Dark-field electron holography for the measurement of geometric phase

Precise positional alignment of atom-resolved HAADF images of heteroepitaxial interface with low signal-to-noise ratio

Heteroepitaxial interfaces are important because they determine the performance of devices such that career mobility is sensitive to the distribution of roughness, strain and composition at the interface. High-angle annular dark field imaging in scanning transmission electron microscopy has been utilized to capture them at an atomic scale. For precise identification of atomic column positions, a technique has been proposed to average multiple image frames taken at a high scanning rate by their positional alignment for increasing signal-to-noise ratio. However, the positional alignment between frames is sometimes incorrectly estimated because of the almost perfect periodic structure at the interfaces. Here, we developed an approach for precise positional alignment, where the images are first aligned by two consecutive images and then are aligned more precisely against the integrated image of the first alignment. We demonstrated our method by applying it to the heterointerface of Si0.8Ge0.2 (Si: silicon, Ge: germanium) epitaxial thin films on a Si substrate.

Geometric phase analysis of magnetic skyrmion lattices in Lorentz transmission electron microscopy images

Magnetic skyrmions are quasi-particles with a swirling spin texture that form two-dimensional lattices. Skyrmion lattices can exhibit defects in response to geometric constraints, variations of temperature or applied magnetic fields. Measuring deformations in skyrmion lattices is important to understand the interplay between the lattice structure and external influences. Geometric phase analysis (GPA) is a Fourier-based image processing method that is used to measure deformation fields in high resolution transmission electron microscopy (TEM) images of crystalline materials. Here, we show that GPA can be applied quantitatively to Lorentz TEM images of two-dimensional skyrmion lattices obtained from a chiral magnet of FeGe. First, GPA is used to map deformation fields around a 5-7 dislocation and the results are compared with the linear theory of elasticity. Second, rotation angles between skyrmion crystal grains are measured and compared with angles calculated from the density of dislocations. Third, an orientational order parameter and the corresponding correlation function are calculated to describe the evolution of the disorder as a function of applied magnetic field. The influence of sources of artifacts such as geometric distortions and large defoci are also discussed.

Relation between sampling, sensitivity and precision in strain mapping using the Geometric Phase Analysis method in Scanning Transmission Electron Microscopy

The sensitivity and the precision of the Geometric Phase Analysis (GPA) method for strain characterization is a topic widely discussed in the literature and is usually difficult to quantify. Indeed, the GPA precision is intricately linked to the resolution of the strain maps defined when masking the periodic reflections in Fourier space. In this study an additional parameter, sampling, is proposed to be analyzed regarding the precision of GPA by developing the concept of a phase noise in the GPA equations. Both experimentally and theoretically, the following article demonstrates how the precision, and the sensitivity of the GPA method is improved when using a larger pixel spacing to record an electron micrograph in Scanning Transmission Electron Microscopy (STEM). The counterintuitive concept of increasing the field of view to improve the GPA precision results is an extension of the application of strain characterization methods in STEM towards low deformation levels.

Reconstruction of Angstrom resolution exit-waves by the application of drift-corrected phase-shifting off-axis electron holography

Phase-shifting electron holography is an excellent method to reveal electron wave phase information with very high phase sensitivity over a large range of spatial frequencies. It circumvents the limiting trade-off between fringe spacing and visibility of standard off-axis holography. Previous implementations have been limited by the independent drift of biprism and sample. We demonstrate here an advanced drift correction scheme for the hologram series that exploits the presence of an interface of the TEM specimen to the vacuum area in the hologram. It allows to obtain reliable phase information up to 2π/452 at the 1 Å information limit of the Titan 80-300 kV environmental transmission electron microscope used, by applying a moderate voltage of 250 V to a single biprism for a fringe spacing of 1 Å. The obtained phase and amplitude information is validated at a thin Pt sample by use of multislice image simulation with the frozen lattice approximation and shows excellent agreement. The presented method is applicable in any TEM equipped with at least one electron biprism and thus enables achieving high resolution off-axis holography in various instruments including those for in-situ applications. A software implementation for the acquisition, calibration and reconstruction is provided.

3D Strain Measurement of Heterostructures Using the Scanning Transmission Electron Microscopy Moiré Depth Sectioning Method

The mechanical properties of micro- and nanoscale materials directly determine the reliability of heterostructures, microstructures, and microdevices. Therefore, an accurate evaluation of the 3D strain field at the nanoscale is important. In this study, a scanning transmission electron microscopy (STEM) moiré depth sectioning method is proposed. By optimizing the scanning parameters of electron probes at different depths of the material, the sequence STEM moiré fringes (STEM-MFs) with a large field of view, which can be hundreds of nanometers obtained. Then, the 3D STEM moiré information constructed. To some extent, multi-scale 3D strain field measurements from nanometer to the submicrometer scale actualized. The 3D strain field near the heterostructure interface and single dislocation accurately measured by the developed method.

Strain Measurement in Single Crystals by 4D-ED

A new method is presented to measure strain over a large area of a single crystal. The 4D-ED data are collected by recording a 2D diffraction pattern at each position in the 2D area of the TEM lamella scanned by the electron beam of STEM. Data processing is completed with a new computer program (available free of charge) that runs under the Windows operating system. Previously published similar methods are either commercial or need special hardware (electron holography) or are based on HRTEM, which involves limitations with respect to the size of the field of view. All these limitations are overcome by our approach. The presence of defects results in small local changes in orientation that change the subset of experimentally available diffraction spots in the individual patterns. Our method is based on a new principle, namely fitting a lattice to (a subset of) measured diffraction spots to improve the precision of the measurement. Although a spot to be measured may be missing in some of the patterns even the missing spot can be precisely measured by the lattice determined from the available spots. Application is exemplified by heavily boron-doped silicon with intended usage as a low-temperature superconductor in qubits.

STEM multiplication nano-moiré method with large field of view and high sensitivity

Strain is one of the important factors that determine the photoelectric and mechanical properties of semiconductor materials and devices. In this paper, the scanning transmission electron microscopy multiplication nano-moiré method is proposed to increase the measurement range and sensitivity for strain field. The formation principle, condition, and measurement range of positive and negative multiplication moiré fringes (PMMFs and NMMFs) are analysed in detail here. PMMF generally refers to the multiplication of field of view, NMMF generally refers to the multiplication of displacement measurement sensitivity. Based on the principle of multiplication nano-moiré, Theoretical formulas of the fringe spacing and strain field are derived. Compared with geometric phase analysis of deformation measurements based on high-resolution atom images, both the range of field of view and the sensitivity of displacement measurements of the multiplication moiré method are significantly improved. Most importantly, the area of field of view of the PMMF method is increased by about two orders of magnitude, which is close to micrometre-scale with strain measurement sensitivity of 2 × 10^-5. In addition, In order to improve the quality of moiré fringe and the accuracy of strain measurement, the secondary moiré method is developed.The strain laws at the interface of the InP/InGaAs superlattice materials are characterised using the developed method.

Assessment of the strain depth sensitivity of Moiré sampling Scanning Transmission Electron Microscopy Geometrical Phase Analysis through a comparison with Dark-Field Electron Holography

In this study, the Moiré sampling Scanning Transmission Electron Microscopy Geometrical Phase Analysis (or STEM Moiré GPA) strain characterization method is compared to the well-established Dark-Field Electron Holography technique on a thin film stack grown by Molecular Beam Epitaxy. While experimental data obtained with the two techniques are, overall, in good qualitative agreement, small statistically relevant differences are locally observed between the two methods. The results obtained from both techniques are further confronted with Finite Element Method (FEM) mechanical simulations modeling the strain relaxation phenomena from a thin lamella. The FEM simulation highlights a non-uniform deformation field along the beam propagation direction with a higher deformation level near the surface of the lamella compared to the center of the same lamella. The center-surface strain differences obtained from modeling are consistent with the experimentally derived differences accounting for the fact that the SMG method is sensitive to the strain state of the surface of the lamella with a very narrow depth-of-field, and the DFEH technique is measuring the strain state of the center of the same lamella averaging over a large section of the thickness. The depth-of-field difference between both methods can be reasonably related to their respective contrast mechanisms (STEM vs Conventional Transmission Electron Microscopy). As the SMG method is using a convergent probe, the narrow depth-of-field might be used to sense the deformation field over different sections of the lamella using the defocus and potentially retrieve the three-dimensional strain field.

Phase detection limits in off-axis electron holography from pixelated detectors: gain variations, geometric distortion and failure of reference-hologram correction

We investigate the effect that recording off-axis electron holograms on pixelated detectors, such as charge-coupled devices (CCD) and direct-detection devices (DDD), can have on measured amplitudes and phases. Theory will be developed for the case of perfectly uniform interference fringes illuminating an imperfect detector with gain variations and pixel displacements. We will show that both these types of defect produce a systematic noise in the phase images that depends on the position of the holographic fringes with respect to the detector. Subtracting a reference hologram from the object hologram will therefore not remove the phase noise if the initial phases of the two holograms do not coincide exactly. Another finding is that pi-shifted holograms are much less affected by gain variations but show no improvement concerning geometric distortions. The resulting phase errors will be estimated and simulations presented that confirm the theoretical developments.

Dark-field electron holography as a recording of crystal diffraction in real space: a comparative study with high-resolution X-ray diffraction for strain analysis of MOSFETs

Diffraction-based techniques, with either electrons or photons, are commonly used in materials science to measure elastic strain in crystalline specimens. In this paper, the focus is on two advanced techniques capable of accessing strain information at the nanoscale: high-resolution X-ray diffraction (HRXRD) and the transmission electron microscopy technique of dark-field electron holography (DFEH). Both experimentally record an image formed by a diffracted beam: a map of the intensity in the vicinity of a Bragg reflection spot in the former, and an interference pattern in the latter. The theory that governs these experiments will be described in a unified framework. The role of the geometric phase, which encodes the displacement field of a set of atomic planes in the resulting diffracted beam, is emphasized. A detailed comparison of experimental results acquired at a synchrotron and with a state-of-the-art transmission electron microscope is presented for the same test structure: an array of dummy metal–oxide–semiconductor field-effect transistors (MOSFETs) from the 22 nm technology node. Both techniques give access to accurate strain information. Experiment, theory and modelling allow the illustration of the similarities and inherent differences between the HRXRD and DFEH techniques.

Study of interfacial random strain fields in core-shell ZnO nanowires by scanning transmission electron microscopy

Imaging strain fields at the nanoscale is crucial for understanding the physical properties as well as the performance of oxide heterostructures and electronic devices. Based on scanning transmission electron microscopy (STEM) techniques, we successfully imaged the random strain field at the interface of core-shell ZnO nanowires. Combining experimental observations and image simulations, we find that the strain contrast originates from dechanneling of electrons and increased diffuse scattering induced by static atomic displacements. For a thin sample with a random strain field, a positive strain contrast appears in the low-angle annular dark-field (LAADF) image and a negative contrast in the high-angle annular dark-field (HAADF) image, but for a thick sample (> 120 nm), the positive contrast always occurs in both the LAADF and HAADF images. Through the analysis of the relationship between strain contrast and various parameters, we also discuss the optimum experimental condition for imaging random strain fields.

Probing local order in multiferroics by transmission electron microscopy

The ongoing trend toward miniaturization has led to an increased interest in the magnetoelectric effect, which could yield entirely new device concepts, such as electric field-controlled magnetic data storage. As a result, much work is being devoted to developing new robust room temperature (RT) multiferroic materials that combine ferromagnetism and ferroelectricity. However, the development of new multiferroic devices has proved unexpectedly challenging. Thus, a better understanding of the properties of multiferroic thin films and the relation with their microstructure is required to help drive multiferroic devices toward technological application. This review covers in a concise manner advanced analytical imaging methods based on (scanning) transmission electron microscopy which can potentially be used to characterize complex multiferroic materials. It consists of a first broad introduction to the topic followed by a section describing the so-called phase-contrast methods, which can be used to map the polar and magnetic order in magnetoelectric multiferroics at different spatial length scales down to atomic resolution. Section 3 is devoted to electron nanodiffraction methods. These methods allow measuring local strains, identifying crystal defects and determining crystal structures, and thus offer important possibilities for the detailed structural characterization of multiferroics in the ultrathin regime or inserted in multilayers or superlattice architectures. Thereafter, in Section 4, methods are discussed which allow for analyzing local strain, whereas in Section 5 methods are addressed which allow for measuring local polarization effects on a length scale of individual unit cells. Here, it is shown that the ferroelectric polarization can be indirectly determined from the atomic displacements measured in atomic resolution images. Finally, a brief outlook is given on newly established methods to probe the behavior of ferroelectric and magnetic domains and nanostructures during in situ heating/electrical biasing experiments. These in situ methods are just about at the launch of becoming increasingly popular, particularly in the field of magnetoelectric multiferroics, and shall contribute significantly to understanding the relationship between the domain dynamics of multiferroics and the specific microstructure of the films providing important guidance to design new devices and to predict and mitigate failures.

Dynamical diffraction effects on the geometric phase of inhomogeneous strain fields

In specimens with an inhomogeneous displacement field in electron beam direction dynamical diffraction effects lead to complex non-linear properties of the diffracted electron wave. Consequently, the diffracted beam's phase contains information about the inhomogeneous displacement field. These phases are experimentally and theoretically investigated under different excitation errors and specimen thicknesses as well as for different depths of the displacement field. An inclined InGaAs layer with a larger lattice constant than the surrounding GaAs matrix serves as controlled displacement field, which is inhomogeneous in electron beam direction with a continuously changing depth. The phase and amplitude of the diffracted beam are measured by dark-field electron holography. The measurements agree with calculations performed by numerical propagation of the electron wave using the Darwin-Howie-Whelan equations. A strong dependency on the excitation conditions is found showing that the interplay between dynamical effects and the strain field must be considered in the interpretation of the geometric phase.

A single slice approach for simulating two-beam electron diffraction of nanocrystals

A simple computational method that can be used to simulate TEM image contrast of an electron beam diffracted by a crystal under two-beam dynamical scattering conditions is presented. The approach based on slicing the shape factor is valid for a general crystal morphology, with and without crystalline defects, avoids the column approximation, and provides the complex exit wave at the focal and the image planes also under weak-beam conditions. The approach is particularly efficient for large crystals and the 3D model required for the calculations can be measured experimentally using electron tomography. The method is applied to show that the shape of a diffracted spot can be affected by shifts, broadening and secondary maxima of appreciable intensity, even for a perfect crystal. The methodology is extended for the case of electron precession diffraction, and to show how can be used to improve nanometrology from diffraction patterns. The method is used also to perform simulations of simple models of crystalline defects. The accuracy of the method is demonstrated through examples of experimental and simulated dark-field images of MgO and ZrO₂ nanocrystals and thin layers of CeO₂.

Strain mapping in a nanoscale-triangular SiGe pattern by dark-field electron holography with medium magnification mode

Recent years have seen a great deal of progress in the development of transmission electron microscopy-based techniques for strain measurement. Dark-field electron holography (DFEH) is a new technique offering configuration of the off-axis principle. Using this technique with medium magnification (Holo-M), we carried out strain measurements in nanoscale-triangular SiGe/(001) Si with (004), (2^-20) and (^-111) diffraction spots. The reconstruction of holograms and interpretation of strain maps in term of strain precision were discussed and the strain distributions in the SiGe/(001) Si patterns were visualized. Based on linear anisotropic elastic theory for strain simulation, the simulated results obtained by the finite element method compared with the experimental results acquired by DFEH. The strain values were found to be 0.9-1.0%, 1.1-1.2% and 1.0-1.1%, for the (004), (2^-20) and (^-111) diffracted beams, respectively, and the strain precisions were determined to be ~2.1 × 10^-3, 3.2 × 10^-3 and 9.1 × 10^-3 for the corresponding diffraction spots. As a result, DFEH is highlighted as a powerful technique for strain measurement, offering high-strain precision, high-spatial resolution and a large field of view.

Differential phase-contrast dark-field electron holography for strain mapping

Strain mapping is an active area of research in transmission electron microscopy. Here we introduce a dark-field electron holographic technique that shares several aspects in common with both off-axis and in-line holography. Two incident and convergent plane waves are produced in front of the specimen thanks to an electrostatic biprism in the condenser system of a transmission electron microscope. The interference of electron beams diffracted by the illuminated crystal is then recorded in a defocused plane. The differential phase recovered from the hologram is directly proportional to the strain in the sample. The strain can be quantified if the separation of the images due to the defocus is precisely determined. The present technique has the advantage that the derivative of the phase is measured directly which allows us to avoid numerical differentiation. The distribution of the noise in the reconstructed strain maps is isotropic and more homogeneous. This technique was used to investigate different samples: a Si/SiGe superlattice, transistors with SiGe source/drain and epitaxial PZT thin films.

Calibration for medium resolution off-axis electron holography using a flexible dual-lens imaging system in a JEOL ARM 200F microscope

In this work the calibration of a medium resolution off-axis electron holography using a dual-lens imaging system in a JEOL ARM 200F is shown. The objective dual-lens configuration allows adjusting the field of view from 35nm to 2.5μm. Subsequently, the parameters used in phase shift reconstruction were calibrated considering biprism voltage versus fringe spacing (σ) and versus fringe width (W). The reliability of the transmission electron microscope performance using these parameters was achieved using gold nanoparticles of known size and adjusting the excitation voltage of the lenses.

Determination of the surface morphology of gold-decahedra nanoparticles using an off-axis electron holography dual-lens imaging system

The purpose of this paper is to show surface irregularities in gold decahedra nanoparticles extracted by using off-axis electron holography in a JEOL ARM 200F microscope. Electron holography has been used in a dual-lens system within the objective lenses: main objective lens and objective minilens. Parameters such as biprism voltage, fringe spacing (σ), fringe width (W) and optimum fringe contrast have been calibrated. The reliability of the transmission electron microscope performance with these parameters was carried out through a plug-in in the Digital-Micrograph software, which considers the mean inner potential within the particle leading a precise determination of the morphological surface of decahedral nanoparticles obtained from the reconstructed unwrapped phase and image processing. We have also shown that electron holography has the capability to extract information from nanoparticle shape that is currently impossible to obtain with any other electron microscopy technique.

Full-Field Strain Mapping at a Ge/Si Heterostructure Interface

The misfit dislocations and strain fields at a Ge/Si heterostructure interface were investigated experimentally using a combination of high-resolution transmission electron microscopy and quantitative electron micrograph analysis methods. The type of misfit dislocation at the interface was determined to be 60° dislocation and 90° full-edge dislocation. The full-field strains at the Ge/Si heterostructure interface were mapped by using the geometric phase analysis (GPA) and peak pairs analysis (PPA), respectively. The effect of the mask size on the GPA and PPA results was analyzed in detail. For comparison, the theoretical strain fields of the misfit dislocations were also calculated by the Peierls-Nabarro and Foreman dislocation models. The results showed that the optimal mask sizes in GPA and PPA were approximately three tenths and one-tenth of the reciprocal lattice vector, respectively. The Foreman dislocation model with an alterable factor a = 4 can best describe the strain field of the misfit dislocation at the Ge/Si heterostructure interface.