TY - JOUR
AU - Shibata,, Naoya
AB - Abstract Differential-phase-contrast scanning transmission electron microscopy (DPC STEM) is a technique to directly visualize local electromagnetic field distribution inside materials and devices at very high spatial resolution. Owing to the recent progress in the development of high-speed segmented and pixelated detectors, DPC STEM now constitutes one of the major imaging modes in modern aberration-corrected STEM. While qualitative imaging of electromagnetic fields by DPC STEM is readily possible, quantitative imaging by DPC STEM is still under development because of the several fundamental issues inherent in the technique. In this report, we review the current status and future prospects of DPC STEM for quantitative electromagnetic field imaging from atomic scale to mesoscopic scale. Introduction Due to the continuous progress in aberration-corrected optics, scanning transmission electron microscopy (STEM) has become an indispensable tool for charactering materials and devices from the atomistic level in both academia and industries. Since atomically sharp electron probe can now stably and routinely be formed onto the specimen, the application of atomic-resolution STEM has been expanded to a wide variety of scientific and technological fields. In the state-of-the-art STEM, it has been demonstrated that sub-0.5 Å spatial resolution (which is smaller than the size of hydrogen Bohr radius) has been experimentally achieved [1], and single-atom spectroscopy using electron energy-loss spectroscopy and energy-dispersive X-ray spectroscopy can also be realized [2, 3]. Imaging of ultralight elements such as H and Li atoms has become capable in STEM using annular bright-field (BF) imaging technique [4–6]. Atomic-resolution STEM is thus rapidly evolving together with the development of aberration-corrected optics, enabling direct characterization of almost all the atoms in the periodic tables within materials and devices at a single-atom level. In recent years, in addition to the development of such atom-by-atom imaging and spectroscopy, electromagnetic field imaging is also becoming popular in STEM. Differential-phase-contrast (DPC) imaging in STEM [7–11] is an imaging technique that can directly visualize local electromagnetic field inside materials. Recent developments of high-speed area detectors (e.g. [12].) have opened the technique to be utilized by many researchers using aberration-corrected STEM. Several applications have been reported in recent years: semiconductors and p–n junctions [13–19], quantum wells [20–23], multiferroics [24, 25], magnetic domains [26–33] and magnetic skyrmions [34–38]. These results have shown that DPC STEM is a very powerful technique for fundamentally understanding the functional properties of materials and devices, because electromagnetic field distribution directly correlates with materials’ magnetic and electrical properties. In addition, it has been shown that DPC imaging can be also effective for image contrast enhancement in organic materials such as polymers [39]. Moreover, applying DPC STEM in atomic-resolution regime may open a new era of atomic-resolution STEM imaging [40]. If we use an atomically sharp electron probe to perform DPC STEM, it is possible to directly visualize the electric field distribution even inside atoms [41]. Direct imaging of the electric field at atomic resolution, such as the electric field between the positively charged atomic nucleus and the negatively charged electron cloud, means that STEM extends its ability to visualize internal structures of atoms with subatomic-size electron probes. Using the real-space electric field distribution inside the atoms, we can also obtain real-space total charge density maps at atomic resolution [42]. Such capability may give us a chance to directly observe atomic bonding information from the very localized volumes in materials, such as surface and interfaces. Indeed, it has been demonstrated that the atomic-resolution DPC STEM may be sensitive to the ionicity of atoms [40, 41, 43]. While the application of atomic-resolution DPC STEM to many different material systems has been intensively reported in recent years [25, 44–50], quantitative methods of DPC image intensity analysis based on the detailed image theory will be indispensable for utilizing atomic-resolution DPC STEM for such purposes. The developments of high-sensitivity segmented- and pixelated-type STEM detectors usable for DPC STEM have rapidly progressed in recent years [12, 27, 51–57]. On the other hand, the quantification of DPC STEM images has been also intensively discussed based on linear imaging theory and detailed image simulation including dynamical scattering effects [58–62]. Such theoretical treatments appear to conclude that the quantification of DPC images can be possible in principle, on condition that the sample is thin enough and the sample structures, electromagnetic field distribution and optical conditions are known a priori. However, in the actual experiments, electromagnetic field information should be extracted from the unknown and complex structures formed inside materials and devices. Moreover, actual materials pose additional complexity for quantifying DPC images such as thickness effect and crystal orientation effect. Therefore, quantification scheme for the experimental DPC images should be established in order to extract the true electromagnetic information from the experimental images. Furthermore, if the true electric field |${\boldsymbol E}({\boldsymbol r})$| can be obtained, the true scalar potential |$v({\boldsymbol r})$| and charge density distribution |$\rho ({\boldsymbol r})$| can be calculated by |$V(r) = - \int {{\boldsymbol E}({\boldsymbol r}){\rm d}{\boldsymbol r}} $| and |$\rho ({\boldsymbol r}) = {\varepsilon _{\rm 0}}\nabla \cdot {\boldsymbol E}({\boldsymbol r})$|⁠, respectively. Integrating the DPC signal is used in the imaging technique called integrated differential phase contrast to visualize atomic structures like a scalar potential [63]. In the present paper, the recent progress and future prospects of quantitative DPC STEM are reviewed. This review consists of the following contents. First, we outline the basic theory of quantitative DPC STEM. Second, we exemplify the major factors affecting the actual experimental DPC STEM image contrasts, which are needed to be settled. Third, we review the recent progress of the development of quantification techniques for DPC STEM images. Finally, the future prospects will be discussed. Theory for quantitative DPC STEM In this section, the basic theory for quantification of electromagnetic fields by DPC is reviewed. In a simplistic picture, incident focused electrons are deflected by electromagnetic fields, and the BF disk is shifted (Fig. 1). The deflection angle can be measured by a segmented or pixelated detector placed in the diffraction plane and can be converted into the magnitude of the electromagnetic fields in some way. Recently, several types of detectors are available, and DPC signals are defined through several different calculations. First, variations of DPC imaging are introduced. Second, the relationship between deflection angle and magnitude of electromagnetic fields is derived by assuming electrons to be classical particles, and then, the quantum mechanical picture is introduced. Finally, quantification via a contrast transfer function is described. Fig. 1. Open in new tabDownload slide Schematic illustration of DPC STEM for the visualization of electric (left) and magnetic (right) fields. Fig. 1. Open in new tabDownload slide Schematic illustration of DPC STEM for the visualization of electric (left) and magnetic (right) fields. Variants of DPC STEM imaging DPC has been first proposed by subtracting two intensities detected in semicircular segments of a split detector. This DPC signal was shown to be related to the partial derivative of the phase in the transmission function along the axis perpendicular to the split direction [8, 9]. To measure electromagnetic fields along arbitrary direction, a quadrant detector (Fig. 2a) is required. In recent years, a segmented detector with 16 segments as shown in Fig. 2b has been developed [12]. An alternative detector for DPC is a pixelated detector. The advantage of the pixelated detector is universality in processing detected intensities; any process carried out for segmented detectors is possible. The disadvantage of the pixelated detectors is slow readout speed, although it has been rapidly improved recently. In the state-of-the-art pixelated detector for STEM, readout speed achieved |$\sim$|100 microseconds per pixel with full dynamic range and 10 microseconds with binary dynamic range [64]. Contrastingly, the readout speed of segmented detectors can go up to |$\sim$|1 microsecond or less per pixel of STEM images with a high dynamic range. High-speed readout is essential especially for imaging at atomic resolution since aberrations significantly affect DPC and should be carefully removed by checking DPC generated in real time with a high-speed detector. The great advantage of the segmented detector has been demonstrated through visualizing atomic electric fields in SrTiO3 (Fig. 3) [41]. Atomic electric fields of both heavy and light atomic columns (in this case, Sr, Ti-O and O columns) are visualized with very high quality. It should be noted that the precise alignment of the BF disk with respect to segmented detectors is also indispensable for obtaining such high-quality atomic electric field images. Fig. 2. Open in new tabDownload slide Schematic illustrations of (a) quadrant detector, (b) 16-segment detector and (c) pixelated detector. Fig. 2. Open in new tabDownload slide Schematic illustrations of (a) quadrant detector, (b) 16-segment detector and (c) pixelated detector. Fig. 3. Open in new tabDownload slide Simultaneously obtained atomic-resolution STEM images of SrTiO3 [001]. (a) Annular dark field image. (b) Electric field vector color map (left side) and electric field strength map (right side). The inset color wheel represents vector direction and magnitude by its color and brightness, respectively. Adopted from [41] with permission. Fig. 3. Open in new tabDownload slide Simultaneously obtained atomic-resolution STEM images of SrTiO3 [001]. (a) Annular dark field image. (b) Electric field vector color map (left side) and electric field strength map (right side). The inset color wheel represents vector direction and magnitude by its color and brightness, respectively. Adopted from [41] with permission. The DPC signal can be defined by several ways. In the quadrant detector, the DPC signal is defined by subtracting intensities detected in two semicircular regions. For this simple DPC signal, a phase contrast transfer function (PCTF) was derived under the weak phase object approximation (WPOA) by Rose [9]. For strong phase objects, a linear imaging theory was developed by Waddell and Chapman [65]. They proposed to detect the first moment, or center of mass (CoM), of the diffraction intensity distribution in the detector plane and showed the derived PCTF to be valid under the phase object approximation (POA). Recently, the CoM measurements has been experimentally demonstrated at atomic resolution and interpreted in a simple picture according to the Ehrenfest’s theorem by Müller-Caspary et al. [44, 59]. The validity of linear imaging theory for strong phase objects is of great advantage in electromagnetic field quantification. To measure the CoM of diffraction intensity distribution, a pixelated detector is basically required. However, using multisegment detector, e.g. the 16-segment detector, the CoM can be approximately measured as discussed in the section ‘CoM measurement’. Classical picture of DPC STEM imaging Figure 4 shows schematic illustrations of electron deflection due to electric or magnetic fields. Here, incident electrons are assumed to be classical particles. The deflection angle can be obtained by calculating the ratio between the momentum of the incident electron |$h/\lambda (h$| and |$\lambda $| denoting Planck’s constant and electron wavelength, respectively) and the momentum transfer during the transmission. For simplicity, it is assumed that the present electromagnetic fields inside specimens do not vary along the thickness direction. The Coulomb and Lorenz forces acting on the incident electrons are |$ - e{{{{\boldsymbol E}}}_ \bot }$| and |$ - e{{{\boldsymbol v}}} \times {{{{\boldsymbol B}}}_ \bot }$|⁠, respectively, where |$e$|⁠, |$v$|⁠, |${{{{\boldsymbol E}}}_ \bot }$| and |${{{{\boldsymbol B}}}_ \bot }$| denote the elemental charge, velocity of the incident electron, the electric and the magnetic field component perpendicular to the optical axis, respectively. The momentum transfer is derived by multiplying the force and the transmission time |$t/v$| (t denoting the thickness of the specimen). Therefore, the deflection angle |${{{\boldsymbol \theta }}}$| can be described as below: $$\begin{equation}{{{\boldsymbol \theta }}} = - {{e\lambda t} \over {hv}}\left( {{{{{\boldsymbol E}}}_ \bot } + {{{\boldsymbol v}}} \times {{{{\boldsymbol B}}}_ \bot }} \right).\end{equation}$$(1) Fig. 4. Open in new tabDownload slide Schematic illustration of electron deflection due to electric (left) and magnetic (right) fields by the classical picture. The triangles represent ratios between the momentum of the incident electron and the transferred momentum. Fig. 4. Open in new tabDownload slide Schematic illustration of electron deflection due to electric (left) and magnetic (right) fields by the classical picture. The triangles represent ratios between the momentum of the incident electron and the transferred momentum. To quantify the electromagnetic fields, the specimen thickness should be known, because the deflection angle is proportional to the thickness as well as the magnitude of the electromagnetic fields. To consider the appropriate experimental conditions to detect electromagnetic fields, let us substitute the specific values into Eq. (1). If an accelerating voltage of 200 kV, a thickness of 100 nm, an electric field of 1 MV/cm, and a magnetic field of 1 T are assumed, deflection angles due to electric and magnetic fields are calculated to be 29.1 and 60.6 μrad, respectively. The assumed magnitude of electromagnetic fields is typical value for mesoscopic electromagnetic field distribution in materials: e.g. electric fields due to polarization in ferroelectric materials, built-in electric field due to carrier redistribution at p–n junctions in semiconductors and magnetic fields in ferromagnetic domains and magnetic skyrmions. It is noted that it is extremely difficult to detect deflection of several tens of μrad with an atomically fine probe whose convergence semiangle is |$\sim$|20 mrad, because the shift of the BF disk with respect to the size of the BF disk is quite small. Therefore, to detect such small electron deflection, a convergence semiangle of |$\sim$|100 μrad to |$\sim$|1 mrad is normally selected, and thus, the spatial resolution is usually limited to the order of several nm. This trade-off between spatial resolution and sensitivity to electromagnetic fields is an important factor in DPC experiments. The required electron dose to detect a specific field strength with a specific probe size can be estimated through the theoretical framework of statistical noise in STEM [66]. The rigid shift model of the BF disk shown in Fig. 1 is correct only when the electromagnetic fields inside the probe are uniform. In general cases, the rigid shift model is too simplistic. Fig. 5 shows a simulated BF disk pattern when a probe is located at a p–n junction [13]. The position is not clearly shifted from the original position denoted by the dashed line. Instead of the disk shift, the intensity distribution inside the BF disk varies. In such case, the deflection angle cannot be defined simply, and a quantum mechanical picture as discussed below is required. Fig. 5. Open in new tabDownload slide Simulated BF disk when a probe is located on a p–n junction. Adopted from [13] with permission. Fig. 5. Open in new tabDownload slide Simulated BF disk when a probe is located on a p–n junction. Adopted from [13] with permission. CoM measurement Although the classical picture is not valid in general cases, it is still helpful owing to the Ehrenfest’s theorem [67]. According to this theorem, quantum mechanical expectation values should obey the classical equations of motion. Thus, Eq. (1) becomes valid even for a quantum mechanical system, if |$\theta $|⁠, |${{{{\boldsymbol E}}}_ \bot }$| and |${{{{\boldsymbol B}}}_ \bot }$| are replaced by their expectation values: |$\left\langle {\boldsymbol \theta} \right\rangle = \lambda \left\langle {\boldsymbol k }\right\rangle = \lambda \int {\Psi ^*({\boldsymbol k},{\boldsymbol R}){\boldsymbol k}\Psi ({\boldsymbol k},{\boldsymbol R}){\rm d}{\boldsymbol k}} $|⁠, |$\langle{{{{\boldsymbol E}}}_ \bot }\rangle = \mathop \smallint {\psi ^*}\left( {{{{\boldsymbol r}}},{{{\boldsymbol R}}}} \right){{{{\boldsymbol E}}}_ \bot }\left( {{{\boldsymbol r}}} \right)\psi \left( {{{{\boldsymbol r}}},{{{\boldsymbol R}}}} \right){{\rm d}}{{{\boldsymbol r}}}$| and |$\langle{{{{\boldsymbol B}}}_ \bot }\rangle = \mathop \smallint {\psi ^*}\left( {{{{\boldsymbol r}}},{{{\boldsymbol R}}}} \right){{{{\boldsymbol B}}}_ \bot }\left( {{{\boldsymbol r}}} \right) \psi \left( {{{{\boldsymbol r}}},{{{\boldsymbol R}}}} \right){{\rm d}}{{{\boldsymbol r}}}$|⁠. The following equation may be derived: $$\begin{equation}\int {{{\boldsymbol k }}}{I_{{{\rm diff}}}}\left( {{{{\boldsymbol k}}},{{{\boldsymbol R}}}} \right){{\rm d}}{{{\boldsymbol k}}} = - {{et} \over {hv}}\mathop \int {\left({{{{\boldsymbol E}}}_ \bot }\left( {{{\boldsymbol r}}} \right) + {{{\boldsymbol v}}} \times {{{{\boldsymbol B}}}_ \bot }\left( {{{\boldsymbol r}}} \right) \right)}{I_{{{\rm probe}}}}\left( {{{{\boldsymbol r}}},{{{\boldsymbol R}}}} \right){{\rm d}}{{{\boldsymbol r}}},\end{equation}$$(2) where |${\boldsymbol k}$| and |${\boldsymbol r}$| respectively denote reciprocal and real coordinates perpendicular to the optical axis, and |${I_{\mathrm{diff}}}({\boldsymbol k},{\boldsymbol R}) = {\left| {\Psi ({\boldsymbol k},{\boldsymbol R})} \right|^2}$| and |${I_{\mathrm{probe}}}({\boldsymbol r},{\boldsymbol R}) = {\left| {\Psi ({\boldsymbol r},{\boldsymbol R})} \right|^2}$| denote normalized intensities (probability densities of an electron) of the diffraction pattern and the electron probe when the probe is located at |${\boldsymbol r} = {\boldsymbol R}$|⁠, respectively. The left-hand side represents the expectation value of the scattered wavevector of the transmitted electron |$\left\langle {\boldsymbol k} \right\rangle = \left\langle {\boldsymbol \theta} \right\rangle /\lambda $|⁠, and the integral in the right-hand side represents the expectation value of the electromagnetic fields (the cross product of velocity and a magnetic field). If the diffraction pattern is recorded by a pixelated detector, the expectation value of deflection angle can be measured by calculating the CoM of the diffraction intensity, which is an analogy with the CoM in classical mechanics. Thus, the electromagnetic fields can be quantified, if the probe function is known. The quantified electromagnetic fields through the CoM measurement is therefore the weighted-average field with the probe intensity profile. The CoM can be measured as deflection angle, wavevector or momentum transfer, and DPC signal can be defined in the several ways. In the present paper, we measure the CoM using the scattered wavevector and define the DPC signal in terms of this wavevector: |${S_{CoM}}({\boldsymbol R}) = \smallint {{\boldsymbol k}{I_{\mathrm{diff}}}({\boldsymbol k},{\boldsymbol R}){\rm d}{\boldsymbol k}} $| While Eq. (2) holds in general cases, it is more practical when the electron probe is not significantly altered by a specimen. Assuming the probe function is not varied during the probe scan, the probe function can be redefined by |${I_{\mathrm{probe}}}({\boldsymbol r} - {\boldsymbol R})$|⁠, Substituting this definition for Eq. (2), the following formula is derived: $$\begin{equation}{S_{{{\rm CoM}}}}\left( {{{\boldsymbol R}}} \right) = - {{et} \over {hv}}\left( {{{{{\boldsymbol E}}}_ \bot }\left( {{{\boldsymbol R}}} \right) + {{{\boldsymbol v}}} \times {{{{\boldsymbol B}}}_ \bot }\left( {{{\boldsymbol R}}} \right)} \right) \otimes {I_{{{\rm probe}}}}\left( { - {{{\boldsymbol R}}}} \right),\end{equation}$$(3) where |$ \otimes $| denotes the convolution operator. While a pixelated detector is required for the CoM measurement, the CoM can be approximately measured by segmented detectors [58]. Here, we define DPC signal through a detector response function as below: $$\begin{equation}S({\boldsymbol R}) = \int {D({\boldsymbol k}){I_{\mathrm{diff}}}({\boldsymbol k},{\boldsymbol R}){\rm d}{\boldsymbol k}.} \end{equation}$$(4) The |$\alpha $| (⁠|$\alpha $| denoting x or y) component of DPC signal based on the CoM measurement is defined by |${D_{\mathrm{CoM},\alpha }}({\boldsymbol k}) = {k_\alpha }$| (⁠|${k_\alpha }$| denoting |$\alpha $| component of |${\boldsymbol k}$|⁠). To measure approximate CoM of the diffraction intensity with a segmented detector, we define a detector response function as below: $$\begin{equation}{D_{{{\rm seg}},{{\rm \alpha }}}}\left( {{{\boldsymbol k}}} \right) = \left\{\begin{array}{ll} k_{\alpha ,j}^{{{\rm CoM}}}& {\rm if}qq(\sim){\boldsymbol k}qq(\sim)\text{lies within the}qq(\sim)j{\rm thqq(\sim)segment},\\ 0 & {\rm otherwise},\end{array}\right.\end{equation}$$(5) where |$k_{\alpha ,j}^{CoM}$| is the |$\alpha $| component of the geometrical CoM of the jth detector segment. Although the approximate CoM measurement enables one to quantify electromagnetic fields by a segmented detector as shown in Fig. 2(b), it leads nonnegligible error in electromagnetic field quantification even in the case of the 16-segment detector. To improve accuracy of the quantification, a segmented detector with more segments is required. Another approach to improve the accuracy is to consider contrast transfer function on detector function as discussed below. Contrast transfer function PCTF in STEM can be derived under the WPOA as below [9, 61]: $$\begin{equation}\mathcal{F}\left[ {S\left( {{{\boldsymbol R}}} \right)} \right] = \delta \left( {{{\boldsymbol K}}} \right)\mathop \int {\left| {T\left( {{{{{\boldsymbol k}}}}} \right)} \right|^2}D\left( {{{{{\boldsymbol k}}}}} \right){{\rm d}}{{{{\boldsymbol k}}}} + {{\Phi }}\left( {{{\boldsymbol K}}} \right)\beta \left( {{\boldsymbol K}} \right)\end{equation}$$(6) $$\begin{equation}\beta \left( {{{\boldsymbol K}}} \right) = {{\rm i}}\mathop \int D\left( {{{{{\boldsymbol k}}}_ \bot }} \right)\left[ {{T^*}\left( {{{\boldsymbol k}}} \right)T\left( {{{{\boldsymbol k}}} - {{{\boldsymbol K}}}} \right)} \right.\left. { - T\left( {{{\boldsymbol k}}} \right){T^*}\left( {{{{\boldsymbol k}}} + {{{\boldsymbol K}}}} \right)} \right]{{\rm d}}{{{\boldsymbol k}}}\end{equation}$$(7) where |$\mathcal{F}$| denotes Fourier transform with respect to R, of which its conjugate coordinate is denoted by K. |$\delta \left( {{{\boldsymbol K}}} \right)$| and |$\beta \left( {{{\boldsymbol K}}} \right)$| denote Dirac’s delta function and the PCTF, respectively. |$T\left( {{{\boldsymbol k}}} \right) = A\left( {{{\boldsymbol k}}} \right){{\rm \chi }}\left( {{{\boldsymbol k}}} \right) $| denotes the lens transfer function normalized so that |$\mathop \smallint {\left| {T\left( {{{\boldsymbol k}}} \right)} \right|^2}d{{{{\boldsymbol k}}}_ \bot } = 1$|⁠, where |$A\left( {{{\boldsymbol k}}} \right)$| and |${{\rm \chi }}\left( {{{\boldsymbol k}}} \right)$| are the aperture function (the top hat function representing the aperture) and the lens aberration function |${{\rm \chi }}\left( {{{\boldsymbol k}}} \right) {=} \left( {{{\rm \pi }}/\lambda } \right)\!\left(\! {\mathop \sum_{m,n} {C_{mn}}{{\left\{ {\lambda \left( {{k_x} \!{-}\! {{\rm i}}{k_y}} \right)} \right\}}^m}{{\left\{ {\lambda \left( {{k_x} \!+\! {{\rm i}}{k_y}} \right)} \right\}}^n}\!/\!\left( {m {+} n} \right) \!{+}\! c.c.} \!\right)$| [68], respectively. |${{\Phi} }\left( {{{\boldsymbol K}}} \right) = \mathcal{F}\left[ {\phi \left( {{{\boldsymbol R}}} \right)} \right]$| denotes Fourier transform of the phase in the specimen transmission function |$\phi \left( {{{\boldsymbol R}}} \right) = \sigma \mathop \smallint V\left( {{{{\boldsymbol R}}},z} \right){{\rm d}}z - e/h\mathop \smallint {A_z}\left( {{{{\boldsymbol R}}},z} \right){{\rm d}}z$|⁠, where |$\sigma = 2\pi me\lambda /{h^2}$| (m denoting the electron’s relativistic mass) is the interaction constant, |$V\left( {{{{\boldsymbol R}}},z} \right)$| is the electrostatic potential, and |${A_z}\left( {{{{\boldsymbol R}}},z} \right)$| is the z component (electron incident direction) of the magnetic vector potential [69]. The integral in Eq. (6) is zero, if the detector response function is antisymmetric with respect to the origin, |$ D\left( { - {{{\boldsymbol k}}}} \right) = - D\left( {{{\boldsymbol k}}} \right)$|⁠, which is normally satisfied in DPC imaging. If the aberration function is also antisymmetric, |$\beta \left( {{{\boldsymbol K}}} \right)$| is purely imaginary. Thus, the point spread function, |$h\left( {{{\boldsymbol R}}} \right) = {\mathcal{F}^{ - 1}}\left[ {\beta \left( {{{\boldsymbol K}}} \right)} \right]$|⁠, is antisymmetric, and this is a reason why DPC images are closely related to the differential coefficient of the phase in the specimen transmission function, namely, electromagnetic fields. For simplicity, we assume that all aberrations are corrected, and the aberration function satisfies the antisymmetry, |${{\rm \chi }}\left( {{{\boldsymbol k}}} \right) = {{\rm \chi }}\left( { - {{{\boldsymbol k}}}} \right) = 0$|⁠. To clarify the relation between DPC images and electromagnetic fields, Eq. (6) is rewritten using the Fourier derivative theorem |$\mathcal{F}\left[ {\partial \phi \left( {{{\boldsymbol R}}} \right)/\partial \alpha } \right] = 2\pi {{\rm i}}{K_\alpha }{{\Phi }}\left( {{{\boldsymbol K}}} \right)$| as below: $$\begin{equation}\mathcal{F}\left[ {{S_{DPC,\alpha }}\left( {{{\boldsymbol R}}} \right)} \right] = \mathcal{F}\left[ {{\partial \over {\partial \alpha }}{{\phi \left( {{{\boldsymbol R}}} \right)} \over {2{{\rm \pi }}}}} \right]{b_\alpha }\left( {{{\boldsymbol K}}} \right),\end{equation}$$(8) where |${b_\alpha }\left( {{{\boldsymbol K}}} \right) = {\beta _\alpha }\left( {{{\boldsymbol K}}} \right)/\left( {{{\rm i}}{K_\alpha }} \right)$| is a real valued transfer function for the partial derivative with respect to |${{\rm \alpha }}$| (normalized by 2 |${{\rm \pi }}$|⁠). We will call |${b_\alpha }\left( {{{\boldsymbol K}}} \right)$| the differential-phase-contrast transfer function (DPCTF) to avoid confusion between |${\beta _\alpha }\left( {{{\boldsymbol K}}} \right)$| and |${b_\alpha }\left( {{{\boldsymbol K}}} \right)$|⁠. Taking the inverse Fourier transform of both sides of Eq. (8), $$\begin{equation}{S_{\mathrm{DPC},\alpha }}\left( {{{\boldsymbol R}}} \right) = {\partial \over {\partial \alpha }}{{\phi \left( {{{\boldsymbol R}}} \right)} \over {2{{\rm \pi }}}} \otimes {h_\alpha }\left( {{{\boldsymbol R}}} \right),\end{equation}$$(9) where |${h_\alpha }\left( {{{\boldsymbol R}}} \right)$| is the point spread function defined by |${\mathcal{F}^{ - 1}}\left[ {{b_\alpha }\left( {{{\boldsymbol K}}} \right)} \right]$|⁠. The point spread function is symmetric with respect to the origin, |${h_\alpha }\left( {{{\boldsymbol R}}} \right) = {h_\alpha }\left( { - {{{\boldsymbol R}}}} \right)$|⁠, because of the real valued function |${b_\alpha }\left( {{{\boldsymbol K}}} \right)$|⁠, and thus, DPC images can be interpreted easily in this form. $$\begin{equation}\left(\begin{array}{l} {\partial _x}\phi \left( {{{\boldsymbol R}}} \right)\\ {\partial _y}\phi \left( {{{\boldsymbol R}}} \right)\end{array}\right) = \left(\begin{array}{l} - \dfrac{2\pi e}{hv}{E_{x,{{\rm prj}}}}\left( {{{\boldsymbol R}}} \right) - \dfrac{e}{h}{B_{y,{{\rm prj}}}}\left( {{{\boldsymbol R}}} \right)\\[4pt] - \dfrac{2\pi e}{hv}{E_{y,{{\rm prj}}}}\left( {{{\boldsymbol R}}} \right) + \dfrac{e}{h}{B_{x,{{\rm prj}}}}\left( {{{\boldsymbol R}}} \right)\end{array}\right),\end{equation}$$(10) where |${E_{\alpha ,{{\rm prj}}}}\left( {{{\boldsymbol R}}} \right) = \mathop \smallint {E_{ \bot ,\alpha }}\left( {{{{\boldsymbol R}}},z} \right){{\rm d}}z$| and |${B_{\alpha ,{{\rm prj}}}}\left( {{{\boldsymbol R}}} \right) = \mathop \smallint {B_{ \bot ,\alpha }}\left( {{{{\boldsymbol R}}},z} \right){{\rm d}}z$| are projected electric and magnetic field components perpendicular to the optical axis. In the present case, the left-handed system is adopted to define x and y axes with respect to z axis along the incident beam direction. Through the contrast transfer function (CTF) approach, electromagnetic fields can be quantified using any detectors, while the applicable domain of this approach (weak phase objects) is more limited than that of the CoM measurements (phase objects). Quantification accuracy by segmented detectors can be improved by combining the CTF approach and CoM measurement as discussed later in the section ‘Segmented detector approximation’. Major factors affecting the quantitative DPC STEM in actual experiments In the previous section, we reviewed the theoretical framework of quantitative DPC STEM, mostly under an ideal sample and optical conditions. However, there are several other factors which could severely affect the quantification of DPC images in the actual experiments. Here, we note these major factors as follows. Sample thickness effects The theory of quantitative DPC STEM basically assumes thin sample thickness, in which the POA as a projection approximation does hold. On the other hand, in actual samples used for DPC STEM experiments, sample thickness often becomes much thicker than the theory assumes. Therefore, thick sample effects are often inevitable in the actual DPC STEM experiments. One main problem due to thick sample is the effect of dynamical scattering. This effect smears the quantitative values of electromagnetic field strengths especially in the atomic-resolution regime [58, 59]. The other problem due to thick sample is the effect of inelastic scattering. This effect has been shown to also smear the quantitative values of electromagnetic field strengths, almost by a factor of two, when using segmented detectors [14]. An attempt to take into consideration the inelastic-scattering effect is discussed in the section ‘Plasmon-scattering effect’. Diffraction contrast effects When observing electromagnetic field distribution inside crystalline samples by DPC STEM, the DPC image intensity not only contains electromagnetic field–induced contrast, but also contains diffraction condition–induced contrast (henceforth diffraction contrast). Therefore, the quantification of DPC images should be severely affected by the diffraction conditions. In single crystalline samples, diffraction contrast may be circumvented by controlling the specimen tilt condition to avoid strong Bragg excitations. On the other hand, for polycrystalline samples or heterostructures, it is almost impossible to find suitable specimen tilt conditions, which sufficiently suppress diffraction contrasts in all the crystalline grains in the field of view. An attempt to suppress diffraction contrast is shown in the section ‘Diffraction contrast’. Superposition of electric and magnetic fields If incident electrons are passing through the region where electric and magnetic field co-exist, incident electron phase will be shifted by the both fields. Therefore, the resultant DPC image should reflect both the electric and magnetic field distributions in the field of view. In order to quantify the electric and magnetic fields in the DPC images separately, we need to differentiate the electric and magnetic phase shift components in the DPC image. Some previous methods developed in electron holography to differentiate electric and magnetic fields [70] may also be usable in DPC (e.g. (i) acquiring two images above and below the Curie temperature and subtract each other, (ii) acquiring two images with the specimen magnetized in opposite directions and subtract each other and (iii) acquiring two images with opposite incidence beam directions on the specimen and subtract each other). However, their applicability to DPC imaging is still under investigation. Stray fields and inactive layers of the sample If we are to quantify the electromagnetic fields of unknown samples by DPC STEM, we need to take good care of stray field contributions to the DPC image. This effect is already recognized in quantitative electron holography [71], but will also be problematic in the DPC quantification. In addition, when quantifying electric fields in semiconductor devices, inactive layers formed at the surface regions of the TEM samples will determine the ‘effective’ sample thickness required for quantifying the electric fields [72–74]. Some previous methods developed in electron holography to measure the inactive layer thickness of the TEM samples (e.g. [75].) may be also usable in DPC imaging. Aberrations and detector performances To quantify the experimental DPC images, it is important to characterize the optical conditions and the detector performance precisely. The optical conditions and residual aberrations can severely affect the DPC images. The detector performances can also affect the quantification of DPC images. Since the CoM of the diffraction pattern is a good measure of local electromagnetic fields at each scan raster, quantification of DPC images strongly depends on the accuracy of CoM measurement by the detectors. Therefore, detector geometry and detection sensitivity will strongly affect the accuracy of experimental DPC images with finite electron dose condition. An attempt to take into account the effect of segmentation geometry is reviewed in the section ‘Segmented detector approximation’. Several on-going attempts for DPC quantification Segmented detector approximation Although (D)PCTF is valid only within the domain of the WPOA, the breakdown of the approximation can be suppressed by combining CoM measurement with the (D)PCTF approach. The point spread function for the CoM measurement is the probe intensity function as in Eq. (3), which is derived according to the Ehrenfest’s theorem and valid within the domain of the POA. The same point spread function should be also derived by the PCTF approach under the WPOA with the detector response function |${D_{{{\rm CoM}},{{\rm \alpha }}}}\left( {{{\boldsymbol k}}} \right) = {k_\alpha }$|⁠, because the WPOA is contained in the POA [61]. Thus, if a detector response function well approximates that of the CoM measurement, the derived DPCTF and point spread function are expected to be a good approximation for strong phase objects. This expectation has been confirmed by systematic image simulations for the 16-segment detector [61]. Fig. 6 shows a comparison between DPCTFs for the segmented detector and a pixelated detector. The DPCTF for the segmented detector in Fig. 2(b) is anisotropic due to the coarse segmentation of the detector plane, in contrast to the isotropic DPCTF for the pixelated detector in Fig. 2(c). The anisotropic feature affects quantification of electromagnetic fields. Fig. 7 shows simulated electric field strength of single atoms: C, Ge and Pb (Z = 12, 32 and 82, respectively). The quantified electric fields based on the approximated CoM measurement (denoted by ‘aCoM’) [58] show anisotropic contrast. On the other hand, the anisotropic contrast can be reproduced by the convolution of the theoretical projected electric fields of the single atoms and the point spread function derived from the DPCTF (denoted by ‘Convolved field’). Their profiles agree well each other as shown in Fig. 7, and it shows the DPCTF is still valid for Pb single atom, which is normally not within the WPOA domain. Therefore, the true electric field distribution should be obtained through deconvolution of the point spread function derived from DPCTF within the bandwidth limit. The deconvolved electric fields and the theoretical projected field low-pass filtered at the information limit are compared in Fig. 7. It is well demonstrated that the anisotropic contrast is removed by the deconvolution, and thus, electric fields can be quantified. Thus, using DPCTF deconvolution method, we can quantify the electric fields within the domain of POA even using segmented detectors. Fig. 6. Open in new tabDownload slide DPCTF of the x component for (a) a segmented detector and (b) a pixelated detector. (c) The geometrical configuration of the segmented detector. The shade represents the BF disk. Adopted from [61] with permission. Fig. 6. Open in new tabDownload slide DPCTF of the x component for (a) a segmented detector and (b) a pixelated detector. (c) The geometrical configuration of the segmented detector. The shade represents the BF disk. Adopted from [61] with permission. Fig. 7. Open in new tabDownload slide Simulated field-strength images for (a) carbon, (b) germanium and (c) lead single atoms. The scale bar represents 0.05 nm. ‘aCoM’ images are based on the DPC signal obtained using the segmented detector with the geometrical configuration as shown in Fig. 6(c). ‘Convolved field’ images are the theoretical projected field convolved with the point spread function on the basis of Eq. (9). ‘Deconvolved CoM’ images are obtained by the deconvolution of the point spread function within the information limit. ‘BWL field’ images are the theoretical projected field bandwidth limited (BWL) with the information limit. The simulations were carried out with an accelerating voltage of 300 kV, a convergence semi-angle of 24 mrad, and all the atomic displacement parameters are set to be 0.0063 Å2. Adopted from [61] with permission. Fig. 7. Open in new tabDownload slide Simulated field-strength images for (a) carbon, (b) germanium and (c) lead single atoms. The scale bar represents 0.05 nm. ‘aCoM’ images are based on the DPC signal obtained using the segmented detector with the geometrical configuration as shown in Fig. 6(c). ‘Convolved field’ images are the theoretical projected field convolved with the point spread function on the basis of Eq. (9). ‘Deconvolved CoM’ images are obtained by the deconvolution of the point spread function within the information limit. ‘BWL field’ images are the theoretical projected field bandwidth limited (BWL) with the information limit. The simulations were carried out with an accelerating voltage of 300 kV, a convergence semi-angle of 24 mrad, and all the atomic displacement parameters are set to be 0.0063 Å2. Adopted from [61] with permission. The concept of the combination of the approximate CoM measurement using a segmented detector and the DPCTF has been experimentally demonstrated through imaging atomic electric fields of single gold atoms dispersed on carbon support film (Fig. 8) [41]. In this case, the error in the CoM measurement is reduced by applying a multiplicative constant to the approximate CoM based on Eq. (5). The constant is determined by least-square fitting of DPCTFs for the ideal CoM and the approximate CoM. The measured and simulated effective CoM profile (denoted by ‘eCoM’) agrees well with the simulated ideal CoM profile (Fig. 8d). Fig. 8. Open in new tabDownload slide Simultaneously obtained STEM images of Au single atoms. (a) Annular dark-field image. (b) Electric field vector color map. The inset color wheel represents vector direction and magnitude by its color and brightness, respectively. (c) Electric field strength map. (d) Comparison between the line profile of electric field strength of Au atom number 2 and simulated line profiles. Adopted from [41] with permission. Fig. 8. Open in new tabDownload slide Simultaneously obtained STEM images of Au single atoms. (a) Annular dark-field image. (b) Electric field vector color map. The inset color wheel represents vector direction and magnitude by its color and brightness, respectively. (c) Electric field strength map. (d) Comparison between the line profile of electric field strength of Au atom number 2 and simulated line profiles. Adopted from [41] with permission. Diffraction contrast When measuring mesoscopic electromagnetic fields using an electron probe larger than the crystal unit cell, atomistic electric fields between nuclei and surrounding electrons inside probe are averaged out and can be neglected within the theoretical framework described in the theory section. However, the periodic structure causes Bragg diffraction and affects the intensity distribution inside the BF disk. For an obvious example, dark lines called high-order Laue zone (HOLZ) lines can appear in the BF disk of a convergent electron beam diffraction pattern [76]. When the HOLZ lines appear in the BF disk nonsymmetrically, the DPC signal will be measured with a nonzero value even if mesoscopic electromagnetic fields are completely absent. In this manner, diffraction condition affects measured DPC signals, and the diffraction contrast is superimposed in DPC image. If observing electromagnetic fields within a single crystalline specimen, the diffraction contrast may be sufficiently suppressed by selecting crystal orientation without strong Bragg excitations. In conventional DPC imaging, diffraction contrast has been typically suppressed by tilting the specimens. However, it is indispensable to tilt the specimens with several degrees from a zone axis to avoid strong Bragg excitation, and it may result in the degradation of spatial resolution. Furthermore, for polycrystalline specimens with randomly oriented crystal grains or severely strained heterostructured materials, it is almost impossible to suppress diffraction contrast in all crystal grains in the field of view simultaneously. Diffraction contrast has been a severe limiting factor for applying DPC STEM in practical material characterizations. To overcome such problems, the specimen-tilt-series averaging method has been devised, and some applications for visualizing electric field in a p–n junction [15] and magnetic domains in an Nd-Fe-B polycrystalline magnet [28] have been reported. The concept of the developed method is schematically shown in Fig. 9. In the case of the polycrystalline magnet, for instance, DPC signal contains electromagnetic field component and diffraction component. Here, for simplicity, we assume that there are only magnetic fields but no electric fields inside the material. In addition, it is assumed that the magnetic field component is robust to the slight tilt of the specimen but the diffraction component sensitively and randomly varies with a slight tilt larger than the convergence angle. If the assumption is valid, diffraction contrast is expected to be suppressed by averaging DPC images obtained with various specimen tilt conditions. The specimen-tilt-series averaging method has been demonstrated in a GaAs p–n junction (Fig. 10) [15] and Nd2Fe14B magnet (Fig. 11) [28]. As expected, diffraction contrasts are significantly suppressed by the averaging method and the p–n junction and magnetic domains are more clearly visualized in the averaged DPC images. However, to obtain one averaged DPC image takes several hours by this technique, because tens of DPC images with various specimen-tilt conditions are required to form the one averaged image. On the other hand, diffraction contrast should be equivalently suppressed by tilting the electron probe instead of specimens. The probe-tilt system or precession system [18] may enable one to obtain averaged DPC images much faster. Fig. 9. Open in new tabDownload slide Conceptual diagram of the tilt-series averaging technique applied to DPC images. Adopted from [28] with permission. Fig. 9. Open in new tabDownload slide Conceptual diagram of the tilt-series averaging technique applied to DPC images. Adopted from [28] with permission. Fig. 10. Open in new tabDownload slide DPC images of a GaAs p–n junction by (a) conventional technique and (b) the tilt-series averaging technique. The color wheel represents direction and strength of electric fields. Adopted from [15] with permission. Fig. 10. Open in new tabDownload slide DPC images of a GaAs p–n junction by (a) conventional technique and (b) the tilt-series averaging technique. The color wheel represents direction and strength of electric fields. Adopted from [15] with permission. Fig. 11. Open in new tabDownload slide DPC images of Nd-Fe-B hot-pressed magnet by (a) conventional technique and (b) the tilt-series averaging technique. The color wheel represents direction and strength of magnetic fields. Adopted from [28] with permission. Fig. 11. Open in new tabDownload slide DPC images of Nd-Fe-B hot-pressed magnet by (a) conventional technique and (b) the tilt-series averaging technique. The color wheel represents direction and strength of magnetic fields. Adopted from [28] with permission. Fig. 13. Open in new tabDownload slide Quantified (a) projected electric field and (b) projected electrostatic potential of a p–n junction. The inset shows a DPC images obtained by the segmented detector. Adopted from [17] with permission. Fig. 13. Open in new tabDownload slide Quantified (a) projected electric field and (b) projected electrostatic potential of a p–n junction. The inset shows a DPC images obtained by the segmented detector. Adopted from [17] with permission. The residual diffraction contrast in the averaged DPC can be quantitatively evaluated from the DPC signals acquired in the specimen-tilt-series method. Under the assumption that the diffraction contrast is assumed to vary randomly with specimen tilts, diffraction component in DPC signal may be regarded as random noise. The residual diffraction contrast can be thus evaluated in the same manner as evaluating errors due to random noise. The details of the evaluation are discussed in the recent report [28]. Plasmon-scattering effect Specimen thickness for DPC observations sometimes has to be several hundreds of nanometers to increase DPC signals on detectors. Especially, for semiconductor materials, specimen thickness has to be thicker than the surface inactive layers, which are tens to hundreds of nanometers. In such thick specimens, inelastic scattering, dominated by plasmon scattering, blurs diffraction patterns and may affect quantification of DPC signals. The blurring effect on diffraction patterns may be removed by an in-column-type energy filter. If an in-column-type energy filter is not equipped, electromagnetic fields can be quantified by incorporating the blurring effect into the DPCTF as shown below. The blurring effect by plasmon scattering can be described by the convolution formula as below [77]: $$\begin{equation}{\tilde I_{{{\rm diff}}}}\left( {{{{\boldsymbol k}}},{{{\boldsymbol R}}}} \right) = {I_{{{\rm diff}}}}\left( {{{{\boldsymbol k}}},{{{\boldsymbol R}}}} \right) \otimes P\left( {{{\boldsymbol k}}} \right)\end{equation}$$(11) where |${\tilde I_{{{\rm diff}}}}\left( {{{{\boldsymbol k}}},{{{\boldsymbol R}}}} \right)$| is a diffraction pattern affected by the plasmon scattering, |${I_{{{\rm diff}}}}\left( {{{{\boldsymbol k}}},{{{\boldsymbol R}}}} \right)$| is a diffraction intensity in the absence of inelastic scattering, and |$P\left( {{{\boldsymbol k}}} \right)$| is a convolution kernel characterizing the plasmon scattering. Using Eq. (11), the plasmon-scattering effect can be incorporated into Eq. (4), and the following formula may be derived [14]: $$\begin{equation}\tilde S\left( {{{\boldsymbol R}}} \right) = \mathop \int {\left[ \left( {D\left( {{{\boldsymbol k}}} \right) \otimes P\left( { - {{{\boldsymbol k}}}} \right)} \right) \right]}{I_{{{\rm diff}}}}\left( {{{{\boldsymbol k}}},{{{\boldsymbol R}}}} \right){{\rm d}}{{{\boldsymbol k}}}\end{equation}$$(12) where |$\tilde S\left( {{{\boldsymbol R}}} \right)$| denotes DPC signal affected by the plasmon scattering. Here, we define an effective detector response function, |$\tilde D\left( {{{\boldsymbol k}}} \right) = D\left( {{{\boldsymbol k}}} \right) \otimes P\left( { - {{{\boldsymbol k}}}} \right)$|⁠, which takes into account the plasmon-scattering effect. Using the effective detector response function, effective PCTF and DPCTF considering the plasmon-scattering effect can be calculated. Plasmon scattering does not affect the quantification by the accurate CoM measurement, because |${D_{{{\rm CoM}},\alpha }}\left( {{{\boldsymbol k}}} \right) \otimes P\left( { - {{{\boldsymbol k}}}} \right) = {D_{{{\rm CoM}},\alpha }}\left( {{{\boldsymbol k}}} \right)$| is shown by assuming an isotropic convolution kernel |$P\left( { - {{{\boldsymbol k}}}} \right)$|⁠. For quantification with a segmented detector, if the convolution kernel |$P\left( {{{\boldsymbol k}}} \right)$| is known, the plasmon-scattering effect can be removed from DPC signals by deconvolution with the effective DPCTF. The convolution kernel |$P\left( {{{\boldsymbol k}}} \right)$| has been formulated by material-specific parameters: the plasmon energy, the plasmon damping coefficient, the plasmon dispersion coefficient and the mean free path for plasmon scattering [17]. If all the parameters are known, |$P\left( {{{\boldsymbol k}}} \right)$| can be obtained. In the case that the mean free path is known, it can be estimated from fitting to an experimental diffraction pattern [14]. Furthermore, even if no parameters is known in advance, |$P\left( {{{\boldsymbol k}}} \right)$| can be obtained by deconvolving two experimentally obtained diffraction patterns corresponding to |${I_{{{\rm diff}}}}\left( {{{{\boldsymbol k}}},{{{\boldsymbol R}}}} \right)$| and |${\tilde I_{{{\rm diff}}}}\left( {{{{\boldsymbol k}}},{{{\boldsymbol R}}}} \right)$| in Eq. (11); one is acquired in the absence of a specimen, and the other in the presence of a specimen without electric field variations. Electric field quantification by the effective DPCTF was demonstrated for a GaAs p–n junction [17]. Fig. 12 shows the two reference diffraction patterns and the extracted convolution kernel. The electric field can be quantified by the deconvolution using the effective DPCTF calculated from the experimentally extracted convolution kernel. Fig. 13 shows the projected electric field and electrostatic potential quantified by the deconvolution of the DPCTF neglecting plasmon scattering (‘DPCTF segment’) and the effective DPCTF (‘Reformed DPCTF segment’). The underestimation when neglecting the plasmon-scattering effect is successfully settled. To confirm the validity of this quantification, the comparisons with the two reference experimental results were made: the CoM measurement by a pixelated detector, which is not affected by the inelastic scattering as discussed above, and electron holography, which is not affected by the inelastic scattering because the interference fringes used for measuring the phase shifts are actually blurred but not shifted by the inelastic scattering. It is clear that all the results agree very well and within the error margins. Fig. 12. Open in new tabDownload slide BF disk patterns and their profiles obtained (a, d) in the presence of the specimen and (b, e) in the absence of the specimen. (c, f) Deconvolved pattern and its profile of (b) from (a). The scale bars denotes 2 mrad. Adopted from [17] with permission. Fig. 12. Open in new tabDownload slide BF disk patterns and their profiles obtained (a, d) in the presence of the specimen and (b, e) in the absence of the specimen. (c, f) Deconvolved pattern and its profile of (b) from (a). The scale bars denotes 2 mrad. Adopted from [17] with permission. Concluding remarks and future prospects As reviewed above, DPC signals of mesoscopic electromagnetic fields can be quantified by the DPCTF combined with the approximate CoM measurement to consider the detector-segmentation effect, by the tilt-series averaging technique to suppress the diffraction contrast and by the BF disk deconvolution technique to consider the inelastic (plasmon) scattering effect. One remaining major task is that when the electric and magnetic fields coexist, they need to be differentiated somehow. This may be possible by adopting the methods used in electron holography or devising new methods optimized for DPC imaging. The applicability should be experimentally evaluated in the actual materials. Quantification of atomistic electromagnetic fields by atomic-resolution DPC is in principle possible for very thin specimens. On the other hand, for thick specimens, the dynamical scattering effect should be taken into account, in order to extract the electromagnetic field distribution at atomic dimensions. While the quantification in atomic-resolution DPC appears to be successful in the image simulation level, experimental DPC quantification for detecting information on chemical bonds is still a very challenging task. Further developments in both instruments and theory must be required, but such ultimate imaging will be one of the promising future goals for the current on-going endeavors in DPC. Funding This work was mainly supported by the SENTAN (grant number JPMJSN14A1), Japan Science and Technology Agency. A part of this work was supported by the KAKENHI (grant numbers 19H05788, JP20H00301, JP20K15014 and 20H05659), Japan Society for the Promotion of Science. A part of this work was conducted in Research Hub for Advanced Nano Characterization, The University of Tokyo, under the support of ‘Nanotechnology Platform’ (project number 12024046) by Ministry of Education, Culture, Sports, Science and Technology, Japan. References 1. Morishita S , Ishikawa R, Kohno Y, Sawada H, Shibata N, and Ikuhara Y ( 2018 ) Attainment of 40.5 pm spatial resolution using 300 kV scanning transmission electron microscope equipped with fifth-order aberration corrector . Microscopy 67 : 46 – 50 . Google Scholar Crossref Search ADS PubMed WorldCat 2. Suenaga K , and Koshino M ( 2010 ) Atom-by-atom spectroscopy at graphene edge . 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TI - Toward quantitative electromagnetic field imaging by differential-phase-contrast scanning transmission electron microscopy
JF - Journal of Electron Microscopy
DO - 10.1093/jmicro/dfaa065
DA - 2021-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/toward-quantitative-electromagnetic-field-imaging-by-differential-PpUtKtWz6A
SP - 148
EP - 160
VL - 70
IS - 1
DP - DeepDyve
ER -