# Automatic software correction of residual aberrations in reconstructed HRTEM exit waves of crystalline samples

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## Abstract

We develop an automatic and objective method to measure and correct residual aberrations in atomic-resolution HRTEM complex exit waves for crystalline samples aligned along a low-index zone axis. Our method uses the approximate rotational point symmetry of a column of atoms or single atom to iteratively calculate a best-fit numerical phase plate for this symmetry condition, and does not require information about the sample thickness or precise structure. We apply our method to two experimental focal series reconstructions, imaging a β-Si_{3}N_{4} wedge with O and N doping, and a single-layer graphene grain boundary. We use peak and lattice fitting to evaluate the precision of the corrected exit waves. We also apply our method to the exit wave of a Si wedge retrieved by off-axis electron holography. In all cases, the software correction of the residual aberration function improves the accuracy of the measured exit waves.

## Keywords

Atomic resolution HRTEM Aberration correction Inline holography Off-axis holography Wavefront sensing## Background

Hardware aberration correction for electron beams in transmission electron microscopy (TEM) is now widespread, substantially improving the interpretable resolution in TEM micrographs [1, 2, 3, 4]. This technology is enabled by the combination of two factors; the ability to accurately measure optical aberrations in the electron beam, and a system of multipole lenses that can compensate for these measured aberrations. Many authors have studied the problem of direct aberration measurement, and most solutions involve capturing a Zemlin tableau [5, 6, 7, 8]. This method requires a thin, amorphous object that can approximate an ideal weak-phase object. Many samples of interest however are partially or fully crystalline. Thus, aberrations must be measured and corrected on an amorphous sample region before micrographs can be recorded on the region of interest. During this delay, the aberrations may drift due to electronic instabilities in the microscope [9], and this factor coupled with imperfect hardware correction can lead to residual aberrations in the resulting electron plane wave measurements.

One possible solution is to reconstruct the complex electron wavefunction via inline holography, by taking a defocus series and employing an exit wave reconstruction (EWR) algorithm such as Gerchberg-Saxton or the Transport of Intensity Equation [10, 11, 12, 13, 14, 15, 16]. Alternatively, an exit wave can be reconstructed by interferometric methods, i.e. off-axis electron holography [17, 18]. We can then estimate the residual aberrations and apply a numerical phase plate to the reconstructed complex wavefunction to produce aberration-free images [19]. These numerical corrections fall into two categories; manual correction, where the operator attempts to determine the aberrations present by trial and error, and automatic correction where the aberrations are directly measured in some manner. While the theory of aberration determination from a thin, amorphous sample is well-understood (and used to calibrate the hardware corrector on a modern TEM) [20, 21, 22], purely crystalline samples are much more difficult to correct due to the sparsity of diffraction space information [23]. If the sample is a low-index zone axis image of a crystal, there is no simple Fourier space technique to measure residual aberrations for a sample of unknown thickness or composition. Some authors have proposed using entropy methods [24] or measuring atomic column asymmetry within Fourier space [25] to measure residual aberrations. However, the first method requires well-separated atomic columns and the second can have difficulty measuring multiple simultaneous aberrations. We also note that some authors have used converged scanning transmission electron microscopy (STEM) probes to directly evaluate the aberration coefficients from crystalline samples [26, 27, 28], but these methods are not directly applicable to plane wave TEM measurements.

In this study, we propose a new method to measure aberrations from TEM images of crystalline samples containing on-axis atomic columns or single atoms. We use these measurements of residual aberrations to iteratively correct the complex exit wave until convergence is reached. Our method requires only a rough guess of the projected crystal structure and a regular (undefected) crystalline region in the image field of view. We test this method on three experimental datasets, focal series reconstructions of a β-Si_{3}N_{4} wedge with O and N doping and a single-layer graphene grain boundary, and an off-axis hologram measurement of a Si wedge.

## Theory

### Calculating images with radial point symmetry

The first two simulations in Fig. 1a, the [001] and [111] zone axes, have equally spaced atomic columns which show local radial symmetry around each peak. The third and fourth simulations in Fig. 1a contain Si dumbbells and appear to have broken radial symmetry at much shorter distances. These images however can be well-described by a sum of identical, radially-symmetric atomic peak shape functions, shown in Fig. 1b–d.

*J*atom types included, \(\mathbf {s}_j(|(x,y)|)\) is the complex atomic shape function for each atom type

*J*, and there are \(K_J\) atoms of type

*J*, located at coordinates \((x_k^j,y_k^j)\).

*x*,

*y*), while the columns represent all possible (rounded) distances to all nearby atomic sites, divided up into different atomic species. This matrix is moderately sparse, where the only non-zero values are ones in the first column (corresponding to \(A_0\)) and ones at the rounded distances of all atoms within some cutoff radius. This formalism allows us to solve for discretized atomic shape function(s) \(\mathbf {s}_j\) using the set of linear equations given by

### Coherent wave aberrations

*x*,

*y*) and \((q_x,q_y)\) represent the real space and Fourier space coordinate systems respectively. The aberration function used here is the basis function

*m*,

*n*) in units of radians, and \(\mathrm {atan2}(q_y, q_x)\) is the arctangent function which returns the correct sign in all quadrants (all combinations of signs of \(q_x\) and \(q_y\)). The radial magnitude of each aberration scales with \(|q|^{2m + n}\) and the rotation symmetry is given by

*n*. Note that when \(n=0\), the aberration is radially symmetric (e.g. constant value, defocus, spherical aberration) and no \(C_{m,n}^y\) term is necessary. Various authors use different conventions for dimensioning the coefficients \((C_{m,n}^x,C_{m,n}^y)\) [7, 19, 31]. We also note that this function describes only coherent wave aberrations that are constant over the field of view (aplanatic).

### Estimating residual aberration coefficients

Next, a symmetrized image is calculated from the aberrated wave and the approximate peak positions, shown in Fig. 2c. The resulting images appear to be approximately aberration free due to the radial symmetry imposed by constructing an exit wave from radially-symmetric point atomic shape functions, and can be used to estimate the aberration function \(\chi (q_x,q_y)\). To generate this estimate, we calculate the windowed Fourier transforms of both the aberrated and symmetrized waves. A window function is used to prevent boundary errors. Next, we measure the difference in phase between the two FFTs and use weighted least squares to fit the aberration coefficients. The weighting function is set to the magnitude of the original exit wave Fourier transform. This ensures that the strongest Bragg components dominate the aberration function fit.

Figure 2d shows the fitted aberration function, including all aberrations up to 6th order. The fits are a good, but not perfect, match to the real aberration functions in Fig. 2a. Applying the fitted aberration functions to the aberrated images produces the images plotted in Fig. 2e. Similar to the fitted aberration function, these images are improved but not yet free of aberrations. This estimation method for the aberration function can be applied iteratively to produce an accurate measurement of the residual aberration functions.

### Iterative algorithm for estimating residual aberrations

Next we calculate the distance matrix \(\mathbf {A}\) between all pixels in the reference region and the atomic coordinates. This procedure is shown geometrically for a single pixel in Fig. 3c. We then use linear regression to solve for the complex atomic shape function for all species present. The distance matrix \(\mathbf {A}\), carrier wave value \(A_0\), and the shape functions \(\mathbf {s}_1 \ldots \mathbf {s}_J\) are then used to calculate a symmetrized exit wave.

Subsequently, we compute a windowed Fourier transform of the current guess for the aberration-free exit wave (in the first iteration the measured exit wave is used) and the symmetrized wave. We measure the phase difference of these Fourier transforms, shown in Fig. 3f. We use weighted least squares to fit the aberration coefficients, where the Fourier transform amplitude of the exit wave is used as the weighting function. These aberration function coefficients are added to the current values from the previous iteration (originally initialized to zero). This fitted aberration function is then applied to the original exit wave as in Fig. 3g, generating an updated guess for the aberration-free exit wave. If the corrected exit wave update is below a user-defined threshold, we assume the algorithm is converged and output the result. If not, we perform additional iterations.

The algorithm described in Fig. 3 has three possible re-entry points for additional iterations, shown by the dashed lines. If we assume the atomic positions are accurate, we do not need to update them or recalculate the distance matrix \(\mathbf {A}\). Since this is the most time-consuming step of the algorithm, skipping it for additional iterations saves most of the calculation time. Alternatively, the atomic positions can be updated by peak fitting or a correlation method, starting the next iteration at the step in Fig. 3b. If the atomic positions are accurate enough, there is one other possible update at the start of each iteration. Each atomic site can be updated with a complex scaling coefficient to approximate slight thickness changes in the reference region. Both of these alternative update steps require updating the distance matrix \(\mathbf {A}\), step Fig. 3c.

### Limitations of the method

The algorithm for measuring and correcting residual wave aberrations described above requires a relatively flat, defect-free region within a portion of the full field-of-view. A small reference region will degrade the accuracy of the measured aberration function. In the experimental results shown below, the size of the reference region was \(\approx\)50 unit cells for the Si_{3}N_{4} sample, \(\approx\)1000 unit cells for the graphene sample, and \(\approx\)150 unit cells for the silicon wedge. The accuracy of the residual aberration function also depends on the signal to noise and accuracy of the exit wave reconstruction or measurement. If the crystalline region of the sample contains random variation of the exit wave due to an amorphous layer on the surface, or systematic variations due to surface reconstruction, the resulting aberration function may contain small errors. This issue can be minimized by using as large a reference region as possible, and with good sample preparation methods.

Another possible source of error is sample mis-tilt. Completely eliminating sample tilt is virtually impossible, and small amounts of sample tilt can mimic some residual aberration functions, in particular axial coma. Similarly, if the sample thickness changes linearly over the reference region, our method may fit a small amount of erroneous axial coma under some circumstances. However, because both of these effects heavily sample-dependent, it is impossible to assign firm numbers to the possible degree or error. In general we recommend using complementary measurements to verify results, such as measurement of mean atomic coordinates or unit cell dimensions or angles from x-ray diffraction.

## Methods

### Simulations

Multislice simulations were performed using Matlab code following the methods of Kirkland [29]. Unless otherwise noted, all simulations were performed at 300 kV, a pixel size of 0.05 \(\mathrm {\AA }\) and 32 frozen phonon configurations. An information limit of 1.5 \(\mathrm {\AA }^{-1}\) was enforced by applying an 8th order Butterworth filter to the exit waves in Fourier space. The exit waves were not further defocused after propagation through the sample, approximating a white atom contrast condition for all amplitude images.

### Experiments

The Si_{3}N_{4} sample was flat polished on one side using an Allied MultiPrep system, then mirror polished with 0.1 μm diamond paper. The second side was dimpled and finished with a 1.0 μm diamond slurry to a thickness of about 20 μm. The sample was then ion milled on a Gatan PIPS at 0 °C using 5 kV Ar ions at an angle of 5° for 3 h, then at 1 kV for 30 min, followed by 0.5 kV for 5 min. This latter sample was not carbon coated and was found to be stable under the beam operated at 300 kV. Focal series of this sample were recorded at 300 kV in the TEAM 0.5, an FEI TITAN-class microsope [3]. The corrector was tuned for bright atom contrast (C_{3} = −6 μm, C_{5} = 2.5 mm) and the monochromater was excited to provide an energy spread <0.15 eV at full width half maximum. The focal series were acquired with a step size of 1.72 nm ranging from −34.4 nm underfocus to 34.4 nm overfocus, recorded on a Gatan Ultrascan 1000.

The graphene sample was grown at 1035 °C by chemical vapor deposition onto a polycrystalline copper substrate. The substrate was held at 150 mTorr hydrogen for 1.5 h before 400 mTorr methane was flowed over it for 15 min to grow single layer graphene [32]. This sample was imaged in the TEAM 0.5 microscope using mochromated, spherical aberration-corrected 80 kV imaging with the monochromater excited to provide an energy spread <0.15 eV at full width half maximum. A focal series of 5 images with a step size of 1.2 nm was recorded on a Gatan OneView detector.

An off-axis hologram of a silicon wedge was recorded in the Cc-Cs-corrected TEAM I microscope operated at 80 kV accelerating voltage using an exposure time of 8 s on a Gatan Ultrascan 1000. The [011]-silicon sample was laser cut from a 3 mm disc down to as 1 mm to fit the TEAM stage geometry [33]. For hologram acquisition, the corrector was tuned to correct all aberrations, up to and including 3rd order, below the measurement accuracy of the aberrations. The exit wave was reconstructed from the hologram using simple numerical Fourier optics [17].

### Focal series reconstruction and data analysis

All focal series reconstructions and data processing except for the off-axis holographic reconstruction were performed using Matlab code. Focal series reconstructions were performed using the Gerchberg–Saxton algorithm [10] where the implementation for HRTEM is described fully in [11, 12]. During reconstruction sub-pixel image alignments were applied using the discrete Fourier transform method given in [34].

## Results and discussion

### Exit wave reconstruction of Si\(_3\)N\(_4\)

The first sample analyzed is a SiAlON wedge sample (isostructural to \(\upbeta\)-Si\(_3\)Al\(_4\) with Al and O doping the Si and N sites to give the composition Si\(_{5.6}\)Al\(_{0.4}\)O\(_{0.4}\)N\(_{7.6}\)), recorded at 300 kV along the [0001] direction. Density functional theory [35] and neutron-scattering studies [36] predict that O might preferentially dope the 2a sites with a nearby Al balancing the extra charge, causing a 21 pm shift in one of three directions [37]. X-ray diffraction by contrast shows no site preference for Al or O [38]. We therefore wish to measure the column positions with as high a precision as possible to evaluate the dopant-ordering hypothesis and its potential local variation at the nanoscale. The SiAlON wedge will be referred to as the Si\(_3\)N\(_4\) sample for the remainder of this paper.

After measuring the residual aberrations from a small reference region, shown in Fig. 4, we have corrected these aberrations on the full image and plotted the amplitude in Fig. 5. The atomic positions appear extremely sharp, and no defects are visible other than the vacuum at the edge of the wedge sample. From multislice simulations we estimate the thickness of the crystalline portion of this sample ranges from 3 to 7 nm.

To quantify the atomic column positions, we used nonlinear least squares to fit the peak positions using a complex, two-dimensional elliptic Gaussian function. For the Si-N dumbbells, two complex elliptic Gaussian functions are fitted simultaneously. The fitted peak positions relative to the ideal Si\(_3\)N\(_4\) lattice positions for a subset of 180 of the peaks are plotted in Fig. 5b, c from the exit waves before and after aberration correction. From the root-mean-square (RMS) displacements plotted, we see that the aberration correction has improved the fitting precision on most of the lattice sites. In particular, the dumbbells with strongly-overlapping peak functions have improved substantially, reaching peak precisions as low as 1.1 and 1.4 pm for the Si and N sites respectively. The isolated 2a N site position precision is not strongly affected by the residual aberrations.

Returning to the original question of measuring atomic shifts due to the doping, we have plotted the bond length distributions of all nearest-neighbor sites that are more than 2 unit cells distance from the vacuum edge and the edge of the full micrograph, in Fig. 5d, e. Before aberration correction, the bond length distribution for the dumbbell Si-N and the 2a N site –Si bonds appears to follow a bimodal distribution. The larger Si-N bond spacing in the hexagonal rings is even more distorted, spreading over approximately 50 pm. However after correcting the residual aberrations, all bond length distributions become monomodal. Therefore we found no evidence of systematic shifts in the 2a N sites. Additionally, no local distortions of the \(\upbeta\)-Si\(_3\)N\(_4\) lattice such as those described in [39] were observed in this experiments. Finally, we note that because the reference lattice contains 14 site locations in each unit cell where measurements are taken, it is highly unlikely that we could be forcing one of the sites (such as a systematic 2a distortion) to be at an incorrect location. The algorithm should select the phase plate function which best minimizes the global aberrations.

### Single-layer graphene grain boundary

The phase of the unobstructed region of the graphene grain boundary is plotted in Fig. 6b, c for the uncorrected and corrected exit waves respectively. Before aberration correction, we observe that the graphene lattices are extremely regular, but contain very little interpretable information. The grain boundary is particularly messy, due to the complex interaction of non-radially symmetric residual aberrations with the various atomic spacings present. By contrast, the corrected phase image in Fig. 6c has very well resolved atomic sites both in the crystalline lattices and along the grain boundary. Almost every site can be identified and the boundary structure can be easily quantified. We have used focal series exit wave reconstruction and the aberration correction algorithm described in this paper to characterize the structure of many different single-layer graphene grain boundary misorientations [42, 43, 44].

### Off-axis hologram of a silicon wedge

The range of phases measured in these images is substantially higher than those in the previous focal series measurements, almost \(2 \pi\) along the thinnest edge of the sample. After aberration correction, the Si dumbbells are more cleanly resolved. To show the dumbbell structure more clearly, we have plotted line traces in Fig. 7c, d, for the uncorrected and corrected phase images respectively. After correction, almost all dumbbells show clear separation between the two Si atomic columns.

## Conclusion

We have developed an algorithm for measuring and correcting residual coherent wave aberrations in complex exit waves of crystalline samples, measured in transmission electron microscopy. Our algorithm relies on creating a synthetic exit wave by applying point-symmetrization to all atomic columns in a reference region, to approximate the aberration-free exit wave. Because our method is objective and automatic, it is not prone to operator errors that could be introduced from manual correction of the residual aberrations. It is important to note that no symmetrization is applied to the final experimental exit wave. We have applied our method to three experimental datasets, focal series reconstructions of a Si\(_3\)N\(_4\) wedge and a single-layer graphene grain boundary, and an off-axis hologram of a silicon wedge. In all cases, the residual aberration correction improved the precision, accuracy and interpretability of the complex exit waves. Our algorithm is simple to implement, and applicable to a large class of experimental exit wave measurements of crystalline samples oriented along a low-index zone axis.

## Notes

### Authors' contributions

JC and HR recorded the focal series from the SiAlON and graphene samples respectively. ML recorded and reconstructed the silicon wedge off-axis hologram. CO reconstructed the exit waves, developed the aberration correction algorithm, applied it to the samples used in this study and performed the analyses. All authors contributed to writing the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

We thank Alain Thorel for providing the SiAlON sample, and Marissa Libbee for preparing a SiAlON wedge. CO thanks Christoph Koch for helpful discussions. We also thank Cory Czarnik for assistance with the Gatan OneView electron detector. Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. HIR and AZ acknowledge support in part by the Director, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, of the U.S. Department of Energy under Contract DE-AC02-05CH11231, within the sp\(^2\)-bonded Materials Program, which provided for detailed TEM characterization.

### Competing interests

The authors declare that they have no competing interests.

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