Until now it has not been possible to image at atomic resolution using classical electron tomographic methods1, except when the target is a perfectly crystalline nano-object imaged along a few zone axes2. The main reasons are that mechanical tilting in an electron microscope with sub-ångström precision over a very large angular range is difficult, that many real-life objects such as dielectric layers in microelectronic devices impose geometrical constraints and that many radiation-sensitive objects such as proteins limit the total electron dose. Hence, there is a need for a new tomographic scheme that is able to deduce three-dimensional information from only one or a few projections. Here we present an electron tomographic method that can be used to determine, from only one viewing direction and with sub-ångström precision, both the position of individual atoms in the plane of observation and their vertical position. The concept is based on the fact that an experimentally reconstructed exit wave3,4 consists of the superposition of the spherical waves that have been scattered by the individual atoms of the object. Furthermore, the phase of a Fourier component of a spherical wave increases with the distance of propagation at a known ‘phase speed’. If we assume that an atom is a point-like object, the relationship between the phase and the phase speed of each Fourier component is linear, and the distance between the atom and the plane of observation can therefore be determined by linear fitting. This picture has similarities with Big Bang cosmology, in which the Universe expands from a point-like origin such that the distance of any galaxy from the origin is linearly proportional to the speed at which it moves away from the origin (Hubble expansion). The proof of concept of the method has been demonstrated experimentally for graphene with a two-layer structure and it will work optimally for similar layered materials, such as boron nitride and molybdenum disulphide.
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We acknowledge discussions with A. Wang, S. Van Aert and I. Lobato. D.V.D. acknowledges financial support from the “Research foundation - Flanders (FWO)” under project nos G.0220.05 and G.0188.08. F.-R.C. would like to acknowledge the support from NSC-100-2120-M-007-005 and NSC-99-2120-M-007-008. J.R.J. thanks E. Yucelen, R. Dunin-Borkowski and Ch. Kisielowski for support, and N. Alem and A. Zettl for the gift of the graphene sample.’’.
This movie file shows the reconstructed tomography from Hubble analysis. It is a movie version for figure (b) and (c) of Supplementary Figures 1-2.