Experimental

Crystal growth

CH₃NH₃PbI₃ single crystals were grown following a published procedure,^[ Reference Dong, Fang, Shao, Mulligan, Qiu, Cao and Huang ^{13 ]} where a temperature gradient was used to grow the crystals on a silicon wafer that was partly immersed in a 57% hydroiodic acid solution of lead iodide and CH₃NH₃I, where the part remaining outside the solution was cooled by an air flow. The resulting tetragonal (pseudo cubic)^[ Reference Dang, Liu, Sun, Yuan, Liu, Lu, Liu, Xia and Tao ^{14 ]} crystals usually expose the (100) and (211) faces (see Fig. 1).

Figure 1.

Scanning electron microscopy images of (left)] (100) and (right) (112) faces of CH₃NH₃PbI₃ crystals.

For the growth of CH₃NH₃PbBr₃ we used anti-solvent (ethyl acetate) slow evaporation into a solution of N,N-dimethylformamide of lead bromide (PbBr₂) and methylammonium bromide (CH₃NH₃Br). Single-crystal growth procedures of CH₃NH₃PbBr₃ and of CH₃NH₃Br have been described previously.^[ Reference Shi, Adinolfi, Comin, Yuan, Alarousu, Buin, Chen, Hoogland, Rothenberger, Katsiev, Losovyj, Zhang, Dowben, Mohammed, Sargent and Bakr ^{4 ,} Reference Tidhar, Edri, Weissman, Zohar, Hodes, Cahen, Rybtchinski and Kirmayer ^{15 ]} The resulting cubic CH₃NH₃PbBr₃ ^[ Reference Zhao, Xu, Dong, Wang, Ren, Bian and Chang ^{16 ]} crystals typically expose the (100) face [see Fig. 2(a)].

Figure 2.

Light-microscope images of (a) as-grown crystal of CH₃NH₃PbBr₃ presenting the (100) face and (b) image after indentation measurements. The spacing between the indentations is ~20 times the penetration depth of 800 nm. (c) Typical nano-indentation load/displacement curve on CH₃NH₃PbBr₃. For load/displacement curves of CH₃NH₃PbI₃ and CsPbBr₃, see supplementary information, Fig. S1.

The growth method for CsPbBr₃ is similar to that used for CH₃NH₃PbBr₃ (see above), where the solvent is dimethyl sulfoxide and the anti-solvent is water or ethanol (to be published). The crystal structure was verified by comparing the powder-x-ray diffraction to literature data.^[ Reference Stoumpos, Malliakas, Peters, Liu, Sebastian, Im, Chasapis, Wibowo, Chung, Freeman, Wessels and Kanatzidis ^{17 ]}

All the precursor preparation and crystal growth were done under ambient conditions with 45%–65% relative humidity. None of the precursors or solvents was specially treated before use. All the crystals were stored in an air-filled desiccator with ~25% relative humidity.

Results and discussion

The Young's modulus and hardness, which are the quantities that can be obtained most directly from the nano-indentation technique, are listed in Table I, for CsPbBr₃, CH₃NH₃PbBr₃, and CH₃NH₃PbI₃, together with other known physical properties. The results for CH₃NH₃PbBr₃ and CH₃NH₃PbI₃ are in quite good agreement with those of a very recent nano-indentation study (on CH₃NH₃PbX ₃; X = Cl, Br, I) that was brought to our attention after submission of this paper.^[ Reference Sun, Fang, Kieslich, White and Cheetham ^{19 ]} Bulk modulus, K, values for CH₃NH₃PbBr₃ ^[ Reference Swainson, Tucker, Wilson, Winkler and Milman ^{20 ]} and CsPbBr₃ ^[ Reference Hirotsu, Suzuki and Sawada ^{21 ]} were previously experimentally derived, and can be related to E through the Poisson's ratio (see Table I). We deduced the linear thermal expansion coefficients, α, of all three APbX ₃ perovskite compounds from literature reports.^[ Reference Kawamura, Mashiyama and Hasebe ^{10 ,} Reference Swainson, Tucker, Wilson, Winkler and Milman ^{20 ,} Reference Rodová, Brožek, Knížek and Nitsch ^{22 ]} Elastic moduli and other physical properties (i.e., density and thermal expansion coefficient) of functionally, compositionally or structurally related materials are also presented in Table I. Broader comparisons, the so-called Ashby two-dimensional maps of Young's modulus versus density or linear thermal expansion coefficient are presented in the supplementary information (Fig. S2 and S3).^[ Reference Ashby ^{23 ]} Where experimental information was missing we used results from theoretical calculations.^[ Reference Egger and Kronik ^{24 –} Reference Murtaza and Ahmad ^{26 ]} For comparison and as guideline for future measurements additional mechanical characteristics, viz., bulk (K) and shear moduli (G), and Poisson's ratio (ν) were derived using fundamental relations that strictly apply only to homogeneous, isotropic, and linear elastic materials.^[ Reference Ashby ^{23 ]}

Table I.

Mechanical constants of halide perovskites and related materials.

The values that are measured in this work are in bold in lines 1–4. The remainder of the mechanical constants were derived ^a,b or taken from experimental or theoretical^[ Reference Feng ^{25 ,} Reference Murtaza and Ahmad ^{26 ]} results from other works, references to which are given in the table. E, Young's modulus; H, (nano) hardness; G, shear modulus; K, bulk modulus; ν, Poisson's ratio; α, linear thermal expansion coefficient; if only the volume value was available, it was divided by 3; ρ, density.

^a Estimated for a homogeneous, isotropic, and linear elastic material, according to^[ Reference Ashby ^{23 ]}: G = E/2(1 + ν).

^b Estimated for a homogeneous, isotropic, and linear elastic material, according to^[ Reference Ashby ^{23 ]}: K = E/3(1 − 2ν).

^c The speed of the sound-based bulk modulus is 26 GPa. This value was not used as it requires an unreasonably large Poisson's ratio (~0.4) to match the data.

^d Estimated value.

^e Estimated from speed-of-sound average value of PbBr₂ (2282 m/s),^[ Reference Weber ^{29 ]} using the formula^[ Reference Kinsler, Frey, Coppens and Sanders ^{49 ]}: $v_s {(p)} = \sqrt {E/\rho \cdot (1 - \nu )/(1 + \nu )(1 - 2\nu )} $ with an estimated Poisson's ratio of 0.33.

Extensive theoretical work on the mechanical properties of several Pb and Sn-based HOIPs,^[ Reference Feng ^{25 ]} which shows fair agreement with our experimental results, suggests that the type and strength of the Pb–X bond determine the elastic properties of the materials. Indeed, in agreement with the reported higher bond enthalpies for Pb–Br than for Pb–I,^[ Reference Luo ^{27 ]} we measure a higher Young's modulus for the Br-based HOIP than for the I-based one [E(CH₃NH₃PbI₃) = 14.2 ± 1.9 GPa and E(CH₃NH₃PbBr₃) = 19.6 ± 0.3 GPa]. Following this logic, the Young's modulus of 17 GPa^[ Reference Veiga and Lepienski ^{28 ]} previously measured along a Pb–I network in PbI₂ —i.e., along the layers that make up the (CdI₂-type) structure, which lie in the c (0001) plane— is comparable with that of APbI₃ crystals measured here. Even though PbBr₂ (average of 23.5 GPa^[ Reference Weber ^{29 ]}) has a STP structure that is very different from that of PbI₂ (PbCl₂-type; not layered), also here we find similar Young's moduli. This is consistent with the idea that the Pb–X framework dominates the mechanical properties of these materials (see Fig. S4).

Comparison of the mechanical properties of the hybrid organic–inorganic with the fully inorganic APbBr₃ perovskite-structured material, reveals a slightly lower elastic modulus for CsPbBr₃ than CH₃NH₃PbBr₃ [the value for CsPbBr₃, derived from speed-of-sound measurements (2775 m/s), was 26 GPa^[ Reference Hirotsu, Suzuki and Sawada ^{21 ]}]. For CH₃NH₃PbI₃ and CsPbI₃ (for which only a value based on theoretical calculations is available), an opposite relation is seen, but, as CsPbI₃ does not have a perovskite(-like) structure at STP, such comparison may be invalid.

The bond length and stiffness trends of “classic” ionic (oxide) perovskite materials are such that the bulk modulus decreases as the ionic radius of the “A” cuboctahedral group increases.^[ Reference Bhadram, Swain, Dhanya, Polentarutti, Sundaresan and Narayana ^{30 ,} Reference Verma and Kumar ^{31 ]} For the oxide perovskites this increase of the A radius is usually accompanied by an increase in the unit-cell volume, which is opposite to the case for APbBr₃ with 0.198 nm³ for CsPbBr₃ and 0.207 nm³ for CH₃NH₃PbBr₃. The difference in the volume is also reflected in different distances between the A cation and X anion (see Table II). A purely electrostatic model would have resulted in a higher elastic modulus for the shorter interatomic bonds, i.e., the CsPbBr₃. As the results show the opposite, there must be additional forces that keep the hybrid structure bound.

Table II.

Average bond lengths at room temperature versus the elastic constants of APbX ₃ bulk materials (A = CH₃NH₃, Cs; X = I, Br). E and K are Young's and bulk moduli, respectively. The average bond distance refers to structures at room temperature with the indicated space groups.

^a Theoretical value^[ Reference Murtaza and Ahmad ^{26 ]}—see Table I.

An interesting similarity to HaPs in terms of their crystal structure (cubic) and surprisingly low thermal conductivity^[ Reference Pisoni, Jaćimović, Barišić, Spina, Gaál, Forró and Horváth ^{32 ,} Reference Nakamura, Yamada and Ohnuma ^{33 ]}, can be found in a structural family called clathrates. Inorganic clathrate compounds (e.g., K₈Al₈Si₃₆ and Sr₈Ga₁₆Ge₃₀) are characterized by an open framework structure with “guest” ions (e.g., K, Al and Sr, Ga) trapped in a well-crystallized “host” framework structure (e.g., Si and Ge). For perovskites, the “host” can be imagined to be as the Pb–X framework and the guest will be the A ion. In terms of the thermal conductivity—for both cases the main contribution to this effect was related to interactions between the collective “host” vibrations with the trapped “guest” atoms/ions.^[ Reference Pisoni, Jaćimović, Barišić, Spina, Gaál, Forró and Horváth ^{32 ,} Reference Sui, He, Bobev, Zhao, Osterloh and Kauzlarich ^{34 ]} As for the elastomechanical properties of clathrates, theoretical work showed that the bulk modulus is logarithmically related to the average interatomic distance of the “host” atoms (see Fig. 3 or^[ Reference Karttunen, Härkönen, Linnolahti and Pakkanen ^{35 ]}). By placing data of the ABX ₃ (B = Sn or Pb) and PbX ₂, which are presented in Table I, and other oxide structures found in the literature on a single plot (see Fig. 3), we find, for perovskite-structured APbX ₃, a logarithmic relation between the B–X bond length and the modulus, but with lower elastic modulus than expected.

Figure 3.

Bulk modulus versus average shortest bond distance. “Average (shortest) bond distance” means: for clathrates—“host”–“host” average bond distance (e.g., for Si-based clathrates this will be the –Si–Si bond); for any ABX ₃ compound it is the average B–X distance; for PbX ₂ structures is the average Pb–X distance. The actual numerical values used for the plot are summarized in Table S1. The black line is a linear fit adapted from^[ Reference Karttunen, Härkönen, Linnolahti and Pakkanen ^{35 ]}

In order to check the influence of these microscopic considerations on behavior of the bulk material, we looked at the relation between Young's modulus and the density or linear expansion coefficient, from the work of Ashby^[ Reference Ashby ^{23 ]} (see Figs. S2 and S3). On these maps the HaP materials are located between polymers, metals, and composite materials.

For a given E, the thermal expansion coefficients of the HaPs are higher than of most metals and composites, smaller than of most polymers and similar to those of rigid polymer foams (and to that of PbI₂; see below). It is remarkable that what is formally an ionic material has such a large expansion coefficient, which may reflect the flexibility of the Pb–X bonds. If we compare both the elastic moduli and the linear expansion coefficients of CH₃NH₃PbX ₃ and PbX ₂ (c-plane in PbI₂—see Fig. S4), it appears that the Pb–X bond plays a dominant role in the elasticity of these materials.

The finding that HaPs are positioned in the diagrams with relatively soft metals and that the Pb–X corner-sharing octahedral framework is quite (thermally) expandable might indicate possible ductility or flexibility, which could ease formation on heterogeneous surfaces with low energetic penalty and good adaptivity. Another implication of enhanced flexibility is that application of external energy (mechanical, electrostatic, and electromagnetic) to such materials might allow facile creation but also ready rearrangement (self-healing) of point defects, which can help atom/ion diffusion in the material. In this context, theoretical calculations gave for the energy barrier to iodide vacancy migration ~0.1 eV,^[ Reference Azpiroz, Mosconi, Bisquert and Angelis ^{36 ]} up to ~0.3 eV for iodide displacement in a vacancy-free surrounding environment (due to proton migration^[ Reference Egger, Kronik and Rappe ^{37 ]}). Confirmation of suggested processes^[ Reference Yang, Gregori, Pellet, Grätzel and Maier ^{6 ,} Reference Eames, Frost, Barnes, O'Regan, Walsh and Islam ^{7 ,} Reference Azpiroz, Mosconi, Bisquert and Angelis ^{36 ]} such as extensive ion migration and “self-healing” in these materials awaits further, direct experimental evidence.

If we look at possible benefits of these mechanical properties for manufacturing, these properties can be beneficial for spreading thin films of perovskites on surfaces with different morphologies, using methods such as “doctor-blading”. However, the ease of plastic deformation (i.e., low hardness), makes these materials less suitable for flexible substrates.

As to any correlation with other lead-based (Pb and PbTe), or photovoltaic materials (CdTe, CIGS, GaAs, and Si) —except for the similar elastic modulus for metallic lead, and some similarity to the hardness of the PbTe and CdTe—the overall mechanical properties do not really correlate.

Summary

The elastomechanical properties of the examined halide perovskites (CH₃NH₃PbI₃, CH₃NH₃PbBr₃, and CsPbBr₃), with respect to their density or linear thermal expansion coefficient, are between those of soft metals, polymers, and composite materials. This combination of properties might be reflected in the (apparent low energies required for) dynamic processes speculated to exist in these materials (e.g., self-healing and ion-migration). Similar to what is the case for “classic” (oxide) perovskites, the elastomechanical properties seem to be dominated by the B–X bond, with the presence of the A group decreasing the elastic coefficients of these large-cage perovskite-structured materials. The lower elastic modulus of CH₃NH₃PbI₃ than of CH₃NH₃PbBr₃ may reflect differences in the Pb–X bond strengths. Comparison between CH₃NH₃PbBr₃ and CsPbBr₃ shows that the organic group actually makes the entire structure stiffer (higher elastic modulus); however, it might also be due [in part] to the difference in the crystallographic orientation between the tested crystals, because, as shown in Sun et al.,^[ Reference Sun, Fang, Kieslich, White and Cheetham ^{19 ]} the (100) is stiffer than (110) orientation in CH₃NH₃PbBr₃. The similarity between the elastic modulus of PbI₂ (c-plane) with that of CH₃NH₃PbI₃ and that of PbBr₂ (which has a structure that is quite different from PbI₂, which cannot easily be related to a perovskite structure) with APbBr₃, suggests that the Pb–X framework dominates the elastic modulus. We do not find an obvious relation between the mechanical and the optoelectronic, especially photovoltaic properties, of the materials.

The conclusions and conjectures presented in this work should mostly serve as guidelines for further investigations, such as growth and mechanics for analogous single crystals (e.g., chloride and formamidinium perovskites).