Dielectric and ferroic properties of metal halide perovskites

Semiconducting halide perovskite materials have attracted intense research interest over the past five years due to the potential for inexpensive solution processing and desirable optoelectronic properties for photovoltaic (PV) and light emission applications.^1–3 Halide perovskites is synthesized with the chemical formula ABX₃, where A is a positively charged cation located in the central cavity created by an extended framework of corner-sharing BX₆ metal-halide octahedra.

Thin-film devices based on the hybrid organic-inorganic perovskite (CH₃NH₃)⁺ PbI₃⁻ (MAPbI₃) were the first to utilize a halide perovskite as the PV absorber layer.^2,4 This material features prominently in the literature due to the rapid increase in power conversion efficiency (PCE), from 3.8% to 23.3%.^5,6 However, MAPbI₃ has been shown to be chemically unstable and contains the toxic element Pb motivating chemical substitution.^7,8 Materials with elemental compositions including A = CH₃(NH₂)₂⁺ (FA), Cs⁺, Rb⁺;^9–12 B = Sn²⁺, and Ge²⁺;^13–15 and X = Br⁻ andv Cl⁻,^16–19 as well as numerous 2-D layered perovskites,^3,20,21 have been widely studied. The best performing PV devices (PCE and lifetime) are based on the mixed-cation, mixed-halide perovskite (Cs,MA,FA)Pb(I, Br)₃ and are currently the most promising route to commercialize the technology.^22–24 

MAPbI₃ has been shown to form in three perovskite structures: a high-temperature cubic Pm$3¯$ m phase stable above 330 K; a tetragonal phase (I4cm or I4/mcm) between 330 K and 160 K; and a final low-temperature orthorhombic phase (Pnma) stable below 160 K.^25,26 Similar phase behavior is observed for other halide perovskites, with the phase stability and transition temperatures being influenced by factors including the radius ratio of the chemical constituents.^27–29 

A number of models have been proposed to explain the origin of the high performance of halide perovskite solar cells, including defect tolerance,³⁰ Rashba splitting,³¹ large polarons,³² and ferroelectric (FE) polarons.³³ Polar nanodomains, observed by piezoforce microscopy (PFM),^34–39 have also been suggested to contribute toward improved charge separation and transport properties.^40,41 However, following the confirmation of ionic transport, current-voltage (J-V) curve hysteresis—typically an indicator of ferroelectricity—has been attributed to the migration of mobile ions rather than spontaneous lattice polarization.^42,43 Photo-enhanced ionic conductivity has also been reported.⁴⁴ The most recent effective-circuit models suggest that coupling between the electronic and ionic charge at interfaces dominates the charge transport and extraction properties of halide perovskite solar devices although a consensus is yet to be formed.^45,46

The dielectric response function describes the long-range response of a material to an external electric field. This can provide information on the underlying physical mechanisms which manifest such diverse behavior. In halide perovskites, differences in the microscopic crystalline properties of samples, experimental or computational methodologies, and nomenclature lead to disparity in dielectric literature which can be confusing to navigate. In this Perspective, we briefly introduce the theory of dielectric polarization in order to critically interpret the polarization mechanisms reported for halide perovskites. Where possible, we highlight publications which employ the “best practice” experimental techniques for investigating the dielectric response. Additionally, we contemplate the ferroic nature of halide perovskites that continues to inspire debate in the field.

A dielectric is defined as a medium which cannot completely screen a static, external, macroscopic electric field from its interior due to physical constraints on charge rearrangement.⁴⁷ The response of a dielectric to an applied electric field is described by the theory of dielectric polarization and is usually framed within classical electrodynamics.^48,49

The dielectric polarization of a solid is often defined as the sum of the induced dipole of a polarizable unit, divided by the volume of the unit.⁵⁶ This approach is well defined for finite ionic systems and forms the theoretical foundation of the Clausius-Mossotti model.⁵⁷ However, it breaks down in the thermodynamic limit (e.g., for extended crystals with delocalized, periodic electronic wavefunctions) as the charge density cannot be unambiguously decomposed into local contributions.^58,59

The modern theory of polarization utilizes the framework of density functional theory (DFT) to compute the response of the electron wavefunctions in terms of a geometric (Berry) phase and not as a charge density.^60,61 Consequently, the mathematical formalism only addresses the dielectric response of electrons and ions. The theoretical formalism describing the orientational and the space-charge dielectric response shall be presented in Sec. III.

The dielectric polarization of a crystal cannot be reliably measured as an intrinsic, bulk property. Generally, one experimentally measures the change in polarization between two polarization states from hysteresis loops that are generated upon switching the direction of polarization currents.⁷⁶ The mathematical formalism introduced above was motivated, in part, by a desire to compare theoretical calculations with such measurements.

The majority of studies investigating polarization phenomena today examine its derivatives, however, such as pyro-/piezo-electric coefficients. In the context of solar cells, measurements of the relative permittivity are frequently reported due to its significance in calculations of physical constants (e.g., absorption coefficient) that impact device performance. The complete device-relevant dielectric response occurs over a large frequency range (Hz–PHz), requiring multiple complementary experimental techniques (on inevitably different devices) to fully characterize.

Materials which feature lower-frequency responses are often labeled “lossy dielectrics” as the associated dielectric mechanisms are not harmonic (resonant) processes but instead totally dispersive (energy attenuating). As such, optical measurements are not an appropriate probe of this behavior. A common interpretation of the relative permittivity is the ratio of the capacitance of a capacitor whose electrodes are separated by vacuum to a capacitor whose electrodes are separated by a dielectric. Impedance spectroscopy, a technique which models the dielectric medium as an effective circuit described by a capacitance, an inductance, and a resistance, is therefore often employed.^90,91 This technique requires many considerations and involves complex analysis, the interface between the dielectric and the electrode contacts being an important factor.^92,93

Optical absorption involves the photoexcitation of electrons from valence to conduction bands, creating either bound electron-hole pairs (excitons) or free carriers. As such, the optical dielectric response is dominated by the optical band gap and other low energy band-edge states. For MAPbI₃, it has been deduced that the valence and conduction bands are composed of hybridized I 5p orbitals and Pb 6p orbitals, respectively.¹⁰¹ The optical band gap corresponds to the VB1 → CB1 transition at the R symmetry point in the first Brillouin zone and has been measured at 1.6 eV (∼400 THz). Further excitonic absorption has been suggested at 2.48 eV (600 THz) and 3.08 eV (745 THz).⁷⁷ 

At room temperature, the potential energy surface for the orientation of the $CH3NH3+$ ion within the lead-iodide octahedra is soft. Consequently, there is computational sensitivity to the choice of electronic structure Hamiltonian and the level of geometry optimization.^102,103 However, DFT calculations applying Eq. (7) within the generalized gradient approximation produce an isotropically averaged value of ϵ_optic = 6.0 for a crystal with MA aligned in the low energy ⟨100⟩ direction.¹⁰⁴ This result was replicated by Ref. 102, which utilized similar methods.

These calculations compare well with the best practice experimental value, ϵ_optic = 5.5, produced from ellipsometry measurements on single crystals.⁷⁷ Leguy et al. also measured the optical response of MAPbI₃ thin films and suggested that the reduction in ϵ_optic, from 5.5 → 4.0, is due to surface effects.⁷⁷ A larger response (6.5) is reported by Hirasawa et al. although we suggest this is due to assumptions taken in data processing and not to an increase in the dielectric response.¹⁰⁵ Furthermore, we posit that Ref. 106 underestimate the electronic response (5.0 ≤ 5.5) due to the classical Lorentz dipole fitting model which is employed. To support this statement, when additional Debye components are included in a later study which employed similar methods, a value closer to the consensus is calculated (5.5).¹⁰⁷ 

The polarizability of a compound, and hence the electronic dielectric response, is inversely proportional to the magnitude of its band gap, as described by second-order perturbation theory.¹⁰⁸ For halide perovskites, a number of general trends related to changes in the band gap can therefore be observed in the published values of ϵ_optic summarized in Table I for different ABX₃ compounds.

The decrease in reported values of ϵ_optic on transition from I (5p) to Br (4p) to Cl (3p) has been understood by considering the higher binding energy of valence electrons and the corresponding increases in optical band gap (⁠$EgI∼1.6$ eV $→EgBr∼2.2$ eV $→EgCl∼2.9$ eV).^109–111 Spin-orbit coupling (SOC) plays an important role in determining the conduction band for compounds which include Pb (Z = 82) in their composition. Exchanging Pb for Sn (Z = 50) leads to a decrease in the optical band gap (⁠$EgPb∼1.6$ eV $→EgSn∼1.2$ eV) and to the expected increase in ϵ_optic.^27,112,113 While the A-site cations do not directly contribute to the band edge states, they do influence the crystal structure and metal-halide bond strength.³¹ The volumetric decrease on exchanging MA for Cs leads to an increase in the optical band gap (⁠$EgMA∼1.6$ eV $→EgCs∼1.7$ eV) and to a decrease in ϵ_optic, again.^10,27,77

A description of the position of ions within the lattice at finite temperature involves an interplay between thermal motion and interatomic restoring forces (e.g., hydrogen bonding and van der Waals forces). It is common to assume that the motion is elastic and non-dissipative—defined as the harmonic approximation. Anharmonic interactions can also occur, however, and have been suggested in MAPbI₃ due to Pb off-centering and octahedral tilting.^131,132 Despite this, the harmonic phonon dispersion has been fully characterized for MAPbI₃ and is found to be dominated by vibrations of the PbI₆ octahedra from 0.5 to 3.2 THz.^117,133 The low energy of these modes can explain why values for ϵ_ion seen in Table I for the halide perovskites are much larger than for tetrahedral semiconductors such as CdTe (ϵ_ion = 2.5).¹³⁴ The strength of the ionic polarization has been suggested to enhance PV performance (e.g., through defect tolerance) and should be an important consideration in future material searches.^135–138 

The averaged BEC tensors for MAPbI₃ have been calculated to be larger than the formal ionic charges (⁠$ZPb*=+4.42$⁠, $ZI*=−1.88ZMA*=+1.22$⁠) such that small ion displacements result in large changes to the polarization.^115,117 Anisotropy in the calculated BEC tensors indicates a preferential direction of vibration for apical and equatorial iodine ions, which has been interpreted as octahedral “breathing.”¹¹⁵ Brivio et al. reported an isotropically averaged value of ϵ_ion = 16.7 for the case when MA is oriented in the low energy ⟨100⟩ direction, which rises to ϵ_ion = 26.4, when MA is oriented in the ⟨111⟩ direction.¹⁰⁴ Whilst it is tempting to average over all possible orientations, as performed in Ref. 111, we report values associated with MA aligned in the ⟨100⟩ direction, when possible.

These calculations compare well with the best practice experimental value of ϵ_ion = 16.5, determined from impedance measurements performed on powder samples.¹²¹ In order to account for the dielectric contribution from interfacial polarization (to be introduced in Sec. III D), the authors introduce additional Maxwell-Wagner terms to the Debye relaxation model with which they perform their spectral fitting.^120,139 The neglect of these effects can explain the larger value for ϵ_ion (23.0) obtained by Poglitsch and Weber in an earlier study.¹²² Sendner et al. reported a value of ϵ_ion = 28.5 (which rises to 31 if ϵ_optic = 5.5 is used) after identifying TO (0.96 THz and 1.9 THz) and LO (1.2 THz and 4.0 THz) phonon modes and applying the Cochran-Cowley expression.¹²⁴ The Cochran-Cowley expression is a generalization of Eq. (13) that accounts for systems with more than two atoms in the unit cell (MAPbI₃ has 48 at room temperature).¹⁴⁰ Whilst this value may improve if a greater number of optical phonon modes are included in the calculation (MAPbI₃ has 141 at room temperature), it exemplifies the limitations of the model for describing complex materials.

The ionic dielectric response is dependent upon the frequency of vibrational modes and the associated ionic charges. Therefore, compositions containing lighter elements may be expected to exhibit a weaker response due to faster vibrations and less polarizable ions. Contemplating the role of the halide, Ref. 122 measured a systematic decrease in ϵ_ion for the sequence MAPb(I)₃ → (Br)₃ → (Cl)₃. The authors attribute this behavior to the blue-shift of vibrations associated with the decrease in mass of the halide, an argument supported by Sendner et al.¹²⁴ Whilst phonon modes greater than 10 THz associated with CH₃NH₃ molecular vibration are influential in phase transitions (as seen in Fig. 2), Bokdam et al. suggested that they have little impact on the ionic dielectric response.¹¹¹ The measurement of similar responses for both MAPbI₃ and CsPbI₃ by Ref. 120 supports this claim. Pb²⁺ is highly polarisable due to its lone pair (6s²) electrons and is therefore expected to dominate the ionic dielectric response.¹³⁷ Lone pair activity with dynamic structural distortions has also been confirmed in Sn²⁺-based halide perovskites.¹⁴¹ 

Unlike the “dynamic” dielectric mechanisms previously introduced, the orientational component is not expected to impact electron transport.¹⁴⁶ Whilst the reasoning for this effect is beyond the scope of this perspective, it can be evidenced by the correspondence between the reported values of charge mobility measured for organic and inorganic halide perovskites.¹⁴⁷ Consequently, the majority of studies contemplating the role A-site dynamics have on PV functionality do so within the context of ferroelectricity (landscape of polar domains) and not dielectrics. However, the orientational component can influence the slower motion of mobile ions and the screening of point defects.^148,149

Application of Onsager theory for a polar liquid estimates the response due to the $CH3NH3+$ dipoles (μ = 2.29 D) to be ϵ_dip = 8.9 at T = 300 K.³² This value corresponds well with a multi-approach measurement on thin films, which reports ϵ_dip ∼ 12.¹¹⁵ We required an effective dipole moment of μ ∼ 0.7 D in order to reproduce a similar value using Eq. (15), however. The reported values of μ = 0.85 D and μ = 0.88 D, derived by fitting spectral data with Eq. (15), are also smaller than theoretical calculations of the static dipole for $CH3NH3+$⁠.^121,122 This disparity can arise from interactions with the environment or inter-molecular correlation that are not properly described by the polar liquid model. In contrast to the values above, Anusca et al. assigned a stronger orientational response (ϵ_dip = 32) from impedance measurements on single crystals.¹⁵⁰ The preceding discussion demonstrates the uncertainty in the field and the need for further investigation and methodological developments.

Beyond the bulk polarization processes previously discussed, understanding of the dielectric response due to the distribution and transport of charged species (molecules, ions, electrons, and holes) is essential to describe and characterize the operational behavior of PV devices. As the name suggests, space-charges are formed by spatial inhomogeneity in the charge distribution which create electrostatic potential gradients. The nature of the space charge formed in a device will depend on the processing history and state of a device (e.g., applied voltage).¹⁵⁴ 

Extended defects—including surfaces, interfaces, and grain boundaries—often act as traps for charged species in semiconductors and can support concentrations of charge carriers and defects well above the bulk equilibrium values. The distribution of charge can enhance the strength of dielectric polarization through the formation of electrical double layers (EDLs)—shown in the inset on the far right of Fig. 1. The EDLs behave as conventional capacitors to induce static fields which screen the interfacial charge and can therefore significantly impact impedance measurements.¹⁵⁵ 

Evidence for space charge polarization instead arrives from observation of Jonscher’s law (ϵ₂ ∝ ω⁻¹) in low-frequency dielectric spectra (ω < 100 kHz).¹⁶² Jonscher’s law can be understood by relating the imaginary dielectric constant to the components of the refractive index, ϵ₂ = 2nk. The real refractive index, n = c/$v$⁠, describes changes in the phase velocity of electromagnetic radiation propagating through a material. The extinction coefficient, k = cα/2ω, describes radiation attenuation due to its proportionality to the Beer-Lambert absorption coefficient, α. Observation of Jonscher’s law in spectroscopic measurements of MAPbI₃ therefore suggests that lossy processes, such as ion migration, dominate at low frequency.^115,123

Presenting an equivalent circuit model, Moia et al. suggested that the accumulation of charged ionic species at the perovskite-electrical contact interfaces modulates the energetic barrier to charge injection and recombination.⁴⁵ It is expected that this effect shall be significantly enhanced under illumination due to an increase in the concentration of free carriers and point defects due to photoexcitation.⁴⁴ As previously mentioned, interfacial charge can impact spectroscopic impedance measurements. Consequently, coupling between electronic and ionic transport has been reported to explain the “photoinduced giant dielectric constant” (⁠$ϵrdark∼103→ϵrillu∼107$⁠) measured at low frequencies under illumination.^46,127

Similar hopping processes may also occur for electron transport. Although electrons and holes exist in the form of diffuse large polarons in the bulk materials,³² localization and hopping through an inhomogeneous potential energy landscape associated with charged point and extended defects is likely as in the case of H and V-centres.¹⁶⁴ Again, further investigation on this topic is necessary.

In analogy to ferromagnets, ferroelectric materials exhibit a spontaneous and reversible polarization below a characteristic Curie temperature. Some of the highest performing ferroelectric materials—as defined by the magnitude of the spontaneous polarization, the Curie temperature, and the coercive electric field—are oxide perovskites.¹⁶⁵ In systems such as BaTiO₃ and PbZr_xTi_1-xO₃, spontaneous polarization arises from a displacement of the A or B species from their ideal (cubic) lattice sites.¹⁶⁶ Credible ferroelectricity has also been reported in certain halide perovskites; below 563 K, CsGeCl₃ adopts a non-centrosymmetric rhombohedral perovskite structure with lattice polarization along the ⟨111⟩ direction and exhibits a non-linear optical response.¹⁶⁷ 

Lossy dielectrics with mobile ions can exhibit apparent ferroelectric signatures. This is exemplified by reports of hysteresis cycles produced from electrical measurements on a banana.¹⁶⁸ We note that stoichiometry gradients (Hebb-Wagner polarization) in mixed ionic-electronic conductors represent a more complicated case.¹⁵⁴ When combined with the molecular dipole, this makes classification of the polar nature of MAPbI₃ difficult. This uncertainty has led to conflicting reports such that the tetragonal phase of MAPbI₃ has been referred to as “superparaelectric, consisting of randomly oriented linear ferroelectric domains” from Monte Carlo simulations,⁴⁰ “a ferroelectric relaxor” from dielectric dispersion measurements,¹⁶⁹ “ferroelectric” on the basis of electrical measurements on single crystals,¹⁷⁰ and “non-ferroelectric” from the analysis of impedance spectroscopy.⁴² Interpreting such disparate results is complicated by differences in the experimental technique and sample quality.

It has been deduced that the CH₃NH₃⁺ dipoles have a fixed, anti-aligned orientation in the low-temperature orthorhombic phase of MAPbI₃ due to strong hydrogen bonding between the amine group and iodine ions.¹⁷¹ The crystal structure is centrosymmetric and anti-polar.¹⁹ Additional degrees of freedom are introduced upon an increase in the temperature, which weakens NH₃-I bonds and that enables rotational motion of the organic cation.¹⁷² Rotational motion induces structural deformations of the inorganic framework due to an asymmetry in the bond strength between iodine and the two functional groups of the CH₃NH₃⁺ molecule. Large polarization currents are induced which dramatically impact the dielectric response, as seen in Fig. 2(a), and which drive the tetragonal-to-orthorhombic phase transition at low temperature.^173,174

In the tetragonal phase, the CH₃NH₃⁺ dipole can occupy several fixed orientations within the lattice, the transition between which may be stimulated by thermal motion, for example. Govinda et al. argued that the 1/T dependency of the dielectric response, observed above 160 K in Fig. 2(a), implies rotational disorder in the ab plane.^120,121 The established dynamic disorder of molecular orientation and inorganic octahedral titling evidences the absence of true ferroelectricity in MAPbI₃.

Following the growing evidence which suggests that polar domains are neither long lived or stable, we suggest that the polar behavior of tetragonal MAPbI₃ exhibits many features common to an electret with a combination of lattice, defect, and surface polarization. An electret is defined as a dielectric that contains a mixture of surface or bulk charge, which may be due to real charges or dipoles, and under the effect of an applied voltage decays slowly over time.¹⁷⁵ Although one can find reports which claim to contradict this statement, the supporting evidence is consistently inconclusive upon close inspection and comparison between reports.

Beyond hybrid organic-inorganic perovskites that contain polar cations, similar behavior is observed in inorganic halide perovskites (such as CsPbBr₃ and CsPbI₃) due to fluctuating rather than permanent electric dipoles.⁵³ 

A ferroelastic material exhibits a spontaneous and reversible strain following a stress-induced phase transition. The formation of domains occurs in order to minimize the total strain within the material, as established for BaTiO₃.^177,178 The most convincing evidence for ferroic behavior in MAPbI₃ thin films comes from observation of domain structures in piezoforce microscopy (PFM) amplitude and transmission electron microscopy (TEM) images, shown in Fig. 3.^151,153 These have been classified as ferroelastic twin domains.

“Ferroelastic fingerprints” were revealed by early PFM measurements on these materials.^37,39 Methodological refinements improved the precision of the PFM technique such that a transformation of the domain structures upon heating and cooling through the cubic-to-tetragonal phase transition can be conclusively observed in Refs. 151 and 179. In analogy to BaTiO₃, this suggests that the cubic phase is prototypic and that the cubic-to-tetragonal phase transition is stress induced.¹⁸⁰ 

Observation of the same structures and behavior in TEM, shown in Figs. 3(d) and 3(f), implies that the domain structures extend into the bulk material.¹⁵³ Moreover, they confirm that the measured PFM signal is not merely due to polarization fields. Rothmann et al. suggested that such structures “eluded observation so far […] due to their very fragile nature under the electron beam” and therefore utilized a low electron dose rate (≈1 eÅ⁻² s⁻¹) during their TEM experiment to mitigate these effects.¹⁵³ Further support for the assignment of MAPbI₃ as ferroelastic arrives from reports of a 0.5% lattice constriction,¹⁸¹ stress induced modifications to domain wall positions,³⁹ and periodic differences in the resonant frequency of MA⁺ dipolar oscillations.^151,182,183

It is clear that whilst there is sufficient evidence to support the assignment of the domains forming in MAPbI₃ as ferroelastic, these alone cannot explain the full range of observed behavior. After all, ferroelastic “domains cannot be probed by PFM,” as Röhm et al. pointed out.³⁸ However, as seen for the case of GaV₄S₈, polar domains in which ferroelastic strain is the primary order parameter can be observed.¹⁸⁴ The ferroic properties of halide perovskites seem characteristic of a ferroelastic electret. A ferroelastic electret can be defined as a dielectric in which the primary order parameter, the spontaneous strain, is coupled to a decaying polarization. The pronounced effect of lattice strain on dipole ordering can be evidenced for MAPbI₃ by the monte-carlo simulation shown in Fig. (4).

Similar twin domains have also been observed in the tetragonal phase of (FA_0.85MA_0.15)Pb(Br_0.85I_0.15)₃ and Cs_0.05(FA_0.85MA_0.15)_0.95Pb(Br_0.85I_0.15)₃ thin film samples.^185,186 Notably, the standard annealing procedure at ca. 100 °C means that all MAPbI₃ thin-films will be subject to this phase transition. However, it has been reported that alloying MAPbI₃ with FA⁺ based halide perovskites reduces the temperature of the cubic-tetragonal phase transition.¹⁸⁷ Therefore, we do not expect that these mixed cation compounds will exhibit the same strain-induced deformation at room temperature.

The nature of a ferroelastic state may be directly probed by the measurement of elastic hysteresis cycles.^188,189 Such measurements do not feature in the literature of halide perovskites, however, as the technique is practically challenging. The majority of experimental evidence instead arrives from the PFM data maps previously discussed. PFM is a complex technique itself as measurements are sensitive to the effect of electrostatic, ionic, and topological artifacts, which makes unambiguous interpretation difficult.¹⁹⁰ Consequently, characterization of the ferroelastic state and its coupling to a polar response has been complicated by conflicting results.

For example, a number of vertical PFM (v-PFM) measurements show lamellar domain contrast between high- and low-response piezoelectric regions—as seen in Figs. 3(a) and 3(c)—which suggest the existence of multiple response mechanisms.^37,39,191 In support of this, Huang et al. reported that electrostatic interactions or ionic activity are responsible for the low piezo-response measured in alternate domains in v-PFM data maps.^151,192,193 The authors spatially correlate these low response regions with domains in lateral PFM measurements which show no piezo-response and subsequently argue that domain boundaries exist between a polar and a non-polar space group.¹⁵¹ 

Reports of a consistently non-zero lateral PFM signal and of variation in the piezoresponse being restricted to domain boundaries—as shown in Fig. 5—seem to contradict this behavior.^34,38,179 Vorpahl et al. suggested instead that inhomogeneity in the PFM amplitude is due to depolarization fields. The authors further argued that their lateral PFM measurement, observed in Fig. 5(c), provides evidence that the polarization lies parallel to the surface and that PFM phase contrast, observed in Fig. 5(d), emerges due to a ∼180° offset in the orientation of electrostatic dipoles.¹⁷⁹ 

Polar domain formation is sensitive to the crystal grain size, morphology, and quality.¹⁹⁴ Strain can lower the energetic barrier to ionic bond dissociation such that the density of point defects may be significantly enhanced within and between domains.^195,196 This effect can explain twin domain contrast observed in $CH3NH3+$ chemical composition data maps.¹⁸³ We suggest that the variation in observed behavior can be linked to differences in initial strain distribution (e.g., due to stoichiometry or substrate effects) and the subsequent response of the crystal.

Hybrid halide perovskites—and organic-inorganic crystals in general—are examples of complex dielectric materials that feature a range of polarization mechanisms. These include the perturbation of the electron clouds around atomic centers, the displacement and rotation of ions and molecules, as well as the transport of charged species (ions and electrons). Based on a survey of the literature for CH₃NH₃PbI₃, we recommend values of ϵ_optic = 5.5 for high-frequency processes and ϵ_ionic = 16.5 for the ionic contribution, which yields a bulk static dielectric response of ϵ_r = 22.0.

A rigorous description of even a homogeneous bulk material is challenging and requires a combination of theoretical (statistical mechanical) and experimental techniques. In his 1949 monograph Theory of Dielectrics, Fröhlich noted “That this application is far from trivial is shown by some of the controversies in the literature.” This is certainly true for the halide perovskites. However, from an assessment of the current literature, we conclude that the intrinsic bulk dielectric response is dominated by the displacement of ions as described by the phonon modes of the crystal.

The ferroic properties of MAPbI₃ and related materials have attracted significant attention over the past five years. It is difficult to separate lattice polarization from charge transport and surface effects. We highlight the growing evidence supporting the assignment of room-temperature domain structures as ferroelastic, arising from the cubic-to-tetragonal phase transition that occurs during the standard annealing process of thin-films. Such domains can be associated with trapped charged defects and charge carriers. We conclude that halide perovskites can be classified as ferroelastic electrets, which raises many exciting possibilities for engineering their polarization states and lifetimes. A significant amount of work remains in physical characterization of these materials and the development of quantitative models to describe the full range of physical processes at play in operating photovoltaic devices.

We thank P. R. F. Barnes, A. P. Horsfield, L. Herz, S. Stranks, and R. W. Whatmore for useful discussion on polarization, piezoresponse, and perovskites. This research has been funded by the EPSRC (Grant No. EP/K016288/1). A.W. is supported by a Royal Society University Research Fellowship. We are grateful to the UK Materials and Molecular Modelling Hub for computational resources, which is partially funded by EPSRC (No. EP/P020194/1). This research was also supported by the Creative Materials Discovery Program through the National Research Foundation of Korea (NRF) funded by Ministry of Science and ICT (2018M3D1A1058536).