The effect of mechanical vibration and drift on the reconstruction of exit waves from a through focus series of HREM images

Abstract

Experimental HREM images can show a limited resolution as a result of mechanical vibration and drift. In this paper the effect of such mechanical vibrations on the accuracy of the through focus exit wave reconstruction method is investigated for different thicknesses of a test structure of La₃Ni₂B₂N₃. A through-focus series of HREM images for this structure is simulated for different kinds of mechanical vibration corresponding to an information limit $\text{g}$ of about 7 nm⁻¹: (1) no mechanical vibration, (2) isotropic mechanical vibration, and (3) several anisotropic mechanical vibrations. From these through-focus series the reconstructed exit wave is calculated (Ultramicroscopy 64 (1996) 109). The above isotropic and anisotropic mechanical vibrations have a large effect on the reconstructed exit waves when compared with the reconstructed exit wave without mechanical vibration, i.e. the range of amplitudes and phases in a reconstructed exit wave decreases and the background intensity increases. The initial thickness and orientation can be obtained using a least-squares refinement procedure (Acta Crystallogr. A 54 (1998) 91) when there is no mechanical vibration present. In the case of isotropic or anisotropic vibration, the refined thickness and orientation are likely to give wrong results depending on the size of the vibrations and on the number of significant reflections (which is related to the size of the unit cell, the thickness and the misorientation).

Introduction

Currently, interest has revived in through-focus exit-wave reconstruction as a method to extract as much information as possible from a focal series of HREM images [1], [2]. The through-focus exit-wave reconstruction (TF-EWR) uses the focus dependence of the image distortion by the electron microscope to restore the distortion of the microscope optics. Using algorithms developed by Van Dyck [2] and Coene [1], all useful information is extracted from a series of high-resolution electron microscope images taken at different defocus values with known focus increments. The result is the exit wave function — or in short: exit wave — which contains amplitude as well as phase information up to the information limit of the microscope.

The exit wave is complex; consequently, it contains amplitude and phase. One of the advantages is that one can obtain information about the local crystal orientation from such complex exit waves [3]. This misorientation can be deduced from the Fourier transform of a given exit wave, since unlike for real images, the absolute value of the Fourier transform is not centrosymmetric. Given the differences between the symmetry-related reflections (including $\text{g}$ and $\text{−}\text{g}$), the misorientation can be determined in a straightforward way. Knowledge of the crystal misorientation is a necessary tool for very precise determinations of atom positions at for instance grain boundaries. Moving the electron beam in nanodiffraction mode, it is almost always observed that — although the crystal is well aligned according to the selected area diffraction — the required orientation is almost never exactly obtained. Fluctuations in orientation of 1–2° over short distances (10–50 nm) should be considered to be the rule rather than the exception.

The misorientation can be refined using a recently developed least-squares refinement procedure in which the dynamical diffraction is taken into account [4]. In this procedure (MSLS) the multislice algorithm is combined with a least-squares refinement algorithm. Normally, the reflections of the diffraction pattern are used as a data set. Alternatively, the Fourier components of the exit wave can be used. Apart from the refinement of the crystal (mis-) orientation, the MSLS program allows for the refinement of the thickness as well as the atomic positions. An essential requirement of the refinement is that the exit wave is reconstructed as accurately as possible. Since, in particular, the reflections with a high scattering angle are susceptible to specimen tilts, it is essential to determine these reflections very accurately, which implies that all damping terms have to be well accounted for.

The resolution of HREM images is damped by several effects as was already described by Glaeser [5]. In the first place, it is damped by the chromatic aberrations. In the TF-EWR, one corrects for the damping envelope corresponding to the chromatic aberrations. The result is an enhancement of the high-frequency components up to a certain frequency which results in a top hat-like aperture function in Fourier space. In the second place, the modulation transfer function of the CCD camera results in a damping. This modulation transfer function can be measured and is included in the TF-EWR software package. In the third place, the HREM images are damped by the mechanically induced specimen vibrations, in short: mechanical vibrations. In the fourth place, stray fields result in a loss of high-frequency information. Their effect on the HREM images is similar to that of the mechanical vibration. Often the damping envelope of the mechanical vibrations is anisotropic.

A distinction can be made between vibrational damping of linear and non-linear interaction terms. The non-linear terms result from an interaction between two diffracted beams, e.g. frequencies $\text{g}$ and $\text{−}\text{g}$ give rise to the non-linear frequency $\text{2}\text{g}$ [6]. The vibrational damping affects the non-linear interaction terms particularly strong since these terms generally expand further into Fourier space (larger frequencies). The non-linear interaction terms are not taken into account in the paraboloid part of exit wave reconstruction (PAM method) but they are used in the maximum likelihood (MAL) part of the reconstruction [6]. Consequently, if these terms are significantly damped by the mechanical vibration of the specimen, the exit wave is unfaithfully reconstructed. The effect of the mechanical damping can be appreciated from Fig. 1 which shows simulated HREM images with and without an isotropic specimen vibration damping of 0.064 nm. In the fourth place, the specimen drift during the recording of a HREM image results in a loss of resolution, except in the direction perpendicular to the drift if the drift is only in one direction. The specimen drift can roughly be estimated from the image shift of two consecutive images.

The specimen vibration and the specimen drift are not yet included in the TF-EWR. In order to evaluate the necessity of including such mechanical vibrations in the reconstruction procedure, we have performed a number of calculations in which the degree of mechanical vibration in the calculated HREM images is varied. These HREM images are subsequently used in a TF-EWR after which the intensities and phases of Fourier transform are compared with those of the original exit wave. It is shown in this paper that it is essential to take the specimen vibration into account. For large mechanical vibrations, the vibration must already be taken into account in the TF-EWR, whereas for smaller mechanical vibrations it is (sometimes) possible to correct the reconstructed exit wave a posteriori for the mechanical vibration.