Statistical Estimation of Oxygen Atomic Positions with Sub Ångstrom Precision from Exit Wave Reconstruction

Abstract

Although high resolution transmission electron microscopy (HRTEM) is considered as the standard technique to study the atomic structure of materials, interpretation of HRTEM images is not always straightforward because of aberrations of the electromagnetic lens system. The information which can be obtained by qualitative HRTEM is therefore limited by the Scherzer point-resolution, but for a coherent electron source (field emission gun) the actual information limit of the microscope is beyond the Scherzer resolution. Using exit wave reconstruction, the information limit can be reached by correcting the phase changes due to the spherical aberration (C s ) of the objective lens [1]. A great advantage of this technique is that the amplitude as well as the phase of the exit wave is reconstructed and since the light atom columns are revealed in the phase of the exit wave [2], it is possible to image oxygen atoms in the presence of heavier atoms.We have applied exit wave reconstruction to determine an atomic model for the new compound Bi 4 Mn 1/3 W 2/3 O 8 Cl of which the space group is determined to be Cm2m [3]. A focal series of 20 images with an equidistant focal decrease is recorded along the [1-10] zone axis and the numerical reconstruction of the exit wave is carried out using the TrueImage software [4]. After reconstruction and correction for the aberrations, the phase is extracted from the complex exit wave. As illustrated in Figure 1, the phase of the exit wave clearly shows the Cl atoms and the projected {WO 6 }/{MnO 6 } octahedra with the oxygen atoms revealed as well. Two types of bismuth atoms can be distinguished in the image, i.e. Bi(1) and Bi(2). Around Bi(2), separate smaller dots corresponding to the oxygen atoms from the basal square plane can be observed, whereas we can observe the Bi(1) blobs with an extension corresponding to the oxygen atoms. Since the projected distance Bi(1)-O is equal to 0.97 Å, which is close to the information limit, it is not possible to observe Bi (1) and O separately.The phase of the exit wave is often considered as the final result, but in this study it is used as a starting point for quantitative refinement of the atom positions. This is done using statistical parameter estimation theory [5,6]. By adapting a parametric model to the experimental phase in the least-squares sense, it is possible to determine the atom positions. Using the Cramér-Rao Lower Bound as a measure of the precision, it can be shown that sub-Ångstrom precision is within reach. The result is presented in Figure 2, where the experimental phase is compared with the model evaluated at the estimated parameters [7].

Atomic Structure of Quantum Gold Nanowires: Quantification of the Lattice Strain

“…To overcome this problem, statistical parameter estimation theory is known to be an extremely powerful technique. 24 The estimated position plots corresponding to the images in (aÀc), demarking the set of atomic columns considered for calculating the mean displacements. The columns bound in red are the outermost layer of atomic columns, those bound in green are in the second outermost layer, those bound in violet make out the innermost core (5 atomic columns wide), and those in yellow mark the 7 neighboring atomic columns chosen to construct the reference lattice.…”

“…23 Before aberration-correction was implemented in TEM, it was necessary to obtain a phase image by exit wave reconstruction, enabling the elimination of, e. g., delocalization effects in the amplitude image due to lens aberrations, to obtain a precise determination of atomic column positions. 24 With the onset of aberration correction, atomic columns can be imaged at their correct positions for extremely thin specimens like nanowires, allowing atomic column positions to be estimated directly from the acquired HRTEM images, without the need for a through focal series and exit wave reconstruction. We exploit this technique to thoroughly investigate the atomic structure of the ∼2 nm thin gold nanowires.…”