Measuring and understanding the nanomechanical properties of halide perovskites and their correlation to structure—A perspective

Halide perovskites, HaP, and especially Pb-based ones with general formula ABX₃, with A = monovalent cation, B = divalent cation, and X = mono-anion halide(s), exhibit a plethora of remarkable properties. Of these, their photovoltaic properties are the most widely studied ones, due to the proven potential these materials hold for significant technological impact. In the context of these studies, particular emphasis has been placed on environmental factors, which can alter, especially degrade, material functionality and device performance. Exposure to humidity, light, and oxygen ranks prominently among these factors.

In addition to photo-response, this material class is characterized by interesting physical properties, of which mechanical properties enjoy special attention. This is due to the important role of mechanical properties for potential applications in flexible devices. Furthermore, from a fundamental angle, the mechanical response is intimately related to dynamic processes and strain-related effects on optoelectronic performance.¹ 

In this perspective, we provide an overview of the special mechanical properties of HaPs and the critical considerations for their measurement. We confine our discussion to insights revealed by fundamental nanoscale investigations. Following early works from our lab² and others,³ a number of studies have been made on trends of elastic modulus and hardness in these materials—both experimental and theoretical ones. In addition, a few more exotic works have investigated how the mechanical behavior is influenced by anisotropy^4–6 and time-dependent phenomena.^7,8 Quite some effort was and remains focused on tangentially related ferroelastic/ferroelectric^9–11 properties and the nuances involved in their measurement.

Whereas the mechanics of the HaPs in the framework of device functionality and stability has been recently comprehensively reviewed,¹² the scope of this perspective is more limited and fundamental, and particular attention is given to the effect of moisture on the mechanical properties of HaPs. The complex, but fascinating interaction of water with these materials is of great interest in its own right, especially for the insights it gives into the behavior of these materials at the atomistic level; its study serves as a platform for applying and interpreting results from nanomechanical measurements. Of necessity, the discussion will delve into some of the details that must be considered for proper nanomechanical measurements, as well as the unique properties of these materials.

With the growing interest and R&D activity in HaPs and their properties, progress in nanomechanical testing of these materials has enabled fundamental understanding of, and insights into the materials. Three main methods have been used to determine nanomechanical properties of the perovskite single crystals, namely, spectral (phonon) analysis, instrumented nanoindentation (INI), and atomic force microscopy (AFM). One condition for the use of these methods is that they need to be applicable to the small and irregularly shaped crystals, which are often the result of preliminary attempts to obtain crystals of new material types.

Instrumented nanoindentation is exploited for its relative ease of use and straightforward interpretation. Because the indenter motion is constrained to the direction perpendicular to the surface, and force can be directly controlled, the data are readily handled by classical contact mechanics relations.

Atomic force microscopy is remarkable for its versatility, being amenable to practically any environment, allowing extreme spatial resolution, and mapping of topography together with mechanical and electrical properties. Nonetheless, the simple and straightforward interpretations quickly break down when trying to account for the nonideal nature of the tip-surface interaction. Since the indenter tip is attached to a cantilevered beam, lateral forces can play a part in addition to those normal to the surface plane.

Spectral (phonon) analysis has been done by neutron scattering,⁵ Brillouin light scattering,⁵ and laser ultrasonics.⁴ Briefly, the phonon relations can be used to determine the sound velocity, which is directly related to the elastic constants.¹³ The number of such constants is significantly reduced for single crystals. Although these measurements cannot be made at sub-micrometer-scale, they have the advantage of returning the entire elastic tensor (providing Young’s modulus, shear and bulk modulus, as well as the Poisson ratio).

Thus, each of these techniques has its advantages and disadvantages. Specifically, AFM is amenable to very thin films and small, irregularly shaped samples and provides an image (and modulus determination) of the sample surface, allowing precise placement of the measurement. INI measures on a somewhat larger scale and in some cases provides imaging, but not of the quality or at the scale of AFM. The significant advantage of INI over AFM is that the indentation is better controlled with respect to force and direction. Both of these provide load versus displacement curves in one direction at a time. Poisson’s ratio is not directly obtainable. Spectral measurements require larger samples and are not amenable to thin films or small, irregular samples but can return the full set of elastic constants. Spectral measurements do not give hardness, as the AFM and INI do.

In any event, the results are subject to some uncertainty arising from the assumptions made in the analysis, as shown in our recent work on mild humidity exposure effects on HaPs. Here, we note that indentation hardness is defined by maximum load applied, divided by the residual imprint area. In Fig. 1(a), we show results for a series of different HaPs. The differences are shown between INI hardness, where the area is determined using a predetermined area function of the indenter and the measured deformation depth, and traditional hardness (TH), measured by AFM, where the imprint area is directly measured from images of the indented region. INI hardness is measured under maximal load while the material is under plastic and elastic deformation [Fig. 1(b)]. This estimation does not include the pile-up contribution to the projected contact area, consisting of the raised area around the imprint circumference, as in Fig. 1(c). The TH quantifies the residual imprint, which excludes elastic deformation while allowing inclusion of the pile-up contribution. Typically, Oliver and Pharr analysis is applied to the INI curves, which include a term to correct for relaxation along the depth direction, but this analysis presumes negligible radial relaxation, an isotropic material and ignores pile-up.¹⁴ These simplifications result in significant discrepancies between INI hardness and TH [Fig. 1(a)]. In particular, for crystals with lower elastic modulus, the INI hardness is lower than the TH, due to the contribution from the elastic component to the calculated area in the former. For crystals with higher elastic modulus, the contribution of pile-up leads to overestimated INI hardness compared to the TH.¹⁵ Nonetheless, if the ratio between sample deformation under load, h_max, and final depth after elastic relaxation, h_f, is considered [Fig. 1(d)], the INI and AFM observations have been shown to be compatible.¹⁶ 

The plethora of tools available for mechanical, structural, and chemical analyses provides an enormous boost to attempt to understand physical and chemical processes at the nanoscale. Contrasting and comparing different types of measurements by AFM studies allows drawing relations between different properties by “correlating” the properties they yield. AFM measurements can record the mechanical properties correlatively and simultaneously with morphology, but morphology cannot always be unequivocally related to a specific chemical or crystallographic condition. The ability to monitor and correlate chemical and structural details with the mechanical response is a key factor in proper interpretation of the data. For instance, FAPbI₃ [FA⁺/FA = formamidinium CH(NH₂)₂] can persist in the metastable cubic α phase in an epitaxial form, which is stabilized due to strain, but converts, upon removing the strain, to the more stable octahedral, non-perovskite δ phase that has both lower E and lower H.^12,17 High resolution x-ray diffraction can identify such transformations. Another example is the decomposition of MAPbI₃ (MA⁺/MA = methylamine CH₃NH₃) after prolonged exposure to water vapor, which leads to changes in the morphology as well as nanomechanical properties. XRD analysis showed that the decomposition led to PbI₂ formation.^18,19 The MAPbI₃-water interaction was also studied carefully, comparing both ex situ powder XRD studies and in situ rapid GI-XRD studies. These were further correlated with optical UV-visible spectral signatures for MAPbI₃ and the decomposition product PbI₂.²⁰ Moreover, morphological features could be assigned to MAPbI₃ and PbI₂ crystallites. Finally, these rapid measurements found a transient hydrated intermediate, which was proposed to comprise the precursor to the degradation process. Direct proof for the existence of an intermediate hydrated phase of MAPbBr₃ came from gas-chromatography, coupled to mass spectrometry and XRD.²¹ 

Young’s modulus is an intrinsic material property, given by the ratio of stress to strain. As noted above, extracting this quantity from indentation experiments typically involves a computation, assuming that the material is isotropic, which in general is not the case with HaPs (even for the time-averaged structure determined by normal diffraction). In fact, very large anisotropy has been identified in MAPbI₃ (Ref. 6) and in CsPbBr₃.⁴ This has been related to a ferroelastic instability leading to a low shear modulus.⁵ Anisotropy in several HaPs has been discussed in a recent review, including the phenomenon of negative Poisson’s ratio.²² The same review also summarizes the related, theoretically predicted negative linear compressibility, which remains to be experimentally verified. Beyond this anomalous anisotropy, these perovskites are on the very low end of E and H values for both typical optoelectronic materials and for materials with a perovskite structure.

Another rare property identified in both inorganic²³ and in a hybrid inorganic-organic halide perovskite (MAPbI₃)²⁴ is ultra-low thermal conductivity. This is unusual as single crystals usually show good thermal conductivity due to the well-ordered lattice structure. For MAPbI₃, this property was linked to phonon interaction with the slow rotations of the MA cation inside the inorganic lattice cell. For the inorganic perovskites CsPbI₃, CsPbBr₃, and CsSnI₃ it is notable that the thermal conductivities are 1–2 orders of magnitude lower than for inorganic oxide perovskites. Explanations for this behavior have invoked the phenomenon of dynamic disorder (i.e., time-dependent changes in the lattice, around well-defined average positions). The influence on thermal behavior has been ascribed to phonon scattering either by “rattling” of the Cs cation inside the cage,²³ or else strong anharmonicity in the B–X bond.²⁵ The dynamic disorder also linked to the remarkable optoelectronic properties of the HaPs is associated with an additional structural factor, tilting of the BX₆ octahedra. Here, the role of the cation is significant, due to the presence of H-bonding in the hybrid organic-inorganic HaPs between the organic cation hydrogen and the halide.¹ Stronger H-bonding is associated with greater tilt, which in turn is associated with higher stiffness.²⁶ While it is not clear which of these drives the other, both stronger H-bonding and octahedral tilt should influence the mechanical properties by softening the inorganic framework bonds. Furthermore, when comparing different A cations, the trade-off between strength of H-bonding (greater for MA than for FA, per available hydrogen) plays against the number of H-bonding sites (for FA twice that for MA).¹⁶ It is important to note that the “softening” effect of FA relative to MA has also been attributed to its larger size, which stretches the B–X bond, thus reducing E.

The positive relationship between hardness and modulus is nearly ubiquitous in nature (see Fig. 2), as well as for many engineered materials and holds for E and H values spanning many orders of magnitude.^27,28 However, for halide perovskites, this elastic modulus-hardness correlation is reversed. While this inverse relation seems counter-intuitive, in fact, hardness and modulus are not identical: modulus follows bond stiffness, whereas hardness is an engineering property that has contributions from several different material features, including elasticity, but also others. Yield stress is one of those that has a key correlation with hardness in some materials, but dependencies exist on other features such as dislocation density. The fact that E and H are anti-correlated in HaPs deserves more fundamental study.

One governing factor in this trend is metal-halide bond strength, which increases the framework stiffness; for example, in MAPbX₃ (MA = methylamine, X = Cl, Br, or I) the modulus values vary with increasing Pb–X bond strength: E_Cl > E_Br > E_I, following the trend of electronegativity of the halide.^2,3 Interestingly, the hardness trend for MAPbX₃ crystals varies with decreasing Pb–X bond strength [see Fig. 1(a)]. It has been suggested that this is due to the reduction of structural symmetry (Cl > Br > I), which generates more dislocations under the applied indentation stress.^3,12 However, below we discuss also the importance of structural dynamics. The A cation has a secondary influence on the Pb–X bond length relative to the inorganic framework. For example, replacement of the MA cation by the larger FA one increases the Pb–X bond length by about 1% and weakens the rigidity of the inorganic framework along all indentation directions.²⁹ 

Dynamic disorder, which has been invoked to explain what appear to be unique electronic property combinations, also influences HaP mechanical properties as it leads to larger than usual (for inorganic compounds) ionic displacements and enhanced anharmonicity.³⁰ This has led to speculation about the importance of dynamic disorder in regulating trends in E and H. Although the association of dynamic disorder with bond softening (lower elastic modulus) is well established,^26,31,32 at this point there is no consensus that the former causes the latter. With respect to how dynamic disorder influences H, the discussion centers around two phenomena—defect tolerance and self-healing.³³ We relate to these two separately.

Self-healing refers both to recovery of charge transport properties after photodamage and to repair of physical cracks in MAPbI₃ films³⁴ and crystals.³⁵ Recently, rapid healing of mechanical damage at single crystal surfaces of CsPbBr₃ has been associated with humidity-induced fluidity of the surface ions, enhancing the diffusion rate by an order of magnitude.³⁶ Interestingly, healing of photodamage follows trends of hardness, which can be seen by comparing our TH measurements [Fig. 1(a)]¹⁶ with results of a study on photodamage of Pb bromide perovskites.³⁷ As noted also in Fig. 3, comparing APbBr₃ for A = Cs, MA, and FA, bulk photodamage recovery was slowest for the Cs cation, followed by MA, and fastest for FA, following the trend of rising TH. This correlation, and other works, suggests the key role of dynamic disorder in the self-healing phenomenon.³⁵ 

Correlations between E, H, and defect tolerance exist in materials such as elastomers that are known to be defect tolerant due to the entanglement of the long polymer chains. Such was recently shown for a synthetic assembly combining a cross-linked skeleton with soft rubber-like matrix.³⁸ The cross-linked skeleton provides a framework with sacrificial bonds which dissipate energy,³⁹ and in combination with the rubbery matrix impedes propagation of cracks. In the HaPs, defect tolerance has been commonly used to explain the outstanding performance of halide perovskites in optoelectronic applications.^40–43 This premise has come under scrutiny lately and in careful surveys of the experimental observations and computational procedures^33,44 it appears that a sweeping general statement about defect tolerance of Halide Perovskites is questionable. Thus, defect-assisted nonradiative recombination can be invoked to explain device properties.

Several studies addressed the influence of water on mechanics of HaPs, in general, and hybrid organic-inorganic HaPs, in particular. Changes in mechanical behavior have been related to formation of the monohydrate, or dihydrate—the latter at longer exposures, and their ultimate decomposition leading to growth of PbX₂ on the surface. These materials are no longer perovskites. Therefore, an in-depth study of water interaction should entail monitoring the concomitant chemical and structural changes taking place. Such multifaceted studies are discussed below.

Interestingly, the earliest hints of influence of humidity on mechanical properties come from atomistic simulations. Using ab initio calculations on water interaction with methyl ammonium lead iodide, Mosconi et al.⁴⁵ found that water initially binds to the Pb ions, increasing the Pb–I bond length, but subsequently enters the cavity between the first and second layers of PbI₆ octahedra, while forming hydrogen bonds with the MA. This process does not deform the Pb–I framework. If the crystal is terminated by MA-I, stronger bonds are formed with water than for Pb–I termination, leading to removal of MAI by dissolution of the respective ions into the liquid water. This work finds that the formation of both monohydrate and dihydrate is reversible, but exposure to liquid water leads to an irreversible process that results in evolution of HI. In contrast to this prediction of an undeformed Pb–I framework, Tong et al.⁴⁶ predicted that water adsorption on the 001 face leads to significant structural changes. Such changes are minimal at low coverage, but increase at higher coverage, and are responsible for the increase in bandgap and, thus, degrade photovoltaic performance.

Yu et al. used molecular dynamics to study both single crystal and polycrystalline MAPbI₃.⁴⁷ Whereas they did not consider interaction with water, it is clear that H-bond formation will compete with and, thus, depend on other ionic interactions in the crystal. Their results suggest that the mechanics are dominated by short-range Coulombic interactions between MA⁺ cations and the octahedral (PbI₆) frame. Furthermore, they propose that plastic deformation occurs by splitting of these octahedra into tetrahedra, which results in atomic-scale disorder that propagates as a crack. Despite the ease of fracture, MAPbI₃ is much more flexible (much lower Young’s modulus) than the oxide perovskites, in line with the low E values found for HaPs, relative to non-perovskite solar PV materials such as Si and GaAs.^12,48

The effect of water on mechanical stability of these materials has also been addressed experimentally. In general, such works involve long exposure times to ambient, which in nearly all studies include not only water vapor but also O₂. It should be noted that exposure to ambient carries additional potential for damage: exposure to O₂, as well as to light, are also known to degrade these materials, a process which has been extensively studied and reviewed.^49,50 Using x-ray diffraction and spectroscopic ellipsometry, LeGuy et al. found that long exposure of single crystal MAPbI₃ to wet N₂ [70% relative humidity (RH)] led to the formation of the monohydrate, which could be reversed in a few hours under dry conditions. However, it appears that the re-dried, formerly single crystal surface has a polycrystalline form.⁵¹ 

Mamun et al. also used very long exposure times (100h), but to 40% relative humidity ambient, while measuring changes in mechanical properties and crystal structure in MAPbI_3−x Cl_x thin (450 nm) films.¹⁹ They monitored both elastic modulus (E) and hardness (H). Here, both E and H rose over the first 20 h of exposure, but then decreased. Concomitantly, a growth in crystal size and appearance of PbI₂ (from characteristic x-ray diffraction peaks) was observed. In this case, the effect of grain size in the thin film (i.e., inverse Hall–Petch effect⁵²), as well as the parallel influence of O₂ exposure, make it hard to assign mechanical changes independently to the reduction of long-range order in the polycrystalline thin film.

Liao et al. used the more surface-sensitive AFM technique to monitor changes in E on polycrystalline MAPbI₃ films with ambient (50%) relative humidity.¹⁸ A complex behavior was revealed, with the modulus originally rising, then dropping, then rising again. Combined XRD and morphological analysis pointed to the formation of a poorly structured PbI₂ film, which eventually formed well-developed PbI₂ crystallites. In contrast to this, Spina et al. observe a monotonic drop in E and H for MAPbI₃ single crystals under ambient (30%–50% relative humidity) conditions.⁵³ Those measurements were made by nanoindentation to a depth of 1 μm. The degradation occurs over many months and can be partially recovered by dipping the crystals in a solution of methylammonium iodide. Comparing these last two works in a review on mechanics-coupled stability of HaPs, Tu et al. suggest that different results could arise from contrast between surface and bulk properties.¹² Indeed, a contrast between bulk and near-surface healing has been observed for photodamage.³⁷ 

Supporting this difference between surface and bulk behavior, our recent results highlighted the difference between surface and bulk in measurements on the influence of humidity on the mechanics. Specifically, polarizability of the X anion anti-correlated with the change of E in the bulk as humidity rose, but showed a positive correlation at the surface. In addition, at the surface the polarization power of the A-site cation correlated positively with E, while no clear trend held for the bulk.¹⁶ The surface behavior can be understood by realizing that polarization at the surface can lead to stronger interactions with water than in the bulk, where the trend is likely due to Pb–X bond length which anticorrelates with X polarizability. Pb–X bond length dependence was also found for mixed halide perovskites: for MAPb(I_1−xBr_x)₃ crystals: larger x correlated with better stability under exposure to 55% relative humidity ambient for a full day. This was claimed to reflect the stronger bonding with shorter Pb–X bond length in more Br-rich samples and the difference in crystal structures, cubic for the bromide-rich and tetragonal for the iodide-rich stoichiometries.⁵⁴ 

To complement the studies of long moisture exposure, it is of interest to study the effects of short exposure times to reveal the first stages of decomposition, knowledge of which may aid in retarding it. Interestingly, an experimental study comparing both optical response and structural changes pointed to the initial reversible formation of hydrated isolated PbI₆⁴⁻ octahedra in MAPbI₃.²⁰ Even after long (24 h) exposure to 98% RH, rapid partial reversibility was observed, indicating that the irreversible acid-base reaction producing lead iodide and volatile CH₃NH₂ is only partially achieved. This result fits well with some of the computational studies mentioned above.

Our recent work compared the reversible humidity effect for different series of Pb-based halide perovskites. In addition to the atypical inverse correlation for trends in E and H (noted above, and see below), also the humidity had an opposite effect on these two properties— humidity exposure led to lower H, but to higher E [as seen by comparing Figs. 4(a) and 4(b)].¹⁶ This trend is also opposite to that found in most materials where hydration lowers both E and H.^27,55

Since device fabrication typically relies on thin films rather than single crystals, the mechanical behavior of such films is of great technological interest. For instance, for wearable devices, this will require matching elastic properties of the film/substrate to that of the skin. This is a consideration for all new developments proposed for wearable devices where a class of materials typically exhibiting high stiffness needs to match the soft, elastic nature of the skin.⁵⁶ Furthermore, when stress on the film is monitored by changes in the film’s optoelectronic properties (see below), it is critical to know if the stress required to obtain sufficient signal is below the damage threshold. Mechanical robustness of devices depends not only on the HaP film, but is strongly governed by the different architectures and methods of film deposition so that the studies done on single crystals may have little relevance here. A careful study of fracture in perovskite-based solar devices showed them to have poor fracture characteristics.⁵⁷ Nonetheless, a couple of newer directions may address some of the drawbacks for applications of the Pb-based perovskites.

Together with intense activity to elucidate the special properties of the lead-based inorganic and hybrid perovskites, there is parallel effort in discovering and developing new materials, which overcome some of the disadvantages noted above. One interesting direction is that of 2D hybrid organic-inorganic perovskites, layered materials for which the A cation is replaced by a larger organic ammonium one, in the out of plane direction, the size of which prevents formation of a 3D perovskite structure. The in-plane structure is then composed of BX₆ octahedra sheets (and of 3D HaP sheets in case of mixed 2D-3D materials), separated by layer(s) of the large organic ammonium cation.^58,59 The rich chemistry provided by altering the organic layer allows for much greater tuning of the optoelectronic properties than is possible with the 3D HaPs.^60,61 Dozens of different organic molecules have been employed,⁶² which, together with variations in the inorganic components, provide enormous optoelectronic tunability, as well as a system to study quantum confinement effects.

These materials have the general structure of (R-NH₃)₂MA_n−1 B_nX_3n+1, where R is the organic moiety and n is the number of ABX₃ octahedra layers between the organic spacer layer. A further distinction is the nature of the linker, common types being Ruddlesden–Popper (RP) materials where the organic layers mate with weak vdW interactions, or Dion–Jacobson (DJ) materials where the organic linker contains a di-amine which binds the successive inorganic octahedral layers more strongly.⁶³ An intermediate case is where this mating is through π-CH₃ interactions.⁶² 

The optoelectrical behavior of these materials is not tuned by composition alone: Mechanical stress can impart profound changes in the functioning of 2D HaP devices. Recently, Ratchford et al. showed that even small (1%) compressive stress enhances emission and charge carrier lifetimes by an order of magnitude.⁶⁴ These studies were done on quasi-2D perovskite films composed of Pb-halides and large organic cations. Tuning of the band gap with moderate pressures (350 MPa) was demonstrated for PbBr₄- and PbI₄-based RP and DJ perovskites with the organic spacer being either benzylammonium or phenylenedimethylammonium, respectively.⁶⁵ Here, compression perpendicular to the plane led to a reversible redshift in the excitonic band. X-ray studies helped to relate changes in the bandgap to changes in interlayer spacing. For the RP series, compression for case of Br halide was much larger than for I and attributed to the spacers penetrating into the Pb–Br layer. In a mixed 2D HaP–(C₆H₅CH₂CH₂NH₂)₂Cs₃Pb₄Br₁₃ for which n = 4 (i.e., four inorganic layers between each organic spacer layer), the photoluminescence was found to blueshift under tensile strain and redshift under compressive strain.⁶⁶ Clearly, these works show the enormous potential for sensing devices, as well as controlled tuning of the wavelength and intensity of emitted light, and device current. This points to the need for deep understanding of the mechanics-structure-optical/optoelectronic relations for this class of materials.

Due to their inhomogeneous composition, the mechanical properties of these 2D HaPs have a complex relation to the nature of both the inorganic part and organic spacers, and the interactions between them. For a series of RP 2D halide perovskites with alkylamine spacer and n = 2, the out of plane E decreased for increasing chain length.⁶⁷ The lowered stiffness was accompanied by increased non-radiative recombination. This work also showed that in contrast to the 3D material, E was lower when the halide was Br than for I, despite the weaker Pb–X bond in the latter. This was explained by inter-digitization of the alkyl chains between the octahedra. In contrast, for a series of n = 1 RP 2D HaPs, the out of plane modulus increased with decreasing X anion size, E_I < E_Br < E_Cl.⁶² Interestingly, in this work, the hardness trend did not follow that of the modulus, with H_Br > H_Cl > H_I as opposed to 3D crystals where H_I> H_Br> H_Cl. Another interesting result of this study was that organic spacers interacting through aromatic π-interactions resulted in a higher modulus and hardness for the material than those interacting covalently. Despite these aforementioned anomalies, the mechanical behavior of these 2D HaPs is intuitive: for n = 1 both (out of plane) E and H are only a small fraction of that of the analogous 3D single crystal with no soft organic spacer (in this case CH₃NH₃PbI₃). As n increases, E and H approach the value of 3D MAPbI₃⁴⁸ This work also found enhanced softening for longer-chain alkyl spacers.

Reyes-Martinez et al. measured a 2D-3D hybrid perovskite where the large layer-separating organic moiety is aromatic, phenethylammonium, with several methyl ammonium lead iodide layers, (C₆H₅(CH₂)₂NH₃)₂(CH₃NH₃)_n−1[Pb_nI_3n+1].⁶⁸ They also calculated the contributions of the different interactions. These calculations showed that for the phenyl-ethyl group the π-interactions did not make a significant contribution to the modulus relative to dispersion interactions, in contrast to the findings of Ref. 62. The out-of-plane E values measured are similar to those found for 3D CH₃NH₃PbI₃, rising modestly with n. They also measured the in-plane E by wrinkling experiments. This value was significantly lower than the out of plane E leading them to conclude that it is dominated by shear. We note here that both in-plane and out of plane moduli need to be properly tuned for flexible devices and more control as well as measurement of the relation between these properties is necessary. These are more readily measured for the 2D HaPs than for 3D crystals. Such studies show high fracture strength, close to the ideal value of E/10.¹² Given the great challenges in preventing fracture of the flexible devices, the importance of self-healing as described in Sec. III should be pursued as a means to maintain their integrity under use.

A pronounced feature of the force curves in some of the 2D materials was “pop-ins” (horizontal jumps in the load versus displacement curves), which were proposed to be due to compression of defects.⁶⁸ Thus, after an initial indent removes such defects, the material stiffens. However, pop-ins were also observed in 3D HaPs where they were ascribed to breaking of H-bonds and in accordance with this they led to a larger jump in the force versus displacement curves for FAPbBr₃ than for FAPbI₃.²⁹ Yet another study on the 3D crystals showed a relation between creep and pop-in behavior, which did not follow the H-bond strength trend.⁷ In this case, the crystal structure, which normally determines whether indentation will be more likely to nucleate defects or activate slip planes was proposed to govern this phenomenon. The different mechanisms for pop-in for 2D or mixed 2D-3D materials can, therefore, be relegated to various other mechanisms, such as slipping of the weak vdW interactions between layers for the RP materials. Modulating the strength of interlayer bonding, thus, provides another degree of tunability to probe the pop-ins. Interestingly, 2D boron nitride (h-BN) shows great defect tolerance and can be strained up to nearly 6% even in the presence of large voids 100 nm in size.⁶⁹ These tensile experiments were done on macroscopic samples and it would be interesting to perform similar tests on inorganic and hybrid HaPs for comparison.

In an attempt to move away from Pb-based materials, double halide perovskites have come under more intense investigation. An investigation of the mechanical properties of (MA)₂AgBiBr₆ found that while Young’s modulus was only about half that of MAPbBr₃, the hardness was quite similar.⁷⁰ This combination could lead to improved wear characteristics.²⁷ Another unique feature of these double perovskites of formula M(MA)₂MBiX₆ with M being Ag, K, Tl, and X either Cl or Br is that E rises with increasing M–X bond length in opposition to what was described above for the Pb–X perovskites.²⁹ 

The correlation between optoelectronic performance and mechanics, briefly mentioned above, is fascinating, but faces time-scale challenges, the former being at orders of magnitude lower time scales than the latter. Whereas carrier dynamics can now be measured at the picosecond time scale,⁷¹ measurements of self-healing are typically made in time scales of minutes/hours/or even days.^34–36 Creep and stress relaxation provide some improvement in measurement speed, but are done on time scales of seconds.^7,8 Although reasoned interpretation and theory can bridge this gap, direct measurement of mechanical processes at fast time scales could bring new insights.

Overall, considering the huge variety of compositions, it is becoming clear that a systematic approach is required, both in synthesizing film/crystals with desired qualities and in linking their structure to their mechanical behavior. Some steps have been made in this direction as outlined above, and a significant effort has been devoted to engineering composition and design of films and full devices to achieve optimal combination of efficiency and stability.¹² Whereas first-principles computations of the mechanical properties of some HaPs can be obtained to predict moduli with reasonable accuracy,^72,73 this is not possible for engineered thin films and full device architectures. In this case, we anticipate that machine learning for directed design can hold great promise,⁷⁴ especially if combined with big-data-driven means to predict optimal materials compositions.⁷⁵ Thus, the burgeoning field of materials by design⁷⁶ should play a large role in the optimization of perovskite materials and devices.

The need to understand and control mechanical behavior of technologically relevant perovskite materials is undisputed. In this perspective, we have briefly outlined some of the main considerations, while stressing the caution required in applying and interpreting experimental measurements. Exposing these materials to the ambient leads to other complications, notably the influence of humidity/hydration. We further suggest a link between dynamic disorder, self-healing, and the intriguing inverse relationship between E and H observed in some Pb-based halide perovskites.

It is important to note that investigation of mechanical robustness of thin film devices introduces complexities not addressed here. While there are some cases where a single parameter such as elastic modulus or elastic energy release rate governs the fracture mechanics, this is not typically the case and the fracture micro-mechanisms must be addressed. This topic is discussed in depth in textbooks.⁷⁷ Failure mechanisms include ductile fracture where voids are nucleated and grow to join together, and cleavage where the fracture is along a specific crystalline plane. For polycrystalline materials as for the materials of interest here, the fracture path crosses grains, but changes direction at each such crossing. The breaking of crystalline bonds in this mechanism means that it will depend on the cohesive bond strength. For highly anisotropic thin film materials, this would mean a large dependence on the orientation of the crystal planes. At low temperatures, or for systems with few dislocations or slip planes, the yield strength is higher, favoring this mode of failure. Finally, if intergrain interactions are weak, failure can occur along the grain boundaries. Another consideration is the ability of the material to retard the crack propagation. This may be expected to occur “intrinsically” by deflection or splitting of the growing crack, which will absorb the energy developed in its formation.²⁷ 

As mentioned at the outset, mechanical properties of these materials are intrinsically related to their optoelectronic properties. This is reflected in the recently studied self-healing properties.^34,35,37,78 As discussed in Sec. V, this linkage is fundamental to implementation and operation of thin film devices. A device that can recover from photobleaching or cracking to regain full functioning would be a boon to device sustainability. Such healing would involve movement of atoms/ions, i.e., must also involve dynamics. All of this points to the need for fast measurements, which can measure mechanical processes or be correlated with them.⁷¹ We, therefore, foresee the growing importance of studies carried out, which probe both short time scales and small volumes.

D.C. thanks the Minerva Centre for Self-Repairing Systems for Energy & Sustainability and the Weizmann Institute’s Sustainability and Energy Research Initiative, SAERI for support.

The authors have no conflicts to disclose.

Irit Rosenhek-Goldian: Conceptualization (equal); Supervision (lead); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal). David Cahen: Conceptualization (supporting); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal). Sidney R. Cohen: Conceptualization (equal); Supervision (lead); Writing – original draft (lead); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.