Quantitative analysis of HOLZ line splitting in CBED patterns of epitaxially strained layers

A SiGe layer epitaxially grown on a silicon substrate is experimentally studied by convergent beam electron diffraction (CBED) experiments and used as a test sample to analyse the higher-order Laue zones (HOLZ) line splitting. The influence of surface strain relaxation on the broadening of HOLZ lines is confirmed. The quantitative fit of the observed HOLZ line profiles is successfully achieved using a formalism particularly well-adapted to the case of a z-dependent crystal potential (z being the zone axis). This formalism, based on a time-dependent perturbation theory approach, proves to be much more efficient than a classical Howie-Whelan approach, to reproduce the complex HOLZ lines profile in this heavily strained test sample.     

Using the Hough transform for HOLZ line identification in convergent beam electron diffraction

The Hough transform as a means for automatic line detection is applied for higher order Laue zone (HOLZ) line identification in convergent beam electron diffraction. The HOLZ line positions are commonly used to measure the strain state in the investigated material. Besides the automation, a very important aspect is the accuracy of the detected line positions. The limits are determined by the amount of noise in the pattern relative to the contrast of the HOLZ lines and the line width. Our investigations show that sub-pixel resolution can be achieved routinely. We have also developed a new strain analysis procedure in which the dynamical shift of each individual line can be taken into account using a model based on several Ewald spheres with different radii. In comparison to the effective high voltage method, finer details of the dispersion surface can be considered, which increases the accuracy of the strain analysis. A measurement of the thermal expansion of aluminium is presented as a model experiment. The lattice constants were determined with an accuracy of about 10⁻⁴.     

Higher-order Laue zone line contrast in large-angle convergent-beam electron diffraction around a dislocation

The physical picture of higher-order Laue zone (HOLZ) line contrast in a large-angle convergent-beam electron diffraction pattern around a dislocation, which is used for determining the Burgers vector, was examined. To evaluate the analytical expression of diffracted wave amplitude, we introduced an approximate form of the atomic displacement field of a dislocation. We showed that the four features of the HOLZ line contrast, that is, splitting, fading, bending and periodical contrast can be explained by analysis of the atomic displacement field. The localized lattice plane bending around a dislocation core made a HOLZ line split, fade and bend. However, we found that the periodical contrast of a HOLZ line was produced by the change of phase difference of the atomic displacement field between the crystals above and below the slip plane across the dislocation line.     

Discrepancies in kinematic calculations of HOLZ lines

We have found significant differences between the results of computer simulations of HOLZ line patterns. The computations in question are made in the kinematical approximation. After trivial errors are eliminated the programs fall into two groups. There is a discrepancy between the two that increases with distance from the zone axis. The difference is small but not negligible at the level of precision used in determining lattice parameters or strain. We show which of the two is correct in the kinematic approximation and that the discrepancy between the two groups is of the order of the error introduced by dynamical interaction. © 1993 Wiley-Liss, Inc.     

HANSIS software tool for the automated analysis of HOLZ lines

A software tool, named as HANSIS (HOLZ analysis), has been developed for the automated analysis of higher-order Laue zone (HOLZ) lines in convergent beam electron diffraction (CBED) patterns. With this tool, the angles and distances between the HOLZ intersections can be measured and the data can be presented graphically with a user-friendly interface. It is capable of simultaneous analysis of several HOLZ patterns and thus provides a tool for systematic studies of CBED patterns.     

Progress toward structure determination

Convergent-beam electron diffraction provides more precise measurements of diffracted intensity than the traditional method of selected-area diffraction. The intensity is recorded at well-defined beam directions for each reflection in the pattern within disks defined by the incident cone of rays. Measurements relating to structure factors or parameters can be arranged in different ways: intensities at the zone axis position; Kossel line profiles or integrated intensities across Kossel lines; conditions for vanishing contrast at a Kossel line (e.g., critical voltage); separation between Kossel line segments at intersections. Examples of application to refinement of structure parameters (zone axis intensities) and structure factor determination (Kossel line methods) are given. The relation of these magnitudes to theory is discussed, especially for the Kossel line methods. These are described in terms of effective Fourier potentials or gaps at the Bloch-wave dispersion surface. Use of the methods for refinement of structure parameters and structure factors is reviewed with special attention to recent developments. This is seen along two lines: (1) extended scope for the more accurate methods in order to cover larger unit cells and (2) better precision in measurements of intensities.     

HOLZ lines splitting on SiGe/Si relaxed samples: Analytical solutions for the kinematical equation

Sample thinning for TEM observation introduces large changes with respect to the initial strain state of the bulk sample and particularly relaxation via the free surfaces which leads to HOLZ lines splitting in the CBED pattern. This phenomenon has been simulated owing to extensive calculations either in the kinematical or the dynamical framework of electron diffraction mainly using displacement fields resulting from finite element modelling of the sample relaxation. HOLZ line splitting is well reproduced and numerical fits can be used to compare experimental and calculated curves. This paper proposes new analytical solutions for the kinematical equation of electron diffraction. Simple mathematical functions are used to approximate the deformation profiles. We showed that, under certain conditions, the rocking curve profile can be analytically calculated, thus providing some clue to separate different contributions to the rocking curves against deformation profile. These simplified analytical expressions are used to extract the maximum amplitude displacement within the sample with about 10% accuracy. This accuracy can even be improved to 1% with a short adjustement routine. The influence of the shape of the displacement profile on the rocking curves is demonstrated.     

An efficient approach to characterize pseudo-merohedral twins by precession electron diffraction: Application to the LaGaO3 perovskite

Pseudo-merohedral twins are frequently observed in crystals displaying pseudo-symmetry. In these crystals, many [u v w] zone axis electron diffraction patterns are very close and can only be distinguished from intensity considerations. On conventional diffraction patterns (selected-area electron diffraction or microdiffraction), a strong dynamical behaviour averages the diffracted intensities so that only the positions of the reflections on a pattern can be considered. On precession electron diffraction patterns, the diffracted beams display an integrated intensity and a “few-beam” or “systematic row” behaviour prevails which strongly reduces the dynamical interactions. Therefore the diffracted intensity can be taken into account. A procedure based on observation of the weak extra-reflections connected with the pseudo-symmetry is given to identify without ambiguity any zone axis. It is successfully applied to the identification and characterization of {1 2 1} reflection twins present in the LaGaO₃ perovskite.     

A dynamical Bragg equation for high-order Laue zone reflections

A dynamically corrected Bragg equation for high-order Laue zone (HOLZ) reflections is derived directly from the Bloch wave formalism instead of the geometric argument used to deduce the kinematical Bragg condition. It differs from the kinematical Bragg equation by replacing the plane wave vector in the kinematical equation with the Bloch wave vectors. This dynamical equation reduces to the kinematical equation when the crystal potential is zero. It also demonstrates the occurrence of dynamical shifts for the HOLZ reflections but their absence for the zero-order Laue zone (ZOLZ) reflections in the symmetrical Laue case.     

An algorithm for refinement of lattice parameters using CBED patterns

A new algorithm for calculation of lattice parameters from convergent beam electron diffraction (CBED) patterns has been developed. Like most of the previous approaches to the problem, it is an optimization procedure matching geometric elements of high order Laue zone (HOLZ) lines in experimental patterns to corresponding elements of kinematically simulated patterns. The procedure uses an original objective function based directly on the underlying algebraic equation of the HOLZ lines. Although the new approach requires crystal orientation parameters to be fitted alongside the strain components, it is easier to implement than methods used previously. It is also straightforward to apply to strain determination from multiple patterns. Numerical tests on dynamically simulated patterns show that in the case of one or two patterns, the new procedure gives results that are more reliable than the established method based on HOLZ line intersections. As an example application, the a and c parameters of a TiAl alloy are determined.     

The relative intensities of electron diffraction maxima in mixed rare-earth oxides

The relative intensities of electron diffraction maxima from yttrium aluminium perovskite were analysed in terms of the atomic scattering contribution to the structure factors and in terms of double diffraction. It has been shown that double diffraction influences the visible intensity between diffraction spots in electron diffraction patterns to a greater extent than does the magnitude of the structure factor. A direct consequence of this is that the intensity of diffraction maxima can increase when a specimen is tilted off an exact zone axis.     

A precession electron diffraction study of α, β phases and Dauphiné twin in quartz

Precession electron diffraction is used to distinguish between the hexagonal β high-temperature and the trigonal α low-temperature phases of SiO₂ quartz. The structures just differ by a kink of the SiO₄ tetrahedra arranged along spiraling chains, which induces a loss of the two-fold axis and subsequent twinning in the low-temperature phase. Conventional selected-area electron diffraction (SAED) does not enable the phases distinction since only the intensity of reflections is different. It becomes possible with precession that reduces the dynamical interactions between reflections and makes their intensity very sensitive to small variations of the electron structure factors. Distinction between the twinned individuals in the low-temperature phase is then easily made and the twin law is characterized using stereographic projections. The actual symmetry of precessed zone axis patterns is also examined in detail. Using dynamical intensity simulations, it is shown that under certain thickness conditions, the diffraction class symmetry can be observed on selected area patterns that are to be used in the case of beam sensitive materials such as quartz.     

The application of convergent beam electron diffraction (CBED) analysis on transformation‐induced plasticity (TRIP) steels

Convergent beam electron diffraction (CBED) in transmission electron microscopy (TEM) was applied to determine local carbon concentrations in low‐carbon transformation‐induced plasticity (TRIP) steels. High‐order Laue‐zone (HOLZ) lines were experimentally obtained for comparison with simulation results. A new procedure for calculating carbon content is thus proposed. Retained austenite (RA) is classified into three types by morphology; the relationship between the carbon content and the corresponding RA morphology is discussed based on CBED results. Furthermore, results of X‐Ray diffractometry measurements are also used for comparison.     

Dynamical electron diffraction simulation for non‐orthogonal crystal system by a revised real space method

In the transmission electron microscopy, a revised real space (RRS) method has been confirmed to be a more accurate dynamical electron diffraction simulation method for low‐energy electron diffraction than the conventional multislice method (CMS). However, the RRS method can be only used to calculate the dynamical electron diffraction of orthogonal crystal system. In this work, the expression of the RRS method for non‐orthogonal crystal system is derived. By taking Na₂Ti₃O₇ and Si as examples, the correctness of the derived RRS formula for non‐orthogonal crystal system is confirmed by testing the coincidence of numerical results of both sides of Schrödinger equation; moreover, the difference between the RRS method and the CMS for non‐orthogonal crystal system is compared at the accelerating voltage range from 40 to 10 kV. Our results show that the CMS method is almost the same as the RRS method for the accelerating voltage above 40 kV. However, when the accelerating voltage is further lowered to 20 kV or below, the CMS method introduces significant errors, not only for the higher‐order Laue zone diffractions, but also for zero‐order Laue zone. These indicate that the RRS method for non‐orthogonal crystal system is necessary to be used for more accurate dynamical simulation when the accelerating voltage is low. Furthermore, the reason for the increase of differences between those diffraction patterns calculated by the RRS method and the CMS method with the decrease of the accelerating voltage is discussed.     

Automated extraction of regular spot arrays from electron diffraction images*

In order to analyse agglomerate particles, we have developed a computer method that automates the analysis of spots in electron diffraction patterns. The method isolates the zone axis diffraction pattern in a polycrystalline diffraction image by isolating the spots from other features of the image and then by selecting the spots that belong to a regular array. The method gives the basis vectors for the array which can then be used to identify the particle by comparison with standard diffraction data.     

Method for determination of the displacement field in patterned nanostructures by TEM/CBED analysis of split high-order Laue zone line profiles

A method to extract accurate information on the displacement field distribution from split high‐order Laue zones lines in a convergent‐beam electron diffraction pattern of nanostructures has been developed. Starting from two‐dimensional many beam dynamical simulation of HOLZ patterns, we assembled a recursive procedure to reconstruct the displacement field in the investigated regions of the sample, based on the best fit of a parametrized model. This recursive procedure minimizes the differences between simulated and experimental patterns, taken in strained regions, by comparing the corresponding rocking curves of a number of high‐order Laue zone reflections. Due to its sensitivity to small displacement variations along the electron beam direction, this method is able to discriminate between different models, and can be also used to map a strain field component in the specimen. We tested this method in a series of experimental convergent‐beam electron diffraction patterns, taken in a shallow trench isolation structure. The method presented here is of general validity and, in principle, it can be applied to any sample where not negligible strain gradients along the beam direction are present.     

Analysis of local strain in aluminium interconnects by energy filtered CBED

Energy filtered convergent beam electron diffraction (CBED) was used to investigate localised strain in aluminium interconnects. The quantitative analysis of the experimental patterns is based on a multi-step evaluation procedure which is the main subject of the present paper. The improvements which were made to the analysis method aim at increasing both the automation and the accuracy. The detection of the higher order Laue zone (HOLZ) line positions is performed by means of the Hough transform. The required sub-pixel resolution can be achieved routinely and the achievable accuracy is only limited by the line width and the amount of noise in the patterns. The determination of the strain state is performed via a refinement algorithm which is based on varying the strain state in the sample coordinate system and simulating the patterns for the individual grains until a best fit with the experiment is obtained. For the simulation we have developed a new correction scheme in which the dynamical effects are treated separately for each individual HOLZ line. The results show that the main source of the observed strains is the difference in thermal expansion coefficients. The strain is substantially reduced underneath a hillock in the interconnect. Asymmetries in the strain distribution around the hillock show that the unidirectional diffusion during electromigration tests causes peak strains in areas next to the hillock which may be possible failure sites.     

Investigation of the local Ge concentration in Si/SiGe nanostructures by convergent-beam electron diffraction

SiGe multi quantum well structures were investigated by convergent-beam electron diffraction (CBED) measurements. Detailed layer characterizations were performed by acquiring series of bright field CBED patterns in the form of a line scan across the nanostructures in scanning transmission electron microscopy (STEM) mode. From the higher order Laue zone (HOLZ) lines the local lattice parameters were deduced. The Ge concentration corresponding to these lattice parameters was determined by means of the elasticity theory. In this work it is shown that the lattice constants can be determined locally with an accuracy of about ±0.001 to ±0.003 Å which leads to an accuracy of the corresponding Ge concentration of about 1–2%. The characteristics of the focused electron probe and its influence on the experimental data were used for an estimation of the spatial resolution of the CBED method. For comparison, experimental values regarding the spatial resolution were determined by investigating the abrupt interface between Si(1 1 1) and AlN(0 0 0 1).     

The assessment of compositional changes in Nimonic PE16 by convergent beam electron diffraction

Lattice parameter changes can be detected by the use of a previously established technique which is based upon computer simulation of HOLZ patterns. This simulation technique is then applied to two precipitation phenomena in Nimonic PE16. Firstly, the growth of gamma prime, γ, precipitates from a solid solution is monitored via the decrease in lattice parameter of the parent phase, and secondly the lattice parameter of coarsened, overaged γ is measured for precipitates sited both at grain boundaries and within grains. The variations thus detected between intergranular and intragranular precipitates are attributed to changes in the Ti/Al ratio due to preferential γ coarsening at the grain boundary. The very subtle compositional changes inferred (～1 at. % Ti) would have been very difficult to establish by chemical analysis, but can be easily measured by convergent beam electron diffraction.