated exceptional capabilities not only as efficient light harvesters but also as highly efficient electronic conductors.7–10 Their remarkable features combine outstandingly small non-radiative recombination losses,11 long electron and hole diffusion lengths,9,10 low exciton binding energies,5 inherent tolerance toward defects,12 and adjustable bandgaps.13 All these positive features have led to one of the most impressive learning curves of all photovoltaic technologies with reported record efficiencies now reaching 26.1%, similar to the reported record for silicon solar cells.

Undoubtedly, efficiency stands as the most compelling advantage of perovskite solar cells in the pursuit of being a part of a sustainable future driven by solar energy. Furthermore, their optical adaptability and adjustability, coupled with a relatively straightforward fabrication process with low cost, serve as added benefits. However, lurking beneath these advantages are challenges that have plagued this technology since its inception. Perovskites, being ionic materials, exhibit a precarious sensitivity to water and humidity.14,15 In addition, the highest-performing perovskites are hybrid organic–inorganic compounds. This impairs the overall stability, as the organic cations are inherently more susceptible to decomposition and commonly relatively volatile.16–18 Numerous degradation pathways involve volatile molecules,19 contributing to the undesirable breakdown of the material. Many of these issues can be attributed to the ionic nature of perovskites, which causes them to behave somewhat like “liquid, although viscous, electrolytes.”

In the year 2017, Rolston and co-workers20 made a stimulating correlation between cohesive energy and degradation rate in materials used for solar cells. With cohesive energies below 1 J/m2, perovskites occupy the lower extremity of an idealized continuum that connects covalent materials, such as silicon and chalcopyrites—exhibiting the utmost stability in the domain of solar cells—down to organic semiconductors and MHPs. Disrupting the lattice structure of solid silicon or polythiophene involves the breaking of strong covalent bonds, such as C–H, C–C, or Si–Si, which possess bond energies on the order of 300–400 kJ/mol. In contrast, the crystalline networks of metal halide perovskites are upheld by relatively “soft” electrostatic interactions.21 Frost et al.22 as early as 2014 attributed the instability of hybrid MHPs to their relatively low Madelung (electrostatic) energies, in comparison with their oxide counterparts. The electrostatic contribution is one of the most relevant factors that explain the very low formation free energies of hybrid MHPs, of the order of a few tenths of kJ/mol.23 

Given the intrinsically low stability and rigidity of MHPs, it is not surprising that one of their most distinctive features is their tendency to show phase segregation and ion migration phenomena. In the year 2015, Eames and co-workers24 claimed that hybrid lead iodide perovskites were “mixed ionic–electronic” conductors and reported an activation energy barrier for iodide ions of 0.6 eV (58 kJ/mol). Evidence of ion migration in MHPs and their solar cell devices is omnipresent in the literature.25–31 Ions play a significant role in the solar cell performance, not only in enhancing the conductivity of the solar cell’s active layer but also in their ability to accumulate at interfaces,31 creating an electrical double or space-charge layer—similar to the classical occurrence of the electrical double layer formation at the electrode/electrolyte interface.32 This accumulation effectively alters the internal electric field in which electrons and holes move,33,34 a critical factor governing the operation of any functional solar cell.

In this Perspective, we examine the present body of evidence concerning the interaction between the electronic attributes of MHPs when employed as the active layer in solar cells, and their ionic characteristics. We give particular emphasis to the experimental analysis and computational simulation of ionic transport, involving not only the material’s intrinsic properties but also their role in the operation of the solar cell device. Crucial inquiries that need addressing include the following: is it feasible to differentiate between ionic and electronic attributes? Considering the blended ionic–electronic nature of MHPs, can we successfully deduce parameters that determine the photovoltaic efficiency, in a similar way to other more conventional solar cells? How can we establish a connection between the ionic aspects, stability, and performance?

In Fig. 1, the typical impedance response of a CH3NH3PbI3 based perovskite solar cell is shown. This particular result was obtained35 by keeping the solar cell at short-circuit upon illumination and applying a small, frequency-modulated voltage perturbation on top of the DC component (V = 0 at short-circuit). Electrochemical impedance spectroscopy (EIS)36–38 is a very popular small-perturbation electrical characterization technique in the emerging photovoltaics community. In basic terms, it consists in measuring the impedance of the device, defined as a voltage-to-current transfer function, for a set of frequencies modulating a small perturbation in the voltage. The frequency-dependent impedance is best described as a complex number,
Z(ω)=VωJ(ω)=Zexpiφ=Zω+iZ(ω),
(1)
where both the modulus, |Z|, and the phase shift, φ, are functions of the angular frequency, ω. A system solely characterized by resistance would not generate any signal within the φ-spectrum. However, the presence of capacitive and inductive components within the electrical attributes of the system under investigation leads to phase shifts between the perturbing voltage and the resulting current. These deviations could manifest as either negative (indicative of capacitance) or positive (indicative of inductance) peaks in the frequency spectrum.
FIG. 1.

Typical impedance spectrum of a perovskite solar cell: phase shift vs frequency of the impedance of a TiO2/CH3NH3PbI3/spiro-OMeTAD perovskite solar cell at short circuit. Features and processes commonly associated with either the slow or fast signals are indicated. Adapted from Ref. 35.

FIG. 1.

Typical impedance spectrum of a perovskite solar cell: phase shift vs frequency of the impedance of a TiO2/CH3NH3PbI3/spiro-OMeTAD perovskite solar cell at short circuit. Features and processes commonly associated with either the slow or fast signals are indicated. Adapted from Ref. 35.

Close modal

The impedance response of MHP solar cells is widely recognized for their intricacy, prompting a huge number of research papers over the past decade.35,38–45 The majority of these studies have concentrated on the fundamental interpretation of the impedance, including the explanation of exotic features such as a negative capacitance and inductive loops. Despite the disparity of observed behaviors, the general impedance response of MHP solar cells can be universally represented by Fig. 1, with two negative signals, one at low frequencies (around 0.1–10 Hz) and another at much higher frequencies, on the order of 0.1–1 MHz.

Figure 1 is a good embodiment of the dual nature of perovskites for photovoltaics. The slow processes that generate the low frequency signal are mainly attributed to ion migration. One of its more noticeable properties is its temperature sensitivity. It is a general result for MHPs that increasing the temperature shifts the low frequency peak toward higher frequencies. Specifically, the shift is larger than one order of magnitude for a temperature variation of 30–40 K. The temperature dependence of the low frequency signal can intuitively be understood as the increasing temperature increases the mobility of the ions and consequently facilitates their migration. As a matter of fact, the low frequency signal fits very well in an Arrhenius-type analysis.27,34,35,46,47 This enables the derivation of an “activation energy.” Typical values obtained from impedance analysis of MHPs fall within the range of 0.35–0.45 eV (35–45 kJ/mol). The characteristic times of the slow component of the impedance can be nicely correlated to the scan rates at which the current–voltage curve shows hysteresis.48,49 Other slow processes, such as degradation50,51 and chemical reactivity,52 also show up in the low frequency region of the spectrum, and the intensity of the capacitive signal can even be used to monitor the extent of physical or chemical degradation.

In contrast, the high frequency signal is generally not very temperature sensitive. This is the region of the spectrum where electronic processes, such as recombination and displacement currents, are expected to manifest. In fact, a solar cell with no ionic or degradation components would only give high-frequency signals (as matter of fact, in many cases, there is just this one signal). The high frequency peak is strongly voltage and light intensity dependent, since these are the parameters that determine the electronic carrier density, and processes such as electron–hole recombination strongly depend on the buildup of electronic charge in the active layer of the device. It is important to note that electron/hole transport cannot be directly observed in a small-perturbation experiment within the 0.1–106 Hz range because it is too fast.

A simple model for the impedance of a conventional solar cell can be derived from the continuity equation for the electron density n in the active layer,
qnt=Jnx+qGqU,
(2)
where q is the elementary charge, t and x are the time and space coordinates, Jn is the current density, and G and U are the optical generation rate and the recombination rate, respectively (in units of electrons volume−1 time−1). The action of a time-dependent frequency modulated small perturbation Vt=VDC+δVeiωt induces a modulation of the electron density (and the current and the recombination rate), which, in the linear approximation, can be expressed as nt=nDC+nVδVeiωt.37,53 Inserting this modulation of n (and Jn and U) into Eq. (2) and subsequent cancellation of all the DC components yields
qiωnV=xJnVqUV,
(3)
which, by integration in the x-direction for the full thickness of the active layer, leads to
A=JnVdx=qUVdx+qiωnVdx,
(4)
where A is the admittance (reciprocal of the impedance). Equation (4) shows that the impedance response of the solar cell can be expressed as the sum of a capacitive contribution, expressed as the voltage derivative of the electron density (a chemical capacitance54), and a recombination resistance, defined by the voltage derivative of the recombination rate. As a matter of fact, the derived impedance can be related to that of a simple parallel RC equivalent circuit,
ZRC=1A=1R+iωC1,
(5)
where R is the recombination resistance and C is the capacitance. In Fig. 2, the simulated impedance of this circuit (with the addition of a series resistance component) is presented. This simple RC model gives a single perfect arc in the Nyquist plot (imaginary part of the impedance vs real part) and a single peak in the frequency plot centered at frequency (RC)−1.
FIG. 2.

Top: Simple equivalent circuit for a solar cell and simulated impedance spectrum for a particular case. Bottom: simulations of a 2 RC equivalent circuit for two combinations of the low frequency resistances and capacitances (data shown correspond to Rser = 12.38 Ω, R1 = 22 Ω, and C1 = 7.6 × 10−7 F). Adapted from Ref. 55.

FIG. 2.

Top: Simple equivalent circuit for a solar cell and simulated impedance spectrum for a particular case. Bottom: simulations of a 2 RC equivalent circuit for two combinations of the low frequency resistances and capacitances (data shown correspond to Rser = 12.38 Ω, R1 = 22 Ω, and C1 = 7.6 × 10−7 F). Adapted from Ref. 55.

Close modal

In MHP solar cells, it is often required to incorporate additional RC elements, thereby introducing additional time constants, in order to accurately simulate the entire spectrum. This expansion increases the range of potential scenarios and, as a result, the number of possible interpretations. In this respect, Todinova et al.55 demonstrated that distinct topologically equivalent circuits, characterized by an equal count of RC elements, could effectively replicate the same experimental spectrum, provided that their associated time constants differ significantly by several orders of magnitude. An illustrative scenario arises when comparing a circuit featuring two or three RC elements in series with that of a nested model, often referred to as a “Matryoshka” model, containing an equivalent number of RC elements (as depicted in Fig. 2). Furthermore, in certain instances, these devices exhibit the presence of a “negative” capacitance or inductive elements, leading to the emergence of loops at positive phase shifts. All of these factors collectively contribute to the substantial level of ambiguity that can arise when attempting to interpret the impedance spectrum accurately, particularly in cases where the use of numerous circuit components in the form of resistances, capacitances, and inductors appears to be indispensable.

All these intricacies have to do with the mentioned presence of the mobile ions in the MHP layer. The tricky part is that the ionic component affects both the capacitive and the recombination terms of the spectrum. The former by adding new ways of storing charge (for instance, by accumulation of ions at interfaces45) or by modifying the internal electric field that drives electronic transport and recombination.33,56,57 In fact, both types of effects can be easily entangled and misinterpreted in perovskites, as pointed out by Jacobs et al.48 An additional characteristic fact for MHP solar cells is that the chemical capacitance cannot be directly observed in the high frequency region of the spectrum (except at very high voltages) because the geometric capacitance of the device is dominant.39,58 This complicates the extraction of electron lifetimes from the corresponding high frequency RC time constant.59 

In the following, a more detailed analysis of this complex interplay and the role of moving ions will be reviewed.

Atomistic modeling of ion migration in MHPs was pioneered by Eames et al.24 and Azpiroz et al.25 in 2015. These authors used density functional theory (DFT) to calculate the energy barriers that perovskite ions have to surmount to migrate in the lattice along specific paths. They found that migration of halide ions, either via a vacancy-assisted mechanism or via interstitials, was the most plausible mechanism to explain the most compelling evidence of ion redistribution in CH3NH3PbI3 solar cells, that is, hysteresis in the current–voltage curve. Assuming a Boltzmann-like hopping mechanism, a diffusion coefficient of 10−12 cm2 s−1 for iodide ions was estimated. In contrast to Eames, who reported an activation energy of 0.6 eV for iodide migration, close to values derived from small-perturbation measurements, Azpiroz et al. predicted an activation energy for vacancy-assisted and interstitial halide migration of 0.1–0.2 eV (10–20 kJ/mol) for CH3NH3PbI3, below the experimental values derived from small-perturbation analysis but close to experimental data obtained from NMR measurements.60 It is important to note at this point that NMR probes the ionic motion at a very short range, for a cation flipping back and forth between two sites, whereas the small-perturbation experiment targets a longer time range for ionic migration, in the order of tenths of seconds, as indicated above. In a further improvement, Meggiolaro and co-workers61 incorporated the formation energy of the defects in their analysis, which also contributes to the observed thermal activation of the low frequency signal associated with ion migration. The group of Islam and co-workers also used DFT to investigate the impact on the activation energy of ion substitution in the crystalline structure.62,63 For instance, Ferdani et al.63 found, and muon spectroscopy experiments confirmed, that replacing the methyl ammonium cation by a larger cation increases significantly the activation energy for halide vacancy migration.

The primary limitation of DFT lies in the computational complexity due to its quantum mechanical nature. As a result, it restricts the analysis to a limited number of atoms and short simulation times. This proves to be disadvantageous when examining slow processes, such as ion migration. In 2016 and 2017, Mattoni and co-workers published several studies64–66 where classical molecular dynamics (CMD) is used to study ion migration in MHPs. As CMD is based on force fields that model in an effective way the interatomic interactions, it is numerically simpler, hence making it possible to run simulations with thousands of atoms over relatively long times. In this way, the ion diffusion coefficient can be directly extracted from the main square displacement of halide ions resulting from the ion dynamics, instead of being estimated from an activation energy. Mattoni’s group obtained diffusion coefficients for iodide migration on the order of 10−6−10−7 cm2 s−1 for CH3NH3PbI3. In contrast to the DFT calculations, the CMD studies provided a lower activation energy for the vacancy-assisted mechanism (0.10 eV) than for interstitials (0.24 eV).

The downside of force fields for CMD is that their parameters need to be specified or optimized in order to provide a realistic description of the intra- and intermolecular interactions of the system. To address this, Mattoni’s team utilized DFT calculations to fine-tune the parameters, ensuring that the classical force field accurately replicated the lattice potential energy as computed by DFT. This approach yielded a relatively accurate prediction of the transition temperatures for the orthorhombic–tetragonal–cubic phase changes of CH3NH3PbI3 based on the resulting force field. Balestra et al.67 and Seijas-Bellido et al.68 extended the strategy of combining DFT and CMD to develop transferable force fields for pure and mixed perovskites of variable compositions CsPb(BrxI1-x)3 and MAxFA1−xPb(BryI1−y)3 (∀x, y ∈ [0, 1]). Instead of using a “manual” fitting algorithm, they devised a genetic algorithm that found the optimized values of the force field parameters that best reproduced the DFT energy for a set of molecular and lattice distortions [Fig. 3(a)]. Figure 3(b) shows a comparison of the predicted and the DFT energy data of reference, which shows the efficacy of the genetic algorithm to model the potential energy of the lattice. The resulting force field also reproduced accurately the experimental XRD diffraction data, the lattice parameters, and the thermal expansion coefficients of both the pure and the mixed MHPs.

FIG. 3.

(a) Molecular and lattice distortions used to optimize force field parameters in mixed MHPs. (b) Shifted lattice energies of the compositions of an inorganic MHP using a genetically optimized interatomic potential vs DFT reference data. (c) Mean square displacement of halide ions in a CMD simulation at 600 K. (d) Arrhenius plot of the ion diffusion coefficient as derived from the calculated mean square displacements for different perovskite compositions. Adapted from Refs. 67 and 68.

FIG. 3.

(a) Molecular and lattice distortions used to optimize force field parameters in mixed MHPs. (b) Shifted lattice energies of the compositions of an inorganic MHP using a genetically optimized interatomic potential vs DFT reference data. (c) Mean square displacement of halide ions in a CMD simulation at 600 K. (d) Arrhenius plot of the ion diffusion coefficient as derived from the calculated mean square displacements for different perovskite compositions. Adapted from Refs. 67 and 68.

Close modal
The CMD simulation with an optimized and transferable force field paves the way for predicting from first principles the ion diffusion coefficients that govern the low frequency behavior of the solar cell and the current–voltage hysteresis. Consistent with the discoveries made by Mattoni’s team, Seijas-Bellido et al.68 and Barboni and De Souza69 also observed that the migration of halide ions only becomes apparent during the simulation timeframe when a specific quantity of neutral vacancies is deliberately introduced into the lattice. This does not imply that migration through interstitial pathways is unfeasible, but rather that it remains beyond reach within the limited simulation duration of a few tenths of nanoseconds. The vacancy-assisted ion diffusion coefficient does, in fact depend on the concentration of the number of vacancies present in the lattice via the following formula:69,
Di=ageomdi2ΓiNvNi,
(6)
where ageom is a geometric factor that depends on the crystalline structure, di is the jumping distance for ions, ΓI is the jumping rate, and Nv and Ni are the vacancy and the halide concentrations, respectively. Equation (6) shows, and the numerical CMD simulation confirms, that ion migration is faster the larger the concentration of vacancies. Indeed, it is found68 that the simulated diffusion coefficient scales quite well with the Nv/Ni ratio, that is, the jumping rate remains relatively constant over a large range of vacancy concentrations. This observation reaffirms that the rate of particle motion remains relatively constant regardless of the number of vacancies present. Consequently, it is possible to extrapolate the diffusion coefficient value to any vacancy concentration, provided that the result for a specific concentration is known. Using an abnormally high concentration of vacancies can lead to a diffusion coefficient several orders of magnitude above the values estimated from the small-perturbation experiments.65,69
An additional complication is that simulations at room temperature produce very slow migration dynamics. Therefore, it is inconvenient to extract the diffusion coefficient from the mean square displacement. Fortunately, the jumping rate fits the Arrhenius’ law very well,
Γi=Γ0expEakBT,
(7)
where kB is Boltzmann’s constant and T is the absolute temperature. Equation (7) makes it possible to obtain by extrapolation the value of the ion diffusion coefficient at room temperature from a set of simulations at larger temperatures. As Eq. (6) can also be used to estimate the diffusion coefficient at any vacancy concentration, both equations considered together allow for the estimation of the ion diffusion coefficient of MHP lattices at room temperature and for a realistic concentration of vacancies. Following this procedure, and assuming a concentration of vacancies corresponding to 4.25% of unoccupied sites70 (Nv/Ni = 0.0425), Seijas-Bellido et al.68 predicted ion diffusion coefficients for the halide ions of the order of Di = 10−11 cm2 s−1 for a pseudo-cubic lattice of CH3NH3PbI3 at room temperature. Larger activation energies upon substitution of iodide by bromide in both methylammonium and formamidinium perovskites were found (from 0.20 to 0.34 eV ≈ 20–35 kJ/mol), in agreement with experimental studies that show a deactivation of ion migration when amounts of bromide are added to the sample.27 They also run long simulations with more than 8600 atoms for MA0.29FA0.71PbBr0.1I2.9 and other mixed compositions. The diffusion coefficient lies in the 1–6 10−11 cm2 s−1 range for various amounts of the Br/I and MA/FA ratios, with activation energies between 0.25 and 0.35 eV.
The values obtained from the CMD simulations can be considered as first-principles estimates of the ion diffusion coefficients acting in MHP solar cells under operation, as they were obtained with force fields whose parameters have been optimized using DFT potential energies. In this respect, it is interesting to see whether the predicted results are consistent with the low frequency signals observed in the small perturbation experiments. The diffusion coefficient can be related to a migration distance d and a characteristic time ti via
Did2ti.
(8)

It can be naïvely assumed that the migration distance that gives rise to the low frequency signals affects the full thickness of the active layer of the device (around 500 nm). However, experimental31 evidence and drift-diffusion modeling56,71 (see below) suggest that ions actually move by charging/discharging the electrical double layers formed in the vicinity of the interfaces. For double layer thicknesses of 10–100 nm and Di = 10−11 cm2 s−1, Eq. (8) gives a characteristic time of 1–10 s, corresponding to 0.1–10 Hz, which agrees with the signal that is commonly observed in the low frequency part of the small-perturbation experiments, as discussed above. This also aligns with the timeframes within which hysteresis is generally observed in typical metal halide perovskite (MHP) solar cells, usually involving scan rates ranging from 0.01 to 0.1 V/s. The consistency between the atomistic description and the observed time signals strongly suggests that vacancy-assisted ion migration is behind the low frequency signals and the related slow phenomena, such as hysteresis, widely observed in this type of device.

Although genetically optimized force fields have provided interesting insights into ion migration phenomena occurring in MHPs, and constituted a useful and accurate tool to describe the atomic/ionic motion of these materials under operating conditions, the availability of a realistic and yet usable (in terms of numerical efficiency) atomistic model remains an open line of active research. In this respect, the use of force fields optimized via machine-learning (ML) tools and protocols have been a turning point in materials science.72–75 DFT calculations for a set of atomic configurations, similar to those indicated in Fig. 3(a), permits to train ML models, such as deep neural networks. These models may predict the potential energy of the system along the dynamic trajectory of the atoms (ions) in the lattice. In a similar fashion, Jinnouchi and co-workers74 used DFT molecular dynamics and Bayesian inference “on the fly” to produce ML-trained force fields [see Fig. 4(a)] and accelerate the calculation of the atomic motion in a perovskite. In Fig. 4(b), it is observed how the predicted error resembles the real error in the first steps of the on-the-fly simulation. Once the force field is accurate enough, the CMD simulation can proceed faster, without the need to readjust with new first-principles calculations. Using this strategy, they were capable of predicting the phase transition temperatures of CH3NH3PbI3 and CsPbI3. Pols et al.75 extended the method to the prediction of ion diffusion coefficient in inorganic perovskites CsPbI3 and CsPbBr3 [Figs. 4(c) and 4(d)].

FIG. 4.

(a) Algorithm for an on-the-fly machine learning (ML) force field generation. (b) The error of the ML force field in the first period of an on-the-fly simulation. (c) Force components of a vacancy trained MF force field for CsPbI3 vs DFT calculations at 750 K. (d) Arrhenius plot of the ion diffusion coefficient as derived from on the fly ML-based CMD for interstitials and vacancy-assisted halide migration in CsPbI3 and CsPbBr3. Adapted from Refs. 73–75.

FIG. 4.

(a) Algorithm for an on-the-fly machine learning (ML) force field generation. (b) The error of the ML force field in the first period of an on-the-fly simulation. (c) Force components of a vacancy trained MF force field for CsPbI3 vs DFT calculations at 750 K. (d) Arrhenius plot of the ion diffusion coefficient as derived from on the fly ML-based CMD for interstitials and vacancy-assisted halide migration in CsPbI3 and CsPbBr3. Adapted from Refs. 73–75.

Close modal

The performance of a solar cell under operational conditions hinges on two critical factors: the efficient generation of electronic carriers, achieved through effective light harvesting and exciton dissociation upon illumination, and the minimization of electrical and energy losses within the active layer and its interfaces, e.g., through recombination. The former primarily determines the short-circuit photocurrent (Jsc) of the device, while the latter governs both the fill factor and the open-circuit photovoltage (Voc). As it stands, the current state of the art76 demonstrates that perovskite solar cells surpass many other photovoltaic technologies in both aspects (see Fig. 5). However, while they have approached the detailed-balance limit concerning the short-circuit photocurrent (Jsc), there is still room for enhancement when it comes to the open-circuit photovoltage (Voc). Therefore, while MHP solar cells exhibit one of the lowest levels of non-radiative photovoltage loss among all solar cell technologies,11 the critical factors for advancing them toward record efficiencies lie in effectively managing carrier transport and recombination, in particular in a scenario of mid- and long-term degradation.

FIG. 5.

Highest efficiency performance parameters for single junction photovoltaic solar cells as a function of the bandgap of the active material. Taken from https://emerging-pv.org/data/.76 

FIG. 5.

Highest efficiency performance parameters for single junction photovoltaic solar cells as a function of the bandgap of the active material. Taken from https://emerging-pv.org/data/.76 

Close modal

Drift-diffusion (DD) numerical simulation for device modeling33,56,77–79 proves to be an invaluable tool for investigating carrier dynamics in solar cells from a fundamental standpoint. In simplified terms, DD modeling involves the numerical solution of the continuity equation for both electrons and holes [as represented in Eq. (2) for electrons] within the active layer, alongside Poisson’s equation to account for electrostatic interactions. The added complexity with mixed ionic–electronic perovskites lies in the necessity to incorporate mobile ions in the game. While ions do not directly contribute to photocurrent, they indirectly influence the photovoltage by regulating the internal electric field in which electronic carriers move.

The simultaneous and self-consistent numerical solution of three continuity equations (one each for electrons, holes, and ions) along with Poisson’s equation presents a formidable mathematical challenge. The encouraging aspect, as highlighted multiple times in this Perspective, is that these entities operate on timescales that differ by several orders of magnitude. While this could potentially be a significant numerical hindrance when dealing with a set of coupled differential equations with very distinct characteristic times, the fact that ions move at a considerably slower pace than electrons and holes permits the introduction of effective approximations. This concept is akin to the well-established Born–Oppenheimer approximation employed to separate electrons and nuclei in classical quantum mechanics. Building upon this philosophy, Richardson and co-workers33 devised an “asymptotic” approach that facilitates the solution of the continuity equation for a CH3NH3PbI3 active layer and provided one of the first fundamental descriptions of hysteresis in MHP solar cells based on numerical simulations.

DD numerical simulation has now become a popular tool to study MHP solar cells with several software packages now available in the market.71,80–84 In all cases, the numerical studies prove the formation of ionic double layers in the vicinity of the perovskite/contact interfaces in agreement with experimental observations.31 These layers can affect the collection of electronic carriers at the contacts, by modifying either the driving force for transport or the rates of recombination in the bulk or across interfaces.34,85 In Fig. 6, the electric potential and the ion density predicted by the DD simulation within the active layer of a state-of-the-art MHP-based device are shown. We can see that cations (ideally corresponding to iodide vacancies as discussed above) accumulate at the interfaces, screening the electric field in the bulk of the perovskite. In accordance with the classical Gouy–Chapman theory, the thickness of the electrical double layer (or space-charge layer) decreases and the screening effect increases with an increase in the overall ion densities. The current literature reports values of this parameter ranging from 1016–1017 cm−3, as inferred by numerical modeling from current and voltage transient measurements86,87 to 1019–1020 cm−3, as derived from the enthalpy of formation of iodide vacancies, obtained in DFT calculations.70 Irrespective of the specific concentration of mobile ions, the existence of electrical double layers introduces a unique element not found in other solar cell types: the shielding of the electric field and the alteration of the driving force responsible for pushing electrons and holes toward the contacts. In this line, Lammar and co-workers34 found that adding deliberately crystalline defects by altering the stoichiometry of the perovskite leads to lower values of Jsc and fill factors. DD modeling demonstrates that this is due to the screening of the electric field, which kills off the driving force for electronic collection for the more defective perovskite.

FIG. 6.

Electrical potential (left) and cationic density (right) at short circuit as derived from DD simulations for a PTAA/Cs0.1FA0.9PbI2.865Br0.135/C60 perovskite solar cell (see Ref. 34 for details) at several values of the total cationic ion density: 1017 (red), 1018 (green), 1019 (blue), and 1020 (violet) cm−3. Vertical dashed lines mark the perovskite boundaries.

FIG. 6.

Electrical potential (left) and cationic density (right) at short circuit as derived from DD simulations for a PTAA/Cs0.1FA0.9PbI2.865Br0.135/C60 perovskite solar cell (see Ref. 34 for details) at several values of the total cationic ion density: 1017 (red), 1018 (green), 1019 (blue), and 1020 (violet) cm−3. Vertical dashed lines mark the perovskite boundaries.

Close modal

One of the most interesting developments of DD modeling is the possibility of simulating the small-perturbation response by introducing a voltage (or optical) modulation of the type Vt=VDC+δVeiωt, as discussed above. Moia et al.83 used this to simulate the impedance response of mixed MHP solar cells and introduced an electrical model that describes the interfaces as transistors where the ion distribution determines the gate potential, hence leading to possible capacitive or inductive behavior at low frequencies, as observed in many experimental papers. Courtier et al.84 extended the DD simulation to consideration of the transport or contact layers (spiro-OMeTAD and TiO2 for hole and electron conduction, respectively) and showed how the doping level and the dielectric constant of the contact materials would affect the ionic distribution in the perovskite layer and, hence, the slow features of the device. Neukom et al.78 provided a full numerical description of a MHP device combining steady-state, time-domain, and frequency-domain features. Their simulations reproduced very nicely the voltage vs light intensity data, the capacitance as a function of frequency, and the IMPS spectrum, among other properties. Riquelme et al.56 proposed a general interpretation of the impedance spectrum in MHP devices based on the analysis of the DD simulation. They showed that a simple model consisting of electrons, holes, and positive ions moving with realistic diffusion coefficients and recombination via bulk routes only was sufficient to reproduce the two characteristic signals of the spectrum as well as their dependence on DC voltage and illumination intensity.

Figure 7 shows a sketch of the evolution of the ion distributions and the electric potential profile in the active layer of a MHP device over a frequency cycle of the voltage modulation in an impedance experiment. It is interesting to compare the behavior of the two magnitudes at high and low frequencies. In the high frequency regime, the double layer profiles do not change over the frequency cycle. This is because ions move so slowly that they cannot follow the rapid external voltage variation. This forces the internal electric field to change from positive to negative values along the frequency cycle. This change of direction of the field drives electrons and holes in favor or against collection, producing a capacitance-recombination signal (ideally modeled using a single RC element) at high frequencies. In contrast, at low frequencies, ions do have time to move and adapt to the external voltage perturbation in such a way that the field is always screened, resulting in a flat potential profile at all times in the cycle. The charging/discharging of the electrical double layer by adding/removing ions over the frequency cycle (to adapt to the external perturbation) is what leads to the low frequency signal. This explains why this signal is more sensitive to temperature and to the ionic properties of the perovskite. The DD simulated spectrum resembles that of a double RC equivalent circuit with two well-separated time constants. It also predicts negative exponential behaviors for both the high and low resistances with respect to voltage, as always observed in the experiments. The charging/discharging of the electrical double layer also explains the same slope but opposite sign commonly found in the impedance measurements for the low frequency resistance and capacitance over a DC voltage sweep.

FIG. 7.

Electric potential (top) and ion density (bottom) profiles over a frequency period as derived from a DD numerical simulation of a TiO2/MAPbI3/Spiro-OMeTAD solar cell. Left: high frequencies. Right: low frequencies. Adapted from Ref. 56.

FIG. 7.

Electric potential (top) and ion density (bottom) profiles over a frequency period as derived from a DD numerical simulation of a TiO2/MAPbI3/Spiro-OMeTAD solar cell. Left: high frequencies. Right: low frequencies. Adapted from Ref. 56.

Close modal

The simulations from Refs. 33, 56, and 84 assumed a relatively large concentration of mobile ions, that is, iodide vacancies, in the perovskite, of the order of 1019 cm−3 (4.25% of vacancies or unoccupied sites70), meaning that the screening effect discussed above is very significant. However, values of 2 and 3 orders of magnitude lower than that have also been reported.86,87 As discussed above, the acting concentration of mobile defects determines the range of the electrical double layers at the interfaces. This, in turn, leads to alterations in the low-frequency characteristics of the device and affects the efficiency of carrier collection. As noted by Meggiolaro et al.,61 distinguishing the impact of ion mobility from ion concentration can prove challenging, particularly in experiments sensitive to temperature variations. Within this framework, it is interesting to explore how these two parameters influence the measurements. Figure 8 contains a representative summary of the different effects that can be observed, as predicted by DD numerical device simulation:

  1. The ion mobility/diffusion coefficient does not affect the steady-state current–voltage curve because ions always equilibrate to the same distribution when given enough time. However, it does affect the kinetics, i.e., the degree of hysteresis for a given scan rate and the low frequency signals. Hysteresis is observed when the voltage scan rate is of the same order of magnitude as the timescale in which ions move84,88 (specifically, within a 10–100 nm distance, corresponding to the charging/discharging of the electrical double layer). If ions move too fast or two slowly, no hysteresis is observed, either because ions get “frozen” in the initial distribution or because they have enough time to equilibrate to the steady-state result. Hence, ion mobility does not affect steady-state performance (but can affect the stability of the device—that is another story).

  2. Faster ion mobility shifts the low frequency signal toward higher frequencies, but it does not modify its intensity. Neither does it affect the position of the high frequency signal because how fast ions move does not affect the electronic capacitive/inductive features of the device. This effect is also observed in the imaginary (Nyquist) plot.34 

  3. Ion concentration or density can modify the steady-state current voltage curve if the ionic screening critically modifies the electrical driving force for electron/hole collection. This happens if electron–hole recombination is fast enough. Hence, mobile ions/crystalline defects can be deleterious for the performance of devices (along with complicating the stability). In this scenario, large ion concentrations can nullify a favorable electric field, hence reducing the photocurrent and the fill factor. On the contrary, if recombination is sufficiently slow, modifications of the internal electrical field by ion redistribution do not affect the electron collection (although they might affect the open circuit voltage). In any case, the ionic double layers always control the concentration of electrons and holes at the interfaces, affecting the overall recombination rate.49 

  4. Larger concentrations of ions can also shift the low frequency signals toward higher values, but, in contrast to the diffusion coefficient case, the intensity of the signals is also altered (this is also observed in the Nyquist plot34). The reason behind this behavior is that the number of mobile ions affects both the resistive and the capacitive properties of the device, especially at low frequencies.49 The concurrent observation of the Nyquist and the frequency plot is, therefore, a good way of discriminating between changes in the mobility of the ions or, alternatively, in their concentration.

FIG. 8.

DD simulations for PTAA/Cs0.1FA0.9PbI2.865Br0.135/C60 perovskite solar cell (see Ref. 34 for details). (a) and (d) Steady-state current–voltage curves. (b) and (e) Imaginary part of the impedance at open-circuit (1 V) at four values of the cationic ion density: 1016 (black), 1017 (red), 1018 (green), and 1019 (blue) cm−3. (c) and (f) Imaginary part of the impedance at open-circuit (1 V) at four values of the cationic ion mobility: 10−10 (red), 5 × 10−10 (green), 10−9 (blue), and 5 × 10−9 (violet) cm2 s−1 V−1. Two bulk recombination regimes are shown: (a)–(c) pseudo-electron lifetime = 5 × 10−9 s; (d)–(f) pseudo-electron lifetime = 5 × 10−8 s (10 times slower recombination).

FIG. 8.

DD simulations for PTAA/Cs0.1FA0.9PbI2.865Br0.135/C60 perovskite solar cell (see Ref. 34 for details). (a) and (d) Steady-state current–voltage curves. (b) and (e) Imaginary part of the impedance at open-circuit (1 V) at four values of the cationic ion density: 1016 (black), 1017 (red), 1018 (green), and 1019 (blue) cm−3. (c) and (f) Imaginary part of the impedance at open-circuit (1 V) at four values of the cationic ion mobility: 10−10 (red), 5 × 10−10 (green), 10−9 (blue), and 5 × 10−9 (violet) cm2 s−1 V−1. Two bulk recombination regimes are shown: (a)–(c) pseudo-electron lifetime = 5 × 10−9 s; (d)–(f) pseudo-electron lifetime = 5 × 10−8 s (10 times slower recombination).

Close modal

For simplicity, the simulations shown in Figs. 7 and 8 were carried out with bulk recombination only. However, in most efficient state-of-the-art perovskite solar cells, interfacial recombination is the dominant mechanism.89 This means that the presence of the ionic double layers affect very particularly the separation of charges at the interface. However, even in this case, the presence of ions is expected to have a deleterious effect, due to the screening of the field and the suppression of the drift-mediated driving force for electron collection.34,85

The presence of mobile ions also brings about a somehow often overlooked result. As first noted by Courtier,57 the traditional diode formalism for solar cells has to be revisited in the case of MHPs. The occurrence of the electrical double layers and the alteration of the internal field mean that the potential drop governing electron–hole recombination in the solar cell has to be redefined. This implies that the classical or apparent ideality factor, nap, typically obtained from a Voc vs light intensity measurement, or from the slope of the recombination resistance vs voltage,90,91 procedure commonly used to establish the dominant recombination mechanism in solar cells,92 is as a matter of fact affected by the ionic screening. Based on this finding, Courtier proposed the use of a so-called ectypal factor, which includes the effect of ion migration.

As a further development of this concept, Bennet et al.53 devised an analytical model to describe the impedance response of a MHP solar cell, including ion migration. This analytical model starts from a linearization of the continuity equations for a small voltage perturbation [in the spirit of the derivation of Eq. (5)]. Bennet’s analytical theory is mainly based on three approximations: (1) ionic charge is much larger than electronic charge, (2) the electron density is given by the Boltzmann approximation, and (3) it is possible to separate the voltage drop produced by the ionic configuration from the drop caused by the external modulation of the field, which depends on the frequency. Within this formalism, a new quantity naturally arises, called the electronic ideality factor,
nel=qRHFJrec(VDC)kBT,
(9)
where RHF is the high frequency component of the resistance (obtainable from a fit of the high frequency signal to a RC element) and Jrec is the recombination current at the DC voltage at which the measurement is performed. For the low-frequency feature of the spectrum, the following expressions are obtained:
RLF=kBT[napVDCnel]qJrec(VDC),
(10)
CLF=qnapVDCJrec(VDC)kBTG+nel[napVDCnel]dQDCdVDC,
(11)
where G+ is the ionic conductance and nap is the apparent ideality factor, obtainable from Voc vs ln(light intensity) measurements. Equation (11) explicitly shows that the low-frequency capacitance arises from the modulation of the recombination current by the variation of the accumulated ionic charge upon voltage variations (dQDC/dVDC). As the recombination current decreases exponentially with VDC, Eqs. (10) and (11) describe well the behavior of the low frequency resistance and capacitance that we have discussed above.

The electronic analogue of the ideality factor holds an advantage in that it remains unaffected by ion migration, thus closely resembling the recombination mechanism for electrons and holes. Consequently, a value of nel = 1 would signify either bulk recombination at the band-to-band level (radiative recombination) or surface recombination, while nel = 2 would indicate bulk trap-limited recombination, and so on, similar to the conventional ideality factor’s common application. This removal of ionic “interference” is easily comprehensible when considering that Eq. (9) depends on the high-frequency component of the impedance only, a regime in which the ion distribution remains stationary, as discussed previously. Figure 9 illustrates the practicality of the electronic ideality factor concept. The apparent ideality factor does not necessarily match the electronic one, especially at relatively large ion concentrations, and only the latter reflects the recombination mechanism. Extracting nel from the experimental impedance spectra allows determining the dominant recombination path without ionic interference.

FIG. 9.

(a) Apparent [obtained from Voc vs ln(intensity) measurement] and electronic [obtained from Eq. (9)] ideality factors as obtained from DD simulation for several recombination pathways. Adapted from Ref. 53. (b) Electronic ideality factors computed via Eq. (9) from impedance measurements at open circuit for varying perovskite compositions, showing the change of recombination mechanism from dominant interfacial to dominant bulk. Adapted from Ref. 34.

FIG. 9.

(a) Apparent [obtained from Voc vs ln(intensity) measurement] and electronic [obtained from Eq. (9)] ideality factors as obtained from DD simulation for several recombination pathways. Adapted from Ref. 53. (b) Electronic ideality factors computed via Eq. (9) from impedance measurements at open circuit for varying perovskite compositions, showing the change of recombination mechanism from dominant interfacial to dominant bulk. Adapted from Ref. 34.

Close modal

The emergence of metal halide perovskites (MHPs) has ushered in a new era in modern photovoltaics. Their distinctive optoelectronic properties and the ease of their fabrication, coupled with the remarkable efficiency records achieved over the past decade, have fueled the prospects of a swift commercialization—whether in the form of monolithic solar cells or as tandem systems alongside silicon. Moreover, their minimal non-radiative recombination losses and good light harvesting efficiency have expanded the field of potential applications to cover lighting, lasers, sensing, and more. However, amid this optimistic landscape, one cannot ignore the significant hurdle that looms on the path to successful industrial implementation: the crucial issue of mid- and long-term stability.

In this Perspective, we have highlighted metal halide perovskites (MHPs) as exceptionally unique materials, primarily because of their hybrid nature, which includes simultaneous electronic and ionic electrical conduction. This duality presents two distinct challenges.

On the one hand, this duality significantly complicates the interpretation and analysis of techniques commonly employed to characterize and diagnose solar cells, such as impedance spectroscopy, in particular, and frequency and time domain techniques, in general. The simultaneous “motion” of electronic charge carriers (electrons and holes) and mobile ions, commonly associated with crystalline defects within the material, such as halide vacancies or interstitials, even though occurring on very different timescales, becomes intricately intertwined. Ion features interfere with the extraction of crucial efficiency-determining parameters, such as the recombination lifetime and the ideality factor. The presence of mobile ionic defects brings about alterations in the internal electric field, which serves as the driving force for collecting electronic charge carriers. This modification of the internal electric field occurs relatively slowly, resulting in the generation of signals in the low-frequency part of the spectrum and causing hysteresis when the voltage sweep rates align with the characteristic time of ionic motion. Although many factors are involved and no general rule can be stated, there are indications that the presence of mobile ions is deleterious for the charge carrier collection, especially under conditions of strong recombination. The existence of the mobile ions also complicates the atomistic description of perovskites. The widely used DFT method is not adequate to describe ionic motion due to computational costs, which limit the time span of the ab initio molecular dynamics and the number of ions considered in the simulations. Classical molecular dynamics is much more convenient in this respect, but realistic force fields are needed to describe the dynamics of the lattice and the time evolution of the mobile defects. In this dilemma, force field parameter optimization plays a central role, with modern techniques, such as genetic algorithms and machine learning tools, offering exciting possibilities for first-principles description of MHP materials and similar systems where mixed electronic–ionic conduction is characteristic.

The second aspect embodied by the ion–electron duality relates to the crucial issue of stability. One of the primary factors, among others, contributing to the perception of metal halide perovskites (MHPs) as delicate and unstable materials is the relative ease with which charged crystalline defects can migrate, accumulate, and redistribute when exposed to illumination or subjected to electrical bias over distances spanning from 10 to 100 nm and timescales extending to up to a few seconds. As has been emphasized throughout this Perspective, the activation energies for ion migration are in the range of a few tenths of kilojoules per mole, remarkably close to the formation enthalpies of the most commonly used photovoltaic perovskites. While the ionic rearrangement and the low defect formation energies may contribute to the renowned defect tolerance of MHPs and their relatively low rates of non-radiative recombination, the buildup of ionic charges at interfaces can trigger processes leading to phase segregation and irreversible degradation. In addition, there are indications that aging processes can be linked to migration of mobile ions.30,93 However, whether the higher concentration of mobile ions is a cause for or an effect of the degradation remains hitherto unknown.

Regardless of the drawbacks and complications associated with the dual nature of MHPs, the truth is that they are fascinating materials. Dealing with the simultaneous influence of electronic and ionic effects, both of which have the potential to impact photovoltaic performance, necessitates the application of principles from a diverse array of disciplines, including quantum mechanics, solid state physics, chemical thermodynamics, computational chemistry, and electrochemistry. The convergence of these disparate fields, coupled with the inherent fragility of the material and devices, may have contributed to the extensive volume of publications and reports, thereby complicating the attainment of a broad consensus regarding fundamental concepts and the potential for exploitation. Nonetheless, to borrow a phrase from one of the most renowned “philosophers” of the twentieth century, Johan Cruyff, “every disadvantage has its advantage.” Metal halide perovskites are complex systems, but they offer the possibility of learning novel and exciting phenomena in systems where “quantum” electrons and “classical” ions seem to be engaged in a confusing and enigmatic dance.

We acknowledge the Ministerio de Ciencia e Innovación of Spain, Agencia Estatal de Investigación (AEI), and EU (FEDER) for the support under Grant Nos. PCI2019-111839-2 (SCALEUP), PID2022-140061OB-I00 (DEEPMATSOLAR), TED2021-129758B-C33 (TransEL), and CNS2022-135694 (IMPRESOL). G. Oskam and P. Pistor acknowledge Ministerio de Universidades and Universidad Pablo de Olavide, Beatriz Galindo Program (Nos. BGP 18/00060 and BG 20/00194, respectively). We also acknowledge European Union’s Horizon 2020 research innovation program under Platform-Zero project (Grant Agreement No. 101058459).

The authors have no conflicts to disclose.

Juan A. Anta: Conceptualization (equal); Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – original draft (lead); Writing – review & editing (equal). G. Oskam: Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). Paul Pistor: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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