Prospects for atomic resolution inline holography for a 3D determination of atomic structures from single projections
 F. R. Chen^{1}Email author,
 C. Kisielowski^{2} and
 D. Van Dyck^{3}
DOI: 10.1186/s4067901700416
© The Author(s) 2017
Received: 17 September 2016
Accepted: 17 January 2017
Published: 6 February 2017
Abstract
It is now established that the 3D structure of homogeneous nanocrystals can be recovered from inline hologram of single projections. The method builds on a quantitative contrast interpretation of electron exit wave functions. Since simulated exit wave functions of single and bilayers of graphene reveal the atomic structure of carbonbased materials with sufficient resolution, we explore theoretically how the approach can be expanded beyond periodic carbonbased materials to include nonperiodic molecular structures. We show here theoretically that the 3D atomic structure of randomly oriented oleic acid molecules can be recovered from a single projection.
Keywords
Inline holography Exit wave function Atomic resolution tomography 3D molecular imagingBackground
The ultimate goal of electron microscopy is to act as a communication channel between structure and properties of materials. Certainly, all material properties are determined by the atom arrangement in three dimensions (3D), which are especially rich if complex atom configurations are considered that are intrinsic to composites such as combinations of catalysts and molecules. There has been significant progress towards electron tomography of crystalline and radiation hard matter using aberrationcorrected scanning transmission electron microscope (STEM) [1, 2] and transmission electron microscope (TEM) [3, 4]. Contributing to these examples, our TEM approach [3, 4] allows for quantitative 3D structure determination with atomic resolution from only one viewing direction if it is chosen close to zone axes orientations of the crystalline matter. This goal is achieved by reconstructing electron exit wave functions from image focal series [5, 6] that capture the dynamic nature of electron scattering and the pronounced dependence of local image contrast on focus. In case of crystalline materials, our interpretation bases on the Channeling Theory that provides the number of atom in a column and on the fact that the zcoordinate of atoms at the exit surface can be determined locally from intensity maxima of propagated wave functions to a precision that exceeds interatomic distances. As a starting point for a 3D characterization of carbonbased materials, we summarize essential features of the tool by modeling a single and bilayer of graphene at atomic resolution [3], which is a material of outstanding radiation hardness unlike the majority of single molecules [7, 8]. Beyond investigations of radiation hard periodic matter, the approach offers intrinsic advantages to study beamsensitive materials such as catalysts and molecules because doserate dependences can be exploited to help reducing beamsample interactions so that atomic resolution and single atom sensitivity may be achieved without altering the pristine structure of radiation sensitive matter [8]. In this paper, we demonstrate the 3D information of the atomic position in encoded in the exit wave function reconstructed from simulated focus series images of single/double layer graphene without considering the influence of the noise. The noise will affect the precision of the focus and mass determination which has been demonstrated in our earlier publication [4] with experimental exit wave functions. We further explore the possibility to recover the nonperiodic, 3D structure of molecules from simulated focus series of the oleic acid molecules.
Methods

Determination of the true “z” height (focus) of the exit surface of a column from the common image plane using the maximum propagation intensity (MPI) as a criterion as shown in Fig. 1b. This is equivalent to determine the distance Δf between the a red dot and the blue dot in the same defocus circle.

Refining the “z” height (Δf) using the BigBang scheme [3].

Correcting the focus of each column by wave propagation to create an image where the red and the blue dot coincide in Fig. 1b and create the mass circle.

Since we evaluate peaktovalley phase values, the focus corrected wave Ψ is calibrated with another wave φ determined in the “valley” surrounding the atom columns. The valley value corresponds to “zero mass” and it is expected to be close to vacuum wave (1,0).$$ \varPsi \left( {\text{norm}} \right) = \psi /\varphi $$
Ψ(norm) is the normalized focus corrected wave function so as to corrected by the mean inner potential of the crystal.

The column mass is given by the phase angle θ’ between the normalized focus corrected wave functions and the vacuum wave which is close to (1,0). See Fig. 2f. The θ’ is proportional to the Et.
In the next section, we first illustrate for the case of single and bilayer graphene that our approach allows resolving interatomic distances in beam direction [3, 4], which is a capability that allows extending the application to reveal the 3D structure of molecular networks if they are captured in low doserate image series that can maintain their pristine structure [13].
Results and discussion
Conclusions
This contribution refines the methodology to reconstruct 3D shape of nanomaterials at atomic resolution from a single projection of electron exit wave functions. Further, it is extended to demonstrate that it is possible to reconstruct nonperiodic, molecular structures in three dimensions that can be recorded in low doserate conditions. Therefore, the approach is exceptionally well suited to image single molecules and reconstruct soft/hard matter components in 3D.
Declarations
Authors’ contributions
DD developed the theory and FRC developed the program codes to test theory and analysis of experimental and simulated data. CK provided evidence for the direct imaging of molecules and suggested to use the Oleic acid molecule as an example. All authors read and approved the final manuscript.
Acknowledgements
Electron Microscopy was performed with the TEAM 0.5 microscope at The Molecular Foundry, which is supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DEAC0205CH11231. D. Van Dyck acknowledges the financial support from the Fund for Scientific Research—Flanders (FWO) under Project Nos. VF04812N and G.0188.08. F.R. Chen would like to thank the support from NSC 962628E007017MY3 and NSC 1012120M007012CC1.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Correspondence and requests for materials should be addressed to F.R.C. (fchen1@me.com), C. K. (cfkisielowski@lbl.gov), and D.V.D.(dirk.vandyck@ua.ac.be).
Consent for publication
All authors drafted the paper and agree to publish.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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