Atomic structure from large-area, low-dose exposures of materials: a new route to circumvent radiation damage.

Beam-induced structural modifications are a major nuisance in the study of materials by high-resolution electron microscopy. Here, we introduce a new approach to circumvent the radiation damage problem by a statistical treatment of large, noisy, low-dose data sets of non-periodic configurations (e.g. defects) in the material. We distribute the dose over a mixture of different defect structures at random positions and with random orientations, and recover representative model images via a maximum likelihood search. We demonstrate reconstructions from simulated images at such low doses that the location of individual entities is not possible. The approach may open a route to study currently inaccessible beam-sensitive configurations.

Introduction

The remarkable developments in electron microscopy over the past few years, in particular the correction of lens aberrations [1,2], have improved resolution to such a degree that practically all atomic distances can be resolved [3-11]. At the same time, the reduction of delocalization effects simplifies the analysis of structures on the level of single atoms [12-16]. However, taking advantage of these developments requires that the structures under investigation remain unchanged under extremely high electron doses. For example, discerning individual light atoms typically requires doses well above [10,16-20].

While radiation damage is a well known bottleneck to the applicability of electron microscopy based methods in biological studies, it has become increasingly relevant also in the study of materials. This is not only because the required signal to noise ratio necessitates higher doses at higher resolution, but also because beam-induced structural changes, which would go unnoticed at a lower resolution, can no longer be tolerated. In the case of low-dimensional materials, such as carbon nanotubes [21], graphene [22], hexagonal boron nitride [23] or mono-layer dichalcogenides [24,25], for which only a single or a few light atoms are present in the projection of a transmission electron microscope (TEM) or scanning-TEM (STEM) image, the problem is at its extreme: the contrast is very low (thus requiring highest doses), the atoms are easily displaced, and at the same time, the position of every single atom (rather than an extended atomic column) is visible and therefore relevant for the analysis.

If knock-on damage dominates the radiation damage, low-voltage microscopy is a viable route to avoid beam-induced changes in the atomic structure [26-32]. Indeed, low-voltage aberration-corrected TEM and STEM have enabled remarkable atomically resolved images of graphene, carbon nanotubes, and other low-dimensional or layered materials [11,15-17,19,20,25,33-49]. Nevertheless, significant beam-driven dynamics are present in all of these studies, especially at defects, edges, or contamination sites. For example, edges of a graphene sheet are still highly dynamic even under 20 kV electron irradiation [30], defects in graphene easily change their shape in 80 kV image sequences [50], and defects are introduced in molybdenum disulfide [25] or hexagonal boron nitride [18,19] under irradiation. It is possible that some configurations completely escape their detection in conventional studies because they decay towards more beam-stable ones within a fraction of the dose needed for high-resolution images. Lower voltages also increase the electron–electron scattering cross section, and hence lead to increased ionization damage. In cases where ionization damage is not suppressed by the high conductivity of a material, it is an order of magnitude larger effect than the atomic displacements [51]. For example, organic molecules typically withstand doses from 10−1 to [52-54,51], which is many orders of magnitude below what is needed for atomic-resolution images of their structure.

An alternative route to circumvent radiation damage is to distribute the dose over many identical copies of an object. This approach is under active development for imaging biological molecules [55-60], where the (very small) tolerable dose limits the available resolution, rather than the instrumental performance of the microscope. Within the single-particle analysis (SPA) [55-60], a large number of images from identical objects are first recorded with very low dose, and then classified into the different orientations (or conformations, if applicable). Finally, averaged images with a sufficient signal to noise ratio are calculated. However, this approach only works if the individual exposures contain enough information for the classification. It was estimated that it requires large biological molecules with a molecular weight above 105 [61]. Very recently, new methods have been developed to reconstruct electron microscopy and X-ray diffraction data even with individual patterns far below the signal level for direct classification [62-67]. These approaches extract information from correlations in the entire data set, rather than sorting and averaging of individual exposures, relying on the fact that different images of identical objects differ only in a small number of hidden parameters (e.g. their orientation). The study in Ref. [67] showed, among other things, the first case of a reconstruction from simulated TEM data for a biological molecule at a dose where orientation assignment by standard classification was not possible, albeit with only one orientational degree of freedom. However, this idea, i.e., to distribute the dose over many identical configurations within a given sample to minimize the exposure of each object, has not been explored so far for non-periodic configurations in a material.

In this study, we consider localized deviations from a regular lattice in high-resolution TEM or STEM images. We show that it is possible to exploit the multiplicity of identical configurations so that the dose on each object can be reduced by approximately two orders of magnitude, compared to similar quality images of individual entities. The key novelty of our approach is that it works even if the dose in the exposures is too low for locating or classifying individual objects (point defects, functional groups, adsorbed molecules, etc.) in the raw data. The method should be applicable to any case where a finite set of deviations from a regular lattice can be expected to occur repeatedly on a sufficiently large area of the sample. Examples include point defects in a material (shown here for a 2-D material) as well as functional groups or small molecules on the surface of the material.

We begin by quantitatively demonstrating the need for this new approach. Fig. 1 shows simulated HRTEM images of a graphene sheet with a randomly distributed mixture of the three frequently observed types of di-vacancy defects [68], for variable doses. The simulation assumes imaging conditions as used previously [68], and a perfect CCD camera. A dose of ca. is needed to detect the presence of a vacancy. However, this dose is already almost sufficient to assign the structure (discernible at about , and clearly visible at ). The set of images in Fig. 1 highlights the key reason why previously developed methods for biological structures are of limited use for the case of point defects in a material: they all require that at least the center of mass of the object is known, and in addition each exposure usually must provide some kind of classification (e.g. orientation or conformation type), prior to averaging. Here, however, very high doses are needed already to detect the defect position. This means that any approach that works by classification of individual exposures and subsequent averaging cannot provide a substantial reduction in dose, compared to direct images of individual entities.

Fig. 1

Simulated HRTEM images of a graphene sheet with a randomly distributed mixture of three di-vacancies, at different doses. (a) Infinite dose simulation, (b–h) finite dose simulation for the same structure, using Poisson noise. From the visual appearance, it becomes difficult to identify the defects at doses below . This can be confirmed by a cross-correlation calculation, using for example the blue box in (a) – the 555777 di-vacancy – as reference: The centers of the circles in (c–e) are placed at the highest three (c,d) and highest five (e) maxima of a cross correlation map that aims to find this structure in the noisy images. Errors occur at , and at it is not possible to locate any of these defects. Scale bar: 1 nm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

We emphasize that the doses needed for imaging light-element low-dimensional samples are generally much higher than those required for bulk crystals of heavy elements. Therefore, for any heavier material, or larger defects or functional groups, the approach shown here is likely to work at significantly lower doses (since the noise level scales with , where D is the dose, increasing the contrast by a factor of e.g. 2× would allow to reduce the dose by a factor of 4, while maintaining the same signal to noise ratio). In addition, we have chosen some of the smallest defect structures or molecules for testing.

Feature recovery from noisy data: overview of the approach

The need to align individual entities can be circumvented by a remarkably simple, however computationally expensive, search for maximum-likelihood model structures or images. In other words, we search for model images for which the probability of obtaining the actual data set, assuming that we are looking at a random mixture of these models, is maximized. Our approach is formally similar to the maximum likelihood reconstruction of molecular structures in Refs. [64,65,69], but leaves object positions, rather than orientations, as the hidden parameters. Most importantly, we show that there is no need to recover the hidden parameter for the individual snapshots.

We begin by explaining the new approach step by step through an example, first without technical details, which are provided subsequently. Our test data set is aimed to simulate imaging of a large sample area with very low doses. The sample is assumed to contain a random mixture from a finite set of defects (or generally, a finite set of deviations from the regular lattice) homogeneously distributed on the area. Hence, all structures of interest occur repeatedly, many times in the field of view, however they cannot be individually recognized. Fig. 2a shows a simulated HRTEM image of graphene with a mixture of three different di-vacancies at random positions, and random orientations with respect to the lattice. Under low dose exposures (Fig. 2b) of the same structure, no individual configurations can be discerned. This image is part of the large noisy data set which is used as the sole input to the reconstruction algorithm. In the following, we refer to this as the “experimental” data set, even though of course it is a simulation, in order to distinguish clearly from the “model” image set. It must be emphasized that the experimental data set is obtained for a large area, in this case . This is a large area compared to atomic dimensions or the typical field of view in a TEM, but is still easily available on a typical suspended membrane (e.g. graphene) sample [17,70-75]. Certainly, an automated acquisition will be needed to acquire such data in a real experiment. Note that the periodic part of the lattice can easily be obtained from the low dose data e.g. by analyzing a Fourier transform of the image (e.g. inset in Fig. 2b). The Fourier transform provides not only the orientation of the lattice, but also the translational offset (via the phases on the diffracted spots), and an accurate lattice spacing. This information is the basis for creating an initial guess for the model set.

Fig. 2

Reconstruction of defect configurations from a large-area, low dose HRTEM data set. (a) Noise-free image of the structure, and (b) image simulation with only . Inset shows a Fourier transform. The whole data set covers about . (c) Initial (left column) and final (right column) model images, and intermediate structures (see text). (d) Convergence of the likelihood value. All scale bars are 1 nm.

Now, we construct sets of model images, which will be tested by comparison to the experimental data. A computer program calculates the probability that the experimental data would be obtained, assuming that we are looking at a random mixture of the model images placed onto the lattice at unknown positions. In statistical terms, this is the likelihood (L) of the model set. The initial guess for the model set is the defect free lattice plus a small amount of random noise (leftmost column in Fig. 2c, showing 4 out of the 20 model images). Its likelihood in this example is , an extremely small number. This means that the probability that we would obtain exactly our experimental data (e.g. if we were to repeat the experiment), by looking at a defect free lattice, is . A base-2 logarithm of L is used during the calculation, and this is also given in Fig. 2c and d for clarity.

Next, we consider the likelihood value as a high-dimensional function of all the grayscale values in a set of model images, and try to find its maximum. We test-modify the images on the level of single pixels, and if the change increases the likelihood value, the change is kept. In this way, the model images are evolved via an iterative coordinate descent in L, and finally arrive at a local maximum. Although there is no guarantee that this is the global maximum, we find that the original structures of the material can be recovered by this approach. Fig. 2d shows the evolution and convergence of the likelihood parameter, along with intermediate structures in Fig. 2c. Each “iteration” in the plot refers to a test of a single-pixel adjustment in one of the model images (in this case, 20 model images of 24×24 pixels, 4 of which are shown). Quite remarkably, the algorithm arrives at the di-vacancy configurations that were originally put into the simulated structure, only on the basis of low-dose exposures and without any a priori assumptions (rightmost column in Fig. 2c; the remaining model frames arrived at rotated versions of the defects, multiple cases of the empty lattice, or partially cut defects). The likelihood of this model set is now . Although this new value does not look much different from the previous one in this representation, it must be recognized that it is larger by a factor of . Hence, it is by far more likely that our “experimental” data was obtained by looking at a mixture of the three different di-vacancies and the empty lattice, rather than by looking at a defect free lattice alone. The difference in L compared to the initial model structure is enormous, due to the very large data set, even though the individual locations of the defects remain unknown.

Feature recovery from noisy data: mathematical background

In statistical terms, the likelihood of our model images (for a given data set) is the probability of measuring the actual data when given or assuming the model. For a single pixel, the probability of observing intensity k when a mean intensity of λ is expected is given by the Poisson probability:

An experimental snapshot can be considered as a matrix of observed pixel intensities k, while each model is a matrix of mean expected intensities λ, , with N being the number of pixels in each snapshot or model. The probability of recording the intensity matrix {k} when assuming model intensities λ is simply the product over all single-pixel probabilitieswhere p is according to Eq. (1), or possibly might be chosen according to other noise models where appropriate (e.g. for considering a non-ideal detector).

Eq. (2) gives the probability that a specific image is obtained from a specific model and at a given position (no offset). It is easy to introduce an offset j between the model and the experimental snapshot, which we write here as

The index in the above formula is to be understood as shifting of the model image with respect to the experimental snapshot by one out of the possible offsets.

We aim to model the data as a mixture of several model images (which includes the discrete set of different orientations) Λ, , with M being the number of model images. Since the model number m and the offset between model and experimental snapshot j are unknown, a summation is carried out over these parameters. We can assume that all positions of the defect structures have the same probability, so that a simple summation over this parameter can be done. For the defect type, we introduce a weight w representing its relative abundance, and carry out a weighted summation over the unknown parameter m. The weight parameters are also subject of optimization (maintaining ), and in result provide a measure of the density of each model structure. The likelihood of a set of model images, given a single experimental snapshot, is thena formula that might be used to quantitatively test a hypothesis with a single experimental exposure. Finally, our simulated experiments consist of a large number of snapshots , where F is the number of frames. The likelihood of the model set can hence be written aswhere the product is over all experimental frames.

In the present implementation, the experimental data is split into F small frames that are of the same size as the model images. This frame or model size is chosen to be somewhat larger than the expected defect structures, large enough to avoid splitting of randomly placed defects into pieces, but small enough to avoid multiple defects in one frame, for a majority of the frames. In evaluating Eq. (3) with an offset between the model and the experimental frame, the data is periodically wrapped. For this reason, the model and frame size is also chosen to be a multiple of the lattice spacing.

An important aspect is the computational power needed to evaluate Eq. (5) in the iterative procedure. Re-evaluating L at each iteration is not practically feasible. Therefore, is only initially evaluated once for all indices, and the values are stored in memory. Then, only the adjustment to this value is calculated when one (or several) pixel values in the model set are changed. This computational optimization is the key to make the approach work at all for realistic examples. A similar optimization appears to be implemented in previous maximum-likelihood reconstructions by a Bayesian updating of L [65].

Technical details

The simulated data was generated as follows. We begin with a script that randomly replaces a section of the graphene lattice in the structure file with atomic coordinates of the defects. We then use the command-line version of the QSTEM software [76] to simulate HRTEM or STEM images, initially without noise, for these structures. In this way, a large number of simulated images (as shown e.g. in Fig. 2a) are generated, all of which contain a mixture of defects at different positions. For the example in Fig. 2, 51 200 images of the size shown in Fig. 2a were used, corresponding to an area of . Poisson noise is applied corresponding to the finite electron dose (assuming an ideal CCD camera or ADF detector, with one count per electron). Now, the positions, types and orientations of the defects are no longer detectable. However, it is important to note that the orientation of the lattice can still be detected, e.g. on the basis of a Fourier transform (Fig. 2b inset). Hence, in an experiment it would be possible to rotate all images to the same lattice orientation. This step was not actually done for our simulation, but instead all images were simulated for the same orientation of the lattice. Finally, the oversampled original images are binned down to a pixel size of 0.53 Å horizontally and 0.61 Å vertically, with the exact values chosen so that the lattice repeats after 8 pixels horizontally and 4 pixels vertically. All the data are split into pieces of 24×24 pixels (same as the model size), and these pieces are the “frames” in Eq. (5).

The initial model set is chosen as multiple copies of the defect free lattice, plus a minimum amount of noise to ensure that they are not numerically identical. We have found that the algorithm also works with a random (pure noise) initial model set but then it frequently does not find all structures and shows more mixed or pure noise solutions (we assume that if the initial choice is closer to the actual solution, the optimization is less likely to end up in a local, non-physical maximum). An initial value for L is calculated according to Eq. (5). Then, small random changes are made to one pixel of one of the model images, the new L value is calculated, and if it has increased the change is kept, otherwise the pixel adjustment is reversed. All pixels of all models, and the weight parameters, are test-adjusted in a repeating serial sequence. In detail, one pixel after another is changed in every model, first by a small random amount in one direction, and if L does not increase the opposite direction is tested. The convergence of L and the appearance of the model images is monitored and the program is stopped manually if no further changes appear (see columns to the right half of Fig. 2c). An ad hoc cluster, made from different PCs and servers (totaling 74 cores and 102 GB of random access memory) was used, on which the most demanding reconstructions shown here took several days.

For easier implementation (avoiding very large exponents), it may be convenient to optimize the logarithm of L (instead of L itself), which can be written as

The above formula is implemented in our code. However, exponents beyond the range of the standard double precision format still occur in , which is taken care of by a separation of exponent and mantissa of these numbers in the code. In order to minimize problems with a finite precision of the numbers, the terms of the sum are separately stored and then summed in the order of increasing exponent. The computational effort for each re-evaluation of L scales as , where F×N is the total number of pixels in the experimental data set. Hence, for a given data set, the computational effort for a single-pixel update is independent of the model size. However, since the number of iterations that is required to build a set of models is proportional to , the overall scaling behavior is proportional to . Moreover, temporary storage of a size proportional to is needed for the values.

Further examples

The approach shown here works similarly well for simulated HRTEM or STEM data. Under optimized imaging conditions, a comparable signal to noise ratio can be expected, for a given dose, by both methods [77]. Fig. 3 shows simulations and reconstructions based on medium-angle annular dark field (MAADF) imaging, with imaging conditions as described in Ref. [10]. Again, the mixture of three different di-vacancies was randomly distributed in a graphene structure and imaged under a low dose. Similar to the HRTEM case shown above, the structures can be recovered from large-area exposures with ; in this case an area of was sufficient. The initial and complete final set of model structures is shown in Fig. 3c and d, respectively, as a typical result of a successful reconstruction. We find all defect types (in all orientations), and empty lattice. Hence, all structures that we put in were recovered. However, it is similarly important to check that incorrect model structures can be identified. First, there are defect structures that are cut off. This is to be expected, due to the segmentation of the data: the frame size is chosen so that “cut” structures do not dominate, but they inevitably exist. They can be identified if also the non-cut structures are found. In addition, spurious solutions of a lattice with a high amount of noise appear, but they can be easily identified by their small weight. The resulting weights for the correct solutions in this example range from 0.019 to 0.064 (considering two identical solutions for structure 3a with weight 2×0.012). At first view, this variation in weights may seem surprising, considering that all defects have the same density. But on closer inspection, it becomes clear that the larger defects are more likely to be cut upon splitting the data set into frames. Correspondingly, the smallest defects (type 1) are recovered with larger weights, and small variation in weight (0.059±0.005), while the largest ones (type 3) appear with lower weights (0.019,…, 0.024) and also the “cut” structures correspond to pieces of the larger defects.

Fig. 3

Reconstructions from low-dose STEM data. (a) Noise-free image, (b) section of the data at a dose of . (c) Initial set of 20 model images (10 are shown), starting as the defect free lattice plus a small amount of random noise. (d) Model images after maximum-likelihood search. Weights that were found in the process are indicated below, those above 0.01 shown in blue. All three vacancy structures (labeled 1–3) with all relative orientations to the lattice (a, b, c) are found (due to the symmetry of the defects, types 1 and 3 appear in three, type 2 in two unique orientations). In addition, “cut” versions of the defects are found (yellow dashed line indicates where it is cut off), and a few noisy frames. The scale bar of 1 nm is valid for all images. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

For MAADF-STEM data, we have carried out further testing to approximately establish a scaling of required data size in dependence on dose and defect density. This is summarized in Fig. 4, as data points for successful and incomplete reconstructions (by varying dose, defect density and data size). Within the present range of parameters, it looks like the required sample area scales as the inverse of the third power of the dose. In addition, two defect densities were considered, one of them being 5× higher than the other. With otherwise identical parameters, the lower density sample required a ca. 10× larger area for successful reconstruction than the higher-density sample. An information theoretic analysis, which would provide the true scaling behavior, a criterion for the existence of a maximum in L at the correct solution, or prerequisites for the convergence, remains as a subject for future work.

Fig. 4

Data area required for reconstructions from noisy MAADF-STEM exposures at different dose and defect density. Circles show successful, crosses incomplete reconstructions at doses between 500 and . Black points show data for a defect density of 0.2 per square nanometer, which is also the density in the example images. Red points are for reconstructions from lower defect density data, 0.04/nm2, or one defect per 25 nm2. The dashed line is a third-power scaling given for comparison. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Whether HRTEM or STEM would be the preferred method for this approach is not clear at this time, and probably depends on details. Some important points can nevertheless be identified. Using HRTEM, at least for carbon, there is still a slight advantage on the signal to noise that can be obtained within a given, limited dose (even if the special imaging conditions of Ref. [77] are used). For STEM, however, there are two more fundamental (potential) advantages: First, a low-dose ADF-STEM image is mostly empty, with only occasionally an electron on the detector (i.e., most pixels are black, only a minority contains one or two electrons). This fact could be used for data compression and more efficient reconstruction, e.g. one could once calculate the values for a completely black (all zero) data image, and then calculate only the “adjustment” to this value for the given set of non-zero pixels in each exposure. Second, the poisson statistics at low counts is advantageous for ultra-thin samples with holes (e.g. graphene hexagons), where close to zero scattered intensity is expected if the beam is on the hexagon center. In this case, a single electron scattered from such a position carries a high amount of information, essentially signaling the presence of an irregularity at this point. The ideal situation would be an imaging mode where zero signal is present at any point of the regular lattice, and non-zero counts only occur at a deviation from the lattice. Dark-field TEM with suited apertures would be a candidate, but efficiency (signal to noise ratio vs. dose on the sample) is usually quite poor. Finding optimized (and realistic) conditions is therefore another aspect for future work.

Molecular structures

The potentially most exciting avenue for low-dose imaging and analysis as described is the study of functional groups or small molecular adsorbates on a 2-D sample like graphene. While the defects in the material are often moderately stable even under higher doses, molecules and fragile functional groups are simply out of reach for direct high-resolution images of individual entities: if ionization damage is the limiting mechanism, the allowed doses range from 10−1 to [51-54]. For molecular structures that are sufficiently large, established methods from structural biology can provide configuration-averaged images. Here, we consider the case of smallest molecules, whose positions would not be recognized in noisy exposures. We assume that these small molecules would be deposited in registry with the underlying lattice, reducing the unknown orientational degree of freedom to a small discrete set of orientations (as is also the case for the defects).

Again, we have selected what appears to be the most challenging example for the present approach: we chose one of the smallest molecules that does not have any symmetry in the projection of the TEM or STEM image. Fig. 5a shows an atomistic model of a guanine molecule adsorbed on graphene. Assuming that the molecule adsorbs in one specific way in registry with the lattice, there are still 12 different orientations that can appear in the projection of the image (6 by rotation of the structure in the plane of the graphene membrane, times the two equivalent ways to put the molecule). A simulated MAADF-STEM image, containing three of these orientations, is shown in Fig. 5b. At (Fig. 5c), it is still possible to recognize the position of the molecule, but it would be difficult to assign the correct orientation. A maximum-likelihood reconstruction easily provides the correct model structures in this case (not shown). At (Fig. 5d), neither the locations nor orientations can be recognized. For this example, Fig. 5e and f shows the result of the maximum-likelihood reconstruction. The algorithm was set to search for 30 model images, starting again with the empty lattice. Ten examples of the model images after convergence are shown in Fig. 5e. The graphene lattice was subtracted from these images in Fig. 5f, and the weights obtained in the reconstruction are given below. Correct model images are recovered for 7 of the 12 orientations (examples marked with blue weight numbers in Fig. 5f). In addition, two cases are obtained that look like a superposition of different orientations (example in Fig. 5f marked with a red weight). We assume that this is due to convergence to a local maximum, since obviously the correct solution would contain multiple separated images instead of one mixed case. We have observed that such mixed solutions appear frequently (also for the defects) if the data set is too small for the given noise level.

Fig. 5

Reconstruction of a molecular adsorbate (guanine) on graphene from simulated low-dose STEM data. (a) Atomistic model. (b) Noise-free STEM image with three molecules in different orientations and at random positions. (c) Image simulation for . The molecule position can be recognized by a higher local intensity, but its orientation cannot be recovered. (d) Image simulation for . (e) Reconstruction from of data. (f) Reconstructed structures, lattice subtracted. Weights obtained via the optimization procedure are given below. All scale bars are 1 nm. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

In the reconstructed images (and also in the original simulated images), after subtraction of the lattice (Fig. 5f), some carbon or nitrogen atoms in the guanine molecule are much brighter than the others. This is not an artefact, but a coherent effect in the medium-angle annular dark-field imaging conditions where two atoms precisely behind each other (from graphene and the molecule) produce more contrast than the sum of the contrast of each atom [78]. Interpretation of the resulting (reconstructed) images follows the same well known paths as for “normal” TEM or STEM images, and is not further discussed here. Again it must be noted that in the present implementation, the reconstruction does not make use of the inherent discrete symmetry of the problem: no use is made of the fact that rotated or mirrored versions of the same entity will always appear with the same probabilities. In other words, the different orientations of the adsorbed molecule with respect to the lattice are currently treated as if they were a mixture of 12 different substances on the substrate. Taking this into account, i.e., linking models of different orientation, is likely to reduce the required dose and amount of data significantly. At the same time, the fact that structures of different orientations are separately recovered demonstrates the huge potential of the present approach, namely that it should be possible to separate the components if indeed a mixture of molecules were present.

It is obvious that the experimental realization of this approach requires an automated approach to record low-dose atomic-resolution images from large sample areas. A discussion on how this might be achieved is beyond the scope of the present work. However, it should provide a key motivation to develop it. Once we achieve an automated acquisition of low-dose images with atomic resolution, the amount of data that can be recorded in a realistic time may easily be sufficient: for example, assuming 5 s per exposure (e.g. 4 s moving and auto-tuning, 1 s exposure) and a typical 30 nm×30 nm field of view, acquisition of a area of data requires less than 2 h. With this estimate, it is interesting to note that the limits in dose and defect density for which this approach works will probably depend on the computing power that is available for the reconstruction, rather than the estimated experimental limitations (i.e., the number of images that can realistically be recorded).

Summary

In summary, we have shown a new approach to extract information from low-dose HRTEM and STEM exposures. For the first time, we demonstrate a reconstruction from images at doses where the locations of individual entities cannot be identified. This is a key for a significant dose reduction in the analysis of material defects. In view of the present high interest in low-dimensional materials, an approach for damage free (low dose) imaging of these radiation sensitive structures is urgently needed. The statistical approach shown here, if achieved with real experimental data, will allow a clear measurement of the structure in any deviations from the ideal periodic configuration that is repeatedly present in the sample sufficiently often and at a sufficient density. This includes point defects, ad-atoms or attached functional groups, adsorbed molecules, or any other deviation from the regular lattice – there is no distinction from the methodological side. The present approach has the potential to circumvent the problems of radiation damage for a wide class of problems where existing methods would fail. If it becomes an experimental reality, the analysis of new materials based on atomically resolved images could be extended from the few examples of radiation-hard substances to many cases where a discrete set of local deviations from the ideal lattice can be expected, with orders-of-magnitude lower dose requirement.