Third-dimension information retrieval from a single convergent-beam transmission electron diffraction pattern using an artificial neural network

We have reconstructed third-dimension specimen information from convergent-beam electron diffraction (CBED) patterns simulated using the stacked-Bloch-wave method. By reformulating the stacked-Bloch-wave formalism as an artificial neural network and optimizing with resilient back propagation, we demonstrate specimen orientation reconstructions with depth resolutions down to 5 nm. To show our algorithm's ability to analyze realistic data, we also discuss and demonstrate our algorithm reconstructing from noisy data and using a limited number of CBED disks. Applicability of this reconstruction algorithm to other specimen parameters is discussed.

Figure 1

CBED imaging process and simulated CBED pattern. (a) Schematic of the CBED imaging process [2]. (b) $(2^{\circ }\times 2^{\circ })$ CBED pattern from 100 nm-thick unstrained Si $\left[110\right]$, on-axis with respect to the beam, simulated at 80 kV with 197 eigenstates from the zero-order-Laue-zone, determined through excitation-error filtering of the on-axis orientation ($s_z=1.225$ nm${}^{-1}$), and a $6.5$ mrad condenser aperture, with $0.{01}^{\circ }$ per simulated pixel. Literature values were used for scattering factors, Debye-Waller factors, and lattice parameters [40, 41, 42]. Inelastic scattering used the Bird-King model [43].

Figure 2

Simulated asymmetric CBED pattern and best shifted-symmetric match. (a) Asymmetric CBED pattern under the same imaging conditions and reflections as Fig. 1, with layer-by-layer specimen tilts in the $\left[001\right]$ direction relative to the zone axis of $\{a,a,b,c,c,d,d,c,b,a\}$, where $a=0.{00}^{\circ }$, $b=-0.{06}^{\circ }$, $c=-0.{15}^{\circ }$, and $d=-0.{20}^{\circ }$. Each layer was 10-nm thick. (b) Best-fitting shifted-symmetric CBED pattern to the asymmetric left pattern, with specimen tilt of $-0.{1269}^{\circ }$ (rounded). The pattern appears shifted, but without the left pattern's asymmetry.

Figure 3

Results from scenarios T${}_{10}$, T${}_{20}$, and T${}_{40}$ (see text) for third-dimension parameter reconstruction in the all-disk case. (a) Scenario T${}_{10}$: ${\alpha }_x$ layer-by-layer results for the first 100 iterations, showing retrieval of correct values. (b) All three scenarios: Mean intensity mismatch $\overline{\Delta E}$ taken over all $p_p$ and mean parameter errors $\overline{\Delta {\alpha }_x},\overline{\Delta {\alpha }_y}$ per iteration. (c) All three scenarios: Parameter error as a function of intensity mismatch, yielding detection thresholds.

Figure 4

Results for disk subsets, graphed in the same manner as Figs. 3 and 3. These simulations all use a $(13\times 13)$-point grid with a $0.{05}^{\circ }$ grid spacing. (a) shows mean parameter error $\overline{\Delta {\alpha }_x}$ (dashed lines) and mean intensity mismatch error $\overline{\Delta E}$ (solid lines) plotted as a function of simulation epoch (i.e., iteration). (b) shows $\overline{\Delta {\alpha }_x}$ as a function of $\overline{\Delta E}$, which is the parameter fit quality at a given intensity mismatch. The key for which subsets of reflections are used is in Table 1; the number indicates the number of included reflections.

Figure 5

CBED patterns with Poisson intensity noise (see text), corresponding to an incident beam intensity of (a) ${10}^2$, (b) ${10}^3$, (c) ${10}^4$, and (d) ${10}^5$ electrons per pixel. Specimen used is the same as that in Fig. 2. The inside edge of the white box added to (d) shows the region used in (e)–(h). (e)–(h) A $20\times 20$-pixel (dot pitch $0.{01}^{\circ }$) excerpt from (a)–(d), showing the central disk, between the left maximum and the central cross. This is not the area used for optimization, but simply included to highlight the noise levels.

Figure 6

Results including noise and using only a few measured CBED disks, graphed in the same manner as Fig. 4. For an explanation for the “waterfall” curves in (b), see text.