HOLZ Lines Research Articles

Prospects of the multislice method for CBED pattern calculation

Abstract We recently showed that the Multisclice method can be used to elucidate the crystal volume contribution to a CBED pattern. Here, we demonstrate in more detail how the directions of each HOLZ line, their Laue zone order, and the specimen thickness determine the volume segment of the Bragg scattering cone of the electron beam and show that secondary rescattering from this volume must take place for deficient line formation in the (000) CBED disk. Moreover, additional prospects of the multislice method are demonstrated such as calculation of line splitting in bent crystals, hollow cone dark-field CBED, large-angle CBED, and whole diffraction patterns including Kikuchi lines.

Reprint of: Inversion of dynamical Bragg intensities to complex structure factors by iterated projections. For Ultramic. 2020. (“Pico” Festschrift, May 2021)

A method for recovering complex structure factors from many simultaneously excited Bragg beam in- tensities is described. The method is applied to simulated transmission electron diffraction data over a wide range of crystal thickness and beam energies. The method is based on iterated projections between structure and scattering matrices, which are related by a matrix unit ary transformation, exponential, which we invert. The algorithm removes multiple-scattering perturbations from diffraction data and might be extended to other fields, including X-ray and neutron diffraction and cryo-electron microscopy. Because coherent multiple scattering involves interference between Bragg beams, the method also solves the phase problem. Unlike dynamical inversion from electron microscope images or ptychography data, the method, which starts with Bragg beam intensities, provides complex structure factors unaffected by focusing errors or resolution limitations imposed by lenses. We provide inversions from simulated data with 441 simultaneously excited Bragg beams over a range of thickness and beam energy. We discuss the retrieval of chirality information from enantiomorphs, the efficient incorporation of symmetry information using the irreducible representation of the group of structure matrices, and the effect of HOLZ lines to provide three-dimensional information.

Particle size and convergent electron diffraction patterns of triangular prismatic gold nanoparticles

Convergent beam diffraction (CBED) patterns of nanoparticles are possible. CBED of triangular prismatic shaped Au nanoparticle with focus on diffraction pattern symmetry and forbidden reflections observed along [111] and [112] zone axes are reported in this work. It is well known that the CBED patterns of nanoparticles of 30 nm or less in size only show bright kinematical discs. The dynamic contrast with Kikuchi and sharp HOLZ lines within the bright discs, as observed in CBED of volumetric materials, is well observed in particles larger of 500 nm in size. In addition, it is shown that the 1/3[422] and 1/2[311] weak forbidden reflections observed in the [111] and [112] electron diffraction patterns of these particles do not modify the symmetry of the CBED patterns, but they disappear as the size of the particle increases. The symmetry of the CBED patterns are always observed in concordance with the space group Fm3m (No. 225) of the Au unit cell. The possible explanations for observing forbidden reflections are the incomplete ABC stacking due to surface termination and the stacking faults in the fcc structure.

Inversion of dynamical Bragg intensities to complex structure factors by iterated projections. For Ultramic. 2020. ("Pico" Festschrift, May 2021)

A method for recovering complex structure factors from many simultaneously excited Bragg beam in- tensities is described. The method is applied to simulated transmission electron diffraction data over a wide range of crystal thickness and beam energies. The method is based on iterated projections between structure and scattering matrices, which are related by a matrix unit ary transformation, exponential, which we invert. The algorithm removes multiple-scattering perturbations from diffraction data and might be extended to other fields, including X-ray and neutron diffraction and cryo-electron microscopy. Because coherent multiple scattering involves interference between Bragg beams, the method also solves the phase problem. Unlike dynamical inversion from electron microscope images or ptychography data, the method, which starts with Bragg beam intensities, provides complex structure factors unaffected by focusing errors or resolution limitations imposed by lenses. We provide inversions from simulated data with 441 simultaneously excited Bragg beams over a range of thickness and beam energy. We discuss the retrieval of chirality information from enantiomorphs, the efficient incorporation of symmetry information using the irreducible representation of the group of structure matrices, and the effect of HOLZ lines to provide three-dimensional information.

Phase and Atomic Displacement Profiles within Crystals Measured and Simulated using the Self-interference of Split HOLZ Lines

Phase and Atomic Displacement Profiles within Crystals Measured and Simulated using the Self-interference of Split HOLZ Lines

Development of splitting convergent beam electron diffraction (SCBED)

Using a combination of condenser electrostatic biprism with dedicated electron optic conditions for sample illumination, we were able to split a convergent beam electron probe focused on the sample in two half focused probes without introducing any tilt between them. As a consequence, a combined convergent beam electron diffraction pattern is obtained in the back focal plane of the objective lens arising from two different sample areas, which could be analyzed in a single pattern. This splitting convergent beam electron diffraction (SCBED) pattern has been tested first on a well-characterized test sample of Si/SiGe multilayers epitaxially grown on a Si substrate. The SCBED pattern contains information from the strained area, which exhibits HOLZ lines broadening induced by surface relaxation, with fine HOLZ lines observed in the unstrained reference part of the sample. These patterns have been analyzed quantitatively using both parts of the SCBED transmitted disk. The fine HOLZ line positions are used to determine the precise acceleration voltage of the microscope while the perturbed HOLZ rocking curves in the stained area are compared to dynamical simulated ones. The combination of these two information leads to a precise evaluation of the sample strain state. Finally, several SCBED setups are proposed to tackle fundamental physics questions as well as applied materials science ones and demonstrate how SCBED has the potential to greatly expand the range of applications of electron diffraction and electron holography.

HOLZ 線図形をもちいた SiGe/Si 界面近傍の格子湾曲および伸張歪みの解析

HOLZ 線図形をもちいた SiGe/Si 界面近傍の格子湾曲および伸張歪みの解析

Coherent Electron Interference of a Split HOLZ Line from a Strained Silicon Crystal

Extended abstract of a paper presented at Microscopy and Microanalysis 2011 in Nashville, Tennessee, USA, August 7–August 11, 2011.

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 adjustment routine. The influence of the shape of the displacement profile on the rocking curves is demonstrated.

Determination of lattice parameters from multiple CBED patterns: A statistical approach

This study deals with the uncertainty of the measurement of lattice parameters by CBED using the kinematic approximation. The analysis of a large number of diffraction patterns acquired on a silicon sample at 93 K with a LaB(6) TEM without energy filter shows the presence of both the systematic and the random parts of errors. It is established that random errors follow the normal statistical distribution and that the precision quantified by the relative standard deviation is about 3-4 x 10(-4) for lattice parameter measurements made from single pattern. The error sources are analyzed, different ways of enhancement are reviewed, and a new approach is proposed. It is shown that both accuracy and precision can be simply improved by taking into account multiple patterns analysis for the determination of the actual voltage, the single lattice parameter "a" or the complete set of lattice parameters. The precision of about 1.5-2 x 10(-4) can be reached using a minimum of three HOLZ line patterns for the single "a" parameter and about 5 x 10(-4) for the complete set of lattice parameters using six diffraction patterns. The use of multiple patterns also allows overcoming the non-uniqueness of solution linked to the CBED studies.

Quantitative evaluation of process induced strain in MOS transistors by Convergent Beam Electron Diffraction

Convergent Beam Electron Diffraction (CBED) experiments and simulations associated with Finite Element calculations were performed in order to measure strain and stress in a complex device such as periodic MOS transistors with a spatial resolution of about 2 nm and a sensitivity that could reach 50 MPa. A lamella of a thickness of about 475 nm was extracted from the wafer with the transistors by Focus Ion Beam (FIB) and was observed in cross-section in a Transmission Electron Microscope (TEM). When approaching the transistors, the HOLZ lines of the CBED patterns acquired in the silicon substrate, become broader and broader. This HOLZ line broadening, which is due to the stress relaxation in the thin foil, was used to determine quantitatively the strain and stress in the lamella and then in the bulk device. We showed that this procedure could be applied to a complex device. Two parameters, the intrinsic material strains – or equivalently the intrinsic material stresses – in the nickel silicide (NiSi) and nitride (Si 3N 4) layers on the top of the transistors gate, were successfully fitted by trial and error, in the procedure.

A simple technique for the estimation of the voltage of a transmission electron microscope

A convergent beam electron diffraction (CBED) technique has been described for the determination of the voltage of a transmission electron microscope (TEM). It involves observing the intersections of HOLZ lines, (1 3 13), (3 1 13), (10 10 8) and (10 10 8), in the transmitted disc of the CBED pattern obtained along the [37 37 4] zone axis of silicon. It is suitable for the TEMs with operational voltage in the range of 200 kV, and is also free from the tilt limitations.

Strain field reconstruction in shallow trench isolation structures by CBED and LACBED

Using a combination of the CBED and the LACBED techniques in the transmission electron microscopy (TEM), we have investigated the strain field in the silicon active region of a shallow trench isolation structure, underlying a TiSi2 layer. Starting from the analysis of the deformation in a sample, thinned for TEM analysis, we have reconstructed the displacement field, simulating the split HOLZ lines visible in the experimental CBED patterns. From the comparison between the experimental LACBED patterns, taken in a suitable sample orientation to evidence the stressors distribution in the polycrystalline silicide layer, and the corresponding dynamically simulated ones, we have reproduced the strain field in the unthinned, bulk sample.

Prospects of the multislice method for CBED pattern calculation

We recently showed that the Multisclice method can be used to elucidate the crystal volume contribution to a CBED pattern. Here, we demonstrate in more detail how the directions of each HOLZ line, their Laue zone order, and the specimen thickness determine the volume segment of the Bragg scattering cone of the electron beam and show that secondary rescattering from this volume must take place for deficient line formation in the (000) CBED disk. Moreover, additional prospects of the multislice method are demonstrated such as calculation of line splitting in bent crystals, hollow cone dark-field CBED, large-angle CBED, and whole diffraction patterns including Kikuchi lines.

On the origin of HOLZ lines splitting near interfaces: multislice simulation of CBED patterns

Splitting of HOLZ lines on CBED patterns is systematically observed at the proximity of interfaces and prevents local strain measurements by monitoring of line shifts. It was previously suggested that such splitting occurs due to interface-strain relaxation in thin TEM lamella. Here we confirm this model by dynamical simulation of CBED patterns using the multislice algorithm.

On the peculiarities of CBED pattern formation revealed by multislice simulation

A modified multislice method has been developed for calculations of Convergent Beam Electron Diffraction (CBED) patterns. The validity of the method for HOLZ- and Kikuchi-line calculations has been proofed by comparison to Bloch-wave calculations. The application of the method leads to the new understanding of CBED patterns formation. Dynamical scattering of weak HOLZ reflections plays the key role in the appearance of deficient lines in the central CBED disk. Different HOLZ lines do have significantly different and extended scattering areas; the central 000 CBED disk, consequently, contains structural information from an area around the primary beam which is determined by the Bragg angle of HOLZ reflections and the thickness of the sample. A variation of lattice parameters, if present within this area, results in artificial symmetry violations of the pattern and in changes of line profiles.

Voltage refinement by deficit HOLZ line geometry

A method of refining electron beam voltage from deficit line intersections, using a quasi-kinematic scattering approximation, is described and applied to a silicon standard. It is shown that dynamical corrections are necessary for higher-index zone axes, such as 310 and 411, where a single deficit line associated with one dominant Bloch state is visible. This leads to a substantial difference in the refined voltage compared with that obtained from a purely kinematic approximation. Neglect of this dynamical correction term effectively invokes a systematic error that may lead to high precision but poor accuracy in higher-order Laue zone measurements of beam voltage or lattice parameters. Although relatively small, differences in the dynamical correction necessary for these two zones are confirmed experimentally for 100-300 keV electrons.

Observation of Splitting HOLZ Lines Around Shallow Trench Isolation Structures by Convergent Beam Electron Diffraction

Extended abstract of a paper presented at Microscopy and Microanalysis 2004 in Savannah, Georgia, USA, August 1–5, 2004.

Measuring the Thin Film Strain Tensor Near Aluminum Grain Boundaries via a New Image Processing Approach to CBED HOLZ Patterns

AbstractHOLZ lines formed in a CBED pattern provide the most accurate means to measure a local strain tensor with high spatial resolution over mesoscopic length scales. With the advent of energy-filtering in a field-emission TEM, the precision of this measurement increases by filtering out the inelastically scattered electrons. This paper presents an alternate approach to obtaining the same increased precision by image processing of CBED patterns formed in a conventional LaB6 microscope. This technique results in the determination of the full strain tensor within ±0.01%. It is based on developing a Wiener filter for CBED patterns, deconvoluting the point spread function of the CCD camera, using the Hough transform to measure distances between HOLZ line intersections, and subtracting out an experimentally determined projector lens distortion. The present technique has been used to measure the strain tensor for the two types of grain boundaries found in MBE grown Al thin films on Si which have the mazed bicrystal microstructure.

A study of interfacial reactions and interface residual strains in K2O·6TiO2w/Al composite

X-ray energy dispersive analyses show that the ratio of titanium to potassium in the interlayer is much higher than that of the whisker K2O·6TiO2, and that titanium and potassium atoms of the whisker are partly dissolved into the matrix Al near the whisker. The interface residual strains in the composite are determined in terms of the shifts of HOLZ lines. To uniquely evaluate the lattice parameters and the lattice strains from an experimental HOLZ line pattern, the plane stress approximation is employed to reduce the number of adjustable parameters in the optimization procedure. The results obtained with this method are in good agreement with other works. The possible explanations for the high von Mises effective stresses in the matrix near the whisker are discussed.