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Picometer-scale atom position analysis in annular bright-field STEM imaging

We study the effects of specimen mistilt on the picometer-scale measurement of local structure by combing experiment and simulation in annular bright-field scanning transmission electron microscopy (ABF-STEM). A relative distance measurement method is proposed to separate the tilt effects from the scan noise and scan distortion. We find that under a typical experimental condition a small specimen tilt (~6 mrad) in 25 nm thick SrTiO3 along [001] causes 11.9 pm artificial displacement between O and Sr/TiO columns in ABF image, which is more than 3 times of scan noise and sample drift induced image distortion ~3.2 pm, suggesting the tilt effect could be dominant for the quantitative analysis of ABF images. The artifact depends the crystal mistilt angle, specimen thickness, defocus, convergence angle and uncorrected aberration. Our study provides useful insights into detecting and correcting tilt effects during both experiment operation and data analysis to extract the real structure information and avoid mis-interpretations of atomic structure as well as the properties such as oxygen octahedral distortion/shift.


Introduction
Local structure distortion at grain boundary, hetero-interface, dislocation and surface can significantly influence on a broad variety of physical properties in complex oxide materials. Precise measurement of atom positions at these defects enables us quantitatively analyze the local strain field, electric dipole, flexoelectric effects, and chemical valence and thus provide new insights into understanding as to how materials properties depend on the local atomic structures and how we can engineer defects to optimize the materials or devices. Traditional crystallographic structure analysis such as x-ray or neutron diffraction is inadequate for the local structure analysis because of the poor spatial resolution. The recent advances in aberration corrected (scanning) transmission electron microscopy (S/TEM), however, can allow us to directly measure the inter-atomic distances with picometer-precision, providing a unique tool to study the local structure distortion. For examples, Bals et al. measured the local atomic structure of Bi 4 W 2/3 Mn 1/3 O 8 Cl by using exit wave reconstruction [1]. Jia et al. directly mapped the electric dipoles in ferroelectric thin films by using negative C s TEM observations [2]. The atomic displacements at ferroelectric domain walls in BiFeO 3 and PbZr 0.2 Ti 0.8 O 3 thin films are studied by measuring annular dark-field (ADF) STEM images from different groups [3,4].
Yankovich et al. reported sub-picometer precision measurement of Si lattice in ADF-STEM imaging [5]. In TEM mode, the exit wave reconstruction or negative C s condition require systematic simulations for the interpretation of image contrast and also very thin specimens (typically less than 5 nm). For the atomic-resolution STEM imaging, the interpretation of image contrast is much straightforward and reliable [6][7][8], and it is also capable of simultaneous spectroscopy, being powerful to characterize the local structural and chemical properties. However, in the ADF-STEM images (Zcontrast, Z is atomic number), the functional light elements such as lithium or oxygen are usually invisible when the compounds contain relatively heavier elements. In contrast, the ABF imaging [8][9][10][11][12] are able to simultaneous visualize both heavy and light element atomic columns over a wide range of thickness (typically more than 50 nm), allowing us to determine the positions of full atomic species from a single image.
Therefore, ABF-STEM imaging is one of best candidate for the quantitative measurement of local structure distortion, especially for the complex oxides analysis [13,14].
Generally, the measurement precision of inter-atomic distance in STEM is limited by scan noise and image distortion originated from the specimen drift (we use 'scan distortion' hereinafter). Several methods, therefore, have been employed to correct scan distortion [5,[15][16][17][18]. In fact, the specimen tilt is another important factor that could influence on the contrast and deformation of the atom-shape, as reported in ADF-STEM [19][20][21][22][23][24][25][26][27] due to the reduction in strength of the electron beam channeling [28]. On the basis of dynamical scattering theory, Van Dyck et al. discussed the effects of different range of specimen tilt [19]. Maccagnano-Zacher et al. pointed out that a specimen tilt reduces the contrast in atomic-resolution ADF-STEM [20]. So et al. reported that a specimen misalignment can cause a shift of atomic columns [21].
In the ABF image, the contrast of light atomic columns are basically related to forward elastic scattering, whereas the contrast of heavy atomic columns are contributed from both thermal diffuse scattering (TDS) and elastic scattering components [8]. Unlike the TDS signal (incoherent), the elastic scattering (coherent) contrast is very sensitive to the specimen conditions. One example is that the intensity of ABF image oscillates along the thickness, which becomes significant for the thin specimen and/or light atomic columns [8]. Therefore, the tilt effect in ABF image contrast is likely more sensitive than in ADF images, which has been firstly pointed out by Findlay et al. based on the SrTiO 3 simulations [8]. Recently, Gao et al. reported that the crystal tilt might cause significant artifact in interpretation of the ferroelectric polarization phenomena in ABF images [13,29]. Zhou et al. studied the deviation of atom positions between ABF and ADF and the effect of tilt on the bond angle measurements in ZrO 2 [30]. Liu et al. reported the effects of crystal tilt on the relative positions in PbTiO 3 in both ADF-and bright filed (BF) STEM images and found that tilt effect strongly depends on the tilt-angle and the specimen thickness [31]. Brown et al. proposed a new method to detect and correct the specimen tilt by simultaneously recording a central bright-field imaging (c-BF) and ABF images [32].
In this paper, we combine experiments and simulations to quantitatively study the small specimen tilt effects on the atom position analysis of ABF images with an emphasis on calculating and correcting the artifact, avoiding mis-interpretation of local atomic structure. Since the tilt is always mixed with scan noise and scan distortion in experimental images, we introduce a new method, relative-distance measurement, to separate the tilt effects from other factors. We choose perovskite discuss the tilt effects on quantitative measurement of atomic column positions in ABF-STEM imaging. In a perovskite structure, the rigid oxygen octahedron exhibits various structural distortions driven by stain and electrostatic conditions, underpinning vast functionalities. Many efforts have been paid to precisely measure the local distortion in perovskite structures by various microscopy techniques [2-4, 26, 33-47] but not much in ABF-STEM imaging [13,14,29]. In our work, we use a probe corrected ARM300CF (JEOL Ltd.) microscope, operating at 300 kV, and we simultaneously record both ADF-and ABF-STEM images, where a convergence semi-angle is 24 mrad, detector collection semi-angles spanning from 65 to 200 mrad for ADF and 12 to 24 mrad for ABF imaging, respectively. We also have performed systematic multi-slice image simulations for quantitative comparison with experiment.
Dynamical image simulations in STEM were carried out by using commercial software of HREM Research, Inc. The Debye-Waller factors for Sr, Ti and O are given in the previous literature [48].
We find that, in the tilted crystal, the atom position shift strongly depends on atom species, leading to significant artifacts in distance measurement between the light anion columns and heavy cation columns, i.e., artificial atomic displacements.
Under typical convergence semi-angle of 24 mrad, even with the small tilt of 6 mrad (0.34°) with 25 nm thick SrTiO 3 viewing along the [001] direction, the artificial displacement is estimated to be 11.9 pm between cation and anions atoms, which must lead to significant mis-interpretation of local atomic structure. This artificial displacement is much larger than that induced by scan noise and distortion (a few picometers). This artificial displacements depend on the tilt angle, defocus, thickness of the specimen, convergence angle and uncorrected aberration. Under some certain experimental conditions, such artifacts can completely dominate the measurement error, even the tilt is as small as 0.5 mrad. Since such small specimen tilt is inevitable even under the deliberate experiment operation, it becomes critical to consider the tilt effect for the precise measurement of atom positions in ABF images. We present in details on mechanism of formation, estimation and correction of the artifact, providing useful insights into local structure measurements of ABF images with specimen tilt.

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The electron transparent specimen of SrTiO 3 were prepared by mechanical polishing the single crystal followed by Ar ion milling (Precision Ion Polishing System, Gatan). Figs. 1a and b are simultaneously and sequentially recorded ADF and ABF images along the [001] direction with sampling rate of 5 pm per pixel [49].
Generally, the coexist of specimen tilt and drift effects are difficult to separate.
However, in the case of sequential imaging, the effect of specimen drift (scan distortion) can be minimized and the tilt effects stand out. Atomistic structure model is overlaid on the images and highlights the positions of Sr, TiO and O columns in  To elucidate such a subtle displacement, all the atom columns in both ADF and ABF images are fitted to two-dimensional (2D) Gaussian function using a homedeveloped MATLAB code [3,[39][40][41][42][43]45] to minimize the effects of scan noise. The 2D Gaussian peaks rather than the apexes are used to precisely determine the atom positions with sub-pixel precision (~ 0.5 pm precision). Note that we used raw atomic-resolution STEM images without any post-filtering. Intensity profiles in Fig. 1g-i show that the fitted 2D Gaussian profiles reproduce the experimental intensity profiles. In this case, D' is equivalent to D, indicating such relative displacement vector is independent of the specimen drift.Therefore, the measured value from experimental STEM images should be mainly due to the specimen tilt. We consider the following displacement vectors with four types of sub-lattices is given in   Note that although we have tried our best to tune the specimen zone axis, a tiny crystal tilt that below the detection limit still exists during operation. The average amplitude of the displacement vectors in cation sub-lattices is (2.9±1.5) pm (red) for ADF image and (4.0±1.7) pm (yellow) for ABF image. The measured deviation in ADF image is slightly better than that of ABF image, because ADF image has higher signal-to-noise-ratio (SNR) than that of ABF. For anion sub-lattice in I-ABF image, the mean value is (7.0±3.8) pm (green) and the standard error becomes larger than that in the case of cation sub-lattice. This lower precision is attributed to the lower SNR in contrast and fewer pixels for fitting O atom columns. Fig. 3e shows that the displacement vectors in ADF image are slightly random (broader distribution) than those in I-ABF image because the former has higher SNR (Note that the orientation of displacement vectors should be random if there is no displacement at all).

Experimental analysis
In contrast, the specimen tilt induced displacement of anion columns respect to the center of the nearest cation columns is clearly recognizable in Fig. 1(c), which is confirmed by the calculated vector map in Fig. 3f. The amplitude (19.0 ± 3.5) pm in For each sub-lattice configurations, the bond length a, bond angle between a and b, and lattice ratio a/b are calculated in Fig. 4. The standard deviation of the measured lattice constant is 3.3 pm (0.83% of lattice constant) for cations sub-lattice, while for anions sub-lattice this value, 6.0 pm, is slightly larger due to the lower SNR as discussed above. The lattice ratio a/b is 1.008 ± 0.012 (corresponding to 3.2±4.8 pm in distance) and the bond angle is (89.6 ± 0.65)° (corresponding to 2.8±4.4 pm in distance) for cations sub-lattice, whereas the distribution of anion sub-lattice is also slightly broader. The relative large standard deviation of lattice ratio and angle suggests a non-uniform lower frequency distortion during recording the entire image.
For the O vs Sr/TiO configuration (mixture sub-lattices), the bond length a', bond angle between a' and b', and ratio a'/b' have no obvious difference from the anion sub-lattice. Overall, the sample drift induced fluctuation in distance is on the level of ~3.2 pm that is relatively small compared to the tilt induced displacement ~19 pm.
Such small specimen drift effect is profit from the use of sequential imaging [49].

Simulation.
It is difficult to distinguish the effects of specimen mistilt from the optical misalignments in STEM. In order to elucidate quantitative information of the net specimen mistilt, a series of STEM images are simulated.  The amplitude of displacement between Sr and TiO columns in the simulated ADF images shown in Fig. 5b is calculated to be 3.3 pm with α=10 mrad, which is on the same level of the experimental scan noise and distortion. Similar result is obtained from the simulated ABF image, suggesting no obvious dependence of artificial displacement in the cation sub-lattices at the collection angle of detector. However, in ABF images, the displacement between cation and anion changes dramatically with specimen tilt. Only when the tilt α, ??≤4 mrad, the displacement is small enough (≤ 2 pm). Once the misalignment increases to 6 mrad, the displacement reaches 11.9 pm that is much higher than that of scan noise level. With α=??=8 mrad, the displacement is 42 pm which is equivalent to ~11% lattice constant. We note that, in the case without specimen tilt, the "background displacement" (systematic error: fitting noise) is only 0.027 pm for the simulated ADF and 0.048 pm for the simulated ABF, indicating the fitting noise is negligibly small compared to the mistilt effect. In Fig. 5c, the displacement vectors in cation sub-lattice and mixture sub-lattice are orientated in different directions. With mistilt > 2 mrad, the vectors calculated from mixture sublattice are orientated in the same direction, indicating the presence of significant artificial displacements. The displacements between different atom species also rely on the defocus and thickness of specimen, because the probe propagation with the tilted crystal shows a dechanneling to the other atomic columns [28].  Fig. 6b and c, respectively. By changing either thickness or defocus, the dependence behavior of amplitude is rather complicated. In our simulations, the relative displacements (>50 pm) for the mixture sub-lattice are maximized in the thickness range of 10-15 nm. The amplitude of displacements in ABF simulations also changes as a function of defocus. The dependence of orientation on thickness is not monotonous. When the thickness increases from 20 to 40 nm, the direction of displacement vectors becomes opposite. The defocus does not influence the orientation significantly. Furthermore, for ADF images, increasing convergence angle has been proposed previously to reduce the impact of the tilt effect [21]. To investigate validity of larger illumination angles to the ABF imaging, we performed systematic image simulation with several convergence angles as well as collection angles (in order to match the 'hollow-cone illumination' conditions [11] to optimize the ABF contrast [8]). The displacements between cation and oxygen are plotted in   [12,24] mrad. Green square: 36 mrad and [18,36].
Since the measured displacement vectors in STEM images are affected by crystal tilt, specimen thickness, defocus, and convergence angle, we could find the practical parameters that can reproduce the artificial displacement in experiments with the aid of image simulations. Fig. 8a  On the basis of the systematic investigations in simulated images, we found that the experimental data in Fig. 3g and h are in good agreement with the histograms of the amplitude and the orientation of displacement vectors in Fig. 8c and d  However, it should be noted that despite good agreement between experimental data and simulation, the experimental conditions are likely (more or less) different from simulation parameters due to the presence of coma, astigmatism, misalignments of aperture and detector, and therefore the simulated results can be considered as equivalent 'effective tilt' effects, which will discuss later. Nevertheless, the simulation can help us to effectively determine the specimen tilt and thus correct the artificial displacements.

Discussion
To reveal why the atom positions in ABF image is sensitive to the specimen tilt, we calculated the probe intensity propagation in the SrTiO 3 in Fig. 9, where the electron beam is focused on the entrance surface of the Sr column. In the case of tilted specimen in Fig. 9b (α=8  is less sensitive to the specimen tilt [21], as shown in Fig. 5a. In the case of specimen tilt, during image recording we usually deliberately introduce coma and astigmatism to partly compensate the tilt effect thus to get better visual appearance of the atom shape. Therefore, the specimen tilt can not only directly affect the positions of atom columns in STEM images but also indirectly influence the contrast via introduction of aberration. To evaluate the effects of aberration, the simulated ABF images with different A 1 , A 2 , B 2 and C 3 are shown in Fig. 10. When the A 2 or B 2 is smaller than 50 nm, the displacement is less than 5 pm that is at the level of experimental error such as scan noise or scan distortion. Note that, in our experiments, we use the automated software to set up the values of A 2 and B 2 less than 20 nm at amorphous regions [52]. In this regard, the net effects of aberration are negligible on the quantitative position analysis. Consequently, most of the quantitative discussion below is based on the direct tilt effect in the simulation, but these conclusions should be qualitatively applicable to experimental data. To quantitatively discuss the crystal tilt effect on precise position measurement, we define an 'effective sub-lattice', within which calculations between any atom columns remain effective. In the case of perfect zone axis (α=0), the entire lattice is an effective sub-lattice. With small tilt α≤4 mrad for ~ 25 nm thick STO, since all types of displacements are small enough (< 2 pm), the entire lattice can still be treated as an effective sub-lattice. Once the mistilt is α>5 mrad, the displacement between O and cations columns is ~11.9 pm for α=6 mrad that is well above the noise level and scan distortion. Therefore, the anion and cation columns in SrTiO 3 are no longer in the same effective sub-lattice, but the Sr and TiO columns can be still considered within the same effective sub-lattice if the tilt is ≤10 mrad. For large crystal mistilt >15 mrad, the Sr and TiO columns are not within the same effective sub-lattice any more. However, respective Sr, TiO and O columns are intrinsically in the same effective sub-lattice regardless of tilt as they have the same shift behavior on tilt, indicating unlike scan noise and distortion the specimen tilt will not affect the measurement of lattice constant, lattice ratio and angle between lattice vectors.

Conclusions
The effects of specimen tilt on the picometer-scale measurement of atom positions for ABF-STEM images are studied by combing the experiments and simulations. By using the relative distance measurement method, we are able to effectively distinguish the effects of crystal tilt from scan noise and distortion. Thus, it is possible to quantitatively evaluate the specimen tilt effects on the atom position analysis. The main conclusions are given below: (1) For ABF imaging, small tilt on the order of ~ 6 mrad can cause an artificial displacement 11.9 pm between anion and cation positions under typical experimental conditions for cubic SrTiO 3 . This value is 3 or 4 times larger than the scan noise and sample drift induced image distortion (3.2 pm), suggesting the specimen tilt is critical for picometre-scale measurement of ABF images because such a small tilt is inevitable during STEM operation.
(2) The tilt-induced artifact relies on the tilt angle, thickness of specimen, defocus and convergence angle. During specimen alignment, changing defocus and using small aperture is helpful to detect and correct the specimen tilt. However, larger aperture is preferable for imaging to minimize the artifact.
(3) Although the residual aberrations can also influence on the atom position analysis, the effects of aberration is difficult to separate from the net mistilt in the experimental images. Generally, small A 2 and B 2 (< 50 nm) would not cause significant artifact on displacements measurement.