Fig. 1. （a） The plot of estimated X-ray penetration depth versus incident angles in FAPbI₃^［19］； （b） diffraction geometries of the depth-dependent strain distribution measurement. The relation between instrument reference （Z） frame and diffraction vector （α and β are the incident and exit angles of X-rays， defined as the angle between the incident/diffracted X-rays and the sample surface）；（c） schematic illustration of the residual strain distribution measurement. The corresponding XRD patterns and lattice structure strain information can be obtained by fixing the test crystal plane and adjusting the instrument tilt angle ψ， where N₀ is the sample normal direction and N_k is the diffraction vector^［11］

Fig. 2. （a） Schematic diagram of surface strain state； （b） GIXRD patterns at different tilt angles at the depth of 50 nm for the tensile-strained film； （c） residual strain distribution in the depth of 50， 200， 500 nm for the tensile-strained film； （d） GIXRD patterns at different tilt angles at the depth of 50 nm for the strain-free film； （e） residual strain distribution in the depth of 50， 200， 500 nm for the strain-free film； （f） GIXRD patterns at different tilt angles at the depth of 50 nm for the compressive strained film； （g） residual strain distribution in the depth of 50， 200， 500 nm for the compressive strained film^［11］

Fig. 3. Schematic diagram of strain. （a） Out-of-plane XRD patterns of the MAPbI₃ annealed film （AF）， scraped powder （SP）， single-crystal powder （SCP）， and non-annealed film （NAF）； （b） in-plane and out-of-plane XRD patterns of AF and out-of-plane XRD pattern of SCP； （c） schematic diagram of the out-of-plane structure； （d） schematic diagram of the in-plane structure^［10］

Fig. 4. （a） The crystal under growing is free of cracks； （b） solution-grown CsPbBr₃ single crystals with cracks， （c） paralleled domains indicate the formation of twin boundaries in CsPbBr₃； （d） a photograph of several as-grown FA _x Cs_1-x PbBr₃ single crystals from different precursor solutions with Cs/（Cs + FA） ratios varying from 0% to 100%^［20］

Fig. 5. Schematic diagram and XRD patterns of the strain formation process. （a） Without substrate， the perovskite forming at 100 °C contracts vertically and laterally during cooling； （b） with the substrate adhesion， the annealed perovskite film only contracts vertically， in situ out-of-plane XRD of scraped powder （c） and annealed film （d） at different temperatures^［10］

Fig. 6. Epitaxial α-FAPbI₃ thin films and structural characterizations. （a） Optical images of the as-grown epitaxial α-FAPbI₃ thin films； （b） a cross-sectional scanning electron microscope （SEM） image of the epitaxial thin film with controlled uniform thickness； （c） high-resolution XRD ω-2θ scan of the （001） peaks of the epitaxial samples on different substrates showing the increasing tetragonality with increasing lattice mismatch； （d） reciprocal space mapping with （104） asymmetric reflection of the α-FAPbI₃， for different lattice mismatches with the substrate^［9］

Fig. 7. （a） Cross-sectional TEM image of device； （b） PL depth profile of confocal fluorescence microscope， the inset represents TOF-SIMS depth profiles of the （FAPbI₃）_0.85（MAPbBr₃）_0.15 perovskite film with tensile strain^［11］

Fig. 8. （a） Residual strain distribution in the depth of 100， 300， 500 nm for PEA₂PbBr₄ single crystal （measured （points） and Gauss fitted （line） diffraction strain data as a function of sin²φ）； （b） the plot of lattice constant versus penetration depth； （c） the results of the layer-decomposed density of state calculation for PEA₂PbI₄ with and without （PEA⁺-I^-） defect and schematics of the model structures vertically aligned with the layers； （d） N 1s XPS of the surface and interior of two-dimensional perovskites PEA₂PbI₄^［28］

Fig. 9. Strain-induced electronic structure analysis. （a） Calculated band structures under biaxial tensile， zero， and compressive strains from first-principle DFT-based approaches， the band structure alignment is made by using the vacuum energy level as reference； （b） the evolution of band-edge energies under gradually increasing tensile strains in perovskite films （left panel）， and the schematic of the band alignment between tensile strain/strain-free film and hole transport layer in solar cell； （c） UV-Vis absorption spectra and PL spectra under tensile strain， strainfree， and compressive strain conditions^［11］

Fig. 10. The energy transfer process between the surface and inside excitons of two-dimensional perovskite. （a） Time-resolved PL kinetics collected in the emission channels of 500~590 nm； （b） normalized PL spectra with the different time delays of 2D-perovskites； （c） normalized PL decay curves monitored at 520 and 559 nm wavelength， respectively； （d） schematic illustration of the energy level and exciton funneling process of the self-wavelength shifting in two-dimensional perovskites^［28］

Fig. 11. （a） Effective masses of the carriers at different strains， and electronic bandstructures under three strain levels （3%， 0% and -3%）； （b） hole mobilities by Hall effect measurements showing that α-FAPbI₃ with strain of -1.2% has the highest hole mobility； （c） transient photocurrent curves of the epitaxial α-FAPbI₃ under different strains； （d） plots of calculated carrier mobilities as a function of the strain magnitudes^［9］