By focusing electrons on probes with a diameter of 50 pm, aberration-corrected scanning transmission electron microscopy (STEM) is currently crossing the border to probing subatomic details. A major challenge is the measurement of atomic electric fields using differential phase contrast (DPC) microscopy, traditionally exploiting the concept of a field-induced shift of diffraction patterns. Here we present a simplified quantum theoretical interpretation of DPC. This enables us to calculate the momentum transferred to the STEM probe from diffracted intensities recorded on a pixel array instead of conventional segmented bright-field detectors. The methodical development yielding atomic electric field, charge and electron density is performed using simulations for binary GaN as an ideal model system. We then present a detailed experimental study of SrTiO3 yielding atomic electric fields, validated by comprehensive simulations. With this interpretation and upgraded instrumentation, STEM is capable of quantifying atomic electric fields and high-contrast imaging of light atoms.
Innovative nanoelectronic devices such as ferroelectric tunnel junctions1,2 in the field of nonvolatile memory technology, or gallium nitride (GaN)-based3,4,5 laser diodes in optoelectronic applications rely on electric properties at the atomic scale. In both fields, differential phase contrast (DPC) experiments6,7,8 are considered as a key for the understanding of tunnelling electroresistance9 and the quantum-confined Stark effect10, respectively. On the basis of its success in the quantification of magnetic fields11,12,13 varying on the micrometre scale, conventional DPC is predicted to be able to map atomic electric fields at the picometre scale by aberration-corrected scanning transmission electron microscopy (STEM), and prospects to detect electron redistributions due to chemical bonding have been given6. Recently, impressive DPC experiments have been performed in which signatures of atomic electric fields, including ionicity, have been observed8 by recording portions of diffraction patterns on a segmented annular bright-field detector14,15,16. The design of this detector is derived from the assumptions that the central part of the diffraction pattern (the ‘ronchigram’) consists of a homogeneously filled disc, and that the Lorentz force causes a constant deflection of the electron beam as a whole.
In retrospect on nearly 100 years of electron diffraction in which elaborate theories have been developed to describe the complexness of multiple scattering of relativistic electrons in crystals17,18, this view on DPC impresses by its striking simplicity. In particular, electron ronchigrams rather appear as an artwork created by the sensitive interplay of scattering geometry, specimen thickness, probe parameters and crystal potential than simply being a shifted copy of a homogeneously illuminated disc. It is thus desirable to enhance DPC microscopy such that it can take intensity variations in diffraction patterns into account, and relate them quantitatively to atomic electric fields. In particular, this would be an important step as to studying electronic properties in nanotechnology with aberration-corrected STEM since Maxwell’s equations, furthermore, allow for the conversion of electric fields to charge- and electron densities.
In this work, we first lay the theoretical basis for the measurement of electric fields independently of the complexness of ronchigrams by exploiting the axioms of quantum theory and Ehrenfest’s theorem. By assuming that propagation and scattering of the STEM probe are negligible in thin specimen, we can relate the quantum mechanical expectation value for the momentum transfer to the electric field convolved with the probe intensity. Second, we develop the full method in a simulation study of GaN to show the validity of this assumption and demonstrate the possibility of mapping atomic electric fields, charge- and electron densities using a crystal where especially the latter exhibit a simple configuration. Finally, we present the electric field determined experimentally in SrTiO3. This is accompanied by comprehensive simulations accounting for the experimental conditions to validate the SrTiO3 results also theoretically. As our quantum mechanical interpretation formally results in a centre of mass calculation to measure the expectation value for the momentum transfer, we conclude that high-resolution DPC can be enhanced significantly by using two-dimensional (2D) pixel arrays instead of conventional segmented detectors.
A quantum mechanical approach to DPC
We contrast the established view on DPC with typical STEM data in Fig. 1. As illustrated by the set-up of conventional DPC experiments in Fig. 1a,b, a non-zero electric field is supposed to cause a global angular deflection of the STEM probe, leading to a characteristic shift of the ronchigram. This translates into a difference signal in opposite detector segments A and B. On the basis of the wave optical picture in Fig. 1c with an airy-shaped profile of a contemporary aberration-corrected probe (white) we raise the question whether the classical picture of Fig. 1a,b is realistic for the investigation of the underlying Coulomb potential of GaN (colour-coded). Exemplarily, the ronchigrams in Fig. 1d have been simulated for the STEM probe at positions labelled 1 and 2 in Fig. 1c. Even at the interstitial position 1 a rich inner structure is observed. The bottom ronchigram in Fig. 1d simulated for an electron probe focused near a gallium site exhibits no shift but a complex intensity transfer to the right, predominantly in the interior of the ronchigram. This explains why a segmented detector as in Fig. 1b is suitable to obtain high contrast at atomic sites8 whereby the quantification of electric fields in terms of ronchigram shifts can be inaccurate.
We solve this problem by looking at diffraction patterns from the standpoint of basic quantum mechanics. As specimen and Fraunhofer diffraction plane are related by Fourier transform , it is instructive to recall that is an equivalent representation of the object exit wave function Ψ with and momentum and real space coordinates perpendicular to the incident electron beam, respectively. By the axioms of quantum mechanics, eigenvalues of the momentum operator are observable with probability . This is the normalized intensity distribution in the diffraction plane for a sufficiently large number of detected electrons. Furthermore, the quantum mechanical expectation value for the momentum is
which equals the average momentum determined from repeated single-electron experiments for which is a solution to the Schrödinger equation. Note that we assume =0 for the incident probe so that is equivalent to the expectation value for the momentum transfer caused by interaction with the specimen. Equation (1) condenses the rich structure of the diffraction pattern into the average momentum , which is necessary and sufficient for our aim, as will be shown below. In order to obtain experimentally, a detector should be capable of performing the respective summation, for example, by using CCDs (charge-coupled devices). As equation (1) equals a centre of mass calculation formally, it becomes obvious that conventional DPC detectors as in Fig. 1b can fail to deliver from the moment the ronchigram exhibits inhomogeneous intensity.
The diffraction patterns in Fig. 1d include momentum transfers calculated from equation (1) as crosses. This yields and for positions 1 and 2, respectively, with Planck’s constant h=6.6261 × 10−34Js. Unlike the conventional DPC technique in Fig. 1b, this quantitatively exploits the detailed intensity redistributions caused by the electric field.
The Lorentz force is equal to momentum transfer per time in classical electrodynamics. According to Ehrenfest’s theorem19, this still holds in quantum mechanics for the expectation value of the momentum transfer for the case considered here. Electrons with velocity v spend the time z/v in a specimen of thickness z from which we can derive the following simplified relation between momentum transfer and measured electric field :
Here, e=1.6022 × 10−19C is the elementary charge and the projection average along the electron beam direction. We note here that the derivation of equation (2) formally involves the integration over the full three-dimensional (3D) electric field in the specimen, weighted with the local intensity of the electron beam. The latter is not known in experiment and changes as the electron wave passes the specimen due to scattering and propagation. To derive equation (2), we therefore simplified this problem by assuming that the intensity of the STEM probe does not change along the z direction. Hence, equation (2) is only valid for small specimen thicknesses (1–2 nm) where a broadening of the electron beam can be neglected (Supplementary Note 1 and Supplementary Figure 1). As the propagation of the electron wave is, furthermore, affected by a curvature of the phase of the incident wave, careful adjustment of aberrations and focus is required for the method. Finally, the measured electric field is a convolution of the true electric field with the normalized probe intensity IProbe,
For the detailed derivation of equations (1, 2, 3) we refer to Supplementary Note 2.
Methodical study using GaN
The power of this concept is initially demonstrated in a theoretical case study of GaN in projection. Here atomic columns are each occupied by a single atomic species with a comparably high difference in atomic number (ZGa=31, ZN=7), making it suitable to study large and small electric fields at respective sites simultaneously. Contrary to SrTiO3 chosen in the experiment below because of the large separation of 195 pm of adjacent columns, GaN exhibits a very clear electronic configuration, which is attractive in view of the derivation of charge- and electron densities.
Multislice17 simulations20 have been carried out at 80 × 80 scan points in the red region of Fig. 1c (see Methods section for details). Ronchigrams at 1.3 nm specimen thickness have been arrayed with respect to the probe positions in Fig. 2a. Atomic sites become evident from the redistribution of intensities inside the ronchigrams that reflects the tendency of electrons to channel along atomic columns. This effect is enhanced near gallium because of a more-than-four times larger nuclear charge compared with nitrogen. If we look at ronchigrams near gallium more closely, we see that intensity is partly also transferred in the opposite direction to form a diffuse halo as indicated in yellow in Fig. 2a. This can be explained by the finite extension of the STEM probe according to Fig. 1c because different parts of the probe experience different electric fields.
However, regardless of the complex intensity patterns inside the ronchigrams, we can rely on equation (1). Performing this integration yields the vector field for the momentum transfer drawn in Fig. 2b, superimposed to the colour-coded Coulomb potential with selected isolines. With Coulomb’s law and equation (2) in mind, this vector field is intuitively satisfying because the momentum transfer is invariably perpendicular to equipotential lines, is antiparallel to the electric field and increases drastically in the vicinity of atomic sites. However, the maximum momentum transfer of 4h nm−1 occurs 30 pm away from the gallium site and decreases again towards the nucleus, which is expected because of the finite probe size as a result of equation (3).
By virtue of equation (2), the electric field has been calculated from Fig. 2b and is depicted in Fig. 2c. Its main characteristic is the superposition of two central fields with different strenghts of up to 2 and 0.7 V pm−1 near the gallium and nitrogen site, respectively. To enable the comparison with the theoretical electric field derived from the negative gradient of the Coulomb potential, was convolved with the probe intensity according to equation (3), resulting in Fig. 2d. The excellent agreement with Fig. 2c as to magnitude and direction of the electric field demonstrates that the quantum mechanical interpretation of diffracted intensities is a key for field quantification at the atomic scale. Furthermore, we found that the integration in equation (1) can be restricted to a region slightly larger than the ronchigram (Supplementary Note 3 and Supplementary Figure 2) so that recording the central part of a diffraction pattern is sufficient in practice. In addition, scattering at phonons, plasmons and core electrons leave the above analysis unaltered for thin specimens (Supplementary Note 4 and Supplementary Figure 3).
Moreover, the divergence of the electric field is proportional to the charge density owing to Maxwell’s theory. Thus, equations (2, 3) reveal that the divergence of the momentum transfer is proportional to the charge density, convolved with the probe intensity. Figure 2e depicts div calculated from Fig. 2b. It shows that atomic sites are sinks of momentum transfer whose magnitudes are determined by the atomic numbers. The data agree perfectly with the inset calculated from the charge density directly using density functional theory (DFT)21, convolved with IProbe.
The overall charge density can be expressed as with and being the densities of electrons and protons, respectively. Compared with electrons, protons are spatially confined to the atomic sites that can be determined with picometre precision from the ronchigrams or from simultaneously acquired Z-contrast STEM images. Thereby, we find
with the vacuum permittivity ε0=8.8542 × 10−12C (Vm)−1. This shows that the measurement of enables mapping of the electron density convolved with the STEM probe, provided that the atomic numbers and the probe intensity are known. The latter can be recorded in practice or it can be simulated as optical parameters for probe formation are well known in aberration-corrected STEM. The proton density is set up by summing over Dirac’s delta functions at atom positions, weighted by the atomic numbers and eventually blurred to a Gaussian to account for thermal vibrations of the atoms. For the formal derivation of equation (4) we refer to Supplementary Note 5.
Figure 2f depicts derived from equation (4), which is obviously composed of two nearly rotationally symmetric atomic densities at the gallium and nitrogen sites, respectively. The inset shows the electron density calculated directly by DFT21, convolved with the STEM probe. Since nearly perfect agreement is found here, it is conceivable to measure the faint redistribution of electrons because of chemical bonding by DPC, as has been shown recently using conventional high-resolution TEM22.
Experimental study of SrTiO3
The applicability of our approach is evidenced in an experimental case study of Strontium titanate in  projection, where the distance of adjacent atomic columns is a factor of 1.8 larger than in GaN. In view of experimental constraints such as specimen drift and scan noise, this material is favourable because of a large ratio of intercolumnar distance to the STEM probe diameter. An aberration-corrected STEM instrument with a semiconvergence angle of 21 mrad was employed to scan over an SrTiO3 unit cell region (see Methods section for details). Ronchigrams of a 20 × 20-pixel scan have been arrayed with respect to the probe positions in Fig. 3a with a STEM dark-field image and atomic sites of the perovskite unit cell as inset. By looking at the intensity redistributions inside the ronchigrams, atomic columns containing heavy atoms, such as Ti at the corners and Sr in the centre, are clearly identified by the eye at first glance. At the second glance, even pure oxygen columns emerge from the ronchigrams, as exemplified by the yellow arrows in Fig. 3a. At this stage, this experiment already confirms that atomic electric fields rather cause complex intensity variations inside the ronchigrams than global shifts.
Next we applied equation (1) to calculate the momentum transfer from the ronchigrams in Fig. 3a to obtain the vector field depicted in Fig. 3b. As expected, a deflection towards atomic sites is observed clearly with a magnitude depending on the atomic numbers. In particular, atomic columns containing Sr or Ti exhibit maximum momentum transfers of ~3h nm−1, which is a factor two to three times larger compared with pure oxygen columns.
We finally proceed with the derivation of the electric field from Fig. 3b by applying equation (2). Using an atom-counting procedure via quantitative high-angle annular dark-field (HAADF) STEM analysis23 the specimen thickness was determined to z=2.5±0.4 nm. Vectors in Fig. 3c show the resulting electric field together with the colour-coded bright-field signal calculated from the ronchigrams in Fig. 3a. Obviously, all atomic columns show up as sources of the electric field with maximum field strengths of 0.4 V pm−1 near pure oxygen and 1 V pm−1 at both Ti/O and Sr columns, respectively. Owing to the extension of the STEM probe, the measured field strength decreases at the exact positions of atomic columns according to equation (3). In the following, these results are discussed with respect to the accuracy of the electric field measurement and the distortions due to scan noise (that is, random deviations of the STEM probe from a regular scan grid) obviously present in Fig. 3b,c.
As to the accuracy of the measured electric field, we calculated the theoretical counterpart for SrTiO3 from the negative gradient of the Coulomb potential and convolved the result with the probe intensity as shown in Fig. 4a. Compared with the experimental field in Fig. 3c, quantitative agreement is found for the columns of pure oxygen. Essentially, this also holds for the Sr and Ti/O columns, except for very few vectors close to the respective sites where the theoretical field takes slightly larger values of up to 1.4 V pm−1. This suggests that a smaller sample thickness would be preferable to enhance accuracy at these positions further as we start to leave the regime of small specimen thickness, at which equation (2) is valid for atomic columns with heavy atoms, such Ti and Sr. In that case, we nevertheless expect quantitative agreement of experimental momentum transfers in Fig. 3b with momentum transfers calculated from simulated ronchigrams at 2.5 nm specimen thickness, which are arrayed in Fig. 4b. Except for scan noise, the ronchigrams exhibit tight analogy to the experimental counterpart in Fig. 3a concerning the observed redistributions of ronchigram intensities towards atomic columns. In addition, the corresponding simulated momentum transfers in Fig. 4c take maximum values of ~3h nm−1 as in the experiment of Fig. 3b.
To complement the discussion on the influence of specimen thickness, Fig. 5 has been added, which shows the electric field that is expected for an experiment performed at slightly lower specimen thickness of 1.6 nm. In particular, ronchigrams have been simulated for that thickness, from which momentum transfer and, subsequently, electric field have been calculated via equations (1) and (2). This field is indeed practically identical with the theoretical one in Fig. 4a with a maximum error of 0.15 V pm−1.
Moreover, momentum transfer, electric field and bright-field intensity in Fig. 3b,c are not exactly rotationally symmetric with respect to atomic sites but exhibit a slight vertical elongation and a displacement of individual scan lines in vertical and horizontal directions. This can be explained by specimen drift during the relatively long acquisition time of 4 min despite the drift correction used, and by scan noise typically present in contemporary STEM data. To verify this interpretation, SrTiO3 ronchigrams have been simulated with positions of the STEM probe on an irregular grid chosen appropriately to the experimental drift rate and typical scan noise parameters24. The resulting momentum transfers in Fig. 6 exhibit the same characteristic distortions as in Fig. 3b, which demonstrates that the observed artefacts must be attributed to residual instabilities of the instrument. Note that the magnitudes of experimental and simulated momentum transfers agree quantitatively.
In addition, the observation that oxygen in Fig. 3b emerges significantly as a sink of momentum transfer despite its low atomic number deserves further attention, since imaging light atoms is still a major challenge in contemporary materials science25,26. As shown by the divergence of in an SrTiO3 experiment with 25 × 25 scan pixels in Fig. 7, all nine atomic columns of the perovskite unit cell in  projection (marked by the dashed black line) are imaged with remarkably high contrast, including columns of pure oxygen indicated by arrows. Note that these columns do not show significant contrast in both the annular dark-field STEM image in Fig. 3a (inset) and the bright-field STEM image in Fig. 3c.
In the strict sense, Fig. 7 ought to be proportional to the charge density in analogy to the GaN counterpart in Fig. 2e. However, calculating the divergence relies on numerical differences between momenta of adjacent scan pixels, which is critical in the presence of scan noise. For this reason we restricted our interpretation to the detection of light atoms, such as oxygen. Quantitative charge- and electron density mapping will therefore profit from ongoing improvements of the stability of the scan coils of STEM instruments, or require subsequent image processing24. Moreover, detection quantum efficiency and readout time of current CCDs impose an upper limit of a few hundred scan points of the STEM probe to minimize drift, specimen damage and contamination. This constraint can be expected to improve drastically with the recent development of ultrafast direct electron detectors27 working at kHz readout rates, which are in prototype state currently. For the present, drift in the range of the STEM probe diameter during several minutes must be accepted. These experimental limitations have been pointed out already in recent ptychography work28, which essentially uses a similar four-dimensional data set to obtain a complex-valued object transmission function by dedicated algorithms29,30,31. However, ptychographic studies have as yet focused on enhancing the microscope resolution beyond the diffraction limit instead of electric field mapping so that we leave a respective methodical comparison as a future task.
The situation is different for DPC microscopy. Being a well-established approach especially for magnetic field mapping, we shall comment on its previous success despite the classical interpretation of recorded data and the current segmented detector design. For fields that are practically constant across the diameter of the STEM probe, interaction with the specimen is indeed well described by multiplication with a phase wedge, leading to a shift of the diffraction pattern as a whole without redistributions of intensity inside the ronchigram. This condition is justified for fields varying on micrometre scale as mentioned in the introduction.
To conclude, we have demonstrated how to push DPC to atomic electric field quantification in a comprehensive simulation study of GaN and both an experimental and simulation study of the important ceramics SrTiO3. By relating diffracted intensities to the expectation value of the momentum transfer via axioms of quantum mechanics, and furthermore converting the result to the electric field by virtue of Ehrenfest’s theorem, this constitutes a fundamentally new view on DPC experiments.
We used the multislice algorithm17 implemented in the StemSim software20 to simulate thickness-dependent diffraction patterns of GaN and SrTiO3 with electron beam incidences along and , respectively (assumed parallel to the z direction). This algorithm especially takes multiple scattering of electrons in the specimen into account.
For the GaN study, a contemporary aberration-corrected STEM probe with a semiconvergence angle of 21 mrad was simulated for 300 keV energy. We assumed a residual spherical aberration of Cs=40 μm and a condenser operated under Scherzer conditions (giving the white intensity profile in Fig. 1c). Multislice results at a regular grid of 80 × 80 positions of the STEM probe in the red patch of Fig. 1c have been converged with respect to the lateral extension of the crystal supercell (being 15 × 15 the yellow patch in Fig. 1c), the slice thickness Δz (being 0.32 nm) and the cutoff in reciprocal space (being 90 nm−1, corresponding to nine times the ronchigram radius). Before the multislice simulations, we used DFT as implemented in the WIEN2K21 software for the calculation of the electron density in GaN to include electron redistribution due to chemical bonding. The result was then converted to the 3D electrostatic Coulomb potential V(r) using the Mott–Bethe relation32 in the framework of the modified atomic scattering amplitudes33 model adapted for GaN. In particular, this approach is able to treat bonding in an atomistic manner that enables us to account for the thermal movement of the atoms in the framework of Debye–Waller damping. The mean squared displacements of 0.0032 and 0.0037 Å2 (according to 300 K) have been used for Ga and N, respectively34.
The SrTiO3 simulations have been performed analogously but using optical parameters measured experimentally as listed in Table 1 and scattering factors according to isolated atoms35 for both the elastic and inelastic parts. A slice thickness equal to the lattice constant of 0.3905, nm was used. The mean squared displacements of 0.01 Å2 for Sr, 0.0071 Å2 for Ti and 0.01 Å2 for O (according to 300 K) have been used. Scan noise in Fig. 5 was accounted for by rastering the STEM probe over an irregular grid given by random deviations from the regular positions corresponding to a Gaussian distribution with a s.d. of 20 pm. This is a rather conservative estimate as literature24 gives 30 pm for a typical s.d.
This is the Coulomb potential of one slice, averaged along the z direction. It is important to note that all other quantities indexed this way (, , , ) analogously refer to their average along the z direction.
All simulations shown in the article have been conducted in absorptive potential approximation by adding an imaginary part35 to the Coulomb potential derived from the mean squared displacements. This accounts for the damping of the elastic signal due to thermal diffuse scattering with increasing specimen thickness and is known to be accurate up to much higher thicknesses than considered here. To verify this, thermal diffuse scattering is explicitly treated in Supplementary Note 4.
Experiments were performed at 300 kV in STEM mode on the Qu-Ant-TEM, an FEI Titan3 with probe and image aberration correctors, a monochromator and an X-FEG. The standard 50 μm condenser aperture was selected as limiting aperture with the monochromator slightly defocused to limit the beam current to ~30 pA. The SrTiO3 sample was thinned in the  direction using an FEI Helios Nanolab focused ion beam instrument with a beam energy of 30 kV. A final cleaning step was performed at low energy (2 kV) to reduce the amorphous layer thickness.
A beam semiconvergence angle of 21 mrad was used. The corrector was tuned such that the aberrations given in Table 1 have been measured.
All ronchigrams have been recorded on CCD camera with a 50-ms frame time and a sampling of 128 × 128 pixels. The SrTiO3 unit cell was scanned with samplings of 20 × 20 and 25 × 25 raster positions of the STEM probe because beam-induced specimen damage was observed for higher sampling rates of 30 × 30 pixels. This has been verified by comparing Z-contrast images recorded before and after acquisition of the ronchigram series, respectively. After each scan line, a drift correction has been performed by cross-correlating with a Z-contrast image of 2 × 2 SrTiO3 unit cells taken at 2 to 5 nm distance from the region of interest.
Specimen thickness was measured from quantitative analysis of an HAADF STEM image of the region of interest using the atom-counting procedure23 taking into account the response of the HAADF detector. In particular, a thickness of 5–7 unit cells (2–3 nm) was found. In addition, we checked this result by comparing the experimental annular bright-field signal with the corresponding signal extracted from simulated ronchigrams.
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K.M. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) under contract No. MU 3660/1-1. This work was also supported by funding from the European Research Council under the 7th Framework Program (FP7), ERC grant no. 246791—COUNTATOMS and ERC starting grant no. 278510—VORTEX. The Qu-Ant-EM microscope was partly funded by the Hercules fund from the Flemish Government. We acknowledge financial support from the European Union under the Seventh Framework Program under a contract for an Integrated Infrastructure Initiative, reference no. 312483-ESTEEM2. A.R. thanks the DFG for funding under contract no. RO 2057/8-1. S.L. and P.S. acknowledge financial support by the Austrian Science Fund (FWF) under grant no. I543-N20.
The authors declare no competing financial interests.
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Müller, K., Krause, F., Béché, A. et al. Atomic electric fields revealed by a quantum mechanical approach to electron picodiffraction. Nat Commun 5, 5653 (2014) doi:10.1038/ncomms6653
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