Reconstructed Exit Wave Research Articles

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.

‘Big Bang’ tomography as a new route to atomic-resolution electron tomography

Until now it has not been possible to image at atomic resolution using classical electron tomographic methods, except when the target is a perfectly crystalline nano-object imaged along a few zone axes. The main reasons are that mechanical tilting in an electron microscope with sub-ångström precision over a very large angular range is difficult, that many real-life objects such as dielectric layers in microelectronic devices impose geometrical constraints and that many radiation-sensitive objects such as proteins limit the total electron dose. Hence, there is a need for a new tomographic scheme that is able to deduce three-dimensional information from only one or a few projections. Here we present an electron tomographic method that can be used to determine, from only one viewing direction and with sub-ångström precision, both the position of individual atoms in the plane of observation and their vertical position. The concept is based on the fact that an experimentally reconstructed exit wave consists of the superposition of the spherical waves that have been scattered by the individual atoms of the object. Furthermore, the phase of a Fourier component of a spherical wave increases with the distance of propagation at a known 'phase speed'. If we assume that an atom is a point-like object, the relationship between the phase and the phase speed of each Fourier component is linear, and the distance between the atom and the plane of observation can therefore be determined by linear fitting. This picture has similarities with Big Bang cosmology, in which the Universe expands from a point-like origin such that the distance of any galaxy from the origin is linearly proportional to the speed at which it moves away from the origin (Hubble expansion). The proof of concept of the method has been demonstrated experimentally for graphene with a two-layer structure and it will work optimally for similar layered materials, such as boron nitride and molybdenum disulphide.

Direct structure inversion from exit waves. Part II: A practical example

This paper is the second part of a two-part paper on direct structure inversion from exit waves. In the first part, a method has been proposed to quantitatively determine structure parameters with atomic resolution such as atom column positions, surface profile and the number of atoms in the atom columns. In this part, the theory will be demonstrated by means of a Au[110] exit wave reconstructed from a set of focal-series images. The procedures to analyze the experimentally reconstructed exit wave in terms of quantitative structure information are described in detail.

High precision measurements of atom column positions using model-based exit wave reconstruction

In this paper, it has been investigated how to measure atom column positions as accurately and precisely as possible using a focal series of images. In theory, it is expected that the precision would considerably improve using a maximum likelihood estimator based on the full series of focal images. As such, the theoretical lower bound on the variances of the unknown atom column positions can be attained. However, this approach is numerically demanding. Therefore, maximum likelihood estimation has been compared with the results obtained by fitting a model to a reconstructed exit wave rather than to the full series of focal images. Hence, a real space model-based exit wave reconstruction technique based on the channelling theory is introduced. Simulations show that the reconstructed complex exit wave contains the same amount of information concerning the atom column positions as the full series of focal images. Only for thin samples, which act as weak phase objects, this information can be retrieved from the phase of the reconstructed complex exit wave.

A method to determine the local surface profile from reconstructed exit waves

Reconstructed exit waves are useful to quantify unknown structure parameters such as the position and composition of the atom columns at atomic scale. Existing techniques provide a complex wave in a flat plane which is close to the plane where the electrons leave the atom columns. However, due to local deviation in the flatness of the exit surface, there will be an offset between the plane of reconstruction and the actual exit of a specific atom column. Using the channelling theory, it has been shown that this defocus offset can in principle be determined atom column-by-atom column. As such, the surface roughness could be quantified at atomic scale. However, the outcome strongly depends on the initial plane of reconstruction especially in a crystalline structure. If this plane is further away from the true exit, the waves of the atom columns become delocalized and interfere mutually which strongly complicates the interpretation of the exit wave in terms of the local structure. In this paper, we will study the delocalization with defocus using the channelling theory in a more systematic way.

Applying an information transmission approach to extract valence electron information from reconstructed exit waves

The knowledge of the valence electron distribution is essential for understanding the properties of materials. However this information is difficult to obtain from HREM images because it is easily obscured by the large scattering contribution of core electrons and by the strong dynamical scattering process. In order to develop a sensitive method to extract the information of valence electrons, we have used an information transmission approach to describe the electron interaction with the object. The scattered electron wave is decomposed in a set of basic functions, which are the eigen functions of the Hamiltonian of the projected electrostatic object potential. Each basic function behaves as a communication channel that transfers the information of the object with its own transmission characteristic. By properly combining the components of the different channels, it is possible to design a scheme to extract the information of valence electron distribution from a series of exit waves. The method is described theoretically and demonstrated by means of computer simulations.

Sub-15 nm beam confinement by two crossed x-ray waveguides

We have combined two high transmission planar x-ray waveguides glued onto each other in a crossed geometry to form an effective quasi-point source. From measurements of the far-field diffraction pattern, the phase and amplitude of the near-field distribution is retrieved using the error-reduction algorithm. In agreement with finite difference field simulations (forward calculation), the reconstructed exit wave intensity distribution (inverse calculation) exhibits a full width at half maximum (FWHM) below 15 nm in both dimensions. Finally, holographic imaging is successfully demonstrated for the crossed waveguide device by translation of a lithographic test structure through the waveguide beam.

Atomic resolution improvement by a new method using the second derivative of the intensity function of the reconstructed exit wave of electrons for a thin β-Si 3N 4 crystal

This report is an extension of a high-resolution transmission electron microscopy study of β-Si 3 N 4 [0 0 0 1] by Ziegler et al. (2002) [1] . The aim of this work was to obtain a better structural image showing complete resolution of the 95 pm spacing for Si–N dumb-bells. This goal has been achieved with a new method using the second derivative of the intensity function of the reconstructed exit wave. The structure of this thin specimen showed an asymmetric aspect, deviating slightly from a hexagonal lattice.

The wrap-around problem and optimal padding in the exit wave reconstruction using HRTEM images

A free fast Fourier transform software FFTW (www.fftw.org) is used in the exit wave reconstruction and shown to be able to handle images of any size effectively. The wrap-around problem commonly encountered in the exit wave reconstruction is investigated, and a new padding scheme is proposed and its performance is investigated. It is shown that while the usual wrap-around problem may be solved via padding the input high-resolution transmission electron microscopy (HRTEM) images with zero or averaged image intensities, the reconstructed exit wave at its boundary is inevitably distorted by the discontinuity of image intensities at the boundary. The proposed new padding scheme is shown to be able to provide a good estimate of the image intensities immediately outside the boundaries of the HRTEM images, reducing the estimated image intensity discontinuity across the boundaries and therefore providing a larger undistorted view of the exit wave.

Pyramidal Defects in GaN:Mg Grown With Ga Polarity

Transmission electron microscopy (TEM) studies show formation of different types of Mg-rich defects in GaN. Types of defects strongly depend on crystal growth polarity. For bulk crystals grown with N-polarity, the planar defects are distributed at equal distances (20 unit cells of GaN). For growth with Ga-polarity (for both bulk and MOCVD grown crystals) a different type of defects have been found. These defects are three-dimensional Mg-rich hexagonal pyramids (or trapezoids) with their base on the (0001) plane and six walls formed on 1123 planes. The defects appear in [1120] and [1100] cross-section TEM micrographs as triangular and trapezoidal with sides inclined at 43 and 47 degrees to the base depending on the above observation directions, respectively. The dimension of these pyramids varies depending on growth method (50-1000 Angstrom), but the angle between the base and their sides remain the same. The direction from the tip of the pyramid to its base (and from the shorter to the longer base for trapezoidal defects) is along the Ga to N matrix bond direction. Analysis of the reconstructed exit wave phase image from the pyramid side indicates a shift of Ga atomic column positions from the matrix to the N position within the pyramid. In this way a 0.6{+-}0.2 Angstrom displacement can be measured on the pyramid side between Ga positions in the matrix and within the pyramid.

Discrete Tomography of Ga and InGa Particles from HREM Image Simulation and Exit Wave Reconstruction

ABSTRACTLow-resolution tomography requires recording images every few degrees. As a consequence, the sample is often degraded after such a procedure. However the required input can be reduced drastically by using knowledge about the position and the number of atoms in each atomic column. This concept has been tested in the present investigation where HREM image simulation (MacTempas) together with exit wave reconstruction (FEI Trueimage) have been performed. A cubeoctahedral nanoparticle is used for the simulation with different compositions i.e., pure solid Ga and In-Ga particles. Six different zone axes ([111], [111], [001], [110], [110], [011]) have been used and the parameters of an aberration corrected microscope (200kV, Cs = 0 mm, resolution = 0.5Å). The discrete grid data were determined by constructing a channeling map from the reconstructed exit wave images. In this special case only three projections [001], [110], [110] were sufficient to find a unique volumetric reconstruction, illustrating the potential of the method. The other projections were used for checking the solution. The comparison between the projected potentials (simulated input) and the final result shows that discrete tomography reconstructs the exact position of all 309 atoms and the three-dimensional shape of the nanocrystal.

Quantitative analysis of surface atom positions in a thick crystal as determined by a reconstructed exit wave.

Atomic structures of an Au (110) 2 x 1 reconstructed surface were analysed quantitatively using an exit wave reconstructed by the three-dimensional Fourier filtering method in high-resolution transmission electron microscopy. To reconstruct the exit wave in a relatively thick crystal, a practical criterion was proposed in the present analysis. In the 'pseudo-exit wave' obtained by the proposed criterion, relaxation of surface atoms was clearly visible in the top three layers. The atoms' displacement was measured to be about 5-20 pm. For quantitative analysis of the atom column positions, image contrast calculations were performed using a structural model of the Au (110) reconstructed surface. Calculations confirmed that the reconstructed pseudo-exit wave could represent the atom column positions directly with an accuracy of several pm, even for a relaxed surface, provided the sample thickness was <7 nm.

Resolution extension and exit wave reconstruction in complex HREM

Direct methods in real and reciprocal space are developed for structural reversion. The direct method in real space involves the use of a novel method to retrieve the phase in the image plane using transport of intensity equation/maximum entropy method (TIE/MEM) and exit wave reconstruction by self-consistent propagation. Since the exit wave is restored from the complex signal in the image planes, no image model between the exit wave and image is assumed. The structural information in the reconstructed exit wave is then further extended by a “complex” maximum entropy method as a direct method in reciprocal space to extrapolate the phase to higher frequencies.

Nano-area electron diffraction pattern reconstructed from three-dimensional Fourier spectrum.

A method to obtain a nano-area electron diffraction pattern in transmission electron microscopy (TEM) was proposed, based on three-dimensional (3D)) image formation theory. This method allows us to reconstruct an electron diffraction pattern from a 3D Fourier spectrum of high-resolution through-focus images. As a test case, an electron diffraction pattern from a tilted Si single crystal was reconstructed using the proposed method and compared with the conventional selected-area electron diffraction pattern. The intensity distribution of the reconstructed electron diffraction pattern was confirmed to be qualitatively equal to that of the selected-area electron diffraction pattern, though the degree of the equivalency between these patterns reduces at the high frequency region and the reproducibility of the intensity degrades when the number of images used in the image processing was decreased. By selecting areas in a reconstructed exit wave field, some electron diffraction patterns were obtained from the nano-areas without the influence of spherical aberration.

Determination of the orientation in small areas of the exit wave

Local differences in thickness and misorientation in a through-focus reconstructed exit wave hamper direct interpretation. These local thicknesses and misorientations can be refined on an atomic scale using the intensities of the Fourier components with the refinement procedure in which a least-squares refinement is combined with a multi-slice calculation. The method was applied to the superconductor La3Ni2B2N3. The crystal tilt and specimen thickness can be determined with an R-value of 8–25%, with a better R-value for thinner areas. Significant differences in the refined tilts, thicknesses and R-values are observed when reconstructed exit waves that are corrected for mechanical vibration are compared with reconstructed exit waves, which are uncorrected.

Atomic model for GP-zones in a 6082 Al–Mg–Si system

Heat treatments of 6082 Al-alloy with 0.6 wt% Mg, 0.9 wt% Si, 0.5 wt% Mn and 0.2 wt% Fe can lead to a considerable increase in hardness. This increase is due to the presence of several metastable phases (in particular β″). To determine the structure of the phase formed before the β″ phase, a detailed high resolution electron microscopy (HREM) study was performed. The pre- β″ phase is needle like, as is the β″ phase. Based on reconstructed exit waves, two models were possible, one of which could be rejected because of the interatomic distances. The model resembles that of the β″ but with different positions for some of the Mg atoms along the needle direction. The structure is more similar to the Al matrix than that of the β″ phase. The Mg sites, and to a lesser extent also the Si sites are partly occupied by Al atoms. The composition is therefore less Mg-rich than β″ (Mg 5Si 6). The content of Al in the structure of the precipitates increases with the degree of coherency in the Al matrix. The space group of the new phase is C2/ m, as for β″.

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

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 La 3Ni 2B 2N 3. A through-focus series of HREM images for this structure is simulated for different kinds of mechanical vibration corresponding to an information limit 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).

The Use of Phase and Amplitude Information of Reconstructed Exit Waves

Exit waves can be reconstructed from through focus series of HREM images or by off-axis holography [1]. We have applied the through focus method to reconstruct exit waves, using algorithms developed by Van Dyck and Coene [2]. Electron microscopy was performed with a Philips CM30ST electron microscope with a field emission gun operated at 300 kV. The high resolution images were recorded using a Tietz software package and a 1024x1024 pixel Photometrix CCD camera having a dynamic range of 12 bits. The reconstructions were done using 15-20 images with focus increments of 5.2 nm. The resulting exit waves were corrected posteriorly for the three fold astigmatism.The exit wave is complex; consequently it contains phase and amplitude. Since in the very thin regions the specimen acts as a thin phase object, such a thin area will show little contrast, an example of which is shown in Figure 1.