Is atomic resolution transmission electron microscopy able to resolve and refine amorphous structures?

Transcription

1 Ultramicroscopy 98 (2003) Is atomic resolution transmission electron microscopy able to resolve and refine amorphous structures? D. Van Dyck a, *, S. Van Aert b, A.J. den Dekker b, A. van den Bos b a Department of Physics, University of Antwerp (RUCA), Groenenborgerlaan 171, 2020 Antwerp, Belgium b Department of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands Received 1 August 2002 Abstract Atomic resolution transmission electron microscopy, even with an aberration free microscope, is only able to resolve and refine amorphous structures at the atomic level for very small foil thicknesses. Then, a precision of the order of 0:01 ( A is possible, but this may require long recording times, especially for light atoms. For larger thicknesses, amorphous structures can in principle only be resolved and refined using electron tomography. r 2003 Elsevier Science B.V. All rights reserved. PACS: r; þy; Dq; Er; Lp Keywords: Amorphous structure; Transmission electron microscopy; Atomic resolution; Precision; Parameter estimation; Electron tomography 1. Introduction As scientists manage to control the structure of materials on an ever finer scale, more and more materials are being developed with interesting properties which are mainly related to their nanostructure. Parallel to this, one sees an evolution in solid-state theory where materials properties are increasingly better understood from first principle theoretical calculations. The merging of these fields will enable materials science to evolve into materials design. However, this is not yet feasible since materials characterization techniques *Corresponding author. Tel.: ; fax: address: dvd@ruca.ua.ac.be (D. Van Dyck). can generally not locally determine atom positions in a nanostructure within sub- ( Angstrom precision. A precision of the order of 0:01 ( A is needed [1 3]. Only atomic resolution transmission electron microscopy (TEM) techniques seem to be appropriate to provide local information to atomic scale for aperiodic objects since the electrons interact sufficiently strongly with materials [4,5]. According to Henderson [6], of all particles, electrons provide the most information for a given amount of radiation damage, even several hundred times more than X-rays. This is in contrast to what is generally believed. Furthermore, the brightness of a focused probe in a modern electron microscope is higher than the brightest synchrotron [7]. In spite of the potential advantages of using electrons, 0:01 ( A precision of the atom positions is /03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi: /s (03)

2 28 D. Van Dyck et al. / Ultramicroscopy 98 (2003) not yet reached. In order to discuss possible future developments and to explore the physical limits of the technique, we will consider the most challenging object to study by atomic resolution TEM: the amorphous object. The resolution of the electron microscope has been investigated in detail in successive steps. In Section 2, Rayleigh and Rayleigh-like resolution criteria are discussed. These criteria are derived from the assumption that the human visual system needs a minimal contrast to discriminate two points in its composite intensity distribution. Therefore, they are expressed in terms of the width of the point spread function of the imaging instrument concerned. Next, in Section 3, deterministic model-based resolution is considered. Here, a priori knowledge about the image intensity distribution is taken into account in the form of a parametric model. However, the images are supposed to be noisefree. The unknown parameters, such as the positions of projected atoms, are measured by means of model fitting. Then, in Section 4, statistical model-based resolution is studied. It is taken into account that the observations fluctuate about their expectations due to the unavoidable presence of electron counting noise in the images. The parameters are measured by means of model fitting using parameter estimation methods. Hence, statistical model-based resolution is discussed in terms of the precision with which the parameters can be estimated. Finally, in Section 5, ultimate model-based resolution is contemplated. It is shown that depending on the observations, the solutions of the position estimates may be exactly coinciding. 2. Rayleigh and Rayleigh-like resolution The most commonly used definition of resolution goes back to Lord Rayleigh in 1874 [8]. Rayleigh estimated the minimal resolvable distance between two points of equal brightness which are imaged by a diffraction-limited imaging system. He states that the two point sources are just resolvable when the maximum of the intensity distribution produced by the first point source coincides with the first zero of the intensity distribution produced by the second point source [8]. Consequently, the Rayleigh resolution is given by the distance between the central maximum and the first zero of the diffraction-limited intensity point spread function of the imaging instrument concerned. The Rayleigh resolution criterion can be generalized to include point spread functions that have no zero in the neighborhood of their central maximum by taking the resolution as the distance for which the ratio of the value at the central dip in the composite intensity distribution to that at the maxima on either side is equal to This corresponds to the original Rayleigh resolution for a rectangular aperture. The Rayleigh resolution, or alternatively, the point resolution, will now be explained in mathematical terms. However, in order to keep the calculations simple, the point spread function will be assumed to be a two-dimensional Gaussian function of the following form: pðrþ ¼ 1 r2 exp 2pr2 2r 2 ; ð1þ where r is the absolute value of the two-dimensional vector r and r is the width of the Gaussian function. According to Rayleigh, the point resolution r p ; that is the smallest distance at which two points can be resolved, is given by the requirement that the value of the cross-section of the composite intensity distribution halfway between these two points is about 0.8 times the value at the maxima. Thus, 2exp r2 p 8r 2! ¼ 0:8; from which it follows that pffiffi r p E2 2 r: ð2þ ð3þ Alternatively, Rayleigh resolution can be discussed in Fourier space. Therefore, the point spread function, given in Eq. (1), will be twodimensionally Fourier transformed. The resulting function, which is called the transfer function, is given by PðgÞ ¼expð 2p 2 r 2 g 2 Þ; ð4þ where g is the absolute value of the two-dimensional spatial frequency vector g: The spatial

3 D. Van Dyck et al. / Ultramicroscopy 98 (2003) frequency corresponding with Rayleigh resolution is g p ¼ 1 E p 1 ffiffi : ð5þ r p 2 2 r At that spatial frequency the modulus of the transfer function, which is given in Eq. (4), is reduced to 8%. Thus, Rayleigh resolution can also be defined as the inverse of the spatial frequency where the transfer function is reduced to 8% of its peak value. Since Rayleigh s days, several other resolution criteria have been proposed which are similar to Rayleigh s. A notable example of such so-called classical criteria for two-point resolution is that of Sparrow [9]. Compared to Rayleigh resolution, which is based on presumed capabilities of the human visual system, the Sparrow resolution is based on a less subjective criterion which is valid for a hypothetical perfect imaging instrument. It states that the smallest resolvable distance between two points is that for which the minimum in the composite image intensity distribution just disappears. For this definition of resolution one has from Eqs. (1) and (3): pffiffiffi 2 rp r s E 2 : ð6þ Thus far, only the diffraction-limited point spread function of the imaging instrument has been taken into account for the derivation of Rayleigh and Sparrow resolution. However, for electron microscopy, this should be extended to include the point spread function describing the effect of thermal vibrations of the atom, the effect of the environment, and the detector [10]. Moreover, it has to be taken into account that the atoms are no point scatterers. Hence, Rayleigh and Sparrow resolution have to be extended from points to objects of finite size. As shown in Fig. 1, each effect contributing to the imaging process can be represented by a transfer function, which acts as a low pass filter. The transfer function of the electron microscope consists of a damping function, which is mainly due to chromatic aberration, and a phase shift, which causes the oscillations. Since there are many ways to get rid of the oscillations, such as, focal series reconstruction Fig. 1. Transfer functions of the different sub-channels of electron microscopic imaging. [11 16] and correction of the spherical aberration [17], the Rayleigh resolution of the electron microscope can be assumed to be given by the so-called information limit, which is proportional to the inverse of the highest spatial frequency that is still transferred with appreciable intensity. For simplicity, it will be assumed that the imaging process is linear. This requires that the interaction between the electron and the object is linear. If the object is a crystal, viewed along a zone axis, the electrostatic potential of all the atoms along the atom column superimpose, which makes the interaction very strong and highly nonlinear, so

4 30 D. Van Dyck et al. / Ultramicroscopy 98 (2003) that this approximation does not apply. In that particular case, due to the focusing effect of the successive atoms, the scattering is increased to higher angles. This effect is explained by the channelling theory [18 21]. However, for amorphous objects, the atoms are stacked in a disordered fashion, so that in projection their cores do not overlap, except by coincidence. As a result, the interaction remains linear for much larger object thicknesses. If the imaging is linear, all transfer functions have to be multiplied, or, equivalently, the point spread functions have to be convolved. If it is assumed that all constituent point spread functions are Gaussian, such as in Eq. (1), the resulting function is a Gaussian as well, with a Rayleigh resolution r p determined by r 2 p ¼ r2 A þ r2 T þ r2 EM þ r2 v þ r2 D ; ð7þ with r A the width of the electrostatic potential of the atom, representing the cross-section for elastic interaction with the atom, r T the Rayleigh resolution limited by thermal vibrations of the atom, r EM the Rayleigh resolution of the electron microscope, r v the Rayleigh resolution limited by the environment (vibrations and stray fields), and r D the Rayleigh resolution limited by the detector. Today, for the best electron microscopes, r EM is somewhat below 1 A ( [22 24]. By future instrumental developments one can improve Rayleigh and Sparrow resolution by improving the resolutions of all different sub-channels. However, a factor that one cannot improve is r A : It is important to notice that beyond a certain point, it will be useless to further improve the Rayleigh resolution of the electron microscope since scattering is dominant. Already now it is difficult to find suitable objects that can be used to demonstrate the true Rayleigh resolution r EM of an electron microscope. Consider, for example, amorphous silicon. From Fig. 1, it follows that r A is about 1 A: ( Therefore, for the best electron microscopes, it follows from Eq. (7) that for amorphous silicon: r p E1 ( A and from Eq. (6) that: r s E0:7 ( A: ð8þ ð9þ From the examples given in this section, it can be noticed that classical resolution criteria are expressed in terms of the width of the point spread function of the imaging instrument. A narrower point spread function corresponds to an improved Rayleigh or Sparrow resolution. However, for the study of amorphous structures in electron microscopy, the atoms themselves limit the classical resolution. One can go a step further using a model-based approach. This will be discussed in the following sections. 3. Deterministic model-based resolution Classical resolution criteria disregard the possibility of using a priori knowledge to extract analytic results from observations by means of model fitting [25]. In this section, a priori knowledge about the model will be taken into account. The model will be assumed to be parametric in the locations. It will be extended from two-peak models to one or more-peak models. Furthermore, the observations will be assumed to be noise-free. Then, instead of classical resolution, we will speak of deterministic model-based resolution. It will be shown that the relevant limits to deterministic model-based resolution are at least computational. However, if the model is inaccurate, which means that it systematically deviates from the exact noisefree observations, it will be shown that the relevant limits are both computational and fundamental. Imagine that Lord Rayleigh would image stars today. First, the image of one star, which can be considered to be a point object, will be observed. Now, there exists a model for the object, namely that it consists of a point. The point spread function of the telescope is also exactly known. So, it is known how an image of one star should look like. Thus, there is no interest in the detailed form of this image, but only in the position of the star. The only objective of the experiment is to determine this position. Obviously, in the absence of noise, numerically fitting the known one-peak model to the image with respect to the position parameter would result in a perfect fit. The resulting solution for this location would be exact,

5 D. Van Dyck et al. / Ultramicroscopy 98 (2003) and despite blurring no limit to location resolution would exist. This line of reasoning can be extended to position measurements of atoms in amorphous structures from noise-free electron microscopical observations. In a sense, one is looking for the optimum value of a criterion in a parameter space whose dimension is equal to the number of parameters to be measured, which is equal to 2n; if n is the total number of atoms. The factor 2 accounts for the x- and y-coordinate of the projected position. Each possible combination of the 2n parameters can be represented by a point in a 2n-dimensional space. The search for the global optimum of the criterion of goodness of fit in this space is an iterative numerical optimization procedure. However, the problem with which one can be faced is of computational kind. The existing optimization methods fail if the dimension of the parameter space is so high that one cannot avoid ending up at a local optimum instead of at the global optimum of the criterion of goodness of fit, so that the wrong structure is suggested. To solve this dimensionality problem, that is, to find a pathway to the global optimum, a good starting structure is required, that is, initial conditions should be available for the parameters. In other words, the structure has to be resolved. This corresponds to X-ray crystallography, where one first has to resolve a structure by using, for example, direct methods, and afterwards one has to refine the structure. Moreover, the computing time needed in order to reach convergence of the iterative procedure increases with the dimension of the parameter space. For the amorphous object, the number of parameters increases with thickness. Therefore, from a certain thickness onwards, it will be difficult to resolve the structure. For example, consider Figs. 2 and 3. In Fig. 2, the amorphous foil is thin, whereas in Fig. 3, it is thicker. Therefore, the number of projected atoms is larger in Fig. 3 than in Fig. 2. It is clear that it will be more likely to resolve the structure for the example given in Fig. 2 than for that corresponding to Fig. 3. In order to resolve the structure, it will be assumed that the distances between neighboring projected atom positions should be larger than or Fig. 2. Amorphous structure containing clearly separable projected atoms. Fig. 3. Amorphous structure containing severely overlapping projected atoms. equal to the Sparrow resolution r s : The reason for choosing this criterion is that the computer will then be able to distinguish the individual atoms, since the observations are assumed to be noisefree. However, it should be noticed that this criterion is not exact and therefore, it will only give rough guidelines. Suppose that the mean

6 32 D. Van Dyck et al. / Ultramicroscopy 98 (2003) concentration of atoms per cubic (angstrom is equal to V: Then, the mean concentration A of projected atoms= A ( 2 is given by A ¼ Vz; ð10þ where z is the thickness of the amorphous foil. On the other hand, if it is assumed that each projected atom occupies a circle with a diameter equal to the average distance d; averaged over distances between nearest-neighbor projected atoms, then AE 1 pðd=2þ 2: ð11þ From Eqs. (10) and (11), it follows that the thickness of the amorphous foil is approximately given by ze 4 pd 2 V : ð12þ In order to resolve the structure and therefore to avoid dimensionality problems, it will be assumed that the following condition is met: dx2r s : ð13þ The factor 2 is arbitrarily chosen. Requiring that d is larger than or equal to r s would not be sufficient. In that case, a substantial part of distances between neighboring atoms would be smaller than r s and hence one cannot resolve the structure. In principle, this can still occur if inequality (13) is fulfilled, but the probability that it occurs is less. Then, it follows from Eqs. (12) and (13), that: zp 1 pr 2 s V: ð14þ For amorphous silicon, it follows from Eq. (9) that r s is approximately equal to 0:7 A ( and furthermore, V is approximately equal to 0:05 atoms= A ( 3 : Hence, it follows from Eq. (14) that the amorphous silicon foil should not exceed thicknesses of the order of 13 A ( so as to avoid dimensionality problems. This thickness is rather small, which means that it is unrealistic to expect that atomic resolution TEM is able to resolve amorphous silicon samples with realistic foil thicknesses from only one projection [26]. It can thus be stated that the structure of a realistic amorphous object cannot be determined from one image alone. However, the situation can be improved drastically by using a tomographic technique in which the sample is tilted and many projections from different viewing directions are combined as shown in Fig. 4. The Fourier transform of a projection yields a section through the origin of the three-dimensional Fourier space. By combining many different projections, one can then reconstruct the whole Fourier space [27]. In this way an ideal microscope can resolve about 1 atom= A ( 3 ; which is sufficient to resolve amorphous structures. In the foregoing, it has been assumed that the model is exact. However, if the model is inaccurate, REAL SPACE 1 FOURIER SPACE 2 F p3 3 f(x,y) f p3 F p2 0 F p1 f p2 f p1 Fig. 4. The principle of tomography.

7 D. Van Dyck et al. / Ultramicroscopy 98 (2003) the estimated position parameters may deviate from the true positions, even if the observations would be noise-free. This is called systematic error. Obviously, systematic errors would put fundamental limits to deterministic model-based resolution. In this section, it has been shown that deterministic model-based resolution is computationally limited. For amorphous structures, studied by atomic resolution TEM, computational problems can only be overcome if the thickness is not too large. For larger thickness, atomic resolution TEM combined with electron tomography is needed. Moreover, if the model is inaccurate, the resolution is also fundamentally limited, since then, a systematic error is introduced. 4. Statistical model-based resolution In Section 3, it has been assumed that the observations are noise-free. However, in any reallife experiment, the observations will contain errors. Then, the resolution depends fundamentally on the signal-to-noise ratio in the detected image. In this section, the approach which is followed to express the resolution is based on statistical parameter estimation theory. In the remainder of this paper, this approach is called statistical model-based resolution. It will be shown that the relevant limits are both computational and fundamental. In accordance with the available literature, in what follows, two kinds of resolution are distinguished: single-source [28] and differential (twosource) [29] resolution. Single-source resolution is defined as the instrument s capacity to determine the position of a point object that is observed in a background of noise. Differential resolution is defined as the instrument s capability to determine the separation of two point objects. Suppose that one has a CCD camera that is able to count the individual photons forming the image of a single point object or of two point objects. The images as measured by this camera appear as in Fig. 5 or as in Fig. 6 for single or two point objects, respectively. The noise on these images stems from the counting statistics. The position number of electron counts N*p(x,y=0) experiment x Fig. 5. Simulation experiment of the image of a point as measured by a CCD camera ( pixels), with pixel size Dx ¼ Dy ¼ 1: The point spread function used is a twodimensional normalized Gaussian: pðx; yþ ¼ð1=pr 2 Þ expð x 2 y 2 =r 2 Þ; with r ¼ 30: The experimental as well as the expectation values N pðx; yþ are shown within the section y ¼ 0: The number of imaging particles N is equal to number of electron counts N*p 1 (x,y=0) N*p 2 (x,y=0) experiment x Fig. 6. The results of a computer simulated image of two neighboring points. The experimental as well as the expectation values of the individual peaks ðn p 1;2 ðx; yþþ are plotted within the section y ¼ 0; where p 1;2 ðx; yþ are two-dimensional normalized Gaussian point spread functions, as in Fig. 5. The number of imaging particles 2N is equal to The peaks are clearly separable. parameters can be estimated by numerically fitting the known parameterized mathematical model to the images with respect to the component positions, in the same way as expressed in Section 3. However, if one would repeat this experiment several times, one would, due to the statistical nature of the observations, find slightly different values for the position or the distance estimates for single or two point objects, respectively. The

8 34 D. Van Dyck et al. / Ultramicroscopy 98 (2003) position and distance estimates are statistically distributed about their mean values. Then, the obvious criterion to quantify statistical modelbased resolution is the precision of the estimate, which is given by the variance of this distribution, or, by its square root, the standard deviation. In a sense, the standard deviation is the error bar on the position or on the distance. Applying statistical parameter estimation theory, the attainable precision can be adequately quantified in the form of the so-called Cram!er Rao lower bound (CRLB) [30]. This is a lower bound on the variance of any unbiased estimator of a parameter. It means that the variance of different estimators, such as, the least squares or the maximum likelihood estimator, can never be lower than the theoretical CRLB on the variance. Fortunately, the maximum likelihood estimator attains the CRLB asymptotically, that is, if the number of observations is sufficiently large. Therefore, the maximum likelihood estimator is asymptotically most precise. The attainable precision with which position and distance parameters of one or two point objects can be measured has been investigated. In Refs. [31,32], it is explained for one- and two-dimensional observations, respectively. Here, the main results will be presented. These results are valid for both one- and twodimensional observations. First, we will consider the position measurement of one isolated point object. If the point spread function is assumed to be Gaussian and defined by Eq. (1) and if the total number of imaging particles is N; the lower bound on the standard deviation s LB of the coordinates estimates of the position, that is, the square root of the CRLB, is given by s LB Ep r ffiffiffiffi; ð15þ N or, from Eq. (3) r p s LB E p ffiffiffiffiffiffiffi: ð16þ 2 2N Thus, the precision with which the position can be determined is a function of both the Rayleigh resolution r p and the number of imaging photons N: If N is large, the precision can be orders of magnitude higher than the point resolution r p : number of electron counts N*p 1 (x,y=0) N*p 2 (x,y=0) experiment x Fig. 7. The results of a computer simulated image of two neighboring points with a Gaussian point spread function, where p 1;2 ðx; yþ are two-dimensional normalized Gaussian point spread functions, as in Fig. 5. The experimental as well as the expectation values of the individual peaks ðn p 1;2 ðx; yþþ are plotted within the section y ¼ 0: The number of imaging particles 2N is equal to The peaks overlap severely. Second, we will consider the distance measurement of two neighboring point objects. Since the imaging is linear, the image consists of the superposition of the two corresponding point spread functions as simulated in Figs. 6 and 7. The total number of imaging photons used in the simulation is equal to 2N: This means that, in agreement with Fig. 5, the number of photons per peak is equal to N: Similar to one peak, the distance d between the two peaks can be measured by numerically fitting the known parameterized mathematical model to the images. Here, the results are briefly sketched. For details we refer to Refs. [31,32]. * d > r p =2 This is shown in Fig. 6. Now, the distance d between the atoms is larger than the half of the Rayleigh resolution. Then, the lower bound on the standard deviation s MIN on the distance d is minimal, independent of d; and given by s MIN E r p ffiffiffiffi: ð17þ 2 N Thus, from Eq. (16) it follows that the CRLB on the variance of the distance is twice the CRLB on the variance of the position of an isolated point object. The reason for this is that

9 D. Van Dyck et al. / Ultramicroscopy 98 (2003) the coordinate estimates of neighboring atoms are uncorrelated. * dpr p =2 This is shown in Fig. 7. Now, the distance d between the atoms is smaller than the half of the Rayleigh resolution. The lower bound on the standard deviation s LB increases inversely proportionally to the distance d; following approximately the relation s LB ðdþe s MINr p ; ð18þ 2d where s MIN is given by Eq. (17). In Fig. 8, the lower bound on the standard deviation of the distance d is shown as a function of the distance. Until now, the model has been assumed to be exact. Then, it is clear that statistical model-based resolution is limited by the lower bound on the standard deviation. This limit is fundamental. Moreover, if the model is inaccurate, the introduced systematic error determines another fundamental limit to statistical model-based resolution. Apart from these fundamental limits, computational limits exist as well, just as for deterministic model-based resolution. The computing time needed in order to fit the model to the images with respect to the locations increases with the total number of parameters to be measured. Furthermore, the optimization methods fail if the structure is not resolved. However, as suggested in Section 3, electron tomography can overcome this problem. Note that for electron tomography, the dose-fractionation theorem states that a statistically significant three-dimensional image can be computed from statistically insignificant twodimensional projections, as long as the total dose that is distributed among these projections is high enough that it would have resulted in a statistically significant projection, if applied to only one image [33,34]. Moreover, the lower bound on the standard deviation of the distance that can be achieved from electron tomography experiments has been investigated in Ref. [32]. Now, we can tackle the question whether the positions of the atoms of an amorphous object can be measured with 0:01 A ( precision by means of atomic resolution TEM. It will be assumed that the model is exact. Moreover, following the conclusions of Section 3, it will be assumed that the thickness of the amorphous object is not too large. This allows one to resolve the structure. In Section 3, an upper bound for the thickness has been derived for noise-free observations from the assumption that the distance between neighboring projected atoms equals r s : If the noise would be taken into account, the upper bound for the thickness would become smaller since then the distance between neighboring projected atoms should even be larger than r s at this thickness. The quantification of this upper bound is outside Lower bound on standard deviation of d s min ρ s ρ p d Fig. 8. The lower bound on the standard deviation of the distance d as a function of the distance.

10 36 D. Van Dyck et al. / Ultramicroscopy 98 (2003) the scope of this paper. However, since the distance between neighboring projected atoms is larger than r s ; it can be shown that the standard deviation s LB of the coordinate estimates of the position of a projected atom is independent of the positions of neighboring projected atoms. Thus, the atoms can be assumed to be isolated. Therefore, we will consider only a single atom. The electrons that interact with the atom are used to form the image. In atomic resolution TEM, there are two imaging modes: dark-field (DF) imaging and bright-field (BF) imaging. In DF imaging, the non-interacting electrons are eliminated from detection, such as in annular dark field scanning transmission electron microscopy (ADF STEM). In BF imaging, the non-interacting electrons contribute to the background intensity in the image, such as in high-resolution transmission electron microscopy (HRTEM). In the weak phase object (WPO) approximation, this background is uniform. In the derivation of the standard deviation s LB of the coordinate estimates of the position of the projected atom, it will be assumed, for simplicity, that the image of the projected atom is a twodimensional Gaussian function without or with background for DF and BF imaging, respectively. The Gaussian function is defined by Eq. (1), where the width r is proportional to the Rayleigh resolution, which for electron microscopy is given by Eq. (7). Notice that r includes the width of the atom itself. Although the obtained results are derived for Gaussian intensity distributions, it will be shown that they provide physical insight for more realistic intensity distributions, which are derived for a WPO. The WPO approximation applies for amorphous structures. In this sense, they are useful rules of thumb. * DF imaging For DF imaging, the standard deviation s LB of the coordinate estimates of the position of the projected atom is given by Eq. (15) or (16): s LB Epffiffiffiffiffiffiffiffiffiffi r E r p p ffiffiffiffiffiffiffiffiffiffiffiffiffi; ð19þ ds DF 2 2ds DF where d is the number of incident electrons per square (angstrom and s DF is the cross-section [35], that is, the effective area of the atom as seen by the incident electrons, which is a measure of the strength of the interaction. Hence N ¼ ds DF is the number of interacting electrons. Moreover, r p is defined by Eq. (7). In Fig. 9, the exactly calculated s LB and the rule of Lower bound on standard deviation of the position S LB ρ/(δσ DF ) aperture radius (Å -1 ) Fig. 9. The lower bound on the standard deviation s LB of the position of an isolated projected silicon atom for ADF STEM imaging as a function of the objective aperture radius. The exactly calculated s LB as well as the rule of thumb, which is given by the right-hand side member of Eq. (19), are computed.

11 D. Van Dyck et al. / Ultramicroscopy 98 (2003) thumb, which is given by the right-hand side member of Eq. (19), are computed for ADF STEM as a function of the aperture angle for a WPO. A realistic ADF STEM model has been assumed, following [36]. For the evaluation of the rule of thumb, the width r is taken equal to the radius of the intensity distribution corresponding to 39% of the total intensity. This choice follows from the analogy for a Gaussian peak, where r can be regarded as such a radius as well. * BF imaging For BF imaging, it can be shown that the standard deviation s LB of the coordinate estimates of the position of the projected atom is given by pffiffiffi 2 r s LB EpffiffiffiffiffiffiffiffiffiffiE r p p ffiffiffiffiffiffiffiffiffiffi: ð20þ ds BF 2 ds BF For a Gaussian peak, defined by Eq. (1), which is added on a constant background 1, r is the width of this peak and s BF is the area under the peak square. Empirically, it has been found that for a more realistic BF model, r and s BF can be derived from the square of the BF intensity distribution above the background. Then, pinffiffi analogy to the Gaussian peak, r is equal to 2 times the radius corresponding to 39% of the total intensity distribution above the background and s BF is the area under the distribution above the background. Following Reimer [35], for HRTEM in the WPO approximation, s BF can be interpreted as being proportional to the cross-section for elastic scattering, but here we include the point spread function of the electron microscope. In Fig. 10, the exactly calculated s LB and the rule of thumb, which is given by the right-hand side member of Eq. (20), are computed for HRTEM as a function of the spherical aberration constant C s for a WPO. The HRTEM model used is given, for example, in Ref. [37]. Although the discrepancy between the exact expression and the rule of thumb is larger for BF than for DF imaging, the behavior as well as the order of magnitude are well described. From the evaluation of Eqs. (19) and (20), it may be discussed which conditions are to be met in order to determine the positions of the atoms of an amorphous object with 0:01 A ( precision. For example, for DF imaging of a silicon atom, using high angle ADF STEM, assuming a cross-section of 0:01 A ( 2 ; and taking r p equal to the width of Lower bound on standard deviation of the position S LB ρ/(δσ BF ) C s (mm) Fig. 10. The lower bound on the standard deviation s LB of the position of an isolated projected silicon atom for HRTEM imaging as a function of the spherical aberration constant C s : The exactly calculated s LB as well as the rule of thumb, which is given by the righthand side member of Eq. (20), are computed.

12 38 D. Van Dyck et al. / Ultramicroscopy 98 (2003) the atom, that is, 1 ( A; it follows from Eq. (19) that, in order to obtain 0:01 ( A precision, the required dose is 10 5 electrons= ( A 2 : For BF imaging of a silicon atom, assuming a cross-section of 0:1 ( A 2 and a resolution of 1 ( A; it follows from Eq. (20), that in order to obtain 0:01 ( A precision, the required dose is 2: electrons= ( A 2 : With present electron microscopes, the required incident electron dose= ( A 2 is, both for DF and BF imaging, only achievable at the expense of long recording times. Finally, an important remark should be made on the accelerating voltage. At the accelerating voltages of kev used for HRTEM, the amorphous structure is heavily damaged by the electron beam during the experiment. The authors believe that much lower voltages are needed for reliable quantitative studies. However, at these low accelerating voltages, correction of both spherical and chromatic aberration are needed by using a spherical aberration (C s ) corrector and either a monochromator or a chromatic aberration (C c ) corrector. The first reason is to improve the Rayleigh resolution of the electron microscope r EM so as to keep the Rayleigh resolution r p close to the width of the electrostatic potential of the atom. The second reason is to improve the precision. This is shown in Fig. 11. The exactly calculated lower bound on the standard deviation s LB of the position of an isolated silicon atom is computed as a function of C s for a 50 kev electron microscope. This is done for an electron microscope without monochromator and without C c corrector, for a microscope with monochromator, and for a microscope with C c corrector. The energy spread, expressed as full width at half maximum, is assumed to be equal to 1:7 ev without monochromator, which is a practical value for a LaB 6 electron gun [10], and equal to 0:2 ev with monochromator. The chromatic aberration constant is assumed to be equal to 1:3 mm without C c corrector and equal to 0:0 mm with C c corrector. At each setting, the recording time is kept constant, assuming that specimen drift puts a practical limit to the experiment. As a consequence the electron dose is higher in the absence of a monochromator. Furthermore, the defocus is Lower bound on standard deviation of the position without monochromator, without C s -corrector with monochromator with C s -corrector C s (mm) Fig. 11. The exactly calculated lower bound on the standard deviation s LB of the position of an isolated projected silicon atom for HRTEM imaging as a function of C s for a 50 kev electron microscope. This is done for an electron microscope without monochromator and without C c corrector, for a microscope with monochromator, and for a microscope with C c corrector. The energy spread, expressed as full-width at half-maximum, is assumed to be equal to 1:7 ev without monochromator and equal to 0:2 ev with monochromator. The chromatic aberration constant is assumed to be equal to 1:3 mm without C c corrector and equal to 0:0 mm with C c corrector. At each setting, the recording time is kept constant and the defocus is adjusted to Scherzer s.

13 D. Van Dyck et al. / Ultramicroscopy 98 (2003) adjusted to Sherzer s. From Fig. 11, it is clear that the precision is highest, that is, s LB is lowest, when using both a C s and a C c corrector. Moreover, the C c corrector obtains better results than the monochromator because of the higher electron dose. In this figure, it is also shown that in the absence of a monochromator or a C c corrector, the precision deteriorates if the spherical aberration is corrected. 5. Ultimate model-based resolution In the preceding section, the Cram!er Rao lower bound on the precision of the estimated distance of two peaks was used as a resolution criterion. Such a criterion makes sense only if an estimator exists that actually achieves this bound. Such an estimator exists in the form of the so-called maximum likelihood estimator. This estimator produces estimates that are normally distributed about the true values of the parameters with a variance equal to the Cram!er Rao lower bound. However, this is true only if the number of observations used by the maximum likelihood estimator is sufficiently large. For small numbers of observations, the estimates have statistical properties that are very different and will now be described. A detailed description is given in Refs. [30,38,39] Two types of observations Central in this section is the space of observations. This is the Euclidean space where every coordinate direction corresponds to a particular observation. Therefore, the dimension of the space of observations is equal to the number of observations and a particular set of observations is represented by a single point in the space. For example, the observations shown in Fig. 7 correspond to a single point in the space of observations. It has been shown in the references mentioned that two types of sets of two-peak observations may occur. If the two-peak model is fitted to the first type of observations, distinct values for the locations are obtained. Then the distance of the peaks is different from zero. If, on the other hand, the two-peak model is fitted to observations of the second type, the estimates for the locations exactly coincide. Then the distance of the peaks is equal to zero. This means that the peaks exactly coincide. It is clear that from the first type of observations both peaks are resolved in the sense that they occur at distinct locations. On the other hand, for the second type of observations, the peaks occur at the same location. Therefore, the peaks add up to form a single peak and cannot be resolved. From these considerations, it follows that the (hyper)surface separating both types of observations in the space of observations may be used as a resolution criterion. If the point representing a particular set of observations is on the one side of the hypersurface the peaks are resolved but if it is on the other side they are not. This perhaps unexpected behavior of the solutions for the locations of the peaks has to do with the structure of criteria of goodness of fit for the estimation of parameters such as the locations. Structure is the pattern of the stationary points of the criterion, where a stationary point is a point where the gradient is equal to zero, such as, a minimum or maximum. The structure is important since the solution of the model fitting problem is represented by a stationary point: the absolute minimum of the criterion of goodness of fit. It has been found that for observations made on overlapping peaks, two different structures may occur and that these correspond to distinct or coinciding solutions for the locations, respectively. Which of both structures occurs, depends on the particular realization of the observations Consequences for resolution If the observations are statistical, they are, in the space of the observations, distributed about the point representing their expectations. This expectation point, therefore, represents ideal exact observations. If the model fitted is correctly specified, it perfectly fits to these exact observations. Then the solutions found for the locations are distinct and equal the hypothetical true locations, no matter how closely located the peaks. Therefore, the expectation point will be located on

14 40 D. Van Dyck et al. / Ultramicroscopy 98 (2003) the two-peak side of the separating hypersurface and not on the one-peak side, since the solutions are distinct. Observations distributed about the expectation point will only with a certain probability be on the two-peak side of the separating hyperplane. It is concluded that resolution should be described in terms of probability of resolution. The distance of the expectation point and the separating hypersurface becomes smaller and smaller as the hypothetical true locations are closer. This means that the probability that a set of observations is located on the one-peak side of the separating hyperplane increases and the probability of resolution decreases correspondingly. The separating hypersurface is computed from and, therefore, specific of the peak model fitted. In this paper, this is the Gaussian peak. The expectation point, however, is specific of the model actually present in the observations. If the model fitted and the model present differ, the expectation point may even be located on the one-peak side of the separating hyperplane. This implies that the peaks are not resolved even if the statistical errors in the observations are very small. It is clear that, ultimately, resolution is limited by errors, both systematic (modelling) errors and non-systematic (statistical) errors. This limit is fundamental: no increase of computer power nor improved software can remove it. It would be present, even if initial conditions would be available for the locations of all peaks and sufficient computer power would be available for fitting them. Removing these limits requires more and preferably different observations of the same peaks. The most obvious way to obtain these is, in the authors opinion, tomography. 6. Discussion and conclusions In this paper, it has been investigated whether atomic resolution TEM is able to resolve and refine amorphous structures from one projected image. It has been shown that this is only possible for very small foil thickness. Then, a precision of the order of 0:01 ( A is possible, but this may require long recording times, especially for light atoms. For larger thicknesses, in the authors opinion, amorphous structures can only be resolved and refined using electron tomography. The results have been obtained from investigating the resolution of the electron microscope in detail in successive steps. First, classical resolution criteria, such as, Rayleigh s and Sparrow s, have been discussed for TEM, which are expressed in terms of the width of the point spread function. The narrower the point spread function, the higher the resolution. However, it has been shown that for the study of amorphous objects by TEM, the width of the electrostatic potential of the atoms, as well as, the effect of thermal vibrations of the atoms, the environment, and the detection have to be taken into account. As a result of these extensions, it should be concluded that the atoms themselves limit the classical resolution. Next, a priori knowledge about the image intensity distribution is taken into account. In fact, classical resolution criteria are concerned with calculated images, that is, noise-free images exactly describable by a known parameterized mathematical model. Then, the unknown parameters, such as the positions of projected atoms, can be measured by means of model fitting. The practical limits to this so-called deterministic model-based resolution are of computational kind. For amorphous structures studied by atomic resolution TEM, these computational problems can only be overcome if the thickness is very small. An upper bound to the thickness is derived, which for amorphous silicon is of the order of 10 ( A: For realistic thicknesses, electron tomography is needed. Moreover, fundamental limits exist if the model is inaccurate, since then, systematic errors are introduced. Then, unavoidable noise in the observations is taken into consideration, that is, detected images instead of calculated images are considered. The fundamental limit to this so-called statistical model-based resolution is determined by the lower bound on the standard deviation with which the positions of, or, the distance between, projected atoms can be measured using parameter estimation methods. Moreover, if the model is inaccurate, systematic errors determine fundamental limits as well. Apart from these fundamental

15 D. Van Dyck et al. / Ultramicroscopy 98 (2003) limits, computational limits exist, just as for deterministic model-based resolution. They can only be overcome if the thickness of the amorphous foil is very small, even smaller than the upper bound which is derived for deterministic model-based resolution. For such small foil thicknesses a precision of the order of 0:01 Ais ( possible, but this may require long recording times, especially for light atoms. Furthermore, it is shown that for low accelerating voltages, which are needed in order to avoid radiation damage, both correction of spherical and chromatic aberration is needed. Finally, ultimate model-based resolution is considered. Depending on the observations, the solutions of the position estimates may be exactly coinciding. It is shown that, ultimately, resolution is limited by errors, both systematic and nonsystematic errors. This limit is fundamental: no increase of computer power nor improved software can remove it. It would be present, even if the structure is resolved and sufficient computer power would be available for fitting them. Removing these limits requires more and preferably different observations of the same structure. Acknowledgements The authors would like to thank Dr. C. Kisielowski for fruitful discussions related to this work. The research of Dr. A.J. den Dekker has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. References [1] D.A. Muller, Core level shifts and grain boundary cohesion, in: G.W. Bailey (Ed.), Proceedings Microscopy and Microanalysis 1998, Atlanta, GA, Vol. 4, Springer, New York, 1998, pp [2] D.A. Muller, Ultramicroscopy 78 (1999) 163. [3] C. Kisielowski, E. Principe, B. Freitag, D. Hubert, Physica B (2001) [4] H. Fujita, N. Sumida, Usefulness of electron microscopy, in: F.E. Fujita (Ed.), Springer Series in Materials Science Physics of New Materials, Vol. 27, Springer, Berlin, 1994, pp [5] J.C.H. Spence, Mater. Sci. Eng. R26 (1999) 1. [6] R. Henderson, Quart. Rev. Biophys. 28 (1995) 171. [7] L.M. Brown, Inst. Phys. Conf. Ser. 153 (1997) 17. [8] J.W. Strutt, Investigations in optics, with special reference to the spectroscope, in: Scientific papers by John William Strutt, Baron Rayleigh, Vol. 1, Cambridge University Press, Cambridge, 1899, pp [9] C.M. Sparrow, Astrophys. J. 44 (1916) 76. [10] A.F. de Jong, D. Van Dyck, Ultramicroscopy 49 (1993) 66. [11] P. Schiske, Image processing using additional statistical information about the object, in: P. Hawkes (Ed.), Image Processing and Computer-aided Design in Electron Optics, Academic Press, London, 1973, pp [12] W.O. Saxton, Computer Techniques for Image Processing in Electron Microscopy, Academic Press, New York, 1978, pp (Chapter 9). [13] D. Van Dyck, W. Coene, Optik 77 (3) (1987) 125. [14] D. Van Dyck, M. Op de Beeck, W. Coene, Optik 93 (3) (1993) 103. [15] W.M.J. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64 (1996) 109. [16] A. Thust, W.M.J. Coene, M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64 (1996) 211. [17] H. Rose, Optik 85 (1) (1990) 19. [18] A. Howie, Philos. Mag. 14 (1966) 223. [19] D. Van Dyck, J. Danckaert, W. Coene, E. Selderslaghs, D. Broddin, J. Van Landuyt, S. Amelinckx, The atom column approximation in dynamical electron diffraction calculations, in: W. Krakow, M. O Keefe (Eds.), Computer Simulation of Electron Microscope Diffraction and Images, The Minerals, Metals & Materials Society, Warrendale, 1989, pp [20] S.J. Pennycook, D.E. Jesson, Ultramicroscopy 37 (1991) 14. [21] D. Van Dyck, J.H. Chen, Solid State Commun. 109 (1999) 501. [22] M.A. O Keefe, C.J.D. Hetherington, Y.C. Wang, E.C. Nelson, J.H. Turner, C. Kisielowski, J.-O. Malm, R. Mueller, J. Ringnalda, M. Pan, A. Thust, Ultramicroscopy 89 (2001) 215. [23] C. Kisielowski, C.J.D. Hetherington, Y.C. Wang, R. Kilaas, M.A. O Keefe, A. Thust, Ultramicroscopy 89 (2001) 243. [24] P.E. Batson, N. Dellby, O.L. Krivanek, Nature 418 (2002) 617. [25] A.J. den Dekker, A. van den Bos, J. Opt. Soc. Am. A 14 (1997) 547. [26] J.M. Cowley, The quest for ultra-high resolution, in: X.F. Zhang, Z. Zhang (Eds.), Progress in Transmission Electron Microscopy Concepts and Techniques, Vol. 1, Springer, Berlin, 2001, pp [27] J. Frank, Electron Tomography Three-Dimensional Imaging with the Transmission Electron Microscope, Plenum Press, New York, [28] O. Falconi, J. Opt. Soc. Am. 54 (1964) [29] O. Falconi, J. Opt. Soc. Am. 57 (1967) 987. [30] A. van den Bos, A.J. den Dekker, Resolution reconsidered conventional approaches and an alternative, in: P.W. Hawkes (Ed.), Advances in Imaging and Electron