A statistical approach for interplanar spacing metrology at a relative uncertainty below 10⁻⁴ using scanning transmission electron microscopy

Atomic-scale metrology in scanning transmission electron microscopy (STEM) allows to measure distances between individual atomic columns in crystals and is therefore an important aspect of their structural characterization. Furthermore, it allows to locally resolve strain in crystals and to calibrate precisely the pixel size in STEM. We present a method dedicated to the evaluation of interplanar spacing (d-spacing) based on an algorithm including curve fitting of processed high-angle annular dark-field STEM (HAADF STEM) signals. By examining simulated data of perovskite cubic SrTiO₃, we confirm that our proposed method is unbiased, and the precision is better than the significant digit of the input value. Then, we study experimental data to learn how electron dose, sampling resolution, and statistical sampling affect the mean and precision values of d₁₁₀. For single d-spacing measurements using a probe corrected STEM, we find that uncertainty ranges between 1 and 3 pm. Here, we measure numerous d-spacings in an automated and statistical approach, resulting in relative uncertainties in mean values ≤ 10⁻⁴. Thus, we propose to calibrate TEMs using this method as it enables measuring lattice parameters at uncertainties comparable to reports of x-ray diffraction measurements, but with a significantly lower sample volume, in this case ∼ 10⁻³ µm³.