High-angle scattering of fast electrons from crystals containing heavy elements: Simulation and experiment

The paper reports on quantitative comparisons of experimental and simulated image intensities in high-angle annular dark-field imaging in scanning transmission electron microscopy of a ${\text{PbWO}}_4$ single crystal. The experimental image intensities are normalized to the incident beam. Provided that the effects of spatial incoherence and multiple thermal diffuse scattering events are taken into account in the simulations, excellent agreement within about 5% is achieved between theory and experiments. The comparisons depend critically on accurate knowledge of the Debye-Waller factors, which were determined by x-ray single-crystal structure refinement, and of the experimental thickness values. The Debye-Waller factors are different for different atomic columns, causing column intensities to not be a simple function of their atomic number. Channeling effects associated with the oxygen columns contribute to intensity variations between the Pb and W columns. The results show that even for single crystals, image simulations are required to correctly interpret the contrast.

Figure 1

(Color online) (Top row) Simulated intensity line scan along [001] in a [100] projection for a 17.5-nm-thick ${\text{PbWO}}_4$ sample without thermal scattering (dashed line) and with thermal diffuse scattering (solid line). The effects of spatial incoherence are not taken into account. The line scan is across the Pb and W column pairs. Note that after including thermal diffuse scattering some of the W columns can appear brighter than the Pb columns, opposite to what might be expected from their atomic number. A [100] projection of the scheelite structure is shown in the bottom row.

Figure 2

(Color online) Frozen phonon simulations of the maximum and minimum (“background”) image intensities for ${\text{SrTiO}}_3$ and ${\text{PbWO}}_4$ as a function of thickness. The signals increase about twice as fast with thickness for ${\text{PbWO}}_4$ compared to ${\text{SrTiO}}_3$. The effects of spatial incoherence are not taken into account.

Figure 3

(Color online) Simulated (top half) and experimental (bottom half) PACBED patterns along [100] for a 14-nm-thick region of ${\text{PbWO}}_4$.

Figure 4

(Color online) Comparison of the mean image intensity for experimental (symbols) and simulated (line) images as a function of thickness. The experimental thickness was determined from PACBED patterns.

Figure 5

(Color online) Comparison of experimental (symbols) and simulated (line) image signals as a function of thickness. Both the maximum signal and minimum (“background”) signal are shown. The dashed and solid lines represent simulations without and with the effects of spatial incoherence taken into account by convolution with a 0.115 nm FHWM Gaussian, respectively. The difference in signals between experiments and simulations after convolution is less than 5% on average.

Figure 6

(Color online) Experimental (left panels) and simulated (right panels) images for different thicknesses (see labels). The simulated images account for finite source size characterized by a 0.115 nm FWHM Gaussian distribution. Note that different intensity scales are used in (a) and (b) for clarity. The uncertainties in the experimental thicknesses are $\pm 2–4\text{nm}$ and the simulated thicknesses are within $\pm 2\text{nm}$ of the value given in the figure.

Figure 7

(Color online) Comparison of simulated 12-nm-thick region of ${\text{SrTiO}}_3$ with different magnitudes of [(a), (b), (d)] the Gaussian FHWM effective source sizes and (c) an experimental image. The right column in (c) shows the original experimental image and the left column after Fourier filtering with a $9.85{\mathrm{nm}}^{-1}$ circular aperture. An effective source size with Gaussian FWHM between (b) 0.11 and (d) 0.12 produces the best fit.