STEM_CELL: A software tool for electron microscopy: Part I—simulations

Ultramicroscopy 125 (2013) 97–111

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Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic

STEM_CELL: A software tool for electron microscopy: Part I—simulations Vincenzo Grillo a,b,n, Enzo Rotunno b a b

S3-NANO, CNR Via Campi 213A, I-41125 Modena, Italy CNR-IMEM Institute Parco Area delle Scienze37/A, I-43124 Parma, Italy

a r t i c l e i n f o

abstract

Article history: Received 27 July 2012 Received in revised form 24 October 2012 Accepted 27 October 2012 Available online 10 November 2012

The software STEM_CELL, here presented, is a useful tool for (S) TEM simulation. In particular innovative solutions are presented in (1) the supercell manipulation and parameters setting (2) simulation execution through the modified Kirkland routines (3) simulation post-processing with extended output and comprehensive graphic tools (4) image contrast interpretation through a strain channeling equation accounting for strain effects in STEM-ADF. & 2012 Elsevier B.V. All rights reserved.

Keywords: Electron microscopy simulation STEM-ADF Crystallographic modeling Channeling

1. Introduction The past decade has seen a huge development of instrumental possibilities in electron microscopy with new aberration corrected microscopes reaching routinely sub-A˚ resolution in both TEM and STEM operation mode [1,2]. The large amount of new information that these instruments are making available has raised the problem of more adequate analysis tools based on data elaboration and comparison with accurate simulations. The two fundamental milestones in image simulation have been the formulation of the Bloch wave theory [3] and of the multislice algorithm [4]. These two approaches have been proven to give equivalent results [5] and have been initially applied to solve the dynamic interaction to simulate HRTEM images, diffraction contrast and convergent beam diffraction patterns [6]. Initially the use of simulation was often as an esthetic or qualitative complement to TEM images and only occasionally it could serve to interpret qualitatively the main features. With the advent of digital imaging the attempt to compare experimental contrast to simulation was strongly frustrated by what is till now called the ‘‘Stobbs factor’’, namely the need to introduce an arbitrary scaling parameter in simulations to reconcile with experiments [7]. Nevertheless the use of normalizations and complex image elaborations has strongly pushed the electron microscopy in the direction of quantitative analysis where simulations had a dominant role [8]. A number of commercial and free text and graphic software has been then created to cope with these [9–15].

n

Corresponding author at: S3-NANO, CNR Via Campi 213A, I-41125 Modena, Italy. E-mail address: [email protected] (V. Grillo).

0304-3991/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2012.10.016

With time multislice has been proven to be quite flexible since it could be extended to include inelastic effects [16] or backscattering [17]. Moreover different ‘‘flavors’’ of the multislice approach have been proposed as the ‘‘real space’’ multislice [18]. However one of the most interesting developments of the multislice approach has been probably the creation of the frozen lattice (or ‘‘frozen phonon’’) [19] that permitted to natively introduce the thermal diffuse scattering (TDS). The idea is that the thermal vibration can be frozen in as a random displacement of ions from the rest of the position and the average intensity is obtained as a Monte Carlo integration over many such atomic configurations. While the original algorithm considered only uncorrelated phonon motion, frozen lattice paradigm has been adapted to include realistic phonon dispersion [6]. Recently an extension of this process has been demonstrated to include other inelastic processes also (i.e. plasmon loss and inner shell ionization) [20]. Meanwhile the increasing interest for STEM was growing thanks to the coupling with EELS and EDX spectroscopy but also as a powerful imaging mode by itself. In spite of the interesting results of the first simulation codes, able to simulate the main elastic processes, the fundamental step has been the creation of the frozen-lattice multislice simulation. Since this is the most precise way to include TDS in STEM imaging (in particular HAADF) the frozen-lattice has become the standard for STEM simulation whereas it can have interesting applications also in TEM and CBED. Thanks to these simulations a quantitative agreement has been obtained between STEM simulations and experiments for the relative contrast between two materials and for the absolute intensity [21,22]. The reference code for the simulation algorithm is the work of Kirkland who created routines for the frozen phonon simulation of HRTEM, CBED and STEM images [10]. The work was based on a

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few monolithic programs for the multislice and separate routines for the calculation of the atomic potential, to account for the image formation through the lens system or to simulate the probe shape. However the original version was a mainly text based serial code with a few basic visualization tools and no instrument for the supercell creation and elaboration. Important progresses have been done to speed up these lengthy simulations by means of parallel computing but there has been little progress so far in the user interface and modeling tools [23–27]. In particular a multislice code presents specific disadvantages and challenges with respect to Bloch waves methods: in fact it disregards the specific symmetry and periodicity of the unit cell and threats the atomic positions independently from a possible unit cell shape. For this reason, for example, it is not easy to switch to an arbitrary zone axis; only zone axes with rectangular symmetry are directly consistent with multislice, at least in the basic multislice implementation [28]. A partial solution is to replicate the unit cell structure in order to reach, where possible, a rectangular symmetry or, at least, to reduce the border effects. The need of accurate cell repetition and tailoring becomes even more important for Frozen Phonon simulations of STEM images. In fact all STEM simulations require that the whole electron probe has to be size matched to the supercell. Frozen phonon additionally requires a very high frequency sampling to account for very high angle diffuse scattering. On the other hand the multislice offers the advantage to simulate extended and strained structures provided they are correctly modeled and sized to match the simulation requirements. All these needs can only be handled with a specific software that works before the actual simulation preparing the cells and the parameters conditions. The STEM_CELL graphical interface, introduced in this article, is proposed to fill this gap and to add tools to help the whole simulation chain. In particular STEM_CELL permits analyzing and graphically rendering the simulation results. In the particular case of STEM imaging we have implemented in STEM_CELL approximate simulation and modeling that helps to design more detailed simulations or better understand simulation results. The most important example in this context is the channeling model that allows predicting the main features of a STEM-ADF image.[29,30] Since there exist very few software programs that try to integrate a full cell handling with simulations and post processing, STEM_CELL is meant to be more ambitious as it aims to integrate the above simulation chain with the main analysis methods in a single tool. Many software solutions are already available (with different access modes) for each of the above operations but usually they perform, alternatively, only the simulation or the analysis and, in most cases, a unitary view is missing. Moreover most of the available solutions are commercial or not completely free: for example they can be implemented on non-free/open platforms (like GATAN Digital Micrograph [31], MATLAB [32]). Finally cell handling and graphics in some codes are partially based on interpreters instead of compiled executable and are therefore intrinsically slower. Some difficulties may also arise in other codes from installation because of the use of different separate libraries. Conversely the aim of the software STEM_CELL is to make available, to the electron microscopy community, a unified software approach to both standard and innovative numerical methods. The software is written in Borland Cþþ and is available as a standalone installation that only includes standard Borland Libraries; all graphical primitives are included. 1

1 Whereas this compiler is quite outdated the main compatibility with future version of Windows is probably not going to be an issue since present operative system still hold compatibility with early 8/16 bits executable.

The software includes pieces of codes from Kirkland suite and numerical recipes. STEM_CELL is not only a software tool collection but a platform to develop innovative methodologies. The list of available functions and methods is therefore continuously in progress and this article is necessarily not a definitive list of the implemented simulation tools. The graphical software is available for free on a web site (http://tem-s3.nano.cnr.it/software). All routines described in the main text are available upon request but part of the routines needs minor debugging and documentation and are excluded by the main distribution. The principal condition of use of the software is the citation in future works making use of the presented routines of the relevant author’s works (including the present paper) in which the methods are explained. This paper is focused on the description of the main simulations tools and possibilities while a companion article will be dedicated to the description of the implementation of analysis methods.

2. The general tools A snapshot of STEM_CELL main graphic windows is shown in Fig. 1. The main windows and the command bar are clearly visible. The image shows that simulated images can be directly rendered and analyzed starting from this point. It is interesting here to appreciate the possibility to produce a full graphic rendering with tunable dynamics (even in color-scale) without spoiling the precision of the simulation. A debug console is always present and gives additional text information about the running algorithms or results. In addition to the main window, other popup windows can be opened for additional tools like GPA analysis [33], profile handling, supercell handling and so on. A few technical details on the software structure are described in Appendix A.

3. Simulation In this section an overview on the main simulation features of the program is given and the principal applications, as well as some examples, are described.

Fig. 1. Snapshot of STEM_CELL main windows.

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Fig. 2. Unit cell geometry used to deduce Eq. 1 in text. The 3 cells’ axis and relevant angles are indicated.

3.1. Unit cells and supercell manipulation The creation of the supercells (SCs) for the multislice simulation and their manipulation is accomplished by an internal class allowing for advanced operations. The native format for opening/ saving SCs is the Kirkland format but it is slightly modified in order to add a short header (compatible with the Kirkland’s autoslice program) that includes information on the cell size and slicing. A conventional extension.krk has been chosen for this kind of files. In addition other basic ASCII xyz formats are also allowed for SCs input/output operations. The program allows a direct creation of a crystal unit cell from space group and base atom coordinates. All the space group symmetries and most of the alternative settings can be used to create a unit cell starting from its symmetry operations. Alternatively it is possible to read JEMS [9], cif [34] or MacTempas [13] (the last so far partially supported) data for the unit cell. An additional internal format containing the crystallographic information has also been introduced to store created unit cells. The creation of the SC is achieved by replicating the input atomic positions of the unit cell according to the symmetry operations of the crystal system. The coordinates are then transformed from the fractional (namely referred to unit cell dimension) coordinates to the Cartesian system through the transformation matrix (an example of derivation of such matrix but with different axis selection is in Ref. [35]) 0 1 a bUcosðgÞ cUcosðbÞ B cosðaÞcosðbÞcosðgÞ C 0 bUsinðgÞ c C m¼B ð1Þ sinðgÞ @ A v 0 0 c sinðgÞ where a, b and c are the lattice parameters defining the unit cell direction and a, b, g are the angles between direction as shown in Fig. 2. The variable v is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ 1cosðaÞ2 cosðbÞ2 cosðgÞ2 þ 2cosðaÞcosðbÞcosðgÞ ð2Þ The unit cell positions can be used to generate a large SC for example by repetition operations, the periodicity axes do not need to be orthogonal but all kinds of symmetry are allowed. The repetition matrix with the repetition vector on its raw is stored separately in order to allow subsequent operations. Special routines have been provided to create semiconductor structures and Mackay construction for icosahedral nanoparticles (main code has been based on Ref. [36]) [37]. The routines for semiconducting structures are described in better detail in Section 3.2.

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Once the SC is ready and stored in the memory it is possible to render it graphically in a separate window called slice_ gen. The graphical output is based on Kirkland’s sliceview routine but graphical commands have been added for interactive change of visualization. The visualization commands permit changing atom radius, rotate the model, change perspective, exclude from visualization part of the atoms or select single atoms. The modification commands permit operating complex cell operations like changing the slicing direction (by changing the axis orientation), operating distortion, shift, rotation or scaling operations, replicating or symmetrizing the given cell, tile different crystals (separately created) and selecting spherical, cylindrical or prismatic section of a unit cell. The selection can be used to remove the atoms inside or outside a region or to operate a restricted partial atom substitution. Given the importance of the selection tool the possibility to take the selection area directly from the silhouette of experimental images after an appropriate threshold is applied has also been implemented. The rotation can be performed by (1) entering Euler angles, (2) free hand rotation by drag and drop mouse operation (3) setting the target crystallographic orientation. The crystallographic directions are not automatically defined for SCs: however the program permits defining a reference unit cell system and its translation matrix as in the case of single unit cell. The axes rotations are therefore defined with respect to this unit cell. After a SC rotation the matrix m can be updated to the new coordinate system and used to replicate the cell or produce further rotation. The rotation operation to defined orientation along with the versatile select-cutting operation is the simplest way to produce faceted crystals. Particular care has been used for partial atom substitution. It can be operated using a full random, a fixed rate or a constrained random substitution. In the full random mode the probability (determined by the species partial occupancy) for each atom to be substituted is independent from other substitutions. The fixed rate mode periodically substitutes an atom in order to fulfill the partial occupancy in each part of the crystal. In the constrained random substitution a fixed number of the substitutional atoms is arranged in the available sites in order to fulfill exact partial occupation while leaving randomness. The last method has been preferred in the article [38] as this permits a more rapid convergence of simulations with different substitutional atoms arrangements. In Fig. 3 a model of a core–shell CdSe/CdS rod structure, with the CdSe core located in the upper part, is shown. The structure is shown down [1–210] with c axis in the horizontal direction. Atoms with a smaller atomic number are coded as smaller; therefore the CdSe core can be seen on the right. Using a different viewing direction and larger atomic spheres the rendering highlights the surface shape (Fig. 3b): it is possible to observe the 3D shape of the object and confirm it has cylindrical shape. The corresponding simulated HRTEM image is shown in Fig. 3c. To create this SC (1) a CdS unit cell has been replicated, (2) a cylindrical region has been selected, atoms outside this region have been removed (3), a smaller cylinder has been selected and a substitution of S with Se has been operated in the selection (4). Another example is shown in Fig. 3d–f where a TiO2 Brookite structure is lattice matched with a Fe3O4 particle. The structure is inspired to an actual binary system of nanoparticles found in experiments [39]. Fig. 3d shows the interface between the two nanocrystals. The Fe2O3 is observed down its [112] direction while Brookite TiO2 is observed in the [010] direction. In Fig. 3e the arrangement of the composite structure can be observed in 3D while the HRTEM simulation is visible in Fig. 3f. In this case the two structures have been created separately and then joined together. To avoid the

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Fig. 3. Atomistic model and HRTEM simulation for a few composite structures (a), (b) and (c) refer to a CdSe/CdS core–shell rod. The CdSe core can be seen in a on the right while the cylindrical section of the rod can be evinced from (b). (d), (e) and (f) refer to the composite structure of a brookite rod and a Fe3O4 magnetite nanoparticle. (e) permits to appreciate the rectangular section of the rod and the spherical shape of the magnetite structure. Part of this sphere has been cut to produce a flat interface region. (g) and (h) refer to a wedge shaped SrTiO3 structure visible in g. The HRTEM image has been rendered in false colors to highlight the contrast changes with thickness.

interpenetration of the two structures the software permits defining a minimal distance (in this case 0.2 nm) between the atoms of the two structures: if the atoms of the structure to be added (in this case the Fe-oxide ball is added to an existing TiO2 rod model) are within this distance to the existing atoms they are excluded. This permits the rapid construction of an interface as in Fig 3e. A further demonstration of the possibilities of the software is the structure in Fig. 3g–h that reproduces a wedge shaped SrTiO3 crystal (g) along with the relevant HRTEM simulation (h) (false colors have been used to highlight the thickness change). In this case the SC created by repetition has been rotated by an angle corresponding to the wedge aperture, a rectangular selection region has been used to perform the cut parallel to the X axis and a rotation opposite to the first has been performed. Fig. 4

shows other interesting examples of simulation where more complex structures, including amorphous material, single crystals and multiple twinned nanoparticles are simulated. The final addition of noise produces a very realistic image. The software also facilitates the selection of important geometry details like the SC dimension. This is particularly important since, in multislice simulations, periodic conditions are tacitly assumed. If the lateral extent of the SC is set to the distance between the extreme atoms, this usually produces errors, namely the artificial proximity of atoms at the border. In general a quite precise notion of the cell size is required to avoid such problems. If instead the exact extent of the cell is unknown, a built-in algorithm can be used; it refines the basic periodicity of the structure by a Fourier analysis in the given direction, on the basis of an initial rough user’s guess. The algorithm just finds the

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Fig. 4. Simulation spherical Au and icosahedral Ni particle on amorphous carbon. (a) and (b) are the model and the HRTEM model for the Au sphere. (c) and (d) are the model and the HRTEM simulation for the icosahedral Ni particle. A random noise has been also added to produce a more realistic effect.

k frequency that maximizes the sum X expði2pkxi Þ A¼

ð3Þ

i ¼ atoms

where atoms are assumed for simplicity as Dirac’s d at x position. The same procedure is applied also in the slicing direction: this procedure permits finding the optimum uniform slice thickness and number of slices for the given SC. The SC size in the slice direction is also slightly changed accordingly. 3.2. Semiconductor special tools As semiconductors are among the most common objects of microscopy analysis, a special tool has been created to produce zincblend and wurzite cells. The advantage of this tool is the possibility to create small layered structures and to automatically account for tetragonal distortion in the case of pseudomorphic growth on mismatched substrates. The program reads a database file with the relevant elastic constants and the lattice parameters and calculates the appropriate elongation in the growth direction. The simplified cell built in this way can be directly fed in the multislice routine of the Kirkland suite or transformed in a real 3D SC. Fig. 5a is an example of a simplified model structure containing partial occupation in a InGaAs structure while Fig. 5b is one of the possible 3D structures that can be constructed by filling the column according to the indicated partial occupation. In this case the column was filled using the constrained random substitution approach. . The semiconductor tool may result useful when performing simulations in alloys as a function of the composition. The simulation can be programmed to schedule automatic simulation

Fig. 5. (a) Schematic semiconductor sample of a layered InGaAs/GaAs alternated in the vertical direction the partial occupation in the In/Ga column is highlighted (blue¼ As, dark green¼Ga, bright green¼ In). (b) 3D view of the atomic columns in a real cell generated from the scheme in (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

series for different alloy compositions. Fig. 5 is an example of statistical evaluation of the top–bottom effect (namely the dependence of the depth position of In atom in the alloy column) in a 10 nm InxGa1  xAs alloy (x¼24%). 24 different In arrangement have been simulated obtaining the histogram visible here. As predicted by [38] the different In arrangements produce different intensity (more than 10% variation in this case) that is

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the intrinsic uncertainty of In quantification for this thickness. The evaluation of this kind of effect has required 24 separate simulations and SC creation but has been completely automated. One of the limits of this approach is that atoms are assumed to lay at the virtual crystal position while in most cases it has been proven to be necessary to introduce static displacement in the alloy to obtain a consistent evaluation at least in the case of the STEM-HAADF intensity [21]. The inclusion of Valence Force Field solver in STEM_CELL is planned in order to fully automate the generation of a complete STEM simulation. 3.3. Strained model creation Strain has important effects in both HRTEM and STEM imaging. For this reason it is very important to account for it in simulations. A special class of STEM_CELL permits storing the component of the displacement field due to elastic forces. These can be calculated analytically in a few cases where a closed formula is available or through the COMSOL software for finite element elasticity modeling (FEM) [40]. At the moment the analytical calculation has been implemented for dislocations, surface strain relaxation at the sides of a quantum well, a simple stacking fault (without the strain field of the partial dislocations at the border) and spherical inclusions. [41–43] In order to import Comsol data it s possible to take advantage of a post processing function of COMSOL 3.4, that exports the component of the displacement on an ASCII file read by STEM_CELL. The coordinates are chosen on a regular grid or directly on a template of atomic positions. This approach permits decorating FEM calculation with atomic positions [44]. An example of such approach has been given in Ref. [29]for the case of a strained quantum well with surface relaxation Fig. 6. In Fig. 7 the FEM calculation for a strained InGaAs/GaAs quantum dot is shown. The color coded map (Fig. 7a) refers to different values of the z (growth direction) component of the strain. The model is obtained assuming a shell like distribution of the In with a larger In and therefore larger mismatch inside. The atomic model in Fig. 7b and c has been produced importing the domain shape from COMSOL in STEM_CELL, this permits importing the strain field but also to select different partial occupation in the different domains. In Fig.7b and c the overall shape of the QD and the strain in the center of the QD can be appreciated. Finally the use of a general class for the strain input permits using more sources for the strain field including experimentally calculated strain maps or to create iterative mechanisms with free parameters able to fit experimental contrast. While a comprehensive

interface for all these operations is still not available, it can be easily implemented in future version. 3.4. Probe simulation and linear STEM-HAADF simulation Before starting a real simulation, in particular a lengthy STEM simulation, a few preliminary operations can be performed to predict the main features of the simulation and design it more efficiently. In particular it is possible to perform a simulation of the electron probe. A dedicated form can be used to set the appropriate optical parameters (defocus, astigmatism, beam energy and condenser aperturey) and the sampling parameters like number of pixels and pixel size in real coordinates. An additional form allows to set aberrations up to fifth order and to use them for simulations. The notation and use is consistent with the most updated version of Kirkland software [45]. The same constants can also be used to plot the corresponding aberration function w. In the above example the effect of each aberration up to the 5th order on the probe is tested. In Fig. 8b and c an aberration function, inspired from the aberration parameters deduced by Sawada [46], and the corresponding probe with the optimized aperture are shown. A list of the aberrations used in the example is given in Table 1 In addition to the aberrations it is possible to design custom apertures and analyze the effects on the corresponding probe. The example below in Fig. 9 is based on the vortex generation [47] by a distorted dislocation-like condenser aperture, Cs ¼0.5 and df¼40 nm are supposed. This special probe will be used for simulation of the peculiar STEM imaging modes, or integrated with the simulation containing the addition of the spin degree of freedom. [48,49] The program also gives the possibility to produce a simplified linear image of the sample which is obtained by convolving the squared probe with the atomic potential (or its square) [10]. Optionally it is possible to use a depth dependent probe accounting for the probe evolution with depth [45]. The latter is useful to give a rough estimation of depth resolution for aberration corrected experiments [50]. The linear imaging is very useful to predict the qualitative appearance of a STEM-ADF image but of course cannot predict any channeling effect. It can be used for example for a first screening between different candidate structures to match experimental pattern [51] or to evaluate the visibility conditions of chemical variations [52]. Fig. 10 is an example of linear simulation obtained with the structures in Fig. 3a and in Fig. 7. The effect of the change of thickness and of the different composition can be qualitatively appreciated. Linear imaging can also be used to check the correct boundary conditions, since if the SC boundary is set incorrectly the atoms at the border can artificially superimpose and produce a meaningless intensity increase. However the main practical advantage of a linear ADF image is that it can be used to select the region to be scanned in the rigorous simulation. A rectangle can be directly plotted in the linear image and this can become the scanning region. In many cases it is important that the edges of the scanning region are located with high precision on the atomic positions: this permits replicating periodically the simulated image. For this reason the image selection can be directly transformed in a selection in the crystal design tool and refined to the average atomic position in a column. 3.5. Parameter setting and sampling conditions

Fig. 6. Example of statistical distribution of the intensity in InGaAs for different atomic arrangements.

One of the most important advantages of having a graphical interface for simulation is that simulation parameters can be easily set, changed and recalled for a different simulation. The set

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Fig. 7. (a)Finite element simulation of a QD with a variable In composition, the In molar faction increases from 0.1 to 0.4 getting toward the center of the QD. (b) and (c) atomistic model of the QD obtained from simulation in (a) viewed in two different directions.

up of the optical/sampling parameters is also simplified by the user interface that can be used to automatically set the typical conditions for each imaging mode (for example the Scherzer defocus and condenser aperture in STEM). The most important automatic setting regards the sampling conditions: once the sample and the optimal parameters are selected the program calculates the optimal conditions for the given imaging mode. Defined modes are HRTEM, diffraction, two beam dark field, STEM and a special mode to image the aberration function. The sampling conditions are those suggested by Kirkland who set the minimal requirement in terms of sampling in the real and diffraction plane [10]. In order to establish an automatic procedure ensuring a correct sampling the minimal requirements have been transformed in a closed formula that certainly satisfies the sampling constraints, although it is not the only possible sampling choice.

The case of STEM will be here considered in more details. In particular for the STEM sampling the selection process is the following: in order to have a large enough sampling of the probe forming aperture a unit cell of size ax ay has to be repeated Nrep x,y times. The number of repetition has been therefore set to the minimum requirement. Nrep_x,y ¼ ceil

 10

U

l

ac ax,y

 ð4Þ

where ceil is a function that approximates its argument to the smallest following integer, ac is the probe semiconvergence and l is the wavelength. This choice ensures that the aperture is sampled with a 10 pix radius circle but this is usually enough also to ensure that a well

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Fig. 8. (a) Simulation of the effect of different aberrations in hierarchic order on the STEM probe, (b) simulation of the aberration function with the parameters in Table 1 and (c) actual probe intensity with the parameters as in (b).

Table 1 Aberration parameters used for the simulations in fig. 7a and b. Aberration

Magnitude

Defocus Astigmatism Astigmatism 3 fold Coma C3 Star aberration Astigmatism 4 fold Coma 4th order 3 lobe aberration Astigmatism 5 fold C5

44.6 A 23.4 A 165 A 1110 A 7.943  3 mm 10300 A 12300 A 46 um 4.1 um 14.8 um 10 mm

required it has been chosen to define a square image with size Angle (1)

96.7 61.7 45.3 11.8  16.4  107.7  20.6 2.7

behaved (i.e. near optimum) probe is smaller than the SC. In fact, as long as the aberration effects are of the same order of the diffractive effects, the size of the central part of the probe is d ¼2 l/ac and the side lobe is distributed according to a Bessel function. As for a Bessel function, the first ripple extends at or 4 ¼4.6 l/ac remaining largely within the above repeated cell limits 10 l/a imposed by the sampling of the aperture. However in special cases where strongly non optimal defocus is used it is possible to increase the sampling by manually changing the graphical parameters that substitute the factor 10 in Eq. 4. A rough estimation of the probe size or 4 in a general case can be gained by assuming or 4 ¼

  1 maxk A ½0, ac =l r k w 2p

ð5Þ

Namely neglecting diffractive effects, the radius or 4 in a direction of the probe roughly corresponds to the maximum assumed by the function in that direction by the function rk w with the wave vector k in the pupil aperture (here ac ) limited range (see Appendix B). This equation makes it even more evident that the size of the probe is related to the derivative of the aberration function. Both are correctly sampled if the pixel size in reciprocal space is sufficiently small. Only after the total SC size is defined it is possible to define the number of pixels. If a special setting is not

  ax,y b Npix ¼ maxx,y 2U1:5UNrep_x,y U

l

ð6Þ

Where b is the maximum detection angle. For the implementation of old type FFT calculation it can be convenient to approximate Npix to the first successive power of 2 but with new implementations of FFT this is not strictly necessary. The maximum sampling angle can be of course set to the experimental detector setting but a wiser recommendation is to ensure that a negligible fraction of the intensity I is scattered beyond b [53] The last constraint in STEM is set for the sampling of the scanned image. This is a very important point since different approaches have been proposed. The initial constraint proposed by Kirkland is to set scanning step at about 1/3 or ¼ resolution. Assuming that the resolution is mainly limited by diffractive effects of the aperture a scanning step of Dx E l/6ac can be assumed. Dweyer has proposed a weaker constraint using the noticeable fact that the STEM images are band width limited with band width¼2ac/l [24]. The importance of this choice cannot be overemphasized since, for example, a factor 2 larger sampling implies a factor 4 slow down of the simulation. The author points out that, adding a pixel size beyond 2ac/l has no meaning since this does not contain information. In reality in a Monte Carlo simulation (such is a Frozen phonon multislice) additional simulations are not redundant since they reduce statistic fluctuation. Especially when the unit cell averaged area is of interest the increase of the number of pixels could reduce the intensity fluctuation. On the other hand, we suggest here that, if a minimal set of pixels has to be used and what matters is only the unit cell integrated intensity, an uneven sampling condition could optimize the estimation of the unit cell integrated intensity by means of the ‘‘Gaussian quadrature’’. The procedure is described in numerical recipes [54]. In addition to the clear sampling problems above, there is a hidden limitation in the calculations when the data regarding each potential are stored and reused. In fact many structures are simply the repetition of a basic structure, therefore appears possible to ‘‘recycle’’ the series of potentials, namely to repeat the calculated potential with its thermal position after a sufficiently long distance.

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This can be done in the Kirkland program stemslic by repeating the sequence of potential to be loaded in the calculation from hard disk, or in the independent parallel version (see Appendix C) directly from RAM memory. The scheme of repetition here proposed is different from the apparently similar scheme in Ref. [19] in the fact that the ‘‘recycled’’ potential are not modified before reusing it. This recycle scheme makes the computation faster but the problem that may arise with this kind of assumption would be that artificial correlations in the thermal displacement may be introduced. The application of this method is based on the assumption that, due to the incoherent nature of the high angle scattering integrated over large annular range, a correlation along z of the displacement should produce minor effects. The effect of a small repetition distance would be to allow a constructive interference between the, otherwise random, scattering of atoms at large distance along z. In practice this means to create artificial HOLZ lines. In order to analyze the size of this phenomenon we simulated GaAs along its [110] zone axis using two schemes: (A) a small (5 nm) cell, sliced in 25 slices, has been repeated 20 times, (B) a

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large (50 nm) cell, sliced in 250 slices, has been repeated only 2 times to reach a total thickness of 100 nm; the simulation has been performed choosing 10 Frozen Phonons (FP) configurations. The angular distribution of the scattered electrons in the two cases is shown in Fig. 11a. Clearly the 2 repetition case is smoother, the difference cannot be credited to statistics effects since we repeated the simulation, for the 5 nm cell, with 20 FP configuration obtaining practically the same angular distribution (dotted red line). The oscillations are therefore due to the artificial repetition in the cell that produces an artificial segmentation of the reciprocal space every 1/C steps where C is the repeated unit cell size. A simple geometric construction (Fig. 11b) demonstrates that the pseudo-HOLZ like lines should appear at a scattering angle y. 

y ¼ arccos 1

nl c

 ð7Þ

with n being an integer in agreement with figure. It is evident that this scheme contains intrinsic errors that increase as the repeated cell size C decreases. The repeated unit cell size C should be selected

Fig. 9. Example of simulation of STEM probe with holographic aperture (a) the effect of distortion in the mask and aberrations can be incorporated in the simulation. (c) Simulated shape of the vortex beams generated with the aperture in (a).

Fig. 10. Linear image model for the structures proposed in Figs. 2a and 6. The effect of a finite source with radius 1A has been added in (a).

Fig. 11. (a) Simulation of the angular distribution of scattering in STEM using a small cell repeated 20 times (black) and a large cell repeated twice (blue) and (b)scheme of the segmentation of the reciprocal space due to the presence of artificial long periodicity C. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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according to the angular precision required: an appropriate binning, filtering or smoothing could indeed reduce the effect. It is also worth noticing that this result confirms that the tiny details of the angular scattering contain actual 3D information.7 3.5.1. Large scale simulations FP algorithm for STEM simulation is not suitable for large scale calculation since the typical simulation time may become prohibitive even when parallel computing is used on a large cluster. For all practical situations it is always advisable to limit to a number of pixels N pix to 1024 or 2048 pixels (or numbers in between in the new versions). In fact the typical computation time scales with approximately N2pix but a strong performance reduction is obtained with too large sampling as a single potential may overcome the typical memory cache of today’s CPU. Depending on the system characteristics it is sometimes possible to choose a different approach: the large SC is divided into smaller patches with partial overlapping of the cells but not of the scanned regions [55]. The size of the patches is chosen in order to be consistent for example with Npix ¼1024 and a scanning region is selected in the center of each patch in order to avoid the order where the boundary conditions produce an incorrect wrap. This approach is also particularly useful in the simulation of complex structures as the QD in Fig. 7. In this case the large empty regions in the corners do not need to be simulated with a net saving in CPU time that would not be possible on a large simulation. A rule of thumb estimation gives a minimum saved time of 20%, moreover the requirements in terms of RAM memory for each single small patch simulation are by far less strict than for a large structure and this makes sometimes mosaic multislice as the only technique of choice. In some cases a more approximated method can be used: the number of pixels sampling the wavefunction can be reduced by reducing the maximum STEM detection angle. The validity of this approach has been demonstrated in a previous work [29]. The approximation is based on the fact that if a single TDS scattering event is assumed, the angular distribution of the diffuse scattering is thickness independent so if only the relative contrast between two regions is relevant the result should be unaffected. The quality of this approximation is only slightly (depending on aspect ratio between the maximum sampled angle and the actual detection angle) better than absorptive potential approximation often used for large scale simulation [56].

installation an environmental variable is set to call the TEMSIM routines from STEM_CELL in any directory. In addition to those programs a new code stemslic3_mpi, largely based on TEMSIM routines, has been written and already published by Carlino and Grillo for the parallel computing computation of STEM images [23]. Some additional details on the parallelization and optimization strategy are reported in Appendix C. In this context it is worth mentioning that the parallel code (the specific routine is called stemslic3_mpi) accepts for back compatibility the same input as stemslic but a number of additional parameters and switches can be used to add functions and inputs, most noticeably the number of frozen lattice configuration. In a recent improvement, the possibility to change the output format has been added in the parallel code stemslic3_mpi. In fact, in the case of stemslic code, it is possible to calculate, within a single multislice calculation, the intensity for different detection conditions and thicknesses. This is an improvement with respect to the Kirkland’s stemslic code that required many tif files to be saved for each thickness/detector. This easily becomes unaffordable if the number of detectors and thickness steps becomes large (for example 100  100). Storing ring-wise the detected intensity at the interval of few mrad may turn useful for trying different detection conditions and different thickness within a single simulation [57]. To simplify the access to these data a single output binary file format has been created that contains all information and conditions. It is a sort of 4D data cube where detection angle, thickness and in plane probe positions are stored. The same data format can be fed in the graphical interface to produce separate information. 3.7. Post-processing tools The analysis of this information can be therefore performed within the graphical interface that allows the post-processing of the simulated data. The post processing can be used in order to (1) include a radial detection efficiency function, (2) create a series of images as a function of thickness with a defined detection acceptance (many intensities on detection rings are summed together), (3) select an area of the image and plot, for this region, the intensity distribution as a function of the detection angle at different thicknesses. Fig. 12 shows a typical output of post processing for a SC of InxGa1  xAs (x¼25%). In this case the simulated STEM images for 3 different angular ranges and 5 different thicknesses are plotted.

3.6. Simulation tools The actual simulation is not performed within STEM_CELL but a fork process is started from the graphical interface. This is necessary since the graphical part reduces the efficiency of the calculation and because independent compiler can be used to optimize the performances of the simulations. In addition a routine of STEM_CELL permits iterating simulation by looping over different parameter conditions. This is used to easily create defocus series, thickness series, tilt series but also more exotic astigmatism, convergence or C3 spherical aberration coefficient series since in principle every parameter can be scanned. For the communication of parameters to the child process an input file is produced in the working directory containing the specific order and formatting required by the Kirkland code. At the moment all simulation programs of the TEMSIM suites including the newly added autostem program are supported [10]. The most recent version of Kirkland’s TEMSIM does not need external files for the execution so they can be started from any point in the file system. Nevertheless at the moment of

Fig. 12. Simulation of STEM images for different thicknesses (X) and detection ranges (Y).

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107

Fig. 13. (a) Simulated HAADF image of 4 InGaAs cells. The positions of a minimum and a maximum are indicated. (b) Simulation of the angular distribution (y) vs. thickness (x) for the intensity in the whole cell, in the maximum and minimum as selected in Fig. a.

An alternative analysis is shown in Fig. 13. Panel a shows a simulated HAADF image for GaAs with thickness 4 nm and angular detection 33–200 mrad. White rectangles indicate reference regions for the analysis in panel b. In fact panel b shows the dependence of the intensity on the thickness and detection angle where the intensity has been integrated in the whole unit cell (first case) or in the indicated region. The low angle region is the forward scattering while a peak of diffuse scattering is visible. The peak is at about 40–45 mrad but its extension increases with thickness. The diffuse scattering distribution can be qualitatively accounted for using the well known diffuse scattering cross section [58] Z ymax  2 2

2 2 sp f ðyÞ 1e2M y =l d y ð8Þ

Fig. 14. Derivative of the intensity vs. thickness of Fig. 12b. The derivative permits to evaluate the probe localization as a function of the thickness.

ymin

where f is the scattering factor and M is the Debye Waller factor. The comparative analysis in Fig. 13 permits appreciating an increase of the diffuse scattering in correspondence with intensity maxima with respect to those obtained in intensity minima. Fig. 14 is another characteristic interesting application of the post processing where the derivative of Fig. 13b along x (thick¨ ness) is plotted in order to highlight the periodic Pendellosung oscillation between small scattering angles [59]. Finally, it is worth mentioning that additional post processing options can be used to operate on multiple ‘‘data cube’’ files from different simulations (for example for different compositions). It is possible to average (for every pixel, detection range and thickness) the results of two simulations. It is also possible to compare simulation results and plot intensity vs. composition at definite thickness and detection range.

4. Special simulations In addition to the standard simulation tools, a few simplified methods have been implemented to get a quick simulation of strained and/or compositionally inhomogeneous objects. A classic example is the implementation of the Howie–Whelan (HW) equation that describes the contrast in two beam conditions [60]. A second somehow similar equation solver has been added to qualitatively account for strain effects in STEM-ADF. 4.1. Howie–Whelan solver As diffraction contrast in two beam conditions is a classical computation problem in electron microscopy, a number of software already exist to cope with classic strain problems like defect analysis.

Fig. 15. Simulation for the contrast of a dark field (200) in an InGaAs/GaAs quantum well.

As explained, within STEM_CELL, it is possible to introduce the results of finite elements (FEM) analysis and simple analytic model to account for strain fields. This approach may result inadequate for a complex dislocation distribution but has been [61,62] used to calculate the contrast from buried QDs or intricate nanoparticles. The solver is based on the simplest HW formulation. The solution is obtained through the Runge-Kutta solver or through the more rough but robust Euler method [61,63]. For the sake of example the contrast for a dark field (200) in an InGaAs/GaAs quantum well is shown in Fig. 15. The example

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shows clearly the dark contrast characteristic of strain effects on the upper barrier of the QW.

4.2. Strain-channeling equation solver As in the case of HW equation a recent attempt has been made to create a contrast equation in the presence of strain in the case of STEM images. Since most of the contrast effects derive from channeling, a simplified equation has been derived to describe it based on an approximate description of the channeling phenomena in the presence of strain [29]. The method tries to solve directly for the effect of strain on the Bloch states [57,29] instead of affording the problem of the strain field effects on each g beam [64]. This operation however can be performed reasonably only for the 1s state that has a small dispersion with the pseudo-momentum K. The derivation of the equation has been done elsewhere [29] and produces the first equation of the system of Eq. (7) where the F function is the excitation of the 1s state averaged over the unit cell. This equation describes the evolution of the F with z (the propagation direction) because of absorption and strain. The equation can be coupled to the other equations in order to describe the overall ADF intensity which is contributed by TDS and an additional term named Huang scattering [65] with intensity IHuang ðx,y,zÞ, namely the additional diffuse scattering in the low angle regime due to the strain effects that disturb the channeling or, more in general, produce a redistribution of intensity between Bloch states [66,67]. This contribution is mainly diffuse and sums to the well known thermal scattering. However, unlike TDS, it contains little or no information about the

atomic number Z of the scattering atoms. 8 2  2 h i d F1s ðx,y,zÞ ðx,y,zÞ > ¼ 2m1s dF1sdz 2 sG 2 F1s ðx,y,zÞ > 2 > dz y > >   > > < dIHuang ðx,y,zÞ ¼ 2H G2 F ðx,y,zÞ 1s 2 dz sy " # > > > R P > > I ¼ ðF1s þ C Þ s d ð zz Þ > i i dz þ IHuang > : i ¼ atoms

ð9Þ

Relevant quantities in this system are the absorption coefficient m the contribution to intensity from non 1s states, here assumed as a constant C, the reciprocal space size of the 1s state sy, the atomic column curvature G and the atomic cross section s in the selected angular regime. The addition of a coefficient H is an angle dependent proportionality factor that can be used to weigh the Huang scattering contribution, described in Eq. (10) with respect to TDS and is strongly dependent on the angular scattering selected in the detection. However in spite of the clear simplifications this approach highlights the role of the curvature of the atomic column to determine the ADF contrast. In the case of a 2D problem the curvature can be written as !2 !2 @2 uy @2 ux 2 G ¼ þ ð10Þ @z2 @z2 Where ux,y are the in plane displacement with respect to an undeformed crystal. In the case of HAADF imaging the intensity would just be the product of F by the atomic cross section while the tuning of the H factor permits obtaining the low angle case and therefore LAADF imaging.

Fig. 16. Simplified simulation based on Grillo’s strain-channeling equation for LAADF images of a screw dislocation viewed end-on (a, b) and viewed on its side. (a) and (b) refer to exact zone axis and with a mistilt of 10 mrad. (c) and (d) report the difference between HAADF and LAADF images.

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109

Fig. 17. (a) Scheme of the deformation of lattice planes at the periphery of a QD. (b) HAADF and (c) LAADF simulation for the contrast of the QD in plan-view.

The non trivial boundary condition is that the initial excitation depends on the angle between the surface and the atomic column directly where the initial column bending is correctly accounted for.

F1s ðz ¼ 0Þ 

1

O

" 1

1

s2y



  2  2 # @ux  1 @uy  yt    2 @z z ¼ 0 sy @z z ¼ 0

These results demonstrate that a very fast account of experimental strain contrast in both HAADF and LAADF imaging can be obtained with a relatively small calculation that takes typically 1 min of a desktop PC.

ð11Þ

where O is the volume of the unit cell. As it can be seen, this equation involves the use of the second derivatives of the displacement. It is clear that a numerical evaluation of the derivative from finite differences would produce large numerical errors. For this reason it is possible to import directly from COMSOL the appropriate derivative which is treated as a separate field. For the sake of example we show in Fig. 16 and Fig. 17 two examples of application of the equation. Fig. 16 is the simulation of the LAADF contrast for a screw dislocation viewed end-on (a,b) and on its side (d). Panel a refers to an exact zone axis condition while in panel b a tilt of 10 mrad has been applied. Both images show a central core of high intensity and a dark ‘‘halo’’ of darker contrast. The dark contrast is due to surface deformation produced by the dislocation, sometimes named Eshelby twist [41]. It is interesting to highlight in particular the similarity of the contrast between the tilted case in Fig. 16b and the case of 2 beams dark field contrast [68], while for zone axis the simulation appears in agreement with rigorous simulations [69]. The simulation in Fig. 16c and d shows the same dislocation viewed on its side using HAADF (c) and LAADF (d) detection condition. In the equations the variation from High angle to Low angle is obtained by varying the H factor from 0 (HAADF) to 1. The appearance of a dark contrast in HAADF is due to the dechanneling effect produced by the dislocation strain field. The Huang scattering adds incoherent intensity to this image and is responsible for the bright contrast in the center of the dislocation. This general description is consistent with other simulations [64] that however produce images apparently with lower resolution: other models do not show, for example, the bright - dark contrast on the side of the dislocation visible in Fig. 16d that can be explained as the antagonism of dechanneling and Huang scattering effects as a function of the local column deformation. In fact it is easy to see from equation that Huang scattering and dechanneling are not exactly complementary. Another interesting example is the case of InAs/GaAs QDs seen in plan-view (Fig. 17). Panel a shows that in this case the curvature of the planes concentrates at the side of the QD. Therefore we expect a minimum of HAADF intensity (16 b) and a maximum of LAADF intensity (Fig. 17c) in those points. This is indeed highly visible in simulations in Fig. 17b and c. The characteristic cross like shape of the contrast is due to the anisotropic elasticity of the materials. Fig. 17b in particular is qualitatively similar to the result obtained on SiGe/Si QDs [70].

5. Conclusions We have introduced the main details of the STEM_CELL software as a useful tool for image simulation. The program facilitates the full simulation chain, from SC tailoring and modification up to the complex analysis of simulation results. STEM_CELL is a comprehensive implementation of image simulation tools and strategies. This software allows some important achievements: (1) the graphical routines for SC handling strongly facilitates the work of creation of complex structures, also in the case of strain; (2) the choice of optical and sampling parameters is strongly facilitated by semi-automatic routines; (3) a full support for probes with high order aberration or unconventional condenser aperture is available; (4) an original post processing approach to simulation of different thickness-detection conditions has been introduced; (5) an improved channeling model that predicts the effect of strain in STEM-ADF images is implemented.

Acknowledgments The authors would like to acknowledge Simone Giustetti for his suggestions on graphical programming, Vittorio Morandi for amorphous calculation and CINECA for the computation time. We would like to thank also Dott. F. Rossi for careful reading of the manuscript and prof. A. Rosenauer for the useful discussions. Part of these calculations have been performed within the HPCEUROPA2 project (Project no. 228398) with the support of the European Commission—Capacities Area—Research Infrastructures.

Appendix A. Details on image and profile management. This appendix describes a few details of the software implementation in particular regarding the way images and profiles are rendered. Images are internally represented as float single precision (4 byte) matrices either real or complex. Each image also holds additional tags containing information on the lateral calibration, electro-optical parameters and possible experimental details. These tags are based on Kirkland’s special TIF image format that included additional tags (at the present version 512) to store simulation parameters [10]. The order in the image parameters has been set consistently to Kirkland choice but a couple of few

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extra parameters originally not defined have been defined for experimental images to store information on the experimental setup. The graphical rendering is obtained either with a 256 gray level scale or with a few custom color scale. The brightness and contrast can be selected directly in the image by selecting the region (the levels are then selected according to the mean and variance in this region) of interest or by selecting explicitly the range. It is possible to select also different visualization modes for complex variables (real part, imaginary part, module and phase), to use logarithmic scale and to swap the quadrants in the case of FFT representation. Finally the software provides a limited support for 3D images which are handled as a collection of 2D images. The native format for input/output of the images is the special Tiff format introduced by Kirkland. Moreover the software permits opening dm3, Tietz,. ser (FEI) images and to import.jpg,.bmp and raw images. It is possible to write dm3 files and to export in jpg,.bmp raw binary and plain tif format. The handling of images includes standard operations like FFT, resize (by linear, spline and sinc interpolation methods), pixelwise operations between images and differentiation. The program allows to draw intensity profiles and, in general, to plot 1d graphics on a separate profile window. The profiles can be directly imported/exported in ASCII but also dm3 profiles can be read. Standard profile operation includes normalization, pointwise operation, fitting with a few common curves, statistical analysis, differentiation and lateral averaging. The built-in classes permit handling, in addition to images and profiles, also SC, strain models and collection of peak positions which are the main objects available in the program.

Appendix B. Probe size In this appendix we estimate the probe size for a given ! aberration function wð k Þ and a given convergence ac limiting

the in plane component of the wavevector to the range k A 0, alc . The probe can be written  2   Z !  ! ! ! i2p!  9 K 9max    ! k r  r probe ! !   2 2 iwð k Þ P r probe  r ,0, D ¼ N  ! e e d k  9K 9¼0    ðB:1Þ where N is an appropriate normalization factor. To find the size of the probe we seek for regions where the total phase of the integral is stationary. Such phase is

f ¼ 2pkðDrÞ þ w

ðB:2Þ

For a given Dr a stationary phase fguarantees a meaningful intensity requiring that the phase derivative with respect to K goes to 0.

r f ¼ 2pDr þ rw ¼ 0

ðB:3Þ

2pDr ¼ rw

ðB:4Þ

Since rw can assume different values as a function of K it is possible to find a solution to this equation in a range of values of Dr. For example if the rw is monotonically increasing with k the main limitation to the range of stationary Dr is due to the aperture angle ac. If only Cs and defocus f are considered 3 3

or 4 ¼ Dr max ¼ 9lf ðkmax Þ þ C s ðkmax Þ l 9  a  a 3   c c ¼ lf þ Cs l3  ¼ 9f ac þ C s ac 3 9 l l

ðB:5Þ

And if Cs is negligible, at high f, this is just a re-statement of the geometric optic principle   o r 4 ¼ f ac ðB:6Þ It is worth mentioning that the above principle can have an extension to non circular aberrations coefficient but a full theory will be exposed in a further coming article.

Appendix C. Parallel STEM software This appendix briefly discusses the parallel stem software stemslic3_mpi referenced in the text in comparison to other parallel software in literature. The code is based on the original Kirkland code of 1998 but has since then departed from the original program. The main improvement with respect to the original codes was the parallelization through the use of MPI and some optimization as described in Ref. [23,24].It is worth mentioning that it exists a similar parallelization of about the same period based on MPI [27]. The use of MPI[71] is most suitable for extended clusters while the present version of Kirkland program implements a form of parallelism based on open MP [26,72] and suitable for shared memory machines that are typically more limited in size. A third way has been demonstrated by C. Dwyer in the use of GPU that nowadays permit large parallelization with limited budget [25].This list is certainly not exhaustive as many other parallel codes are now being published (e.g. [56,73] ). In the present version stemslic3_mpi presents as additional features with respect to the past versions an optional form of parallelism on potential calculation, the possibility to save results in a single file for post processing (see the main text) and the possibility to save intermediate results in order to recover from possible anticipate exit of the code. References [1] C. Kisielowski, B. Freitag, M. Bischoff, H. van Lin, et al., Detection of single atoms and buried defects in three dimensions by aberration-corrected electron microscope with 0.5-A˚ information limit, Microscopy and Microanalysis: the Official Journal of Microscopy Society of America, Microbeam Analysis Society, Microscopical Society of Canada 14 (2008) 469–477. [2] P.D. Nellist, M.F. Chisholm, N. Dellby, O.L. Krivanek, M.F. Murfitt, Z.S. Szilagyi, A.R. Lupini, A. Borisevich, W.H. Sides Jr., S.J. Pennycook, Direct sub-angstrom imaging of a crystal lattice, Science 305 (2004) 1741. [3] H.A. Bethe, Theorie der Beugung von Elektronen an Kristallen, Annals of Physics 87 (1928) 55. [4] A.F. Cowley, J.M. Moodie, The scattering of electrons by atoms and crystals. I. A new theoretical approach, Acta Cystallography 10 (1957) 609. [5] P.G. Self, M.A. O’Keefe, P.R Buseck, A.E.C. Spargo, Practical computation of amplitudes and phases in electron diffraction, Ultramicroscopy 11 (1983) 35. [6] B. Edwards, E.J. Kirkland, J. Silcox, D.A. Muller, Simulation of thermal diffuse scattering including a detailed phonon dispersion curve, Ultramicroscopy 86 (2001) 371. [7] M.J. Hytch, W.M. Stobbs, Quantitative comparison of high resolution TEM images with image simulations, Ultramicroscopy 53 (1994) 191. [8] S. Kret, P. Ruterana, A. Rosenauer, D. Gerthsen, Extracting quantitative information from high resolution electron microscopy, Physica Status Solidi B 227 (2001) 247. [9] P. Stadelmann, EMS—a software package for electron diffraction analysis and HREM image simulation in materials science, Ultramicroscopy 21 (1987) 131. [10] E.J. Kirkland, Advanced Computing in Electron Microscopy, Plenum Press, New York, 1998. [11] K. Ishizuka, A practical approach for STEM image simulation based on the FFT multislice method, Ultramicroscopy 90 (2002) 71. [12] C Koch. /http://www.christophtkoch.com/stem/index.htmlS. [13] R. Kilaas, in: Proceedings of the 45th Annual Meeting of the Microscopy Society of America, 1987, p. 66. [14] P.R. Buseck, O’ Keefe, Transactions of the American Crystallographic Association 15 (1979) 27. [15] /http://www.totalresolution.com/S. [16] Z.L. Wang, Dynamical inelastic scattering in high-energy electron diffraction and imaging: a new theoretical approach, Physical Review B 41 (1990) 12818.

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[47] [48] [49]

[50]

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[56]

[57]

[58] [59] [60]

[61] [62]

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[68]

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