Extended Data Fig. 9: Quantitative estimation of the effect of mistilt on the analysis of c/a.
From: Freestanding crystalline oxide perovskites down to the monolayer limit
To quantitatively estimate the effect of the degree of mistilt on the analysis of c/a, we calculated the simulated HAADF-STEM images of a hypothetical cubic BiFeO3 crystal as a function of the tilting angle around the [100] axis (a), and [010] axis (b), respectively, using a multislice simulation code called QSTEM. The thickness of the BFO model was about 20 nm (50 unit cells). As shown in a and b, within a few degrees of the tilt, the atomic columns in STEM images are noticeably elongated along the direction perpendicular to the rotation axis. Using the same two-dimensional Gaussian fitting procedure as that one used for analysing the experimental data in Fig. 3, the deviations of the fitted lattice constants from the nominal ones are negligible (c). The maximum deviation of the calculated c/a ratio is less than 0.002, which is two orders of magnitude smaller than the increment of the c/a ratio (1.22) observed in the ultrathin freestanding BFO films. The electron energy was 300 kV, the probe convergence angle was 25 mrad, and the angular range of the HAADF detector was 79.5 mrad to 200 mrad in the simulation, values consistent with the experiment. The error bars represent the fitting error of the lattice constants.
