Probing the Internal Atomic Charge Density Distributions in Real Space

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Probing the Internal Atomic Charge Density Distributions in Real Space Gabriel Sánchez-Santolino,† Nathan R. Lugg,† Takehito Seki,† Ryo Ishikawa,† Scott D. Findlay,‡ Yuji Kohno,∥ Yuya Kanitani,§ Shinji Tanaka,§ Shigetaka Tomiya,§ Yuichi Ikuhara,†,¶ and Naoya Shibata*,†,¶ Downloaded via UNIV OF SUSSEX on August 3, 2018 at 22:52:16 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

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Institute of Engineering Innovation, School of Engineering, The University of Tokyo, 2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan ‡ School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia ∥ Electron Optics Division JEOL Limited, Tokyo 196-8558, Japan § Advanced Technology Research Division, SONY Corporation, 4-14-1, Asahi, Atsugi-shi, Kanagawa 243-0014, Japan ¶ Nanostructures Research Laboratory, Japan Fine Ceramic Center, 2-4-1 Mutsuno, Atsuta-ku, Nagoya 456-8587, Japan S Supporting Information *

ABSTRACT: Probing the charge density distributions in materials at atomic scale remains an extremely demanding task, particularly in real space. However, recent advances in differential phase contrast-scanning transmission electron microscopy (DPC-STEM) bring this possibility closer by directly visualizing the atomic electric field. DPC-STEM at atomic resolutions measures how a sub-angstrom electron probe passing through a material is affected by the atomic electric field, the field between the nucleus and the surrounding electrons. Here, we perform a fully quantitative analysis which allows us to probe the charge density distributions inside atoms, including both the positive nuclear and the screening electronic charges, with subatomic resolution and in real space. By combining state-of-the-art DPC-STEM experiments with advanced electron scattering simulations we are able to map the spatial distribution of the electron cloud within individual atomic columns. This work constitutes a crucial step toward the direct atomic scale determination of the local charge redistributions and modulations taking place in materials systems. KEYWORDS: aberration-corrected STEM, differential phase contrast, electric field imaging, charge density, GaN he development of aberration correction1,2 has pushed the limits of transmission electron microscopy down to sub-angstrom imaging of crystalline lattices3−5 and the identification of single atoms.6−10 Furthermore, the electron microscope has been used to obtain information about the electronic charge density, determining the chemical bonding between atoms.11−14 However, the direct visualization of charge structures inside atoms has been a long-standing challenge.15 Recently, atomic-scale electric fields (atomic electric fields) have been directly probed in the scanning transmission electron microscope (STEM) by precisely measuring the deflection in the trajectory of the incident electrons resulting from the Coulomb−Lorentz force interactions with the electrostatic field of the material.16−18 This advance may allow the direct visualization of the nuclear and electronic charge densities with subatomic detail, as the atomic electric field contains the information on how both are distributed in real space. In the STEM, in order to quantify the atomic electric fields and charge densities it is necessary to

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measure the electron-diffraction pattern at each probe position. Based on the first-moment detector:19 the “center of mass” of the intensity distribution at the diffraction plane (ICoM) can be directly related to the projected atomic electric field.20,21 It is important to note though that an accurate quantitative measurement of the atomic electric fields and charge densities cannot be always obtained from the ICoM deflections. Dynamical electron-scattering effects depending on sample thickness and probe size have to be taken into account in order to apply the necessary tools such as the phase object approximation.22 It is a must then to ensure that the proper experimental conditions are met so we can infer the absolute values of the projected atomic electric fields and charge densities from the measurements of the ICoM. In this context, we make use of differential phase contrast STEM (DPCReceived: May 17, 2018 Accepted: July 30, 2018

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DOI: 10.1021/acsnano.8b03712 ACS Nano XXXX, XXX, XXX−XXX

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Figure 1. Visualization of crystalline GaN by aberration-corrected DPC-STEM. (a) Simulation depicting the intensity redistribution in the diffraction plane when the electron probe passes close to a heavy Ga column within a crystalline GaN specimen. The segmented area detector geometry is superimposed over the diffraction plane. (b) Experimental atomic resolution Z-contrast image for a GaN single crystal down the [112̅ 0] axis. The scale bar corresponds to 10 Å. (c) Orthogonal ICoMy (left), ICoMx (right) images obtained by weighting the signals acquired by each of the detector segments by their respective geometric center of mass. (d) Projected electric field vector map calculated from the ICoM components in (c). The inset color wheel indicates how the color and shade denote the electric field orientation and strength. (e) Electric field strength map obtained from the ICoM components in (c). All experimental images were acquired simultaneously. The superimposed atomic models depict the Ga (purple) and N (yellow) atomic column positions.

theoretical28,29 reports. An experimental atomic-resolution Zcontrast STEM image of GaN viewed along the [112̅ 0] axis is shown in Figure 1b. Simultaneously, using a recently developed segmented detector,30 we acquire the orthogonal components of the ICoM (ICoMx and ICoMy) which are shown in Figure 1c. These images are obtained by weighting the signals acquired in each of the detector segments by their respective geometric center of mass coordinates. The black (white) contrast indicates an upward/leftward (downward/rightward) shift of the center of mass, as shown previously in atomicresolution DPC-STEM images.16 This fact is more clearly illustrated in the projected electric field vector map of Figure 1d. This image is obtained from the components of the ICoM and denotes the direction and magnitude of the atomic electric fields, as indicated by the inset color wheel. This image depicts how the intensity at the diffraction plane is redistributed as the probe is scanned around the atomic columns; i.e., the atomic electric fields point outward from the nucleus. The magnitude of the atomic electric fields is given in Figure 1e. The atomic columns show a local minimum at the center where the incident electron beam is parallel to the atomic electric field and is therefore not deflected. In order to proceed with the quantitative analysis of the atomic electric fields and charge densities, it is necessary first to measure with high precision the specimen thickness for the observed area. To do so, we quantitatively compared the measured ICoM deflection with a series of systematic simulations for varying thicknesses. Figure 2a shows the modulus of the ICoM deflection at the diffraction plane obtained from the experimental orthogonal components in Figure 1c and from scattering simulations for thicknesses between four and nine unit cells. The experimental values are compared with those from simulation in the profiles of Figure 2b and in the maximum deflection values in Figure 2c. Notably, Figure 2c shows that the ICoM deflection increases linearly with the

STEM) in combination with advanced electron-scattering simulations to perform a fully quantitative analysis of the atomic electric fields and charge densities. We use a state-ofthe-art aberration-corrected STEM (JEOL ARM-300cF) equipped with a high-speed segmented detector to measure the ICoM as a sub-angstrom electron beam is scanned over a crystalline specimen. We carry out a quantitative comparison between experiment and simulations and demonstrate how our experimental setup and specimen fulfill the requirements in order to attain a quantitative measurement of the projected atomic electric fields and subsequently derive the nuclear and electronic charge densities. By these means we are able to push the limits of the technique to map the electron cloud charge distribution in real space within individual atomic columns with subatomic detail.

RESULTS AND DISCUSSION Figure 1a shows the simulated intensity distribution at the diffraction plane, and thus at the segmented area detector, when a 300 kV electron probe passes 0.2 Å to the left of a gallium atomic column in a 1.6 nm thick GaN crystal oriented along the [112̅ 0] direction. GaN is a common material in a wide variety of optoelectronic devices, such as blue lightemitting diodes,23 and has been extensively studied using DPC-STEM at nanometer resolutions, exploring the possibility of detecting the piezoelectric fields in optoelectronic devices.24−26 The probe intensity at the diffraction plane is shifted toward the Ga column due to the interaction between the incident electrons and the atomic electric field: the direction of deflection shows that the attractive force exerted on the incident electrons by the nucleus is dominant, although it is moderated by the screening electrons. The simulation exposes how the traditional picture of a rigid deflection of the probe due to the atomic electric field is only a simplification of the real effects, as described in recent experimental27 and B


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specimen thickness can be determined with one unit cell precision for thin samples: for the current data set, the thickness was determined to be five unit cells (or 1.59 nm). That our experimental specimen thickness is in the range for which the signal is linear with thickness justifies applying the phase object approximation, i.e., assuming that the GaN specimen is a single multiplicative phase object with all scattering effectively occurring in a single plane.22 Thus, with the current experimental conditions we can obtain a quantitative measurement of the atomic electric fields and charge densities. With the specimen thickness accurately determined, we can proceed to quantify the atomic electric fields and charge densities. Figure 3a shows a unit-repeated-averaged31 projected electric field image extracted from the data set in Figure 1d. Applying the differential form of Gauss’s Law to the projected electric field map, we obtain the projected total charge density for GaN shown in Figure 3b. Relatively lighter nitrogen atomic columns, which are not visible in the conventional Z-contrast imaging, are evident in the field images and even more clearly evident in the projected total charge density map. It is important to note that both the measured projected electric fields and charge densities constitute the true fields and densities convolved with the probe intensity. The experimental projected total charge density is displayed on the same scale for comparison with the similarly constructed image obtained from full dynamical scattering simulations (Figure 3c) and the direct calculation from the isolated atomic form factors (Figure 3d). Both the simulated and the directly calculated images are convolved with the effective source size (see the Supporting Information). The experimental and simulated images are normalized by the number of unit cells along the beam direction, and therefore, the values in Figure 3b−d correspond to the projected charge density distribution of single Ga and N

Figure 2. Sample thickness determination by comparing experimentally measured ICoM deflections with a series of systematic simulations for varying thicknesses. (a) Modulus of the ICoM for GaN viewed along the [112̅ 0] axis comparing the experimental unit-repeated-averaged image (top) obtained from the data set in Figure 1 and images reconstructed from scattering simulations assuming a segmented detector for a series of varying thicknesses (four to nine unit cells; below). The scale bar corresponds to 2 Å. (b) Line profiles taken along the white arrow in (a) for the experimental and simulated images. (c) Maximum value of the ICoM deflection at the Ga column with respect of the number of unit cells. The gray line serves as a guide to the eye to depict the linear behavior of the deflection up to seven unit cells.

number of unit cells up to a thickness of seven unit cells, beyond which this linearity breaks down. Consequently, the

Figure 3. Quantitative mapping of the GaN projected total charge density in real-space. (a) Unit-repeated-averaged image of the projected electric field for GaN down the [112̅ 0] direction obtained from the experimental data set in Figure 1d. The color and brightness, respectively, denote the direction and magnitude of the field. (b) Projected total charge density map calculated from (a). The scale bars correspond to 2 Å. (c) Projected total charge density map obtained from the scattering simulation shown in Supplementary Figure S1. (d) Projected total charge density calculated directly from the isolated-atom form factors, convolved with the probe intensity. All (b), (c), and (d) images are shown on the same quantitative color scale given on the right. (e) Line profiles taken along the white arrow in (c) showing the normalized Z-contrast signal profile (dashed black line) and the experimental (red dots), simulated (blue) and calculated (light green) projected total charge density profiles. Negative (positive) values represent negative (positive) charge densities. C


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Figure 4. Scattering simulations and projected nuclear and electronic charge density reconstructions for a single Ga atom. Montage of the intensity distribution on the diffraction plane (up to 32 mrad) as the probe is scanned over a 2 Å2 area with the negative (a) and positive (b) charge density distributions corresponding to a single Ga atom. Adjacent probe positions are spaced 0.2 Å apart. The insets on the bottom right show the reconstructed projected charge density maps using the center of mass calculation with a virtual segmented area detector. (c) Line profiles along the positive and negative reconstructed projected charge densities. (d) Projected total charge density of a single Ga atom obtained by adding the separated positive and negative simulations. A squarelike shape is just barely visible due to the fidelity of the ICoM measurement being different in different directions.22 (e) Z-contrast simulated image of a single Ga atom. (f) Line profiles comparing the projected total charge density values in (d) along with the normalized nuclear charge and Z-contrast intensity.

scattering factor for the full atomic potential is given by the Mott formula

atoms. The net-positive charge distribution for both atoms is clearly visible in these images. Though the theoretical spatial distribution of the nuclear charge can be considered as a δfunction, the experimentally obtained distribution is blurred by both the thermal smearing vibrations (finite temperature) and the effective source of the incident electron probe. Interestingly, one can observe a dark-blue area surrounding the Ga atomic columns indicating negative charge density values: the electron cloud around the positive nucleus of the Ga atoms. A close inspection on the numerical values for the total charge densities is provided by the line profiles of Figure 3e. The simulation reproduces well the experimental values, but both differ slightly from the calculated distribution, mainly in the heavier Ga column. On the other hand, the positive charge density values at the lighter N column and the surrounding electron densities of both the Ga and N columns agree with the direct calculation. It is worth emphasizing the good agreement in the absolute values between the calculation and the experiment illustrating how a segmented area detector can be used for quantitative studies even for strong phase objects, as suggested recently.32 Moreover, the graph shows the profile along the adjacent Ga and N columns for the simultaneously acquired Z-contrast signal, in which, surprisingly, the full width at half-maximum (fwhm) at the Ga atomic column is evidently broader than that of total charge density. This is despite the fact that the contrast transfer function, which describes how a specific spatial frequency in the imaged object is transferred to the image, is the same for both incoherent Z-contrast imaging and DPC-STEM: the Fourier transform of the probe intensity.32−35 To gain greater insight, we perform systematic image simulations for a single Ga atom in vacuum. The electron

me 2 ijj Z − fx (q) yzz ijj me 2 Z yzz jj zz = jj z z j 2π ℏ2ε q2 zz q2 2π ℏ2ε0 jk 0 { { k 2 ji me fx (q) zyz z ≡ fe(n) (q) + fe(e) (q) − jjjj 2 z z π ℏ ε 2 q 0 k {

fe (q) =

in which m is the relativistic electron mass, e is the electron charge, ℏ is the Dirac constant, ε0 is the permittivity in vacuum, Z is the atomic number, and f x(q) is the X-ray scattering factor as a function of reciprocal space coordinate q. To simulate the contribution to the total charge density image from the electron or nuclear charge components separately, we split the scattering factor into two components: one for the nuclear charge f(n) e (q) and a second for the electron charge f(e) e (q). The elastic scattering potentials used in the image simulations (which are the inverse Fourier transform of the scattering factors) are therefore calculated using either the electron or nuclear component only. Parts a and b of Figure 4 show montages of the simulated intensities at the diffraction plane at every raster position within a scanning area of 2 Å2 obtained from the electron or nuclear charge scattering potentials of a single Ga atom, respectively. The simulation in Figure 4a shows that the electron beam is repelled via the Coulomb−Lorentz force interaction with the negative charge distribution: the intensity at the diffraction plane is shifted outward from the atom position. The complementary simulation for the positive charge, Figure 4b, shows the opposite behavior: the intensity at the diffraction plane is shifted inward toward the atom position. The insets of Figure 4a,b show the projected negative and positive charge density D


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METHODS

maps, reconstructed from the pure negative and positive charge distribution scattering image simulations assuming our segmented detector configuration. Line profiles across these reconstructed charge densities are shown in Figure 4c. We add the separately reconstructed negative and positive charge densities to obtain the total projected charge density shown in Figure 4d. As in the experimental results, it is possible to detect a dark-blue area corresponding to the net-negative charge surrounding the Ga atomic nucleus: the electron cloud. Moreover, the quantitative values for a single Ga atom in this simulation lie within the same range as the experimental measurements, which were normalized by sample thickness. This finding strongly supports our quantitative analysis of the total projected charge density for atoms within a crystalline specimen. Figure 4e shows the simulated Z-contrast image for the single Ga atom, which is compared with the total charge density in the profiles of Figure 4f. As in the experimental images for GaN, the fwhm of the Z-contrast signal is wider than that of the net-positive charge density. The fwhm of the Z-contrast signal is 73 pm, which is ∼38% bigger than that of the net-positive charge density distribution (53 pm). For Zcontrast imaging, the annular detector collects the electrons that pass close to the nucleus and are scattered to high angles and discards the ones interacting with the atomic electron cloud that are scattered to small angles and pass through the hole of the detector. Consequently, one can neglect the electron−electron interactions, and the scattering becomes essentially Rutherford scattering.36 Therefore, for Z-contrast imaging, it is possible to effectively replace the full atomic scattering potential for a nuclear scattering potential.37 This is illustrated in the profiles of Figure 4f, in which the normalized Z-contrast signal obtained from a simulation using the full atomic potential follows that of the nuclear charge distribution obtained from only the nuclear scattering potential. We find similar results for other imaging techniques such as annular bright field,38 as shown in Figure S2. On the other hand, using DPC-STEM we are sensitive to both the atomic nucleus and the screening electrons. Therefore, when subtracting the broader negative charge distribution of the screening electrons (83 pm fwhm) from the narrower nuclear one, the tails of the positive distribution are suppressed, making the width of the net-positive contribution smaller than that of the conventional Z-contrast image. These results illustrate that we are indeed sensitive to both the charge distribution of the negative electron cloud and the positive nucleus, probing the internal atomic structure.

Sample Preparation. GaN film was grown on a sapphire (0001) substrate by metal organic vapor-phase deposition (Tgrowth = 1050 °C). STEM specimens were prepared by a focused ion beam using FEI Helios and Ar+ milling using Technoorg Linda Gentle Mill. DPC-STEM Experiments. Aberration-corrected differential phase contrast imaging was carried out using a new-generation segmented annular all field (SAAF) detector30 installed in an aberrationcorrected JEOL JEM-ARM300CF operated at 300 kV. The detector is composed of four rings divided in four quadrants for a total of 16 segments. For an accurate determination of ICoM, the 24 mrad brightfield disk is placed in the middle of the second ring of the SAAF detector, and the center of mass of the intensity is calculated using the first two rings.22 The collection angle for Z-contrast imaging was 70− 200 mrad. The experimental images are obtained by aligning 10 rapidly recorded 1024 × 1024 pixel frames acquired with an exposure time of 4 μs/pixel for a total acquisition time of 42 s, making it possible to obtain high signal-to-noise ratio images while reducing sample damage and drift effects.39 Scattering Simulations. Full dynamical multislice simulations40 using code developed in-house were performed for crystalline GaN down the [1120] axis for a thickness series ranging from 10 to 30 Å. Neutral scattering potentials for isolated atoms were assumed for both the image simulation and the direct calculation. Reproducing the experimental conditions, a 300 keV acceleration voltage and a 24 mrad probe forming aperture were used. The simulations produce convergent beam electron diffraction (CBED) patterns for every probe position as the beam is scanned over the sample. The full dynamical simulation for all probe positions and a thickness of 15.9 Å can be found in Figure S1. To compare with our experimental segmented detector images, we applied a virtual segmented detector to the simulated CBED patterns, obtaining 16 simulated images. The same analysis procedure as with the experimental images is then used to quantify the ICoM and obtain total charge density maps. To account for the finite source size and scan instabilities, simulated images were convolved with a 55 pm fwhm Gaussian, this size having been determined by comparison of the simulations with the simultaneously acquired experimental Z-contrast STEM image.

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.8b03712. Supplementary figures S1 and S2 (PDF)

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Gabriel Sánchez-Santolino: 0000-0001-8036-707X Yuichi Ikuhara: 0000-0003-3886-005X Naoya Shibata: 0000-0003-3548-5952

CONCLUSION In summary, we have shown that aberration-corrected DPCSTEM is capable of quantitatively mapping the projected total charge density distribution for a thin crystalline specimen. This technique allows us to reveal the internal atomic structure, visualizing the distribution of both the positive nuclear and negative electron charges in real space and thus discerning subatomic details. The next challenge will be to map charge redistributions in between the atoms, which is to visualize bonding valence electrons in real space. Moreover, the capability of DPC-STEM to operate with fast acquisition times and in real space makes it a promising technique to directly explore the electronic reconstructions taking place at heterointerfaces, grain boundaries and surfaces of materials at the atomic scale.

Author Contributions

N.S. and G.S.-S. designed and conceived the experiment. G.S.S. performed the STEM experiments and wrote the manuscript. N.L. performed the image simulations. T.S. and R.I. contributed to the analysis of the experimental results and simulations. Y.Ko. contributed to the development of the aberration-corrected DPC-STEM system and software. Y.Ka., S.Ta., and S.To. carried out the sample growth and specimen preparation. S.D.F. and Y.I. contributed to the discussion and commented on the manuscript. N.S. directed the entire study. Notes

The authors declare no competing financial interest. E


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ACKNOWLEDGMENTS This work was supported by the Research and Development Initiative for Scientific Innovation of New Generation Batteries (RISINGII) project of the New Energy and Industrial Technology Development Organization (NEDO), SENTAN, JST, and the JSPS KAKENHI Grant No. JP17H01316 and JP17H06094. A part of this work was conducted in the Research Hub for Advanced Nano Characterization, The University of Tokyo, under the support of “Nanotechnology Platform” (Project No.12024046) by MEXT, Japan. This research was supported under the Discovery Project funding scheme of the Australian Research Council (Project No. DP160102338). We would thank Dr. Hamish Brown for several helpful discussions. REFERENCES (1) Haider, M.; Uhlemann, S.; Schwan, E.; Rose, H. Electron Microscopy Image Enhanced. Nature 1998, 392 (6678), 768−769. (2) Krivanek, O. L.; Dellby, N.; Lupini, A. R. Towards Sub-Å Electron Beams. Ultramicroscopy 1999, 78, 1−11. (3) Nellist, P. D.; Chisholm, M. F.; Dellby, N.; Krivanek, O. L.; Murfitt, M. F.; Szilagyi, Z. S.; Lupini, A. R.; Borisevich, A.; Sides, W. H.; Pennycook, S. J. Direct Sub-Angstrom Imaging of a Crystal Lattice. Science 2004, 305, 1741. (4) Erni, R.; Rossell, M. D.; Kisielowski, C.; Dahmen, U. AtomicResolution Imaging with a Sub-50-pm Electron Probe. Phys. Rev. Lett. 2009, 102, 96101. (5) Sawada, H.; Shimura, N.; Hosokawa, F.; Shibata, N.; Ikuhara, Y. Resolving 45-pm-Separated Si−Si Atomic Columns with an Aberration-Corrected STEM. Microscopy 2015, 64, 213−217. (6) Crewe, A. V.; Wall, J.; Langmore, J. Visibility of Single Atoms. Science 1970, 168, 1338−1340. (7) Batson, P. E.; Dellby, N.; Krivanek, O. L. Sub-Ångstrom Resolution Using Aberration Corrected Electron Optics. Nature 2002, 418, 617−620. (8) Varela, M.; Findlay, S. D.; Lupini, A. R.; Christen, H. M.; Borisevich, A. Y.; Dellby, N.; Krivanek, O. L.; Nellist, P. D.; Oxley, M. P.; Allen, L. J.; Pennycook, S. J. Spectroscopic Imaging of Single Atoms Within a Bulk Solid. Phys. Rev. Lett. 2004, 92, 95502. (9) Shibata, N.; Findlay, S. D.; Azuma, S.; Mizoguchi, T.; Yamamoto, T.; Ikuhara, Y. Atomic-Scale Imaging of Individual Dopant Atoms in a Buried Interface. Nat. Mater. 2009, 8, 654−658. (10) Krivanek, O. L.; Chisholm, M. F.; Nicolosi, V.; Pennycook, T. J.; Corbin, G. J.; Dellby, N.; Murfitt, M. F.; Own, C. S.; Szilagyi, Z. S.; Oxley, M. P.; Pantelides, S. T.; Pennycook, S. J. Atom-by-Atom Structural and Chemical Analysis by Annular Dark-Field Electron Microscopy. Nature 2010, 464, 571−574. (11) Zuo, J. M.; Kim, M.; O’Keeffe, M.; Spence, J. C. H. Direct Observation of D-Orbital Holes and Cu−Cu Bonding in Cu2O. Nature 1999, 401, 49−52. (12) Meyer, J. C.; Kurasch, S.; Park, H. J.; Skakalova, V.; Künzel, D.; Gross, A.; Chuvilin, A.; Algara-Siller, G.; Roth, S.; Iwasaki, T.; Starke, U.; Smet, J. H.; Kaiser, U. Experimental Analysis of Charge Redistribution due to Chemical Bonding by High-Resolution Transmission Electron Microscopy. Nat. Mater. 2011, 10, 209−215. (13) Zhou, W.; Kapetanakis, M.; Prange, M.; Pantelides, S.; Pennycook, S.; Idrobo, J.-C. Direct Determination of the Chemical Bonding of Individual Impurities in Graphene. Phys. Rev. Lett. 2012, 109, 206803. (14) Borghardt, S.; Winkler, F.; Zanolli, Z.; Verstraete, M. J.; Barthel, J.; Tavabi, A. H.; Dunin-Borkowski, R. E.; Kardynal, B. E. Quantitative Agreement between Electron-Optical Phase Images of WSe2 and Simulations Based on Electrostatic Potentials That Include Bonding Effects. Phys. Rev. Lett. 2017, 118, 86101. (15) Shibata, N.; Findlay, S. D.; Matsumoto, T.; Kohno, Y.; Seki, T.; Sánchez-Santolino, G.; Ikuhara, Y. Direct Visualization of Local F


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