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1 Ultramicroscopy 109 (2009) Contents lists available at ScienceDirect Ultramicroscopy journal homepage: Quantitative atomic resolution mapping using high-angle annular dark field scanning transmission electron microscopy S. Van Aert a,, J. Verbeeck a, R. Erni b, S. Bals a, M. Luysberg c, D. Van Dyck a, G. Van Tendeloo a a Electron Microscopy for Materials Science (EMAT), University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium b National Center for Electron Microscopy, Ernest Orlando Lawrence Berkeley National Laboratory, 1 Cyclotron Road, MS 72R0150, Berkeley, CA 94720, USA c Institute of Solid State Research and Ernst Ruska Center for Microscopy and Spectroscopy with Electrons, Helmholtz Research Center Jülich, Jülich, Germany article info Article history: Received 21 January 2009 Received in revised form 7 May 2009 Accepted 13 May 2009 PACS: Rr Pj þs Keywords: High angle annular dark field scanning transmission electron microscopy (HAADF STEM) Statistical parameter estimation theory Compositional mapping abstract A model-based method is proposed to relatively quantify the chemical composition of atomic columns using high angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) images. The method is based on a quantification of the total intensity of the scattered electrons for the individual atomic columns using statistical parameter estimation theory. In order to apply this theory, a model is required describing the image contrast of the HAADF STEM images. Therefore, a simple, effective incoherent model has been assumed which takes the probe intensity profile into account. The scattered intensities can then be estimated by fitting this model to an experimental HAADF STEM image. These estimates are used as a performance measure to distinguish between different atomic column types and to identify the nature of unknown columns with good accuracy and precision using statistical hypothesis testing. The reliability of the method is supported by means of simulated HAADF STEM images as well as a combination of experimental images and electron energy-loss spectra. It is experimentally shown that statistically meaningful information on the composition of individual columns can be obtained even if the difference in averaged atomic number Z is only 3. Using this method, quantitative mapping at atomic resolution using HAADF STEM images only has become possible without the need of simultaneously recorded electron energy loss spectra. & 2009 Elsevier B.V. All rights reserved. 1. Introduction It is generally known that high angle annular dark field scanning transmission electron microscopy (HAADF STEM) images show Z-contrast meaning that the intensities of the atomic columns scale with the mean atomic number Z. One of the advantages of this imaging mode is therefore the possibility to visually distinguish between chemically different atomic column types. However, if the difference in atomic number of distinct atomic column types is small or if the signal-to-noise ratio is poor, direct interpretation of HAADF STEM images is inadequate. Moreover, if the tails of the electron probe have considerable contributions on neighboring atomic columns, intensity transfer from one atomic column to neighboring atomic columns may occur. Such effects also hamper the direct interpretation of the images, especially when studying interfaces [1,2]. Then, quantitative methods are needed. Unlike qualitative STEM methods, which are based on visual interpretation of the images, quantitative STEM allows the extraction of local structural and chemical information with good accuracy and precision [3]. Corresponding author. Tel.: ; fax: address: sandra.vanaert@ua.ac.be (S. Van Aert). In the past, several approaches have been proposed for quantitative compositional mapping using HAADF STEM [4 8]. In general, the starting point of quantitative (S)TEM is the notion that one is not so much interested in the images as such, but rather in the structural and chemical information of the object under study. Therefore, images are to be considered as data planes, from which structure parameters have to be determined as accurately and precisely as possible. From these parameters, structural and chemical information can be extracted. In the methodology used in this paper, the key to successful quantitative HAADF STEM image analysis is the availability of a pertinent parametric model of the images. The model includes all ingredients needed to perform a computer simulation of the image. It is parametric in quantities, such as the location, width and height of the image intensity peaks, which are usually unknown beforehand and have to be estimated from the experiment. The quantities are estimated by fitting the model to the experimental data using a criterion of goodness of fit, such as least squares, least absolute values or maximum likelihood. This search for the optimum of the criterion of goodness of fit is usually an iterative numerical procedure. Recently, the same methodology has been used to extract quantitative structural and chemical information from electron microscopy data obtained using high resolution transmission electron microscopy, exit wave /$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi: /j.ultramic

2 S. Van Aert et al. / Ultramicroscopy 109 (2009) reconstruction or electron energy-loss spectroscopy (EELS) [9 14]. In this paper, the methodology will be used and expanded in order to obtain chemical information from HAADF STEM images. In principle, the model should accurately describe all effects contributing to the HAADF STEM image intensities [8,15], such as thermal diffuse scattering and dynamical electron diffraction effects. This generally requires the use of a multislice method within the frozen phonon framework [15,16] or alternative methods including TDS [17,18]. Although such methods can be considered as the most accurate ones, they are very time consuming. Therefore, it is rather unrealistic to use such a model in an iterative optimization procedure of a criterion of goodness of fit. Instead of a time consuming model, a simplified, empirical incoherent imaging model to describe HAADF STEM images is proposed. This simplified model is then used to estimate the positions as well as the total intensity of the scattered electrons for every atomic column from a HAADF STEM image. Although this model cannot be used to absolutely quantify the chemical composition, it will be shown by means of accurate HAADF STEM image simulations that it is very useful to extract relevant structural and chemical information which can then be used for relative quantification of the chemical composition. In order to relatively quantify the chemical composition, the total intensity scattered from an unknown atomic column type can be compared with the intensities scattered from the neighboring known atomic column types. In other words, a decision among possible hypotheses about the atomic column types can be made based on the estimated intensities provided that the composition of the unknown columns is similar to the composition of the known columns and that the thickness of neighboring columns is approximately the same. Of course, due to noise, the estimated intensities are inherently random in nature so that a statistical approach is necessary. Chemical composition analysis thus becomes a statistical detection problem. Note that an absolute instead of a relative quantification of the chemical composition would be possible given an exact relation between the totally scattered intensity and the atomic number Z. However, this relation is only known approximately and needs further investigation in the framework of the proposed model-based method including an analysis of sample thickness effects and the effect of noise on the measurement results. As a practical example, the method is used to investigate multilayer structures in order to characterize the interfaces quantitatively. For thin samples, the scattered intensities will not only scale with the mean atomic number Z but also scale with the thickness of a column [5]. Therefore, the method will be useful as well to help determine, for example, the three-dimensional shape at atomic resolution of nanoparticles or to quantify chemical inclusions in crystalline materials [7] although it should be mentioned that channeling effects may complicate this analysis. The outline of this paper is as follows. In Section 2, the method will be explained. Next, the usefulness of the method will be shown by means of accurate simulations of HAADF STEM images including probe formation, thermal diffuse scattering, and image detection. This will be the subject of Section 3. In Section 4, the method will be applied to experimental images. In addition, the results will be compared with simultaneously recorded electron enerey loss spectra to proof the validity of the method. Finally, in Section 5, conclusions are drawn. 2. Methodology The proposed method for quantitative mapping at atomic resolution is based on a comparison of the total intensity of the scattered electrons of an unknown atomic column with the intensities of known atomic columns. Such a method can be used, for example, to study interfaces in multilayer structures. By comparing the total intensity scattered from unknown atomic columns at an interface with the intensities scattered from known atomic columns in the layers, the interfaces can be characterized. The key to the method is thus an accurate and precise measurement of the total intensity of the scattered electrons at atomic resolution. Therefore, use has been made of statistical parameter estimation theory. In order to apply this theory, a parametric model is required describing the expectations of the pixel values of a HAADF STEM image Expectation model Assuming a high-angle annular detector, an incoherent STEM image will be formed, which can be written as a convolution between an object function and the probe intensity [19 21]. The model describing the expectations of the image intensities at the pixels ðk; lþ, corresponding to the STEM probe at position r k;l ¼ðx k y l Þ T, is therefore given by the following incoherent expression: Z f kl ðyþ ¼f ðr k;l ; yþ ¼Oðr k;l ; yþpðr k;l Þ¼ Oðr; yþpðr k;l rþ dr (1) with Oðr; yþ the object function depending on a set of unknown structure parameters y and PðrÞ the probe intensity depending on a set of probe parameters including the acceleration voltage, the objective aperture semi-angle, defocus, spherical aberration constant, and higher order aberration coefficients. Note that in a non-dedicated scanning transmission electron microscope, the condenser lens and its aperture are the electron-probe-forming optical elements that correspond to the objective lens and aperture, respectively, in a dedicated scanning transmission electron microscope. The object function is sharply peaked at the atomic column positions [21]. Here, it is modeled as a superposition of Gaussian peaks. The width of the peaks of a certain column type, defining the peakedness, is considered as one of the unknown parameters. The object function is thus given by the following expression: 0 1 Oðr; yþ ¼z þ XI X M i ðx b xmi Þ 2 ðy b ymi Þ 2 a mi exp@ A (2) 2r i 2 i¼1 m¼1 where z is a constant background, r i is the width of the Gaussian peak of a particular column type i, a mi is the height of the Gaussian peak and b xmi and b ymi are the x- and y-coordinate of the m i th atomic column, respectively. The unknown parameters of the model are thus given by the parameter vector y ¼ðb x11 b xmii by11 b ymii a 11 a MII r 1 r I zþ T. From this parameterized object function, the volume under a peak above the background can be calculated as follows: V mi ¼ 2pa mi r 2 i (3) These numbers are proportional to the total intensity of electrons scattered toward the ADF detector for every atomic column. The illuminating STEM probe PðrÞ is given by PðrÞ ¼jpðrÞj 2 SðrÞ (4) with jpðrþj 2 the coherent point source contribution and SðrÞ representing incoherent extended source size effects [22]. The function pðrþ is given by the inverse Fourier transform of the transfer function of the objective lens TðgÞ: pðrþ ¼I 1 g!rtðgþ (5)

3 1238 S. Van Aert et al. / Ultramicroscopy 109 (2009) The transfer function TðgÞ is given by squares criterion: TðgÞ ¼ AðgÞ expðiwðgþþ (6) where AðgÞ is the aperture function, which usually is a circular top-hat function with unity height and radius g ap. Notice that the objective aperture semi-angle a o is equal to g ap l with l the electron wavelength. The phase shift wðgþ is controlled by the objective lens aberrations. It depends on rotationally symmetric aberrations including the defocus and the spherical aberration of the third and the fifth order. Also non-symmetric aberrations such as astigmatism and coma can be included in this phase shift to obtain a more accurate calculation of the probe profile [23]. The intensity distribution of the extended source image SðrÞ can be assumed to be Gaussian [24]: SðrÞ ¼ 1 exp r2 (7) 2pd S 2 2d S 2 with r ¼jrj and d S describing the size of the extended source image. Straightforward calculations show that the relationship between d S and the full width at half maximum height of the source image distribution described by Eq. (7) is given by p FWHM ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffi 2ln2d S 2:35d S (8) Although some components of the incoherent imaging model proposed in this section lack a solid physical basis, it will be shown in Sections 3 and 4 that it may describe the experimental contrast adequately. The simplicity of the model has the additional advantage that it can be used very effectively in an iterative optimization procedure to estimate the unknown model parameters. Since the expectations of the image intensities are modeled as a superposition of Gaussian peaks (Eq. (2)) blurred with the probe intensity profile (Eq. (4)), overlap between neighboring columns is taken into account. As such, the effect that probe intensity may be transferred from one atomic column to a neighboring atomic column is incorporated in the model. Moreover, the model incorporates a constant background z, to account for the variable black level setting of the experiment. Of course, care has to be taken that no clipping occurs [25]. Experimental HAADF STEM images, however, often give the impression of the presence of a varying background, especially in the study of interfaces. In Section 4, it will be shown that such effects can be very well described by overlap of the images of neighboring columns, which is here taken into account. Despite these strengths of the model, it should be mentioned that thermal diffuse scattering and dynamical diffraction effects are not incorporated in the proposed model. Fluctuations of the intensity with thickness are here described by the object parameters a mi.as such, a good agreement between the model and the experiment can be obtained by estimating the unknown a mi. However, since the dependence of a mi on the thickness is not explicitly incorporated, structural or chemical information cannot directly be derived from these parameters. Therefore, the model can thus only be used to relatively quantify the composition of unknown columns assuming the thickness of neighboring columns to be the same Statistical parameter estimation The parameters y of the model as presented in Section 2.1 are usually unknown beforehand and have to be estimated from the experiment. Therefore, use has been made of the uniformly weighted least squares estimator. The estimates b y are then given by the values of t that minimize the uniformly weighted least b X K X L y ¼ arg min ðw kl f kl ðtþþ 2 (9) t k¼1 l¼1 with w kl the value of the recorded ADF STEM image at the pixel ðk; lþ and the function f kl given by Eq. (1). We note that for pixel values which are independent and identically normally distributed about the expectations, the uniformly weighted least squares estimator is equal to the maximum likelihood estimator, which is known to have optimal statistical properties [11]. As shown in Miedema et al. [26], this assumption about the statistical distribution of the pixel values is a reasonable one. It has been assumed that the probe parameters are known with good accuracy. In principle these parameters could be estimated together with the object parameters y. This will be a subject for future work. Using the invariance property of the maximum likelihood estimator, estimates of the total intensity of scattered electrons can be obtained from Eq. (3). For example, the estimated scattered intensity for atomic column m i is given by bv mi ¼ 2pba mi br 2 i (10) In Section 2.3 it will be explained how these estimates can be used to distinguish between different atomic column types and to identify the nature of unknown columns. Note that the quantities which are estimated here in fact correspond to measurements of the volume under the individual peaks after a proper deconvolution with the probe intensity profile. In this sense, the proposed method can thus be regarded as an alternative for deconvolution methods [27]. It should be mentioned, however, that this deconvolution method assumes prior knowledge about the object function. If, for example, the presence of an interstitial atom is not incorporated in the object function, the position of this interstitial atom will not be revealed Statistical hypothesis testing It is generally known that the contrast of HAADF STEM images is sensitive to compositional changes [19]. The scattered intensity toward the ADF detector from an atomic column scales with the mean atomic number Z. Hence the name Z-contrast. For thin objects, it is often assumed that the scattered intensity scales with Z n, with n a constant, which strongly depends on the HAADF detector geometry [28]. However, a detailed relation between the scattered intensities and Z, which is also valid for thick objects, has not yet been established due to the complicated dependence on the sample thickness, the atomic number, the crystal structure, the crystal orientation, and the detector geometry. In principle, for large thicknesses, the intensity of high Z columns could even become less than the intensity of lower Z columns. Therefore, an absolute quantification of the chemical composition from HAADF STEM images is impossible. However, since the parameter values V mi are expected to be identical for columns with the same chemical composition, whereas these are expected to be different for columns with different composition or different thickness, relative quantification of the chemical composition is possible. Assuming the thickness of the sample to be rather constant, this principle can indeed be used to identify the composition of unknown columns by comparing the corresponding volumes under a peak with the volumes of known columns. This method is useful when applied to, for example, multilayer structures where one usually has prior knowledge about the layer composition, provided that the layers are not too thin, but where the interface composition is unknown.

4 S. Van Aert et al. / Ultramicroscopy 109 (2009) However, the parameter values V mi cannot be measured exactly from a real-life experiment. They can only be estimated from the observations w k;l using, for example, Eqs. (9) and (10). Due to, for example, counting noise, scanning noise, sample imperfections, and thermal vibrations, the estimated parameters bv mi will be different for different atomic columns even if the underlying atomic column types are identical. Due to this inherently random nature of the parameter estimates, a statistical approach is necessary for quantitative composition analysis. Therefore, use has been made of statistical hypothesis testing. First, tolerance intervals for the population of estimated parameters V b mi are computed for the atomic columns of which the composition is known from prior knowledge about the specimen and whose thicknesses are assumed to be identical. If these populations are assumed to have a normal distribution, which is usually true for maximum likelihood estimated parameters, tolerance intervals can be constructed using the following expression [29]: ðv i ks V i ; V i þ ks V i Þ (11) The sample mean V i from the set of estimated values V b mi corresponding to column type i is given by V i ¼ 1 X V b mi (12) M i m i The standard deviation s V i is the square root of the sample variance s 2 V i, which is defined by s 2 V i ¼ 1 X ðv M i 1 b mi V i Þ 2 (13) m i The k factors in Eq. (11) are determined so that the intervals cover at least a proportion p of the population with confidence g: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm i 1Þ 1 þ 1 u z 2 M ð1 pþ=2 k ¼ t i (14) w 2 g;m i 1 where w 2 g;n 1 is the critical value of the chi-square distribution with M i 1 degrees of freedom that is exceeded with probability g and z ð1 pþ=2 is the critical value of the standard normal distribution which is exceeded with probability ð1 pþ=2. Next, the column types of the unknown atomic columns are determined using hypothesis testing. Suppose that we have a column for which the estimated scattered intensity is equal to V b u. We then want to test the null hypothesis H 0 against the alternative hypothesis H 1, with H 0 : the column type of this column is i H 1 : the column type of this column is not equal to i. As a test statistic the tolerance interval for the estimated parameters V b mi as given by Eq. (11) is chosen with parameters p and g. These parameters have been chosen equal to 0.9 in the experimental example that will be shown in Section 4. The hypothesis H 0 is rejected at the significance level 1 p if this tolerance interval does not contain V b u.ifh 0 is true, the maximum probability of rejecting H 0 is 1 p and, therefore, the minimum probability of accepting H 0, making the correct decision, is p. In other words, if the column type of the unknown column is i, there is a probability of 1 p to observe the estimated scattered intensity V b u of this column outside the tolerance interval corresponding to column type i. Based on these considerations, one could suggest that if significantly more than 1 p of the estimated scattered intensities V b u falls outside a specific tolerance interval, these intensities cannot be considered as outliers. In that case, it can be concluded from the scaling relation between the scattered intensity and Z that a column, for which the estimate V b u falls in between the tolerance intervals of the columns with averaged atomic number Z i and Z j, will have an atomic number in between Z i and Z j assuming the thickness of the sample to be constant. However, in that case, an exact quantification of Z of this unknown column would require an accurate expression of the dependence of V on Z. In this hypothesis test it has been assumed that the tolerance intervals of the known atomic columns are not overlapping. Although more elaborate hypothesis tests exist which can be used in case of overlapping tolerance intervals [30], the performance of the decision will greatly decrease with increasing overlap. It is therefore recommended to measure the scattered intensities from the atomic columns as accurately and precisely as possible in order to avoid overlap. The accuracy and precision of these estimates will generally improve if the signal-to-noise ratio of the images increases or if the model describing the expectations of the pixel values of an HAADF STEM image is further improved. Fig. 1. (Color online) (a) Simulated HAADF STEM image of a LaAlO 3 /SrTiO 3 multilayer structure along the [0 0 1] zone axis with simulation parameters as listed in Table 1. (b) An overlay indicating the different types of atomic columns. The column types of the columns at the interfaces have been assumed to be unknown in their quantification. (c) Model for the HAADF STEM image evaluated at the estimated parameters. (d) Estimated peak volumes for the corresponding layers in (b). The columns at the interfaces can clearly be identified by comparing their estimated peak volumes with the peak volumes of the TiO and AlO columns in the layers.

5 1240 S. Van Aert et al. / Ultramicroscopy 109 (2009) Simulation study The method has first been tested on simulated HAADF STEM images. Therefore, a multislice calculation within the frozen phonon framework using an Einstein approximation has been performed in order to accurately simulate thermal diffuse scattering. Use has been made of the STEMsim program introduced in Rosenauer and Schowalter [31]. As a test structure, a LaAlO 3 /SrTiO 3 multilayer structure has been chosen in which an alternation of AlO and TiO columns at the successive interfaces along the growth direction has been assumed in agreement with the results found in Huijben et al. [14]. Fig. 1a shows a simulated image along the [0 0 1] zone axis. Simulation parameters were chosen as listed in Table 1 where the Scherzer conditions for incoherent imaging have been assumed [19]. Note that incoherent extended source size effects are not taken into account in this simulation. This means that the size of the extended source image is set equal to 0 nm. In Fig. 1b the simulation is shown together with an overlay indicating the positions of the different types of atomic columns. The columns indicated by the symbol X are the columns whose composition would be assumed to be unknown in the characterization of the interfaces from a real experiment. Next, the parameters y of the empirical model given by Eq. (1) have been estimated in the least squares sense using Eq. (9) with w kl corresponding to the pixel values of the simulation. The refined model is presented in Fig. 1c showing a good visual comparison with the simulation (Fig. 1a). From this, it could be concluded that the simple incoherent model given by Eq. (1) is, perhaps surprisingly, adequate to describe most of the image contrast obtained from a much more elaborate multislice calculation including phonon scattering. As shown in Nellist and Pennycook [32], a simple incoherent model is indeed valid provided that the inner collection angle of the annular detector is sufficiently larger than the angle defining the circular objective aperture. Next, from the estimated parameters, the volumes under the peaks of the object function have been computed at atomic resolution using Eq. (10). The results arepresented in Fig. 1d. In order to find out if the method works well for interface characterization, the estimated peak volumes of the columns at the interfaces have been compared with the estimated volumes of the columns in the layers. Note that in contrast to real experiments, which are prone to noise, there is no need to compute the tolerance intervals given by Eq. (11) for simulated experiments unless the number of phonon configurations chosen in the simulation is too small. In this simulation, convergence of the estimated peak volumes has been reached when the intensities of 10 phonon configurations have been incoherently averaged. From Fig. 1d it is clear that the estimated peak volumes are identical for columns of the same type. Moreover, the interfaces can unambiguously be identified by comparing their estimated peak volumes with the peak volumes of the TiO and AlO columns in the layers. This example thus shows that the method works for simulated experiments. The advantage to test the method on a simulated image is that the results can directly be compared with the input for the simulations. In the next section, the method will be applied to experimental data where it becomes much harder to judge whether the results are in agreement with the unknown true chemical composition of the sample. In order to compensate for this disadvantage, the results will be compared with atomic resolution EELS results although one has to note that the latter technique has its interpretation problems as well at atomic resolution. 4. Experiment In order to demonstrate its practical applicability, the proposed model-based method has been used for quantitative characterization of the interfaces in a La 0.7 Sr 0.3 MnO 3 SrTiO 3 multilayer structure grown on a (0 0 1) SrTiO 3 substrate by pulsed-laser deposition. The specimen has been prepared by mechanical polishing followed with ion milling under low angle and low acceleration voltage (8, 5 kv). This should lead to relatively flat surfaces as suggested by the homogeneous contrast observed in Fig. 2, where an HAADF STEM image is shown obtained from an area close to the STO substrate. The averaged atomic numbers Z of the atomic columns when the material is viewed along the [0 0 1] zone axis are equal to 51, 38, 33, and 30 for the (La 2/3,Sr 1/3 ), Sr, MnO, and TiO columns, respectively. Note that the difference in Z between the TiO and MnO columns is only 3. The challenge of this experiment is thus to measure the scattered intensities from the individual TiO and MnO columns with a precision that allows us to clearly distinguish between these two column types and to identify the column composition at the interfaces. In addition, the results have been compared with data obtained from simultaneously recorded electron energy loss spectra in order to test the validity of the proposed model-based method applied to HAADF STEM images. Table 1 Simulation parameters used in Section 3. Acceleration voltage 300 kv Defocus C 1 0nm Spherical aberration constant C 3 2:275 mm Spherical aberration of the fifth order C 5 1mm Objective aperture angle a o 30 mrad Standard deviation d s of the source image 0nm ADF detector range mrad Number of phonon configurations 10 Specimen thickness 30 nm Number of unit cells per supercell 6 6 Pixels per unit cell Pixel size Å Fig. 2. HAADF STEM image of the La 0.7 Sr 0.3 MnO 3 SrTiO 3 multilayer structure along the [0 0 1] zone axis. The area of the image indicated by the white rectangle is used for model-based quantification of the columns at the interfaces.

6 S. Van Aert et al. / Ultramicroscopy 109 (2009) Fig. 3. (Color online) (a) Enlarged section of the selected area in Fig. 2. (b) Refined model. (c) Experimental data (a) and refined model (b) averaged along the horizontal direction. (d) Overlay indicating the estimated positions of the columns together with their atomic column types. The columns whose composition is unknown are indicated by the symbol X. EELS and HAADF STEM experiments were carried out using an FEI Titan microscope operated at 300 kv. This microscope is equipped with a high-resolution spectrometer and an illumination aberration corrector that corrects coherent aberrations up to third order and partially corrects for coherent aberrations up to fifth order [33]. The HAADF STEM images that have been analyzed were recorded using an inner semi-detection angle of about 80 mrad. The probe semi-convergence angle was 25 mrad. The demagnification of the condenser system was adjusted such that for HAADF STEM images a probe size (in terms of full width at half maximum) of 0.08 nm and for the EELS measurements a probe size of 0.15 nm was measured from the Fourier transform of the images. Although the image shown in Fig. 2 has been obtained using a state-of-the-art microscope, no visual conclusions could be drawn concerning the sequence of the atomic planes at the interfaces. Therefore, a quantitative analysis of the data is necessary. The method as explained in Section 2 has been applied to the area as indicated by the white rectangle. Fig. 3a shows an enlargement of this selected area. Next, the parameters y of the expectation model given by Eq. (1) have been estimated using Eq. (9). The probe parameters which have been used in this estimation procedure are listed in Table 2. The refined model is shown in Fig. 3b. Visually comparing Fig. 3b with 3a does not reveal any significant systematic deviations of the refined model from the experimental observations. This is illustrated more clearly in Fig. 3c where the data of the experiment (Fig. 3a) and refined model (Fig. 3b) are shown together after averaging along the horizontal direction. From this figure, it is also clear that the impression of a varying background is very well described by the proposed expectation model in which a constant background is assumed. Clearly, the background trend may Table 2 Values for the microscope settings of the experiment as described in Section 4. Acceleration voltage Defocus C 1 Spherical aberration C 3 Spherical aberration of the fifth order C 5 Objective aperture angle a o Standard deviation d S of the source image Pixel size 300 kv 0nm 2:275 mm 1mm 25 mrad nm 0.31 Å equally well be due to the overlap of neighboring columns at the interface. This shows that the simple incoherent model is able to describe the image contrast adequately. Assuming that the model is correct, Fig. 3b may be regarded as an optimal image reconstruction. Fig. 3d shows the experimental observations together with an overlay indicating the estimated positions of the columns together with their atomic column types. The composition of the columns away from the interfaces is assumed to be known whereas the composition of the columns in the planes close to the interface (indicated by the symbol X ) is unknown. In this experiment, more than one plane of unknown columns has been assumed since the composition from one layer to the next layer may change gradually rather than abruptly due to intermixing or diffusion [34]. The goal is now to identify the unknown column types using the statistical hypothesis testing method from Section 2.3. The estimated peak volumes, which are required in this method, can be obtained from Eq. (10) after refinement of the model. Histograms of the estimated peak volumes of the known columns are presented in Fig. 4 and show the statistical nature of the result. The colored vertical bands correspond to tolerance intervals

7 1242 S. Van Aert et al. / Ultramicroscopy 109 (2009) Fig. 4. (Color online) Histograms of the estimated peak volumes of the columns in Fig. 3 whose composition is assumed to be known. The colored vertical bands represent the corresponding tolerance intervals. The unknown columns can then be identified by comparing their estimated peak volumes with these tolerance intervals. Single-colored dots will be used to indicate columns whose estimated peak volume falls inside a tolerance interval whereas pie charts will be used to characterize columns whose estimated peak volume falls outside the tolerance intervals. Fig. 5. (Color online) Results of the quantification of the unknown column types of the areas on the top and at the bottom of the STO layer. covering at least a proportion p ¼ 0:9 with confidence g ¼ 0:9 and have been computed using expression (11). It is important to note that these tolerance intervals are not overlapping meaning that columns, even for which the difference in averaged atomic number is only 3 (TiO and MnO), can clearly be distinguished. Next, in order to identify an unknown column, one needs to compare its estimated peak volume with the colored confidence intervals. Based on the proposed statistical hypothesis test, its column type is equal to the column type corresponding to the tolerance interval containing the estimated peak volume of the unknown column. As discussed in Section 2.3, there is a probability of only 1 p ¼ 0:1 to observe the estimated peak volume outside a tolerance interval. In this experiment, significantly more than 10% of the estimated peak volumes are found outside the calculated tolerance intervals and can therefore not be considered as outliers. In that case, these columns will have an averaged atomic number in between Z i and Z j if the estimated peak volume falls in between two tolerance intervals corresponding to atomic numbers Z i and Z j. In this example, such an effect might be the result of intermixing or diffusion of different types of atoms. As indicated in the bottom of Fig. 4, single-colored dots will be used to indicate columns whose estimated peak volume falls inside a tolerance interval whereas pie charts will be used to characterize columns whose estimated peak volume falls outside the tolerance intervals. The size of the two segments is an indication of the relative position of the estimated peak volume with respect to the nearest tolerance intervals and give as such an idea of the degree of intermixing. Note that currently an absolute quantification of the averaged atomic numbers is impossible due to the absence of an accurate expression of the dependence of V on Z. The results of this quantification of the unknown column types of the areas on the top and at the bottom of the STO layer are shown in Fig. 5. The lower planes of the quantified areas consist of purely TiO and MnO columns, respectively. However, intermixing of La and Sr and of Ti and Mn is observed in the upper two planes. The degree of intermixing seems more important for the interface on top of the STO layer than for the interface at the bottom of the STO layer. In order to check these results, they have been compared with the conclusions obtained from an experimental EELS line scan across the STO layer. The plot in Fig. 6 shows the integrated signals of the Ti L 3,2 absorption edge, the Mn L 3,2 and the La M 5,4 edges which were simultaneously recorded along a line as displayed in the STEM micrograph on top of the plot. A probe size of 0.15 nm was used in this EELS measurement in order to have a sufficient signal-to-noise ratio. Note that a probe size of 0.08 nm was used in the acquisition of the HAADF STEM image shown in Fig. 2. Although the model-based method could be applied to any HAADF STEM image, the accompanied improvement of resolution with reduced probe size leads to more reliable results. This switch between different probe sizes used in the EELS and HAADF STEM measurements probably implies that slightly different areas have been analyzed. However, no specimen changes are expected on this small length scale. The characteristics of the Ti, Mn and the La signal obtained from HAADF STEM analysis reflect to a large extent the expected atomic structure within the STO and the LSMO layers. This suggests that the modulations of the individual EELS signals at the interfaces similarly reveal the chemical nature of the material on a length scale that is comparable with the unit cell size. From the EELS profiles it follows that the left interface (corresponding to the interface at the bottom of the STO layer in Fig. 5) is sharper than the right interface (corresponding to the interface at the top of the STO layer in Fig. 5). Moreover intermixing of Ti and Mn is suggested for both interfaces. Also here, the degree of intermixing is more important for the right than for the left interface. Note that comparable conclusions have been obtained from experimental EELS line scans in Muller et al. [34]. Furthermore, it should be noted that in the analysis of the HAADF STEM images it has been tacitly assumed that the thickness of the sample is constant. This assumption could in principle be checked from a low-loss EELS analysis. In fact, it is currently impossible to rule out the possibility of preferential milling during TEM sample preparation at the interface as an explanation for the estimated peak volumes which have been found outside the tolerance intervals. However, preferential milling would result in a decrease of the sum of the Mn and Ti EELS signal, which is not observed here. This proves that intermixing can indeed be detected by using the approach explained in this paper. Here, the EELS results can be more directly compared with the quantitative HAADF STEM results by plotting the sizes of the segments of the pie charts shown in Fig. 5 when averaged along the planes parallel with the interfaces. If, for simplicity, the dependency of V on Z in the intervals between two atomic numbers Z i and Z j is assumed to be linear, these sizes are, in agreement with the integrated EELS signals, a measure of the relative concentration of a specific element. The estimated concentrations are shown in the bottom of Fig. 6. Note that the EELS signals of La M 5,4 and Mn L 3,2 are alternating with atomic resolution on the right-hand side of Fig. 6 which is consistent with the perovskite stacking. However, this is not true for the Ti L 3,2 signal in the STO layer which appears to peak on the SrO columns. This effect can be expected in high spatial resolution EELS experiments and can be understood as the result of the interaction between elastic and inelastic interactions

8 S. Van Aert et al. / Ultramicroscopy 109 (2009) Fig. 6. (Color online) Integrated signals of the Ti L 3,2 absorption edge, the Mn L 3,2 and the La M 5,4 edges which were simultaneously recorded along a line as displayed in the STEM micrograph on top of the plot. The profile on the bottom shows the estimated relative concentrations averaged along the planes parallel with the interfaces as obtained from the HAADF STEM analysis of which the results are shown in Fig. 5. in the crystal [35,36]. Therefore, the results of our method can only be compared with EELS in a qualitative way. It can be observed that both methods show a similar trend and that they both show a stronger intermixing on the right interface as opposed to the left. The results obtained from the quantitative analysis of the HAADF STEM images are thus in good correspondence with the results obtained from an EELS line scan proving that the method works well in practice. 5. Conclusions In this paper, a model-based method has been proposed to relatively quantify the chemical composition of atomic columns using HAADF STEM images. The strength of the method comes from an accurate and precise measurement of the scattered intensity at atomic resolution. Therefore, use has been made of statistical parameter estimation theory. This requires the parametric model describing the expectations of the pixel values of a HAADF STEM image to be known. An empirical model has been used depending on a set of unknown parameters, such as, the location, width, and height of the image intensity peaks. The model has been composed in such a way that it is very effective in an iterative optimization of the unknown model parameters while keeping the essential physics. Next, the unknown atomic columns are identified by comparing its scattered intensity with the scattered intensity from known columns using statistical hypothesis testing. Such a method is very useful, for example, to characterize interfaces of multilayer structures. The method leads to very accurate results when applied to simulated HAADF STEM images. Moreover, the success of the method has been demonstrated on experimental HAADF STEM images of a La 0.7 Sr 0.3 MnO 3 SrTiO 3 multilayer structure. Statistically meaningful information on the composition of the columns could be obtained even though the difference in averaged atomic number between the TiO and MnO columns is

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