- Email: [email protected]
HREM: ultimate performances
Thin Solid Films 319 Ž1998. 148–152
HREM: ultimate performances D. Van Dyck
)
Department of Physics, UniÕersity of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
Abstract The possibilities and limitations of HREM are discussed with respect to accurate structure derivation. It is shown that direct holographic methods are useful to yield approximate structure models that can be used as a start for final quantitative refinement. q 1998 Published by Elsevier Science S.A. Keywords: Holographic; Electron; Particle
1. The electron microscope as a communication channel The microstructure of matter can only be studied by interaction with particles, which carry information from the object to the observer. For this purpose, electrons are extremely useful since, as compared to other particles, they are easy to generate, easy to accelerate, easy to deflect and easy to detect. Information from the bulk of an object can be obtained by scattering with high energy electrons ŽG 100 keV. as is the case in high-resolution electron microscopy ŽHREM. and scanning transmission electron microscopy ŽSTEM.. In an ideal scattering experiment, the state of the electron is carefully determined immediately before and after the interaction with the object Ži.e., in the planes A and B in Fig. 1.. From the change in the electron state one can deduce information about the interaction and hence about the object itself. The plane A characterizes the illumination condition and the plane B the detecting condition. The states of the electron can be determined either in real space or reciprocal space. Full flexibility for converting the electron states in the planes A and B from real to reciprocal space or vice versa can be obtained by placing two lenses Žor two lens combinations., one at each side of the object, i.e., the condenser and objective lens ŽFig. 1.. All existing combinations of illumination and detection are listed in Table 1. Due to reciprocity Žsymmetry of time reversal. HREM and STEM are equivalent. In a sense, both techniques have the same configuration ŽFig. 1. if the z-axis is inverted.
)
The electron microscope can thus be described as a communication channel in which each electron carries two degrees of spatial information. The most performant high resolution electron microscopes ŽHREM. nowadays achieve an interpretable resolution of about 0.2 nm so that structural details of matter can be visualised almost up to the atomic scale. If a resolution of 0.1 nm could be reached, most of the individual columns of a structure could be resolved, which would make HREM an extremely useful technique for the study of new materials. However, the power of the technique is still severely limited by the problem of the interpretation of the images. Due to the complex imaging process, details in the observed images do not necessarily correspond to features in the structure of the objects. In the past, reliable information was obtained by comparing simultaneously the experimental images with computer simulated ones for known structure models. However, this trial and error technique
Corresponding author.
0040-6090r98r$19.00 q 1998 Published by Elsevier Science S.A. All rights reserved. PII S 0 0 4 0 - 6 0 9 0 Ž 9 7 . 0 1 1 1 1 - 5
Fig. 1. Ideal scattering experiment.
D. Van Dyck r Thin Solid Films 319 (1998) 148–152
requires a large amount of prior information obtained by other techniques. Recently, new direct methods have been developed that allow to obtain quantitative structural information of solids at the atomic level.
2. Image formation In general, the effect of imperfect imaging can be described using a transfer function which has two major effects: the spatial frequencies are altered and above a certain threshold they are damped. The highest spatial frequency that is still transferred above the noise determines the effective resolution of the instrument. In electron microscopy, the image formation process is somewhat more complicated. Since the imaging in an electron microscope is coherent, the object function, transfer function and image function are all complex valued, while the recorded images only show intensities. In the microscope, the electron suffers phase shifts due to the spherical aberration Cs of the lens and due to the defocus d f. For a special focus, the effect of the spherical aberration can be compensated somewhat by defocus so that for a range of spatial frequencies the phase shift is nearly constant and equal to pr2. The phase transfer function is depicted in Fig. 2. Two types of resolution can be defined. First, the cut-off point where the phase transfer function becomes zero. Up to this point, the phases are nearly constant so that the information is transferred in a forward direction. In this way, the image of a very thin object can be interpreted directly in terms of the structure of the object up to this resolution. Another definition of resolution is the point where information becomes comparable to the noise content. This is called the information limit. It is mainly determined by incoherent effects such as the spatial and temporal incoherence. When one uses a field emission source, which can be considered as an electron laser gun,
Fig. 2. Imaginary part of a typical transfer function at Scherzer focus for a traditional LaB 6 filament and a Field Emission Gun. Both have the same point resolution, but the information limit, i.e., the maximum transferable frequency, is greatly improved with the FEG source.
149
the information limit can be reduced up to 0.08 nm, while the point resolution is of the order of 0.17 nm. At this limit, the ultimate resolution is not limited by the microscope but rather by the object. However, the high resolution details are in the rapidly oscillating part of the transfer function so that image processing is required to recover this information.
3. Interpretation of the images The ultimate goal of high resolution electron microscopy is to determine quantitatively the atomic structure of an object. High resolution images are then to be considered as data planes from which the structural information has to be extracted. However, this structural information is usually hidden in the images and cannot easily be assessed. Therefore, a quantitative approach is required in which all steps in the imaging process are taken into account. Ideally, quantitative extraction of information should be done as follows: one has a model for the object, for the electron object interaction, microscope transfer and for the imaging process. The object model that describes the interaction with the electrons consists of the assembly of the electrostatic potentials of the constituting atoms. Since for each atom type the electrostatic potential is known, the model parameters then reduce to atom numbers and coordinates, Debye Waller factors, object thickness and orientation Žif inelastic scattering is neglected.. The imaging process is characterised by a small number of unknown Žor not exactly known. parameters such as defocus, spherical aberration, etc. These model parameters now have to be determined from the experiment. The parameters can be estimated from the fit between the theoretical images and the experimental images. The goodness of the fit is evaluated using a criterium such as likelihood, mean square difference or R-factor Žcf. X-ray crystallography.. For each set of parameters of the model, one can calculate this goodness of fit, so as to yield a fitness function in parameter space. The parameters for which the fitness is optimal then yields the best estimates that can be derived from the experiment. In a sense, one is searching for a maximum Žor minimum depending on the criterium. of the fitness function in the parameter space, the dimension of which is equal to the number of parameters. The best estimates for the model parameters are obtained by using a Maximum Likelihood criterium ŽML.. It is known Že.g., Ref. w1x. that if there exists a Minimum Variance Bound Žor Cramer–Rao Bound. estimator, it is given by the Maximum Likelihood method. ŽThe least squares estimator is sub optimal and can only be used under specific assumptions.. The probability that the model parameters are a n 4 given that the experimental outcomes are n i 4 can be calculated from Bayesian statistics. It is
D. Van Dyck r Thin Solid Films 319 (1998) 148–152
150
Fig. 3. Scheme of the refinement procedure.
called the likelihood function. pŽ n i 4 r a n 4. is the probability that the measurement yields the values n i 4 given that the model parameters are a n 4 . This probability is given by the model. pŽ a n 4. is the prior probability that the set of parameters a n 4 occurs. If no prior information is available all pŽ a n 4. are assumed to be equal. For instance, in case of HREM, pŽ n i 4 r a n 4. represents the probability that n i electrons hit the pixel x i in the image if all the parameters of the model Žobject structure and imaging parameters. i.e., n i then represents the measured intensity, in number of electrons, of the pixel i. L can then be considered as a fitness function. In principle, the search for the best parameter set is then reduced to the search for optimal fitness in parameter space. This search can only be done in an iterative way as schematised in Fig. 3. First one has a starting model, i.e., starting value for the object and imaging parameters a n 4 . From this one can calculate the experimental outcome pŽ n i 4 r a n 4.. This is a classical image simulation. ŽNote that the experiment can also be a series of images andror diffraction patterns. From the mismatch between experimental and simulated images one can obtain a new estimate for the model parameters Žfor instance using a gradient method. which can then be used for the next iteration. This procedure is repeated until the optimal fitness Ži.e., optimal match. is reached.
parameters is small, as is the case for small unit cell crystals. In some very favourable cases, the number of possible models, thanks to prior knowledge, is discrete and very small so that visual comparison is sufficient. These cases were the only cases where image simulation could be meaningfully used in the past. In high resolution electron microscopy, the dimensionality problem can be solved by deblurring the information, so as to unscramble the influence of the different object parameters in the image. In this way, the structural parameters can be uncoupled and the dimension of the parameter space reduced. This can be achieved in different ways: high voltage microscopy, correction of the microscopic aberrations or direct holographic methods for exit wave and structure reconstruction. Therefore, direct holographic reconstruction methods will provide an approximate model that can be used as a starting point for a final refinement by fitting with the original images that is sufficiently close to the global maximum so as to guarantee convergence. Direct methods aim at deblurring the information in the images. Since only the intensity is recorded, the phase has been lost. For the problem of the phase retrieval, the focus variation method has been proposed, in which the focus is used as an extra parameter. Images are captured at very close focus values so as to collect all information in the 3D image space. Each image contains linear and nonlinear information. By Fourier transforming all 3D image space, the linear information of all images is superimposed onto a sphere, while the nonlinear contribution is distributed all over the 3D frequency space. By separating the information on this so-called Ewald sphere, the imaging problem is linearised to a great extent Žsee Fig. 4.. Therefore, in a first approximation, the phase can be retrieved using a
4. Direct methods A major problem now is that the structural information of the object can be strongly delocalised by the image transfer in the electron microscope so that the effect of the structural parameters is completely scrambled in the high resolution images. Due to this coupling, one has to refine all parameters simultaneously which poses a combinatorial problem. Indeed, the dimension of the parameter space becomes so high that even with advanced optimisation techniques such as genetic algorithms, simulated annealing, tabu search, etc., one cannot avoid ending in local optima. The problem is only manageable if the number of
Fig. 4. Schematical representation of the phase retrieval procedure using a through focus series of images, recorded with a Slow Scan CCD camera and processed on a fast computer system. The paraboloid which contains the linear information is also shown.
D. Van Dyck r Thin Solid Films 319 (1998) 148–152
Wiener-type of deconvolution technique w2,3x. Nonlinear corrections can be calculated consistently on this sphere, allowing to converge to the perfect complex electron wavefunction. In this approach, only the information on the sphere is taken into account, which allows to speed up the calculations enormously w10,11x. On the other hand, not all information in the imaging process is used optimally. Therefore, further convergence can be achieved using a Maximum Likelihood iteration scheme w4,9x, in which the Picard iterator conveys all information, yet fully optimising the signal to noise characteristics. The focus variation method described above allows to retrieve the phase in the image plane. In order to return from the image plane to the exit face of the object, one has to compensate for the microscope aberrations. This requires a very accurate knowledge of these aberrations, one of them is the absolute focus. In a sense, the reconstructed image plane is not perfectly conjugated to the exit face of the object but to another fictitious plane in object space. The distance between both planes is the absolute focus. In a crystalline object, the exit wavefunction is generated by dynamical diffraction of the electrons in the crystal. Using the single column channelling theory w5x, we are able to show that the electron wave at the exit face of an atom column parallel to the electron beam is most sharply peaked if the focus is close to zero. This conclusion holds for all columns so that we can use criteria based on the sharpness of the columns to estimate the zero focus condition. A very useful criterion in this respect is the entropy as originally introduced by Shannon w6x in communication theory. Based on this theory, we have proposed a robust focus determination procedure based on the physical properties of the amplitudes and phases in electron diffraction w7x, allowing the defocus to be better than 5 nm. The final step consists in retrieving the projected structure of the object from the wavefunction at the exit face. Is the object thin enough to act as a phase object, the phase is proportional to the electrostatic potential of the structure, projected along the beam direction so that the retrieval is
151
Fig. 6. Reconstruction from an experimental series of an edge dislocation in Si. The array indicates the edge between the perfect structure at the left and the tilted area at the right. Here, channelling conditions do not apply so that different columns cannot be discriminated.
straightforward. If the object is thicker, the problem is much more complicated. However, if the crystal object is perfectly oriented along a zone axis, the electrons are trapped in the positive potential of the columns. The columns then, in a sense, act as channels for the electrons w5x. If the distance between the columns is not too small, a one-to-one correspondence between the wavefunction at the exit face and the column structure of the crystal is established. Within the columns, the electrons oscillate as a function of depth without however leaving the column. It is important to note that channelling is not a property of crystal, but occurs even in an isolated column and is not much affected by the neighbouring columns, provided the distance is not too close.
5. Experimental results In the framework of a Brite–Euram project, the proposed method has been implemented on a fully computercontrolled Philips CM30 UltraTwin electron microscope equipment with Field emission source, a cooled 1024 = 1024 Slow Scan CCD camera and a fast on-line image processing computer. The alignment of the microscope is done by an automatic and highly accurate procedure. Examples of simulated and experimental results are shown in Figs. 5 and 6. The resolution obtained allows to visualise the atom columns. Quantitative filtering enables to determine the structure with high accuracy w8x.
References
Fig. 5. Computer simulation of a core dislocation in Si. ŽCourtesy A. Thust. Left: best image showing the Si dumb bell structure at the outside of the image. The core of the dislocation is totally obscured by the complex transfer of the microscope. Right: reconstructed structure from a series of 2D images clearly revealing the atomic structure.
w1x A. Van den Bos, in: P.H. Sydenham ŽEd.., Handbook of Measurement Science, Vol. 1. w2x M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64 Ž1996. 153–165. w3x D. Van Dyck, M. Op de Beeck, W. Coene, Optik 93 Ž1993. 103–107. w4x W. Coene, G. Janssen, M. Op de Beeck, D. Van Dyck, Phys. Rev. Lett. 29 Ž1992. 3743–3746. w5x D. Van Dyck, M. Op de Beeck, Ultramicroscopy 64 Ž1996. 99–107.
152
D. Van Dyck r Thin Solid Films 319 (1998) 148–152
w6x C.E. Shannon, The Mathematical Theory of Communication, Univ. of Illinois Press, Chicago Ž1949.. w7x D. Tang, H.W. Zandbergen, J. Jansen, M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64 Ž1996. 265–276. w8x H.W. Zandbergen, D. Tang, J. Jansen, R.J. Cava, Ultramicroscopy 64 Ž1996. 231–247.
w9x W.J.M. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64 Ž1996. 109–135. w10x M. Op de Beeck, D. Van Dyck, W. Coene, Ultramicroscopy 64 Ž1996. 167–183. w11x H.W. Zandbergen, S. Anderson, J. Jansen, Science 277 Ž1997. 1221–1225.
HREM: ultimate performances D. Van Dyck
)
Department of Physics, UniÕersity of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
Abstract The possibilities and limitations of HREM are discussed with respect to accurate structure derivation. It is shown that direct holographic methods are useful to yield approximate structure models that can be used as a start for final quantitative refinement. q 1998 Published by Elsevier Science S.A. Keywords: Holographic; Electron; Particle
1. The electron microscope as a communication channel The microstructure of matter can only be studied by interaction with particles, which carry information from the object to the observer. For this purpose, electrons are extremely useful since, as compared to other particles, they are easy to generate, easy to accelerate, easy to deflect and easy to detect. Information from the bulk of an object can be obtained by scattering with high energy electrons ŽG 100 keV. as is the case in high-resolution electron microscopy ŽHREM. and scanning transmission electron microscopy ŽSTEM.. In an ideal scattering experiment, the state of the electron is carefully determined immediately before and after the interaction with the object Ži.e., in the planes A and B in Fig. 1.. From the change in the electron state one can deduce information about the interaction and hence about the object itself. The plane A characterizes the illumination condition and the plane B the detecting condition. The states of the electron can be determined either in real space or reciprocal space. Full flexibility for converting the electron states in the planes A and B from real to reciprocal space or vice versa can be obtained by placing two lenses Žor two lens combinations., one at each side of the object, i.e., the condenser and objective lens ŽFig. 1.. All existing combinations of illumination and detection are listed in Table 1. Due to reciprocity Žsymmetry of time reversal. HREM and STEM are equivalent. In a sense, both techniques have the same configuration ŽFig. 1. if the z-axis is inverted.
)
The electron microscope can thus be described as a communication channel in which each electron carries two degrees of spatial information. The most performant high resolution electron microscopes ŽHREM. nowadays achieve an interpretable resolution of about 0.2 nm so that structural details of matter can be visualised almost up to the atomic scale. If a resolution of 0.1 nm could be reached, most of the individual columns of a structure could be resolved, which would make HREM an extremely useful technique for the study of new materials. However, the power of the technique is still severely limited by the problem of the interpretation of the images. Due to the complex imaging process, details in the observed images do not necessarily correspond to features in the structure of the objects. In the past, reliable information was obtained by comparing simultaneously the experimental images with computer simulated ones for known structure models. However, this trial and error technique
Corresponding author.
0040-6090r98r$19.00 q 1998 Published by Elsevier Science S.A. All rights reserved. PII S 0 0 4 0 - 6 0 9 0 Ž 9 7 . 0 1 1 1 1 - 5
Fig. 1. Ideal scattering experiment.
D. Van Dyck r Thin Solid Films 319 (1998) 148–152
requires a large amount of prior information obtained by other techniques. Recently, new direct methods have been developed that allow to obtain quantitative structural information of solids at the atomic level.
2. Image formation In general, the effect of imperfect imaging can be described using a transfer function which has two major effects: the spatial frequencies are altered and above a certain threshold they are damped. The highest spatial frequency that is still transferred above the noise determines the effective resolution of the instrument. In electron microscopy, the image formation process is somewhat more complicated. Since the imaging in an electron microscope is coherent, the object function, transfer function and image function are all complex valued, while the recorded images only show intensities. In the microscope, the electron suffers phase shifts due to the spherical aberration Cs of the lens and due to the defocus d f. For a special focus, the effect of the spherical aberration can be compensated somewhat by defocus so that for a range of spatial frequencies the phase shift is nearly constant and equal to pr2. The phase transfer function is depicted in Fig. 2. Two types of resolution can be defined. First, the cut-off point where the phase transfer function becomes zero. Up to this point, the phases are nearly constant so that the information is transferred in a forward direction. In this way, the image of a very thin object can be interpreted directly in terms of the structure of the object up to this resolution. Another definition of resolution is the point where information becomes comparable to the noise content. This is called the information limit. It is mainly determined by incoherent effects such as the spatial and temporal incoherence. When one uses a field emission source, which can be considered as an electron laser gun,
Fig. 2. Imaginary part of a typical transfer function at Scherzer focus for a traditional LaB 6 filament and a Field Emission Gun. Both have the same point resolution, but the information limit, i.e., the maximum transferable frequency, is greatly improved with the FEG source.
149
the information limit can be reduced up to 0.08 nm, while the point resolution is of the order of 0.17 nm. At this limit, the ultimate resolution is not limited by the microscope but rather by the object. However, the high resolution details are in the rapidly oscillating part of the transfer function so that image processing is required to recover this information.
3. Interpretation of the images The ultimate goal of high resolution electron microscopy is to determine quantitatively the atomic structure of an object. High resolution images are then to be considered as data planes from which the structural information has to be extracted. However, this structural information is usually hidden in the images and cannot easily be assessed. Therefore, a quantitative approach is required in which all steps in the imaging process are taken into account. Ideally, quantitative extraction of information should be done as follows: one has a model for the object, for the electron object interaction, microscope transfer and for the imaging process. The object model that describes the interaction with the electrons consists of the assembly of the electrostatic potentials of the constituting atoms. Since for each atom type the electrostatic potential is known, the model parameters then reduce to atom numbers and coordinates, Debye Waller factors, object thickness and orientation Žif inelastic scattering is neglected.. The imaging process is characterised by a small number of unknown Žor not exactly known. parameters such as defocus, spherical aberration, etc. These model parameters now have to be determined from the experiment. The parameters can be estimated from the fit between the theoretical images and the experimental images. The goodness of the fit is evaluated using a criterium such as likelihood, mean square difference or R-factor Žcf. X-ray crystallography.. For each set of parameters of the model, one can calculate this goodness of fit, so as to yield a fitness function in parameter space. The parameters for which the fitness is optimal then yields the best estimates that can be derived from the experiment. In a sense, one is searching for a maximum Žor minimum depending on the criterium. of the fitness function in the parameter space, the dimension of which is equal to the number of parameters. The best estimates for the model parameters are obtained by using a Maximum Likelihood criterium ŽML.. It is known Že.g., Ref. w1x. that if there exists a Minimum Variance Bound Žor Cramer–Rao Bound. estimator, it is given by the Maximum Likelihood method. ŽThe least squares estimator is sub optimal and can only be used under specific assumptions.. The probability that the model parameters are a n 4 given that the experimental outcomes are n i 4 can be calculated from Bayesian statistics. It is
D. Van Dyck r Thin Solid Films 319 (1998) 148–152
150
Fig. 3. Scheme of the refinement procedure.
called the likelihood function. pŽ n i 4 r a n 4. is the probability that the measurement yields the values n i 4 given that the model parameters are a n 4 . This probability is given by the model. pŽ a n 4. is the prior probability that the set of parameters a n 4 occurs. If no prior information is available all pŽ a n 4. are assumed to be equal. For instance, in case of HREM, pŽ n i 4 r a n 4. represents the probability that n i electrons hit the pixel x i in the image if all the parameters of the model Žobject structure and imaging parameters. i.e., n i then represents the measured intensity, in number of electrons, of the pixel i. L can then be considered as a fitness function. In principle, the search for the best parameter set is then reduced to the search for optimal fitness in parameter space. This search can only be done in an iterative way as schematised in Fig. 3. First one has a starting model, i.e., starting value for the object and imaging parameters a n 4 . From this one can calculate the experimental outcome pŽ n i 4 r a n 4.. This is a classical image simulation. ŽNote that the experiment can also be a series of images andror diffraction patterns. From the mismatch between experimental and simulated images one can obtain a new estimate for the model parameters Žfor instance using a gradient method. which can then be used for the next iteration. This procedure is repeated until the optimal fitness Ži.e., optimal match. is reached.
parameters is small, as is the case for small unit cell crystals. In some very favourable cases, the number of possible models, thanks to prior knowledge, is discrete and very small so that visual comparison is sufficient. These cases were the only cases where image simulation could be meaningfully used in the past. In high resolution electron microscopy, the dimensionality problem can be solved by deblurring the information, so as to unscramble the influence of the different object parameters in the image. In this way, the structural parameters can be uncoupled and the dimension of the parameter space reduced. This can be achieved in different ways: high voltage microscopy, correction of the microscopic aberrations or direct holographic methods for exit wave and structure reconstruction. Therefore, direct holographic reconstruction methods will provide an approximate model that can be used as a starting point for a final refinement by fitting with the original images that is sufficiently close to the global maximum so as to guarantee convergence. Direct methods aim at deblurring the information in the images. Since only the intensity is recorded, the phase has been lost. For the problem of the phase retrieval, the focus variation method has been proposed, in which the focus is used as an extra parameter. Images are captured at very close focus values so as to collect all information in the 3D image space. Each image contains linear and nonlinear information. By Fourier transforming all 3D image space, the linear information of all images is superimposed onto a sphere, while the nonlinear contribution is distributed all over the 3D frequency space. By separating the information on this so-called Ewald sphere, the imaging problem is linearised to a great extent Žsee Fig. 4.. Therefore, in a first approximation, the phase can be retrieved using a
4. Direct methods A major problem now is that the structural information of the object can be strongly delocalised by the image transfer in the electron microscope so that the effect of the structural parameters is completely scrambled in the high resolution images. Due to this coupling, one has to refine all parameters simultaneously which poses a combinatorial problem. Indeed, the dimension of the parameter space becomes so high that even with advanced optimisation techniques such as genetic algorithms, simulated annealing, tabu search, etc., one cannot avoid ending in local optima. The problem is only manageable if the number of
Fig. 4. Schematical representation of the phase retrieval procedure using a through focus series of images, recorded with a Slow Scan CCD camera and processed on a fast computer system. The paraboloid which contains the linear information is also shown.
D. Van Dyck r Thin Solid Films 319 (1998) 148–152
Wiener-type of deconvolution technique w2,3x. Nonlinear corrections can be calculated consistently on this sphere, allowing to converge to the perfect complex electron wavefunction. In this approach, only the information on the sphere is taken into account, which allows to speed up the calculations enormously w10,11x. On the other hand, not all information in the imaging process is used optimally. Therefore, further convergence can be achieved using a Maximum Likelihood iteration scheme w4,9x, in which the Picard iterator conveys all information, yet fully optimising the signal to noise characteristics. The focus variation method described above allows to retrieve the phase in the image plane. In order to return from the image plane to the exit face of the object, one has to compensate for the microscope aberrations. This requires a very accurate knowledge of these aberrations, one of them is the absolute focus. In a sense, the reconstructed image plane is not perfectly conjugated to the exit face of the object but to another fictitious plane in object space. The distance between both planes is the absolute focus. In a crystalline object, the exit wavefunction is generated by dynamical diffraction of the electrons in the crystal. Using the single column channelling theory w5x, we are able to show that the electron wave at the exit face of an atom column parallel to the electron beam is most sharply peaked if the focus is close to zero. This conclusion holds for all columns so that we can use criteria based on the sharpness of the columns to estimate the zero focus condition. A very useful criterion in this respect is the entropy as originally introduced by Shannon w6x in communication theory. Based on this theory, we have proposed a robust focus determination procedure based on the physical properties of the amplitudes and phases in electron diffraction w7x, allowing the defocus to be better than 5 nm. The final step consists in retrieving the projected structure of the object from the wavefunction at the exit face. Is the object thin enough to act as a phase object, the phase is proportional to the electrostatic potential of the structure, projected along the beam direction so that the retrieval is
151
Fig. 6. Reconstruction from an experimental series of an edge dislocation in Si. The array indicates the edge between the perfect structure at the left and the tilted area at the right. Here, channelling conditions do not apply so that different columns cannot be discriminated.
straightforward. If the object is thicker, the problem is much more complicated. However, if the crystal object is perfectly oriented along a zone axis, the electrons are trapped in the positive potential of the columns. The columns then, in a sense, act as channels for the electrons w5x. If the distance between the columns is not too small, a one-to-one correspondence between the wavefunction at the exit face and the column structure of the crystal is established. Within the columns, the electrons oscillate as a function of depth without however leaving the column. It is important to note that channelling is not a property of crystal, but occurs even in an isolated column and is not much affected by the neighbouring columns, provided the distance is not too close.
5. Experimental results In the framework of a Brite–Euram project, the proposed method has been implemented on a fully computercontrolled Philips CM30 UltraTwin electron microscope equipment with Field emission source, a cooled 1024 = 1024 Slow Scan CCD camera and a fast on-line image processing computer. The alignment of the microscope is done by an automatic and highly accurate procedure. Examples of simulated and experimental results are shown in Figs. 5 and 6. The resolution obtained allows to visualise the atom columns. Quantitative filtering enables to determine the structure with high accuracy w8x.
References
Fig. 5. Computer simulation of a core dislocation in Si. ŽCourtesy A. Thust. Left: best image showing the Si dumb bell structure at the outside of the image. The core of the dislocation is totally obscured by the complex transfer of the microscope. Right: reconstructed structure from a series of 2D images clearly revealing the atomic structure.
w1x A. Van den Bos, in: P.H. Sydenham ŽEd.., Handbook of Measurement Science, Vol. 1. w2x M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64 Ž1996. 153–165. w3x D. Van Dyck, M. Op de Beeck, W. Coene, Optik 93 Ž1993. 103–107. w4x W. Coene, G. Janssen, M. Op de Beeck, D. Van Dyck, Phys. Rev. Lett. 29 Ž1992. 3743–3746. w5x D. Van Dyck, M. Op de Beeck, Ultramicroscopy 64 Ž1996. 99–107.
152
D. Van Dyck r Thin Solid Films 319 (1998) 148–152
w6x C.E. Shannon, The Mathematical Theory of Communication, Univ. of Illinois Press, Chicago Ž1949.. w7x D. Tang, H.W. Zandbergen, J. Jansen, M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64 Ž1996. 265–276. w8x H.W. Zandbergen, D. Tang, J. Jansen, R.J. Cava, Ultramicroscopy 64 Ž1996. 231–247.
w9x W.J.M. Coene, A. Thust, M. Op de Beeck, D. Van Dyck, Ultramicroscopy 64 Ž1996. 109–135. w10x M. Op de Beeck, D. Van Dyck, W. Coene, Ultramicroscopy 64 Ž1996. 167–183. w11x H.W. Zandbergen, S. Anderson, J. Jansen, Science 277 Ž1997. 1221–1225.