How precise can atoms of a nanocluster be positioned in
3D
from a tilt series of scanning transmission electron microscopy images?
Authors: Alania Marcos , Backer Annick De , Lobato Ivan , Krause Florian F. , Dyck Dirk Van , Rosenauer Andreas , Aert Sandra Van 
Published: December 2016

Nanoclusters play key roles in a wide range of materials and devices because of their unique physical and chemical properties. These properties are determined by the specific three‐dimensional (3D) morphology, structure and composition. It is well known that extremely small changes in their local structure may result in significant changes of their properties. Therefore, development of techniques to measure the atomic arrangement of individual atoms down to (sub)‐picometer precision is important. This allows one to fully understand and greatly enhance the properties of the resulting materials, increasing the number of applications.
Electron tomography using aberration‐corrected scanning transmission electron microscopy (STEM) is considered as one of the most promising techniques to achieve atomic resolution in 3D. Although this is not yet a standard possibility for all structures, significant progress has recently been achieved using different approaches [1,2]. Once the atoms can be resolved in 3D, the next challenge is to refine the atom positions in order to locate them as precisely as possible. However, the answer to the question how precise these measurements are, is still open. Here, we investigate the theoretical limits with which atoms of a nanocluster can be located in 3D based on the acquisition of a tilt series of annular dark field (ADF) STEM images.
A parametric model, describing the expectations of the intensities observed when recording a tilt series of ADF STEM images, is needed in order to derive an expression for the highest attainable precision [3,4]. Although the multislice method is more accurate to describe the electron‐object interaction, it is very time‐consuming, especially when simulating a tilt series of images. Therefore, a Gaussian approximation model has been used as well in order to perform fast, albeit approximate simulations that allow us to get insight into the precision that can be attained to locate atoms in 3D. The precision has been computed for locating the central atom of four gold nanoclusters of different sizes with a Mackay icosahedral morphology. A cross‐section of such a nanoparticle is shown in Fig. 1(a) indicating the x‐, y‐, and z‐axis.

In Fig. 1(b), the attainable precision is shown for the x‐, y‐ and z‐coordinate of the central atom computed taking all the atoms into account, the atoms of the central plane (orange atoms and red atom in Fig. 1), or the central atom only (red atom in Fig. 1(a)) based on simulations using the Gaussian approximation model. From this figure, it can be seen that the precision is not significantly affected by neighbouring atoms, and therefore, it is allowed to use only the central atom to evaluate the attainable precision. In figure 2(a), 2(b) and 2(c) the attainable precision is illustrated as a function of the number of projections, the tilt range, and the incident electron dose. The precision increases with increasing number of projections, tilt range, and incident electron dose. Using optimal parameters for the number of projections, the tilt range and electron dose determined based on the calculation of the precision using the Gaussian approximation model, realistic STEM simulations have been performed using the multislice method. The precision has been evaluated for a dose of 8680 e
−
/Å
2
as a function of the inner detector radius of the annular STEM detector (Fig. 3(a)). The optimal inner angle equals the semi‐convergence angle. Next, the precision to locate the central atom is determined for the different cluster sizes using all optimised settings (Fig. 3(b)). Here, it is shown that theoretically, a precision of a few picometers can be attained for locating atoms in 3D using a tilt series of ADF STEM images.

In conclusion, it is shown that the attainable precision for locating atoms in 3D can be optimized as a function of the number of projections, tilt range, electron dose, and inner radius of the STEM detector. It is demonstrated that a precision in the picometer range for positioning atoms in 3D is feasible.