The complete understanding of the mechanical and thermal responses to strain in hybrid organic–inorganic perovskites holds great potential for their proper functionalities in a range of applications, such as in photovoltaics, thermoelectrics, and flexible electronics. In this work, we conduct systematic atomistic simulations on methyl ammonium lead iodide, which is the prototypical hybrid inorganic–organic perovskite, to investigate the changes in their mechanical and thermal transport responses under uniaxial strain. We find that the mechanical response and the deformation mechanisms are highly dependent on the direction of the applied uniaxial strain with a characteristic ductile- or brittle-like failure accompanying uniaxial tension. Moreover, while most materials shrink in the two lateral directions when stretched, we find that the ductile behavior in hybrid perovskites can lead to a very unique mechanical response where negligible strain occurs along one lateral direction while the length contraction occurs in the other direction due to uniaxial tension. This anisotropy in the mechanical response is also shown to manifest in an anisotropic thermal response of the hybrid perovskite where the anisotropy in thermal conductivity increases by up to 30% compared to the unstrained case before plastic deformation occurs at higher strain levels. Along with the anisotropic responses of these physical properties, we find that uniaxial tension leads to ultralow thermal conductivities that are well below the value predicted with a minimum thermal conductivity model, which highlights the potential of strain engineering to tune the physical properties of hybrid organic–inorganic perovskites.

Hybrid organic–inorganic perovskites are endowed with remarkable physical properties, which derive from their unique chemical and structural makeup.1,2 Since their first successful demonstration as materials for photovoltaic applications little more than a decade ago,3 these materials have demonstrated exceptional optical, electrical, and thermal properties, placing them as one of the premier materials for emerging technologies, such as in thermoelectrics, electrochemical energy storage, and solar cells.4–11 More recently, these materials have been integrated as thin film absorbers on flexible polymer substrates and in shape recoverable device architectures for flexible electronics.12,13 As such, a comprehensive understanding of their mechanical and thermal properties that inherently set the limitations of hybrid organic–inorganic perovskites as materials for deformable electronics is quintessential for their realization and further improvement in efficiencies in such applications.

Recent studies have focused on understanding heat transfer and lattice dynamics of hybrid perovskites from both experimental and computational perspectives.14–26 Experimental measurements report thermal conductivities in the range of 0.34–0.73 W m−1 K−1 for various three-dimensional single crystal hybrid perovskites. The variation in the ultralow thermal conductivities between different hybrid perovskites has been attributed to changes in the speed of sound, which are mainly dictated by the elastic properties of the inorganic framework.19 The ultralow room temperature thermal conductivity of these materials and the hot phonon bottleneck have been attributed to low group velocities and strong anharmonic phonon–phonon scattering that are prevalent in these types of structures mainly due to their low elastic stiffnesses.17,18,25,27 Therefore, these findings suggest that the thermal properties of hybrid perovskites are strongly associated with their mechanical and structural properties.

In terms of the mechanical properties, nanoindentation studies and laser-based experiments have highlighted the unique anisotropy in elastic properties of hybrid perovskites with the measured Young’s modulus in the range of 7–20 GPa.12,28–30 Computational studies conducted with first-principles calculations have corroborated the anisotropy in Young’s modulus and have also demonstrated that the low shear modulus in hybrid organic–inorganic perovskites can be beneficial for applications in compliant devices where large deformations are demanded.31 Furthermore, polycrystalline CH3NH3PbI3(MAPbI3) have been shown to possess nanoductility surpassing their single crystal counterparts, which was attributed to the extensive and continuous amorphization in the polycrystalline structure.32 

Another unique mechanical property predicted by computational studies of hybrid perovskites (and other oxide based perovskites) is the possibility of negative Poisson’s ratio in certain directions.31,33,34 Based on the elastic constant tensor from first-principles calculations, Ji et al.34 have shown that the orthorhombic phase of MAPbI3 and other similar hybrid perovskites can demonstrate a negative Poisson’s ratio in certain directions due to the rotational motion of the PbI6 octahedron.34 In most materials, a uniaxial tension in the orthogonal direction leads to the shrinking in the lateral directions; these materials are characterized by a positive Poisson’s ratio. However, there are a certain class of materials, although rare, that have a negative Poisson’s ratio (also known as auxetic materials)35–41 and are characterized with an expansion in the lateral directions when stretched in the orthogonal direction such as that predicted by Ji et al.34 for hybrid perovskites in certain directions. These materials are often accompanied by enhanced physical properties that are beneficial for different kinds of applications.42–47 For example, auxetic materials are used in a range of applications, such as in medicine and tissue engineering, flexible photovoltaics, and aerospace and defense.48–51 Auxetic behavior has been shown for honeycomb structures and open cell foams,44,52 cubic metals strained along nonaxial directions,46,53 two-dimensional materials,54–56 select types of polymers,37–39,57,58 and metal organic frameworks.59,60 However, direct observation of auxeticity in hybrid organic–inorganic perovskites has not yet been demonstrated, and only inferences based on the elastic tensor predicted from first-principles calculations have been made.34 Therefore, a systematic investigation from an atomistic perspective of the mechanical as well as thermal responses of hybrid organic–inorganic perovskites under uniaxial strain would shed light on the microscopic mechanisms dictating these physical properties and would also be beneficial for a range of applications that are reliant on these novel materials.

Through atomistic simulations, in this work, we show that the mechanical response and the deformation mechanism in hybrid perovskites are highly dependent on the direction of the applied uniaxial strain with a characteristic ductile deformation under uniaxial tension along [100] or [010] directions, whereas brittle failure occurs when the tensile loading is applied along the [001] direction. With uniaxial tension along the [100] or [010] orthogonal direction, an anisotropic mechanical response is observed in the two lateral directions with a length contraction in the [001] direction, while a negligible change in the length occurs along the other direction. We also find an anisotropic thermal response to uniaxial strain, where the thermal conductivity along the length contraction direction remains unchanged, whereas the thermal conductivity in the plane of the uniaxial tensile loading [001] decreases monotonically, thus increasing the anisotropy in thermal conductivity by up to 30% before plastic deformation occurs.

We study the prototypical hybrid organic–inorganic perovskite, MAPbI3, via molecular dynamics (MD) simulations. The interatomic potential utilized in our MD simulations is the ab initio-based potential (MYP force field) developed by Mattoni et al.61 This potential has been shown to predict the correct vibrational physics (including thermal properties of MAPbI3)18,23 and also shown to correctly predict their elastic properties.32 Moreover, the potential was developed specifically to replicate the energy profile as the molecular constituents reorient themselves with respect to the deforming inorganic framework, which further validates the use of this potential for investigation of mechanical response to strain of these materials. We use the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package for all of our simulations.62 

Using the MYP potential,61 the initial computational domain for MAPbI3 is equilibrated for a total of 1 ns at 0 bar pressure with a time step of 1 fs under the Nose–Hoover thermostat (at room temperature) and barostat;63 this is the NPT integration where the number of particles, pressure, and temperature are held constant during the simulation. Following the NPT integration, an additional equilibration under the NVT integration (where the volume and number of particles are held constant during the simulation) is performed. Note that during equilibration and the subsequent simulation to investigate the thermal and mechanical properties, periodic boundary conditions are applied in all directions and a time step of 1 fs is utilized. The size of the computational domain is 26.45 × 26.55 × 112.90 Å3. The cross section of the equilibrated computational domain for MAPbI3 is shown in Fig. 1(a). The x-, y-, and z-directions, as shown in Fig. 1(a), reflect the [100], [010], and [001] directions, respectively.

FIG. 1.

(a) Schematic cross section of the equilibrated computational domain for tetragonal CH3NH3PbI3 at room temperature. (b) Green–Kubo predicted thermal conductivity of MAPbI3 as a function of the integration time at 300 K. (Inset) HCACF vs time for MAPbI3. (c) An error of ∼8% to 11% is calculated based on predictions from ten independent simulations (κ = 0.52 ± 0.04 W m−1 K−1).

FIG. 1.

(a) Schematic cross section of the equilibrated computational domain for tetragonal CH3NH3PbI3 at room temperature. (b) Green–Kubo predicted thermal conductivity of MAPbI3 as a function of the integration time at 300 K. (Inset) HCACF vs time for MAPbI3. (c) An error of ∼8% to 11% is calculated based on predictions from ten independent simulations (κ = 0.52 ± 0.04 W m−1 K−1).

Close modal

To assess the mechanical properties, a uniaxial deformation in the three principle (x-, y-, and z-) directions is applied at a strain rate of 108 s−1. During the uniaxial loading, the other periodic boundaries are held under “stress-free” conditions (at 0 bars) with the NPT integration. To generate the stress–strain relationships for our MAPbI3, the stress and strain along the three principle directions are calculated every 0.1 ps.

To understand how the structural and mechanical anisotropy affects the thermal properties of MAPbI3, the thermal conductivities along the three principle directions under different strain conditions are predicted via the Green–Kubo (GK) approach under the equilibrium molecular dynamics (EMD) framework. The thermal conductivity within this framework is calculated as

κx,y,z=1kBVT20Jx,y,z(t)Jx,y,z(0)dt,
(1)

where t is the time, T and V are the temperature and volume of the systems, respectively, and ⟨Jx,y,z(t)Jx,y,z(0)⟩ is the component of the heat current autocorrelation function (HCACF) in the prescribed three principle directions and is given as

J=1Viviεi+iSivi,
(2)

where, vi, εi, and Si are the velocity, energy, and stress of atom i, respectively.64 The total correlation time period for the integration of the HCACF is set to 40 ps. This ensures that the HCACF decays to zero, as shown in the inset of Fig. 1(b). The heat current is computed every ten time steps followed by the integration of the HCACF to calculate the converged thermal conductivity for the MAPbI3 domain. The converged thermal conductivity is determined from the integration from 10 to 40 ps, as shown in Fig. 1(b) (dashed line). Utilizing this procedure, we conduct ten independent simulations under different initial conditions to determine the uncertainty of ∼8% to 11%, as exemplified in Fig. 1(c), showing the distribution of thermal conductivities of our unstrained MAPbI3 (with κ = 0.52 ± 0.04 W m−1 K−1).

Recently, Surblys et al.65 and Boone et al.66 have shown that the implementation of the calculation for per atom stress tensor in the LAMMPS package can produce erroneous results for several multi-body force fields in predicting the correct heat flux. As such, to check the veracity of our results, we perform additional simulations based on the approach-to-nonequilibrium molecular dynamics (AEMD) method. The details of the calculations are provided in Ref. 67. Briefly, for our AEMD calculations, the simulation domains (with simulation domain lengths ranging from ∼10 to 190 nm in the z-direction) are divided into equal halves. A step-like temperature profile is then created in the z-direction, and the systems are evolved in a microcanonical ensemble to observe how the temperature of the atoms relaxes due to the application of the step-like temperature profile. From this, the thermal conductivities are extracted by relating the time evolution of the temperature profile to the thermal diffusivity of our MAPbI3 computational domains. This approach results in considerable size effects in the predicted thermal conductivity with increasing computational domain length (d) in the direction of the applied temperature profile.67 Therefore, the “bulk” thermal conductivity of our MAPbI3 is calculated from the extrapolation to 1/d = 0 from the plot of 1/κ vs 1/d, as shown in Fig. 2. Using this approach, we predict a thermal conductivity of ∼0.61 W m−1 K−1, which agrees reasonably well with our GK-predicted thermal conductivity, as shown in Fig. 2. Furthermore, to check that our GK-predicted thermal conductivity is also independent of size effects, we calculate the thermal conductivity with our GK approach for simulation domains with d varying from 3.7 to 19 nm, as shown in the inset of Fig. 2. The GK-predicted thermal conductivities are similar within uncertainties for all the simulation domain lengths, which shows that our results are independent of the size of the computational domain.

FIG. 2.

The thermal conductivity extrapolated to 1/d = 0 from the approach-to-equilibrium method, which predicts a value of ∼0.61 W m−1 K−1 in agreement with our GK-predicted thermal conductivity. (Inset) Our GK-predicted thermal conductivity for MAPbI3 as a function of computational domain length.

FIG. 2.

The thermal conductivity extrapolated to 1/d = 0 from the approach-to-equilibrium method, which predicts a value of ∼0.61 W m−1 K−1 in agreement with our GK-predicted thermal conductivity. (Inset) Our GK-predicted thermal conductivity for MAPbI3 as a function of computational domain length.

Close modal

Figures 3(a) and 3(b) show the representative stress–strain curves from the uniaxial tensile loading along the y- and z-directions, respectively. Note that the stress–strain relationships in the x- and y-directions are similar; therefore, only the results for the uniaxial loading along the y-direction are shown. The slopes of the linear regions in the stress–strain curves, as shown by the dashed lines, are used to predict Young’s modulus. In agreement with previous results from nanoindentation measurements and first-principles calculations,28,34,68 Young’s modulus for MAPbI3 under uniaxial tensile loading along the y- and z-directions is 13.6 and 8.4 GPa, respectively. It is interesting to note that although plastic deformation initiates at similar strain levels in the two directions, the ultimate tensile strength in the y-direction is higher as compared to that along the z-direction. This anisotropy in the mechanical response can be largely attributed to the anisotropic structure of the inorganic framework, where the lead atoms are bonded to four iodide atoms in the xy-plane, whereas the lead atoms are bonded to only two iodide atoms within the plane parallel to the z axis, as illustrated in the inset of Fig. 3(b). The relative changes in the bond environments along the two orthogonal planes lead to a more compliant mechanical response under uniaxial tension in the z-direction in comparison to that in either the x-direction or the y-direction.

FIG. 3.

Characteristic stress–strain curves under uniaxial tensile loading along (a) y- and (b) z-directions. Note that stress–strain curves for uniaxial loading along x- and y-directions are similar; therefore, we only show the results for uniaxial tension along the y-direction. The slope of the linear elastic region as represented by the dashed line corresponds to Young’s modulus. The resultant strain in the two orthogonal directions vs the applied strain in the (c) y- and (d) z-directions.

FIG. 3.

Characteristic stress–strain curves under uniaxial tensile loading along (a) y- and (b) z-directions. Note that stress–strain curves for uniaxial loading along x- and y-directions are similar; therefore, we only show the results for uniaxial tension along the y-direction. The slope of the linear elastic region as represented by the dashed line corresponds to Young’s modulus. The resultant strain in the two orthogonal directions vs the applied strain in the (c) y- and (d) z-directions.

Close modal

More interestingly, the uniaxial tensile loading simulations reveal another unique anisotropic mechanical behavior of MAPbI3 where the length of the computational domain in the x-direction remains constant, while the length in the z-direction is decreased under uniaxial tensile loading in the y-direction. This is quantitatively shown in Fig. 3(c) where we plot the strain in the two lateral directions as a function of the applied strain in the y-direction. In contrast, when tension is applied in the z-direction, the computational domain shrinks in both the x- and y-directions in response to the applied strain, as shown in Fig. 3(c).

From the stress–strain curves shown in Figs. 3(a) and 3(b), it can be inferred that the mechanism of deformation under uniaxial tensile loading in the two orthogonal directions is also different. While ductile failure occurs under uniaxial tension along the x- or y-directions, a more brittle-like failure is observed at ∼10% strain level under uniaxial loading along the z-direction. This is schematically shown in Figs. 4(a) and 4(b), which shows the computational domains under uniaxial deformation when strain is applied in the y- and z-directions, respectively. To highlight the local deformation mechanisms, the atoms are colored in terms of the associated von Mises strain as calculated in Ref. 69. Figure 4(a) shows the local strain under uniaxial tension along the y-direction at different strain levels. At 5% strain level, no observable stress localization occurs. However, as the strain is increased beyond 10%, stress localization (as represented by the red colored atoms) spreads throughout the structure, which ultimately leads to a ductile failure. As shown by the corresponding figure highlighting the octahedral tilts for ɛy = 0.22 in Fig. 4(a), the ductile behavior in MAPbI3 can be ascribed to the rotation and tilting of the octahedral cages due to the applied strain. This deformation mechanism makes the crystal more compliant and leads to the unique strain response, as shown in Figs. 3(a) and 3(c); although our results do not directly support the hypothesized auxetic behavior in MAPbI3,34 the unique mechanical response where the lateral contraction occurs in one direction, while the length in the other lateral direction does not change, separates these materials from other crystalline solids where it is usually observed that length contractions along both lateral directions occur due to uniaxial tensile force. We note that the discrepancy between our MD results and the negative Poisson’s ratio predicted from first-principles calculations may arise due to the use of our MYP potential to describe the interactions between the atoms in our simulations. It could also be due to the relatively larger simulation domain utilized in our calculations in comparison to that used in the first-principles calculations; the larger simulation domain is able to capture the disorder resulting from the octahedral tilting mechanisms, as shown in Fig. 4(a), where it could potentially result in strain relaxation along the orthogonal direction due to the applied uniaxial tensile force. In contrast, when uniaxial strain is applied in the z-direction, stress localization occurs along a specific plane of atoms parallel to the xy-plane, leading to a more brittle-like failure, as shown in Fig. 4(b). Note that we check for size effects on the mechanical response by considering a domain with a 2× bigger cross-sectional area along with another computational domain with ∼1.7× longer length in the z-direction. For all simulation domain sizes, we obtain identical stress–strain relationships, which suggest that the results shown in Fig. 3 are independent of the size of the computational domain.

FIG. 4.

Snapshots of the cross sections of the CH3NH3PbI3 computational domain showing calculations of atomic level strain relative to the relaxed computational domain at (a) various ɛy and (b) ɛz. Ductile failure dominates beyond the elastic region when the applied uniaxial tensile strain is along either the x- or y-direction. The ductile deformation originates from the distortion of the PbI6 octahedron. However, brittle failure occurs when uniaxial strain is applied along the z-direction.

FIG. 4.

Snapshots of the cross sections of the CH3NH3PbI3 computational domain showing calculations of atomic level strain relative to the relaxed computational domain at (a) various ɛy and (b) ɛz. Ductile failure dominates beyond the elastic region when the applied uniaxial tensile strain is along either the x- or y-direction. The ductile deformation originates from the distortion of the PbI6 octahedron. However, brittle failure occurs when uniaxial strain is applied along the z-direction.

Close modal

Next, to understand how the structural and mechanical anisotropy affect the thermal properties of MAPbI3, the thermal conductivity is calculated with the GK formalism. For the equilibrated computational domain without the application of uniaxial strain, the thermal conductivities along all three principle directions are similar within the 8%–11% uncertainties that are associated with our GK predictions (κ ∼ 0.52 ± 0.04 W m−1  K−1), as shown in Fig. 5(a), and agree well with the experimentally measured room temperature thermal conductivity for MAPbI3.15,19 However, as shown in Fig. 5(a), the application of tensile strain along the x-direction leads to a monotonic reduction in thermal conductivity in both the x- and y-directions (solid circles), whereas the change in the thermal conductivity in the z-direction is negligible (in the elastic region of the stress–strain relationship, which is up to ∼8% strain, hollow squares). Beyond the strain where the ultimate strength of the material is reached, thermal conductivity decreases for all three directions, as shown in Fig. 5(a).

FIG. 5.

(a) Thermal conductivity of MAPbI3 along the xy-plane and z-direction as a function of applied uniaxial strain along the y-direction, ɛy. For comparison, the dashed line shows predictions from the minimum thermal conductivity model. Separate contributions from the (b) inorganic and (c) organic constituents to the total thermal conductivity at ɛy = 0. Separate contributions from the (d) inorganic and (e) organic constituents to the total thermal conductivity at ɛy = 0.08.

FIG. 5.

(a) Thermal conductivity of MAPbI3 along the xy-plane and z-direction as a function of applied uniaxial strain along the y-direction, ɛy. For comparison, the dashed line shows predictions from the minimum thermal conductivity model. Separate contributions from the (b) inorganic and (c) organic constituents to the total thermal conductivity at ɛy = 0. Separate contributions from the (d) inorganic and (e) organic constituents to the total thermal conductivity at ɛy = 0.08.

Close modal

In Fig. 5(a), we also plot the prediction from a minimum thermal conductivity model (κmin, as detailed in previous works in Refs. 24 and 70) for comparison. This model is usually applied (and often correctly predicts the thermal conductivity) for pure amorphous solids where non-propagating vibrations (namely, diffusons and locons) are the dominant heat carriers.71 The main assumptions in calculating κmin for a disordered solid are that the “mean-free-paths” of vibrations in the solid are limited to the spacing between adjacent atoms and the lifetimes of these heat-carrying oscillations are one half the period of vibration (see Refs. 24 and 70 for details). The GK-predicted thermal conductivities for all strain levels, as shown in Fig. 5(a), are well below the prediction from this minimum limit model, suggesting that the explanation of energy propagation through thermal interactions of the order of the vibrational wavelength cannot explain the heat conduction mechanism in MAPbI3 for both the unstrained and strained cases. Instead, the ultralow thermal conductivity of unstrained MAPbI3 has been attributed to strong acoustic-optical phonon scattering that results from the significant overlap in energies of the acoustic and optical phonons.18,72 The application of strain further enhances these scattering mechanisms and leads to an overall decrease in thermal conductivity.

To better understand the effect of uniaxial strain on the anisotropic thermal response in MAPbI3, we separate the contributions from the inorganic and organic constituents [as shown in Figs. 5(b)5(e)] by analyzing the total heat flux. Using Eq. (2), the thermal conductivity contributions from the inorganic and organic constituents can be easily separated by calculating the heat flux for the atoms of the methylammonium cations separately from the atoms forming the inorganic framework (Pb and I atoms). This is shown in Figs. 5(b) and 5(c) for the inorganic and organic constituents, respectively, at 0% strain level. The contributions to the total heat conduction from the inorganic and organic constituents are similar, as shown in Figs. 5(b) and 5(c), for the respective framework. However, at 8% strain, the anisotropy in thermal conductivity originates from the inorganic framework leading to an increased contribution to the total thermal conductivity from the inorganic framework along the z-direction, whereas the contributions from the organic constituents are unaffected by the strain [see Fig. 5(e)].

The change in thermal conductivity upon mechanical loading (up to the elastic region) can be used in applications such as thermal switches and diodes.73 Usually when tensile force is applied to a crystalline solid, the thermal conductivity decreases due to phonon softening.74,75 Upon compression, phonon hardening generally leads to enhanced thermal conductivity. The uniaxial tension along the x- or y-directions in MAPbI3 leads to compression in the z-direction [see Fig. 3(c)]. However, the thermal conductivity in the z-direction remains unchanged in the linear elastic region, which is likely due to the competing effects of phonon hardening that increase thermal conductivity and phonon scattering that leads to a reduction in thermal conductivity. These competing mechanisms can lead to as much as 30% increase in thermal conductivity anisotropy in MAPbI3 as the thermal conductivity monotonically decreases in the x- and y-directions due to uniaxial tension along one of these directions [see Fig. 5(a)]. Taken together with the mechanically compliant nature of hybrid perovskites along certain directions, enhancement in thermal conductivity anisotropy due to strain engineering in these novel materials could be beneficial for applications such as in flexible electronics where preferential energy transfer along selected directions is required.76 

In summary, the mechanical and thermal responses under uniaxial strain of the prototypical hybrid organic–inorganic perovskite, MAPbI3, are investigated via molecular dynamics simulations. It is found that the mechanical response and the deformation mechanism are highly dependent on the direction of the uniaxial strain with a characteristic ductile deformation under uniaxial strain along [100] or [010] directions, whereas brittle failure occurs when the tensile loading is applied along the [001] direction. Furthermore, when uniaxial tension is applied along the [100] or [010] direction, an anisotropic mechanical response is observed in the two lateral directions with a length decrease in the [001] direction and a negligible change in the length along the other lateral direction. The anisotropy in the mechanical response also manifests in an anisotropic thermal response where the thermal conductivity along the length contraction direction remains unchanged during the elastic response under uniaxial tensile strain, whereas the thermal conductivity in the plane of the uniaxial tensile loading [001] decreases monotonically, thus increasing the anisotropy in the thermal conductivity by up to 30%. The combination of these anisotropic physical properties positions hybrid organic–inorganic perovskites as an emerging class of multifunctional materials with potential applications in the development of sensors for pressure detection and shock absorbing materials, such as in “smart” body armors.77,78

This study is based on the work supported by the Office of Naval Research, Grant No. N00014-21-1-2622. The work is also partially supported by the National Science Foundation (NSF Award No. 2119365).

The authors have no conflicts to disclose.

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

1.
J.
Huang
,
Y.
Yuan
,
Y.
Shao
, and
Y.
Yan
, “
Understanding the physical properties of hybrid perovskites for photovoltaic applications
,”
Nat. Rev. Mater.
2
,
17042
(
2017
).
2.
G.
Grancini
and
M. K.
Nazeeruddin
, “
Dimensional tailoring of hybrid perovskites for photovoltaics
,”
Nat. Rev. Mater.
4
,
4
22
(
2019
).
3.
A.
Kojima
,
K.
Teshima
,
Y.
Shirai
, and
T.
Miyasaka
, “
Organometal halide perovskites as visible-light sensitizers for photovoltaic cells
,”
J. Am. Chem. Soc.
131
,
6050
6051
(
2009
).
4.
L.
Dou
,
A. B.
Wong
,
Y.
Yu
,
M.
Lai
,
N.
Kornienko
,
S. W.
Eaton
,
A.
Fu
,
C. G.
Bischak
,
J.
Ma
,
T.
Ding
 et al, “
Atomically thin two-dimensional organic-inorganic hybrid perovskites
,”
Science
349
,
1518
1521
(
2015
).
5.
M.
Saliba
,
T.
Matsui
,
J.-Y.
Seo
,
K.
Domanski
,
J.-P.
Correa-Baena
,
M. K.
Nazeeruddin
,
S. M.
Zakeeruddin
,
W.
Tress
,
A.
Abate
,
A.
Hagfeldt
 et al, “
Cesium-containing triple cation perovskite solar cells: Improved stability, reproducibility and high efficiency
,”
Energy Environ. Sci.
9
,
1989
1997
(
2016
).
6.
J.
Burschka
,
N.
Pellet
,
S.-J.
Moon
,
R.
Humphry-Baker
,
P.
Gao
,
M. K.
Nazeeruddin
, and
M.
Grätzel
, “
Sequential deposition as a route to high-performance perovskite-sensitized solar cells
,”
Nature
499
,
316
319
(
2013
).
7.
H.
Zhu
,
Y.
Fu
,
F.
Meng
,
X.
Wu
,
Z.
Gong
,
Q.
Ding
,
M. V.
Gustafsson
,
M. T.
Trinh
,
S.
Jin
, and
X.-Y.
Zhu
, “
Lead halide perovskite nanowire lasers with low lasing thresholds and high quality factors
,”
Nat. Mater.
14
,
636
642
(
2015
).
8.
H. J.
Snaith
, “
Perovskites: The emergence of a new era for low-cost, high-efficiency solar cells
,”
J. Phys. Chem. Lett.
4
,
3623
3630
(
2013
).
9.
Y.
Fang
,
Q.
Dong
,
Y.
Shao
,
Y.
Yuan
, and
J.
Huang
, “
Highly narrowband perovskite single-crystal photodetectors enabled by surface-charge recombination
,”
Nat. Photonics
9
,
679
686
(
2015
).
10.
M.
Yuan
,
L. N.
Quan
,
R.
Comin
,
G.
Walters
,
R.
Sabatini
,
O.
Voznyy
,
S.
Hoogland
,
Y.
Zhao
,
E. M.
Beauregard
,
P.
Kanjanaboos
 et al, “
Perovskite energy funnels for efficient light-emitting diodes
,”
Nat. Nanotechnol.
11
,
872
877
(
2016
).
11.
B.
Saparov
and
D. B.
Mitzi
, “
Organic–inorganic perovskites: Structural versatility for functional materials design
,”
Chem. Rev.
116
,
4558
4596
(
2016
).
12.
M.
Park
,
H. J.
Kim
,
I.
Jeong
,
J.
Lee
,
H.
Lee
,
H. J.
Son
,
D.-E.
Kim
, and
M. J.
Ko
, “
Mechanically recoverable and highly efficient perovskite solar cells: Investigation of intrinsic flexibility of organic–inorganic perovskite
,”
Adv. Energy Mater.
5
,
1501406
(
2015
).
13.
P.
Docampo
,
J. M.
Ball
,
M.
Darwich
,
G. E.
Eperon
, and
H. J.
Snaith
, “
Efficient organometal trihalide perovskite planar-heterojunction solar cells on flexible polymer substrates
,”
Nat. Commun.
4
,
2761
(
2013
).
14.
M. A.
Haque
,
S.
Kee
,
D. R.
Villalva
,
W. L.
Ong
, and
D.
Baran
, “
Halide perovskites: Thermal transport and prospects for thermoelectricity
,”
Adv. Sci.
7
,
1903389
(
2020
).
15.
A.
Pisoni
,
J.
Jaćimović
,
O. S.
Barišić
,
M.
Spina
,
R.
Gaál
,
L.
Forró
, and
E.
Horváth
, “
Ultra-low thermal conductivity in organic–inorganic hybrid perovskite CH3NH3PbI3
,”
J. Phys. Chem. Lett.
5
,
2488
2492
(
2014
).
16.
T.
Hata
,
G.
Giorgi
, and
K.
Yamashita
, “
The effects of the organic–inorganic interactions on the thermal transport properties of CH3NH3PbI3
,”
Nano Lett.
16
,
2749
2753
(
2016
).
17.
X.
Qian
,
X.
Gu
, and
R.
Yang
, “
Lattice thermal conductivity of organic-inorganic hybrid perovskite CH3NH3PbI3
,”
Appl. Phys. Lett.
108
,
063902
(
2016
).
18.
M.
Wang
and
S.
Lin
, “
Anisotropic and ultralow phonon thermal transport in organic–inorganic hybrid perovskites: Atomistic insights into solar cell thermal management and thermoelectric energy conversion efficiency
,”
Adv. Funct. Mater.
26
,
5297
5306
(
2016
).
19.
G. A.
Elbaz
,
W.-L.
Ong
,
E. A.
Doud
,
P.
Kim
,
D. W.
Paley
,
X.
Roy
, and
J. A.
Malen
, “
Phonon speed, not scattering, differentiates thermal transport in lead halide perovskites
,”
Nano Lett.
17
,
5734
5739
(
2017
).
20.
T.
Liu
,
S.-Y.
Yue
,
S.
Ratnasingham
,
T.
Degousée
,
P.
Varsini
,
J.
Briscoe
,
M. A.
McLachlan
,
M.
Hu
, and
O.
Fenwick
, “
Unusual thermal boundary resistance in halide perovskites: A way to tune ultralow thermal conductivity for thermoelectrics
,”
ACS Appl. Mater. Interfaces
11
,
47507
47515
(
2019
).
21.
R.
Heiderhoff
,
T.
Haeger
,
N.
Pourdavoud
,
T.
Hu
,
M.
Al-Khafaji
,
A.
Mayer
,
Y.
Chen
,
H.-C.
Scheer
, and
T.
Riedl
, “
Thermal conductivity of methylammonium lead halide perovskite single crystals and thin films: A comparative study
,”
J. Phys. Chem. C
121
,
28306
28311
(
2017
).
22.
Y.
Wang
,
R.
Lin
,
P.
Zhu
,
Q.
Zheng
,
Q.
Wang
,
D.
Li
, and
J.
Zhu
, “
Cation dynamics governed thermal properties of lead halide perovskite nanowires
,”
Nano Lett.
18
,
2772
2779
(
2018
).
23.
C.
Caddeo
,
C.
Melis
,
M. I.
Saba
,
A.
Filippetti
,
L.
Colombo
, and
A.
Mattoni
, “
Tuning the thermal conductivity of methylammonium lead halide by the molecular substructure
,”
Phys. Chem. Chem. Phys.
18
,
24318
24324
(
2016
).
24.
A.
Giri
,
A. Z.
Chen
,
A.
Mattoni
,
K.
Aryana
,
D.
Zhang
,
X.
Hu
,
S.-H.
Lee
,
J. J.
Choi
, and
P. E.
Hopkins
, “
Ultralow thermal conductivity of two-dimensional metal halide perovskites
,”
Nano Lett.
20
,
3331
3337
(
2020
).
25.
H.
Ma
,
C.
Li
,
Y.
Ma
,
H.
Wang
,
Z. W.
Rouse
,
Z.
Zhang
,
C.
Slebodnick
,
A.
Alatas
,
S. P.
Baker
,
J. J.
Urban
 et al, “
Supercompliant and soft (CH3NH3)3Bi2I9 crystal with ultralow thermal conductivity
,”
Phys. Rev. Lett.
123
,
155901
(
2019
).
26.
T.
Zhu
and
E.
Ertekin
, “
Mixed phononic and non-phononic transport in hybrid lead halide perovskites: Glass-crystal duality, dynamical disorder, and anharmonicity
,”
Energy Environ. Sci.
12
,
216
229
(
2019
).
27.
A. C.
Ferreira
,
A.
Létoublon
,
S.
Paofai
,
S.
Raymond
,
C.
Ecolivet
,
B.
Rufflé
,
S.
Cordier
,
C.
Katan
,
M. I.
Saidaminov
,
A. A.
Zhumekenov
 et al, “
Elastic softness of hybrid lead halide perovskites
,”
Phys. Rev. Lett.
121
,
085502
(
2018
).
28.
S.
Sun
,
Y.
Fang
,
G.
Kieslich
,
T. J.
White
, and
A. K.
Cheetham
, “
Mechanical properties of organic–inorganic halide perovskites, CH3NH3PbX3 (X = I, Br and Cl), by nanoindentation
,”
J. Mater. Chem. A
3
,
18450
18455
(
2015
).
29.
P.-A.
Mante
,
C. C.
Stoumpos
,
M. G.
Kanatzidis
, and
A.
Yartsev
, “
Directional negative thermal expansion and large Poisson ratio in CH3NH3PbI3 perovskite revealed by strong coherent shear phonon generation
,”
J. Phys. Chem. Lett.
9
,
3161
3166
(
2018
).
30.
A.
Létoublon
,
S.
Paofai
,
B.
Rufflé
,
P.
Bourges
,
B.
Hehlen
,
T.
Michel
,
C.
Ecolivet
,
O.
Durand
,
S.
Cordier
,
C.
Katan
 et al, “
Elastic constants, optical phonons, and molecular relaxations in the high temperature plastic phase of the CH3NH3PbBr3 hybrid perovskite
,”
J. Phys. Chem. Lett.
7
,
3776
3784
(
2016
).
31.
T.
Feng
and
X.
Ruan
, “
Prediction of spectral phonon mean free path and thermal conductivity with applications to thermoelectrics and thermal management: A review
,”
J. Nanomater.
2014
,
206370
.
32.
J.
Yu
,
M.
Wang
, and
S.
Lin
, “
Probing the soft and nanoductile mechanical nature of single and polycrystalline organic–inorganic hybrid perovskites for flexible functional devices
,”
ACS Nano
10
,
11044
11057
(
2016
).
33.
L.
Dong
,
D. S.
Stone
, and
R. S.
Lakes
, “
Softening of bulk modulus and negative Poisson ratio in barium titanate ceramic near the Curie point
,”
Philos. Mag. Lett.
90
,
23
33
(
2010
).
34.
L.-J.
Ji
,
S.-J.
Sun
,
Y.
Qin
,
K.
Li
, and
W.
Li
, “
Mechanical properties of hybrid organic-inorganic perovskites
,”
Coord. Chem. Rev.
391
,
15
29
(
2019
).
35.
E.
Kittinger
,
J.
Tichý
, and
E.
Bertagnolli
, “
Example of a negative effective Poisson’s ratio
,”
Phys. Rev. Lett.
47
,
712
(
1981
).
36.
K. E.
Evans
,
M. A.
Nkansah
,
I. J.
Hutchinson
, and
S. C.
Rogers
, “
Molecular network design
,”
Nature
353
,
124
(
1991
).
37.
K. E.
Evans
,
A.
Alderson
, and
F. R.
Christian
, “
Auxetic two-dimensional polymer networks. An example of tailoring geometry for specific mechanical properties
,”
J. Chem. Soc., Faraday Trans.
91
,
2671
2680
(
1995
).
38.
J. N.
Grima
,
R.
Jackson
,
A.
Alderson
, and
K. E.
Evans
, “
Do zeolites have negative Poisson’s ratios?
,”
Adv. Mater.
12
,
1912
1918
(
2000
).
39.
N.
Pour
,
L.
Itzhaki
,
B.
Hoz
,
E.
Altus
,
H.
Basch
, and
S.
Hoz
, “
Auxetics at the molecular level: A negative Poisson’s ratio in molecular rods
,”
Angew. Chem., Int. Ed.
45
,
5981
5983
(
2006
).
40.
J.-W.
Jiang
,
S. Y.
Kim
, and
H. S.
Park
, “
Auxetic nanomaterials: Recent progress and future development
,”
Appl. Phys. Rev.
3
,
041101
(
2016
).
41.
C.
Huang
and
L.
Chen
, “
Negative Poisson’s ratio in modern functional materials
,”
Adv. Mater.
28
,
8079
8096
(
2016
).
42.
A.
Yeganeh-Haeri
,
D. J.
Weidner
, and
J. B.
Parise
, “
Elasticity of α-cristobalite: A silicon dioxide with a negative Poisson’s ratio
,”
Science
257
,
650
652
(
1992
).
43.
K. L.
Alderson
,
A. P.
Pickles
,
P. J.
Neale
, and
K. E.
Evans
, “
Auxetic polyethylene: The effect of a negative Poisson’s ratio on hardness
,”
Acta Metall. Mater.
42
,
2261
2266
(
1994
).
44.
R.
Lakes
, “
Foam structures with a negative Poisson’s ratio
,”
Science
235
,
1038
1041
(
1987
).
45.
N. R.
Keskar
and
J. R.
Chelikowsky
, “
Negative Poisson ratios in crystalline SiO2 from first-principles calculations
,”
Nature
358
,
222
224
(
1992
).
46.
R. H.
Baughman
,
J. M.
Shacklette
,
A. A.
Zakhidov
, and
S.
Stafström
, “
Negative Poisson’s ratios as a common feature of cubic metals
,”
Nature
392
,
362
365
(
1998
).
47.
L. J.
Hall
,
V. R.
Coluci
,
D. S.
Galvão
,
M. E.
Kozlov
,
M.
Zhang
,
S. O.
Dantas
, and
R. H.
Baughman
, “
Sign change of Poisson’s ratio for carbon nanotube sheets
,”
Science
320
,
504
507
(
2008
).
48.
P.
Mardling
,
A.
Alderson
,
N.
Jordan-Mahy
, and
C. L.
Le Maitre
, “
The use of auxetic materials in tissue engineering
,”
Biomater. Sci.
8
,
2074
2083
(
2020
).
49.
Y. J.
Park
and
J. K.
Kim
, “
The effect of negative Poisson’s ratio polyurethane scaffolds for articular cartilage tissue engineering applications
,”
Adv. Mater. Sci. Eng.
2013
,
853289
.
50.
K. E.
Evans
and
A.
Alderson
, “
Auxetic materials: Functional materials and structures from lateral thinking!
,”
Adv. Mater.
12
,
617
628
(
2000
).
51.
L.
Rothenburg
,
A. A.
Berlin
, and
R. J.
Bathurst
, “
Microstructure of isotropic materials with negative Poisson’s ratio
,”
Nature
354
,
470
472
(
1991
).
52.
L. J.
Gibson
and
M. F.
Ashby
, “
Frontmatter
,” in
Cellular Solids: Structure and Properties
, Cambridge Solid State Science Series, 2nd ed. (
Cambridge University Press
,
1997
), pp.
i
vi
.
53.
F.
Milstein
and
K.
Huang
, “
Existence of a negative Poisson ratio in fcc crystals
,”
Phys. Rev. B
19
,
2030
(
1979
).
54.
G.
Qin
and
Z.
Qin
, “
Negative Poisson’s ratio in two-dimensional honeycomb structures
,”
npj Comput. Mater.
6
,
51
(
2020
).
55.
J. W.
Jiang
and
H. S.
Park
, “
Negative Poisson’s ratio in single-layer black phosphorus
,”
Nat. Commun.
5
,
4727
(
2014
).
56.
J.-W.
Jiang
,
T.
Chang
,
X.
Guo
, and
H. S.
Park
, “
Intrinsic negative Poisson’s ratio for single-layer graphene
,”
Nano Lett.
16
,
5286
5290
(
2016
).
57.
C.
He
,
P.
Liu
, and
A. C.
Griffin
, “
Toward negative Poisson ratio polymers through molecular design
,”
Macromolecules
31
,
3145
3147
(
1998
).
58.
R. H.
Baughman
and
D. S.
Galvão
, “
Crystalline networks with unusual predicted mechanical and thermal properties
,”
Nature
365
,
735
737
(
1993
).
59.
E.
Jin
,
I. S.
Lee
,
D.
Kim
,
H.
Lee
,
W.-D.
Jang
,
M. S.
Lah
,
S. K.
Min
, and
W.
Choe
, “
Metal-organic framework based on hinged cube tessellation as transformable mechanical metamaterial
,”
Sci. Adv.
5
,
eaav4119
(
2019
).
60.
M. R.
Ryder
,
B.
Civalleri
,
G.
Cinque
, and
J.-C.
Tan
, “
Discovering connections between terahertz vibrations and elasticity underpinning the collective dynamics of the HKUST-1 metal–organic framework
,”
CrystEngComm
18
,
4303
4312
(
2016
).
61.
A.
Mattoni
,
A.
Filippetti
,
M. I.
Saba
, and
P.
Delugas
, “
Methylammonium rotational dynamics in lead halide perovskite by classical molecular dynamics: The role of temperature
,”
J. Phys. Chem. C
119
,
17421
17428
(
2015
).
62.
S.
Plimpton
, “
Fast parallel algorithms for short-range molecular dynamics
,”
J. Comput. Phys.
117
,
1
19
(
1995
).
63.
W. G.
Hoover
, “
Canonical dynamics: Equilibrium phase-space distributions
,”
Phys. Rev. A
31
,
1695
(
1985
).
64.
A. P.
Thompson
,
S. J.
Plimpton
, and
W.
Mattson
, “
General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions
,”
J. Chem. Phys.
131
,
154107
(
2009
).
65.
D.
Surblys
,
H.
Matsubara
,
G.
Kikugawa
, and
T.
Ohara
, “
Application of atomic stress to compute heat flux via molecular dynamics for systems with many-body interactions
,”
Phys. Rev. E
99
,
051301
(
2019
).
66.
P.
Boone
,
H.
Babaei
, and
C. E.
Wilmer
, “
Heat flux for many-body interactions: Corrections to LAMMPS
,”
J. Chem. Theory Comput.
15
,
5579
5587
(
2019
).
67.
K. R.
Hahn
,
M.
Puligheddu
, and
L.
Colombo
, “
Thermal boundary resistance at Si/Ge interfaces determined by approach-to-equilibrium molecular dynamics simulations
,”
Phys. Rev. B
91
,
195313
(
2015
).
68.
J.
Feng
, “
Mechanical properties of hybrid organic-inorganic CH3NH3BX3 (B = Sn, Pb; X = Br, I) perovskites for solar cell absorbers
,”
APL Mater.
2
,
081801
(
2014
).
69.
F.
Shimizu
,
S.
Ogata
, and
J.
Li
, “
Theory of shear banding in metallic glasses and molecular dynamics calculations
,”
Mater. Trans.
48
,
2923
2927
(
2007
).
70.
D. G.
Cahill
,
S. K.
Watson
, and
R. O.
Pohl
, “
Lower limit to the thermal conductivity of disordered crystals
,”
Phys. Rev. B
46
,
6131
(
1992
).
71.
P. B.
Allen
and
J. L.
Feldman
, “
Thermal conductivity of glasses: Theory and application to amorphous Si
,”
Phys. Rev. Lett.
62
,
645
(
1989
).
72.
H.
Ma
,
Y.
Ma
,
H.
Wang
,
C.
Slebodnick
,
A.
Alatas
,
J. J.
Urban
, and
Z.
Tian
, “
Experimental phonon dispersion and lifetimes of tetragonal CH3NH3PbI3 perovskite crystals
,”
J. Phys. Chem. Lett.
10
,
1
6
(
2019
).
73.
N.
Li
,
J.
Ren
,
L.
Wang
,
G.
Zhang
,
P.
Hänggi
, and
B.
Li
, “
Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond
,”
Rev. Mod. Phys.
84
,
1045
1066
(
2012
).
74.
K. D.
Parrish
,
A.
Jain
,
J. M.
Larkin
,
W. A.
Saidi
, and
A. J. H.
McGaughey
, “
Origins of thermal conductivity changes in strained crystals
,”
Phys. Rev. B
90
,
235201
(
2014
).
75.
A.
Giri
,
J. L.
Braun
, and
P. E.
Hopkins
, “
Reduced dependence of thermal conductivity on temperature and pressure of multi-atom component crystalline solid solutions
,”
J. Appl. Phys.
123
,
015106
(
2018
).
76.
Z.
Cheng
,
M.
Han
,
P.
Yuan
,
S.
Xu
,
B. A.
Cola
, and
X.
Wang
, “
Strongly anisotropic thermal and electrical conductivities of a self-assembled silver nanowire network
,”
RSC Adv.
6
,
90674
90681
(
2016
).
77.
A. L.
Goodwin
,
D. A.
Keen
, and
M. G.
Tucker
, “
Large negative linear compressibility of Ag3[Co(CN)6]
,”
Proc. Natl. Acad. Sci. U. S. A.
105
,
18708
18713
(
2008
).
78.
W.
Li
,
M. R.
Probert
,
M.
Kosa
,
T. D.
Bennett
,
A.
Thirumurugan
,
R. P.
Burwood
,
M.
Parinello
,
J. A. K.
Howard
, and
A. K.
Cheetham
, “
Negative linear compressibility of a metal–organic framework
,”
J. Am. Chem. Soc.
134
,
11940
11943
(
2012
).