TY - JOUR
AU1 - Yamamoto, Kazuo
AU2 - Sugawara, Yoshihiro
AU3 - McCartney, Martha R.
AU4 - Smith, David J.
AB - Abstract Phase-shifting electron holography was used to reconstruct the object-wave function of high-spatial-frequency specimens of HgCdTe, and the requirements for precise measurements were investigated. Fresnel fringes due to the electrostatic biprism caused serious calculation errors during the phase-shifting reconstruction. Uniform interference fringes, obtained by adjusting the biprism voltage to cancel out the Fresnel fringes, were needed to minimize these errors. High-resolution holograms of a HgCdTe single crystal were recorded with coarse interference fringes and a high visibility of 65% and then used to reconstruct the atomic-scale object wave. Although the spatial resolution (0.25 nm) of the transmission electron microscope was worse than the separation (0.16 nm) between Hg (or Cd) and Te columns, the crystal polarity was determined from the aberration-corrected object wave. phase-shifting electron holography, high-resolution electron microscopy, Fresnel fringes Introduction Electron holography was originally proposed by Gabor [1] as a means to overcome the spherical aberration of the objective lens and improve the spatial resolution of the transmission electron microscope (TEM). Since the development of TEMs with field emission guns and electrostatic biprisms [2], aberration-correction studies using electron holography have been conducted by several groups of researchers [3–9]. One of the most serious problems in object-wave reconstruction using Fourier transformation of off-axis electron holograms is the limited coherence of the incident electron wave. To obtain high spatial resolution in the reconstructed wave, fine interference fringes are essential. Generally, the spacing of interference fringes should be narrower than one-third of the spatial resolution available in the reconstructed wave [10]. Under such conditions, however, the fringe contrast is drastically reduced because of the coherence limitation. Thus, it is difficult to improve both the spatial resolution and the phase measurement precision at the same time. Phase-shifting electron holography was developed by Ru et al. [11–13] to overcome such problems. The reconstructed object wave is obtained from a series of holograms in which the sets of interference fringes are displaced one from another. The amplitude and phase values are then retrieved pixel by pixel from intensity changes in the holograms. Therefore, the spatial resolution is determined either by the pixel size or the TEM resolution rather than by the fringe spacing. It is thus not necessary to use fine interference fringes, i.e. coarse fringes with high contrast can be used for more precise reconstruction. Moreover, since many holograms are normally used in the reconstruction process, the object wave can be reconstructed with higher signal-to-noise ratio. Thus, the phase-shifting reconstruction method should be particularly effective for aberration correction when reconstructing high-resolution images. Ru et al. [12] confirmed that it was possible to reconstruct high-resolution images, i.e. amplitude images reconstructed from six holograms having coarse interference fringes, which corresponded well with the TEM images. However, the aberration correction was not attempted with the reconstructed object waves. Here, we focus on reconstructing more precise high-resolution images using the phase-shifting method for aberration correction. The most serious practical problem in using this approach is Fresnel diffraction caused by the electrostatic biprism. The resulting Fresnel fringes that are present in recorded holograms cause additional errors during the reconstruction. We have previously proposed an image-processing method termed ‘Fresnel correction’ to remove these Fresnel fringes [14,15]: we describe here the requirements for using Fresnel correction for atomic image reconstruction. We then demonstrate a simple application of aberration correction of the reconstructed wave and determine the polarity of a HgCdTe single crystal. Methods Phase-shifting electron holography with Fresnel correction Figure 1 shows the schematic setup of off-axis electron holography. The intensity of interference fringes can be expressed as:  (1) where A and Φ are the amplitude and phase of the object wave, respectively, k is 2π / λ where λ is the wavelength of the electron wave, α is the deflected angle due to the electrostatic biprism and Φi is the initial phase difference between the object wave and the reference wave. Further details of phase-shifting electron holography have been described in our previous papers [14,15]. Fig. 1 View largeDownload slide Principle of phase-shifting electron holography. (a) Experimental setup used when taking holograms, the interference fringes are shifted sequentially. (b) Intensity variation at point Ix,y in (a) as a function of initial phase difference Φi. Reconstructed amplitude and phase values are determined using the cosine curve which shows the intensity variation. (c) One-dimensional intensity profile of typical interference fringes. The non-uniform fringes modulated by Fresnel diffraction cause calculation errors during reconstruction. The non-uniform distribution is normalized by the top and bottom envelopes, Etop and Ebottom (Fresnel correction). Fig. 1 View largeDownload slide Principle of phase-shifting electron holography. (a) Experimental setup used when taking holograms, the interference fringes are shifted sequentially. (b) Intensity variation at point Ix,y in (a) as a function of initial phase difference Φi. Reconstructed amplitude and phase values are determined using the cosine curve which shows the intensity variation. (c) One-dimensional intensity profile of typical interference fringes. The non-uniform fringes modulated by Fresnel diffraction cause calculation errors during reconstruction. The non-uniform distribution is normalized by the top and bottom envelopes, Etop and Ebottom (Fresnel correction). First, we record a series of holograms with the sets of interference fringes shifted sideways. Shifting the fringes corresponds to changing the initial phase difference Φi. Tilting of the incident electron wave is the simplest way to shift the fringes. The tilt angle of one step is ∼10−6–10−4 rad, depending on the operating conditions of the TEM lens system. However, tilting the electron wave might negatively impact the quality of high-resolution images. Other ways of shifting the fringes, such as displacing the biprism filament, or shifting the sample and then fitting the sample image position using image-processing software give equivalent results because the fringes and the sample image are still being shifted relative to one another. In the experiments described below, we simply used thermal drift to shift the sample, and then the sample image positions of all the holograms were fitted to one another. The second step is reconstruction from the series of holograms. The object wave, namely, the amplitude and phase values, is obtained in each pixel. For example, the intensity of point Ix,y in Fig. 1a changes cosinusoidally as the interference fringes are shifted sideways. Figure 1b illustrates the plot of intensity as a function of the initial phase Φi. The initial phase in each hologram is obtained from the phase value at the centre of the sideband in the Fourier transform of the hologram. The plots are fitted using a cosine curve and the object wave, namely, Ax,y and Φx,y, is finally determined from the curve, as shown in Fig. 1b. The major problem here is Fresnel diffraction from the electrostatic biprism. This Fresnel diffraction modulates both the amplitude and phase of the electron wave. The amplitude modulation results in non-uniform interference fringes, as presented in Fig. 1c. When these fringes are shifted, the plots shown in Fig. 1b cannot be fitted using a cosine curve, leading to calculation errors in the reconstructed object wave. In the Fresnel correction method, which is used to avoid these errors, the non-uniform fringes are normalized by the top and bottom envelopes of the fringes (Etop and Ebottom in Fig. 1c), which can be obtained using Lagrange’s interpolation function. This process corresponds to removing 1 + A2 and 2A from Eq. (1), resulting in the uniform fringes having only the cosine term. Moreover, the phase modulation caused by Fresnel diffraction is corrected by subtracting the phase of the reference holograms. These two types of Fresnel correction have been successfully used to observe electrostatic or magnetic fields that have low spatial frequency [14–17]. Aberration correction by electron holography Here, we define object waves ψo and ψi in the object plane and image plane as:  (2)  (3) where the wave ψi takes lens aberration and defocus effects into account. In electron holography, aberration correction is normally carried out in Fourier space. Fourier transformation of the wave ψi is expressed as:  (4) where:  (5) Cs is the coefficient of spherical aberration, ΔF is the objective-lens defocus, λ is the electron wavelength and Es and Ec are the damping envelope functions due to spatially and temporally partial coherence [7]. In the rotationally symmetric approximation, i.e. no axial astigmatism and no axial coma cases, the exponential term is given by the spherical aberration and the defocus effects as shown in Eq. (5) and is the so-called ‘phase plate’. Moreover, assuming the complete coherence of electron waves, we can omit the terms Es and Ec. Consequently, the aberration-corrected wave, namely, the reconstructed wave in the object plane ψo is thus retrieved:  (6) In the following, we used this equation for aberration correction. Results and discussion Simulation of phase-shifting reconstruction for objects with high spatial frequency We constructed a simple model of an object wave with high spatial frequency. Figure 2a and b shows the amplitude and phase distributions, respectively. The line profile is inset in each image. A series of holograms and reference holograms were then calculated wave-optically using the object wave [18]. Fresnel diffraction caused by the biprism and reduction in fringe contrast due to finite electron illumination angle were taken into account, but quantum noise was not. The typical parameters of electron optics used in the hologram simulation are listed in Table 1. Figure 2c shows one of the holograms corresponding to a biprism voltage of 27 V. The non-uniform fringes caused by Fresnel diffraction are clearly visible in the hologram. Figure 2d shows the hologram after it has been corrected using the interpolation envelopes. The fringes are slightly bent at the points where phase shift occurs. Using five corrected holograms, we obtained the reconstructed phase image (shown in Fig. 2e). However, the reconstructed white spots do not match with the original pattern in Fig. 2b. We calculated another series of holograms using the lower biprism voltage of 23 V where Fresnel diffraction effects were almost cancelled out and uniform fringes occurred. Although the contrast of the object model modulates the uniform fringes as shown in Fig. 2f, the intensity at each pixel changes uniformly like Fig. 1b because the object image does not shift during fringe shifting. Thus, clear reconstructed images without Fresnel correction can be obtained. Figure 2g shows the phase image directly reconstructed. Although some white spots are slightly distorted, the pattern matches well with the original positions of the model structure (shown in Fig. 2b). The vignetting effect caused by the biprism [19,20], which is caused by the lack of information on the biprism filament, does not affect the lattice reconstruction. The spurious lattice images visible in the vacuum area are caused by the modulated reference wave split in the biprism plane, i.e. the defocused plane. Fig. 2 View largeDownload slide Influence of Fresnel correction on the reconstructed phase (simulation). (a) Amplitude and (b) phase distribution of model structure with high spatial frequency. (c) Hologram of model with Fresnel fringes, calculated with biprism voltage Vb of 27 V. (d) Hologram normalized using Fresnel correction. (e) Reconstructed phase image after Fresnel correction. (f) Hologram of model without Fresnel fringes, calculated at Vb = 23 V. (g) Directly reconstructed phase image without Fresnel correction. Fig. 2 View largeDownload slide Influence of Fresnel correction on the reconstructed phase (simulation). (a) Amplitude and (b) phase distribution of model structure with high spatial frequency. (c) Hologram of model with Fresnel fringes, calculated with biprism voltage Vb of 27 V. (d) Hologram normalized using Fresnel correction. (e) Reconstructed phase image after Fresnel correction. (f) Hologram of model without Fresnel fringes, calculated at Vb = 23 V. (g) Directly reconstructed phase image without Fresnel correction. Table 1 Electron-optical conditions used in the hologram simulation in Fig. 2 Acceleration voltage with relativistic correction [kV]  171.87  Electron wave length [nm]  0.00251  Distance between biprism filament plane and image plane [m]  2.2 × 10−3  Illumination angle of electron waves [rad]  5 × 10−7  Radius of prism filament [m]  0.3 × 10−6  Distance between prism filament and ground plate [m]  1.2 × 10−3  Acceleration voltage with relativistic correction [kV]  171.87  Electron wave length [nm]  0.00251  Distance between biprism filament plane and image plane [m]  2.2 × 10−3  Illumination angle of electron waves [rad]  5 × 10−7  Radius of prism filament [m]  0.3 × 10−6  Distance between prism filament and ground plate [m]  1.2 × 10−3  View Large We found that the errors due to Fresnel correction shown in Fig. 2e occur when spatial frequencies of the object are greater than that of the interference fringes. The periodic contrast of the object is mixed with the interference fringes, leading to calculation errors in the top and bottom fringe envelopes. Finer fringes are needed to minimize these errors. However, such finer fringes would reduce the advantage of using the phase-shifting method. Therefore, for reliable reconstruction of objects with high spatial frequency, uniform interference fringes are essential. This can be achieved either by adjusting the biprism voltage or by using two biprism filaments with a special lens condition if permitted by the TEM hardware [21,22]. Reconstruction of HgCdTe structure and aberration correction A Philips CM200 TEM (200 kV) equipped with a Schottky-type field emitter and an electrostatic biprism was used in these experiments. The point resolution of this TEM was 0.25 nm. Figure 3a shows a high-resolution image of Hg0.8Cd0.2Te single crystal oriented to the [011] projection, recorded at a magnification of 900 kX. This material has the zincblende structure, and thus Hg (or Cd) and Te atomic columns correspond to the dark image contrast. However, the separation between atomic columns in this projection is 0.16 nm, so that individual columns cannot be distinguished. Figure 3b shows one of a series of holograms for this specimen. Reference holograms were also recorded in vacuum without the sample. The biprism voltage was adjusted to 24 V to obtain uniform and high-contrast interference fringes. The contrast of the fringes in the reference holograms was then 65%. The fringe spacing was almost the same as the lattice spacing of the sample, and thus it would have been impossible to reconstruct the object wave using the conventional Fourier transformation method because the sidebands would not be separated from the centreband in Fourier space. Fig. 3 View largeDownload slide High-resolution TEM image of Hg0.8Cd0.2Te single crystal and its electron hologram. (a) Direction of the incident beam is [011]. Dimer arrangement of Hg (or Cd) and Te columns is shown in the dark area. (b) One of the series of holograms taken at Vb = 24 V. Fig. 3 View largeDownload slide High-resolution TEM image of Hg0.8Cd0.2Te single crystal and its electron hologram. (a) Direction of the incident beam is [011]. Dimer arrangement of Hg (or Cd) and Te columns is shown in the dark area. (b) One of the series of holograms taken at Vb = 24 V. The object wave in the region shown in Fig. 3b was reconstructed from 24 pairs of holograms and reference holograms. The reconstructed amplitude and phase images are shown in Fig. 4a and b, respectively. Atomic-scale details are visible in both images in spite of using coarse interference fringes. In high-resolution electron microscopy, most of the specimens can be considered to be phase objects. Thus, if the aberration and the defocus effect were completely corrected, the contrast of the corrected amplitude image should disappear. We calculated the standard deviation of the corrected amplitude images as functions of Cs and ΔF and obtained the condition to show minimum standard deviation at Cs = 1.29 mm and ΔF = − 62 nm [8]. These values are close to the nominal Cs (1.3 mm) and Scherzer defocus (ΔF = − 67 nm) of this microscope. Figure 4c shows the corrected amplitude image and Fig. 4d shows an enlargement of the boxed area in Fig. 4c. Figure 4e and f shows line profiles of the dark areas shown in Fig. 4a and c, respectively. These profiles demonstrate the change before and after aberration correction. Asymmetric contrast appears after aberration correction, which is indicative of the presence of separate Hg (or Cd) and Te columns. Fig. 4 View largeDownload slide Reconstructed wave and aberration-corrected amplitude. (a) Amplitude and (b) phase images reconstructed with 24 pairs of holograms and reference holograms. (c) Amplitude image and (d) enlarged area aberration-corrected with Cs = 1.29 mm and ΔF = − 62 nm. (e) and (f) Line profiles of the lines indicated in (a) and (c), respectively. Asymmetrical contrast appears in the dark area after aberration correction, which shows Hg (or Cd) and Te columns. Fig. 4 View largeDownload slide Reconstructed wave and aberration-corrected amplitude. (a) Amplitude and (b) phase images reconstructed with 24 pairs of holograms and reference holograms. (c) Amplitude image and (d) enlarged area aberration-corrected with Cs = 1.29 mm and ΔF = − 62 nm. (e) and (f) Line profiles of the lines indicated in (a) and (c), respectively. Asymmetrical contrast appears in the dark area after aberration correction, which shows Hg (or Cd) and Te columns. A high-resolution image simulation was performed to confirm the polarity of the projected crystal structure. Images were calculated using the program of MacTempas Ver. 1.8 software (Total Resolution). Figure 5a and b shows the amplitude and phase images simulated with Cs = 1.3 mm, ΔF = − 62 nm and crystal thickness d = 11.8 nm, respectively. The upper-left position corresponds to a Hg (or Cd) column. Both images resemble the experimental images shown in Fig. 4a and b. When the aberration phase plate was corrected, as shown in Fig. 5c and d, asymmetrical contrast appeared in the amplitude image, but not in the phase image, presumably because this region of the sample was too thick for high-resolution electron microscopy. The line profile along the dark area is similar to the experimental one shown in Fig. 4f. Therefore, the Hg (or Cd) column must be located in the upper-left part of the area of dark contrast, as shown in Fig. 5c. Fig. 5 View largeDownload slide High-resolution image simulation of Hg0.8Cd0.2Te zincblende structure in [011] projection. (a) Amplitude and (b) phase images calculated with Cs = 1.3 mm, ΔF = − 62 nm and d = 11.8 nm. Circle signs correspond to the positions of Hg (or Cd) and Te dimer. Line profiles along the line are inset in each figure. (c) Amplitude and (d) phase image with Cs = 0 mm, ΔF = 0 nm and d = 11.8 nm. Crystal polarity of Hg (Cd) and Te columns can be determined from the asymmetrical contrast in the amplitude image. Fig. 5 View largeDownload slide High-resolution image simulation of Hg0.8Cd0.2Te zincblende structure in [011] projection. (a) Amplitude and (b) phase images calculated with Cs = 1.3 mm, ΔF = − 62 nm and d = 11.8 nm. Circle signs correspond to the positions of Hg (or Cd) and Te dimer. Line profiles along the line are inset in each figure. (c) Amplitude and (d) phase image with Cs = 0 mm, ΔF = 0 nm and d = 11.8 nm. Crystal polarity of Hg (Cd) and Te columns can be determined from the asymmetrical contrast in the amplitude image. Conclusions The application of phase-shifting electron holography to objects with high spatial frequency, such as high-resolution images of HgCdTe, has been investigated. The Fresnel correction method, when used to remove overlapping Fresnel fringes, can result in additional errors when applied to objects with high spatial frequency. If finer interference fringes are used for reconstruction, then the Fresnel correction approach is effective. However, the fringe contrast decreases, and shifting the fine fringes becomes more difficult, resulting in less precise measurements. Adjusting the biprism voltage to form high-contrast holograms without Fresnel fringes is the simplest solution for using the phase-shifting method. A series of high-resolution holograms of a HgCdTe single crystal were taken under the above non-Fresnel fringe conditions using a TEM having a point resolution of 0.25 nm. The interference fringe contrast was 65%, which is considerably higher than that available for holograms using the conventional Fourier transformation method. Atomic-scale amplitude and phase images were reconstructed. After aberration correction, the polarity of the Hg (or Cd) and Te columns with 0.16 nm separation were confirmed by comparison with the high-resolution image simulations. This work was supported by the Research Fellowship of Japan Society for the Promotion of Science for Young Scientists. We are grateful to Professor Takayoshi Tanji for his valuable comments about aberration correction using electron holography. We also thank Dr Changzhen Wang for providing the HgCdTe TEM sample. We gratefully acknowledge the use of facilities within the John M. Cowley Center for High-Resolution Electron Microscopy at Arizona State University. 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TI - Phase-shifting electron holography for atomic image reconstruction
JF - Journal of Electron Microscopy
DO - 10.1093/jmicro/dfq033
DA - 2010-06-11
UR - https://www.deepdyve.com/lp/oxford-university-press/phase-shifting-electron-holography-for-atomic-image-reconstruction-bWRdPx8sAN
SP - S81
EP - S88
VL - 59
IS - S1
DP - DeepDyve
ER -