Exit Wave Function Research Articles

Recording low and high spatial frequencies in exit wave reconstructions

Aberration corrected Transmission Electron Microscope (TEM) images can currently resolve information at significantly better than 0.1 nm. Aberration corrected imaging conditions seek to optimize the transfer of high-resolution information but in doing so they prevent the transfer of low spatial frequency information. To recover low spatial frequency information, aberration corrected images must be acquired at a large defocus which compromises high spatial frequency information transfer. In this paper we present two a posteriori solutions to this problem in which the information bandwidth in an exit wave reconstruction is increased. In the first we reconstruct the electron exit wavefunction from two focal series datasets, with different, uniform focal steps, experimentally demonstrating that the width of the transfer interval can be extended from 0.2 nm⁻¹ (∼5 nm) to better than 10 nm⁻¹ (0.1 nm). In the second we outline the use of a focal series recorded with a non-uniform focal step to recover a wider range of spatial frequencies without the need for a large number of images. Using simulated data we show that using this non-uniform focal step the spatial frequency interval for a five image data set may be increased to between 0.25 nm⁻¹ (4 nm) and 8.3 nm⁻¹ (0.12 nm) compared to between 0.74 nm⁻¹ (1.4 nm) and 8.3 nm⁻¹ (0.12 nm) for the standard focal series geometry.

Phase retrieval from image intensities: Why does exit wave restoration using IWFR work so well?

The iterative wave function reconstruction (IWFR) is one type of in-line holography and retrieves a complex wave function from a set of through-focus images. We have verified that the IWFR provides an extremely good estimate of an atomic-resolution exit wave function (EWF) simply from the Fourier transforms of observed intensities. Thus, the first guess of the EWF using only five images gives all the features of the final result, and the convergence of the IWFR is very quick. The IWFR accepts a wide variety of defocus step, and the total defocus span or the defocus step is not essential. The absolute defocus can be estimated by propagating the EWF to the plane where the propagated wave function gives the minimum amplitude variation. Even when there is some error in the spherical aberration coefficient, the EWF suffers from only the aberrations due to the estimation error. The residual error may be adjusted on the reconstructed complex wave function. With the development of a stable microscope it becomes more realistic to routinely record multiple images with good quality, which allows advanced image processing such as the IWFR. Such an exit wave reconstruction is also desirable to investigate a phase object using a Cs-corrected microscope, since the intentionally introduced aberrations to amplify the phase object contrast are desirable to be eliminated by post-processing.

Direct imaging of atomic clusters in an amorphous matrix: A Co-C granular thin film

The atomic structure of extremely small cobalt (Co) nanoparticles embedded in an amorphous carbon (C) matrix has been studied by spherical aberration (Cs) corrected high-resolution transmission electron microscopy and focal-series restoration. The Co nanoparticles, 1–3 nm in diameter, are crystalline with the face centered cubic structure, while the radial distribution function analysis revealed the existence of a Co–C bond. The reconstructed phase images of the exit-wave function clearly show the projected potential distribution within the Co nanoparticles. The Cs-correction has hence a benefit to visualize embedded crystalline clusters unambiguously, which are responsible for the magnetotransport properties of the Co-C films.

Optimization and stability of the contrast transfer function in aberration-corrected electron microscopy

The Contrast Transfer Function (CTF) describes the manner in which the electron microscope modifies the object exit wave function as a result of objective lens aberrations. For optimum resolution in C₃-corrected microscopes it is well established that a small negative value of C₃, offset by positive values of C₅ and defocus C₁ results in the most optimal instrument resolution, and optimization of the CTF has been the subject of several studies. Here we describe a simple design procedure for the CTF that results in a most even transfer of information below the resolution limit. We address not only the resolution of the instrument, but also the stability of the CTF in the presence of small disturbances in C₁ and C₃. We show that resolution can be traded for stability in a rational and transparent fashion. These topics are discussed quantitatively for both weak-phase and strong-phase (or amplitude) objects. The results apply equally to instruments at high electron energy (TEM) and at very low electron energy (LEEM), as the basic optical properties of the imaging lenses are essentially identical.

Atomic Resolution Phase Contrast Imaging and In-Line Holography Using Variable Voltage and Dose Rate

The TEAM 0.5 electron microscope is employed to demonstrate atomic resolution phase contrast imaging and focal series reconstruction with acceleration voltages between 20 and 300 kV and a variable dose rate. A monochromator with an energy spread of ≤0.1 eV is used for dose variation by a factor of 1,000 and to provide a beam-limiting aperture. The sub-Ångstrøm performance of the instrument remains uncompromised. Using samples obtained from silicon wafers by chemical etching, the [200] atom dumbbell distance of 1.36 Å can be resolved in single images and reconstructed exit wave functions at 300, 80, and 50 kV. At 20 kV, atomic resolution <2 Å is readily available but limited by residual lens aberrations at large scattering angles. Exit wave functions reconstructed from images recorded under low dose rate conditions show sharper atom peaks as compared to high dose rate. The observed dose rate dependence of the signal is explained by a reduction of beam-induced atom displacements. If a combined sample and instrument instability is considered, the experimental image contrast can be matched quantitatively to simulations. The described development allows for atomic resolution transmission electron microscopy of interfaces between soft and hard materials over a wide range of voltages and electron doses.

Study of coherence strain of GP II zones in an aged aluminum composite

Strain mapping using the geometric phase analysis (GPA) technique was applied to Al–GP II (Guinier–Preston) nanoscale precipitates, using both high resolution transmission electron microscopy (HRTEM) micrographs as well as the exit wave function (EWF) obtained by focal series reconstruction. The experimental strain results were compared with strain maps obtained from an atomic model which consisted of an Al supercell containing a GP II precipitate. It was built as a reference from literature data. The experimental results demonstrate a complex strain distribution and larger fluctuations than the reference strain maps. These differences were found to be partly a consequence of image artifacts produced by the technique as well as complex microstructural events which were present at the development stage studied.

Electron Image Series Reconstruction of Twin Interfaces in InP Superlattice Nanowires

The twin interface structure in twinning superlattice InP nanowires with zincblende structure has been investigated using electron exit wavefunction restoration from focal series images recorded on an aberration-corrected transmission electron microscope. By comparing the exit wavefunction phase with simulations from model structures, it was possible to determine the twin structure to be the ortho type with preserved In-P bonding order across the interface. The bending of the thin nanowires away from the intended 110 axis could be estimated locally from the calculated diffraction pattern, and this parameter was successfully taken into account in the simulations.

Imaging the Active Surfaces of Cerium Dioxide Nanoparticles

Watching the changes: Aberration-corrected transmission electron microscopy is used to image CeO2 nanoparticles during an electron-beam-induced structure transformation. Analysis of the phase of the computationally restored exit wavefunction (see Figure, a) allows the nature of the different surfaces (b) to be characterized for the first time and provides direct experimental verification of previously reported theoretical surface structure calculations.

Automated phase retrieval of a single-material object using a single out-of-focus image

Phase retrieval is widely used in phase contrast microscopy. Here we present an autofocus algorithm that allows the phase of the exit wave function, from a single-material object, to be reconstructed at medium resolution from a single phase contrast image without any a priori knowledge of the imaging system or object. The algorithm is demonstrated on coherent out-of-focus electron micrographs of 30 nm latex sphere calibration standards, giving <10% RMS error over a large defocus range.

Applications of the Oxford-JEOL aberration-corrected electron microscope

A review is provided of the use of aberration-corrected electron microscopy at Oxford from 2003 to the present day. In particular, examples of work carried out in Oxford during this period are described. Aberration-corrected exit wave function reconstruction is used to examine the surface structure of a catalyst nanoparticle. It is shown that tilt-series reconstruction can retrieve higher spatial resolution information. The Z-contrast nature of annular dark field scanning transmission electron microscopy can be used to quantitatively locate dopant atoms in an intergranular film. The recording of annular dark field images on an absolute intensity scale is demonstrated and, finally, it is shown that confocal trajectories can dramatically improve the depth resolution for optical sectioning.

Finding phase information in the darkness

High resolution local structure information is revealed when the complex exit wavefunction is restored from a series of transmission electron microscope (TEM) images. Aberration corrected image data sets allow high spatial frequency information to be restored but this is often at the expense of lower spatial frequencies. In this paper we present a new approach to exit wavefunction restoration using a novel five image focal series with non-uniform focal steps. This data set extends the spatial frequency range over which information can be successfully restored. We present simulation studies comparing this approach to a conventional focal series and demonstrate the benefit of including low spatial frequencies in the restoration. We propose that this approach will be important for the study of large unit cell materials or those with long range ordering.

Atomic-scale characterization of theSrTiO3Σ3(112)[1¯10]grain boundary

The strontium titanate $\ensuremath{\Sigma}3\phantom{\rule{0.3em}{0ex}}(112)\phantom{\rule{0.3em}{0ex}}[\overline{1}10]$ grain boundary has been investigated using aberration-corrected high-resolution transmission electron microscopy and exit wave-function reconstruction. By quantitatively comparing the experimental exit wave phase with that expected from relaxed model structures obtained by density-functional theory, it has been possible to differentiate between two proposed structures not significantly different in their calculated energies, and to deduce information about three-dimensional atomic arrangements near the boundary using the reconstructed exit wave-function phase.

Object wave reconstruction by phase-plate transmission electron microscopy

A method is described for the reconstruction of the amplitude and phase of the object exit wave function by phase-plate transmission electron microscopy. The proposed method can be considered as in-line holography and requires three images, taken with different phase shifts between undiffracted and diffracted electrons induced by a suitable phase-shifting device. The proposed method is applicable for arbitrary object exit wave functions and non-linear image formation. Verification of the method is performed for examples of a simulated crystalline object wave function and a wave function acquired with off-axis holography. The impact of noise on the reconstruction of the wave function is investigated.

Reconstruction of the projected electrostatic potential in high-resolution transmission electron microscopy including phenomenological absorption

The projected electrostatic potential is reconstructed from a high-resolution exit wave function through a maximum-likelihood refinement algorithm. The theory of an already existing algorithm [1] is extended to include the effects of phenomenological absorption. Various tests with a simulated exit wave function of YBa 2Cu 3O 7 in [1 0 0] orientation used as a source show that the reconstruction is successful, regardless of the strongly differing scattering power of atomic columns, even for the case of strong dynamical diffraction. Object thickness, the amount of absorption, and a residual defocus aberration of the wave function—parameters often unknown or difficult to measure in experiments—can be determined accurately with the aid of the refinement algorithm in a self-consistent way. For the next generation of instruments, with information limits of 0.05 nm and better, reconstruction accuracies of better than 2% can be expected, which is sufficient to measure and display the structural and chemical information with the aid of an accurate projected potential map.

Optimal tilt magnitude determination for aberration-corrected super resolution exit wave function reconstruction

Transmission electron microscope (TEM) images recorded under tilted illumination conditions transfer higher spatial frequencies than axial images. This super resolution information transfer is highly directional in a single image, but can be extended in all directions through the use of complementary beam tilts during exit wave function reconstruction. We have determined the optimal experimental tilt magnitude for aperture synthesis in an aberration-corrected TEM. It is shown that electron-optical aberration correction allows the use of larger tilt angles and reduces the constraints that are imposed on experimental data acquisition in an uncorrected microscope. We demonstrate that, in many cases, the resolution improvement achievable is now limited by the sample and not by instrumental parameters. An exit wave function is presented that has been successfully reconstructed from a dataset of aberration-corrected images, including images acquired at a beam tilt of 18 mrad, which clearly demonstrates a resolution improvement from 0.11 nm to better than 0.08 nm at 200 kV.

Computational comparison of the conventional multislice method and the real space multislice method for simulating exit wavefunctions

The simulation of exit wavefunctions of a TEM sample is carried out by both the conventional multislice (C-MS) method and the real space multislice (RS–MS) method, with a series of different slice thicknesses. Surprisingly, the results calculated by the RS–MS method remain constant to seven significant figures when the slice thicknesses are changed within a wide range, while those calculated by the C-MS method change significantly. From the mathematical point of view, the C-MS method is only a first order approximation of the RS–MS method. Thus, the calculation precision of the RS–MS method is much higher than that of the C-MS method. Or to achieve the same precision, the slice thickness for the RS–MS method can be taken as two orders of magnitude larger than for the C-MS method in the simulation. As a result, the calculation speed for the RS–MS method may be even faster than that for the C-MS method in some cases. For the two advantages stated above, the RS–MS method may be more suitable for precise calculations.

Aberration corrected tilt series restoration

Aberration correction can be achieved using either direct electron optical correction or indirect image restoration. In the past focal series restoration has been applied to aberration corrected images in order to improve the quality of aberration correction and retrieve the complex specimen exit wavefunction. Image restoration can also be performed from a number of images with differing illumination tilts (a tilt series) instead of the more common focal series datasets where only the defocus value is varied. Here we apply tilt series image restoration to aberration corrected images and discuss the advantages of this approach. Preliminary results demonstrate the potential of this technique to provide interpretable structural information at resolutions beyond the axial information limit of the microscope.

Measurement of specimen thickness by phase change determination in TEM

A non-destructive method for measuring the thickness of thin amorphous films composed of light elements has been developed. The method employs the statistics of the phase of the electron exit wave function. The accuracy of this method has been checked numerically by the multislice method and compared with that based on the mean inner potential.

An Accurate Image Simulation Method for High-Order Laue Zone Effects

A completely different formulation for simulation of the high order Laue zone (HOLZ) diffractions is derived. It refers to the new method, i.e. the Taylor series (TS) method. To check the validity and accuracy of the TS method, we take polyvinglidene fluoride (PVDF) crystal as an example to calculate the exit wavefunction by the conventional multi-slice (CMS) method and the TS method. The calculated results show that the TS method is much more accurate than the CMS method and is independent of the slice thicknesses. Moreover, the pure first order Laue zone wavefunction by the TS method can reflect the major potential distribution of the first reciprocal plane.

Half analytical method with application to the high order Laue zone effects in monoclinic and triclinic crystals

The simulation of the high order Laue zone (HOLZ) diffractions of monoclinic and triclinic crystals is a very complicated problem. The conventional multislice (MS) method suffers from the problem of infinity of the phase grating size. To avoid the difficulty of the infinity, Chen et al. arranged the phase grating plane along the ab plane of the crystal lattice. In this paper we introduce a new method which allows the incident and exit wavefunction planes perpendicular to the zone axis without the difficulty of infinity. We name the method half analytical method. The results calculated by half analytical method are in good agreement with those calculated by the conventional MS method.