Fig. 4: Preservation of the single crystallinity of the SRO flexible membranes.

a X-ray diffraction θ−2θ scans show the (002) diffraction peaks of SRO near 2θ = 45.9−46.1°, as guided by a dashed line, for \({{{\mathrm{SRO}}}}_{{{{\mathrm{PET}}}}}^{30{{{\mathrm{nm}}}}}\), \({{{\mathrm{SRO}}}}_{{{{\mathrm{SAO}}}}/{{{\mathrm{STO}}}}}^{30{{{\mathrm{nm}}}}}\), and \({{{\mathrm{SRO}}}}_{{{{\mathrm{STO}}}}}^{30{{{\mathrm{nm}}}}}\); SAO and STO indicate a Sr₃Al₂O₆ sacrificial layer and the SrTiO₃ substrate, respectively. b Full widths at half maximum of the normalized ω scan increased from 0.14° \(({{{\mathrm{SRO}}}}_{{{{\mathrm{STO}}}}}^{30{{{\mathrm{nm}}}}})\) to 2.33° \(({{{\mathrm{SRO}}}}_{{{{\mathrm{PET}}}}}^{30{{{\mathrm{nm}}}}})\). For convenience, we used the normalized intensity \(( = \frac{I}{{I_{{{{\mathrm{max}}}}}}})\) and Δω (= ω − ω_c), where I_max is the maximum intensity at ω_c. c Scanning transmission electron microscopy images show few mechanical fractures over a wide area of \({{{\mathrm{SRO}}}}_{{{{\mathrm{PET}}}}}^{30{{{\mathrm{nm}}}}}\); energy-dispersive X-ray spectroscopy measurements indicate a uniform distribution of Sr and Ru.