Suppressing Geometric Phase Shift Owing to Antiphase Boundaries in Dark-Field Electron Holography

Abstract

Antiphase boundaries (APBs) formed in Fe₇₀Al₃₀ alloy have attracted significant attention because they intensify ferromagnetic spin order by atomic disordering. Electron holography can be a useful tool for examining the magnetism in APB regions, although the observations suffer from an undesired contribution from an additional geometric phase shift in the incident electron wave. This paper proposes a method based on dark-field electron holography to suppress the unwanted geometric phase shift owing to APBs in electron holograms. The results of this study yield useful information for examining APBs and other planar defects in ordered crystals.

Fig. 5 Electron holograms and reconstructed phase images obtained from the rectangular region (object region) shown in Figs. 4(a)–4(b). (a)–(c) Electron holograms acquired by using 000, 100, and 200 spots, respectively. (d)–(f) Reconstructed phase images obtained from the holograms shown in (a)–(c), respectively.

1. Introduction

Engineering of interfaces, grain boundaries, and other planar defects is important for improvements in material functions, such as deformability,¹⁾ magnetism,²⁾ and electrical conductivity.³⁾ Antiphase boundaries (APBs) formed in ordered alloys are one such subject of intensive studies relevant to planar defects in alloys. APBs in ferromagnetic systems, for instance, can induce intriguing phenomena, such as the pinning of magnetic domain walls⁴⁾ and magnetoresistance.⁵⁾ These phenomena are a result of the significant suppression of ferromagnetic spin order in the APB region, in which the degree of atomic ordering is reduced. This relationship has been observed in several systems.^6–10) An exceptional case is B2-type Fe–Al alloys in which magnetization is not suppressed but rather enhanced in the APB region.¹¹⁾

There have been several comprehensive studies on the magnetism and structure of APBs in B2-type Fe–Al alloys.^12–20) According to a transmission electron microscopy (TEM) study,¹¹⁾ thermally-induced APBs show a finite width, approximately 2 nm, in which the degree of atomic order is suppressed as compared to the matrix region that is free of APBs. Magnetism in the narrow APB region has been examined using electron holography¹¹⁾ which measured the phase shift of an incident electron wave traversing a thin-foil specimen;^21,22) refer to Fig. 1. As demonstrated in the plot of phase shift in Fig. 1(b), in which the slope changes in the APB region, the magnetic flux density is amplified in the narrow APB region.¹¹⁾

Fig. 1

APBs produced in a B2-type Fe₇₀Al₃₀ alloy. (a) Bright-field image of a thin-foil specimen showing APBs. (b) Plot of phase shift measured along a line crossing the APB indicated by arrows in (a). Both matrix 1 and matrix 2 represent regions free from APBs. Reprinted from Ref. 11 with permission.

Although electron holography can provide essential information on magnetism in the APB regions, there is one crucial issue associated with data acquisition and analysis. In crystalline specimens, an undesirable geometric phase shift due to structural imperfection (e.g., irregular atomic sequence in ordered systems having APBs) may be superposed on the intrinsic phase shift by a magnetic flux. For example, according to kinematical diffraction theory,²³⁾ an APB produced in a B2-type crystal results in a significant geometric phase shift of π. Image processing using a set of observations collected at different temperatures below and above the Curie temperature allows for the extraction of temperature-dependent magnetic information, as demonstrated in Fig. 1(b); refer to Ref. 11 for further details. However, this method cannot be applied to specimens showing high Curie temperature and/or those showing structural deterioration upon heating. Thus, a method is needed that allows for the suppression of the undesired contribution of the geometric phase shift. The aim of this study was to demonstrate a process based on dark-field electron holography (DFEH)²⁴⁾ that enables suppression of the geometric phase shift caused by APBs in holography observations.

2. Methods

2.1 Specimen preparation and electron microscopy

An ingot of Fe₇₀Al₃₀ alloy was heat-treated at 1473 K for 12 h followed by quenching in ice water. TEM study revealed that this quenched alloy was in the B2 phase showing well-defined APBs related to the B2/A2 phase transformation.¹¹⁾ The average size of the antiphase domains was approximately 100 nm. Refer to the previous report for details of the crystallographic microstructure of this alloy.¹¹⁾

Thin-foil specimens for TEM studies were prepared by electrochemical polishing. Electron holograms were collected from the area of the specimen where the thickness was approximately 100 nm or smaller; in thicker regions, the contrast for the electron hologram was insufficient.¹¹⁾ Electron holography and conventional TEM observations were carried out using a 300 kV electron microscope (HF3300S, Hitachi). Dark-field electron holograms were collected using the Bragg reflection (selected by a conventional objective aperture) from the back-focal plane beneath the objective lens. The holograms were produced using two electron biprisms in the HF3300S microscope, which enabled removal of unwanted Fresnel fringes. Details of the double-biprism electron holography optics are described elsewhere.²⁵⁾ The images were recorded using a charge-coupled device camera (USC 4000, Gatan).

2.2 Dark-field electron holography

Electron holography is briefly explained to illustrate the difference between DFEH and conventional electron holography. In both methods, an electron hologram is generated via interference of an object electron wave with a reference electron wave, as shown in Fig. 2. The phase information (i.e., phase shift in the object wave) can be obtained using Fourier transform with the electron hologram.²¹⁾ As the phase shift is caused, in principle, by the magnetic field (or electric field) in a thin-foil specimen, electron holography is a useful tool for revealing magnetic nanostructures. An additional phase shift in the electron wave arises from the geometric imperfection induced by lattice defects, and an APB is one such source resulting in a significant geometric phase shift. Note that the geometric phase shift can be carried by electron diffraction. More importantly, the magnitude of the geometric phase shift in the electron wave depends on Bragg reflections, which are excited, as described later in further detail.

Fig. 2

Schematic representation of (a) conventional electron holography (CEH) and (b) dark-field electron holography (DFEH). While CEH uses the transmitted beam to produce holograms, DFEH uses a Bragg reflection which carries information about the geometric phase shift.

In conventional electron holography [Fig. 2(a)], the object wave is the transmitted beam (not the diffracted beam), which traverses the thin-foil specimen, whereas the reference wave is in a vacuum region outside the specimen. For DFEH,²⁴⁾ an electron hologram is produced by using the diffracted beam [e.g., such as the 200 Bragg reflection shown in Fig. 2(b)] instead of the transmitted beam. Accordingly, the electron hologram is a type of dark-field image. DFEH reveals the phase difference of the diffracted beams which have traversed the object region and the reference region in a thin-foil specimen as shown in Fig. 2(b). Assuming a simple configuration in which a metallic thin-foil specimen is magnetized in one direction and the thickness variation is negligible, DFEH is sensitive to the APB (showing a change in the phase) present in the object region [Fig. 2(b)] where the reference region is free from APBs.

Notably, DFEH enables mapping of elastic strain in the crystal as the diffracted beam carries the phase shift information owing to lattice deformation.²⁴⁾ This strain mapping has been successfully applied to semiconductors,²⁶⁾ ferroelectrics,²⁷⁾ and permanent magnets.²⁸⁾ For the APB region in a Fe₇₀Al₃₀ alloy, the phase shift due to strain can be negligibly small (i.e., smaller than the uncertainty of the phase measurement). Thus, the specimen configuration in this study facilitated intensive exploration of the geometric phase shift due to APBs, which is much larger than the strain contribution.

3. Results and Discussion

3.1 Geometric phase shift in APB region

In this study, for collecting electron holograms, the specimen was placed in the objective lens, which produced a strong magnetic field (∼1.6 × 10⁶ A/m) in the direction of the incident electrons (z direction). Because of the small magnetocrystalline anisotropy in the B2-type Fe–Al alloy (3.5 × 10⁴ erg/cm³),¹⁶⁾ the specimen could be magnetized approximately in the z direction. As electron holography is insensitive to the z component of the magnetic flux,²¹⁾ the magnetized specimen provided only a negligible phase shift due to the magnetic field. Moreover, the specimen thickness only changed gradually over the area of data acquisition, hence the phase shift due to the electric field (the mean inner potential) could be sufficiently small. Under these simple experimental conditions, the geometric phase shift due to the APBs is the major signal in the electron holography observations.

Strictly speaking, the specimen surface is the singular region in which the in-plane (x-y plane) component of magnetization can remain intact.²⁹⁾ The residual in-plane magnetization, which may exist only in a limited region near the surface, can provide an additional phase shift. However, as will be discussed in greater detail later using Figs. 5 and 6, we could not recognize an appreciable phase shift due to surface spin textures within the field of view. In other words, only the geometric phase shift due to the APBs was pronounced when the superlattice 100 reflection was used in the acquisition of the dark-field holograms in this study.

As already mentioned, the magnitude of the geometric phase shift depends on the Bragg reflections used to generate the dark-field electron hologram. In other words, the undesired geometric phase shift can be suppressed by selecting a Bragg reflection that is insensitive to APBs. Within the framework of kinematical electron diffraction theory (particularly, in the two-beam excitation condition),²³⁾ the magnitude of the phase shift (α) due to an APB is given by   

\begin{equation} \alpha = 2\pi \boldsymbol{{g}}_{\boldsymbol{{hkl}}} \cdot \boldsymbol{{R}}, \end{equation} (1)

where g_hkl represents a reciprocal lattice vector relevant to the hkl plane that excites a Bragg reflection, and R is a vector representing the displacement by the lattice imperfection; see Fig. 3, which schematically shows the displacement vector R in a B2-type crystal. For APBs produced in a B2-type ordered crystal, the magnitude of R corresponds to 1/2⟨111⟩.^11,23) Accordingly, with reference to the superlattice reflections (e.g., the 100 Bragg reflection), there is a significant phase shift of π between the diffracted waves ϕ₁ and ϕ₂, which traverse the left (matrix 1) and right (matrix 2) sides of the APB, respectively; see the illustration in Fig. 3 showing an APB slightly tilted off the incident electron wave. However, for fundamental reflections (e.g., the 200 Bragg reflection), the phase shift between ϕ₁ and ϕ₂ is zero. Thus, we expect that the geometric phase shift due to the APBs is negligible in a dark-field electron hologram produced using a fundamental reflection. Conversely, the geometric phase shift can be maximized by using a superlattice reflection.

Fig. 3

Schematic illustration of an APB. Open and closed circles represent the two atomic sites in a B2-type ordered structure. Thermally-induced APB showing a finite thickness is simply depicted using a solid line.

3.2 Relationship between geometric phase shift and reflections in Fe₇₀Al₃₀ alloy

Figure 4 shows conventional TEM images obtained in the (a) bright-field mode and (b), (c) dark-field mode of imaging. When collecting these images, h00 Bragg reflections were systematically excited, as shown in the insets, although other weak diffraction spots were also observed. The spots used in the bright-field and dark-field imaging are indicated by circles in the figure. As mentioned in the previous section, APBs could be imaged by the dark-field method using the 100 superlattice reflection (the meandering lines tracing APBs in Fig. 4(b)). However, the APBs were obscured in the dark-field image produced using the 200 fundamental reflection, as shown in Fig. 4(c). It should be noted that a very weak contrast remains in some positions of APBs in Fig. 4(c). It is likely that this residual contrast is due to the deviation from the ideal two-beam excitation condition (a simple condition within the framework of kinematical diffraction theory) in the presence of bending in the thin-foil specimen. A weak contrast tracing APBs was observed in the bright-field image as well [Fig. 4(a)].

Fig. 4

Images using distinct types of diffraction spots. (a) Bright-field image using the 000 transmitted beam. (b) Dark-field image using the 100 superlattice reflection. (c) Dark-field image using the 200 fundamental reflection. (d) Illustration showing the object region and reference region used in the collection of electron holograms in Figs. 5(a)–5(c). Insets in Figs. 4(a)–4(b) show the diffraction condition and the spots used in the imaging. A small aperture was used so that only one diffraction spot was selected in each case.

The phase shift in the APB region was evaluated for all three imaging conditions shown in Figs. 4(a)–4(c). Figures 5(a) and 5(d) show an electron hologram and a reconstructed phase image, both obtained using a transmitted beam (the 000 spot). The hologram was collected from a rectangular region (object region) containing an APB as delineated by the white lines in Fig. 4(a). Note that, as illustrated in Fig. 4(d), the hologram was obtained via interference with the neighboring reference region (not vacuum, but in the specimen) free from APBs. The phase shift owing to the APBs was unclear with reference to the reconstructed phase image in Fig. 5(d). The other panels in Fig. 5 present DFEH observations obtained from the same object/reference regions but using distinct reflections. The 100 superlattice reflection produced the dark-field electron hologram and reconstructed phase image shown in Figs. 5(b) and 5(e), respectively. As expected from the well-defined diffraction contrast in Figs. 4(b) and 5(b), the reconstructed phase image [Fig. 5(e)] showed a significant change in phase in the APB position. However, the phase shift owing to the APB became obscure in the other set of DFEH observations [Figs. 5(c) and 5(f)], which were generated using the fundamental 200 reflection.

Fig. 5

Electron holograms and reconstructed phase images obtained from the rectangular region (object region) shown in Figs. 4(a)–4(b). (a)–(c) Electron holograms acquired by using 000, 100, and 200 spots, respectively. (d)–(f) Reconstructed phase images obtained from the holograms shown in (a)–(c), respectively.

For deeper understanding of the relationship between phase shift and reflections, Fig. 6 plots the phase shift measured along the X–Y line crossing the APB; see Fig. 5. With reference to the plot obtained using the 100 superlattice reflection [Fig. 6(b)], a significant change in the phase (Δϕ ∼ 2.3 rad) was observed in the APB position. The observed phase shift due to the APB was smaller than the value predicted by the kinematical diffraction theory assuming an ideal B2-type structure (i.e., π rad). This deviation can be attributed to several factors, including excitation errors in the two-beam condition (i.e., effect of more complex multiple electron scattering), off-stoichiometry in the B2-type Fe₇₀Al₃₀ alloy causing a deviation from the ideal B2-type structure, and the degree of atomic order in the matrix, which depends on the heat treatment used for the specimen. Despite the deviations in magnitude, the plot in Fig. 6(b) demonstrates that the geometric phase shift due to the APB obtained using the superlattice reflection is pronounced.

Fig. 6

Plots of phase shift measured along the X–Y line indicated in Figs. 5(d)–5(f). (a) Phase plot obtained using the 000 transmitted beam. (b) Phase plot obtained using the 100 superlattice reflection. (c) Phase plot obtained using the 200 fundamental reflection.

In contrast, the observation using the 200 fundamental reflection [Fig. 6(c)] did not show a step in the APB region. There was no appreciable change in phase greater than the measurement uncertainty (±0.3 rad) evaluated by the deviations between the observation and background fitting as indicated by the dotted line in the figure. This indicates that DFEH using a fundamental reflection provides a route for suppressing the undesired geometric phase shift in the reconstructed phase image.

The plot in Fig. 6(a), obtained using the 000 transmitted beam, showed only a small change in phase (i.e., 0.3 rad or smaller) near the APB. From the contrast variations in the original holograms shown in Fig. 5(a) (owing to surface roughness of the thin-foil specimen and other imperfections), some artificial phase shift could be added to the reconstructed phase image/profile. Accordingly, it is difficult to determine the major source of the small phase shift of 0.3 rad observed near the APB in Fig. 6(a). Despite the ambiguity, when the transmitted beam is selected for generating a hologram, the geometric shift due to APBs is sufficiently small as compared to the significant value of Δϕ in Fig. 6(b). This is useful information for various experiments of phase retrieval that employ systematic excitations of Bragg reflections.

3.3 DFEH in APB studies

DFEH using a fundamental reflection provides a way to reduce unwanted geometric phase shifts due to APBs, as indicated earlier. For simplicity, the present study used a specimen approximately magnetized in the z direction, which produced only a negligible in-plane component of magnetic flux. Thus the magnetic information in the holograms was negligibly small. However, the magnetic anomaly in APBs (e.g., significant increase in magnetization for APBs produced in the Fe₇₀Al₃₀ alloy¹¹⁾) can, in principle, be examined using DFEH when a specimen is magnetized in the in-plane direction. A dark-field electron hologram generated by a fundamental reflection is thus suitable for the study of the magnetic nanostructure in APBs as the undesired geometric phase shift can be suppressed.

One technical issue encountered in our DFEH observations was the poor signal-to-noise ratio, since the dark-field hologram is produced by using a weak Bragg reflection. As indicated by Lichte et al.,³⁰⁾ the precision in phase measurements depends on the factors relevant to the image quality of a hologram — i.e., factors representing the signal-to-noise ratio, the number of electrons per pixel in the digitized image, and the image contrast of the interference fringes. Due to the poor image quality of the dark-field electron holograms, as shown in Fig. 5(c), the uncertainty in the phase measurement was ±0.3 rad in this study. This value appears to be insufficient for high-precision studies of the magnetic anomaly in APBs and other planar defects. Accordingly, we need a complementary technique that improves the image quality of electron holograms when the method based on DFEH is applied to intensive studies of APBs and other planar defects.

For this essential problem, a plausible method is noise reduction in dark-field electron holograms. One promising noise reduction technique is wavelet transformation using digitized holograms, since it facilitates noise filtering in various frequency ranges. Figure 7 provides an example of noise reduction using the dark-field hologram shown in Fig. 5(b). For the criterion of the noise threshold, we employed the hidden Markov model,³¹⁾ which enables a reasonable separation of weak signal from noise. Conventional noise reduction with wavelet transformation employs a threshold which simply depends on the sampling number and standard deviation. Because of this threshold, a weak signal may be lost in the process of noise filtering. The hidden Markov model is advantageous for separating a weak signal from noise; see Ref. 31 for further details on noise reduction based on the wavelet transformation combined with the hidden Markov model.

Fig. 7

Noise reduction in the electron hologram by using wavelet transform combined with the hidden Markov model. (a) Original dark-field hologram [identical to Fig. 5(b)] including high-frequency noise. (b) Intensity profile measured along the X–Y line in (a). (c) Denoised dark-field hologram. (d) Intensity profile measured along the X–Y line in (c).

As shown in Fig. 7(b), which represents the intensity profile measured along the X–Y line in Fig. 7(a), high-frequency noise was significant in the original dark-field electron hologram. In the presence of noise, the intensity profile is blurred, particularly for the APB region, which showed poor contrast (poor peak-to-background ratio in the fringe pattern). Figures 7(c) and 7(d) show a dark-field electron hologram and an intensity line profile, respectively, subjected to noise reduction by using the wavelet transformation combined with the hidden Markov model. As a result of the removal of high-frequency noise, details in the interference pattern were improved, including those in the APB region. Thus, this method is promising for further applications of DFEH studies, which will be discussed in greater detail elsewhere.

With regards to improving image quality, averaging over many images provides satisfactory results. However, for many specimens including ordered alloys, radiation damage and/or surface contamination induced by electron exposure hamper the collection of many images from a specific region. Noise reduction is essential for such specimens sensitive to electron exposure.

Diffraction effects in the crystal, which offer undesired phase shifts in electron holography observations, are another technical concern. For example, precise phase measurement due to APBs can be hampered by bend contours which are widely observed diffraction contrast features in TEM images and/or electron holograms. It should be noted that in this study, when the dark-field electron holograms were acquired, the specimen was tilted appropriately so that bend contours were not superposed on the APBs, thereby avoiding this issue.

4. Summary

The impact of DFEH in suppressing undesired geometric phase shift due to APBs was examined through systematic experiments using a B2-type Fe₇₀Al₃₀ alloy. Pairs of electron holograms and reconstructed phase images were obtained using a (1) 000 transmitted beam, (2) 100 superlattice reflection, and (3) 200 fundamental reflection. The observations demonstrated that the geometric phase shift was significant (∼2.3 rad) when the superlattice reflection was used for generating the hologram, whereas an appreciable phase shift was not observed in the APB position when using the fundamental reflection. The results indicate that DFEH can be a potential tool for reducing the contribution of geometric phase shift carried by electron holograms.

Acknowledgments

The authors are grateful to Professor R. Kainuma for useful discussions about the Fe₇₀Al₃₀ alloy. This study was supported by JST, CREST (JPMJCR1664), and JSPS KAKENHI (JP18H03845).

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