New approach for the dynamical simulation of CBED patterns in heavily strained specimens

A new method for the dynamical simulation of convergent beam electron diffraction (CBED) patterns is proposed. In this method, the three-dimensional stationary Schrödinger equation is replaced by a two-dimensional time-dependent equation, in which the direction of propagation of the electron beam, variable z, stands as a time. We demonstrate that this approach is particularly well-suited for the calculation of the diffracted intensities in the case of a z-dependent crystal potential. The corresponding software has been developed and implemented for simulating CBED patterns of various specimens, from perfect crystals to heavily strained cross-sectional specimens. Evidence is given for the remarkable agreement between simulated and experimental patterns.     

Mapping inelastic intensities in diffraction patterns of magnetic samples using the energy spectrum imaging technique

We present the quantitative measurement of inelastic intensity distributions in diffraction patterns with the aim of studying magnetic materials. The relevant theory based on the mixed dynamic form factor (MDFF) is outlined. Experimentally, the challenge is to obtain sufficient signal for core losses of 3d magnetic materials (in the 700-900eV energy-loss range). We compare two experimental settings in diffraction mode, i.e. the parallel diffraction and the large-angle convergent-beam electron diffraction configurations, and demonstrate the interest of using a spherical aberration corrector. We show how the energy spectrum imaging (ESI) technique can be used to map the inelastic signal in a data cube of scattering angle and energy loss. The magnetic chiral dichroic signal is measured for a magnetite sample and compared with theory.     

Convergent-beam and small-area-parallel-beam electron diffraction of icosahedral quasicrystals of a melt-quenched Al-Mn alloy

Intensity distribution in convergent-beam electron diffraction (CBED) patterns obtained from icosahedral quasicrystas of a melt-quenched Al-Mn alloy reveal that the quasicrystals do not have fivefold, threefold and twofold rotation axes and have no inversion center, although ordinary diffraction patterns obtained thus far showed these rotation symmetries. CBED patterns taken from specimen areas of about 3 nm in diameter show a deviation in geometry in spot positions from the fivefold rotation symmetry. Ring patterns due to higher-order Laue zone reflections are not observed in CBED patterns. Kikuchi bands are composed of two sub-bands in the five equivalent directions, and each band has a different intensity profile. Parallel-beam (3 × 10^-5 rad) electron diffraction patterns obtained from specimen areas less than 100 nm in diameter also show a deviation from the fivefold symmetry in spot positions and make clear that each Bragg reflection consists of many fine spots which show no fivefold symmetry. It is proven experimentally that all the observed reflections occur already in the approximation of kinematical diffraction, although their intensities may be modified by dynamical diffraction effect.     

Convergent-beam electron diffraction study of transverse stacking faults and dislocations

Transverse stacking faults and dislocations have been studied by convergent-beam electron diffraction (CBED). Stacking faults and dislocations induce splitting in some reflections in the CBED patterns. The splitting and unsplitting of the reflections correspond to the visibility and invisibility of the defect in the kinematic theory of diffraction contrast of imperfect crystals of Hirsch, Howie and Whelan. This method provides a powerful means for the study of crystal defects.     

Seeing atoms with aberration-corrected sub-Angström electron microscopy

High-resolution electron microscopy is able to provide atomic-level characterization of many materials in low-index orientations. To achieve the same level of characterization in more complex orientations requires that instrumental resolution be improved to values corresponding to the sub-Ångström separations of atom positions projected into these orientations. Sub-Ångström resolution in the high-resolution transmission electron microscope has been achieved in the last few years by software aberration correction, electron holography, and hardware aberration correction; the so-called “one-Ångström barrier” has been left behind. Aberration correction of the objective lens currently allows atomic-resolution imaging at the sub-0.8 Å level and is advancing towards resolutions in the deep sub-Ångström range (near 0.5 Å). At current resolution levels, images with sub-Rayleigh resolution require calibration in order to pinpoint atom positions correctly. As resolution levels approach the “sizes” of atoms, the atoms themselves will produce a limit to resolution, no matter how much the instrumental resolution is improved. By arranging imaging conditions suitably, each atom peak in the image can be narrower, so atoms are imaged smaller and may be resolved at finer separations.     

Effect of electron beam parameters on simulated CBED patterns from edge-on grain boundaries

Convergent beam electron diffraction (CBED) at vertical grain boundaries (parallel to the electron beam) can be applied to determine the symmetry of bicrystals. It can also be used to investigate the structure of the boundary region itself when subnanometre probe sizes are employed. In this paper it is shown that (sub)nanometre-probe CBED patterns are largely influenced by the electron-beam geometry. In particular, simulations of coherent CBED patterns based on the multislice algorithm show that the CBED pattern of an edge-on interface depends on the defocus distance between the probe position and the specimen midplane, the probe size and the beam-convergence angle. The pattern symmetry may be lower than the theoretically predicted symmetry in case of large spherical aberration. This effect increases with smaller accelerating voltages. An increase in the beam-convergence angle also increases the possibility of a non-optimum symmetry due to spherical aberration of a coherent probe. Thus, for the determination of an interface structure using subnanometre (coherent) probes, the imaging conditions play an important role.     

Fortran source listing for simulating three-dimensional convergent beam patterns with absorption by the Bloch wave method

The FORTRAN source code is given for a computer program that calculates the two-dimensional intensity distribution in convergent-beam transmission electron microdiffraction (CBED) patterns from perfect crystals. The program uses the eigenvalue or Bloch-ware method. It allows three-dimensional dynamical diffraction, and so includes all higher-order Laue zone effects without approximation. No symmetry reduction is included. The program accepts noncentrosymmetric or centrosymmetric crystal structures and allows absorption corrections to be included. It uses the “EISPACK” subroutines for the diagonalisation of a general complex matrix. Up to 100 CBED disks may be included. The code is also available via “Bitnet”.     

Effect of sample bending on diffracted intensities observed in CBED patterns of plan view strained samples

Convergent beam electron diffraction is used to study the effect of the sample bending on diffracted intensities as observed in transmission electron microscopy (TEM). Studied samples are made of thin strained semiconductor Ga(1-)(x)In(x)As epitaxial layers grown on a GaAs substrate and observed in plan view. Strong variations of the diffracted intensities are observed depending on the thinning process used for TEM foil preparation. For chemically thinned samples, strong bending of the substrate occurs, inducing modifications of both kinematical and dynamical Bragg lines. For mechanically thinned samples, bending of the substrate is negligible. Kinematical lines are unaffected whereas dynamical lines have slightly asymmetric intensities. We analyse these effects using finite element modelling to calculate the sample strain coupled with dynamical multibeam simulations for calculating the diffracted intensities. Our results correctly reproduce the qualitative features of experimental patterns, clearly demonstrating that inhomogeneous displacement fields along the electron beam within the substrate are responsible for the observed intensity modifications.     

A comparison of techniques for obtaining convergent beam electron diffraction patterns with a JEOL 200CX

Three different techniques for obtaining convergent beam electron diffraction (CBED) patterns using a JEOL 200CX transmission electron microscope will be described and compared. The first technique, described by Goodman, will be shown to yield clear, undistorted patterns, but only with relatively large camera length, and a limited field of view. A second technique, which is a modification of Goodman's technique, will be shown to yield CBED patterns of both large camera length and small camera length with a much larger angular coverage, but the magnitude of the beam convergence is limited by distortion in the pattern. A third technique will then be presented which permits the formation of small camera length, relatively undistorted CBED patterns with large angular coverage and greatly increased beam convergence; high quality large camera length CBED patterns can also be obtained by simply increasing the strengths of the diffraction lenses.     

Zone-axis diffraction patterns by the “Tanaka” method

The “Tanaka” method is one of several techniques that make it possible to obtain zone-axis electron diffraction patterns in a transmission electron microscope without the restriction in the field of view that limits normal convergent-beam diffraction patterns. The method employs a convergent-beam of electrons focused to a probe in a plane that does not coincide with the specimen. The selected area aperture can then be used to eliminate all but one of the diffracted beams to obtain the desired pattern. Practical details of operation and values of operating parameters are discussed. The Tanaka method is a useful addition to the techniques available to the electron microscopist, especially since no instrumental modification is required.     

Convergent-beam electron diffraction from faulted crystals of ZrS3

Convergent-beam electron diffraction has been applied to faulted crystals of ZrS₃. The CBED patterns were found to give more readily available crystallographic information than the spot patterns which are complicated by streaks produced by faults normal to the beam. Twins were identified, mirror symmetry was confirmed and the two modifications of ZrS₃ could be distinguished from CBED patterns. This example shows convergent-beam diffraction to be an essential method for obtaining crystallographic information from faulted crystals.     

Prospects for 3D, nanometer-resolution imaging by confocal STEM

Confocal STEM is a new electron microscopy imaging mode. In a microscope with spherical aberration-corrected electron optics, it can produce three-dimensional (3D) images by optical sectioning. We have adapted the linear imaging theory of light confocal microscopy to confocal STEM and use it to suggest optimum imaging conditions for a confocal STEM limited by fifth-order spherical aberration. We predict that current or near-future microscopes will be able to produce 3D images with 1 nm vertical resolution and sub-Angstrom lateral resolution. Multislice simulations show that we will need to be cautious in interpreting these images, however, as they can be complicated by dynamical electron scattering.     

Crystal structure determination by direct inversion of dynamical microdiffraction patterns

The intensity at points where coherent convergent-beam transmission diffraction discs overlap is shown to be described by interference between elements of the same row but different columns of the dynamical scattering matrix for an axial orientation. These intensities are used as the basis for an exact, nonperturbative inversion of the multiple electron scattering problem, allowing crystal structure factors to be obtained directly from the intensities of multiply scattered Bragg beams. Eigenvectors of the structure matrix are obtained using coherent CBED patterns from many crystal orientations. Unique eigenvalues are obtained from these patterns recorded at two accelerating voltages. The inevitable variation in electron probe position at different crystal tilts is considered. The analysis applies to centrosymmetric crystals with anomalous absorption, to centrosymmetric projections of acentric crystals and to acentric crystals if the mean absorption potential only is included. The method would allow the direct synthesis of charge-density maps of unknown crystal structures at high resolution from multiple scattering data, using a scanning transmission electron microscope (STEM). The resolution of this map may be much higher than the first-order d -spacing; however, the STEM need only be capable of resolving this first-order spacing. Such a charge-density map provides fractional atomic coordinates and the identification of atomic species (as in X-ray crystallography) from microcrystalline samples and other multiphase inorganic materials for which large single crystals cannot be obtained or X-ray powder patterns obtained or analysed. In summary, we solve the inversion problem of quantum mechanics for the case of electron scattering from a periodic potential, described by the nonrelativistic Schrödinger equation, in which the scattering is given as a function of some parameter, and the potential sought.     

Crystallographic electron microscopy

We present a family of techniques for the transmission electron microscope that generate surface zone-axis patterns. These patterns display the variation of the diffracted-beam intensity as a function of the angle of the incident electrons. The conditions of the experiments are those of reflection high-energy electron diffraction at near grazing incidence. The techniques are: surface convergent-beam diffraction, a surface analogue of the Tanaka method and a modified double-rocking scheme. Experimental results are presented for diffraction from surfaces of MgO and MoS₂. We anticipate that surface zone-axis patterns (surface ZAPs) will become established as an important tool for surface characterization, especially when used in conjunction with high-resolution surface imaging and surface energy loss spectroscopy; surface ZAPs may be expected to play, in surface analysis, a role analogous to that played by convergent-beam diffraction in normal transmission electron microscopy.     

Refinement of the 200 structure factor for GaAs using parallel and convergent beam electron nanodiffraction data

We present a new method to measure structure factors from electron spot diffraction patterns recorded under almost parallel illumination in transmission electron microscopes. Bloch wave refinement routines have been developed to refine the crystal thickness, its orientation and structure factors by comparison of experimentally recorded and calculated intensities. Our method requires a modicum of computational effort, making it suitable for contemporary personal computers. Frozen lattice and Bloch wave simulations of GaAs diffraction patterns are used to derive optimised experimental conditions. Systematic errors are estimated from the application of the method to simulated diffraction patterns and rules for the recognition of physically reasonable initial refinement conditions are derived. The method is applied to the measurement of the 200 structure factor for GaAs. We found that the influence of inelastically scattered electrons is negligible. Additionally, we measured the 200 structure factor from zero loss filtered two-dimensional convergent beam electron diffraction patterns. The precision of both methods is found to be comparable and the results agree well with each other. A deviation of more than 20% from isolated atom scattering data is observed, whereas close agreement is found with structure factors obtained from density functional theory [A. Rosenauer, M. Schowalter, F. Glas, D. Lamoen, Phys. Rev. B 72 (2005), 085326-1], which account for the redistribution of electrons due to chemical bonding via modified atomic scattering amplitudes.     

CBED and LACBED: characterization of antiphase boundaries

Convergent-beam electron diffraction (CBED) and large-angle convergent-beam electron diffraction (LACBED) techniques are well adapted to the characterization of several types of crystal defects. In fact, dislocations, grain boundaries and stacking faults have already been successfully characterized with these methods. In the present paper, we describe the CBED and LACBED characterization of another type of crystal defect showing a special interest in materials science: antiphase boundaries (APBs). The first part of the paper is devoted to the determination of the effects of antiphase boundaries on CBED and LACBED patterns that could be expected from a theoretical point of view. It indicates that the superlattice excess lines present on these patterns are split into two lines with equal intensity when the incident beam is located on an APB. In the second part, we experimentally test these theoretical predictions on a specimen showing two different known types of antiphase boundaries. In a third part we indicate how these methods could be used to identify unknown APBs in a specimen. Finally, the advantages and disadvantages of both methods for the characterization of antiphase boundaries are discussed.     

A practical method to detect and correct for lens distortion in the TEM

A practical, offline method for experimental detection and correction for projector lens distortion in the transmission electron microscope (TEM) operating in high-resolution (HR) and selected area electron diffraction (SAED) modes is described. Typical TEM works show that, in the simplest case, the distortion transforms on the recording device, which would be a circle into an ellipse. The first goal of the procedure described here is to determine the elongation and orientation of the ellipse. The second goal is to correct for the distortion using an ordinary graphic program. The same experimental data set may also be used to determine the actual microscope magnification and the rotation between SAED patterns and HR images. The procedure may be helpful in several quantitative applications of electron diffraction and HR imaging, for instance while performing accurate lattice parameter determination, or while determining possible metrical deviations (cell edges and angles) from a given symmetry.