Metal-halide perovskites under pressure: Effect of anisotropic deformation on optoelectronic properties

Organic lead halide perovskite materials have garnered significant attention for their promising applications in optoelectronics, including perovskite solar cells (PSCs) and photodetectors.^1–4 The power conversion efficiency (PCE) of PSCs has increased from 3.8% to over 25% in the past decade, approaching that of the commercial single-junction silicon-based solar cells.^5–8 This rapid advancement is due to their unique characteristics, including a soft lattice structure⁹ and sensitivity to external forces such as pressure and temperature, distinguishing them from traditional semiconductors.^10–12 The ability to manipulate these materials under varying pressure conditions is particularly relevant, as such pressures can arise in real-world applications, including impacts and extreme environmental conditions. Investigating the physical properties of organic lead halide perovskite materials under external pressure is, therefore, crucial for exploring their application prospects and developing new optoelectronic devices. MAPbI₃, as a representative of perovskites, has been widely studied. However, its relatively large bandgap of around 1.50 eV greatly limits its potential for high-efficiency photovoltaic performance.¹³ Among these materials, FAPbI₃ stands out due to its optimal bandgap of ∼1.43 eV,^14,15 which is closer to the Shockley–Queisser limit of 1.40 eV for maximum photovoltaic efficiency.¹⁶ Despite its advantages, the inherent instability of FAPbI₃ remains a significant barrier to its commercial application.¹⁴ Previous studies have shown that FAPbI₃ undergoes phase transitions and exhibits changes in photovoltaic performance under hydrostatic pressure.^17–19 Jiang et al. reported that at pressures below 0.1 GPa, the cubic phase FAPbI₃ irreversibly transforms into the hexagonal phase δ-FAPbI₃.¹⁷ This transformation follows a phase transition sequence from $Pm3̄m$ → P4/mbm → $Im3̄$⁠, ultimately resulting in a polycrystalline state where the original structure does not fully recover after decompression.¹⁷ These stress-induced phase transitions in FAPbI₃ have been further supported by theoretical simulations, offering a detailed understanding of this behavior.^20–23 To explore these phenomena, first-principles calculations can provide detailed insights into the structural, electrical, optical, and mechanical properties of FAPbI₃ under pressure. Previous research indicates that pressure can induce significant changes in the material’s properties, potentially enhancing its photovoltaic performance by modulating the bandgap and increasing PCE. Our results show that applying pressure of less than 5 GPa can appropriately improve the photoelectric properties of FAPbI₃, making the bandgap transition from direct to indirect. In addition, we observed that the flexibility of FAPbI₃ increases with the application of hydrostatic pressure while its structural stability decreases. In summary, our research results provide theoretical insights and suggestions for understanding the physical processes of phase transitions and stability degradation of FAPbI₃ under hydrostatic pressure.

Figure S1 illustrates the unit cell of cubic FAPbI₃ crystal. The optimized lattice parameters are a = 6.41 Å, b = 6.25 Å, c = 6.31 Å, and α = β = γ = 90° (see Table S1), which are compatible with experimental results: a = 6.361 Å, b = 6.508 Å, c = 6.323 Å, and α = β = γ = 90°.²⁴ The small variation in the calculated lattice parameters compared to the experimental data is attributed to the reduced symmetry of the crystal structure caused by the organic molecule FA⁺. The stable FAPbI₃ crystal structure was obtained by optimizing the lattice constant. Upon comparison, the optimized FAPbI₃ structure parameters closely matched experimental and theoretical predictions. This alignment reinforces our confidence in the comprehensive and accurate cubic-FAPbI₃ structural parameters we have established, ensuring their rigorous validation and their reliable use in subsequent predictions of pressure-induced properties.^25–28 However, since the isometric system construction method was used, resulting in equal lattice constants and perpendicular central axes. A pseudo-cubic structure of FAPbI₃ was confirmed, potentially leading to anisotropic optoelectronic properties.²⁹ An asymmetric structure and polarity from the organic cation (FA⁺) result in unequal force in all directions. To investigate the anisotropic properties of FAPbI₃, the total energies were calculated for different FA⁺ rotation angles around the crystal directions of a (100), b (010), and c (001), as shown in Figs. 1(a) and 1(b). Our results indicate that the rotation energy of the FA⁺ cation exhibits a periodic variation of 180° in all crystal directions, implying that FAPbI₃ can revert to its original structure by rotating the FA⁺ cation by 180° [Fig. 1(b)]. Meanwhile, we observed that the rotation energy of FA⁺ differs across the three crystal directions, with the highest energy observed in the FA⁺ (001) direction and lower energies in the FA⁺ (100) and FA⁺ (010) directions. These findings support previous research on the impact of the FA⁺ direction on the anisotropy and symmetry of the crystal structure.^30–32 The anisotropy of the FA⁺ cation in FAPbI₃ perovskite may result in inconsistent stability in all directions. To better understand the stability, we present a schematic of the effect of hydrostatic pressure on the FAPbI₃ structure in Fig. 1(a). Our results indicate that the lattice constants, Pb–I bond lengths, and Pb–I–Pb angles of the FAPbI₃ perovskite vary differently under different directions due to the incomplete symmetry of the FA+ cation, as illustrated in Figs. 1(c)–1(e). Notably, all attenuation trends exhibit a curvature inflection point at a hydrostatic pressure of 5 GPa, suggesting that the FAPbI₃ structure undergoes a phase transition at this pressure (Table S2). Similar phenomena were observed in pressure-related experiments on FAPbI₃ perovskite.^33–35 The phase transition of the FAPbI₃ structure was further confirmed by the changes in the slope of the normalized volume with increased hydrostatic pressure (Fig. S2). Our findings indicate that the FAPbI₃ structure undergoes lattice distortion and decreased structural symmetry under increased hydrostatic pressure, leading to a phased transition from cubic to syncline square structure. The phase transition was also calculated using existing semi-empirical volume parameter methods.³⁶ These lattice distortions affect the orientation of atoms and the degree of octahedral torsion, ultimately resulting in a phased structural transition when the hydrostatic pressure exceeds a particular value. To further analyze the torsional angles of the octahedrons, we calculate the included angles between octahedrons along different crystal directions under hydrostatic pressure in Fig. 1(f). The attenuation trend of the octahedral torsion angle is consistent with that of the Pb–I–Pb bond angles in Fig. 1(e). The inflection points of these angles are at hydrostatic pressures of 0.3, 0.6, and 2 GPa, respectively, especially when the hydrostatic pressure exceeds 5 GPa. The octahedral structure, belonging to the equiaxial crystal system, has the highest symmetry. The change in the octahedral angle destroys the high symmetry and causes the crystal to transform into a low-symmetry structure.³⁷ In summary, it is demonstrated that cubic FAPbI₃ experiences structural imbalance and eventual breakdown under extreme conditions, such as high pressure, due to the failure of crystal symmetry. The Pb–I–Pb angles and octahedral angles indicate that the degradation of the FAPbI₃ crystal structure does not follow a strictly linear relationship with increasing pressure. Instead, the crystal structure first changes gradually and then accelerates after a critical point (5 GPa), leading to structural imbalance.

The anisotropy and phase transition of FAPbI₃ are further depicted in Fig. 2, which displays the 2D charge density diagram along different directions.⁹ An apparent mixture of ionic and covalent bonds is observed in the charge distribution around the Pb or I atoms. The strength of Pb–I bond and structural anisotropy can be identified by the contour line in the charge density diagram. A slightly different in charge transport was observed between the Pb–I bonds in the FA⁺ (010) direction compared with the other directions. Under the absence of hydrostatic pressure, the contour line expended more in the (100) direction than in the other directions. The Pb–I bond lengths were optimized along the a (100), b (010), and c (001) directions and were found to be 3.21, 3.12, and 3.16 Å, respectively. This implies that the stronger Pb–I bond occurred in the b (010) direction due to the shorter Pb–I bond length. The anisotropy of the FAPbI₃ structure is indicated by the differences in Pb–I bond lengths and contour lines in three directions. With increased hydrostatic pressure, stronger covalent bond properties occur as the Pb–I contour lines begin to coincide, resulting in the overlap of their electron clouds. Notably, the Pb–I contour lines joined when the hydrostatic pressure reached 3 GPa. The increase in Pb–I bond strength in the FA⁺ (010) direction was slower than in the other directions due to the reduction of FAPbI₃ structural symmetry with the increasing hydrostatic pressure. The decreased symmetry leads to unequal results in three crystal planes, consistent with the previous conclusion that hydrostatic pressure causes the cubic structure to change to a less symmetrical structure.

The electronic properties of FAPbI₃ are influenced by changes in the lattice structure under hydrostatic pressure. The analysis of the electron distribution at the Fermi energy level was based on the density of states (DOS) of the FAPbI₃ structure, as shown in Fig. 3(a). The conduction band minimum (CBM) of FAPbI₃ was found to be contributed by Pb–p orbitals and a small number of I–s orbitals, while their valence band maximum (VBM) was contributed by Pb–s and I–p orbitals [Figs. 3(a) and 3(b)]. The corresponding projected band structures of FAPbI₃ were calculated using the Brillouin zone path of G–X–M–G–R–X [Fig. 3(d)]. At the higher symmetry point R, a direct bandgap of 1.31 eV was found in cubic FAPbI₃, consistent with an experimental value of 1.43 eV [Fig. 3(c)]. The introduction of hydrostatic pressure from 0 to 5 GPa resulted in a decrease in the bandgap of FAPbI₃ from 1.31 to 0.69 eV with a direct bandgap at the R point, as shown in Fig. 3(e) and Fig. S3. This decrease in bandgap was due to the enhanced orbitals coupling effect with the increase in hydrostatic pressure, resulting in a stronger sp orbitals coupling effect.³⁸ The bandgap increased with the weaker orbital coupling effect because the impact of the Pb–I–Pb bond angle dominated when the hydrostatic pressure exceeded 5 GPa. As a result, the band structure of FAPbI₃ transformed from a direct to an indirect bandgap [Fig. 3(e)].

To accurately describe atomic charge transport, the Bader charges of FAPbI₃ with the increasing hydrostatic pressure were listed in Table S3. It was observed that both FA⁺ and Pb²⁺ lost electrons, while the I⁻ gained electrons with the increase in hydrostatic pressure.³⁹ This indicates that the Pb–I interactions weaken, leading to a decreased bandgap and reduced stability of FAPbI₃. Furthermore, the optical absorption properties of the FAPbI₃ under different hydrostatic pressures were calculated in Fig. 3(f) to reflect their bandgap change. The absorption curve for FAPbI₃ without hydrostatic pressure covered the visible spectrum and the near-infrared region, indicating its photo active nature. With the increase in hydrostatic pressure, the absorption strength was enhanced due to the increased absorption coefficient caused by the reduced Pb–I bond length and increased overlap of atomic electron clouds. In addition, the lattice contraction due to the increase in hydrostatic pressure caused a slight red shift in the absorption edge under the range of 0–5 GPa. This was because the bandgap was reduced with the increased s orbitals closed to the VBM by the effect of Pb–I bond length. However, when the hydrostatic pressure exceeds 5 GPa, the increase in bandgap leads to a blue shift of the absorption edge. This is attributed to the rise in the bandgap value of FAPbI₃, transforming its electronic structure from a direct to an indirect bandgap—an undesirable outcome. Moreover, factors influencing the position of the absorption edge can also be attributed to the effects of hydrogen bonding and orbital hybridization. The increase in hydrostatic pressure leads to a decrease in the N–H bond length within the organic cation FA⁺, thereby enhancing intramolecular hydrogen bonding. This effect results in a reduction in stretching frequency and significantly impacts the spectral peak positions. Orbital hybridization, considered a means of altering the electronic structure of the system, can induce a redshift (shift toward longer wavelengths) or blueshift (shift toward shorter wavelengths) in the absorption peaks. In summary, applying a certain level of hydrostatic pressure to the FAPbI₃ system can be an effective means of controlling optical absorption. This, in turn, has the potential to further enhance the efficiency of FAPbI₃ as the absorber layer in perovskite solar cells.

The geometric structure of FAPbI₃ has a significant impact on its thermal and mechanical properties, both of which directly reflect the material’s stability. Although some experimental and theoretical studies have reported certain thermodynamic properties of the cubic phase of FAPbI₃,^40,41 current reports and analyses have not delved deeply enough into a comprehensive understanding. Therefore, this study focuses on the decomposition enthalpy of FAPbI₃ under different hydrostatic pressure conditions to provide an in-depth analysis of its thermal stability. As shown in Table S2, under zero hydrostatic pressure, the decomposition enthalpy is negative, indicating that the decomposition of FAPbI₃ is exothermic. This suggests that, after structural optimization, FAPbI₃ exhibits a pseudo-cubic phase structure, which may undergo spontaneous decomposition. From a thermodynamic perspective, this pseudo-cubic phase of FAPbI₃ exhibits poor long-term stability and inevitable phase separation, consistent with many experimental and computational results.^42,43 With the application of hydrostatic pressure, the decomposition enthalpy of the FAPbI₃ phase reaction becomes more negative, indicating that hydrostatic pressure reduces the system’s stability and accelerates the phase decomposition of FAPbI₃. This conclusion aligns with previous research, emphasizing the influence of hydrostatic pressure in regulating the stability of FAPbI₃. In summary, through an in-depth study of thermodynamic properties and elastic constants, we can comprehensively understand the stability changes in FAPbI₃ under different conditions, providing crucial insights for optimizing material performance.

Mechanical stability depends on elastic constants, reflecting a material’s bond strength, flexibility, stiffness, and stability. The accurate calculation of these constants is critical for practical applications and device applications in extreme environments. The cubic phase FAPbI₃ structures has three independent matrix elements (C₁₁, C₁₂, and C₄₄), detailed in Table S4 of the supplementary material. C₁₁ represents the resistance to linear compression along the a-axial, and C₄₄ represents the resistance to shear stress along the same a-axis in this constant elastic matrix. It is noted that optimized FAPbI₃ structures exhibit a pseudo-cubic phase, resulting in distinct values of C₁₁, C₂₂, and C₃₃ and unequal values of C₄₄, C₅₅, and C₆₆. Figure 4(b) illustrates the dependence of the elastic constants C₁₁, C₂₂, and C₃₃ of FAPbI₃ structures on pressure. All elastic constants (C_ii) associated with linear compression resistance increase with hydrostatic pressure from 0.5 to 3.0 GPa. Importantly, C₁₁ exhibits a positive linear correlation with hydrostatic pressure, adhering to Hooke’s law. In contrast, C₂₂ and C₃₃ exhibit a declining trend with hydrostatic pressure from 2.0 to 3.5 GPa. Notably, when hydrostatic pressure is above 2.5 GPa, C₂₂ surpasses both C₁₁ and C₃₃, and the reverse is observed for pressures below C₁₁. These observations imply that FAPbI₃’s compression is more robust along the b-direction than along the a and c directions under milder hydrostatic pressures. For the cubic phase FAPbI₃ structures, the inequality of C₁₁ and C₄₄ indicates directional dependence and anisotropic mechanical properties. These elastic modulus vary in different directions and can be evaluated by the anisotropy constant (A). When the anisotropy index A is 0, the material is more anisotropic, resulting in a steeper peak in the 3D Young’s modulus diagram. Figure S4 depicts 3D diagrams of Young’s modulus, shear modulus, and Poisson’s ratio for the FAPbI₃ structure without hydrostatic pressure, demonstrating the anisotropy of the material. The 3D Young’s modulus was projected from three directions to draw the surface profiles in their 2D anisotropic, as shown in Fig. 4(a). The optimized pseudo-cubic FAPbI₃ is an un-equiaxed crystal, leading to more obvious anisotropy in mechanical properties between the plane perpendicular to (010) and the other two.

To illustrate the anisotropy of the FAPbI₃ structure under hydrostatic pressure, various material parameters were calculated, including the elastic constants (C_ii), bulk modulus (B), shear moduli (G), Young’s moduli (E), Poisson’s ratio (ν), and the anisotropy constant (A), as detailed in Table S4. The Voigt, Reuss, and Voigt–Reuss–Hill approximations were employed to determine the upper limit (B_V G_V), lower bound (B_R G_R), and average of the actual effective modulus (B_H G_H), respectively. The B of the FAPbI₃ structure increased with hydrostatic pressure from 0 to 3.5 GPa, indicating enhanced resistance to volume change under pressure loading [Fig. 4(c)]. However, when hydrostatic pressure exceeds 0.5 GPa, the elastic constants C₄₄ and C₆₆ become negative, indicating structural instability. Therefore, Fig. 4(d) displays only vertical G and E graphs in the 0–1 GPa range to investigate the relationship between the resistance to reversible deformation and shear stress. Both the shear modulus and elastic modulus decreased with increasing pressure. Negative values of G_R for FAPbI₃ further indicated its unstable structural characteristics at 0.5 and 1 GPa. Moreover, the anisotropy of the FAPbI₃ structures was significantly affected by hydrostatic pressure, as shown in Fig. 4(e). As the hydrostatic pressure increased from 0 to 2 GPa, the A₀₁₀ along the (010) direction did not change significantly but increased beyond 2 GPa. The shear anisotropy factors along (100) and (001) directions were smaller than those along the (010) direction, suggesting a weaker anisotropy degree of bonds from atoms along the (010) direction. Considering the crucial importance of ductility and brittleness for material application, the Poisson’s ratios (υ) of FAPbI₃ under different hydrostatic pressures were calculated and displayed in Fig. 4(f). The critical value of ∼0.26 distinguishes ductile from brittle materials. All υ values for the FAPbI₃ structure under different hydrostatic pressures satisfied the ductile criterion with values greater than 0.26. Moreover, the υ of the FAPbI₃ decreased as the hydrostatic pressure increased from 2 to 2.5 GPa but increased in other cases, suggesting that the ductility of FAPbI₃ can be improved by applying hydrostatic pressure. In summary, the interaction among hydrostatic pressure, band structure, and mechanical properties highlights the complex and diverse nature of FAPbI₃ single crystals. This comprehensive understanding, revealing the interdependence of electronic, mechanical, and thermodynamic aspects, contributes to unveiling the interesting behavior of the material under varying pressure conditions.

The crystal structure, electron configuration, optical absorption characteristics, and stability of FAPbI₃ were investigated using first-principles calculations. A pseudo-cubic phase of FAPbI₃ was obtained through the optimization of its cubic phase. With increasing hydrostatic pressure, discernible reductions were observed in the lattice constant, bond length, and octahedral angle of FAPbI₃. This pressure-induced effect facilitated structural phase transitions, leading to an alteration in the bandgap from direct to indirect. A decrease in charge transfer for Pb–I and FA–I was identified with increased hydrostatic pressures, indicating a weakening of their interaction. The optical performance of FAPbI₃ exhibited a redshift under hydrostatic pressures ranging from 0 to 5 GPa, resulting in a corresponding enhancement in material absorption. In addition, the pseudo-cubic structure of FAPbI₃ induced deviations in C₁₁, C₂₂, and C₃₃. Critically, the elastic constants of C₄₄ and C₆₆ exhibited negativity beyond 0.5 GPa, indicating a reduction in structural stability under pressure. The anisotropy and ductility of FAPbI₃ demonstrated an increase with escalating hydrostatic pressure. In conclusion, FAPbI₃ exhibits significant tunability within moderate pressure ranges, with increased anisotropy and ductility, highlighting its tremendous potential for optoelectronic applications in extreme environments. The pressure-induced changes in its properties provide new perspectives and possibilities for developing highly efficient and stable optoelectronic materials under harsh conditions.

The supplementary material associated with this article can be found on the online version of the website.

This work was financially supported by the National Key Research and Development Program of China (Grant Nos. 2021YFA0715600, 2023YFB3609900), the National Natural Science Foundation of China (Nos. 52192610, 61704131, 62304171, 62374128, and 61804111), the Joint Research Funds of Department of Science and Technology of Shaanxi Province and Northwestern Polytechnical University (No. 2020GXLH-Z-018), the Postdoctoral Science Foundation of China (No. 2022M722500), Key Research and Development Program of Shaanxi Province (Grant 2024GX-YBXM-512), and the State Key Laboratory of Advanced Technology for Materials Synthesis and Processing at Wuhan University of Technology (No. 2024-KF-12).

The authors have no conflicts to disclose.

Siyu Zhang: Software (equal); Writing – original draft (lead); Writing – review & editing (lead). Mengyu Liu: Writing – review & editing (equal). Jie Su: Data curation (lead); Writing – original draft (supporting). Zhenhua Lin: Data curation (supporting); Writing – review & editing (supporting). Haidong Yuan: Conceptualization (supporting); Writing – original draft (supporting). Lixin Guo: Conceptualization (equal); Data curation (equal); Writing – original draft (equal). Yue Hao: Funding acquisition (equal). Jingjing Chang: Conceptualization (equal); Data curation (equal); Writing – original draft (equal).

The data that support the findings of this study are available within the article and its supplementary material.