Supplementary Figure 1. 2-Fold astigmatism. (a-f) Multi-slice image simulations of graphene structure with variation of the level of 2-fold

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1 Supplementary Figure 1. 2-Fold astigmatism. (a-f) Multi-slice image simulations of graphene structure with variation of the level of 2-fold astigmatism increasing from 0 to 1.0 nm with an interval of 0.2nm. (g-o) direction studies of the 2-fold astigmatism with a value of 0.2 nm. The angle of the directions increases from 0 o to 315 o with an interval of 45 o (0 o is an equivalent to 360 o in practice.) The inset to the right of each figure demonstrates the direction of the astigmatism which the simulation adopts.

2 Supplementary Figure 2. 3-Fold astigmatism correction with a pre-set 2-fold astigmatism value of 0.2 nm a direction of 225 o. Multi-slice image simulations of graphene structure with (a-f) different levels of 3-fold astigmatism increasing from 0 to 50 nm with an interval of 10nm. (g-o) direction studies of the 3-fold astigmatism with a value of 40 nm. The angle of the directions increases from 0 o to 315 o with an interval of 45 o (0 o is an equivalent to 360 o in practice.) The inset to the right of each figure demonstrates the direction of the astigmatism which the simulation adopts.

3 Supplementary Figure 3. Axial coma correction with a pre-set 2-fold astigmatism value of 0.2nm and a 3-fold astigmatism value of 40 nm. (a-f) Level of axial coma increasing from 0 to 150 nm with an interval of 30nm. (g-o) direction studies of the axial coma with a value of 60 nm. The angle of the directions increases from 0 o to 315 o with an interval of 45 o (0 o is an equivalent to 360 o in practice.) The inset to the right of each figure demonstrates the direction of the astigmatism which the simulation adopts.

4 Supplementary Figure 4. (a) A frame of AC-TEM image with the bond length measurements of the interested region superimposed on the lattice. (b) Multi-slice image simulation of the DFT optimised geometry with appropriate defocus and defocus spread values, the bond length measurements are superimposed on the lattice. (c) Multi-slice image simulation of the same geometry as (b) but with 2- fold, 3-fold and axial coma astigmatism values set to represent the actual AC-TEM image. The bond length measurement values are superimposed on the lattice. (d) Parameters of the astigmatisms used to generate the multi-slice image simulation in (c).

5 Supplementary Figure 5. (a) DFT relaxed graphene edge structure include the triangle structure. (b)- (e) Multislice image simulation of the atomistic structure shown in panel (a) with defocus spread values of (b) 3nm, (c) 5nm, (d) 7nm, and (e) 9nm respectively. (f)-(i) Lookup table color coded images of (b)-(e).

6 Supplementary Figure 6. (a) Phase plate due to residual aberrations for the JEOL 2200MCO HRTEM at 80 kv just before image acquisition, using an enhanced tableau setting with outer tilt angle set to 36 mrad. (b) Tableau of 21 diffractograms taken at different incident beam tilts.

7 Supplementary Figure 7. Illustrating how boxed line profiles were determined. Line profiles were taken using a box profile across 3 different directions of the lattice, shown as black (1), green (2) and white (3). The CCD counts are summed for all pixels in the y direction, shown on the black box (1), and then plotted as a function of distance x.

8 Supplementary References 42. Krivanek, O. L., Dellby, N. & Lupini, a. R. Towards sub-å electron beams. Ultramicroscopy 78, 1 11 (1999). 43. Merkle, K. L., Csencsits, R., Rynes, K. L., Withrow, J. P. & Stadelmann, P. A. The effect of three-fold astigmatism on measurements of grain boundary volume expansion by high-resolution. 190, (1998). 44. Krivanek, O. L. Three-fold astigmatism in high-resolution transmission electron microscopy. Ultramicroscopy 55, (1994). 45. Krivanek, O.. & Stadelmann, P.. Effect of three-fold astigmatism on high resolution electron micrographs. Ultramicroscopy 60, (1995). 46. Haider, M. et al. A spherical-aberration-corrected 200 kv transmission electron microscope. 75, (1998). 47. Ishizuka, K. Coma-free alignment of a high-resolution electron microscope with threefold astigmatism. Ultramicroscopy 55, (1994). 48. Haider, M. et al. Towards 0. 1 nm resolution with the first spherically corrected transmission electron microscope. 405, (1998).

Aberration corrected tilt series restoration

Journal of Physics: Conference Series Aberration corrected tilt series restoration To cite this article: S Haigh et al 2008 J. Phys.: Conf. Ser. 126 012042 Recent citations - Artefacts in geometric phase