IMAGING AND ELECTRON PHYSICSAberration–Corrected Electron Microscopy 978-0-12-374220-9

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Advances in

IMAGING AND ELECTRON PHYSICS VOLUME

153 Aberration-Corrected Electron Microscopy

EDITOR-IN-CHIEF

PETER W. HAWKES CEMES-CNRS Toulouse, France

HONORARY ASSOCIATE EDITORS

TOM MULVEY BENJAMIN KAZAN

Advances in

IMAGING AND ELECTRON PHYSICS VOLUME

153 Aberration-Corrected Electron Microscopy Edited by

PETER W. HAWKES CEMES-CNRS, Toulouse, France

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 84 Theobald’s Road, London WC1X 8RR, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA First edition 2008 Copyright © 2008, Elsevier Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http: //elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material. Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-374220-9 ISSN: 1076-5670 For information on all Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 08 09 10 10 9 8 7 6 5 4 3 2 1

CONTENTS

Contributors Preface Future Contributions

xi xiii xvii

PART 1: Historical background

1

1. History of Direct Aberration Correction

3

Harald Rose I. Introduction II. Birth of Aberration Correction III. Other Early Correction Efforts IV. The Darmstadt Correction Project V. The Chicago and EMBL Quadrupole-Octopole Probe Correctors VI. Evolution of the Hexapole Corrector VII. Aberration Correction of Systems with Curved Axis VIII. Revival of the Quadrupole-Octopole Corrector IX. Conclusion and Outlook Acknowledgments References

3 5 9 12 18 19 25 29 34 36 36

PART 2: Aberration corrector design

41

2. Present and Future Hexapole Aberration Correctors for High-Resolution Electron Microscopy

43

Maximilian Haider, Heiko Müller, and Stephan Uhlemann I. Introduction II. Present Hexapole Correctors III. Hexapole Aplanats IV. Conclusion Acknowledgments References

44 70 101 114 116 117

v

vi

Contents

3. Advances in Aberration-Corrected Scanning Transmission Electron Microscopy and Electron Energy-Loss Spectroscopy

121

Ondrej L. Krivanek, Niklas Dellby, Robert J. Keyse, Matthew F. Murfitt, Christopher S. Own, and Zoltan S. Szilagyi I. Introduction II. Aberration Correction by Non-Round Lenses III. Performance of Aberration-Corrected Instruments IV. New Applications V. Conclusions Acknowledgments References

121 124 129 140 154 155 155

PART 3: Results obtained with aberration-corrected instruments

161

4. First Results Using the Nion Third-Order Scanning Transmission Electron Microscope Corrector

163

P. E. Batson I. Introduction II. The IBM High-Resolution STEM/EELS System III. First Results IV. Calculation of Probe Properties V. Imaging of Crystalline Objects VI. AlN/GaN/AlN Quantum Well Structure VII. Hafnium Oxide Gate Stack VIII. Multislice Simulations IX. Atomic Movement Under the Electron Beam X. Conclusions Acknowledgments References

5. Scanning Transmission Electron Microscopy and Electron Energy Loss Spectroscopy: Mapping Materials Atom by Atom

163 164 168 174 176 179 183 187 190 192 193 193

195

Andrew L. Bleloch I. Introduction II. Experimental Heuristics

195 196

Contents

III. Case Studies IV. Discussion References

6. Aberration Correction With the SACTEM-Toulouse: From Imaging to Diffraction

vii

199 221 221

225

Florent Houdellier, Martin Hÿtch, Florian Hüe, and Etienne Snoeck I. Introduction II. Aberration Correction and Strain Mapping III. Aberration-Corrected Convergent-Beam Electron Holography IV. Aberration-Corrected Electron Diffraction V. Pseudo-Lorentz Mode for Medium-Resolution Electron Holography VI. Conclusions Acknowledgments References

7. Novel Aberration Correction Concepts

225 232 240 245 252 256 256 256

261

Bernd Kabius and Harald Rose I. Introduction II. Concepts for Improving Resolution for In Situ TEM III. Cc Correction IV. Summary Acknowledgments References

8. Aberration-Corrected Imaging in Conventional Transmission Electron Microscopy and Scanning Transmission Electron Microscopy

261 262 265 279 280 280

283

Angus I. Kirkland, Peter D. Nellist, Lan-Yun Chang, and Sarah J. Haigh I. Introduction II. The Wave Aberration Function III. Coherence Effects in CTEM and STEM IV. Aberration-Correction Imaging Conditions V. Conclusions Acknowledgments References

284 286 297 308 319 320 320

viii

Contents

9. Materials Applications of Aberration-Corrected Scanning Transmission Electron Microscopy

327

S. J. Pennycook, M. F. Chisholm, A. R. Lupini, M. Varela, K. van Benthem, A. Y. Borisevich, M. P. Oxley, W. Luo, and S. T. Pantelides I. Introduction II. Key Instrumental Advances III. Measurement and Definition of Resolution IV. Complex Oxides: Manganites V. High-Temperature Superconductors VI. Complex Dislocation Core Structures VII. Understanding Structure-Property Relations in Ceramics VIII. Semiconductor Quantum Wires IX. Catalysts X. Future Directions Acknowledgments References

10. Spherical Aberration-Corrected Transmission Electron Microscopy for Nanomaterials

327 328 348 355 361 363 367 371 374 377 378 378

385

Nobuo Tanaka I. II. III. IV. V.

Introduction Imaging Theories of HRTEM Using Aberration Correctors Actual Advantages for Observation by Cs -Corrected TEM Actual Advantages for Observation by Cs -Corrected STEM Three-Dimensional Observation of Atomic Objects in Cs -Corrected STEM VI. Conclusion and Future Prospects Acknowledgments References

11. Atomic-Resolution Aberration-Corrected Transmission Electron Microscopy

386 387 406 424 430 433 434 434

439

Knut Urban, Lothar Houben, Chun-Lin Jia, Markus Lentzen, Shao-Bo Mi, Andreas Thust, and Karsten Tillmann I. Introduction II. Fundamentals of Atomic-Resolution Imaging in a Transmission Electron Microscope III. Atomic-Resolution Electron Microscopy and the Inversion of the Scattering and Imaging Problem

440 440 450

Contents

IV. Selected Materials Science Applications V. Conclusions References

12. Aberration-Corrected Electron Microscopes at Brookhaven National Laboratory

ix

457 477 478

481

Yimei Zhu and Joe Wall I. Introduction II. Environmental Requirements and Laboratory Design for Aberration-Corrected Electron Microscopes III. The BNL Aberration-Corrected Instruments IV. A Brief Comparison of the Three Instruments V. Evaluation and Applications of STEM VI. Outlook Acknowledgments References

481 483 492 507 510 520 521 521

Contents of Volumes 151 and 152

525

Index

527

Color Plate Section

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CONTRIBUTORS Harald Rose Technical University Darmstadt, 64289 Darmstadt, Germany Maximilian Haider, Heiko Müller, and Stephan Uhlemann Corrected Electron Optical Systems GmbH, 69126 Heidelberg, Germany Ondrej L. Krivanek, Niklas Dellby, Robert J. Keyse, Matthew F. Murfitt, Christopher S. Own, and Zoltan S. Szilagyi Nion Co., Kirkland, WA 98033, USA

3 43

121

P. E. Batson IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA

163

Andrew L. Bleloch SuperSTEM Laboratory, Daresbury and University of Liverpool, United Kingdom

195

Florent Houdellier, Martin Hÿtch, Florian Hüe, and Etienne Snoeck CEMES-CNRS, 29 rue Jeanne Marvig, 31055 Toulouse, France

225

Bernd Kabius Argonne National Laboratory, Argonne, Illinois 60439, USA

261

Angus I. Kirkland, Peter D. Nellist, Lan-Yun Chang, and Sarah J. Haigh University of Oxford, Department of Materials, Parks Road, Oxford, OX1 3PH, United Kingdom

283

S. J. Pennycook, M. P. Oxley, W. Luo, and S. T. Pantelides Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

327

Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA M. F. Chisholm, A. R. Lupini, M. Varela, and A. Y. Borisevich Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

327

K. van Benthem Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

327

Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

xi

xii

Contributors

Nobuo Tanaka EcoTopia Science Institute, Nagoya University, Nagoya 464-8603, Japan Knut Urban, Lothar Houben, Chun-Lin Jia, Markus Lentzen, Shao-Bo Mi, Andreas Thust, and Karsten Tillmann Institute of Solid State Research and Ernst Ruska Centre for Microscopy and Spectroscopy with Electrons, Helmholtz Research Centre, 52425 Jülich, Germany Yimei Zhu and Joe Wall Brookhaven National Laboratory, Long Island, New York, 11973 USA

385

439

481

PREFACE

It is just over ten years since the first spherical aberration correctors were incorporated into commercial high-resolution scanning and conventional transmission electron microscopes. The effect on the electron microscopy community was dramatic and not only have such correctors been installed into VG STEMs and new TEMs but research has continued to develop correctors of chromatic aberration as well as spherical aberration. At the same time, more complex devices, capable of correcting all third-order aberrations have been proposed, for the needs of electron-beam lithography in particular. The present generation of correctors is also able to cancel the many parasitic aberrations, most of which were of no consequence earlier. It therefore seemed timely to assess the results obtained with these newly-equipped instruments and the present volume is the outcome. It begins, appropriately, with a short history of aberration correction by means of multipoles or mirrors by H. Rose, who has worked on electron lens aberrations for many years. The need for correction may be dated to 1936, when Otto Scherzer (1909–1982) demonstrated that the spherical aberration coefficient can never be cancelled by skilful lens design. In 1947, he listed several possible corrector designs and, a year later, Dennis Gabor proposed the two-stage method of correction that we call holography (among the early names given to the procedure was gaboroscopy!). Of Scherzer’s suggestions, only those involving the use of multipole correctors and electron mirrors have survived, though Japanese attempts to use thin-foil correctors continued until very recently. Rose’s historical account covers the multipole and mirror projects and contains much information that only an ’insider’ could give us. The other methods, now of historical interest only, are described in my chapters on aberrations and their correction in Science of Microscopy (Springer, New York 2007/8) and the Handbook of Charged Particle Optics (edited by J. Orloff, CRC Press, Boca Raton 2008). Part 2 contains contributions by O. Krivanek and colleagues at his firm, Nion, and M. Haider and members of his company, CEOS (Corrected Electron Optical Systems). Although efforts to use quadrupole–octopole correctors go back to the 1950s, as Rose reminds us, the first real breakthrough came in 1997 when O. Krivanek, N. Dellby, A.J. Spence, R.A. Camps and L.M. Brown, working in the Cavendish Laboratory in Cambridge, succeeded in using such a corrector to improve the performance of a VG STEM. In the following year, the first publications of M. Haider, B. Kabius, H. Rose, E. Schwan, S. Uhlemann and K. Urban showing that a sextupole corrector was capable of correcting the spherical aberration of the objective lens of a Philips transmission electron microscope appeared; this date is, however, misleading, for their Nature

xiii

xiv

Preface

paper was first submitted in 1997 and contained images obtained on 24 June 1997. The paper by S. Uhlemann and M. Haider, which likewise did not appear in Ultramicroscopy until 1998, was also submitted in 1997. Nion now offers a newly designed STEM of which a much improved quadrupole–octopole corrector is an integral part. CEOS has equipped many commercial TEMs with sextupole correctors. The challenge of correcting both the spherical and chromatic aberrations has blurred this neat division into quadrupoles for STEMs and sextupoles for TEMs, for quadrupoles appear to be indispensable for chromatic correction. Part 3 consists of progress reports from most of the laboratories around the world that are equipped with corrected microscopes: P.E. Batson at IBM Yorktown Heights; A. Bleloch who is responsible for the British SuperSTEM project; F. Houdellier, M. Hÿtch, F. Hüe and E. Snoeck in the CEMES-CNRS in Toulouse, the former Laboratoire d’Optique Electronique; B. Kabius and H. Rose, representing the TEAM project, discussed further below; A.I. Kirkland, P.D. Nellist, L.-y. Chang and S.J. Haigh from the Department of Materials in Oxford; S.J. Pennycook and colleagues in Oak Ridge National Laboratory; N. Tanaka, who writes about developments in Japan; K. Urban and colleagues, who describe the work at Jülich where the first results with the CEOS corrector were obtained; and Y. Zhu and J. Wall at Brookhaven National Laboratory. These provide, if any justification was needed, ample evidence that all the efforts to build correctors have been worthwhile, results are pouring out from these new instruments and we are certainly only at the beginning. The flavour of the article by Kabius and Rose in this section is rather different from that of its neighbours. Their chapter originally began as an account of the TEAM project, first discussed at a workshop at Argonne National Laboratory in 2000 (see ncem.lbl.gov/team/TEAM%20Report%202000.pdf) at which time TEAM stood for Transmission Electron Achromatic Microscope; the objective was to attain a resolution a resolution of 0.5 Å in both the TEM and the STEM modes, with the relatively large gap between the polepieces that is required for tomography and in situ experiments. A second workshop was held in Berkeley in 2002 (ncem.lbl.gov/team/TEAM%20Report%202002.pdf) and the meaning of the acronym was changed to Transmission Electron Aberration-corrected Microscope. In 2003, the TEAM project became part of a 20-year plan of the American Department of Energy for Facilities for the Future of Science. TEAM is a collaborative project involving groups from laboratories in Berkeley, Argonne, Oak Ridge, Brookhaven and the University of Illinois in Urbana–Champaign. Subsequent workshops were held at the ’Microscopy and Microanalysis’ meetings of the Microscopy Society of America in San Antonio (2003), Savannah (2004) and Honolulu (2005), see ncem.lbl.gov/team3.htm. More details are to be found in a status report by U. Dahmen (Microscopy and Microanalysis 13, Supplement 2, 2007,

Preface

xv

1150–1CD), which predicted that the first instrument due to emerge from the project would be operational (though as yet without chromatic correction) in 2008 and the CEOS website confirmed in April 2008 that this first objective had been met. This claim is further justified in a paper by C. Kisielowski et al. in Microscopy and Microanalysis 14 (2008) 469–477 that reached me as I was correcting the proofs of this Preface (October 2008). The instrument, installed in the National Center for Electron Microscopy in Lawrence Berkeley National Laboratory, should be in full working order in 2009. All this to explain why the chapter by Kabius and Rose contains not results but a critical examination of the design of correctors. Two other topics might have been included in this collection: monochromators and holography with corrected microscopes. In the absence of a satisfactory chromatic aberration corrector, several electron microscope manufacturers incorporate a monochromator in their instruments to reduce the effect of this aberration though at the cost of reducing the beam current. This is of course not ‘correction’, perhaps we might see it as truce! Finally, however, the subject was abandoned as being in too premature a state for a satisfactory survey and I hope to include a review of these devices in a year or two. Holography is not included partly because the book was threatening to become unmanageably large but also because a chapter on the subject is already promised for a future volume of these Advances by H. Lichte. Finally, what about the microscope manufacturers? Certainly I should have liked to invite contributions from the principal companies involved in this adventure but it was clear that this would require a large fourth section, which would not only have made the book unwieldy but might not have been easy to orchestrate. I therefore decided against such a section but have not completely abandoned the idea. I always conclude my Preface with a word of thanks to the contributors and this is especially well deserved here. For the business authors, the AIEP deadline was not the only one they had to meet while the research microscopists doubtless found it hard to leave their new, corrected microscopes for their desks. I am all the more grateful to them and I am convinced that this appraisal of the first decade of aberration-corrected microscopy will be eagerly read, especially in 2009, the centenary of Scherzer’s birth. Peter W. Hawkes

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FUTURE CONTRIBUTIONS

S. Ando Gradient operators and edge and corner detection V. Argyriou and M. Petrou (Vol. 156) Photometric stereo: an overview W. Bacsa Optical interference near surfaces, sub-wavelength microscopy and spectroscopic sensors C. Beeli Structure and microscopy of quasicrystals C. Bobisch and R. Möller Ballistic electron microscopy G. Borgefors Distance transforms Z. Bouchal Non-diffracting optical beams F. Brackx, N. de Schepper, and F. Sommen (Vol. 156) The Fourier transform in Clifford analysis A. Buchau Boundary element or integral equation methods for static and time-dependent problems B. Buchberger Gröbner bases T. Cremer Neutron microscopy N. de Jonge (Vol. 156) Electron emission from carbon nanotubes A. X. Falcão The image foresting transform R. G. Forbes Liquid metal ion sources B. J. Ford The earliest microscopical research C. Fredembach Eigenregions for image classification A. Gölzhäuser Recent advances in electron holography with point sources

xvii

xviii

Future Contributions

D. Greenfield and M. Monastyrskii (Vol. 155) Selected problems of computational charged particle optics H. F. Harmuth and B. Meffert (Vol. 154) Dirac’s difference equation and the physics of finite differences M. I. Herrera The development of electron microscopy in Spain J. Isenberg Imaging IR-techniques for the characterization of solar cells K. Ishizuka Contrast transfer and crystal images A. Jacobo Intracavity type II second-harmonic generation for image processing L. Kipp Photon sieves G. Kögel Positron microscopy T. Kohashi Spin-polarized scanning electron microscopy R. Leitgeb Fourier domain and time domain optical coherence tomography B. Lencová Modern developments in electron optical calculations H. Lichte New developments in electron holography M. Matsuya Calculation of aberration coefficients using Lie algebra S. McVitie Microscopy of magnetic specimens M. A. O'Keefe Electron image simulation D. Oulton and H. Owens Colorimetric imaging N. Papamarkos and A. Kesidis The inverse Hough transform K. S. Pedersen, A. Lee, and M. Nielsen The scale-space properties of natural images E. Rau Energy analysers for electron microscopes

Future Contributions

xix

E. Recami and M. Zamboni-Rached (Vol. 156) Superluminal solutions to wave equations R. Shimizu, T. Ikuta, and Y. Takai Defocus image modulation processing in real time S. Shirai CRT gun design methods T. Soma Focus-deflection systems and their applications I. Talmon Study of complex fluids by transmission electron microscopy M. E. Testorf and M. Fiddy Imaging from scattered electromagnetic fields, investigations into an unsolved problem N. M. Towghi Ip norm optimal filters E. Twerdowski Defocused acoustic transmission microscopy Y. Uchikawa Electron gun optics K. Vaeth and G. Rajeswaran Organic light-emitting arrays M. van Droogenbroeck and M. Buckley Anchors in mathematical morphology M. Yavor Optics of charged particle analysers

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PART 1

Historical background

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CHAPTER

1 History of Direct Aberration Correction Harald Rose*

Contents

I II III IV V

Introduction Birth of Aberration Correction Other Early Correction Efforts The Darmstadt Correction Project The Chicago and EMBL Quadrupole-Octopole Probe Correctors VI Evolution of the Hexapole Corrector VII Aberration Correction of Systems with Curved Axis VIII Revival of the Quadrupole-Octopole Corrector IX Conclusion and Outlook Acknowledgments References

3 5 9 12 18 19 25 29 34 36 36

I. INTRODUCTION Aberration correction in electron microscopy dates back to Otto Scherzer, who was a professor of theoretical physics at the Darmstadt University of Technology from 1936 until his retirement in 1976. Scherzer had been a physics student at the University of Munich completing his thesis at the institute of Theoretical Physics headed by the famous Arnold Sommerfeld. He received his Ph.D. degree in 1931. After receiving his doctorate, he shared his room for some time with Hans Bethe, who was writing his renowned review article, “Elektronentheorie der Metalle” published in the Handbuch der Physik. After Black Friday in 1929, financial support for research was almost impossible to obtain. At the end of 1931, Sommerfeld ran out of funds because the German Research Foundation went bankrupt. To find some *Technical University Darmstadt, 64289 Darmstadt, Germany Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01001-x. Copyright © 2008 Elsevier Inc. All rights reserved.

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Harald Rose

support for his post-doctoral colleague and assistant Otto Scherzer, Sommerfeld recommended him to Professor Ernst Brueche, who was director of research at the company AEG in Berlin at that time. Brueche’s main research activities concerned the development of electron optical components for novel instruments and electronic devices. At the beginning of 1932, Scherzer joined Brueche’s group. His task involved the theoretical investigation of the properties of electron lenses, deflection elements and other components used in oscilloscopes, image converter tubes, and electron microscopes. The results of his two-year investigations at AEG culminated in the book, Geometrische Elektronenoptik, which he published with Brueche in 1934. This first comprehensive treatise on geometrical electron optics was the standard book on the subject for many years. Aberrations of light-optical lenses are characterized by aberration coefficients. Using different procedures Walter Glaser (1937) (eikonal method) and Otto Scherzer (1937) (trajectory method) showed independently that the coefficients of electron-optical aberrations of round lenses can be expressed as integrals of the form

z1

f (B(z), (z), u(z))dz.

(1)

z0

The integrand f is composed of terms consisting of the axial magnetic field B(z), the axial electrostatic potential (z), the paraxial trajectory u(z), and their derivatives. In the case of rotational symmetry, the primary axial aberrations are determined by the paraxial trajectory u = ωuα , which is given by the axial fundamental ray uα (z) and the complex initial slope ω = α + iβ. This ray intersects the centers of the object plane zo and of the conjugate image plane zi . Different forms of the integrand can be obtained by partial integrations. The contributions at the boundary planes vanish because the axial fundamental ray is zero at these planes (uα (zo ) = 0, uα (zi ) = 0). In 1936, Scherzer proved that chromatic and spherical aberrations of standard electron lenses are unavoidable by transforming each integrand of the corresponding aberration integrals in a sum of positive squared terms. His finding was so important that it was named the “Scherzer theorem.” The validity of this theorem requires that several conditions are satisfied. Object and image must both be real, the electromagnetic field must be static and rotationally symmetric, the electric potential and its derivatives must be continuous (no space charges within the region of the electron beam), and the mirror mode has to be excluded. Walter Glaser (1940) tried to find a loophole in Scherzer’s proof by transforming the integrand of the integral for the spherical aberration coefficient of a magnetic lens in a product of two terms such that one term consists only of the magnetic field and its derivatives. By putting this term equal

History of Direct Aberration Correction

5

to zero, he derived a nonlinear differential equation for the axial magnetic field B(z). Unfortunately, as shown by Recknagel in 1941, the solution does not provide a real image, thus violating one of the constraints. In the case of a virtual image, the upper boundary terms of the partial integrations do not vanish, contrary to the assumption made by Glaser. Nevertheless, Glaser’s field distribution proved to be useful in β-ray spectroscopy, because it served Kai Siegbahn (1946) as the guide for the design of his high-performance spectrometer. The dispute between Scherzer and Glaser culminated in the problem of finding the minimum value for the coefficient Cs of the spherical aberration of real electron lenses that cannot be surpassed due to physical limitations. The problem finally was solved in 1959 by W. Tretner, who established minimum values for Cs as functions of various lens parameters such as the maximum field strength or maximum field gradient, respectively. His results showed among others that the Glaser bell-shaped field for magnetic lenses closely approximates the optimum field distribution if the object plane is placed at the maximum field. The symmetric condenser-objective lens designed by Riecke and Ruska (1966) was the first lens with this property. Today, the field of all high-resolution objective lenses closely approximates the optimum field for minimum spherical aberration. The resolution limit of electron microscopes equipped with these objective lenses is ∼ 100 λ, where λ denotes the wavelength of the image-forming electrons. Hence, the only method to further improve the resolution seemed to be to decrease the wavelength by using higher voltages. However, at that time many electron microscopists already knew that radiation damage was a decisive obstacle preventing a significant gain in specimen resolution. In order to minimize radiation damage, the electron microscope must be operated at voltages below the threshold voltage for atom displacement.

II. BIRTH OF ABERRATION CORRECTION In 1947, Scherzer (1947a) found an ingenious way to enable aberration correction. He demonstrated in a famous article that it is possible to eliminate chromatic and spherical aberrations by lifting any one of the constraints of his theorem, either by abandoning rotational symmetry or by introducing time-varying fields, or space charges. At about the same time Dennis Gabor invented holography as a means to compensate retrospectively for the unavoidable spherical aberration of rotationally symmetric electron lenses by light-optical reconstruction using glass lenses with negative spherical aberration. Gabor (1968) states in an article that he dedicated to Otto Scherzer on the occasion of his sixtieth birthday: “Holography owes its birth to Professor O. Scherzer. It was his brilliant proof, in 1936, of

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Harald Rose

the theorem that the spherical aberration can never be eliminated in any axially symmetrical optical system which induced me, in 1947, to look for a way around this fundamental difficulty of electron microscopy.” Moreover, Gabor emphasized in his letter to the organizers of the Festschrift that he was impressed by Scherzer’s ingenious idea to relax one or other of the constraints in order to enable aberration correction. Nevertheless, Gabor was awarded the Nobel Prize, because holography became an important part of light optics after the invention of the laser. At the beginning of electron microscopy experimenters encountered an obstacle that affected the resolution more severely than the third-order spherical aberration. This deleterious aberration is the first-order axial astigmatism resulting from misalignment, mechanical imperfections, and from magnetic inhomogeneities within the pole pieces of the objective lens. Unfortunately, the parasitic axial astigmatism could not be reduced sufficiently by mechanical means. To compensate for this resolution-limiting aberration, in 1947 Scherzer designed a correcting octopole element, which he called stigmator (meaning “point maker”) (Scherzer, 1947b). Such an element allows adjustment of both the strength and the azimuthal orientation of its quadrupole field compensating for the parasitic astigmatism of the objective lens. The stigmator was successfully realized by Otto Rang (1949) and improved significantly the resolution of the electron microscope at that time. Today the stigmator is an indispensable element of every electron microscope. Scherzer always believed that departure from rotational symmetry offered the most promising approach for correcting the axial aberrations of electron lenses. Therefore, he allocated to his student Rudolf Seeliger the task of building and testing the electrostatic corrector shown in Figure 1 within the frame of his doctoral thesis.

y z x

O

St

Zx

Ky

R

Kx

Zy

Kxy

FIGURE 1 Scheme of the Scherzer corrector consisting of a stigmator St, two electrostatic cylinder lenses Zx and Zy , a round lens R, and three octopoles Kx , Ky , and Kxy compensating for the spherical aberration of the objective lens O.

History of Direct Aberration Correction

FIGURE 2 Figure 1.

7

View of the Seeliger corrector based on Scherzer’s design depicted in

Starting in 1948, Seeliger built and tested the Scherzer corrector during a period of about five years. He aligned the constituent elements mechanically by means of adjustment screws. However, this approach was a major obstacle preventing an improvement of the resolution due to insufficient stability of the mechanical alignment. Although Seeliger could not improve the resolution limit of the basic electron microscope, he could demonstrate that the corrector provided a negative spherical aberration, which could be adjusted to compensate for the spherical aberration of the objective lens (Seeliger, 1951). The prime reason for the failure to improve the resolution resulted from the mechanical and electrical instabilities, which limited the information limit rather than the aberrations of the objective lens. Despite this disappointing result, Gottfried Moellenstedt was convinced that aberration correction was feasible by means of Scherzer’s corrector suggestion. Scherzer, who appreciated very much the experimental

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Harald Rose

expertise of his colleague, agreed that the Seeliger corrector should be moved to the Moellenstedt Institute in Tuebingen in 1953. Moellenstedt (1956) found an ingenious method to demonstrate the correction of the spherical aberration corrector. By using critical illumination with a large cone angle of 2 × 10−2 rad, he enlarged the spherical aberration to such an extent that it became by far the dominant aberration, which strongly blurred the image. After compensating for the spherical aberration by means of the corrector octopoles, the resolution improved by a factor of ∼ 7, accompanied by a striking increase in contrast. Although Seeliger failed to improve the experimental resolution by eliminating the spherical aberration, Scherzer was convinced that aberration correction was the most appropriate procedure for improving the performance of electron microscopes. In order to find the reasons for the failure of the experiment, in 1957 he charged W. E. Meyer with the task to investigate for his thesis the obstacles limiting the actual resolution of the spherically corrected microscope and to find means for eliminating these perturbations. The results of Meyer’s intensive and painstaking investigations revealed that two groups of parasitic perturbations must be considered (Meyer, 1961). Meyer defined the first group as alignment aberrations. They are produced by static imperfections and misalignment of the objective lens and the constituent elements of the corrector. In order to elucidate the effect of these perturbations, he investigated theoretically the axial parasitic aberrations up to the third order inclusively by using a multipole expansion for the parasitic electromagnetic field. Nowadays, we characterize the axial misalignment aberrations of second order as axial coma and threefold astigmatism, and those of third order as star aberration and fourfold astigmatism. It follows from Meyer’s representation of the corresponding aberration coefficients that all axial astigmatisms can be eliminated by means of multipole stigmators. These results remained largely unnoticed because only the first-order astigmatism affected the resolution at that time. This astigmatism is compensated for by means of a quadrupole stigmator. This stigmator was sufficient as long as the instrumental resolution did not approach the Scherzer limit. Only when this limit was in reach was the degree of the deleterious effect of the parasitic second-order aberrations realized. Since it is only possible to eliminate the threefold astigmatism by means of a sextupole stigmator, one a method is also needed to compensate for the axial coma. Friedrich Zemlin et al. (1977) achieved this correction by means of the coma-free alignment procedure. This method uses the so-called diffractogram tableau, which represents a set of diffractograms obtained from micrographs of an amorphous object taken with different tilt angles of the illumination. The tableau allows the optic axis to be matched with the coma-free axis.

History of Direct Aberration Correction

9

The second group of aberrations originates from time-dependent aberrations caused by charging, alternating external electromagnetic fields and mechanical instabilities. Meyer defined the deleterious effect of these incoherent parasitic aberrations as image blur. At that time, contrast transfer theory was not yet established in electron microscopy. Within the frame of notation of this theory, the resolution limit resulting from the incoherent aberrations is the so-called information limit. Meyer investigated the effect of the individual perturbations in great detail and designed and tested means for stabilizing the microscope and for effectively shielding the electron beam from the external electromagnetic fields. His results and findings served as guidelines for the second correction attempt at the Technical University of Darmstadt by Scherzer and Rose, which started in 1971 with the design of an aplanatic corrector (Rose, 1971b). Before describing this project, the correction efforts made at other institutes following the suggestions of Scherzer and the experimental work of Seeliger are detailed.

III. OTHER EARLY CORRECTION EFFORTS Scherzer’s ingenious idea to compensate for the unavoidable spherical aberration of round lenses by means of a corrector consisting of nonrotationally symmetric elements and the experiments of Seeliger and Moellenstedt initiated correction efforts at other places, predominantly in England. To simplify the arrangement of the Scherzer corrector, Archard (1954) replaced the two cylinder lenses and the round lens of the Scherzer corrector shown in Figure 1 by four quadrupoles forming an antisymmetric quadrupole quadruplet. The quadrupole fields were excited within octopole elements, which also produced octopole fields compensating for the combined spherical aberration of the round lens and the corrector together with the twofold and fourfold aperture aberrations introduced by the quadrupole fields of the corrector. In 1964, Deltrap used Archard’s suggestion for the construction of a nonsymmetric telescopic quadrupoleoctopole corrector to eliminate the spherical aberration of a probe-forming lens. With this corrector Deltrap nullified the spherical aberration, confirming the earlier results of Seeliger and Moellenstedt. But like his predecessors, he failed to improve the actual resolution of the uncorrected probe-forming system. At about the same time Dymnikov and Yavor (1963) proposed a symmetric quadrupole quadruplet with antisymmetric excitation of the quadrupole fields. Several versions of this prototype served later as correctors designed by Rose (1971a), Beck and Crewe (1974), Zach and Haider (1995), and Krivanek et al. (1998) to reduce the spot size of scanning electron microscopes. Figure 3 illustrates the arrangements of the quadrupoles

10

Harald Rose

MQ

MQ

MQ

MQ

FIGURE 3 Course of the principal rays and nodal rays in the x–z and the y–z sections of a telescopic quadrupole quadruplet; astigmatic line images of the infinitely distant plane are formed at the centers of the inner quadrupoles.

and the course of the fundamental rays in the x–z and y–z sections of the telescopic version proposed by Rose. Although Scherzer had shown that the departure from rotational symmetry suffices to eliminate chromatic aberration, feasible achromatic systems or elements were not found until Kelman and Yavor (1961) demonstrated that the chromatic aberration coefficient of a combined electrostatic-magnetic quadrupole can have negative sign. If the strengths of the electric and magnetic components are chosen such that their combined action nullifies for electrons with nominal energy, the element acts as a first-order Wien filter, which can be used to correct chromatic aberration in one section. However, the aberration will be doubled in the orthogonal section, if the filter is placed at a position free of astigmatism (Figure 4). In order to achieve a decoupled correction in both sections, Rose (1971a) proposed to use two first-order Wien filters, each of which is superposed onto the inner quadrupoles of the telescopic quadrupole quadruplet shown in Figure 3. If a round lens is placed in front of this system, it serves as a chromatic corrector. The combined system exhibits two orthogonal astigmatic images of the object plane, each of which is located at the center of one of the two Wien filters. The correcting force of the first-order Wien filter is proportional to the distance of the electron from the axis. Accordingly, each Wien filter affects only one of the x and y components of the chromatic aberration. Hence, by matching the correcting section of each Wien filter with the direction of the astigmatic line image, the two components of the axial chromatic aberration can be corrected independent of each other.

History of Direct Aberration Correction

x

11

DE . 0 DE 5 0 DE , 0 z

Object plane

Image plane

y

z

FIGURE 4 Action of a first-order Wien filter placed on the far side of an image-forming round lens. (See color insert).

The first experimental study with electric-magnetic quadrupoles was performed by Hardy (1967). He demonstrated in a proof-of-principle experiment that these elements enable chromatic correction. Figure 5 shows the combined electric-magnetic quadrupole used by Hardy for his experimental studies proofing the achromatic condition. Although the work of Deltrap and Hardy demonstrated that the techniques for correcting spherical and chromatic aberration were sound, they were never used in a standard scanning electron microscope. Owing to the complexity of the multi-element correctors, the difficulties in precisely aligning the system and the insufficient mechanical and electric stabilities, further work on correction was abandoned in England.

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Harald Rose

FIGURE 5 View of the combined electrostatic-magnetic quadrupole constructed and tested by Hardy (dissertation, 1967, Cambridge).

IV. THE DARMSTADT CORRECTION PROJECT Despite the frustrating failures to improve the performance of a welldesigned objective lens by means of a corrector, Scherzer strongly believed that the technological obstacles, which had prevented a gain in resolution of all efforts up to this date, could be overcome. For this purpose, he and his co-workers investigated the sources of the numerous perturbations preventing an improvement of the actual resolution. Moreover, to largely

History of Direct Aberration Correction

13

eliminate the effect of mechanical vibrations, he intended to incorporate the corrector in the Siemens 102 high-resolution electron microscope (Siemens AG). At that time, this instrument had the highest mechanical stability of all commercial transmission electron microscopes (TEMs) due to its large column diameter of 30 cm. It is quite interesting to note that all present atomic-resolution TEMs are equipped with such columns. Aside from eliminating the parasitic perturbations, it proved necessary to design a novel corrector for the TEM because the multipole fields of all previous correctors introduced large third-order off-axial comas and a twofold chromatic distortion that prevented a sufficiently large field of view. In addition, the concatenation of the comas with the axial aberrations introduced by the octopoles for correcting the spherical aberration resulted in large fifth-order aperture aberrations. To nullify the off-axial aberrations, it was necessary to increase the number of quadrupoles, thus further increasing the apparent complexity of the corrector. Intensive studies by Rose (1971b) revealed that a symmetric telescopic quadrupole quintuplet is free of coma and chromatic distortion if two of the four fundamental paraxial rays are symmetric and two antisymmetric with respect to the midplane of the central quadrupole. These symmetries had the additional advantage of facilitating the alignment of the corrector because deviations from symmetry can be detected very precisely and, hence, eliminated with high accuracy. The arrangement of the quadrupoles and their strength G2 are schematically depicted in Figure 6 together with the course of the axial fundamental rays xα , yβ and the field rays xγ , yδ . Astigmatic images of the object plane are formed at the centers of the inner three quadrupoles. In addition, an astigmatic image of the front nodal plane z−1 of the corrector is formed at the center of the long central quadrupole. The location of the objective lens is that of the test system. Due to mechanical constraints it was not possible for the test of the corrector to match its coma-free front nodal plane with the coma-free plane of the objective lens, which resulted in a nonvanishing round-lens coma. The preliminary theoretical studies also gave detailed information on the required stabilities for the currents and voltages applied to the constituent elements of the corrected system. Because the necessary stabilities could not all be achieved with commercial power supplies, novel supplies were designed and built to satisfy the requirements (Fey, 1980; Heinzerling, 1976). Work on the experiments started in 1972 supported by funds from the German Research Foundation. The individual components of the corrector were tested in a modified Zeiss TEM (Zeiss), which served as an electron optical bench. The first task was to demonstrate the correction of chromatic aberration in the image of a real object since this proof had not yet been furnished. To minimize the complexity of the experiment, G. Kuck used only the three inner electric-magnetic quadrupoles of the aplanator for his thesis

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Harald Rose

103 B U

V T

Frontlinse 4 2

103G2 mm2 Korrektiv

10 I

5

V III

50

0

II

100

z /mm 250

150

IV

25

10

mm

1 0 ZG ZF Z21 f 210

S2

I1 I2 S1

I3

S2

I2 I1 S1

x␣ ZK

ZM

y␦

f/2

ZE

Z1

Z

y␤

220

x␥ f/2

20 mm

FIGURE 6 Field distribution of the objective lens and of the quadrupoles of the aplanator (upper drawing) and course of the fundamental rays in the x–z section and the y–z section, respectively.

work (Kuck, 1979). This symmetric quadrupole triplet forms an undistorted stigmatic image with unit magnification and corrects simultaneously the chromatic aberration of two projector lenses. The simple setup was sufficient for demonstrating chromatic correction in the image of a real object. In order to increase the chromatic aberration, Koops et al. (1977) added an alternating voltage of 130 V to the cathode potential, thereby strongly blurring the image. The image became sharp again after he corrected the chromatic aberration by properly adjusting the electric and magnetic quadrupoles, as shown in Figure 7. Thanks to the convincing demonstration of chromatic correction by Gerd Kuck, Scherzer received further funds that enabled him to continue with the experiments. In the next step, W. Bernhard constructed the aplanator by adding the missing outer quadrupoles of the aplanator to those of the triplet. Additional coils were applied to each octopole element for producing octopole fields to enable the correction of spherical aberration. Subsequently, the aplanator was placed behind the objective lens of the test microscope, (Figure 8). Appropriate stigmator coils were added to compensate for the unavoidable parasitic second-order axial coma and the threefold astigmatism. The astigmatic course of the paraxial rays within

History of Direct Aberration Correction

15

FIGURE 7 Image of a holey carbonaceous foil taken by an electric-magnetic quadrupole triplet with unit magnification and two projector lenses. The images on the left were obtained without chromatic correction; those on the right-hand side were obtained with chromatic correction. The images on the bottom were formed by adding an alternating voltage of 130 V to the 40 kV of the cathode potential.

the corrector makes it possible to eliminate the threefold astigmatism and the axial coma by means of the threefold stigmator fields. The experiments proved the second-order stigmators to be indispensable because it is impossible to match sufficiently accurately the axes of all quadrupole and octopole fields with the optic axis defined by the objective lens. The reasons for this inaccuracy are mechanical imperfections, misalignment, and magnetic inhomogeneities within the pole pieces. After eliminating the second-order aberrations Bernhard succeeded in correcting the third-order aperture aberrations together with the chromatic aberration (Bernhard, 1980). Bernhard’s experiments showed that the resolution of the corrected microscope was limited primarily by external stray fields and mechanical instabilities of the test microscope. In his thesis work, Hely (1981, 1982) largely reduced the deleterious effect of the external parasitic electromagnetic fields by incorporating a novel shielding device designed by Pejas (1978). This effective shielding improved the resolution by a factor of 3. Nevertheless, the attained resolution still was limited by the insufficient stability of the multipole fields and the mechanical instabilities of the test microscope. Moreover, the central octopole element introduced a very large fifth-order aperture aberration coefficient of ∼ 55 cm.

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Harald Rose

1

2

H

O

K

S

5 cm

Z

FIGURE 8 Drawing of the aplanator placed behind the objective lens O of the test electron microscope.

History of Direct Aberration Correction

17

It was hoped that the remaining obstacles could be overcome by incorporating the corrector into the rather stable Siemens 102 TEM and by using sufficiently stable voltage and current supplies designed by Fey (1980). In addition, Haider, Bernhardt, and Rose (1982) designed, constructed, and successfully tested an electric-magnetic 12-pole element, which was substituted for the central octopole element of the aplanator. The excitation of a quadrupole field within an octopole element is accompanied by a dodecapole field, which produces a fifth-order aperture aberration. Because this aberration was very large only for the central octopole, it was sufficient to replace this octopole element by the newly designed 12-pole element. This element allows excitation of the quadrupole and octopole fields without introducing a dodecapole field. In 1981 the dodecapole element was inserted into the aplanator. First experiments showed an appreciable reduction of the fifth-order aberration. The cross section of this final version of the aplanator is shown in Figure 9.

FIGURE 9 Cross section of the objective lens and the final version of the aplanator of the Darmstadt corrected electron microscope.

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Harald Rose

Because the dodecapole element is also very suitable for producing a pure hexapole field with arbitrary azimuthal orientation about the optic axis, Max Haider investigated this excitation mode too. The results showed that the dodecapole was a very suitable element for the novel hexapole corrector proposed by Rose (1981).

V. THE CHICAGO AND EMBL QUADRUPOLE-OCTOPOLE PROBE CORRECTORS The Darmstadt project was abandoned after Scherzer died in 1982, although it was successful as far as it went. Fortunately, Haider was offered a position at the European Molecular Biology Laboratory (EMBL) at Heidelberg where he was able to continue his experimental work on novel electron optical elements. The knowledge and the experience he had gained from his thesis work combating the obstacles of aberration correction proved invaluable for his later work on the TEM hexapole corrector. Apart from the Darmstadt project, another attempt to correct the spherical aberration of a scanning transmission electron microscope (STEM) was started in 1972 by Albert Crewe and Vernon Beck at the University of Chicago (Beck and Crewe, 1974). They built and tested over a period of six years a magnetic quadrupole-octopole corrector consisting of a symmetric quadruplet with combined quadrupoles and octopoles. Because the corrector is designed for the STEM, the difficulties encountered by the Darmstadt project of correcting a sufficiently large field of view do not arise. Although Beck and Crewe did introduce many stigmator coils for producing weak dipole and hexapole fields, they were unable to find a suitable setting. As stated by Crewe (2002), the reasons for their failure to align the corrector with the required accuracy were very likely magnetic hysteresis effects and inhomogeneities within the iron pole pieces. Moreover, the tools necessary to determine the state of alignment, the parasitic geometrical aberrations, and to adjust the many multipole fields with the required accuracy were not yet available. Because of the continuous failure to achieve any real gain in resolution, the experimental work on aberration correction stopped until Joachim Zach tackled the problem experimentally anew in the early 1990s. In his thesis, he had designed a chromatic and spherically corrected low-voltage scanning electron microscope (LVSEM) using a symmetric dodecapole quadruplet (Zach, 1989), because the work of Haider had shown the advantages of dodecapole elements. After he finished his thesis work at the University of Darmstadt, Zach joined Haider’s group at the EMBL in Heidelberg. Because the chromatic aberration determines primarily the resolution of the LVSEM rather than the mechanical and electromagnetic instabilities, the chances to improve the instrumental resolution by correcting the aberrations seemed high enough to justify

19

History of Direct Aberration Correction

QPmag

2.5

QPel&mag

QPel&mag

QPmag

Objective lens dE 5 0 dE . 0 dE , 0

2 1.5 1 0.5

Y – Z plane

0 X – Z plane

20.5 21 21.5 22 22.5

0

50

100

150

200

FIGURE 10 Schematic arrangement of the elements of the corrected LVSEM and course of the axial trajectories for different energy deviations illustrating chromatic correction.

another correction attempt. Therefore, a new Cc/Cs quadrupole octopole corrector was designed and built by Lanio and Haider at the EMBL. Unfortunately, the first tests of the corrector revealed that the hysteresis causes cross-talk between magnetic multipole fields that are combined in a single element. The unavoidable cross-talk prevented a suitable adjustment of the corrector, as had been the case for the Chicago corrector. To eliminate this difficulty, Zach substituted electric multipoles for the magnetic multipoles apart from the two inner magnetic quadrupoles, which are mandatory for eliminating the chromatic aberration. With the redesigned corrector, Zach and Haider (1995) achieved for the first time a real improvement in resolution of an actual electron microscope. The chromatic correction is schematically illustrated in Figure 10.

VI. EVOLUTION OF THE HEXAPOLE CORRECTOR For a long time sextupoles were not considered suitable candidates for correcting the aberrations of round lenses because they introduce in first approximation a threefold second-order path deviation. It was first pointed

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Harald Rose

out by Peter Hawkes in 1965 that in second approximation the sextupoles introduce additional third-order aberrations equivalent to those of round lenses. In the beginning of the 1970s, Rose and Plies (1973) investigated in detail the aberrations of electron optical systems with curved axis consisting of magnetic round lenses, dipoles, quadrupoles, sextupoles, and octopoles. In particular, they derived formulas for the coefficients of the third-order aberrations and showed that the spherical aberration can have negative sign even if the index of refraction is rotationally symmetric in the paraxial domain (Plies, 1973). However, they did not explore further the properties of sextupoles in combination with round lenses because the dominant second-order aberrations of the sextupoles seemed to rule out their usefulness for correcting the much smaller third-order spherical aberration of a good objective lens. It was not until 1979 that Beck showed that a combination of a round lens and two sextupoles has a negative spherical aberration and can be made free of second-order aperture aberration. Unfortunately, his systems and those suggested by Crewe (1980) introduce a large threefold fourthorder aperture aberration. This aberration results from the combination of the large uncorrected second-order field aberrations with the third-order axial aberrations and prevents a significant improvement in resolution. Moreover, the axes of the sextupoles must match the axis of the round lens with an extreme accuracy that can barely be realized in practice. The problem was solved in 1981 by Rose, who proposed a sextupole or hexapole corrector consisting of a telescopic round lens doublet and two identical sextupoles—one centered at the front focal point of the first round lens and the other at the back focal point of the second lens (Figure 11). To eliminate all second-order aberrations introduced by the hexapole fields, Rose imposed symmetries on the hexapole fields and on the course of the fundamental paraxial rays because this method had been extremely powerful in earlier work for eliminating the primary aberrations of deflection systems, imaging energy filters in particular (Rose and Plies, 1974; Rose, 1978). The magnetic lenses of the telescopic round-lens doublet have equal and opposite excitation, which guarantees that the Larmor rotation of the paraxial rays cancels out at the far side of the second lens. The telescopic doublet forms a so-called 4f-system, which images the front focal plane of the first lens with magnification −1 onto the backfocal plane of the second lens. The corresponding focal points are conjugates, which coincide with the nodal points N1 and N2 of the doublet. Figure 11 demonstrates that the fundamental rays uα and uγ of the paraxial trajectory u(1) = ωuα + uo uγ exhibit double symmetry with respect to the hexapole field of the sextupoles; uo = xo + iyo defines the starting point of the trajectory at the object plane. The parameters ω1 = α + iβ and ω2 = xo + iyo are complex linear combinations of the slope components α, β and of the position coordinates xo , yo

History of Direct Aberration Correction

21

Corrector

Sextupole

Round-lens doublet

Sextupole Axial ray

u␣ z u␥

f

N1

2f

Field ray

f

N2

FIGURE 11 Schematic arrangement of the elements of the hexapole corrector and course of the fundamental rays, N1 and N2 , indicate the location of the object and image nodal points, respectively.

of the trajectory at the object plane. Conjugate complex quantities will be indicated by a bar. The axial fundamental ray uα is symmetric within the field of each sextupole but antisymmetric about the mid-plane zm of the entire hexapole field, while the field ray uγ has the opposite symmetry. The hexapole fields introduce a second-order deviation u(2) = ω12 u11 + ω1 ω2 u12 + ω22 u22 of the general trajectory from its Gaussian approximation. Owing to the imposed symmetry, the second-order fundamental rays u11 , u12 , u22 vanish in the region outside the corrector (Figure 12). Therefore, the second-order deviation vanishes for any ray after it passes through the corrector. The front nodal point N1 of the telescopic round-lens doublet coincides with its coma-free point. Each round lens has a coma-free plane whose center represents the coma-free point. If a beam-limiting aperture is placed at this plane, the image will be free of isotropic coma. The same holds true for the STEM, if the scan pivot point is matched with the coma-free point of the objective lens. For magnetic lenses, the coma-free plane falls within the lens field upstream from the diffraction plane. The coma-free plane always exists in the presence of spherical aberration. However, if we eliminate the spherical aberration, the coma no longer depends on the location of the beam-limiting aperture or the image of the effective source, respectively.

22

Harald Rose

Transfer doublet Sextupole

Sextupole

u11

u22

zm

z u12

FIGURE 12 Course of the second-order fundamental rays u11 , u12 , and u22 within the hexapole corrector.

Hence, in order to eliminate the isotropic coma too, the coma-free point of the objective lens must be matched with that of the corrector. Unfortunately, each point is located in the interior of one of these components. Therefore, it is not possible to place these points in a common plane of a real system. Because of this shortcoming, the general opinion prevailed up until 1990 that hexapole correctors will only be usable, if ever, for the STEM. In 1990, Rose found an optical approach to satisfy the coma-free condition by using the property of the telescopic round-lens doublet to image the front-nodal plane onto the back-nodal plane without coma. He placed another transfer doublet between the objective lens and the corrector such that the doublet images the coma-free plane of the objective lens onto the equivalent plane zN1 of the hexapole corrector. The entire system (shown in Figure 13) forms a semi-aplanat, whose field of view is limited by the small anisotropic coma of the objective lens. Due to the ongoing failure and the enormous technological difficulties in improving the performance of an actual high-resolution electron microscope by means of a corrector, a group of experts declared in 1988 at a meeting in the United States that the successful realization of aberration correction was unthinkable. As a result, funding was stopped worldwide. After becoming aware of the novel hexapole corrector, Haider was convinced that it would work in practice because the hexapole fields need only a relative stability of 10 ppm, which is two orders of magnitude less than that required for the quadrupoles of the correctors built in the past. In order

23

History of Direct Aberration Correction

Corrector

Magnetic objective lens

Transfer doublet

Sextupole

Round-lens doublet

Sextupole Axial ray

u␣ z u␥ f

2f

2f

f

Field ray

8f

Object Coma-free plane plane 5 nodal plane N0

N1

N2

FIGURE 13 Semi-aplanat obtained by matching optically the coma-free point N0 of the objective lens with the corresponding point N1 of the corrector by means of the coma-free transfer doublet.

to receive funding, it was necessary to convince the electron microscopists that this corrector was a realistic and promising alternative because the required technology was available in 1990. Moreover, it was necessary to demonstrate the advantage of aberration correction in a high-performance TEM operating on a routine basis. At the end of 1990, Haider, Rose, and Urban submitted a joint grant application to the Volkswagen Foundation to obtain the necessary funds. All other granting agencies refused to fund a “nonfeasible” project. Fortunately, the Volkswagen Foundation was willing to take the risk and approved funding in 1991. The hexapole corrector was built by Haider at the EMBL in Heidelberg, The hexapole fields are excited in dodecapole elements shown in Figure 14. These elements allow adjustment of the strength and the azimuthal orientation of the hexapole field. The first tests by Haider, Braunshausen, and Schwan (1995) demonstrated that the corrector worked satisfactorily. However, in order to achieve an actual improvement of the resolution by correcting the spherical aberration, two additional major obstacles had to be overcome. The failures of the previous correction attempts had shown that the correction of the spherical aberration does not suffice to improve the resolution of the electron microscope. For the correction to improve the resolution, the information limit of the basic microscope must be significantly smaller than the so-called Scherzer resolution limit defined by the spherical aberration. In addition to reducing the incoherent aberrations, which determine the information limit, the coherent

24

Harald Rose

FIGURE 14 Dodecapole elements producing the hexapole fields of the spherical aberration corrector.

geometrical aberrations resulting from misalignment and inaccuracies of the elements also must be kept sufficiently small. Uhlemann and Haider (1998) solved this problem by developing an iterative computer-aided procedure, that precisely determines all residual axial aberration coefficients up to the fourth-order inclusively by using the diffractogram tableau method introduced by Zemlin et al. (1977). Subsequently, the disturbing residual aberrations are eliminated by aligning the optic axis at appropriate positions within the system. About three iteration steps suffice to align the system with the required accuracy by means of microprocessors. The fast alignment procedure was the key to the success of the project, because the system can be aligned sufficiently accurately only if the required time is appreciably shorter than the time during which the system remains sufficiently stable. The inability to satisfy this requirement was very likely a decisive reason for the failure of all previous correction attempts. After the successful test of the corrector, it was incorporated in a highresolution 200-kV TEM. In order to reduce the resolution limit from the Scherzer limit of 0.24 nm by a factor of one half down to 0.12 nm, it was necessary that the information limit of the basic microscope not surpass this limit. Although the manufacturer had guaranteed to meet this requirement, the actual instrument did not fulfill the specifications. It took Haider more than a year to reduce the information limit from about 0.2 nm to 0.12 nm,

History of Direct Aberration Correction

25

which was done primarily by lowering the gun temperature and replacing the tip of the field emission gun. The final information limit of ∼ 0.12 nm is determined by the chromatic aberration rather than by mechanical and electrical instabilities. The corrector was incorporated in the upgraded microscope in 1995. The breakthrough was achieved in 1997 (Haider et al., 1998) when a resolution limit of ∼ 0.14 nm was reached. This remarkable result represents a milestone in the struggle to improve the resolution of a high-performance electron microscope by correction of the spherical aberration. The lifetime of the corrected state, during which the phase shift varies less than π/4, lasted 90 minutes. After this time only the axial coma had to be adjusted. The twofold astigmatism and the defocus were less stable and had to be adjusted for each exposure. The images furnished by the aplanatic objective lens exhibit a surprisingly high contrast. Moreover, the correction of spherical aberration strongly reduces the formation of artifacts resulting from delocalization. These artifacts arise predominantly in the image of nonperiodic object details such as dislocations or interfaces. The coherent superposition of the waves originating from randomly distributed atoms produces at the image plane positions with constructive and destructive interference of these waves. The resulting artifacts are known in light optics as speckles. By correcting the spherical aberration, the extension of each scattered wave at the image plane is confined to the Airy disc whose radius equals the resolution limit. Hence, only few discs overlap when the spherical aberration has been corrected. After the successful test of the corrected microscope, it was transferred to Juelich and there further improved. Routine operation started at the beginning of 2000. Since this date, the corrected microscope operates in the same way as any standard commercial high-resolution TEM. The impressive results obtained by Urban and co-workers (Urban, 2008) with the aberration-corrected microscope have received great attention and recognition, especially among the members of the materials science community.

VII. ABERRATION CORRECTION OF SYSTEMS WITH CURVED AXIS The task of correcting aberrations in electron-optical systems with curved axis started in 1973 during a workshop on high-resolution electron microscopy in biology held at the small village of Geiss in the Swiss Canton Appenzell. The meeting was organized by the late Edward Kellenberger

26

Harald Rose

from Basel. One evening, while drinking some local red wine after the lectures, Peter Ottensmeyer asked Harald Rose what he could do to improve the performance of his imaging Castaing–Henry filter, which he had incorporated in a Zeiss TEM. His main objective was to increase the voltage and to reduce the disturbing primary second-order aberrations, which limit the field of view, distort the image, and prevent isochromatic filtering. The Castaing–Henry filter consists of a magnetic prism and an electrostatic electron mirror. The accelerating voltage is limited to ∼ 80 kV to avoid electric discharges within the mirror. In order to eliminate this obstacle, Rose substituted two 90-degree homogeneous deflection magnets for the mirror. Because there was no simple immediate answer of how to compensate for the disturbing second-order aberrations, Rose started intensive investigations on the problem with E. Plies after he returned home. Their studies revealed that imposing symmetries on the fields and the paraxial rays is a very effective means of eliminating the primary aberrations of systems with curved axis. A single midplane symmetry suffices to eliminate half of the second-order geometrical aberrations. Utilizing this behavior, they designed the first symmetric 90-degree omega filter free of second-order aperture aberration and distortion (Rose and Plies, 1974). The remaining aberrations can be eliminated by introducing an additional symmetry plane for each half of the filter. However, this procedure also eliminates the dispersion outside the filter. Although the resulting achromatic system cannot be used for spectrum imaging, it is well suited as a monochromator with internal energy selection. The dispersion-free CEOS monochromator designed by Kahl and Rose (1998) uses double symmetry to eliminate all second-order aberrations outside the monochromator. The aperture aberrations at the energy-selection plane outside the filter prevent isochromatic energy filtering. Since these aberrations cannot be eliminated by imposing symmetry conditions without canceling out the dispersion as well, another correction procedure is needed. To gain an insight into the structure of the integrands of the aberration coefficients of systems with curved axis, Plies and Rose (1971) reformulated the theory of electron-optical systems with curved axis introduced by Cotte (1938). They expanded the electromagnetic potential in a series of multipoles centered about an arbitrarily curved and twisted axis, whereupon the integrands become functions of the paraxial rays and the multipole strengths along the curved optic axis. The dipole and quadrupole strengths and the axial potential define the course of the paraxial rays. Hence, additional higherorder multipoles can be incorporated to compensate for the aberrations without affecting the paraxial rays. The structure of the integrands reveals the most suitable locations for placing the correcting multipoles, which are sextupoles, for the primary second-order aberrations. Hexapole fields had

History of Direct Aberration Correction

27

been generated previously by curving the entrance and/or exit pole faces of homogeneous deflection magnets used in electron spectroscopy. Scherzer did not anticipate that it would be possible to align an imaging energy filter with the required accuracy because of the difficulties in aligning the deflection magnets, the curved optic axis, and the required course of the paraxial rays. Fortunately, Dieter Krahl at the Fritz-HaberInstitut in Berlin did not share this pessimistic opinion and was convinced that it would be possible to realize a corrected magnetic imaging energy filter. In his first setup he generated the correcting hexapole fields by curving the pole faces of the magnets. However, the experimental tests were very disappointing because it was impossible to precisely align the system. Detailed theoretical investigations by Rose showed that the curvature of the pole faces creates chaotic behavior during the alignment of systems consisting of several deflection magnets. To avoid this insurmountable difficulty, an improved system was designed by Lanio, Rose, and Krahl (1986) consisting of four deflection magnets with straight pole faces and seven adjustable sextupoles. In addition, over a period of three years Kujawa et al. (1990) developed an efficient adjustment procedure enabling a precise alignment of the magnets and the sextupole elements. The improved filter can be precisely aligned and its aberrations are corrected within a short period of time. This filter has become an essential part of the Zeiss LIBRA analytical 200-kV TEM (Carl Zeiss SMT AG). Omega filters are straight-vision systems that are placed within the column of a TEM. In order to convert a conventional TEM retroactively into an energy-filtering TEM, Krivanek, Gubbens, and Dellby (1991) developed an attachable imaging energy filter placed beneath the viewing screen. This post-column filter consists of a single 90-degree deflection magnet with curved pole faces followed by a sequence of six quadrupoles and seven sextupoles. The precise alignment and excitation of the many elements of the post-column filter is performed by means of a computer. The sextupoles compensate for the second-order distortion and the chromatic aberration of magnification. These aberrations are eliminated by symmetries in the omega filter. The method of correcting aberrations by imposing symmetry conditions was also used by Mueller, Preikszas, and Rose (1999) to eliminate the dispersion and the second-order aberrations of the beam separator for the mirror corrector of the SMART (Spectro-Microscope for All Relevant Techniques). This spectroscopic microscope (Fink et al. 1997) can operate as a low-energy electron microscope (LEEM) and as a photo-emission electron microscope (PEEM). The schematic arrangement of the constituent elements is shown in Figure 15. The doubly symmetric course of the axial fundamental rays and the grooves for the coils are shown in Figure 16 for one quarter of the magnetic beam splitter, which separates the reflected beam from the beam propagating toward the mirror.

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Harald Rose

Mirror corrector

Objective Electric/ magnetic Specimen deflector Objective lens

Transfer optics

Energy filter

Electron source Deflector D1 Transfer lens T1 Field aperture

Projector/Detector

Energy selection slit

Projector

Condenser D2

L3

L4

D3

L5 T2

Dipole P1

D4

Quadrupole

T3 D5

P4

H6 Field lens L1

Optic axis

Beam seperator

L2 X-ray mirror

H2

Hexapole H1

H5 H4

H3 P2

X-ray illumination

Camera system

Apertures

Electron mirror

P3

Dodecapole

Electric-magnetic multipole deflector elements 0

100

200

300

400

Electrode surfaces Energy selection slit for x-ray illumination

500

600 mm

Axial ray

Polepieces

Field ray

Coils

Dispersive ray

FIGURE 15 Schematic depiction of the SMART (Hartel, Preikszas, Spehr, Mueller and Rose (2002)). (See color insert).

S2 S1

S2

Pole piece plate Coil Axial electron bundle

FIGURE 16 One quarter of the SMART beam separator showing the grooves for the coils and the doubly symmetric course of the axial rays, S1 and S2 , indicate the symmetry planes.

History of Direct Aberration Correction

29

VIII. REVIVAL OF THE QUADRUPOLE-OCTOPOLE CORRECTOR The successful alignment of the many elements of the post-column filter convinced. Krivanek (1994) that it also should be possible to precisely align the many elements of a quadrupole-octopole corrector by means of a computer. He started work on aberration correction around 1995 at the Cavendish Laboratory in Cambridge with an improved version of the Deltrap quadrupole-octopole corrector aimed to compensate for the spherical aberration of a VG STEM operating at an accelerating voltage of 100 kV. Krivanek and co-workers constructed the corrector in such a way that apart from the basic magnetic quadrupole and octopole fields for the correction of the spherical aberration, additional multipole fields could be excited under computer control to compensate for deleterious parasitic geometrical aberrations caused by misalignment, magnetic inhomogeneities, and mechanical inaccuracies. The experiments showed that cross-talk between the quadrupole and octopole fields posed a major difficulty for the alignment, as had been the case with the Chicago corrector. To avoid this obstacle, Krivanek, Dellby, and Lupini (1999) designed a new corrector such that the quadrupole and octopole fields are excited in separate elements. The corrector was incorporated in a 100-kV VG STEM. To adjust the system and to determine the resolution-limiting aberrations, they developed a procedure based on the evaluation of Ronchigrams. With these improvements they succeeded in rapidly performing the many adjustment steps in a systematic way. The results demonstrated for the first time a genuine improvement in resolution of a STEM by correction of the spherical aberration. After leaving Cambridge, Krivanek and Dellby set up the company NION to manufacture and improve their STEM corrector. Because they did not intend to provide the corrector to the electron microscope manufacturers, they decided to develop themselves a high-performance STEM aiming for a resolution limit of 0.5 Å. The realization of aberration correction was achieved at research laboratories and not by the manufacturers of electron microscopes. The reasons for this failure by the manufacturers have been the lack of expertise and the belief that aberration correction would not be possible in a commercial electron microscope operating on a routine basis. The situation changed after Haider and Zach established the company CEOS in 1996 with the goal of providing feasible correctors for high-resolution electron microscopes. Since the hexapole corrector was first used in the TEM and the quadrupoleoctopole corrector in the STEM, it was widely misbelieved that the hexapole corrector would not be appropriate for the STEM. This was probably the reason that FEI (FEI Company) developed its own quadrupole-octopole Cs corrector. The design of the corrector and first results were reported by

30

Harald Rose

van der Zande et al. (2003) at the Third TEAM Workshop during the M&M meeting in San Antonio. The corrector was incorporated into a modified Tecnai F20 Super-Twin. First experiments had shown that by correcting the spherical aberration of the objective lens, the optimum opening angle increased more than twofold. However, further work on this corrector was suspended without any comments. Owing to the successful correction of spherical aberration, in 1999 Murray Gibson at the Argonne National Laboratory had the visionary idea to realize a sub-angstrom and sub–electron volt in situ electron microscope. He intended to convert the standard electron microscope into a materials science laboratory with adequate space to conduct experiments under various environmental conditions. The dynamic microlaboratory requires an increased space within the pole piece of the objective lens. However, increasing the gap between the pole pieces enlarges the focal length and the chromatic aberration, thus preventing sub-angstrom resolution. Hence, atomic resolution can only be achieved for the proposed microscope by correcting spherical and chromatic aberration. In order to establish a national project for the development of a transmission electron achromatic microscope (TEAM), Gibson organized a workshop on this topic in July 2000. After extensive discussions, the numerous experts came to the conclusion that the project was feasible. Four national laboratories decided to develop a mutual proposal for the construction of the microscope with Gibson as coordinator. A crucial part of the project was the design of a novel corrector compensating for spherical and chromatic aberrations and for off-axial coma to obtain a sufficiently large field of view of at least 2000 equally resolved object points per image diameter. In order to find an appropriate corrector design, Gibson invited Harald Rose by the end of 2000 to tackle this challenging task during a stay at the Argonne National Laboratory. At that time, Rose was involved in the problem of finding a suitable corrector for projection electron lithography. In order to image all points of a large mask with equal resolution onto the wafer, it is necessary to remove chromatic aberration, image curvature, and field astigmatism. This problem initiated the question of whether it is possible to compensate simultaneously for the chromatic aberrations and all third-order geometrical aberrations by means of a suitable corrector. Correction of aberrations necessitates departure from rotational symmetry. However, the incorporation of multipole elements increases significantly the number of aberrations. Hence, the inherent aberrations of the round lenses and the additional nonrotationally symmetric aberrations introduced by the multipole fields must be eliminated. In order to solve this formidable problem, Rose again used symmetry considerations. By imposing symmetry conditions on the multipole fields and the course of the fundamental paraxial rays, a large number of aberrations cancel out. The largest reduction is achieved by imposing

31

History of Direct Aberration Correction

symmetry conditions on the system as a whole and on each half of it. Using this procedure, Rose found a feasible corrector during his stay at Argonne in 2002. This ultracorrector eliminates all primary chromatic and geometrical aberrations. One version consists of two identical telescopic multipole septuplets. The quadrupole fields are symmetric with respect to the center plane of each septuplet and antisymmetric with respect to the plane midway between the two subunits. A strongly anamorphotic image of the front-nodal plane N1 is formed at the center of each septuplet (Rose, 2004). By splitting the central element of each septuplet into two separate quadrupoles, each anamorphotic image is placed midway between the central quadrupoles of the resulting octuplet (Figure 17). The third-generation NION corrector also uses a multistage quadrupole octopole arrangement, as discussed in detail in the chapter by Krivanek. In 2002, Uli Dahmen from the Lawrence Berkeley National Laboratory became the coordinator of the TEAM project when Gibson became director of the Advanced Photon Source at Argonne National Laboratory. The proposal for funding of the project was submitted to the Department of Energy (DOE) and was approved in 2003. However, the original aim and its name were changed from the achromatic in situ electron microscope to

O1

O2

O1

y␤

N1

N2

ZM

x␣

y␦

x␥ Q1

Q2

Q2

Q1 Q1

Q2

Q2

Q1 Q1

Q2

Q2

Q 1 Q1

Q2

Q2

Q1

FIGURE 17 Schematic arrangement of the quadrupoles and course of the fundamental rays within the ultracorrector, the octopoles (O1 and O2 ) compensate for the spherical aberration. (See color insert).

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the “Transmission Electron Aberration-Corrected Microscope,” although the acronym TEAM remained unchanged. In order to realize the required resolution limit of 0.5 Å, the information limit of the basic instrument must stay below 0.4 Å because the corrector also introduces incoherent aberrations, which enlarge this limit. Because of the enormous technical difficulties to obtain the required information limit at 200 kV, the accelerating voltage was raised to 300 kV after the company FEI received the contract to manufacture the basic instrument in 2004. At the same time the company CEOS obtained the contract to provide the corrector compensating for chromatic and spherical aberrations and the off-axial coma of the round lenses of the basic instrument. To keep the number of corrector elements as small as possible, Mueller, Uhlemann (CEOS), and Rose designed an alternative for the ultracorrector by replacing each sextupole element of the hexapole corrector with a telescopic quadrupoleoctopole quintuplet (Figure 18). The resulting TEAM corrector is shown schematically in Figure 19. Detailed calculations proved the feasibility of the corrector concept. By optimizing the design, Mueller and Uhlemann obtained an achromatic aplanatic system, which satisfies theoretically all requirements. The most stringent constraints are the extremely high stabilities of ∼ 2 × 10−8 for the electric and magnetic quadrupole fields correcting for the chromatic aberration. These fields are excited within the central element of each multipole quintuplet. Fortunately, in 2006 CEOS reached the unprecedented relative stability of 1 × 10−8 for the necessary currents and voltages. Q1

Q2

Q3

Q2

Q1

y␯ Principal ray

x␯

Z zN

zN

y␲

Nodal ray

x␲ ZS

FIGURE 18 Schematic course of the x- and y-components of the principal ray and the nodal ray, respectively, within the telescopic quadrupole-octopole quintuplet.

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History of Direct Aberration Correction

Multipole quintuplet y␤

Round-lens transfer doublet

Multipole quintuplet x␣

Field ray z

zm

zN

zN Axial ray

x␥

z1

y␦ z2

FIGURE 19 Schematic arrangement of the TEAM corrector and course of the fundamental rays, strongly anamorphotic images of the diffraction plane are located at the center planes z1 and z2 of the multipole quintuplets.

Placing an anamorphoptic image of the diffraction plane at the symmetry plane zs of the quintuplet requires matching the principal ray with the field ray uγ and the nodal ray with the axial ray uα (Figure 19). The axial ray starts from the center of the object plane, and the field ray intersects the center of the diffraction plane. The central element of each multipole quintuplet is an electric magnetic dodecapole enabling independent excitation of electric and magnetic quadrupole fields and an octopole field. The mixed electric and magnetic quadrupole fields compensate for the axial chromatic aberration and the octopole field for the spherical aberration of the round lenses without introducing any appreciable field aberrations. The remaining fourfold axial astigmatism can be eliminated by an additional octopole element placed at an distortion-free image of the diffraction plane, either at the plane zm midway between the round lenses of the transfer doublet or at the conjugate plane behind the corrector. The latter location is more favorable because it induces smaller fifth-order combination aberrations than the other location. The comparison of Figure 11 with Figure 19 demonstrates that the double symmetry of the fundamental rays with respect to the multipole fields of the hexapole corrector is also preserved in the TEAM corrector, although the x- and y-components of the fundamental rays differ from each other within the multipole quintuplets. The ray components exhibit exchange symmetry with respect to the midplane zm because the quadrupole fields are excited antisymmetric with respect to this plane.

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IX. CONCLUSION AND OUTLOOK The invention and the development of the microscope has been one of the greatest achievements of mankind. By using electrons instead of light and aberration-corrected electron lenses, it is now possible to explore the microscopic structure of objects down to their smallest constituents: the atoms. At the beginning of light microscopy, about 200 years ago, it was widely believed that the improvement of resolution was merely a matter of eliminating the aberrations of the lenses because it was known from experience that the resolution is directly related to the size of the usable aperture angle. Therefore, all efforts were directed at compensating for the aberrations by designing systems composed of an increasing number of lenses. At the end of the 18th century, Ernst Abbe (1878) achieved the best possible performance of the light microscope by reaching a usable numerical aperture somewhat larger than 1. Simultaneously he discovered that diffraction was the ultimate limit that prevents pushing the resolution limit beyond about half the wavelength λ. Decreasing the resolution limit of the microscope beyond the wavelength of visible light, required finding some other radiation with a much shorter wavelength than that of light. The electron provided the appropriate radiation. After Busch had shown in 1927 that an axially symmetric magnetic field focuses electrons in the same way as a glass lens the light, Ruska (1980) developed the electron microscope. Within a short time the resolution of the electron microscope surpassed that of the light microscope (Figure 20). This success resulted primarily from the extremely small wavelength of the electrons rather than from the quality of standard electron lenses, which limit the attainable resolution to ∼ 100 λ. Therefore, shortening the wavelength by increasing the voltage was the most convenient method for improving the resolution. However, radiation damage by knock-on displacement of atoms severely limits the application of high-voltage electron microscopes. In addition, the so-called delocalization caused by spherical aberration prevents an unambiguous interpretation of images of nonperiodic objects such as interfaces and grain boundaries. The correction of the spherical aberration eliminates this deleterious effect. The successful correction of the spherical aberration can be considered a quantum step in the development of the electron microscope because it enables sub-Angstrom resolution at voltages below the threshold for atom displacement. The threshold voltage depends on the composition of the object and lies in the region between 60 and 300 kV for most materials. The ultimate goal of aberration correction is the development of a subangstrom and sub–electron volt analytical electron microscope operating at voltages below the threshold for atom displacement. Such a microscope can be realized in principle by incorporating the MANDOLINE

History of Direct Aberration Correction

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Aberrationcorrected EM

Å21

TEAM

Stuttgart (1.2 MV)

1

Haider et al. (200 kV)

Dietrich (200 kV)

1021 Zach, Haider (1 kV, SEM)

22

Resolution

10

1023

Borries and Ruska, Marton, Ardenne (100 kV)

Electron microscope

Light microscope Ruska (75 kV) Abbe Amici

1024 Ross 1025 1800

1850

1900

1950

2000

2050

Year

FIGURE 20 Increase of resolution in diffraction-limited microscopy as a function of time.

filter (Uhlemann and Rose, 1994) into the TEAM microscope. At present only a single MANDOLINE filter exists; it is incorporated into the 200-kV SESAME microscope at the Max Planck Institute in Stuttgart. This filter allows isochromatic energy filtering of large object areas with an energy resolution below 0.1 eV. The history of aberration correction demonstrates the necessity of longterm research funding for achieving real breakthroughs. The success of the seemingly fruitless correction efforts provides a lesson showing that real advancement in science requires endurance, devotion, and team work.

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The failures of the many previous correction attempts have not been in vain because they paved the route for the final success. The results of the early unsuccessful experiments showed that aberration correction requires high technology and fast computers, which were not available at the time. About ten years ago the technological tools reached the necessary standard for achieving sub-angstrom resolution. Using these aids, Uhlemann and Haider (1998) developed computer-controlled procedures that precisely determine the state of alignment and provide the means for eliminating the resolution-limiting residual geometrical aberrations. Pushing the information limit of the microscope below 0.5 Å, the aim for the TEAM project—requires extreme mechanical and electrical stabilities. Thanks to the intensive cooperation of many experts, these requirements can now be met. Therefore, no insurmountable physical and technological obstacles exist any longer that prevent achieving the goal of the TEAM project. Moreover, future advancements in technology will further improve the resolution and enable atomic resolution at voltages down to ∼ 50 kV.

ACKNOWLEDGMENTS I want to thank Drs. M. Haider, H. Mueller, and J. Zach (CEOS, Heidelberg) for assistance in editing and for providing figures 8, 9, 10, and 14. Graphical support by Mrs. A. Zilch is gratefully acknowledged.

REFERENCES Abbe, E. (1878). Die optischen Hilfsmittel der Mikroskopie. In Report on scientific instruments at the London International Exhibition 1876 (A. W. Hofmann, ed.), pp. 383–240. Vieweg & Sohn, Braunschweig. Archard, G. D. (1954). Requirements contributing to the design of devices used in correcting electron lenses. Br. J. Appl. Phys. 5, 294–299. Beck, V., and Crewe, A. V. (1974). A quadrupole octopole corrector for a 100 kV STEM. Proc. Ann. Meeting EMSA 32, 426–427. Beck, V.D. (1979). A hexapole spherical aberration corrector. Optik 53, 241–255. Bernhard, W. (1980). Erprobung eines sphaerisch korrigierten Elektronenmikroskops. Optik 57, 73–94. Brueche, E., and Scherzer, O. (1934). “Geometrische Elektronenoptik.” Springer, Berlin. Busch, H. (1927). Berechnung der Bahn von Kathodenstrahlen im axialsymmetrischen elektomagnetischen Felde. Ann. Phys. 4, 974–993. Cotte, M. (1938). Recherches sur l’optique électronique. Ann. Phys. (Paris) 10, 333–405. Crewe, A.V. (1980). Studies on sextupole correctors. Optik 57, 313–327. Crewe, A.V. (2002). Some Chicago aberrations. Microsc. Microanal. 8(Suppl. 2), 4–5. Deltrap, J.M.H. (1964). Correction of spherical aberration with combined quadrupoleoctopole units. In “Proceedingsof the Third European Regional Conference on Electron Microscopy, Prague” (M. Titlbach, ed.), vol. A, 45–46. Czechoslovak Academy of Sciences, Prague. Dymnikov, A.D., and Yavor, S.Y. (1963). Four quadrupole lenses as an analogue of an axially symmetric system. Sov. Phys. Tech. Phys. 8, 639–643.

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Fey, G. (1980). Elektrische Versorgung eines elektronenoptischen Korrektivs. Optik 55, 55–65. Fink, R., Weiss, M.R., Umbach, E., Preikszas, D., Rose, H., Spehr, R., Hartel, P., Engel, W., Degenhardt, R., Wichtendahl, R., Kuhlenbeck, H., Erlebach, W., Ihrmann, K., Schloegel, R., Freund, H.-J., Bradshaw, A.M., Lilienkamp, G., Schmidt, T., Bauer, E., and Benner, G. (1997). SMART: a planned ultrahigh-resolution spectromicroscope for BESSY II. J. Electr. Spectrosc. 84, 231–250. Gabor, D. (1969). The outlook for holography. Optik 28, 437–441. Glaser, W. (1937). Elektronenbewegung als optisches problem. In “Beitraege zur Elektronenoptik” (H. Busch and E. Brueche, eds.), pp. 24–33. Barth, Leipzig. Glaser, W. (1940). Ueber ein von sphaerischer Aberration freies Magnetfeld. Z. Physik 116, 19–33; 734–735. Haider, M., Bernhardt, W., and Rose, H. (1982). Design and test of an electric and magnetic dodecapole lens. Optik 63, 9–23. Haider, M., Braunshausen, G., and Schwan, E. (1995). Correction of the spherical aberration of a 220 kV TEM by means of a hexapole corrector. Optik 99, 167–179. Haider, M., Rose, H., Uhlemann, S., Kabius, B., and Urban, K. (1998). Towards 0.1 nm resolution with the first spherically corrected transmission electron microscope. J. Electron Microscopy 47, 395–405. Hardy, D. F. (1967). Combined magnetic and electrostatic quadrupole electron lenses. Dissertation, Cambridge. Hartel, P., Preikszas, D., Spehr, R., Mueller, H., and Rose, H. (2002). Mirror corrector for low-voltage electron microscopes. Adv. Imag. Electr. Phys. 120, 41–133. Hawkes, P.W. (1965). The geometrical aberrations of general electron optical systems, I and II. Philos. Trans. Roy. Soc. A 257, 479–552. Heinzerling, J. (1976). A magnetic flux stabilizer for the objective lens of a corrected electron microscope. J. Physics E 9, 131–134. Hely, H. (1981). Ein verbessertes korrektiv zur Beseitigung des sphärischen und chromatischen Fehlers einer Elektronenlinse. Dissertation, Technical University Darmstadt. Hely, H. (1982). Technologische Voraussetzungen fuer die Verbesserung der Korrektur von Elektronenlinsen. Optik 60, 307–326. Kahl, F., and Rose, H. (1998). Outline of an electron monochromator with small Boersch effect. In “Proceedings of the 14th International Conference on Electron Microscopy,” (H. A. Calderon and M. J. Yacaman, eds.), vol. 1, 71–72. Institute of Physics, Bristol and Philadelphia. Kelman, V. M., and Yavor, S. Y. (1961). Achromatic quadrupole electron lenses. Zh. Tekh. Fiz. 31, 1439–1442. Koops, H., Kuck, G., and Scherzer, O. (1977). Erprobung eines elektronenoptischen Achromators. Optik 48, 225–236. Krivanek, O. (1994). Private communication, 1994 Skalský Dvúr seminar on electron microscopy. Krivanek, O. L., Gubbens, A. J., and Dellby, N. (1991). Development in EELS instrumentation for spectroscopy and imaging. Microsc. Microanal. Microstruct. 2, 315–332. Krivanek, O. L., Dellby, N., and Brown, L. M. (1998). Spherical aberration corrector for a dedicated STEM. In “Proceedings of EUREM-11, the 11th European Conference on Electron Microscopy,” Dublin 1996 (CESEM, ed.), vol. 1, 352–353. CESEM, Brussels. Krivanek, O. L., Dellby, N., and Lupini, A. R. (1999). Towards sub-Å electron beams. Ultramicroscopy 78, 1–11. Kuck, G. (1979). Erprobung eines elektronenoptischen Korrektivs fuer Farb- und Oeffnungsfehler. Dissertation, Technical University Darmstadt. Kujawa, S., Krahl, D., Niedrig, H., and Zeitler, E. (1990). Second-rank aberrations of a magnetic imaging filter: measurement and correction. Optik 86, 39–46.

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Lanio, S., Rose, H., and Krahl, D. (1986). Test and improved design of a corrected imaging magnetic energy filter. Optik 73, 1986, 56–68. Meyer, W. E. (1961). Das praktische Aufloesungsvermoegen von Elektronenmikroskopen. Optik 18, 101–114. Moellenstedt, G. (1956). Elektronenmikroskopische Bilder mit einem nach O. Scherzer sphaerisch korrigierten Objektiv. Optik 13, 209–215. Mueller, H., Preikszas, D., and Rose, H. (1999). A beam separator with small aberrations. J. Electr. Microsc. 48, 191–204. Pejas, W. (1978). Magnetische Abschirmung eines korrigierten Elektrononmikroskops. Optik 50, 61–72. Plies, E. (1973). Korrektur der Oeffnungsfehler elektronenoptischer Systeme mit krummer Achse und durchgehend astigmatismusfreien Gausschen Bahnen. Optik 38, 502–518. Plies, E. and Rose, H. (1971) Über die axialen Bildfehler magnetischer Ablenksysteme mit krummer Achse. Optik 34, 171–190. Rang, O. (1949). Der elektrostatische Stigmator, ein Korrektiv fuer astigmatische Elektronenlinsen. Optik 5, 518–530. Recknagel, A. (1941). Ueber die sphaerische Aberration bei elektronenoptischer Abbildung. Z. Physik 117, 67–73. Riecke, W. D., and Ruska, E. (1966). A 100-kV transmission electron microscope with singlefield condenser objective. Proceedings of the 6th International Congress on Electron Microscopy 1, 19–20. Rose, H. (1971a). Abbildungseigenschaften sphaerisch korrigierter elektrononoptischer Achromate. Optik 33, 1–24. Rose, H. (1971b). Elektronenoptische Aplanate. Optik 34, 285–311. Rose, H. (1978). Aberration correction of homogeneous magnetic deflection fields. Optik 51, 15–38. Rose, H. (1981). Correction of aperture aberrations in magnetic systems with threefold symmetry. Nucl. Instr. Meth. 187, 187–199. Rose, H. (2004). Outline of an ultracorrector compensating for all primary chromatic and geometrical aberrations of charged-particle lenses. Nucl. Instr. Meth. Phys. Res. A 519, 12–27. Rose, H., and Plies, E. (1973). Correction of aberrations in electron optical systems with curved axis. In “Image Processing and Computer-aided Design in Electron Optics” (P. W. Hawkes, ed.), pp. 344–370. Acad. Press, London. Rose, H., and Plies, E. (1974). Entwurf eines fehlerarmen magnetischen Energie-Analysators. Optik 40, 336–341. Ruska, E. (1980). The Early Development of Electron Lenses and Electron Microscopy. Hirzel Verlag, Stuttgart. Scherzer, O. (1936). Ueber einige Fehler von Elektronenlinsen. Z. Physik 101, 593–603. Scherzer, O. (1937). Berechnung der Bildfehler dritter Ordnung nach der Bahnmethode. In “Beitraege zur Elektronenoptik” (H. Busch and E. Brueche, eds.), pp. 33–41. Barth, Leipzig. Scherzer, O. (1947a). Sphaerische und chromatische Korrektur von Elektronenlinsen. Optik 2, 114–132. Scherzer, O. (1947b). Private communication. Seeliger, R. (1951). Die sphaerische Korrektur von Elektronenlinsen mittels nicht rotationssymmetrischer Abbildungselemente. Optik 8, 311–317. Siegbahn, K. (1946). A magnetic lens of special field form for β- and γ-ray investigations; designs and applications. Philos. Mag. 37, 162–184. Tretner, W. (1959). Existenzbereiche rotationssymmetrischer Elektronenlinsen. Optik 16, 155–184. Uhlemann, S., and Rose, H. (1994). The MANDOLINE filter—a new high-performance imaging filter for sub-eV EFTEM. Optik 96, 163–178.

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Uhlemann, S., and Haider, M. (1998). Residual wave aberrations in the first aberrationcorrected transmission electron microscope. Ultramicroscopy 72, 109–119. Urban, K. (2008). Studying atomic structures by aberration-corrected transmission electron microscopy. Science 321, 505–510. Van der Zande, M. J., Mentink, S. A. M., Kok, C., Tiemeijer, P. C., Kujawa, S., and Van der Stam, M. A. J. (2003). Development of a Cs probe corrector for FEI TEM/STEM systems. In “Abstracts of the Third TEAM Workshop,” San Antonio. Zach, J., and Haider, M. (1995). Aberration correction in a low-voltage SEM by a multipole corrector. Nucl. Instrum. Meth. Phys. Res. A 365, 316–325. Zach, J. (1989). Design of a high-resolution low-voltage scanning electron microscope. Optik 83, 30–40. Zemlin, F., Weiss, K., Schiske, P., Kunath, W., and Herrmann, K. H. (1977). Coma-free alignment of high-resolution electron microscopes with the aid of optical diffractograms. Ultramicroscopy 3, 49–60.

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PART 2

Aberration corrector design

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CHAPTER

2 Present and Future Hexapole Aberration Correctors for High-Resolution Electron Microscopy Maximilian Haider, Heiko Müller, and Stephan Uhlemann*

Contents

I Introduction A Aberrations and Information Transfer B Advent of Hexapole Correctors C The EMBL Project D The EMBL Corrector E Commercialization F Further Progress of Hexapole Correctors G Future Hexapole Correctors II Present Hexapole Correctors A Hexapole Elements B Aberrations of Hexapole Fields C Hexapole Corrector D Higher-Order Aberrations III Hexapole Aplanats A Semi-Aplanat Versus Aplanat B Advancing the CTEM Hexapole Cs Corrector C Aplanats Without Axial Fourth-Order Aberrations D Feasibility and Prediction of Properties IV Conclusion Acknowledgments References

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* Corrected Electron Optical Systems GmbH, 69126 Heidelberg, Germany Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01002-1. Copyright © 2008 Elsevier Inc. All rights reserved.

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I. INTRODUCTION Ever since the invention of the electron microscope by Ruska (Knoll and Ruska, 1932), endeavors to improve the resolving power of this new type of microscope were continuously ongoing. At first the attempt was to achieve a resolution superior to that of light microscopes. As soon as this goal was attained, theoreticians started to support the efforts of experimental physicists. The goal was twofold: First, to develop mathematical methods to calculate the electron ray path and the electron optical properties of electron lenses and, second, to understand the image formation in an electron microscope. Very soon after the theoreticians started their investigations it became clear that for fundamental reasons the quality of charged-particle lenses must be rather poor compared to high-quality lenses in light optics. In charged-particle optics the primary lens aberrations cannot be compensated by just a combination of various lenses (Scherzer, 1936). Hence, in the early days of charged-particle optics the search for aberration correctors was already started (Scherzer, 1947). A survey of the early history is provided by Harald Rose in the first chapter of this volume.

A. Aberrations and Information Transfer An electron microscope transfers spatial information from the illuminated object to the magnified image by means of electromagnetic lenses. This information transfer cannot be perfect. The performance of a chargedparticle optics instrument is characterized by two different properties: the optical aberrations and the instrumental information limit. The aberrations of the optical system cause residual phase shifts that deteriorate the phases of the scattered waves in a conventional transmission electron microscope (CTEM) or of the probe-shaped illumination in the scanning transmission electron microscope (STEM). As a result of interference with falsified phase information the image recorded by the detector is fudged or at least artifacts are introduced. Aberrations are classified in two main categories. The effect of axial aberrations depends only on the aperture and affects the on-axis image point in the same manner as the off-axis image points. Off-axial aberrations are only visible at off-axis image points and do not affect the on-axis image point. At the image plane aberrations can be described quantitatively by aberration coefficients. From the aberration coefficients both the residual phase shift and the delocalization can be calculated. In geometrical optics it is common practice to classify the aberration coefficients according to their Seidel order. The Seidel order of an aberration determines the dependency on the beam parameters. Actually, it is the sum of exponents of the aperture angle and the image coordinate in the delocalization monomial.

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For high-resolution imaging primarily axial aberrations and coma-type off-axial aberrations are important. The latter depend linearly on the image coordinate and limit the number of equally well-resolved image points, whereas the axial aberrations determine the achievable optimum point resolution of the instrument. For example, for phase-contrast imaging in a CTEM the point resolution  1/4 at Scherzer defocus C1 = −1.2 (C3 λ)1/2 can be defined as d = 0.7 C3 λ3 , where C3 (or Cs ) denotes the third-order spherical aberration of the objective lens and λ the wavelength of the electrons. In general, the definition of instrumental resolution is a very difficult matter. What we actually can see depends strongly on the contrast mechanism and also on the specimen and its preparation. Whenever resolution numbers are given or compared, it is prudent to ask what assumptions have been made and which definition of point resolution has been used for the assessment. For a TEM equipped with an aberration corrector the achievable optimum instrumental resolution is at best determined by the higher-order residual intrinsic aberrations of the optical system. The residual intrinsic aberrations are present for fundamental reasons even for the idealized instrument. They can be assessed very precisely during the optical design of the instrument. For a real microscope unavoidable manufacturing tolerances and misalignments cause additional parasitic aberrations. In order to gain the full benefits from aberration correction all parasitic aberrations must be compensated or at least minimized by an appropriate alignment of the system. For this purpose every aberration corrector has a considerable number of alignment deflectors and stigmators in addition to the principal optical elements. For a well-designed and well-corrected instrument the residual parasitic aberrations should not limit the attainable spatial resolution. Hence, for any aberration-corrected instrument the definition of a complete and efficient set of alignment tools and the implementation of robust and precise methods for aberration measurement are the most crucial ingredients. In addition to the coherent effects discussed previously, incoherent effects on the optical performance must also be considered. The axial chromatic aberration of a TEM makes the defocus energy-dependent. Since the electron beam is not monochromatic, the recorded image intensity results from an incoherent average over the chromatic focus spread. This is the most prominent incoherent effect that damps the information transfer in an electron microscope for high spatial frequencies. For bright-field imaging in a CTEM the instrumental information limit is governed by the phase-contrast envelope function as a result of chromatic focus spread and lateral incoherence. In a real system, additionally high-voltage ripple, instability of the lens currents, and noise-induced image spread impair the information limit.

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The instrumental information limit is the most important parameter for a Cs -corrected TEM. With an aberration corrector the point resolution can be improved up to the information limit but never beyond. Because any multipole Cs corrector consists of focusing elements the corrector slightly increases the total chromatic aberration. On the other hand, because Cs is corrected the delocalization is minimized and, hence, the influence of the lateral incoherence (determined by the effective size of the electron source) on the information limit is largely suppressed. Hence, the information limit after Cs correction should be as good as for the uncorrected instrument.

B. Advent of Hexapole Correctors An important period for the development of hexapole correctors was in the 1970s when several researchers investigated the optical action of hexapole fields but did not yet notice the advantageous character of hexapoles for the compensation of the third-order spherical aberration. In 1978, Beck for the first time proposed the use of hexapole fields to compensate for the spherical aberration of an objective lens (Beck, 1979). In the following years Albert Crewe in Chicago worked on proposals for STEM hexapole correctors (Crewe, 1980; Crewe and Kopf, 1980; Crewe, 1982), and in Germany Rose analyzed different corrector setups and proposed a new design free of fourth-order aperture aberrations (Rose, 1981). Crewe finally could convince the Department of Energy (DOE) and IBM to fund the development of a hexapole corrector for a dedicated STEM aiming for 0.05 nm resolution. This attempt never did succeed. According to Crewe’s own perception funding ran out too early and the project was stopped before the prototype could be finalized (Crewe, 2002).

C. The EMBL Project When Scherzer died in 1982, the TEM corrector project at Darmstadt, Germany—aiming for simultaneous Cc and Cs correction by means of a quadrupole-octupole corrector—was stopped (Koops, Kuck, and Scherzer, 1977). However, the participants of the Darmstadt project presevered in their strong conviction that aberration correction is feasible and its successful realization should only be a matter of better understanding, patience, and improved technologies for machining, corrector control, and alignment. With this attitude a new correction project was started at the European Molecular Biology Laboratory (EMBL) at Heidelberg, Germany. The new goal was to set up a Cc - and Cs -corrected low-voltage scanning electron microscope (LVSEM) for biological applications (Zach, 1989). This project started in 1987 with the design and construction of a quadrupoleoctupole corrector (Haider, 1990). During this time discussions with Harald

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Rose were ongoing to determine how to stimulate a correction project for high-resolution electron microscopy. At the EUREM meeting in York in 1988, the idea was born to search for a possibility to correct only the spherical aberration in CTEM (or STEM, but less favorable at those days) and to combine it with a field-emission gun to avoid the necessity of a Cc corrector for TEM. At that time field-emission guns were just under development at various places and almost available for commercial instruments. In 1989, at the Dreiländertagung in Salzburg, Austria, the atomicresolution microscope (ARM) project was introduced. This project at Stuttgart, Germany, aimed for highest resolution by means of a 1.25-MeV CTEM (Phillipp et al., 1994). Rose, who attended this conference, stated during the presentation of this project vituperating that it would be far better to invest a much smaller amount of money in the development of a corrected TEM instead for such a microscope (Rose, 1989). In 1990 he published the outline of a Cs -corrected 300-kV CTEM for sub-angstrom resolution (Rose, 1990). Rose could convince Urban to move forward for such a project and to evaluate the benefits of a Cs -corrected TEM for materials science. Finally, a grant proposal was submitted to the German Volkswagen-stiftung at the end of 1990. This was the birth of a successful joint project of three groups: the theory group of Rose at Darmstadt, the physical instrumentation group of Haider at the EMBL, and the materials science group of Urban at Jülich, Germany. By the end of 1991 the funding for this project was ensured and the development of the first Cs -corrected 200-kV CTEM could be started. A Philips CM200F was selected as the base instrument because of the demand for a modern high-resolution microscope with field-emission gun. The assumption was that the performance of this system is limited mainly by the spherical aberration and not by chromatic focus spread. The point resolution of this CTEM, equipped with the so-called super-twin objective lens, was given by 0.24 nm at Scherzer focus and an information limit of at least 0.16 nm was guaranteed by the manufacturer. The phase-contrast transfer function (PCTF) for the uncorrected instrument at Scherzer focus is shown in Figure 1. The point resolution is given by the first zero of the PCTF while the information limit is the spatial frequency for which the PCTF envelope is reduced to 1/e2 due to the chromatic aberration and the lateral incoherence. For the uncorrected instrument the effect of the lateral incoherence on the information limit is sensitive to the illumination conditions and the choice of the defocus. The PCTF in Figure 1 represents typical conditions for high-resolution imaging. Shifting from Scherzer to Lichte focus (for minimized delocalization) or reducing the semi-convergence angle of the illumination and, hence, the current density at the specimen further could slightly improve the nominal instrumental information limit.

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PCTF of TEM @ 200 kV, semi conv. 0.3 mrad, dE 5 0.7 eV (FWHM)

1

200 kV, Cs 5 1.25 mm, Cc 51.3 mm damping envelope function

0.8 0.6

Phase contrast

0.4 1/e2

0.2 0 20.2 Point resolution

20.4 20.6 20.8 0

2

4

6 1/nm

8

10

12

FIGURE 1 The phase contrast transfer function of an uncorrected 200-kV CTEM at Scherzer focus. The point resolution and the contrast level used to define the information limit are indicated.

In 1991 the director general of the EMBL, Lennart Philipson, accepted the project even though the main research of the EMBL is molecular biology and not applied physics. Therefore, it was agreed that the benefits of Cs -corrected CTEM in structural biology would be evaluated during the course of the project. The project was divided into a two-step process: During the first phase, an old 200-kV TEM was used with the goal to achieve a proof of principle. After this first milestone, in a second project phase the improvement of the point resolution by means of a hexapole corrector integrated in a new CM200F needed to be demonstrated. For the new 200-kV TEM we asked for an improved high-tension supply and an increased overall stability of the instrument. Our goal for the final Cs -corrected instrument was to come close to the optimum information limit of IL = 9/nm due to the total chromatic aberration and the energy width of the field-emission source.

D. The EMBL Corrector The design of the EMBL hexapole corrector consists of two modules: (1) an upper part with a double transfer lens and additional image deflectors

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and stigmators and (2) a lower part with two multipole stages to generate the strong hexapole fields, two transfer lenses, and a final adapter lens. The lower part of the original corrector is shown in Figure 4. The transfer lens doublet in the upper part is necessary to match the coma-free plane of the objective lens with the coma-free plane of the hexapole corrector (Rose, 1990). The coma-free plane of the objective lens is situated very close to its back-focal plane; without the transfer system, off-axial third-order coma and spherical aberration of fifth-order C5 would be too large to achieve a substantial improvement in high-resolution. This matter is discussed in depth in Section II. It is technologically demanding to insert two transfer lenses into the bore of the lower iron circuit of the objective lens without changing the design of the objective lens itself (see Figure 2). In order to have sufficient room for the additional lenses we used the larger objective lens iron circuit of a 300-kV lens for the 200-kV system and designed a water-cooled double-gap lens with just one coil and a floating inner pole piece instead of two separate lenses (see Figure 3). The original image deflectors and stigmators of the CM200F were removed and replaced by miniaturized versions integrated in the bore of the transfer system. A few iterations were needed to find a solution that provided the necessary alignment tolerances and the mechanical and thermal stability required for high-resolution imaging. A compact design is essential

Specimen Objective lens

SAD

FIGURE 2 The unmodified objective lens of the CM200F. The first transfer doublet of the corrector must be inserted in the bore of the lower iron circuit.

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Gap 1

Coil

Floating pole piece

Gap 2

FIGURE 3 Drawing of first transfer system for the EMBL corrector with floating double pole piece, two gaps, and just one coil.

to leave sufficient room between the housing of the coil and the soft iron material to prevent a transfer of heat from the high-current density (finally ∼20 A/mm2 ) within the coil to the iron circuit of the objective lens and to avoid mechanical vibrations caused by the flow of cooling water very close to the objective lens. The core module of the hexapole corrector consisting of the lower and the upper multipole element with two transfer lenses in between and additional alignment deflectors was also designed and constructed. We opted for a twelve-pole design to allow for additional weak multipole fields on the principal correction elements for alignment purposes. The twelve-pole elements were made of one cylinder of soft iron to avoid asymmetries of the magnetic flux (see Figure 14). The corrector module with all the wires to excite the various multipole fields, deflectors, and round lenses is depicted in Figure 4 (Haider et al., 1995). In addition to the corrector hardware itself, we developed all necessary electronics drivers and the software required for aberration assessment, corrector control and alignment. Fast personal computers to calculate a fast Fourier transformation (FFT) of a full image recorded by the charge coupled device (CCD) within a short time were not yet available; therefore, a workstation with an additional image processor board was used. For online calculations new routines have been implemented so that the state of alignment could be measured within about one minute by a fully automatized Zemlin tableau technique. For the acquisition of a Zemlin tableau (Zemlin et al., 1978) the illumination is tilted and guided around

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FIGURE 4 The inner part of the EMBL hexapole corrector with the two stages of twelve-pole elements at the top and bottom and between the two lenses of the transfer doublet.

a hollow cone. As objects we used amorphous films of strongly scattering material (e.g., tungsten). In the diffractograms—the modulus of the twodimensional (2D) discrete Fourier transform of the image—the so-called Thon rings show the phase shift as a function of the scattering angle. The phase shift depends on the aberrations within the imaging system. Depending on the state of alignment (e.g., which aberrations are dominant) an outer tilt angle between 6 and 30 mrads is used and 12 or more images are taken at different tilt positions. If axial aberrations are present, the diffractograms are disturbed by tilt-induced twofold astigmatism and defocus. We developed computer routines that measure defocus and twofold astigmatism very precisely by a new online correlation method with a library of theoretical diffractograms. Given the full set of defocus and astigmatism measurements of the individual diffractograms of the tableau, the full set

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of axial aberrations can be calculated from a linear system of equations. The new methods worked semi-automatically. The only requirements were choosing some parameters and starting the measurement. The result is a list of the axial aberrations that characterize the state of correction (Uhlemann and Haider, 1998). In 1995 the EMBL group demonstrated the improvement of resolution by means of a quadrupole-octupole Cc /Cs corrector in a low-voltage SEM (Zach and Haider, 1995). This success—for the first time ever the improvement of instrumental resolution by means of an aberration corrector had been shown—also encouraged the hexapole corrector project. After the successful implementation and an appropriate alignment of the Cs -corrected CM200F in February 1997 an information limit of ∼0.18 nm was measured. Since the first transfer lens system was now considered sufficiently stable this indicated the presence of another limitation that had to be detected and eliminated. Meanwhile, the director general at the EMBL had changed and physical instrumentation was not considered appropriate and was no longer supported at the laboratory. As a consequence, Haider’s group (EM development and STEM application) was given notice that their main field of research had to be stopped. With this decision it was clear that only a timely breakthrough with respect to the information limit could turn the project into a success because in case of failure we could hardly expect a second chance outside the EMBL to continue the project until an improvement of resolution could be demonstrated. From our investigations it was clear that the restriction of the information limit was caused by either a perturbation near the selected area (SA) plane or within the projector system. Since the magnification within the hexapole corrector is adjustable within a certain range by a variation of the first transfer lens doublet, we could obtain a better information limit if the intermediate magnification in the hexapole corrector was increased or if an additional intermediate image was positioned between the last hexapole and the SA plane. Hence, either the projection system or the electron path between the lower hexapole and the SA plane caused the limitation. Unfortunately, with this beam path in the corrector Cs correction was no longer possible. The projector system could not be changed; therefore, we decided to design a new much stronger double-gap adapter lens with an additional intermediate image and increased the magnification at the SA plane by a factor of three. This decision was made four weeks before the turn-off switch had to be used at the CM200F because the microscope had to be transferred to the group of Urban at Jülich, Germany. We then needed only three weeks for the design, the construction of the iron circuit (including the necessary annealing process of the soft iron), and for making a new coil. Just 8 days before the Cs -corrected TEM was to be moved we incorporated this

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(b)

(a)

Cs 5 1.2 mm

53

(c)

Cs 5 0.07 mm

7.1/nm

FIGURE 5 First proof of the correction of the third-order spherical aberration Cs without a deterioration of the information limit. Three diffractograms with the same calibration of the spatial frequency are shown. (a) The uncorrected mode at Scherzer focus, (b) the corrected mode near Gauss focus, and (c) the corrected mode superimposed by Young fringes. For the Cs -corrected CM200F the information limit is of 8.3/nm and 7.1/nm, respectively, for two perpendicular directions. (From Uhlemann and Haider, 1998.)

new lens and one day later we started with the alignment. The first images obtained with this modification were disappointing because the images were very unstable; with this result we lost hope of succeeding with this development. Nevertheless, half a day later the new lens had stabilized and the information limit could be measured in the good direction at 0.12 nm and in the not-so-good at ∼0.14 nm. We also took images of amorphous tungsten to illustrate at the diffractogram the modified characteristic of the contrast transfer function (CTF) as shown in Figure 5 (Uhlemann and Haider, 1998). This result was the hoped-for breakthrough (Haider et al., 1998). The first Cs -corrected CTEM with an improvement of the point resolution by a factor of about two was in place. The exact alignment of the instrument was achieved the following night by means of many Zemlin tableaus. The result of the alignment is shown in Figure 6. In the diffractograms for an outer tilt angle of 10.8 mrad (shown at the right-hand side) almost no induced astigmatism is visible. This illustrates vividly that the Cs -corrected microscope can focus electrons scattered into large angles without introducing additional disturbing phase shifts. Within the remaining six days before the CM200F had to be moved to its new site on July 31, 1997, Bernd Kabius demonstrated the superior resolution with seven samples he had prepared for the examination of the first Cs -corrected CTEM. At GaAs in [110] orientation (Figure 7) he could resolve the dumbbell-shaped structure of the pairs of Ga and As atoms with a spacing of 0.14 nm in the raw image. At an interface of CoSi2 /Si he demonstrated the vanishing delocalization when working very close to the Gaussian focus with Cs ≈ 0. He compared the result with a defocus series of images of the same interface (Figure 8) taken without hexapole corrector

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FIGURE 6 Zemlin tableau of the uncorrected (left half) and the corrected CTEM (right half). The outer tilt angle is 10.8 mrad. (From Haider et al., 1998.)

[100]

1 nm

FIGURE 7 Image of GaAs [110] orientation. The distance between the two atom columns in the dumbbell structure in [100] direction is 0.14 nm. (From Rose, Haider, and Urban, 1998.)

Present and Future Hexapole Aberration Correctors

Cs 5 1.2 mm / 42 nm

55 nm

Scherzer

82 nm

96 nm

110 nm

123 nm

136 nm

150 nm

164 nm

177 nm

190 nm

204 nm

218 nm

Lichte

Cs 5 0.07 mm / 14 nm

55

CoSi

Si

FIGURE 8 Focus series of a CoSi2 /Si interface without Cs correction. The last image is taken with Cs corrected. In Scherzer focus the delocalization is very pronounced. At Lichte defocus the delocalization can be reduced but it is still present; only in the Cs -corrected mode is the image of the interface atomically sharp.

where the strong delocalization due to Cs is clearly visible even at Lichte focus. The benefit of vanishing delocalization was also demonstrated at an YBa2 Cu3 O7−x /SrTiO3 substrate interface shown in Figure 9. The original CM200 equipped with the EMBL hexapole corrector was reinstalled at Jülich in August 1997. Since that time it is—apart from some downtime for service—operating routinely as a user instrument.

E. Commercialization Urged by the anticipated closure of the electron optics instrumentation group at the EMBL and motivated by the successful realization of the first aberration corrected LVSEM, Haider and Zach founded the Corrected Electron Optical Systems (CEOS) company in Heidelberg in 1996. In the

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FIGURE 9 Early attempt to image the substrate interface of the high-temperature superconductor YBa2 Cu3 O7−x /SrTiO3 . The left side of the figure shows strong delocalization due to C3 (left), and no delocalization is seen in C3 corrected state (right).

following two years, two more people from the EMBL group joined CEOS. The first project at CEOS was the development of a commercial prototype of the LVSEM Cc /Cs corrector in co-operation with the JEOL company. This development was completed successfully from 1996 to 1999 and the corrector later became commercially available from JEOL (Kazumori et al., 2004). In 1997 the unique success of the first Cs -corrected CTEM with improved resolution generated a strong demand for commercial availability of TEM Cs correctors in the scientific community. Therefore, CEOS decided to further develop the hexapole corrector and to transform it into a commercial product. For routine applications several components of the system had to be revised to simplify operation and to improve reliability. In addition, we aimed for an optimization of the electron optical properties and added more alignment flexibility for bright- and dark-field imaging and diffraction mode. Finally we achieved the following: 1. Reduced the number of necessary current drivers from 56 to 29 2. Modified the first transfer lens doublet by implementing two independent coils and iron circuits 3. Added additional stigmators between the objective lens and the first transfer lens for a more precise compensation of twofold and threefold astigmatism without hysteresis 4. Used hexapoles consisting of six separate pole pins and a yoke instead of the monolithic twelve-pole elements and optimized the magnetic flux to avoid saturation and remanence effects,

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5. Increased the magnification between objective lens and corrector to reduce the Cc contribution of the transfer lenses from 25–30% down to 15–20% of the Cc of the objective lens. With this improved design of the hexapole corrector, CEOS started co-operations with the EM manufacturers Carl Zeiss (LEO at that time), JEOL, Philips (now FEI), and Hitachi in the 200-kV TEM market. The hexapole corrector for the conventional TEM has been named CETCOR (CEOS TEM CORrector). The first commercial CETCOR was shipped to JEOL in October 2002. At this time the CEOS company had grown to 15 people, several of whom are former members of Rose’s theoretical charged-particle optics group at Darmstadt. Shortly after the first commercial system for CTEM was finished it became obvious that the hexapole corrector is also very suitable as a probe corrector in STEM (Haider, Uhlemann, and Zach, 2000). In order to improve the resolution in STEM the illumination angle has to be increased. This additionally provides a higher beam current for a Cs -corrected STEM. The CEOS hexapole corrector for STEM is called CESCOR (where the S stands for STEM). From 2001 to 2003 the first commercial hexapole corrector for STEM was developed for a JEOL 2010F (S)TEM. This system, finally installed at Oxford University (Hutchison et al., 2005), is shown in Figure 10. It is the first double-corrected (S)TEM, equipped with two hexapole correctors: one in the illumination system and one in the imaging system. With this system high-resolution TEM imaging can be combined with the analytical capabilities offered by a Cs -corrected STEM (Sawada et al., 2005). The precise Zemlin tableau technique based on the analysis of diffractograms could not be used for aberration measurement for a STEM Cs corrector. Diffractograms of STEM images do not provide the same information as for CTEM images. We finally implemented a combination of the tableau technique with an analysis of the STEM point-spread function. The latter method was originally developed for the LVSEM corrector by Joachim Zach. To analyze the aberrations of the probe-forming system the illuminating beam is tilted between the gun and the corrector and the point-spread function is deconvoluted from STEM images in overfocus, Gaussian focus, and underfocus. Then defocus and twofold astigmatism are calculated from the shape of the point-spread function. The evaluation of the tableau afterward uses the same methods as used for CTEM. For the CTEM hexapole corrector two transfer lenses are used in front of the first hexapole element. For STEM a transfer system between the lower hexapole and the objective lens is also necessary to avoid the strong spherical aberration of fifth-order C5 as a combination aberration. To avoid scan coma in STEM, additionally, the scan unit must be placed between the corrector and the objective lens. This could be achieved by using a

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FIGURE 10 First double-corrected 200-kV (S)TEM. JEOL 2010 F equipped with CESCOR integrated between condenser and objective lens and CETCOR between objective lens and projector.

single transfer lens or, preferentially, by using the most often available condenser mini-lens close to the objective lens in addition to a further transfer lens to maintain flexibility when adjusting the correction strength for different high voltages. Over the years hexapole correctors have been adapted to several objective lenses with different pole pieces. However, the preferential choice is still a high-resolution small-gap pole piece with small C3 and Cc . This system has minimum chromatic focus spread and small fifth-order residual intrinsic aberrations. With the advent of commercially available correctors most TEM manufacturers started a redesign of their columns by concentrating more on the overall mechanical and electrical stability. Zeiss, for example, introduced

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FIGURE 11 CESCOR for dedicated STEM Hitachi HD2700 (Kimoto et al., 2007). The filter box for the electronics jacks and the connectors for the cooling water are visible. The total column extension after integration of the corrector between condenser and objective lens is ∼300 mm.

a hanging column for their high-resolution TEM (Essers et al., 2002), and FEI developed the Titan 80-300 series (Van der Stam et al., 2005). For the new FEI platform CEOS developed the first 300-kV hexapole correctors for CTEM and STEM. During this time the stability requirements increased, and several design improvements were necessary also for the hexapole correctors. The noise-induced image spread caused by the alignment deflectors has been further reduced. We improved the stability of the current drivers and reduced the maximum strength of the deflectors. The attainable machining tolerances for the electron optical elements has a direct influence on the required relative stability of the power supplies. This is due to the fact that a poor mechanical precision of the electron optical components requires larger excitation ranges for the current drivers of the alignment dipoles in order to counterbalance the parasitic aberrations. The more precise the mechanical alignment is, the lower is the contribution of the alignment elements to the total noise budget. Therefore, the precise construction of the corrector helps to improve the performance of the TEM, although the actual electron ray path can be aligned up to the required precision with the available deflectors and stigmators. The

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available noise budgets for the various components can be analyzed for a given goal of resolution when designing a new corrector (Haider et al., 2008). The required precision of the mechanics as well as strength and stability for the power supplies must be defined at the same time. With the advanced TEM platforms equipped with gun monochromators (Tiemeijer, 1999; Uhlemann and Haider, 2002; Walther and Stegmann, 2006) in order to reduce the chromatic focus spread information and a CTEM hexapole corrector information transfer in the diffractogram at spatial frequencies above 12/nm at 200 kV was demonstrated (Schlossmacher et al., 2005; Freitag et al., 2005). A Young fringe pattern recorded for an amorphous sample is shown in Figure 12. This indicates the improved stability of the commercial correctors. The precise alignment attainable with the latest hexapole correctors for CTEM can be observed with a Zemlin tableau with an outer tilt angle of 50 mrad (Figure 13).

0.07nm

0.10nm

0.08nm 0.10nm 0.07nm

0.09nm

800 k, 1 sec, ␤-Ta sample

FIGURE 12 Young fringe pattern obtained with a Zeiss UHRTEM consisting of a CETCOR and the CEOS gun monochromator (Uhlemann and Haider, 2002). It demonstrates an information limit of better than 0.08 nm (or 12.5/nm) at 200 kV. The diffractogram has been calculated from the image of an amorphous β-tantalum sample. (From Schlossmacher et al., 2005.)

Present and Future Hexapole Aberration Correctors

25 mrad

B2

51 mrad

C5

61

FIGURE 13 Zemlin tableau obtained with a CETCOR for an outer tilt angle of 50 mrad. Although the tilt angle is very large the induced twofold astigmatism and defocus are almost not observable. With this alignment almost aberration-free imaging down to 0.1 nm at 200 kV is possible.

This enables almost aberration-free imaging down to 0.1 nm at 200 kV. Also in the STEM the CESCOR very much improved resolution and analytical capabilities. At 200 kV it became possible to clearly resolve adjacent atomic columns in Ge [211] with a dumbbell spacing of 82 pm. Also the [426] reflection corresponding to a lattice spacing of 76 pm is clearly visible in the diffractogram (Figure 16). With the same 200-kV instrument the 78 pm dumbbell spacing of Si in [211] orientation and the 89 pm dumbbell spacing of Diamond in [110] orientation have been resolved in the image (Sawada, 2008). This demonstrates the particular benefits of Cs correction for TEM with a high tension of 200 kV and below. To avoid drift effects after changing thermal load in the corrector we have introduced a constant power design for the transfer lenses. A constant power lens uses a bifilar coil driven by two separate power supplies such that the focal length can be changed while the total power dissipation in the two lens coils is kept constant.

F. Further Progress of Hexapole Correctors In 2004 the transmission electron aberration-corrected microscope (TEAM) project set the goal of 50 pm resolution in STEM. To achieve this ambitious goal the residual phase shifts for large aperture angles of 40 mrad must be controlled. This requirement made a partial redesign of the CEOS STEM hexapole corrector necessary. For the advanced hexapole corrector (D-COR) special care had been taken to reduce the intrinsic sixfold astigmatism A5 introduced by the corrector and to offer the possibility to compensate for the fourth-order

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parasitic aberrations by appropriate alignment tools (Müller et al., 2005; Müller et al., 2006b). This was possible because our understanding of the detailed mechanisms of how the intrinsic combination aberrations and residual parasitic aberrations are produced in the system had improved substantially over the years. By design we reduced the coefficient of the intrinsic sixfold astigmatism by more than one order of magnitude from 2—3 mm down to below 200 µm (Müller et al., 2006a). Additionally, by tuning the transfer lenses between objective lens and corrector the coefficient of the intrinsic fifth-order spherical aberration C5 can be corrected (Hartel et al., 2004). For details, refer to Section II of this chapter. The magnetic circuit of the hexapole elements as used in CESCOR and CETCOR (compare with Figure 23) has been changed to further improve the mechanical and magnetic precision. For the D-COR the multipole elements are made of just one piece of soft iron. The inner shape of the pole pieces is crafted by wire erosion and the coils were wound up on the pole pins. For this purpose the coil bodies are constructed of two halve-shells and mounted on the pole pins. With this design we came back to the original monolithic solution as used for the first multipole elements at the EMBL. The two designs are compared in Figure 14. For easier operation additional alignment methods were defined and implemented in software. For the D-COR alignment tools for automatic compensation of all axial aberrations up to and including the fourth-order now exist. In total eighteen real-valued aberration coefficients can be controlled during the user alignment. Ascreen shot of the D-COR user interface is shown in Figure 15 (Hartel et al., 2007).

FIGURE 14 Pictures of the multipole elements used for the excitation of the principal hexapole fields. The left-hand photo shows the first multipole constructed at the EMBL as a dodecapole element and the right-hand side shows the design of the hexapole element for the D-COR-type corrector.

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FIGURE 15 Screen shot of the graphical user interface for control and alignment of the CEOS D-COR hexapole corrector for STEM with auto-alignment tools for all fourth-order parasitic aberrations. The measured residual aberrations up to fifth order are visualized by a simulated phase plate image.

The alignment of the first hexapole corrector at the EMBL took several months. Today the factory alignment for a corrector can be done in about one week thanks to more experience and computer-assisted alignment methods. The factory alignment needs to be performed just once during the lifetime of a corrector. The result of the coarse alignment is a fingerprint of the actual manufacturing tolerances and misalignments of an individual system. From our experience we can state that even if a corrector is dismantled and reassembled, most of these data are still valid and can be reused for the precise alignment. Any changes that occur over time due to drift are mainly lower-order aberrations such as twofold and threefold astigmatism and axial coma. Residual aberrations in third- and higherorder are stable over a long time, typically days or weeks. Occasionally the

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1 nm

FIGURE 16 Raw STEM image of Ge in [211] orientation obtained with a JEOL 2100F (S)TEM equipped with CEOS hexapole corrector CESCOR operated at 200 kV. The dumbbell structure has a spacing of 82 pm and is well resolved (left). The reflections for the [444] direction (82 pm) and [426] direction (76 pm) are clearly visible in the diffractogram (right). (Image courtesy of Hidetaka Sawada, JEOL Ltd.)

auto-alignment tools must be used to maintain the well-corrected state of the instrument. One impressive example of this stability was the shipping of a highresolution TEM equipped with a Cs corrector—showing sub-angstrom resolution—from Europe to the annual Microscopy and Microanalysis Conference in Hawaii in the summer of 2005. The TEM arrived in large crates. After installation and electronics test the hexapole corrector was turned on and just slight tuning of second-order coma was necessary to reduce the coma from 100 nm to ∼20 nm and sub-angstrom resolution could be observed again. The D-COR design further pushed the limits for STEM resolution. With the first prototype installed at CEOS, Heidelberg in a standard FEI Titan 80-300, we were able to demonstrate 0.082 nm dumbbell resolution in an image of Ge in [211] orientation at 300 kV (Hartel et al., 2007). With the final TEAM 0.5 instrument equipped with a D-COR the 63 pm Ga dumbbells of GaN in [211] orientation have been resolved with clear

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FIGURE 17 STEM image of the dumbbell structure of GaN in [211] orientation (a) obtained with an D-COR hexapole corrector incorporated into the FEI TEAM 0.5 instrument operated at 300 kV. The dumbbell structure has a spacing of 63 pm and is well resolved (c). A reflection for the [555] direction corresponding to a spacing of 49 pm is clearly visible in the diffractogram (b). At an Au sample a weak signal for the [660] direction (48 pm) could be demonstrated (d).

reflections in the diffractogram corresponding to 49 pm for the 555 direction as shown in (Figure 17). Additionally, with an Au sample 48 pm reflection, in the diffractogram were demonstrated. These results (in the summer of 2007) approved the ambitious goal of the TEAM project for STEM at 300 kV and, finally, set a new world record in STEM resolution (Kisielowski et al., 2008). With the D-COR-type design it is possible to use very large STEM apertures. The ronchigram shown in Figure 18 shows a phase-flat area (sweet spot) of more than 50 mrad. Nevertheless, the aperture used for these systems is much smaller since the STEM d50 probe size (e.g., the diameter of the disc that contains 50% of the total probe current) is primarily governed by the chromatic focus spread. For a standard Schottky-field

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FIGURE 18 In-focus ronchigram with an aperture radius of 80 mrad taken with a CEOS D-COR at 300 kV. The phase-flat area amounts to more than 50 mrad.

emission gun (FEG) at 300 kV the optimum aperture amounts to ∼25 mrad. To exploit the full benefits of the advanced hexapole STEM corrector it would be advantageous to combine the corrector with a cold FEG or a high-brightness Schottky-FEG equipped with a gun monochromator. With an energy spread reduced down to 0.3 eV full-width half-maximum (FWHM) the optimum aperture angle (for minimum d50 ) would increase to ∼40 mrad at 300 kV. The attainable d50 STEM probe size in the zero-current limit (zero size virtual source) versus the aperture angle for different Cs -corrected systems is plotted in Figure 19 (Müller et al., 2006a). To obtain the finally achievable STEM probe size the results must be convoluted with the shape of the geometrical image of the virtual source at the specimen plane. The calculations compare two systems: a reduced-gap 200-kV objective lens with small C3 and small Cc and a standard-gap 300-kV objective lens. With a Schottky-type FEG with E = 0.7 eV (FWHM) both systems are limited by chromatic focus spread and not by fifth-order aberrations. If the energy width is reduced to E = 0.3 eV (FWHM), the aperture angle can be increased accordingly. For a CESCOR-type design the sixfold astigmatism now becomes the limiting aberration for both systems. In this regime the advanced D-COR-type design is most appropriate and can be

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U ⫽ 200 kV, fOL⫽ 1.8 mm, C3,OL⫽ 0.5 mm, CC,tot ⫽ 1.4 mm

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A5 ⫽ 3.0 mm A5 ⫽ 3.0 mm A5 ⫽ 0.5 mm A5 ⫽ 0

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FIGURE 19 Theoretically attainable d50 STEM probe size at the zero-current limit for a 200-kV and 300-kV instrument with different energy length E·Cc and different residual sixfold astigmatism coefficients A5 (Müller et al., 2006a) and (Hartel et al., 2007).

used to push the STEM probe size to d50  50 pm and to increase the probe current without loss of spatial resolution for an acceleration voltage of 200 kV or 300 kV. A larger illumination angle increases the probe current, which is favorable not only for a better signal-to-noise ratio but first of all for analytical work such as electron energy-loss spectroscopy (EELS). For EELS a strong localization of the probe current is important to correctly correlate spatial and analytic information. By increasing the aperture angle depth resolution can also be improved, because the depth of focus depends quadratically on the aperture angle. Depth resolution will become more important in the future if the expectations with respect to tomography and depth sectioning at atomic scale can be fulfilled (Kisielowski et al., 2008). An illustrative example is shown in Figure 21. In STEM individual layers of the specimen can be imaged selectively by focusing the beam at different depth sections. During the past years the application of hexapole correctors in TEM produced many exciting scientific results. Cs correction improved the visibility of light atoms in bright-field images. Here, imaging with small negative C3 is a very useful technique to optimize the contrast in the images. This is illustrated by an image of SrTiO3 in Figure 20. In this image the atomic columns are visible as bright spots on dark ground (Jia, Lentzen, and Urban, 2004). The reduction of the delocalization becomes very obvious in CTEM images taken at lower accelerating voltages. When working at 80 kV, because of the strong interactions of electrons with the objects at this energy the electrons are scattered more often in large angles and, hence, the delocalization is more pronounced. This can be demonstrated by images taken of single carbon nanotubes (SCNTs) in uncorrected and corrected mode (Figure 22).

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FIGURE 20 SrTiO3 sample in [110] orientation imaged with negative third-order spherical aberration C3 ≈ −40 µm and positive defocus C1 = 8 nm. The atomic columns are visible as bright spots on dark ground. The oxygen columns situated between the titanium columns in [001] direction are clearly visible; it is even possible to measure the variable oxygen occupancy quantitatively by comparing image simulations with the recorded contrast. (From Jia, Lentzen, and Urban, 2003.)

(a)

(b)

e2

e2

Sample

Sample

Boundary

Boundary

1 nm

1 nm

FIGURE 21 STEM depth sectioning with different defocus applied reveals structural boundary in bulk material.

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FIGURE 22 Image of a single carbon nanotube taken at 80 kV without (left) and with Cs correction (right). The delocalization due to the spherical aberration is very strong in the case of an uncorrected CTEM.

G. Future Hexapole Correctors After reviewing the developments of hexapole aberration correctors over the past 10 years the question remains: What will be required in the future and which developments can be anticipated? The spatial resolution in CTEM and STEM has improved considerably compared with the situation 10 years ago. Therefore, it cannot be expected that the coming 10 years will improve the resolving power by the same factor again. The present limit in high-resolution imaging is no longer due to the coherent axial aberrations but is related more to the chromatic aberration and other incoherent disturbances. Hence, the further development of hexapole correctors will focus more on new fields of applications such as far-field lenses for in situ electron microscopy, Lorentz microscopy for imaging of magnetic domains, and applications such as projection optics in which a very large field of view is mandatory (Rose, 2002). Structural biology with low-dose imaging and single-particle reconstruction also could become an exciting new field for the application of aberration correctors. The increasing availability of Cs -corrected instruments will initiate the invention of new imaging techniques and applications and these innovations most probably will generate further demands for improved instrumentation. In general, a hexapole corrector is a useful choice whenever for a certain instrument or application the information limit is considerable better than the Cs -limited point resolution. For example, one could imagine designing a fully electrostatic hexapole corrector for low-voltage or ion-beam applications. An electrostatic hexapole corrector would work essentially like a magnetic one but the design could be very compact.

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A reduction of the energy spread is essential for improved resolution in TEM. This can be achieved by using either a cold FEG or a gun monochromator. For the latter a design that is fully dispersion-free seems preferential to keep the brightness of the Schottky-type field emitter as high as possible (Uhlemann and Haider, 2002). For electrons scattered in larger angles in CTEM then higher-order axial and coma-type off-axial aberrations gain importance. The latter limit the number of equally well-resolved image points and, hence, effectively the field of view for a given target resolution (Haider et al., 2008). The dominant coma-type aberration of a present CTEM is the azimuthal or anisotropic off-axial coma of a magnetic objective lens. This is caused by the Larmor rotation and cannot be avoided for a single-gap magnetic lens. For this reason the present CTEM instruments equipped with a hexapole-corrector are called semi-aplanatic. An aplanatic system would be completely free of third-order off-axial coma. In Section III we propose novel designs for hexapole correctors with three and more multipole stages that can be used to compensate for the spherical aberration and the third-order off-axial coma of the objective lens simultaneously. The correction of the chromatic aberration is the next major step to achieve highest resolution in STEM and CTEM. The TEAM project has recognized this necessity and aims for the development of a Cc -/Cs -corrected CTEM with the goal to allow for an information limit of 0.05 nm at 200 kV. Nevertheless, a hexapole Cs corrector could be combined with a Cc corrector; for this purpose a design based on electric-magnetic quadrupoles and octupoles seems to be the more appropriate choice (Rose, 1971a; Rose 2004). With this type of corrector it is possible to correct simultaneously for the chromatic aberration, the spherical aberration, and the azimuthal coma of the objective lens (Haider et al., 2008). A first prototype quadrupoleoctupole Cc -/Cs -corrector using a quadrupole Wien filter for chromatic correction has been developed by CEOS on behalf of Argonne National Lab (Hartel et al., 2008). If this novel design proves successful, it will certainly start an new area in aberration-corrected electron microscopy.

II. PRESENT HEXAPOLE CORRECTORS A. Hexapole Elements In a standard electron microscope all optical elements except for alignment deflectors and stigmators are cylinder symmetric. For systems equipped with a multipole aberration corrector this is no longer true. Hexapole Cs correctors use strong magnetic fields with threefold symmetry to correct for the spherical aberration of the objective lens. The optical elements producing these fields are called hexapoles or sextupoles, since the arrangement of their pole pieces has sixfold symmetry.

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FIGURE 23 Magnetic hexapole element as used in a hexapole Cs corrector. The diameter of the outer yoke is 152 mm and the bore is 8 mm. The length of the element in z-direction amounts to LHP = 30 mm. The liner tube placed inside the bore and the field clamps are not shown. (From Müller et al., 2006a.)

Figure 23 shows an embodiment of such an element. The hexapole consists of a cylindrical outer yoke with six pole pins pointing toward the optic axis. The tips of the pole pins have an optimized shape to produce a strong and accurate hexapole field in close vicinity to the optic axis. The coils mounted at the pole pins are excited by currents whose directions alternate from pin to pin.

1. Magnetic Field The magnetic flux density B near the optic axis can be described by the gradient of a magnetic scalar potential ψ3 . Sufficiently far away from the entrance and exit faces of the element it adopts the form of a plane hexapole field

      ψ3 = Re 3 w3 = 3c x3 − 3xy2 + 3s 3x2 y − y3 , B = −∇ = −2∂ w 3 = −3 3 w2 .

(1) (2)

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To simplify the mathematics we use a complex notation where the lateral distance from the optic axis is w = x + iy and its complex conjugate w = x − iy. The hexapole strength along the optic axis is described by two real-valued components, 3c and 3s , corresponding to a cosinelike and sine-like azimuthal modulation of the potential ψ3 , respectively. In the complex notation we have 3 = 3c + i 3s . The azimuthal orientation of the hexapole is determined by an angle 0◦ ≤ ϕ ≤ 30◦ . For ϕ = 0◦ the hexapole strength is real for ϕ = 30◦ imaginary.

2. Hexapole Strength A rather simple argument based on Ampere’s law and the assumption of infinite permeability µ → ∞ of the magnetic material can be used (Haider, Bernhardt, and Rose, 1982) to relate the maximum hexapole strength along the optic axis to the current driving the coils of the element

| 3 | = FHP

µ0 NI 3 RHP

,

(3)

where RHP denotes the bore radius, NI the current in ampere turns, and FHP is a Fourier factor in the order of one depending on the exact shape of the pole tips. More detailed investigations consider not only the correct Fourier factor FHP but also the exact fringe field at the entrance and exit face of the element. For this purpose numeric field calculations must be performed. The result of such a calculation (Müller et al., 2006b) using the boundary element method for an unsaturated magnetic hexapole element is shown in Figure 24. The course of the hexapole strength along the optic axis can be approximated roughly by a box-shaped function as long as fringing field effects can be neglected. This sharp cut-off fringing field (SCOFF) approximation is a useful tool for initial theoretical investigations. Very often the principal behavior of a corrector system can be described by rather simple analytical relations based on SCOFF calculations. The SCOFF length L of the hexapole field is larger than the physical length of the element LHP ; typically the FWHM of the axial multipole strength is used.

B. Aberrations of Hexapole Fields Before the advent of aberration correctors in electron microscopy hexapole elements—most often simply air coils—have been used as stigmators for threefold astigmatism A2 . Hexapole fields do not influence the paraxial path of rays since paraxial optics considers only constant and linear fields coming from deflectors, quadrupoles, and round lenses. Effectively, a beam transversing a hexapole field is shaped threefold.

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0.02

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0.015

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210

0 z [mm]

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FIGURE 24 Course of the axial hexapole strength for a magnetic hexapole element with length LHP = 30 mm and bore radius RHP = 4.05 mm enclosed by field clamps in front and behind the element. The hashed area indicates the coil with unit excitation NI = 1 At. The data were obtained by a 3D magnetic boundary element method based on the indirect reduced magnetic potential approach for unsaturated magnetic material with permeability µ/µ0 ≈ 104 . (From Müller et al., 2006b.)

1. Equation of Motion To describe this quantitatively we add the hexapole field as a nonlinear perturbation at the right-hand side of the linear paraxial ray equation. To simplify the mathematics we neglect the influence of the fringe fields and assume a box-shaped hexapole strength 3 (z) along the optic axis. Electrons traveling through a constant hexapole field experience a Lorentz force perpendicular to the direction of the magnetic field

d2 η2 B(z)2 u = 3iη 3 (z) u2 , u + 4 dz2

|e| , 2m0 U0⋆

(4)

where e denotes the charge of the electron, m0 the mass   of rest, c the speed of light, and U0⋆ = U0 1 + 2m|e|c2 U0 the relativistically 0 modified acceleration voltage. The left-hand side of the equation of motion [Eq. (4)] describes the focusing action of the round lenses and the free-space propagation in between. with η =

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The hexapole strength causes a small deviation of the electron trajectory u from the course of the paraxial ray. Equation (4) is not exact because it neglects third-order terms form the lens fields and fourth-order terms from the hexapole fields. Nevertheless, all beam plots shown in the subsequent figures are based on the exact theory, which also includes round-lens and fringe-field effects.

2. Successive Approximation To solve the nonlinear equation of motion [Eq. (4)], we perform a perturbation expansion of the electron path u = u(z) with respect to the complex-valued axial and off-axial beam parameters at the object α = ϑx + iϑy and γ = x + iy, respectively

u(z) =

∞

u(i) (z).

(5)

i=1

Here, u(n) denotes the path deviation of order n. The paraxial trajectory u(1) = αuα + γuγ is the general solution of the homogeneous linear equation at the left-hand side of Eq. (4). This paraxial solution is cylindersymmetric, and the two fundamental rays are real-valued with respect to the Larmor frame of reference even if magnetic round lenses are present. By variation of the coefficients we find an integral equation for the general solution of the dynamic problem [Eq. (4)] including the hexapole perturbation



u(z) = α +



z

zγ



 3iη 3 u uγ dz · uα + γ − 2

z zα

2



3iη 3 u uα dz · uγ . (6)

For a system without aperture the lower bounds of the integration are the object plane zα = zγ = zo . In this case αu′α (zo ) is equal to the slope of the exact trajectory at the object plane z = zo . For the canonical fundamental rays with initial values

uα (zo ) = 0,

uγ (zo ) = 1,

u′α (zo ) = 1,

u′γ (zo ) = mγ ,

(7)

the Helmholtz-Lagrange invariant is unity

u′α uγ − uα u′γ = 1,

for all z ≥ zo .

(8)

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In this mixed representation the axial ray uα has the dimension of a length while the field ray is dimensionless. The slope parameter mγ describes the convergence of illumination in CTEM. For parallel illumination we must set mγ = 0. Formally convergent or divergent illumination corresponds to a linear transformation of the off-axial ray uγ → uγ + mγ uα . In STEM this operation corresponds to a change of the position of the aperture plane z = za with uγ (za ) = 0. For a system with aperture a constant multiple of the axial ray must be added at the right-hand side of Eq. (6) to make the path deviation u − u(1) vanish at the aperture plane z = za . This is identical to choosing zγ = za for the lower bound of the first integral. In this case, αuα (za ) is equal to the aperture coordinate of the exact trajectory.

3. Primary Aberrations The integral equation [Eq. (6)] can be solved iteratively by successive approximation. Substituting u → u(1) at the right-hand side yields

2  u2 = αuα + γuγ = α2 u2α + 2αγuα uγ + γ 2 u2γ ,

and the second-order path deviation adopts the form

u(2) = α2 uαα + αγuαγ + γ 2 uγγ .

(9)

This shows that due to the structure of Eq. (6) only a very limited number of index combinations are allowed by symmetry—in this case 3 of 10. In the following we chose a fixed orientation for the hexapole field 3 = i 3s . In this case, the right-hand side of Eq. (4) and, therefore, all aberration rays become real-valued. With this choice the evaluation of the integral [Eq. (6)] results in an explicit representation for the second-order aberration rays

uαα = −uα · uαγ = −uα · uγγ = −uα ·



z

−∞  z −∞  z −∞

3η 3s u2α uγ dz + uγ

·

6η 3s uα u2γ dz + uγ · 3η 3 u3γ dz + uγ ·



z



z

−∞  z −∞

−∞

3η 3s u3α dz 6η 3s u2α uγ dz

(10)

3η 3s uα u2γ dz.

By choosing the fundamental rays uα = f0 and uγ = −z/f0 according to Figure 25 we obtain a piecewise representation of the second-order

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FIGURE 25 Single hexapole element with a certain choice for the paraxial fundamental rays uα = f0 and uγ = − fz . At the left the course of the second-order 0 aberration rays is depicted, and at the right the corresponding aberration coefficients are shown. The parameters are L = 30 mm and f0 = 2 mm.

aberration rays in SCOFF approximation

z < − L2 uαα = 0 uαγ = 0 uγγ = 0

L − L2 ≤ z ≤ L2 2 0 of an objective lens. We further see that the corrector itself is free of off-axial coma Cααγ and geometrical distortion Cγγγ . Up to third order the hexapole doublet behaves exactly like a round lens with negative spherical aberration. This means that with this corrector device the rays corresponding to large-aperture angles are refracted less strongly than the paraxial rays. For any round lens the opposite is true. The correction principle of the hexapole doublet is illustrated vividly by a 3D plot of the shape of the axial beam through the corrector (Figure 27). All path deviations up to third-order have been considered. The threefold shape of the beam between the hexapole elements is clearly visible. Although at the entrance the beam is cylindrical, it is shaped like a divergent cone at the exit. The angle of divergence increases with the third power of the lateral distance from the optic axis. This is exactly the effect of the negative third-order spherical aberration.

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FIGURE 27 Three-dimensional view of the second-order and third-order approximation of a thin sheet of the beam with |α| = constant. Entrance view (top), side view (center), and exit view (bottom) of the corrector.

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The intermediate amount of threefold astigmatism A2 required to correct for a certain C3 is quantified by

A2 = f0



3 |C3 | . 2L

(28)

The derivation of the approximate relations for the third-order coefficients does not consider contributions of the transfer lenses and of the fringe fields. In contrast, all beam plots are based on semi-analytical calculations considering the realistic field shape and the rigorous theory. The results shown in Figure 28 verify that the round-lens contribution to C3 is negligible. Off-axial coma and distortion at the image are not affected by the symmetric arrangement of the transfer lenses as seen in Figure 29 and Figure 31. Nevertheless, the threefold beam shape interacts with third-order aberrations of the transfer lenses. The exact calculations shown in Figure 30 show that the field astigmatism [Eq. (25)] is not purely real valued. The weak imaginary component is due to the magnetic transfer lenses; hence, for field astigmatism and field curvature we must consider small additional contributions—FA3,TL and FC3,TL , respectively—which deteriorate the one-to-one imaging from plane S1 to plane S2 . Both coefficients depend

Rays and coefficient functions [mixed units]

3 2 1 0 21 22

u␣ u␣␣ ¯ ¯ /10 u␣␣␣¯ / 10 C␣␣␣¯

23 24 2100 280 260 240 220

0 20 z [mm]

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FIGURE 28 Coefficient function of third-order spherical aberration C3 = Cααα with the axial fundamental rays uα and the axial aberration rays of second and third-order uαα and uααα , respectively.

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Rays and coefficient functions [mixed units]

3 2 1 0 21 22 Re C␣␣␥¯ Im C␣␣␥¯ x␣␣␥¯ /30 y␣␣␥¯

23 24 2100 280 260 240 220

0 20 z [mm]

40

60

80

100

FIGURE 29 Coefficient function of the third-order off-axial coma K3 = Cααγ and the corresponding aberration ray uααγ . Both the real (x) and imaginary (y) parts are shown.

Rays and coefficient functions [mixed units]

3 2 1 0 21 22

C␣␥␥¯ /10 Re C␣␥␥ /10 ¯ Im C␣␥␥ ¯

23 24 2100 280 260 240 220

0 20 z [mm]

40

60

80

100

FIGURE 30 Coefficient functions of the third-order field curvature FC3 = Cαγγ and the field astigmatism FA3 = Cαγγ . For the field-astigmatism the real and imaginary parts are shown.

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Rays and coefficient functions [mixed units]

3 2 1 0 21 22 Re C␥␥␥¯ / 30 Im C␥␥␥¯ / 30 Re c␥␥␥¯ / 100 Im c␥␥␥¯ / 100

23 24 2100 280 260 240 220

0 20 z [mm]

40

60

80

100

FIGURE 31 Coefficient functions of the third-order distortion D3 = Cγγγ and the third-order spherical aberration at the diffraction plane E3 = cγγγ along the optic axis. Both the real and imaginary parts are shown.

only on the refraction power and the gap geometry of the transfer lens doublet and not on the intermediate magnification.

3. Correction Strength The expression in Eq. (22) for C3 derived shows that the correction strength depends on the square of the hexapole excitation and on the fourth power of the beam radius f0 inside the hexapole elements. Additionally, we observe that C3 correction demands extended hexapole elements.  The limit L → 0 yields C3 → 0 even if the integral hexapole strength η 3s dz = η 3s L is kept constant. This behavior is due to the fact that the negative C3 is a secondary aberration of the hexapole field (or in other words, a combination aberration). So far we have discussed the hexapole corrector independent of the objective lens. Using Eq. (3) we can summarize our findings by a useful rule of thumb for the current driving the hexapole coils,

ηµ0 NI ≈

3 RHP

f02



C3,OL , 6L3

(29)

where C3,OL denotes the spherical aberration of the objective lens to be corrected. To estimate the performance of the entire aberration-corrected instrument it is essential to consider the optical coupling between the

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Rays and coefficient functions [mixed units]

3 2 1 0 21 22 23 24 250

u␣ u␥ /10 C␣␬ C␣␣␣¯ 0

50

100

150

200 250 z [mm]

300

350

400

450

FIGURE 32 Hexapole corrector integrated in CTEM with objective lens, transfer lens system, and adapter lens. The course of the fundamental rays uα and uγ and of the coefficient functions of the axial chromatic Cc = Cακ and third-order spherical aberration C3 = Cααα are also plotted.

corrector and the objective lens. At least one additional transfer lens is necessary between the aberration corrector and the objective lens. Most systems actually use a transfer lens doublet to gain further flexibility in alignment. A typical beam path for a CTEM equipped with a hexapole Cs corrector with two transfer lenses between the objective lens and the first hexapole is depicted in Figure 32. Below the corrector an additional adapter lens is used to focus the telescopic beam at an intermediate image plane; typically the selected area diaphragm can be inserted there. In STEM the corrector is integrated between the condenser system and the objective lens. In this case, the STEM aperture is situated above the corrector. Again one or two transfer lenses are required between the corrector and the objective lenses. By adjusting the objective lens excitation and the focal length of the transfer lenses the coma-free aperture plane of the objective lens field (usually this is roughly the focal plane of the objective lens) can be imaged at or near the center S1 of the first hexapole element while the magnification between these conjugated planes can be tuned over a certain range (most often between 0.5 ≤ MTL ≤ 1.0). We should carefully distinguish this angular magnification from the lateral magnification at the mid-plane S0 of the f corrector, which is actually M TLf ≈ 15 − 30, where fOL and fTL denote TL OL the focal length of the objective lens and of the transfer lenses between the

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Rays and coefficient functions [mixed units]

3 2 1 0 21 22 23 24 250

u␣ u␥ /10 Re C␣␣␥¯ Im C␣␣␥¯ 0

50

100

150

200 250 z [mm]

300

350

400

450

FIGURE 33 Coefficients of radial (isotropic) and azimuthal (anisotropic) off-axial coma Cααγ for a CTEM equipped with hexapole corrector. The transfer lenses are adjusted such that the semi-aplanatic condition is fulfilled.

hexapole stages, respectively. Changing the magnification MTL effectively changes the height of the axial ray

(30)

f0 = MTL fOL

inside the hexapoles. By this means the correction strength of the hexapole corrector is tuned according to Eq. (29). The correction strength stays constant if the hexapole current is changed simultaneously keeping the product NI · f02 constant. If the magnification MTL is kept constant and only the hexapole excitation is changed, we can tune the total spherical aberration C3 between overcompensation and undercompensation

C3 = −2C3,OL

I . I

(31)

This is very useful in allowing for the optimization of phase contrast in the CTEM, which requires alternating signs (e.g., C1 > 0, C3 < 0, and C5 > 0).

4. Chromatic Aberration The transfer lenses of the corrector increase the total chromatic aberration of the system. Their contribution to the coefficient Cc is clearly visible in

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Figure 32. This agrees with the estimate

Cc,TL =

2 f02 2 fOL = MTL fTL fTL

(32)

for the chromatic aberration of a single transfer lens. We observe that the corrector adds about 15–25% to the chromatic aberration of the objective lens. The additional Cc is proportional to the square of the intermediate magnification MTL . Hence, high excitations of the hexapole elements are preferable to minimize the total chromatic aberration.

5. Semi-Aplanatic Condition For CTEM imaging the course of the field ray between the objective lens and the corrector is essential. Only if the coma-free plane of the objective lens is matched with the center of the hexapoles at S1 and  S2 does the third-order radial (isotropic) part of the off-axial coma, Re Cααγ , vanish. If this condition is fulfilled, the corrected system is called semi-aplanatic (Rose, 1990). It is not fully aplanatic since the azimuthal off-axial coma of the magnetic objective lens caused by Larmor rotation is still present. The course of the off-axial coma for a semi-aplanatic system is shown in Figure 33. If the coma-free aperture planes are not matched, residual real-valued off-axial coma is introduced. The amount of off-axial coma is proportional to the distance HP between S1 and the image of the coma-free aperture plane of the objective lens

Cααγ = − HP

C3,OL f02

.

(33)

For any C3 -corrected CTEM the off-axial coma is independent of the convergence of the illumination, since changing the illumination angle changes the course of the rays in the total system. In this case, the total C3 = 0 is relevant in Eq. (33) and, hence, no off-axial coma is induced.

6. Field Aberrations For a semi-aplanat the only residual intrinsic third-order aberrations at the image are field curvature and field astigmatism. For an objective lens with focal length fOL and spherical aberration C3,OL , we find the relation

Cαγγ = −Cαγγ =

1 C3,OL L2 . · 10 (MTL fOL )4

(34)

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This contribution of the hexapole fields dominates the intrinsic contribution of the objective lens. The parameters L and MTL could be used to minimize field curvature and field astigmatism, or a different corrector design with more then two hexapole stages could be used to eliminate this aberration completely (Rose, 2002). Nevertheless, field curvature and field astigmatism are an issue only at low magnification. In this case, the Nyquist frequency of the CCD detector becomes so small that Cs correction is no longer useful; therefore, it seems most appropriate to disable the corrector for low magnification.

D. Higher-Order Aberrations The next steps in the successive approximation scheme reveal further higher-order aberrations. Here, we restrict our attention to the axial aberrations only.

1. Fourth-Order Aberrations For the fourth-order path deviation we find

u(4) = α4 uαααα + αα3 uαααα .

(35)

This follows by substituting u(3) = αuα + α2 uαα + α2 αuααα in the iteration formula [Eq. (6)] after collecting the fourth-order terms. The image coefficients are

Cαααα = Cαααα =





2 uα dz, 3η 3s uαα

(36)

6η 3s uααα u2α dz.

(37)

The second coefficient can be evaluated further after inserting the explicit representation of the spherical aberration ray. We obtain by partial integration the relation

Cαααα = 4Cαααα .

(38)

According to this result the three-lobe aberration at the image has the form

u(4) /uγ = α4 D4 + 4αα3 D4 .

(39)

Since 3s is a symmetric function and uα is anti-symmetric, we find from Eq. (36) that the three-lobe aberration D4 = Cαααα vanishes at the image

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plane but not the corresponding slope coefficient

cαααα = −



2 uγ dz. 3η 3s uαα

(40)

As a direct effect the latter is visible as a threefold distortion at the diffraction plane. Residual three-lobe aberration shows up also at the image if the intermediate image plane is moved away from the mid-plane S0 of the corrector. For this reason focusing through the corrector should be avoided. To shift the position of an image plane (e.g., the SA plane below the corrector in CTEM) always excitation of the adapter lens should be changed; the objective lens should never be tuned. Also in STEM the probe semi-angle should be tuned by changing lenses above the corrector only. From Eq. (40) we can calculate a SCOFF approximation for the slope coefficient of the three-lobe aberration

cαααα =

9 3 3 3 6 η 3s f0 L . 10

(41)

In Figure 34 the fourth-order aberration rays uαααα and uαααα are plotted according to an exact calculation. Although the three-lobe aberration becomes complex-valued due to the influence of the transfer lenses, it is still point-corrected at the image. The contribution of the transfer lenses

Rays and coefficient functions [mixed units]

3 2 1 0 21 22 23 24 250

u␣ u␥ /20 5 Re C␣␣␣␣ x␣␣␣␣ /5 5 y␣␣␣␣

0

50

100

150

200 250 z [mm]

300

350

400

450

FIGURE 34 Fourth-order axial aberration ray uαααα = xαααα + iyαααα and coefficient function of the three-lobe aberration D4 = Cαααα for a CTEM with hexapole corrector.

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to D4 cancels by symmetry. It is important to note that inside the second hexapole the aberration ray uαααα ≈ f0 cαααα is almost constant. The slope coefficient cαααα is affected by the third-order aberrations of the transfer lenses. Actually their residual field astigmatism FA3,TL makes the uαααα ray complex-valued. This effect can be estimated by considering an off-axial image point u = αf0 at plane S1 with slope u′ = α2 A2 /f0 . At the conjugated plane S2 a path deviation is induced by the field astigmatism of the transfer lenses. The result adds to the aberration ray of the three-lobe aberration inside the second hexapole field

uαααα = −

9 3 3 6 4 η 3s L f0 + 3η 3s Lf04 FA3,TL . 10

(42)

This effect is clearly visible in Figure 34. Although it is rather small, it has important consequence for the residual fifth-order aberrations. At this stage, we can summarize that for a system equipped with a hexapole Cs corrector, no residual axial aberrations up to and including fourth order occur. The first nonvanishing axial aberrations of the hexapole corrector are of fifth order.

2. Fourth-Order Off-Axial Aberrations Here we note that the hexapole corrector has coma-type off-axial aberrations only in fourth order, which depend linearly on the radius of the field of view. Nevertheless, the number of equally well-resolved image points for a present Cs -corrected CTEM is limited by the azimuthal third-order coma of the magnetic objective lens and not by the corrector. This matter and how a hexapole corrector can be used to correct for azimuthal off-axial coma are discussed in greater detail in Section III.

3. Fifth-Order Aberrations The next step of the iteration procedure results in the fifth-order path deviation

u(5) = α3 α2 uααααα + α5 uααααα .

(43)

Evaluated at the image we obtain

u(5)/uγ = α3 α2 C5 + α5 A5 .

(44)

with the coefficients of the fifth-order spherical aberration C5 and the sixfold astigmatism A5 . The hexapole corrector has an intrinsic positive C5 ,

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which adds to that of the objective lens. Additionally, a strong combination effect can occur between objective lens and corrector, which then contributes to the total C5 of the system.

4. Tuning Fifth-Order Spherical Aberration Since the aberration ray of the spherical aberration between objective lens and corrector has the form

uααα = uγ C3,OL ,

(45)

the combination effect vanishes if the nodal plane of the field ray uγ = 0 coincides with the coma-free aperture plane of the corrector. This is the standard situation as depicted in Figure 35. If the transfer lenses between the objective lens and the corrector are changed such that C3 correction is maintained but the coma-free aperture plane of the corrector is separated by the distance HP from the plane conjugated to the coma-free aperture plane of the objective lens, a combination contribution C5 is intentionally introduced

C5 = −3 HP

C3,OL f0

2

.

(46)

Rays and coefficient functions [mixed units]

3 2 1 0 21 22 23 24 250

u␣ u␥ /20 C5 /3 Re A5/3 Im 3 * A5 0

50

100

150

200 250 z [mm]

300

350

400

450

FIGURE 35 Coefficient function of the fifth-order spherical aberration C5 = Cααααα and sixfold astigmatism A5 = Cααααα for a CTEM with hexapole corrector. For the sixfold astigmatism A5 both the real and imaginary parts are plotted.

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Rays and coefficient functions [mixed units]

3 2 1 0 21 22 23 24 250

u␣ u␥ /10 Re C␣␣␥Im C␣␣␥C5/3 0

50

100

150

200 250 z [mm]

300

350

400

450

FIGURE 36 Transfer lenses adjusted for C5 = 0. The fifth-order spherical aberration C5 and both components for the third-order off-axial coma Cααγ are plotted.

This tunable contribution can be used to counterbalance the intrinsic contribution of the objective lens and of the corrector with the effect that the total fifth-order spherical aberration becomes zero C5 = 0. Figure 36 shows a system in which the transfer lenses have been tuned for C5 -free imaging. It is immediately clear that in this case some off-axial coma must be accepted. By combining Eqs. (33) and (46) the residual amount of off-axial radial coma can be estimated as

Cααγ =

1 C5 ∈ R, · 3 C3,OL

(47)

where C5 is the sum of the primary fifth-order spherical aberration of the objective lens and of the corrector. Unfortunately, Eq. (47) shows that the off-axial coma introduced for C5 -free alignment is rather pronounced, since C5 is typically considerably larger than C3 . Therefore, C5 correction is most useful in STEM, where the residual off-axial coma does not matter since the scan is performed between the corrector and the objective lens. The possibility of tuning the fifth-order spherical aberration in a system with C3 corrector has been discussed by Rose (1971b). He already states Eq. (46). For a STEM equipped with hexapole corrector this relation is also mentioned by Crewe (1980) and Shao (1988). Later Hartel et al. (2004) demonstrated the possibility to tune C5 experimentally (see Figure 37).

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C5 tuning at a JEOL 2010F URP with CETCOR

C5 ⫺ free

3

Aberration coefficients

4 3

2

2

1

1

0

0

⫺1 ⫺2 B3c⫺ free ⫺3 ⫺4 ⫺9 ⫺8 ⫺7 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1

⫺1

OL TL12 C5/mm B3c B3s |A5|/mm 0

1

2

3

⫺2

Change of lens excitation dl/l [%]

4

IHP ⫽ 144 mA, Cc ⫽ 1.42 mm, C3 ⫽ 0.51 mm, f ⫽ 1.81 mm

⫺3

4

⫺4

Change of TL11 excitation dl/l [%]

FIGURE 37 Experimental verification of semi-aplanatic (here K3 = B3c + i B3s ) and C5 -free alignment for a CTEM hexapole corrector with small gap objective lens taken. (From Müller et al., 2007.)

After it had been demonstrated that a system with C3 = C5 = 0 is feasible, the sixfold astigmatism remained the only residual axial aberration in fifth-order and the question arose how A5 could be corrected or at least minimized.

5. Minimizing Sixfold Astigmatism The dominant contribution to the sixfold astigmatism is caused by the combination of the fourth-order aberration ray uαααα with the hexapole field. With the approximation from Eq. (42) we can estimate the sixfold astigmatism by

A5 = −

27 |η 3s |4 f06 L7 + 18 |η 3s |2 f06 L2 · FA3,TL . 5

(48)

This result is in good agreement with more accurate calculations. The influence of the transfer lenses causing the second term in Eq. (48) typically

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Rays and coefficient functions [mixed units]

3 2 1 0

21

22

23

24 250

u␣ u␥ /10 C5 /3 Re 3A5 Im 3A5 0

50

100

150

200 z [mm]

250

300

350

400

450

FIGURE 38 Axial fifth-order coefficient functions C5 and A5 for a hexapole corrector with optimally chosen length Lopt of the hexapole elements.

is rather small. By using the expressions from Eq. (22) we can recast this expression

 2

 MTL fOL C3,OL C3,OL 2 3 A5 = − L · FA3,TL . +3 20 MTL fOL L

(49)

For most corrector systems the first term strongly dominates (see Figure 38). Therefore, the most obvious way to reduce A5 is to use an objective lens with a narrow gap. For such a system we have C3,OL = 0.5– 0.6 mm and, hence, the residual A5 after C3 correction can be well below 1 mm, and Cc is sufficiently small for optimum high-resolution performance. The alternative measure to increase MTL is not attractive since this also increases the Cc contribution of the corrector. The second complex-valued term in Eq. (49) due to FA3,TL becomes important if the beam inside the hexapoles is large or if the hexapoles are short. Since the real part of field astigmatism for the anti-symmetric transfer doublet is positive a certain length Lopt exists such that the real part A5x of the sixfold astigmatism vanishes. By setting A5x = 0 in Eq. (49)

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we derive

Lopt =



20

2 FA3x,TL  MTL fOL > 0. C3,OL

(50)

This shows that the most efficient way to reduce A5 is to reduce the length of the hexapole elements if feasible down to the optimum where A5x = 0. Additionally, it can be advantageous to maximize the field astigmatism FA3x,TL of the transfer doublet by reducing the gap and bore dimensions to avoid infeasibly small values for Lopt . Since the imaginary part FA3y,TL of the field astigmatism is non-zero, this procedure introduces a residual

A5y

2  MTL fOL C3,OL FA3y,TL . = −3 L

(51)

This effect sets an upper limit for the tolerable FA3y,TL . Fortunately, for a magnetic transfer doublet the real part FA3x,TL strongly dominates the imaginary part FA3y,TL of the field astigmatism. Hence, the unwanted increase of A5y is small Eq. (50) shows that the magnitude of the sixfold astigmatism |A5 | adopts a minimum if L is varied for constant MTL or if MTL is varied for constant L. Figure 38 plots the course of the fifth-order aberration coefficients through the corrector for a system with reduced L and optimized transfer lens geometry. For a certain hexapole excitation both coefficients A5x = 0 and C5 = 0 are corrected. The data are based on an exact calculation. The residual sixfold astigmatism in this system A5y is in good agreement with Eq. (51). Figure 38 shows that the transfer lenses additionally contribute directly to A5x . This minor effect is due to the third-order field curvature of the transfer lenses and has been neglected in the derivation of Eq. (48). For this reason the exact calculation results in a slightly larger optimum length Lopt than predicted from Eq. (50). Figure 39 shows the result of the minimization of |A5 | performed for the design of the CEOS advanced hexapole STEM corrector D-COR (Müller et al., 2006a). For the D-COR systems the aim was to reduce the residual A5 to such an extent that STEM resolution better than 50 pm with a semi-aperture angle of ϑ = 38 mrad at 300 kV can be achieved. For the first and second prototype of this corrector the residual A5x and A5y have been measured. Figure 40 shows the behavior of both coefficients during a tuning of the intermediate magnification MTL . To keep C3 corrected the hexapole excitation must be adjusted accordingly. With increasing excitation we find a linear decrease of A5x and finally a change of sign while A5y stays almost constant and close to zero. The slightly different results

99

Present and Future Hexapole Aberration Correctors

C3s |A5| A5y A5x

3

8 6 4

1

2

0

0

21

22

22

24

A5x , A5y , |A5| [mm]

2

23

5

10

15

20 25 30 Length of hexpoles LHP [mm]

35

40

Hexapole strength C3s/␮0 [At/mm3]

4

26 45

FIGURE 39 Variation of A5x and A5y with the length of the hexapole elements for a typical hexapole corrector according to Müller et al. (2006a). 300

A5x A5y A5x A5y

(1. (1. (2. (2.

prototype) prototype) prototype) prototype)

Sixfold astigmatism A5 [um]

200

100

0

2100

2200 125

130

135 140 145 Hexapole current [mA]

150

155

FIGURE 40 Measured values of the sixfold astigmatism for the first two prototypes of the advanced CEOS STEM hexapole corrector (D-COR). (From Hartel et al., 2007.)

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for the two prototypes can be attributed to manufacturing tolerances and non-ideal alignment. Additionally, the true quantitative assessment of fifthorder aberrations on the level of only a few hundred micrometers is questionable with the currently available methods for aberration measurement in STEM. Within the TEAM project the advanced hexapole STEM corrector has been used to set the new milestone of STEM resolution better than 50 pm as mentioned in Section I.

6. Elimination of Scan Coma The D-COR-type hexapole STEM corrector with minimized A5 and C5 free alignment as depicted in Figure 38 is the optimum system for a highresolution STEM equipped with a cold-FEG or gun monochromator. To obtain a large scan field the scan-induced axial coma in second order must be carefully considered. The usual practice of aligning the pivot point of the scan with the front focal plane or with the center of the coma-free aperture plane of the objective lens is not sufficient in this case. The scan coils situated between the corrector and the objective lens must be operated such that both components of the scan coma, the isotropic and the anisotropic (or azimuthal) part, are compensated simultaneously. Effectively, this makes the scan skew with respect to the optic axis.

7. Further Axial Aberrations The optical performance of TEM instruments equipped with hexapole Cs correctors typically is not limited by residual intrinsic aberrations but by incoherent effects such as focus spread or image spread. The sum of all incoherent effects finally determines the information limit of the instrument. An advanced hexapole design with C3 /C5 correction and strongly suppressed sixfold astigmatism A5 is necessary if the chromatic focus spread in a TEM is strongly reduced by using a cold FEG or a gun monochromator. For every aberration-corrected system it is also mandatory to correct for the parasitic aberrations caused by manufacturing tolerances of the optical elements and misalignments. This requires precise methods for aberration assessment and highly sophisticated alignment strategies. Under optimum conditions even the residual parasitic aberrations should not limit the attainable optical resolution. The next higher-order intrinsic aberrations of a hexapole corrector occur in order six and seven. By our iteration method we find for the path deviation at the image

u(6) + u(7) = 2α5 αD6 + 5α2 α4 D6 + α4 α3 C7 + α7 G7 + 7αα6 G7 .

(52)

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The coefficients are the three-lobe aberration in sixth order D6 , the seventh-order spherical aberration C7 , and the sixfold chaplet aberration G7 . For a carefully designed corrector system, none of these aberrations are limiting.

III. HEXAPOLE APLANATS A. Semi-Aplanat Versus Aplanat About 30 years ago Vernon Beck discovered the negative spherical aberration of a hexapole doublet (Beck, 1979). Ten years later, Rose pointed out that it is essential to adapt the hexapole Cs corrector to the objective lens by a transfer lens system, in order to make it applicable to the conventional TEM (Rose, 1990). The reason for the transfer system is twofold. First, it avoids a large fifth-order spherical aberration that would otherwise arise as a strong combination between the objective lens and the corrector. On the other hand, the transfer lens system allows imaging of the coma-free plane of the objective lens into the coma-free plane of the corrector. The isotropic off-axial coma of the system vanishes only if this condition is met. We would call an optical system that is free of spherical aberration and off-axial coma aplantic. However, unfortunately, the magnetic objective lens also introduces anisotropic off-axial coma that still remains even if the coma-free planes are matched. Rose suggested the term semi-aplanat for a spherical aberration-corrected system that is free of isotropic coma and still has anisotropic off-axial coma (Rose, 1990). The delocalization area of both the isotropic (radial) and the anisotropic (azimuthal) off-axial coma are illustrated in Figure 41. Today hexapole semi-aplanats consisting of a magnetic round lens and a hexapole doublet are commercially available options for both the CTEM and the STEM operation mode of the microscope. Optimized and stable current drivers and high-voltage supplies and the increasing availability of monochromators have dramatically improved the information limit of the microscopes. Sub-Ångstroem resolution is now readily available, and hence, apertures in the range of 25 mrad to even 50 mrad become usable. Considering these large apertures, the remaining anisotropic part of the off-axial coma seriously limits the field of view that can be imaged with same, nearly constant phase shift. We can draw a circle of radius R around a perfectly corrected axial point, which indicates that an additional π/4−phase shift due to the off-axial coma is surpassed:



  3 π 3 2π   < . · R · Cααγ · ϑ 1 − λ 4 4

(53)

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FIGURE 41 The delocalization area of both the isotropic/radial (light grey) and the anisotropic/azimuthal (dark grey) off-axial coma. For increasing circles in the image plane, the shape of the comet-tail–like delocalization figure is depicted. The diameter of the shape increases linearly with the distance to the aberration-free axial point. If a semi-aplanat is perfectly aligned, only the azimuthal component remains.

Here ϑ denotes the aperture semi-angle that corresponds to the highest spatial frequency in consideration gmax = ϑ/λ (e.g., the information limit). The factor (1 − 3/4) = 1/4 accounts for the compensation of the third-order phase shift of the coma by a first-order phase shift induced by a change of the image magnification and/or rotation that one would immediately recognize.   Assuming a typical dimensionless coefficient of Cααγ  = 2/3 for the anisotropic off-axial coma the π/4−radius can be illustrated as shown in Table II. According to this criterion, only 600 . . . 2400 points of size 1/gmax can be resolved. In this chapter, we discuss advanced hexapole correctors that are also capable of eliminating the anisotropic part of the off-axial coma. Combining these correctors with a magnetic objective lens results in a true aplanatic optical system, which we will consequently call an aplanat.

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TABLE II Field of view limited by the off-axial coma. The radius R π = λ2 /( Cααγ  · ϑ3 ) of a circle in the image plane around an axial point for 4   which the additional phase shift of an anisotropic off-axial coma Cααγ  = 2/3 is less than or equal π/4. Here N denotes the number of pixels of size 1/gmax in the diameter 2R π . 4

λ[pm]

2.5 2.5 2 2

ϑ[mrad]

25 50 25 40

gmax =

ϑ λ

[1/nm]

10 20 12.5 20

R π [nm]

N = 2R π · gmax [1000 px]

120 15 96 23

2.4 0.6 2.4 0.9

4

4

B. Advancing the CTEM Hexapole Cs Corrector Various options are available to correct the off-axial coma in quadrupoleoctupole correctors (and eventually the coma correction there is even mandatory). However, in this chapter we want to adhere to the simplicity and the advantages of the hexapole corrector with rotational symmetric Gaussian rays. Hence, the only focusing elements are (transfer-)round lenses. We tailor an appropriate set of combination aberrations induced by the strong threefold astigmatism, which in turn travels through hexapole fields.

1. Requirements Since we do not want to compromise any of the advantageous properties of the original semi-aplanat, the advanced hexapole corrector must meet a set of constraints. It must 1. Be completely free of second-order geometrical aberrations at any plane outside the corrector for that of the objective lens 2. Produce a negative Cs to compensate  3. Produce a negative Im Cααγ to compensate for that of the objective lens 4. Be free of fourth-order axial aberrations at the image plane (at least point correction). Moreover, a preferable aplanat would additionally 1. Have a Cc that is not (much) higher than that of the semi-aplanat 2. Have a small fifth-order sixfold astigmatism A5 , at least below the π/4−limit 3. Have an optimally chosen spherical aberration C5 that enables phase contrast.

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2. Integral Formulation of Essential Requirements for Aplanats As discussed for the basic setup of any hexapole corrector, all second-order aberration coefficients in [Eq. (20)] must vanish behind the corrector; the aberration rays are zero in every plane. However, within the corrector the second-order aberration rays are present and intentionally chosen. Again, the aberration ray of the threefold astigmatism

uαα (z) = uα (z) ·

z

−∞

3iη u2α uγ 3 dz − uγ (z) ·

z

3iη u3α 3 dz

−∞

= uα (z) · a2 (z) + uγ (z) · A2 (z) plays an important role, because it carries two α parameters. Since the corrector should correct mainly axial aberrations, we further restrict the selection of suitable correctors to systems that fulfill “locally”



u2α uγ 3 dz = 0.

(54)

local

This means that hexapole elements are placed at aperture planes or anti-symmetric around intermediate image planes. Hence, the coefficient function a2 (z) of the aberration ray uαα does not arise outside the hexapole element and will not produce combination aberrations with other elements. The aberration ray of the threefold astigmatism then can be written as

uαα (z) = −uγ (z) ·

z

−∞

3iηu3α 3 dz = + uγ (z) · A2 (z)

(55)

We now generalize the results from Sections II.C.2 and II.D for complex 3 and A2 with varying orientation (arguments) along the z-axis. The orientation is always measured with respect to the rotating Larmor frame of reference. The results are written as:

Cααγ = − D4 = Cαααα = −





6iηuα u2γ A2 3 dz 2

3iηuα u2γ A2 3 dz.

and

(56) (57)

The existing semi-aplanat (consisting of a magnetic objective lens and the two-stage hexapole corrector) suffers from the remaining anisotropic

Present and Future Hexapole Aberration Correctors

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  part Im Cααγ of the off-axial coma. Hence, the corrector should produce only





Im Cααγ = −



  6ηuα u2γ · Re A2 3 dz,

(58)

and no real part. Even harder to accomplish is the requirement that the coma correction must not introduce an accompanying axial fourth-order aberration

Im{D4 } = −



 2  3ηuα u2γ · Re A2 3 dz.

(59)

As can be seen from Eqs. (58) and (59), the integral kernels for the coma and the three-lobe aberration are nearly identical. They have the same signature in the fundamental rays. Hence,  most of the conceivable hexapole  correctors will produce neither Im Cααγ nor Im {D4 } or, even worse, both.

C. Aplanats Without Axial Fourth-Order Aberrations The first finding from the last section is that the hexapole elements of the corrector must be rotated with respect to each other. Otherwise it would not be possible to obtain a non-zero contribution to the azimuthal     coma [Eq. (58)] since we can deduce Re A2 3 ∼ Re i 3 3 = 0. A coma correction essentially needs the following two steps: 1. It must produce A2 in a first hexapole element of the corrector and, 2. Let this aberration combine with a second hexapole element, which is rotated against the first. This immediately results in the second finding, that we need at least three separate hexapole elements. This is obvious since the total A2 must be compensated behind the corrector, and astigmatism A2 of two rotated hexapole fields cannot cancel each other. We will now further reduce the variety of still possible hexapole-type correctors by requiring them to have a mid-plane symmetry. This means that the fields (and hence the geometry of the field producing optical elements), the fundamental rays, and the primary aberration functions of the corrector shall be symmetric or anti-symmetric functions with respect to the mid-plane zm of the corrector. In detail, these requirements are stated as

uα (zm + z) = ±uα (zm − z), 3c (zm + z) = ± 3c (zm − z), 3s (zm + z) = ± 3s (zm − z),

uγ (zm + z) = ±uγ (zm − z), A2x (zm + z) = ±A2x (zm − z), A2y (zm + z) = ±A2y (zm − z).

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These symmetry requirements are the most obvious way to tailor the aberration integral kernels such that contributions of the first and the second half of the corrector either can compensate for each other or can add up to twice the contribution of one half. (As mentioned previously, Hosokawa suggested (Mitsuishi, 2006) that it is also possible to apply scaling rules between both halves of the corrector, which to some extent results in the same cancellation of unwanted integrals. Such generalized, scaled symmetry rules eventually also could be applied here to some extent.) We indicate the chosen symmetry of functions by attaching a sign top-right to the function (e.g., the function u− α would have the symmetry uα (zm + z) = −uα (zm − z), and u+ the symmetry uα (zm + z) = α + uα (zm − z), respectively). The integral kernel of the imaginary coma [Eq. (58)] now is written as a symmetric function

  uα u2γ · (A2y 3s + A2x 3c )

zm +z

  = +uα u2γ · (A2y 3s + A2x 3c )

zm −z

,

while the kernel of the accompanying three-lobe aberration [Eq. (59)] should be anti-symmetric to cancel      uα u2γ · 2A2x A2y 3s + A22x − A22y 3c  zm +z      = −uα u2γ · 2A2x A2y 3s + A22x − A22y 3c 

zm −z

.

Here we again have set A2 (z) = A2x (z) + iA2y (z) and 3 (z) = 3c (z) + i 3s (z), assuming four different real functions for the respective real and imaginary parts. Now we are able to find all possible symmetries, which provide correction of the anisotropic coma without introduction of Im {D4 }. The solutions are listed in Table III. Note that the symmetry type of uγ does not matter due to the square. However, it is always opposite to that symmetry type chosen for uα .

1. Type (a) corrector with u+ α : three hexapoles only This type of corrector seems to allow the shortest systems with a minimum of three hexapole elements, since one can be placed at the symmetry plane zm , which is an aperture plane for type (a) correctors (see Figure 42). The anti-symmetry of A2x is achieved by exactly overcompensating that induced by the first hexapole by a factor of two. The other component A2y is introduced in the first and compensated in the last hexapole. In other terms, the hexapoles have the orientations ϕ − ϕ, ϕ, ϕ + ϕ. In the first and third hexapole 3 and A2 are perpendicular; no coma correction

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TABLE III All possible choices for the symmetry of fundamental rays, hexapole field, and threefold astigmatism, if a corrector is to produce anisotropic coma and no three-lobe aberration. Type

uα

A2

(a)

u+ α

A− 2x

A+ 2y

− 3c

+ 3s

(b)

u− α

A− 2x

A+ 2y

+ 3c

− 3s

(c)

u− α

A− 2x

A− 2y

+ 3c

+ 3s

(d)

u+ α

A− 2x

A− 2y

− 3c

− 3s

3

occurs. The center hexapole, however, is rotated by ± ϕ against both the entrance and exit A2 , and therefore produces coma with





Im Cααγ = −



2 + + 6η u+ α uγ A2y 3s dz.

center hexapole

− The summand with A− 2x 3c is zero in the center hexapole. All three hexapoles produce negative contributions to Cs . A disadvantage of the simple setup in Figure 42 is the lack of flexibility. First, if coma and Cs correction should be adjusted separately, the first and third hexapole field would have to be rotated. A separate adjustment is mandatory, however, if the magnification between the objective lens and the corrector is changed for some reason and preservation of both corrections is desired. Second, even if rotatable hexapole fields are provided, a magnification change and the subsequent adjustment of the Cs correction will destroy the elimination of the second-order threefold distortion, since this can only be done once for one excitation by choosing the length of the central hexapole element. The other second-order integral, which is not compensated locally within each hexapole, is A2x , see Eq. (10). This compensation, however, can always be achieved by balancing the strength of the center hexapole against the two outer, which is required to make A2x anti-symmetric. However, this lack of flexibility might not be too bothersome, since the optimization for minimum A5 requires a fixed (optimum) excitation of the hexapoles. Moreover, this optimized operation mode also fixes the magnification between objective lens and corrector and, hence, the fundamental rays. Thus, this corrector combined with an objective lens can be considered as an optimized aplanat for a given high-resolution mode of the microscope, and as a semi-aplanat for all other operation conditions.

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Ray, coefficient [mixed units]

1

ua/10 u␥ /100 A2x A2y

0.5

0

20.5

21

2100

250

0 z-zm [mm]

50

3

Im(C␣␣␥) 10*Im(D4) Cs

2 Ray, coefficient [mixed units]

100

1

0

21

22

23

2100

250

0 z-zm [mm]

50

100

  FIGURE 42 Minimum setup for a hexapole corrector that corrects Cs and Im Cααγ simultaneously without introducing fourth-order axial aberrations. Both coefficients to be corrected start with the initial values of the objective lens on the left-hand side (bottom). The choice of the symmetric fundamental and primary aberration rays is also shown (top).

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2. Type (a) corrector with u+ α : three hexapoles plus two weak hexapole pairs More flexibility provides a second solution of the same symmetry type (a) as shown in Figure 43. Here the coma correction is provided by two hexapole pairs positioned around the intermediate images between the strong hexapoles. The coma integral is

  Im Cααγ = −



2 − − 6η u+ α uγ A2x 3c dz ,

which is much more efficient here, since uγ is about maximal and constant and A2x is the larger astigmatism component. Complementary to the + minimal setup we find here A+ 2y 3s ≈ 0, since A2y is about three orders of magnitude smaller than A2x . Rose suggested using the original two-stage hexapole corrector, completed by two pairs of weak hexapole elements centered around the images before and after the first hexapole (Rose, 2005). This is approximately the first half of the corrector in Figure 43. For the situation depicted there, one would expect about D4y = −60 µm, which would vary, depending on the Cs to be corrected and on the intermediate magnification. However, as derived above, this unwanted D4y cannot be avoided if a two-stage corrector produces imaginary off-axial coma. The solution of Figure 43 also adds mid-plane symmetry to A2y (and a third hexapole plus its lens doublet) and enables the cancellation of D4y .

3. Type (b) corrector with u− α : a minimum of four hexapoles Figure 44 shows the simplest solution with an anti-symmetric axial fundamental ray. As shown in Table III, the threefold astigmatism has the same symmetry as in the above solutions, while the hexapole components + − exchange their symmetries with 3c and 3s . In contrast to the solution of Section III.C.1, this solution fulfills all constraints for the primary secondorder aberrations by symmetry relations; therefore, it is more flexible. The intermediate magnification can be adjusted, and hence the Cs /Cααγ correction without violating the second-order constraints of Eq. (20). Only four hexapole fields are needed; however, the inner two must be rotatable to allow for separate adjustments of the spherical aberration and the anisotropic coma. The second disadvantage of this system is the necessary third round lens doublet and the resulting additional length of 4f . The coma correction this time is given by the integral

  Im Cααγ = −



  − + + − 2 A + A 6η u− u α γ 2x 3c 2y 3s dz,

where both terms contribute equally.

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Ray, coefficient [mixed units]

1

ua/10 u␥ /100 A2x 1000*A2y

0.5

0

20.5

21

2100

250

0 z-zm [mm]

50

3

Im(C ␣␣␥) 10*Im(D4) Cs

2 Ray, coefficient [mixed units]

100

1

0

21

22

23

2100

250

0 z-zm [mm]

50

100

FIGURE 43 More flexible  setup for a hexapole corrector of symmetry type (a) that corrects Cs and Im Cααγ simultaneously without introducing fourth-order axial aberrations. It consists of three strong hexapoles with 3 ≡ i 3s (large boxes) and two locally anti-symmetric pairs of weak hexapole elements with 3 ≡ 3c (small boxes). The choice of the symmetric fundamental and primary aberration rays is depicted (top). The correction of the coma and the spherical aberration can be assigned to the weak and the strong hexapole fields, respectively (bottom).

Present and Future Hexapole Aberration Correctors

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1

Ray, coefficient [mixed units]

ua/10 u␥ /100 A2x A2y

0.5

0

20.5

21 2200

2150

2100

250

0 z-zm [mm]

50

100

3

200

Im(C ␣␣␥) 10*Im(D4) Cs

2 Ray, coefficient [mixed units]

150

1

0

21

22

23 2200

2150

2100

250

0 z-zm [mm]

50

100

150

200

− FIGURE 44 Minimum setup for   a hexapole corrector with anti-symmetric axial ray uα that corrects Cs and Im Cααγ simultaneously without introducing fourth-order axial aberrations. The choice of the symmetric fundamental and primary aberration rays is depicted (top). The orientation of the hexapole fields is ϕ, −ϕ + ϕ, ϕ − ϕ, −ϕ. The correction of the coma can be assigned to the second and third hexapole, while all four contribute to the Cs correction (bottom).

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4. Type (c) and (d) correctors Finally the type (c) and (d) correctors of Table III remain. They do not exist. This is due to the following contradiction: From Eq. (55) we again deduce the integral for A2x with an arbitrary orientation of the hexapoles along the z-axis

A2x (z) =

z

3η u3α 3s dz ,

and

−∞

 d  A2x (z) = 3η u3α 3s  . z dz

According to Table III, type (c) correctors shall have

A− 2x

→

d A2x dz

+

 + → u3α 3s ,

+ which is in contradiction to u− α and 3s . The same argument holds for A2y , and hence type (c) correctors cannot be constructed. If we exchange the symmetries of uα and 3s for type (d), we find the same contradiction.

D. Feasibility and Prediction of Properties After the elimination of the anisotropic coma of the objective lens, the isoplanar field of view is considerably enlarged. The area that can be imaged without an additional phase shift is now limited by higher-order, generalized coma-type aberrations, which are generated by the corrector itself—namely, Cαααγ and Cαααγ , the off-axial variants of the third-order axial star aberration and the third-order fourfold astigmatism. These coefficients can be calculated. Here we again restrict our considerations to the combination aberrations of the hexapole. If we insert two second-order aberration rays and the off-axial coma ray in the iteration formula [Eq. (6)], we eventually find

2  u2 = . . . αuα + α2 uαα + αγuαγ + α2 γuααγ = . . . + 2α3 γuαα uαγ + 2α3 γuα uααγ .

Considering again Eq. (54) we can write HP Cαααγ HP Cαααγ







  6ηuα Im A2 uαγ 3 dz − i 6ηuα Re A2 uαγ 3 dz       = 6ηu2α Im uααγ 3 dz − i 6ηu2α Re uααγ 3 dz.

=



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Interestingly, only the two-stage semi-aplanat with 3 ≡ i 3s suffers from strong real hexapole contributions and the imaginary parts vanish, since it + − + uses the symmetries u− α , A2 , uαγ , and 3 . These real components vanish for all aplanats in Section III.C since they use the symmetries in Table III. However, the aplanats also have these off-axial generalized coma coefficients, but only their imaginary part. Here we note without proof that these imaginary aberrations are caused mainly by combination aberrations of the second-order aberrations with the transfer lenses. However, they can be calculated to be typically one order of magnitude smaller than the strong real coefficients. We use them to determine the size of the enlarged isoplanatic field of view. It is given by the radius of a circle, for which an additional phase shift resulting from the fourth-order off-axial aberrations is still below π/4:

R π4 =

−1

 √    λ 1 · 3 − 8 Cαααγ  · ϑ4 + Cαααγ  · ϑ4 . 8 4

(60)

√ Here, by the factor 3 − 8 ≈ 1/5.8 we take into account that the phase shift of a twofold star aberration can be counterbalanced by a twofold astigmatism A1 , which has the off-axial variant Cαγ . In the same manner, as the phase-shift of Cs (∼ ϑ4 ) can be counterbalanced by a defocus C1 (∼ ϑ2 ) at Lichte focus, here we find for the resulting phase shift of the two coefficients at some off-axial point γ

S(α, γ) =

  1 2π 2π · χ(α, γ) = · Re γα2 · αα · Cαααγ + · Cαγ . λ λ 2 opt

As discussed in Section II.C, the necessary aberration Cαγ = √ 2ϑ2 (2 − 8) Cαααγ can be easily produced in a hexapole corrector, if we slightly tune the transfer lenses to adjust the integral for Cα γ rf. to Eq. (10). For the off-axial four-fold astigmatism no such compensation can be achieved. Additionally, the optimization of the fifth-order sixfold astigmatism A5 by choosing the appropriate length of the hexapole fields can be done as for two-stage STEM corrector (see details in Section II.D.5). Here we note that a (sufficiently) small rest will remain (Table IV). Also the fifth-order spherical aberration eventually can be chosen optimally for phase contrast, since the eigen aberration value of the corrector depends on the setup of the corrector and on the geometry of hexapole fields. The results for the examples in Section III.C.4 are listed in Table IV for illustration.

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TABLE IV Properties of the correctors described in section III.C, assuming the parameters λ = 2 pm and an aperture of α = 40 mrad. Type

   Cαααγ  Cαααγ 

[1]

CETCOR 13 3 hexapoles 3.0 3 + 4 hexapoles 1.3 4 hexapoles 0.81

[1]

Rπ 4

[µm]

16 0.02 (0.01∗ ) 0.3 0.17 3.6 0.09 0.14 0.56

N = 2R π · gmax |A5 | 4

Ctot 5

Ctot c

[1000px]

[mm] [mm] [mm]

0.37∗ 6.6 3.5 22

3.0 6.7 2.3 0.6 + 4.5 2.5 0.15 − 5.0 2.4 0.13 + 4.0 2.7

The objective lens has the typical coefficients Cs = 1.3 mm, Cααγ = 0.66, and Cc = 2 mm. Note that the necessary information limit of gmax = 20/nm in the example would require a cold field emission gun or a monochromator. The values marked with ∗ of the conventional two-stage hexapole corrector include the anisotropic coma of the objective lens.

IV. CONCLUSION Starting any new endeavor is always difficult. Several circumstances must “come together” to create the opportunity to take off. First, someone must be convinced that the time for an idea has come. In the case of the hexapole corrector, the applicants for funding believed strongly in the success of their endeavor—although a first attempt by Crewe and co-workers had failed at Chicago. A second vital requirement is an organization that is willing to provide funding, is sufficiently patient, and will even provide additional funding if the project runs out of time. The hexapole corrector project received such support from the Volkswagen-Stiftung, especially in the person of Herbert Steinhardt. Third, the project needs a physical home. The open-minded environment of the EMBL at Heidelberg was the optimum breeding ground. It attracted people from all over the world and it provided excellent resources, including technical support and workshops for constructing hardware. Thus the “physical instrumentation group” with its group leader Max. Haider had very good starting conditions for the realization of the hexapole corrector. However, even this favorable situation did not ensure success. A complex system such as a “high-resolution electron microscope plus corrector” has a large number of parameters. There is always a good chance to get lost, to optimize the wrong thing, or to do an experiment that provides more questions than answers. From today’s point of view, the crucial point was identifying the most important parameters. Even then we often found ambiguous answers, both reasonable, but mutually contradictory, as listed below: • Compact design ⇔ no thermal drift • Low upper frequency limit ⇔ no remanence

Present and Future Hexapole Aberration Correctors

• • • •

Low sensitivity and chromatic aberration ⇔ no saturation Relaxed mechanical tolerances ⇔ small parasitic aberrations Large alignment ranges ⇔ low dipole noise Small number of coils ⇔ small residual parasitic aberrations

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A deeper understanding of the non-ideal hexapole corrector with manufacturing tolerances and alignment elements attained by computeralgebraic perturbation methods was helpful to trim the tree of possibilities. Of equal importance was the increasing availability of hardware for numerical image processing to characterize the state of alignment in an efficient and reliable manner and to generate the required feedback for the corrector control. After several failures and disappointments, the group at the EMBL finally succeeded as reviewed in section I. The consolidation phase of the hexapole corrector in the years after 1997 at CEOS led to robust and reproducible Cs correctors in different flavors. Again theoretical methods were used for optimization, simplification, and adaption. A completely new mechanical design and new construction methods were developed and we experienced the difference between academic and industrial research. In the following years, the hexapole corrector became a usable tool also for nonexpert users. Reliable recipes for alignment “from scratch” have been developed and refined. Today, these recipes allow trained service engineers to perform the factory alignment within a few days after a new corrector has been assembled and integrated into the microscope column. Over the years, the theoretical understanding of the hexapole corrector constantly grew. We developed efficient and compact algebraic models for both the ideal and the non-ideal perturbed corrector. For the former, many relations among the geometry, the excitation of the elements, and the intrinsic aberrations were derived. These rules of thumb (discussed in section II) provide insight into how the corrector works. They allow fairly good estimations of what happens if one or more parameters of the corrected microscope are changed. The most surprising results of this further theoretical work are the following two: First, the strong fifth-order sixfold astigmatism of the hexapole corrector, which was considered to be its handicap and ultimate limit, can almost be eliminated. The hexapole corrector, therefore, can be used also for very large apertures of ≥50 mrad, provided the electron source is sufficiently monochromatic. These large apertures were infeasible for the EMBL corrector, and an advanced hexapole corrector based on a new design of the principal correction elements has been developed for the TEAM project. Second, it is very surprising that rather small modifications of the correction concepts additionally allow for the correction of the anisotropic coma of the objective lens. The rotation of the hexapoles with respect to

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each other enables a true aplanat with no linear off-axial aberrations in third order. The resulting aplanatic system maintains all other favorable properties of the hexapole semi-aplanat. The most flexible solution for an aplanat is that proposed in Section III.C.2. Although it consists of seven hexapoles, probably only three of them require the support of a ferromagnetic circuit. For the weak pairs near the intermediate images planes, wire coils should be sufficient. The correction of the spherical aberration and the anisotropic coma can be adjusted independently and subsequently. The strong hexapoles require six poles only and have a fixed orientation. The most promising solution, however, seems to be the four hexapole solution of Section III.C.3. It immediately enlarges the isoplanatic field of view by a factor of more than 20, compared to the field of view, which is restricted by the anisotropic coma of the objective lens alone. The minimization of A5 (see Section II.D.5) requires the operation at a given hexapole excitation. Therefore, the required rotation of the inner two hexapole elements might not matter too much since it can be fixed by design. However, this version of an aplanat has two more transfer lenses and therefore is longer than the other solutions. It seems very likely that an aplanatic system will be the next step in the future development of the hexapole corrector for applications that demand a large isoplanatic field of view.

ACKNOWLEDGMENTS The funding of the Cs -correction project by the Volkswagen-Stiftung, additional financial and technical support by Philips, Eindhoven, NL, and the indirect support of the development by the DFG SATEM-project for the Research Center Jülich are gratefully acknowledged. Especially for the important Cs -correction project, the authors would like to thank the former reviewers of this project who took the risk and supported the idea by recommending this project to the Volkswagen-Stiftung. At the Volkswagen-Stiftung we would like to emphasize the support in the background by the former officer of the Volkswagen-Stiftung, Dr. H. Steinhardt. At the EMBL we would like to thank the former director Prof. L. Philipson, who opened the opportunity to perform the Cs correction project at the EMBL. We also would like to thank the former members of the EMBL who contributed to the project, including H. Wittman (drawing office), P. Raynor (electronics), Dr. E. Schwan (setup of instrument), D. Mills (maintenance), Dr. R. Wepf (samples), and for the critical and indispensable discussions Dr. J. Zach, and at the CEOS company we would like to thank K. Hessenauer for the fast design and construction of the new adapter lens. The provision of samples and the continuous support of the project by B. Kabius (former FZ Jülich and now at Argonne National Laboratory) is gratefully acknowledged. We also would like to thank Dr. G. Benner, Dr. C. Jia, Dr. B. Kabius, Dr. H. Sawada and Prof. Dr. U. Kaiser for the provision of images. The development of the advanced hexapole corrector for STEM was conducted as part of the TEAM project funded by the U.S. Department of Energy, Office of Science. The support for high-resolution STEM imaging by Dr. Rolf Ernie was indispensable and is gratefully acknowledged.

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The authors are very grateful to their colleagues at CEOS company for their contributions to research and development of the hexapole correctors and to the user community and the manufacturers of the many Cs -corrected instruments installed worldwide who stimulate and support the improvement of the existing and the development of new corrector systems. Finally, we would like to acknowledge our esteemed teacher, Prof. Dr. Harald Rose, who laid the basis for all our work and who constantly supports us with his helpful advice.

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Kimoto, K., Nakamura, K., Aizawa, S., Isakozawa, S., Matsui, Y. (2007). Development of dedicated STEM with high stability. J. Electron Microsc. 56 (1), 17–20. Kisielowski C., Freitag B., Bischoff M., van Lin H., Lazar S., Knippels G., Tiemeijer P., van der Stam M., von Harrach S., Stekelenburg M., Haider M., Uhlemann S., Müller H., Hartel P., Kabius B., Miller D., Petrov I., Olson E. A., Donchev T., Kenik E. A., Lupini A. R., Bentley J., Pennycook S. J., Anderson I. M., Minor A. M., Schmid A. K., Duden T., Radmilovic V., Ramasse Q. M., Watanabe M., Erni R., Stach E. A., Denes P., and Dahmen U. (2008). Detection of single atoms and buried defects in three dimensions by aberration-corrected electron microscope with 0.5 Åinformation limit. Microsc. Microanal. 14, 454–462. Knoll, M., and Ruska, E. (1932). Das Elektronenmikroskop. Z. Physik 78, 318–339. Koops, H., Kuck, G., and Scherzer, O. (1977). Erprobung eines elektronenoptischen Achromators. Optik 48, 225–236. Müller, H., Uhlemann, S., Hartel, P., and Haider, M. (2005). Optical design, simulation and alignment of present-day and future aberration correctors for the TEM. Proceedings of Microscopy Conference, Davos, Switzerland. Müller, H., Uhlemann, S., Hartel, P., and Haider, M. (2006a). Advancing the hexapole Cs -corrector for the the STEM. Microsc. Microanal. 12, 442–455. Müller, H., Uhlemann, S., Hartel P., and Haider M. (2006b). Aberration corrected optics: from an idea to a device. In Proceedings of 7th International Conference on Charged Particle Optics (Munro, E., Rouse, J., Eds.) Physics Procedia 1(1), 167–178. Phillipp, F., Höschen, R., Osaki, M., Möbus, G., and Rühle, M. (1994). New high-voltage atomic resolution microscope approaching 0.1 nm point resolution installed in Stuttgart. Ultramicroscopy 56, 1–10. Rose, H. (1971a). Abbildungseigenschaften sphärisch korrigierter elektrononoptischer Achromate. Optik 33, 1–24. Rose, H. (1971b). Elektronenoptische Aplanate. Optik 34, 285–311. Rose, H. (1981). Correction of aperture aberrations in magnetic systems with threefold symmetry. Nucl. Instr. Meth. 187, 187–199. Rose, H. (1989). Private communication. Rose, H. (1990). Outline of a spherically corrected semi-aplanatic medium-voltage transmission electron microscope. Optik 85, 19–24. Rose, H. (2002). Advances in electron optics. In “High-Resolution Imaging and Spectroscopy of Materials” (F. Ernst and M. Rühle, Eds.), pp. 189–269. Springer Verlag, Berlin. Rose, H. (2004). Outline of an ultracorrector compensating for all primary chromatic and geometrical aberrations of charged-particle lenses. Nucl. Instr. Meth. A 519, 12–27. Rose, H. (2005). Private communication. Rose, H., Haider, M., and Urban, K. (1998). Elektronenmikroskopie mit atomarer Auflösung. Phys. Bl. 54 (5), 411–416. Sawada, H. (2008). Private communication. Sawada, H., Tomita, T., Naruse, M., Honda, T., Hambridge, P., Hartel, P., Haider, M., Hetherington, C., Doole, R., Kirkland, A., Hutchison, J., Titchmarsh, J., and Cockayne, D. (2005). Experimental evaluation of a spherical aberration-corrected TEM and STEM. J. Electron Microsc. 54, 119–121. Scherzer, O. (1936). Über einige Fehler von Elektronenlinsen. Z. Phys. 101, 593–603. Scherzer, O. (1947). Sphärische und chromatische Korrektur von Elektronenlinsen. Optik 2, 114–132. Schlossmacher P., Matjevic M., Thesen A., and Benner G. (2005). Breaking through the Barrier, Imaging & Microscopy 2/2005, 50–52. Shao, Z. (1988). On the fifth-order aberration in a sextupole corrected probe forming system. Rev. Sci. Instrum. 59, 2429–2437.

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Tiemeijer, P. C. (1999). Measurement of Coulomb interactions in an electron beam monochromator. Ultramicroscopy 78, 53–62. Uhlemann, S., and Haider, M. (1998). Residual wave aberrations in the first spherical aberration corrected transmission electron microscope. Ultramicroscopy 72, 109–119. Uhlemann, S., and Haider, M. (2002). Experimental set-up of a fully electrostatic monochromator for a 200 kV TEM. In Proceedings 15th International Congress on Electron Microscopy, vol. I. (J. Engelbrecht, T. Sewell, M. Witcomb, R. Cross, and P. Richards, eds.), pp. 327–328. Durban: Microscopy Society of Southern Africa. Van der Stam, M., Stekelenburg, M., Freitag, B., Hubert, D., and Ringnalda, J. (2005). A new aberration-corrected transmission electron microscope for a new era. Microscopy and Analysis 19 (4), 9–11. Walther, T., and Stegmann, H. (2006). Preliminary results from the first monochromated and aberration corrected 200 kV field-emission scanning transmission electron microscope. Microsc. Microanal. 12, 498–505. Zach, J. (1989). Design of a high-resolution low-voltage scanning electron microscope. Optik 83, 30–40. Zach, J., and Haider, M. (1995). Correction of spherical and chromatic aberration in a low voltage SEM. Optik 98, 112–118. Zemlin, F., Weiss, K., Schiske, P., Kunath, W., and Herrmann K.-H. (1978). Coma-free alignment of high-resolution electron microscopes with the aid of optical diffractograms. Ultramicroscopy 3, 49–60.

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CHAPTER

3 Advances in Aberration-Corrected Scanning Transmission Electron Microscopy and Electron Energy-Loss Spectroscopy Ondrej L. Krivanek, Niklas Dellby, Robert J. Keyse, Matthew F. Murfitt, Christopher S. Own, and Zoltan S. Szilagyi*

Contents

I Introduction 121 II Aberration Correction by Non-Round Lenses 124 III Performance of Aberration-Corrected Instruments 129 A Probe Size and Probe Current 129 B Complications Resulting From Aberration Correction 138 IV New Applications 140 A Atomic-Resolution Imaging and Analysis at Low Primary Energies 140 B Atomic-Resolution Spectroscopy and Elemental/Chemical Mapping 142 C Single Atom Spectroscopy and Mapping 147 V Conclusions 154 Acknowledgments 155 References 155

I. INTRODUCTION The properties of electron lenses depend on the distributions of magnetic or electric fields in the space between the lens pole pieces. The fields are subject to the constraints of the Laplace equation and cannot be shaped arbitrarily. This results in conventional round lenses that have unavoidable * Nion Co., Kirkland, WA 98033, USA Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01003-3. Copyright © 2008 Elsevier Inc. All rights reserved.

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spherical and chromatic aberrations, whose coefficients are of the same order of magnitude as the focal length of the lenses. This much—plus the fact that abandoning cylindrical symmetry, putting charge on axis, using electron mirrors, or working with time-varying fields can lead to optical systems able to correct the aberrations—has been known for some 60 years. Much of the progress was due to the work of Scherzer (1936, 1947); see also Zworykin et al. (1945) for a description of early work in aberration correction. Aberration-corrected instruments built from 1947 to 1990 typically verified the correction principle, but they failed to improve on the resolution of the best uncorrected instruments. This was the case for correctors built by Seeliger (1953) and Möllenstedt (1956), which followed Scherzer’s original designs; correctors that used quadrupoles and octupoles as proposed by Archard (1955) and constructed by Deltrap (1964), Hardy (1967), and the Darmstadt group (Koops, 1978); and the first proof-of-principle sextupolebased corrector of Chen and Mu (1990) that built on the proposals of Beck (1979), Crewe and Kopf (1980), and Rose (1981). In retrospect, it is clear that several major difficulties impeded improved resolution even after the correction principles were verified experimentally. First, the corrected systems were always more complicated than uncorrected ones and used many more power supplies. The precision needed for these supplies was high, and practical electronics was not able to achieve it until the 1970s. Second, not enough attention was paid initially to parasitic aberrations, which arise due to imperfect materials, limited precision of machining and assembly, and imperfect alignment. The result was that the parasitic aberrations often limited the attainable resolution more severely than the principal aberrations. Third, even when it was realized that parasitic aberrations needed to be characterized and fixed, diagnostic procedures able to quantify them easily and preferably automatically (without direct human intervention) took time to develop. Without such procedures, it was beyond most human operators to set up the instruments as needed. The situation turned more favorable in the 1990s, largely as a by-product of three developments: (1) the introduction of charge-coupled device (CCD) cameras in electron image recording (Mochel and Mochel, 1986), which made high-quality image data available for immediate computer analysis; (2) the increasing power of personal computers, which allowed the data to be analyzed in real time; and (3) the development of efficient aberration-diagnosing algorithms (e.g., Krivanek et al., 1992, 1993, 1994, 1997), which allowed the aberrations to be determined rapidly and accurately. These advances made possible the development of the first successful correctors for the three main types of electron microscopes (EMs) in the space of just 3 years: scanning electron microscopes (SEMs) (Zach and

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Haider, 1995a, 1995b); transmission electron microscopes (TEMs) (Haider et al., 1998a, 1998b); and scanning transmission electron microscopes (STEMs) (Krivanek et al., 1997; Krivanek, Dellby and Lupini, 1999). The new instruments ushered in a new, “aberration-corrected” era of electron optics. The era is characterized by rapid advances in both the attainable spatial resolution and the analytical sensitivity. Starting from about five aberration-corrected EMs at the turn of the millennium, there are now about 100 such instruments in the world, and their numbers are growing rapidly. Several reviews have been published that describe the principles and history of aberration correction (e.g., Rose, 2003; Crewe, 2004; Hawkes, 2004, 2007, 2008; Krivanek, Dellby and Murfitt, 2008) and the abilities of the corrected instruments (e.g., Lentzen, 2006; Nellist, 2005; Smith, 2008; Varela et al., 2005, and this volume). As key examples of the new capabilities, aberration correction has produced the first directly interpretable sub-Ångstrom resolution EM images (Batson, Dellby and Krivanek, 2002), as well as deep sub-Ångstrom resolution images (Nellist et al., 2004). Many new applications using the improved resolution have followed since, as documented in this volume. The correction has also improved the attainable contrast levels and thus the visibility of light elements. This has allowed light atoms such as oxygen to be imaged in a CTEM (Jia, Lentzen, and Urban, 2003) and in a STEM using both bright-field (BF) and high-angle annular dark-field (HAADF) modes (Chisholm et al., 2004). Correction has also led to greatly increased currents in atom-sized probes. This has resulted in the acquisition of electron energy-loss spectra from single atoms even in an instrument with non-optimized collection optics (Varela et al., 2004), and the acquisition of atomic-resolution chemical maps in less than 1 minute in a more recent instrument (Muller et al., 2008). The increased probe current and improved resolution have similarly lowered the detection limits of energy-dispersive X-ray spectroscopy (EDXS) close to a single atom (Watanabe et al., 2006). Aberration correction has also initiated a drive toward new EM columns whose stability and flexibility are improved to be in line with the increased performance made possible by the correction (van der Stam et al., 2005; Sawada et al., 2007; Krivanek et al., 2008a). One of the columns was designed “from the ground up” and incorporates major innovations, such as a sample stage that is compensated for temperature variations and is more vibration- and drift-free, and better suited for computer control than typical side-entry stages (Own et al., 2006; Krivanek et al., 2008b). This chapter reviews the principles used by aberration-corrected instruments and the performance they should be able to attain. We provide application examples from areas we consider especially promising, and discuss what may be next for aberration-corrected STEM.

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II. ABERRATION CORRECTION BY NON-ROUND LENSES The account of aberration correction optics presented here is a brief one that concentrates on matters of practical interest to scanning transmission electron microscopy. For more thorough reviews of the optics behind aberration correction, see Krivanek, Dellby, and Murfitt (2008) and Hawkes (2008), plus Hawkes (2007) and Hawkes and Kasper (1996) for additional information. There are many aberrations to correct. Figure 1 shows a graphic representation of the distortions of the electron wavefront that are produced by axial geometric aberrations of up to fifth order, together with the symbols denoting the aberrations. The n index in the Cn,m notation system refers to the order of the aberration: the wavefront distortion due to an aberration of nth order increases as the distance of an electron ray from the optic axis in the aperture plane to the (n + 1)th power, and it displaces the corresponding ray in the image plane as the nth power of the distance. The m index refers to the angular multiplicity: the wavefront distortion due to an aberration of multiplicity m goes through m maxima (and m minima) when the coordinate system is rotated through 360 degrees. The a and b indices account for the fact that azimuthally varying aberrations (all aberrations with m > 0) have two orthogonal components, rotated by π/2m with respect to each other. The magnitude of each Cn,m,a or Cn,m,b aberration stands for the strength of the corresponding aberration coefficient. Quantitatively, the wavefront distortions are described by the aberration function χ(θ, φ), which is defined as the physical distance between the actual wavefront converging on the sample, and an ideal spherical wavefront as

χ (θ, φ) =



n

m

{Cn,m,a θ n+1 cos (mφ) + Cn,m,b θ n+1 sin (mφ)}/ (n + 1), (1)

where the sum over n is taken from 0 to the highest order of aberrations of interest, and the sum over m is taken from 0 (or 1) to n + 1 for each order n, subject to the additional constraint that m + n is odd. The angles θ and φ are the polar angular coordinates of a ray converging on the sample. The standard location for defining the aberration function is the front-focal plane of the final probe-forming lens of the STEM (i.e., its objective lens). This plane lies a focal length f in front of the sample, and the transverse positions (x, y) in it are defined, using the paraxial approximation, as x = f θ cos(φ), y = f θ sin(φ). The aberration function also describes the phase shift of each partial ray relative to the central ray traveling along the optic axis, which is given by η = 2πχ(θ, φ)/λ. See Krivanek, Dellby, and Murfitt (2008) for a more detailed explanation of the notation and of the equation defining the aberration function, and

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C0,1,b

C0,1,a

C1,0 C1,2,b

C1,2,a

C2,1,a

C2,3,a

C2,3,b

C3,0 C3,2,a

C4,1,a

C3,4,a

C4,3,a

C3,4,b

C4,5,a

C5,0 C5,2,a

C5,4,a

C5,6,a

FIGURE 1 Contributions to the aberration function due to aberrations from the zeroth to the fifth order.

Hawkes (2008) for a table comparing the notation used here with other notation systems currently in use. Note also that notation systems that use different letters and different pre-factors for different aberrations do not provide a simple expression for χ(θ, φ) for aberrations up to an arbitrarily high order (e.g., compare Eq. (1) above with equation (2) of Haider et al. (2008)) and that expressing the rules linking different aberrations that are discussed below is much more cumbersome in the alternate systems.

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In the first electron microscopes, defocus C1,0 was the only parameter that could be adjusted freely. The main resolution limit came from the next important aberration: simple twofold astigmatism C1,2 . This was nulled by the addition of an adjustable stigmator (Bertein, 1947; Hillier and Ramberg, 1947; see Haguenau et al., 2003). Axial coma C2,1 , which arises when systems with spherical aberration are misaligned, emerged as a concern in mainstream electron microscopy in the 1970s (Krivanek, 1978; Zemlin et al., 1978). Threefold astigmatism C2,3 became a concern in the 1990s after procedures for determining it rapidly were developed (Ishizuka, 1994; Krivanek, 1994; Krivanek and Stadelmann, 1995). With aberration correction, all the terms up to C3,4 became a concern for the first practical correctors. The latest correctors now either directly control, or at least keep acceptably small, all aberrations up to C5,6 (Krivanek et al., 2003; Dellby, Krivanek, and Murfitt, 2006; Müller et al., 2006). This illustrates how accelerated the progress has become in the past 10 years. Figure 1 does not show field-dependent (off-axial) aberrations, because these are usually not limiting in the STEM. When they need to be considered in depth, such as for conventional broad-beam CTEM imaging, they can be readily described as the variation of the axial aberrations across the field of view (Dellby, Krivanek, and Murfitt, 2006; Krivanek, Dellby, and Murfitt, 2008). Despite the large number of aberrations that need to be dealt with, designers of aberration correctors for TEMs have but a few types of optical elements in their “tool kit”: round lenses and various multipoles. The multipoles include dipoles and weak quadrupoles, which were already used in non-aberration-corrected optics, and further strong quadrupoles, sextupoles, and octupoles, and probably higher-order multipoles in future. Round lenses typically only increase the aberrations of the total system. This leaves the task of correcting the many possible aberrations depicted in Figure 1 up to the multipoles. When multipoles are working in isolation, they can change only a rather limited number of aberrations. A quadrupole, sextupole, and an octupole acting on a non-aberrated round beam only produce twofold astigmatism C1,2 , threefold astigmatism C2,3, and fourfold astigmatism C3,4 , respectively, as their principal effect. These are only indirectly related to spherical aberration C3,0 , which limits the resolution in most uncorrected instruments, and to aberrations such as axial coma of second and fourth order (C2,1 and C4,1, respectively) and twofold astigmatism of spherical aberration (C3,2 ), which are likely to arise as parasitic aberrations. Precise control of aberrations other than those readily produced by the multipoles acting on a round beam is nevertheless possible because of two fundamental effects produced by optical elements separated along the direction of the electron travel: combination and misalignment aberrations.

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Combination aberrations arise when a beam of electrons traverses an optical element producing aberration Cn,m and then enters another optical element producing aberration Cn′ ,m′ . When the aberrations are small (i.e., the originally round beam is barely distorted), this combination of events produces aberrations Cn+n′ −1,m+m′ and Cn+n′ −1,|m−m′ | , except when both the aberrations are the highest multiplicity ones allowed for a given order (m = n + 1, m′ = n′ + 1), in which case only the lower-multiplicity |m − m′ | aberration is produced. As a practical example, combining the effect of two thin sextupoles, each of which produces threefold astigmatism C2,3 , produces adjustable spherical aberration C3,0 (n = 2 + 2 − 1, m = 3 − 3) as a combination aberration. The same applies to an extended sextupole, which can be viewed as a succession of thin sextupoles and thus also produces C3,0 . This is the principle behind sextupole correctors, which use the resultant negative spherical aberration to counteract the positive C3,0 (= Cs ) of the rest of the optical system. (Note that we prefer to adhere to Latin nomenclature and call six-pole optical elements sextupoles. If we were to follow Greek nomenclature instead by calling six-pole elements hexapoles, for consistency we should then have to call four-pole elements “tetrapoles.”) As another practical example, combining C3,0 produced by a corrector with C3,0 produced by the objective lens of a microscope produces an adjustable fifth-order spherical aberration C5,0 as a combination aberration. The size of the aberration can be adjusted by moving the two elements relative to each other along the optic axis, or much more practically, by coupling them electron-optically via a set of lenses such that an image of the first element is produced in the vicinity of the second element, and then adjusting the exact position of the image. This principle, proposed by Shao (1988), is used to correct or minimize C5,0 in all present-day instruments that correct aberrations higher than third-order. It is interesting to note that combining octupoles has the same effect: C3,4 combined with C3,4 also gives adjustable C5,0 . Correction of C5,0 by octupoles was indeed suggested by Rose (1981) and Shao, Beck and Crewe (1988). But because the same effect comes essentially for free in any C3 corrector properly coupled by first-order optics to an objective lens, using octupoles for C5,0 correction has not been explored experimentally. As two more practical examples, combining C3,4 produced in one stage of a correction apparatus with C2,3 produced in another stage of the apparatus gives adjustable C4,1 , and this can be used to adjust fourth-order coma. Combining C3,0 with C2,3 produces C4,3 , and it can be used to adjust fourthorder threefold astigmatism. In similar ways, controls for all important parasitic aberrations can be devised and implemented. As a final example, combining C2,3 produced by the strong sextupoles in a sextupole corrector with C3,0 that is itself produced as a combination

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aberration in an extended sextupole (and is also present in the round coupling lenses between the sextupoles) gives C4,3 . Combining this mostly double-combination aberration with the strong C2,3 of the sextupoles once more produces C5,0 and C5,6 as triple-combination aberrations. C4,3 is canceled along with C2,3 by the second sextupole in a symmetrical sextupole corrector, whereas C5,0 and C5,6 are doubled. C5,0 can be canceled by misprojecting the second sextupole into the objective lens, leaving C5,6 as the most important intrinsic aberration of this type of corrector (Shao, 1988; Rose, 1990a; Müller et al., 2006). When the beam distortion produced by the first optical element is no longer slight, several combination aberrations of the order n + n′ − 1 are produced. This principle is used by quadrupole-octupole correctors, in which a quadrupole at the corrector’s entrance typically produces either a line crossover or an elliptical beam in a subsequent octupole. n + n′ − 1 = 3 for this case, and an octupole with an elliptical beam in it gives adjustable C3,0 , C3,2, and C3,4 . The exact proportions of the aberrations depend on the aspect ratio of the beam ellipse. For a line crossover, the ratios for the three aberrations are 1, 4/3, and 1/3, respectively (Krivanek, Dellby, and Brown, 1996)—that is, their magnitudes are comparable. A line crossover rotated by 90 degrees incident on a similar octupole gives coefficients of the same magnitude, with the sign of C3,2 is reversed. A pair of octupoles with the same aspect ratio ellipses, but rotated by 90 degrees to each other, therefore gives adjustable C3,0 and C3,4 , whose ratio is determined by the aspect ratio of the ellipse. Adding a third octupole to this system, at a location in which the beam is round, allows the fourfold astigmatism C3,4 to be nulled, resulting in a corrector that provides adjustable spherical aberration C3,0 . This is the correction principle used by quadrupole-octupole correctors, including quadrupole-octupole correctors that also correct fifth-order aberrations (Krivanek et al., 2003; Rose, 2004; Dellby, Krivanek, and Murfitt, 2006). Misalignment aberrations arise when the electron beam enters an optical element miscentered with respect to its optic axis. They can be viewed more generally as combination aberrations caused by a dipole (which gives rise to a simple deflection C0,1 ) and a more complicated element, but because they are rather ubiquitous, we prefer to put them in a class of their own. For a small miscentering on an element producing Cn,m aberration, the principal misalignment aberrations are Cn−1,|m−1| , plus, when m < n, Cn−1, m+1 . A miscentered quadrupole acting on a round beam produces an additional deflection (C0,1 ), a miscentered sextupole produces twofold astigmatism C1,2 , a miscentered octupole produces threefold astigmatism C2,3 , and a miscentered round lens with spherical aberration C3,0 produces axial coma C2,1 . These effects are used in practice: centering on a sextupole with a round beam in it is used to null twofold astigmatism C1,2 in sextupole correctors, centering on an octupole is used to null threefold astigmatism

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C2,3 in quadrupole-octupole correctors, and centering on a C3,0 –producing element such as a round lens is used to null axial coma. Centering on the strong sextupoles used in sextupole–round lens correctors must be particularly precise and stable. In a typical corrector of this type, miscentering the beam on a sextupole by just 3 nm leads to twofold astigmatism that is too large for 50 pm resolution high-angle dark-field imaging and 80 pm resolution BF imaging (Rose, 1981; Krivanek, Dellby, and Murfitt, 2008). This is a direct consequence of the fact that sextupoles excite threefold astigmatism C2,3 directly and strongly, whereas C3,0 arises only as a second-order effect in them. The threefold astigmatism produced by the first sextupole in a sextupole corrector is rather strong, typically of the order of C2,3 = 0.5 mm (referred to the objective lens of the microscope). Removing it with sufficient precision thus requires a very accurately aligned second sextupole. No such strong lower-order aberration is first excited and then precisely removed in quadrupole-octupole correctors, in which the astigmatic focusing of the quadrupoles is simply first-order optics arranged differently in two orthogonal directions. It produces alignment tolerances that are no more serious than the tolerances for round lenses in other parts of the system. Quadrupole-octupole correctors of geometric aberrations are therefore less sensitive to misalignment and hence potentially more stable.

III. PERFORMANCE OF ABERRATION-CORRECTED INSTRUMENTS A. Probe Size and Probe Current In an aberration-corrected STEM, the smallest attainable probe size depends on a number of factors: the remaining (uncorrected) principal aberrations, the strength of parasitic aberrations (often determined by the precision of the tuning), and the dominant instabilities. The current available in a probe of a given size depends on these factors too, plus the brightness of the electron source. The attainable resolution depends on the probe size, the probe current, and the nature of the sample. Given this degree of complication, it is often difficult to provide an exact expression for what the resolution and/or the probe current will be. Nevertheless, simple formulas for the main factors governing the probe size and current provide useful insights into the limits of aberration-corrected performance. To work out the resolution limited by geometric aberrations, we adopt the criterion that the optimum convergence semi-angle α of the STEM probe is one for which the wavefront converging on the sample does not deviate by more than ±λ/8 from the ideal spherical wavefront. This criterion is slightly stricter than necessary under ideal conditions, but because there

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are many aberrations to consider and various small effects will not be taken into account by the present discussion, adopting a stricter criterion is likely to lead to a better overall agreement with experiment. Before aberration correction, the aberration function χ(θ) was dominated by spherical aberration Cs and defocus z:

χ (θ) = Cs θ 4 /4 + zθ 2 /2,

(2)

where we are using the conventional symbols for spherical aberration Cs and defocus z rather than the equivalent C3,0 and C1,0 used in the preceding section. The aberration function is rotationally invariant in this simple case. The largest usable probe half-angle αgeom for which the aberration function χ(θ) stays within a band λ/4 wide is obtained when

z = − (Cs λ)1/2 , αgeom =

√

2 (λ/Cs )1/4 .

(3)

The maximum angle admitted into the probe then determines the geometric probe size (full width at half-maximum, FWHM), as described by the familiar expressions for diffraction-limited resolution:

dgeom = 0.61λ/αgeom 1/4

= 0.43Cs λ3/4 .

(4a) (4b)

Equation (4b) is the same as the expression for BF resolution limited by spherical aberration derived by Scherzer (1949), except that the numerical factor is 1.6× smaller. Note that if we do not compensate for the spherical aberration by adjusting the defocus to optimally oppose it, the aberration function will reach the √ limiting value already at α = (λ/Cs )1/4 , and the probe size will then be 2× worse. Choosing the optimal defocus therefore √ has a similar effect to leaving the defocus at zero and reducing Cs by ( 2)4 = 4×. Cs was typically between 0.5 and 1.0 mm for high-performance uncorrected microscopes, which represents a variation of only 20% when it appears in the fourth root. This meant that the attainable probe size was determined much more by the electron wavelength λ and thus the primary beam energy than by the exact size of the spherical aberration coefficient. Going to a higher energy was therefore the only practical way to attain atomic resolution in uncorrected instruments. With aberration correction, some things have changed, and some have stayed the same. Cs can now be set to an arbitrarily low value, and other aberrations determine the resolution. There is a large number of aberrations to consider, as shown by Figure 1. The effects of many of the aberrations can

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be reduced by adjusting lower-order aberrations to optimum values, similar to the way defocus is used to reduce the effect of spherical aberration. The general rule is that a higher-order aberration can be compensated by lower-order aberrations of the same multiplicity. As an example, twofold astigmatism of seventh-order C7,2 can be compensated by C5,2 , C3,2 , and regular astigmatism C1,2 . Not surprisingly, such a large number of lowerorder terms allows one to compensate C7,2 rather accurately, reducing its effect by up to 110× (Krivanek, Dellby, and Murfitt, 2008). The reduction factor is smaller when fewer lower-order aberrations of the same multiplicity are available for the compensation, and no compensation is possible for maximum multiplicity aberrations such as C1,2 , C2,3 , etc., for which no lower-order aberrations of the same multiplicity exist. The compensation is best described by introducing weighing factors Fn,m that show how much the effect of a properly compensated aberration of type Cn,m on the aberration function is reduced relative to aberrations that cannot be compensated. The weighing factors also take into account that azimuthally invariant aberrations (m = 0) only change the aberration function in a unipolar way, and the range of wavefront deviations they cause is therefore only half of those due to azimuthally varying aberrations (with m > 0). The weighing factors are listed in Table I for aberrations up to seventh order. Note that the different aberrations used to balance each other are typically controlled by separate power supplies, and each one of the aberrations needs to be kept stable enough so that its variation does not distort the wavefront unacceptably. In other words, properly adjusted compensation can be used to accommodate larger aberration coefficients than would be admissible without it, but it has no effect on the stability requirements for each aberration. Fortunately, the stability requirements for higher-order TABLE I

Weighing Factors Fn,m for Different Aberration Coefficients

Coefficient

C0,1 C1,0 C1,2 C2,1 C2,3 C3,0 C3,2 C3,4

Weighing factor Fn,m

Coefficient

Weighing factor Fn,m

Coefficient

Weighing factor Fn,m

1 0.5 1 0.244 1 0.125 0.167 1

C4,1 C4.3 C4,5 C5,0 C5,2 C5,4 C5,6

0.0588 0.132 1 0.0313 0.04 0.103 1

C6,1 C6,3 C6,5 C6,7 C7,0 C7,2 C7,4 C7,6 C7,8

0.0172 0.0263 0.0909 1 0.0078 0.0091 0.0189 0.0769 1

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aberrations, some of which can be compensated very successfully, are typically much looser than the requirements for lower-order ones. The requirements therefore almost never present a practical problem even without the help of the compensation. The maximum probe half-angle for which the range of wavefront distortions remains smaller than λ/4 in the presence of optimally compensated aberrations of a particular order n is given by

αgeom



 1/(n+1) = (n + 1)λ 8 Fn,m Cn,m , 

(5)

m

which leads to a probe size of



dgeom = 0.61 (8/(n + 1))



Fn,m Cn,m

m

1/(n+1)

λn/(n+1) .

(6)

For a corrector limited by C5,4 (such as the Nion second-generation one, Dellby et al., 2001), F5,4 = 0.103, and the probe size limit due to geometric aberrations becomes 1/6

dgeom = 0.44 C5,4 λ5/6 .

(7)

For a C5,6 -limited sextupole corrector, F5,6 = 1, and the geometric aberration limit on the probe size is 1/6

dgeom = 0.64 C5,6 λ5/6 .

(8)

For quadrupole-octupole correctors that have C7,0 to C7,8 of comparable magnitudes as the lowest-order uncorrected principal aberrations (Dellby, Krivanek, and Murfitt, 2006), the resolution is mainly limited by C7,8 as 1/8

dgeom = 0.61 C7,8 λ7/8 .

(9)

All the above expressions have a much stronger dependence on the electron wavelength than on the aberration coefficients. This means that decreasing λ by increasing the primary energy Eo will remain an effective path to higher resolution in all aberration-corrected instruments, with the possible exception of geometrically and chromatically corrected ones, for which an optimum primary energy of ∼200 keV may exist (Haider et al., 2008). The electron wavefront is also affected by chromatic aberration effects. The contributions to the probe intensity for electrons of different energies

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add up incoherently, and chromatic aberration therefore often preserves a sharp central maximum of the probe while increasing the intensity of the probe tails. A more detailed discussion of this effect is given by Krivanek et al. (2008a). The tail increase is largely avoided when the probe semi-angle is limited to

  0.5 , αchrom = 1.2 λ/ Cc δE/Eo

(10)

dchrom = 0.5 (λCc δE/E0 )0.5 .

(11)

where δE is the FWHM of the energy spread, and E0 is the primary energy. This leads to a chromatic limit on the probe size of

The electron wavelength λ is given by the familiar expression

   0.5 λ = h/ 2mo eEo 1 + eEo / 2mo c2 ,    0.5 , λ/pm = 1226/ Eo 1 + 0.9785∗ 10−6 Eo  = 1226/ E∗o ,

(12a) (12b) (12c)

where Eo is measured in electron volts and E∗o is the “relativistically corrected” primary energy. The chromatic resolution limit is therefore 3/2 proportional to 1/Eo at low primary energies and to 1/Eo at high energies. Using a higher primary energy is thus also an effective strategy for decreasing the chromatic resolution limit. Because both the geometric and chromatic aberrations require that the angular range of the wavefront incident on the sample be restricted, we simply set the illumination half-angle αaber to correspond to the optimum value for whichever aberration is more limiting. The expected probe size due to the combined effect of aberrations and the projected source size is then calculated as

0.5  2 2 , dp = daber + dsource

(13)

2 Ip = Bπ2 dsource α2/4,

(14)

where the sum of squares is appropriate because the two resolutiondetermining factors combine incoherently. To determine the projected size of the electron source dsource that is needed for a given beam current, we start by noting that the current in the electron probe is given by

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where B is the brightness of the source, specified as current per unit area per unit solid angle. For a particular desired current Ip , the size of the source projected onto the sample therefore needs to be

or alternately

 0.5 dsource = 2 Ip /B /(πα),

(15a)

 0.5 dsource = 2 Ip /Br Vo∗ /(πα),

(15b)

  0.5 dp = 1 + 1.1 Ip / Bλ2 daber 0.5  = 1 + 7.3 × 1017 Ip /Br daber .

(16a) (16b)

Ip > 1.4 × 10−18 Br .

(17)

where Br is the “reduced” (or “normalized”) brightness Br = B/Vo∗ , and Vo∗ is the relativistically corrected accelerating voltage. In the absence of deleterious effects of instabilities or space charge effects, Br is invariant throughout the illumination system, from the electron source all the way to the sample. Combining Eqs. (13), (15), and (4a) then gives the probe size for a desired current Ip :

These expressions show that the minimum attainable probe size is determined by the aberration characteristics of the optical column, and further that this probe size can only be reached in the limit of zero probe current. At any probe current > 0, the probe size also depends on the actual current value and the gun brightness. Note also that the probe size expressed as above does not explicitly depend on the primary energy, although daber is usually strongly dependent on it. At high current values, the brightness-dependent term in Eq. (16) becomes as important as the optical performance of the microscope in determining the probe size. This occurs at probe currents for which 7.3 × 1017 Ip /Br > 1, that is for

This simple expression shows that the transition from probes limited mainly by aberrations to probes limited by aberrations and brightness together occurs at a probe current that is the same for all instruments with a given reduced source brightness, i.e. that the transition is independent of the quality of the instrument’s optics and of its primary energy. At this point it is useful to note that the experimental measurement of the electron brightness B (or Br ), that is, the determination of the probe current Ip , illumination half-angle α, the size of the projected source dsource , and the

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primary voltage Vo , so that B or Br can be determined using Eq. (14), is not completely straightforward. Determining Ip , α, and Vo is not difficult, but dsource can be easily overestimated when using an insufficiently large solid angle at the source, which makes the measured source size appear larger as a result of diffraction-limited imaging, when aberrations arise in the gun part of the optical system that are not accounted for, and when highfrequency vibrations and other instabilities broaden the projected source. The source size can be underestimated when the sharpest detail in a probe is confused with the overall size of the probe, i.e. when the current that ends up in the probe tail is considered as a part of the central maximum. The generally accepted brightness value for 100 kV cold field emission guns (CFEGs) is B ∼ 1 × 1013 A/(m2 sr), that is, Br ∼ 1 × 108 A/(m2 sr V). However, we typically measure Br ∼ 2 × 108 A/(m2 sr V) for the VG (310) W CFEG. The difference probably arises because we have increased the stability of the gun by improving the stability of its high voltage and deflector coil currents, and because we are careful to align the gun mechanically close to its brightest direction. For Schottky electron guns, the generally accepted values are B ∼ 1 − 5 × 1012 A/(m2 sr) at 100 kV, that is, Br ∼ 1 − 5 × 107 A/(m2 sr V). For example, Müller et al. (2006) give B = 8 × 1012 A /(m2 sr) for a 300-kV Schottky gun, which is equivalent to Br = 2.1 × 107 A/(m2 sr V). Figure 2 shows the probe size as a function of the probe current, calculated according to Eq. (16b) for two values of reduced brightness: Br = 1.5 × 108 A/(m2 sr V) and 3 × 107 A/(m2 sr V), taken here as 6

Probe size/ daber

5

250 Br

4

⫽ 3 ⫻ 107

200

3

150 Br ⫽ 1.5 ⫻ 108

2

100

1.41 daber

1

50

0 0

0.2

0.4 0.6 Probe current/nA

0.8

1

Probe size (for daber ⫽ 50 pm)/pm

300

0

FIGURE 2 STEM probe size dp as a function of the probe current for two sources of different reduced brightnesses. Br = 1.5 × 108 A/(m2 sr V) is representative of CFE sources, Br = 3 × 107 A/(m2 sr V) of Schottky ones. The left-side vertical scale is shown in units of the aberration-limited probe size daber , the right-side one is in pm for daber = 50 pm.

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representative of CFE and Schottky electron sources, respectively. We have plotted the probe size normalized by daber , that is, as a general curve describing the growth of the probe size at nonzero currents. The figure shows that the probe size is determined more by the aberration performance of the column than by the source brightness for probe currents less than ∼40 pA for Schottky systems and less than ∼200 pA for CFEG ones. For probe currents greater than these values, the probe size is determined by the brightness and daber together. The figure demonstrates that producing small probes at nonzero probe currents is a two-part undertaking: the aberration-limited probe size daber needs to be minimized by optimizing the optics design, and the source brightness needs to be maximized by optimizing the electron source. Users of practical instruments will point to the importance of a third part: optimizing the (in)stabilities of the total system so that they become less limiting than its fundamental optical performance. Laborious though it is, this part can be improved even after the system is installed and is therefore less stringent than the other two. In the rest of this chapter, we assume that the stabilities have been pushed to the required level. Table II shows the resolutions and probe currents calculated using the formulas given above for various CFEG STEMs that Nion has worked with. The first three columns show data for VG STEMs before correction. These uncorrected instruments were clearly much more limited by spherical rather than chromatic aberration. The analytical VG HB501 was able to put a current of 0.3 nA into a probe just smaller than 0.5 nm. The “highresolution” VG HB501 improved on this slightly and could reach a probe size of 0.21 nm at a much reduced beam current. The high-resolution VG HB603 improved on both these figures, with a probe size of 0.12 nm at a small beam current, and an ability to deliver 0.3 nA of current into a 0.22 nm probe. With a C3 -only aberration corrector and doubled gun brightness, the theoretical performance of the VG HB501 improved significantly: the smallest probe size became 84 pm at 100 keV, and the probe sizes for a current of 0.3 and 1 nA became 122 and 184 pm, respectively. The corrected VG HB603 improved by a similar margin, to a 51 pm small-current probe, and a 0.3 nA, 90 pm probe. Because of the small energy spread of their CFEG guns and relatively small values of Cc , both these instruments were still limited more by geometric than by chromatic aberrations. The C3 /C5 corrector of the Nion UltraSTEM changed the situation. The geometric aberrations are now corrected so well that the chromatic aberration becomes more limiting. With the UltraSTEM column mounted on top of the VG 100 keV gun, the theoretical performance for small currents is worse than the corrected VG 603 (66 pm probe vs. 51 pm for the 300 keV instrument), but it improves at larger beam currents—the UltraSTEM is

TABLE II

Theoretical Probe-Forming Performance of Various CFEG STEMs VG 501 high resolution

VG 603 high resolution

Corrected VG 501

Corrected VG 603

UltraSTEM 100

UltraSTEM 200

100 3.70 C3,0 3.5 260 3.5 0.35 106 260 8.7 1 × 1013 270 348 478 777

100 3.70 C3,0 1.3 203 1.3 0.35 65 203 11.1 1 × 1013 211 272 373 606

300 1.97 C3,0 1.0 118 1.5 0.35 29 118 10.2 4 × 1013 123 158 218 354

100 3.70 C5,4 60 83 1.4 0.35 67 83 27.3 2 × 1013 84 98 122 184

300 1.97 C5,4 60 49 1.7 0.35 31 49 24.6 4 × 1013 51 65 90 146

100 3.70 C7,8 20 37 1.3 0.35 65 65 34.8 2 × 1013 66 77 96 145

200 2.51 C7,8 20 26 1.5 0.35 41 41 37.7 4 × 1013 41 48 60 90

The resolution limit that determines the aberration-limited performance of each microscope is shown in bold italics.

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Eo (keV) λ(pm) Limiting geometric aberration Aberration value (mm) Geometric resolution limit (pm) Cc (mm) δE (eV) Chromatic resolution limit (pm) Aberration-limited resolution (pm) Optimum aperture semi-angle (mr) Brightness (A/(m2 sr)) Probe size for 10 pA (pm) Probe size for 100 pA (pm) Probe size for 300 pA (pm) Probe size for 1 nA (pm)

VG 501 analysis

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able to pack 0.3 nA beam current into a 96 pm probe, and 1 nA into a 145 pm probe. Mounted on top of a 200 keV CFEG source that we are building and testing, the UltraSTEM is expected to have a small-current probe of 41 pm and deliver 1 nA into a 90 pm probe. With the exception of the Nion 200-keV STEM, which remains to be thoroughly tested, most of the cited numbers are closely borne out in practice. This includes the large-current performance of the UltraSTEM 100, which has proved itself able to put 0.3 nA current into a probe of ∼100 pm, and 0.7 nA into a probe of ∼130 pm (Muller et al., 2008). When estimating probe sizes for currents > 1 nA, the gun aberrations typically need to be taken into account, and this has not been done here. However, a corrector can correct aberrations originating in the gun just as easily as those originating in the objective lens, so the influence of the gun aberrations will be much less than in pre–aberration-correction days. Note also that when a large dsource is being used, the illumination half-angle can be opened up slightly relative to the low-current optimum without a significant additional deterioration in resolution. This strategy is especially useful when the optics is limited by chromatic aberration, since in this case the aberration-limited probe size increases only linearly with the illumination angle. It was, for instance, used by Muller et al. (2008). It would be interesting to estimate the performance of other STEMs using the above formulas, particularly STEMs with Schottky guns. However, because the published range of Schottky brightness and energy spread values is rather broad and we have not measured the values ourselves, it is not possible for us to assess these instruments accurately. Using Müller et al.’s (2006) values (Br = 2.1 × 107 A/(m2 sr V), energy spread δE = 0.7 eV and Cc = 2.4 mm), the above formulas indicate that the corresponding instrument is likely to be limited by chromatic aberration and that it should be able to reach a probe size of just over 50 pm at 300 keV for small beam currents. The probe size, however, will increase rapidly at larger beam currents, to over 0.3 nm at 1 nA. At 100 keV, the large-current (0.3 and 1 nA) probe sizes for such an instrument are likely to be similar to those of the uncorrected VG HB501.

B. Complications Resulting From Aberration Correction Major gains often come with new complications, and this holds for aberration correction, too. One standard complication for aberration-corrrected STEM is that the available dark field signal becomes weaker for higher resolutions. As the angular range of the probe is increased to attain a smaller probe size, a progressively larger fraction of elastically scattered primary electrons remains within the BF cone and is thus not available for DF imaging. Moreover, the low-angle HAADF cutoff needs to increase in proportion to the increase in the angular range of the probe, otherwise the

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HAADF images become progressively more nonlinear (Hartel, Rose, and Dinges, 1996). Detailed simulations of this effect for aberration-corrected probes remain to be performed, but it is clear that either the HAADF signal available from each atom will grow progressively weaker as the STEM resolution is improved (if the low-angle HAADF cutoff is increased), or the HAADF images will become more coherent and nonlinear, and therefore harder to interpret (if the lower angular limit is not changed). These effects will be particularly important for light atoms whose scattering is more forward-peaked. The electrons that reach the DF detector are scattered from the deep and partially screened potential well surrounding the atomic nucleus (e.g., Kirkland, 1998). It is useful to note that the atomic scattering center is much smaller than the atomic size defined by the valence electrons, and that this causes the angular distribution for elastic scattering to extend to large angles. As a result, the HAADF resolution has been able to progress well below the separation of near-neighbor atoms in solids, and beyond the resolution of imaging techniques that probe the distribution of valence electrons, such as scanning tunneling microscopy (STM). Depending on the Z of the atom, the size of the scattering center is of the order of 10–30 pm. The HAADF image is further broadened by the thermal vibrations of the atomic nucleus, which have a root-mean-square magnitude ∼10 pm at room temperature (Loane, Xu, and Silcox 1991). Even so, atomic images less than 50 pm wide should ultimately become available. At probe sizes below 100 pm, however, the apparent size of the atomic scattering center needs to be explicitly taken into account in estimating the HAADF resolution, especially for lighter atoms such as carbon and oxygen. It even makes a significant contribution to the size of the images of heavy atoms such as gold (Batson, 2006). Another readily foreseeable complication is that in a crystal, electron probes with the high convergence needed for sub-angstrom resolution spread onto neighboring atomic columns. This can be modeled by simulating the propagation of the probe through the crystal (Ishizuka, 2001; Dwyer and Etheridge, 2003; Peng, Nellist, and Pennycook, 2004; Allen et al., 2006). A very simple approximation is that spreading onto neighboring columns for highly convergent probes that are not strongly bound to individual atomic columns will occur for sample thicknesses greater than

  tmax ∼ dcol / 2α ∼ dcol dres /λ,

(18)

where dcol is the distance between the neighboring atomic columns and dres the resolution corresponding to the probe half-angle α. For dcol = 0.2 nm and dres = 0.05 nm, Eq. (18) gives tmax ∼ 3 nm for a 100 keV beam and tmax ∼ 4 nm for a 200-keV beam. Focusing the probe so that it is the smallest in the center of the sample rather than at the entrance face can probably

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double the thickness limits to 6 and 8 nm, respectively, but even the larger figures are smaller than the typical thickness of a crystalline sample of 10–30 nm. For an inter-columnar spacing of 0.2 nm in a 20 nm thick crystalline sample, the probe will start spreading onto neighboring columns at HAADF resolutions better than 100 λ. In aberration-corrected instruments the probe size is typically 20–50 λ, which means that probe spreading onto neighboring columns in samples of typical thickness is likely to be the norm in all advanced EMs. A detailed calculation of the probe propagation though the sample will then be needed to interpret both HAADF images and spectroscopic data. The above effect is the “other side of the coin” of improved z-depth resolution, which can be used to assign atoms to different heights in a sample without needing to use a series of tilts (van Benthem et al., 2006; Borisevich et al., 2006). Such capabilities will become significantly enhanced as the resolution grows better, especially at lower primary energies (larger electron wavelengths).

IV. NEW APPLICATIONS The improved performance of aberration-corrected instruments has produced many new applications for STEM imaging and analysis. Numerous examples are described in this volume. As instrument designers and makers, we admire the inventiveness of the users of the new instruments and the elegance of the results. At the same time, we need to look toward the future and concentrate development efforts on areas not yet fully explored. Presently we see three areas that hold exceptional promise. We describe them in the following subsections.

A. Atomic-Resolution Imaging and Analysis at Low Primary Energies Even though the probe-forming capabilities of an aberration-corrected microscope will typically be better when operating at a higher primary energy than at a lower one, the optics improvements are now such that even low-energy operation can provide atomic resolution (Dellby et al., 2008). Low energies have several advantages: increased cross-sections for elastic and inelastic scattering and thus a higher signal from each atom, decreased radiation damage due to the knock-on mechanism, and slightly better localization of inelastic scattering and thus potentially higher resolution in elemental maps [see Eq. (19) below]. Figure 3 shows a HAADF image of Si acquired at 60 keV with the Nion UltraSTEM100. The Si dumbbells that are separated by 0.136 nm are resolved. Figure 4 shows HAADF and STEM BF images of a carbon

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0.5 nm

FIGURE 3 < 110 > Si STEM HAADF image recorded at 60 keV primary energy, Fourier-filtered to remove statistical noise. Nion UltraSTEM, 0.1 nA probe.

(a)

(b)

2 nm

FIGURE 4 Bright-field (a) and HAADF (b) images of a carbon nanotube obtained at 60 keV primary energy. (Sample courtesy Dr. Mathieu Kociak.)

nanotube also acquired at 60 keV. Operating at this primary energy avoids radiation damage by knock-on displacement in materials such as boron nitride nanotubes, in which the knock-on threshold energy is 74 keV (Zobelli et al., 2007) and in carbon nanotubes, for which the theoretical knock-on threshold energy is 86 keV (Smith and Luzzi, 2001), but in which the intense electron beam of a STEM in fact produces weak but observable damage at 80 keV (Kociak, 2007). The resolution at low primary energies is limited much more by chromatic aberrations than by geometric ones. Correction of the chromatic aberration would remove this limit. Unfortunately, simple combined

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correctors of spherical and chromatic aberration (Zach and Haider, 1995a, 1995b) produce fifth-order aberrations of the order of several centimeters. Correctors that combine third-order, fifth-order, and chromatic correction (Rose, 2004; Haider et al., 2008) are rather complicated even by aberration correction standards, and it remains to be seen whether they can be made to operate on a routine basis. Decreasing the energy spread of the electron beam to ∼0.1 eV by an electron monochromator (Tiemeijer, 1999) can also improve the spatial resolution, especially if the loss of brightness is minimized by using a zero-final-dispersion design (Rose, 1990b; Kahl and Rose, 2000; Martinez and Tsuno, 2004). The monochromation approach also has the advantage that the energy resolution of electron energy-loss spectroscopy (EELS) is improved, thereby providing new information about the sample (Kimoto et al., 2005; Lazar et al., 2006, Browning et al., 2006). This is the approach we plan to adopt in the future. Applying the resolution relations of the previous section to the Nion C3 /C5 -corrected CFEG STEM shows that even without monochromation, this column is capable of atomic resolution (d < 150 pm) at 40 keV. Producing atom-sized electron probes at primary energies ∼10 keV will only be possible using one of the approaches outlined above.

B. Atomic-Resolution Spectroscopy and Elemental/Chemical Mapping Elemental mapping in the EM can potentially use emitted X-rays, emitted Auger electrons, or transmitted electrons which have lost energy as the collected signal. Because the initial excitation resulting from an energy loss event is usually more likely to decay by the emission of an Auger electron than of an X-ray (e.g., Reimer, 1997), the characteristic cross-sections for X-ray emission are typically much smaller than those for energy loss. The signal collection is also typically less efficient for the X-rays, at ∼1–3% vs. > 50% for the EELS. These factors generally lead to more favorable signal-tonoise ratios (SNRs) for the EELS signal than for the X-ray signal, especially for low-Z elements (Leapman and Hunt, 1991; Leapman, 2004). The Auger signal has a limited escape depth and is not suitable for subsurface analysis. This makes the EELS signal the most promising one for elemental mapping at atomic resolution. Three main requirements must be met if elemental mapping with atomic resolution by EELS is to be reached: 1. the instrumental resolution (i.e., the probe size in the STEM) must be sufficiently small, 2. the selected EELS signal must be sufficiently localized on the atomic columns of interest, and

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3. the signal strength and collection efficiency must be such that a sufficiently large SNR is reached before radiation damage or instrumental drift invalidate the experiment. The first requirement has been discussed extensively above. The second requirement accounts for the delocalization of inelastic scattering, i.e., for the fact that an electron passing some distance from an atom can still ionize it, and that the effective interaction region can therefore be much larger than the atom itself. A simple formula describing the delocalization that is in good agreement with experimental evidence was given by Egerton (1996)

dd = 0.5λ/θE 3/4 ,

(19)

where dd is the diameter encompassing 50% of the inelastic interactions, θE the characteristic angle for inelastic scattering given by θE = E/2E∗o , E the energy loss, and E∗o is the “relativistically corrected” primary energy Eo . For an energy loss of 500 eV, Eq. (19) predicts dd = 0.18 nm at 100 keV and 0.24 nm at 300 keV primary energy. In reality, the situation is more complicated, and the localization of the detected inelastic signal can be enhanced significantly by opening up the collection angle (Muller and Silcox, 1995), especially if low-angle inelastic scattering is excluded by using an annular aperture in front of the EELS (Rafferty and Pennycook, 1999). With wide collection angles, it can also be improved by using energy losses higher than the edge threshold, that is, by enhancing the contribution of the electrons scattered into the Bethe ridge. These enhancements, however, come at the cost of a diminished signal, and it therefore remains to be confirmed by experiment whether they will be of practical use. The third requirement arises because a feature is only identifiable if its SNR exceeds a lower limit, usually taken to be 3. Because of the weak signals involved, such a limit may not always be reached. The cross-sections of the characteristic inner-shell loss edges used for analysis are typically in the range of 10−8 to 10−5 nm2 i.e., 102 to 105 times weaker than cross-sections for elastic scattering (Egerton, 1996), and they appear on a continuous background that must be subtracted sufficiently accurately. To make the SNRs of EELS maps approach the SNRs available in HAADF images, ∼103 times stronger incident dose is needed per pixel for mapping a pure element, and a higher dose still is needed for mapping a minority element in a matrix. At the same time, the collection efficiency for the EELS signal needs to approach 100%. Two instrumental solutions exist for EELS elemental mapping: energyfiltered TEM (EFTEM) using an energy filter acting on a broad-beam image (e.g., Reimer, 1995; Grogger et al., 2005), and the STEM approach of collecting an EEL spectrum at every pixel in the scanned image (Jeanguillaume and Colliex, 1989; Tencé, Quatuccio, and Colliex, 1995).

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EFTEM is capable of collecting data in parallel from image areas of up to the size of its two-dimensional (2D) detector (typically a CCD with 1k × 1k or 2k × 2k pixels, but often binned down to 512 × 512). The large field of view and the fact that only a few energy-selected images are needed to produce a map of each element in the sample make EFTEM fast and convenient. Compositional maps of many elements can be acquired in a few seconds or tens of seconds, at ∼0.4–2 nm resolution, provided that the studied area is illuminated with an incident beam current of the order of 10–100 nA (Krivanek, Kundmann, and Kimoto, 1995; Freitag and Mader, 1999; Grogger et al., 2005). However, when attempting to reach atomic resolution (better than 0.2 nm) in elemental maps, EFTEM runs into problems with all three requirements listed above. The EFTEM image is formed by the microscope’s post-sample optics, whose aberrations therefore act on an electron beam that is considerably spread both in angle and energy by scattering from the sample. The chromatic resolution limit is especially severe for wide energy windows (>10 eV). Avoiding much-worsened resolution resulting from electronoptical limits thus demands that both the energy width and the range of scattering angles admitted through the objective aperture be restricted. This reduces the collection efficiency and typically worsens the SNR. Moreover, restricting the range of angles accepted into the energy-selected image eliminates the high-angle energy-loss signal, which contains the high spatial resolution information about the sample. This limits the resolution attainable with small acceptance angles even more than the unavoidable diffraction limit due to the aperture. Finally, EFTEM data along the energy axis must be acquired serially—a separate exposure must be made for each new energy point. Energy-loss electrons not admitted into the energyfiltered image are simply lost. This lowers the SNR per given illumination dose level relative to more efficient acquisition methods. Correction of the aberrations of the imaging column, in particular of its chromatic aberration, would make the EFTEM instrumental resolution limitation less severe. But even when chromatic aberration correction does become available, the overall efficiency of the EFTEM technique will still be limited by the serial nature of the acquisition of information along the energy axis. Another limitation for the EFTEM approach arises in the quantification of minority or trace elements, which needs many data points along the energy axis if the weak signal is to be extracted as reliably as possible. Acquiring the energy data serially is then especially inefficient. Because the doses required for elemental mapping at atomic resolution are much higher than those for imaging with elastically scattered electrons, the increased dose and hence larger radiation damage due to the EFTEM approach is likely to be a major concern. The STEM/EELS spectrum-imaging approach is better suited to all the three principal requirements for atomic-resolution elemental mapping. The

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instrumental resolution is determined by the pre-sample optics, which acts on a beam that has not been spread in angle and energy by the sample. With optimized collection optics, it is possible to accept electrons scattered by 50 mr or more into the energy loss spectrometer. Such relatively large angles improve both the localization of the signal and the collection efficiency. A full spectrum is available at each pixel, making it possible to use techniques such as principal component analysis (e.g., Trebbia and Bonnet, 1990; Bonnet, Herbin, and Vautrot, 1997; Stone, 2004; Bosman et al., 2006) and multiple least squares fitting (e.g., Leapman, 2004) to optimally extract weak signals in the presence of statistical noise. The STEM approach is prone to one major limitation: speed. Acquiring a complete EEL spectrum with sufficient SNR with an incident beam current of 10–100 pA and EELS collection optics of ∼10–30% efficiency (as was the case prior to aberration correction and optimized collection optics) requires an acquisition time of ∼0.1–1 s per pixel. An elemental map of 256 × 256 pixels acquired and processed at 0.1 s per pixel takes 1.8 hours, which is beyond the patience of most operators. If the acquisition time could be reduced, then the intrinsic advantages of STEM spectrum-imaging would become very clear. Speeding up the acquisition to 1 ms per pixel would reduce the total acquisition time to a much more acceptable 66 s per 256 × 256 map. An improvement factor of 100× is not impossible: a 33× improvement in the incident current plus a 3× improvement in the EELS collection efficiency would reach it. This amounts to increasing the probe current to ∼1 nA while maintaining its size close to 0.1 nm and improving the collection efficiency to nearly 100%. As the instrumental performance has been approaching such levels, interest in atomic-resolution STEM-EELS mapping has grown correspondingly. Several groups have demonstrated the potential of the technique: Bosman et al. (2007) on a sample of Bi0.5 Sr0.5 MnO3 ; Kimoto et al. (2007) and Kimoto, Ishizuka, and Matsui (2008) on La1.2 Sr1.8 Mn2 O7 , and Si3 N4 ; Watanabe et al. (2007) on SrTiO3 , and Muller et al. (2008) on a La0.7 Sr0.3 MnO3 /SrTiO3 multilayer sample. The acquisition conditions have varied widely. Kimoto et al. (2007) used a 200-keV CFEG STEM that did not have an aberration corrector and produced only a 7 pA current in a 0.12 nm probe. Their collection semi-angle was 31 mr, and the acquisition time 2 s per pixel. Drift was compensated by recording and re-registering a small HADF image close to the area of interest, once every 30 s, and the total acquisition time for a 24 × 61 pixel map was 61 minutes. Bosman et al. and Watanabe et al. used C3 -corrected STEMs with more probe current, probably of the order of 100–200 pA, less optimized EELS collection optics (acceptance angles of the order of 15–25 mr), and acquisition times of 0.1– 0.2 s per pixel. Muller et al. used a C3 /C5 -corrected CFEG STEM with 780 pA current in a 0.14 nm probe with a 40 mr convergence angle and a 45 mr collection angle. The acquisition time was only 7 ms per pixel, no drift

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correction was performed, and a 64 × 64 pixel map needed just 29 s net acquisition time. The SNR of Muller et al.’s spectra was good enough for probing bonding variations across the sample. The La, Mn and Ti elemental maps obtained by Muller et al. are shown in Figure 5 together with a red-green-blue composite image combining the three elemental maps. Table II shows that 1 nA current is presently available in probes ∼0.15 nm in diameter, and that 0.1 nm probes containing 1 nA of current should be available soon. With present CFEG designs, the probe current can probably be increased 2× to 5× higher still while maintaining atomic resolution before gun aberrations decisively limit further increases. At the

(a)

(b)

(c)

(d)

1 nm

FIGURE 5 Color-coded elemental maps of a La0.7 Sr0.3 MnO3 /SrTiO3 multilayer. (a) La; (b) Ti; (c) Mn; and (d) RGB representation of the sample obtained by combining the individual maps. Note how Ti and Mn occupy the same type of sites in the lattice, and that the purple dots at multilayer boundaries indicate intermixing of Ti and Mn within each column. (Courtesy Lena Fitting-Kourkoutis and David A. Muller, with permission from Science.) (See color insert).

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increased current, atomic-resolution elemental mapping should be possible with per-pixel acquisition times of the order of 1 ms. With readout rates >1 MHz for the EELS CCD detector, which are now standard, it should be readily possible to record the intensity in 1000 energy channels per pixel at this pixel advance rate. No present-day EEL spectrometer is capable of such overall speed yet, but the limitation is likely to be overcome within the next few years. When this state of development is reached, atomicresolution EELS maps of 256 × 256 pixels should be obtainable in just over 1 minute. An attendant complication for achieving atomic resolution in elemental maps is likely to be that the channeling of the probe down the atomic columns will have to be simulated to ensure that the initially sharply focused probe does not spread excessively in thicker samples when high probe angles are used. Radiation damage will have to be monitored carefully, and the beam will have to be blanked whenever no data are being acquired. Even with these factors, it would be surprising if the ability to map the composition of solid samples at atomic resolution on a time scale of one to a few minutes did not find widespread use.

C. Single Atom Spectroscopy and Mapping The requirements for single-atom spectroscopy are slightly different from atomic-resolution mapping. Resolution of 0.2 nm or better will help to optimize the detection limits, but it is not an absolute requirement—atoms with poorly localized low-energy edges may be detectable even though their EELS images may be >0.5 nm wide. On the other hand, the SNR of the EELS signal collected from the single atom is paramount. In order to identify a single atom unequivocally, the SNR resulting from the atom should ideally be much greater than the simple detection threshold of SNR = 3 (Krivanek et al., 1991). In a pioneering study of the detection limits by EELS, Isaacson and Johnson (1975) predicted that a single atom of fluorine in a 2 nm thick carbon matrix should be detectable in a per-pixel dwell time of 0.01 s with an electron probe of 1 nA current and 0.3 nm diameter and a 100% efficient parallel-detection EELS. Probes of comparable and greater energy density and efficient EEL spectrometers are now available, and it is therefore not surprising that single atoms are being detected spectroscopically and mapped. Suenaga et al. (2000) used an uncorrected analytical VG with a 0.6 nm, 0.65 nA probe at 100 keV, an illumination half-angle of 8 mr, and probably a similar detection half-angle. They formed EELS spectrum-images of single Gd atoms encapsulated in fullerene spheres that were stuffed inside a single-wall carbon nanotube, using the gadolinium N4,5 edge with 140-eV threshold energy and a high cross-section but moderate localization of

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∼0.4 nm. The Gd atoms were spaced by 1–3 nm. A spectrum-image 128 × 32 pixels in size was recorded at 35 ms per pixel (i.e., a net acquisition time of 145 s). Quantifying the spectrum-image into an elemental map produced an SNR just high enough to identify single Gd atoms. A CCD detector, for which detective quantum efficiency (DQE) ∼1 for a wide range of operating conditions (Krivanek and Mooney, 1993; Hunt et al., 2001; Tencé et al., 2006), was used, but the probe characteristics could have been much better optimized if an aberration-corrected STEM were available. As a result, the SNR for the Gd M4,5 edge at 1217 eV, whose cross-section (per 1-eV energy interval) is ∼100× weaker than the N4,5 cross-section, was far from sufficient, and this edge could not be used. Radiation damage was severe enough at 100 keV that many of the Gd atoms left their fullerene cages. Even so, a clear elemental map showing single Gd atoms was obtained. It would be very interesting to repeat this experiment at lower primary energy, with aberration-corrected probe-forming optics, and an optimized EELS collection and detection setup to determine whether the radiation damage could be minimized, and whether electronic and magnetic information about single atoms of Gd might become available with a sufficiently strong Gd M4,5 signal. The detection of single atoms of Ca in a biological sample has been demonstrated by Leapman (2003). The sample consisted of isolated Ca atoms and Ca aggregates on a carbon film just 4 nm thick. An uncorrected analytical VG was used at 100 keV, with a probe of ∼1 nm in diameter and 1 nA current. The EELS collection half-angle was 20 mr, and the sample was held at −160◦ C. The per-pixel dwell time was 0.1–0.4 s, the energy dispersion 0.3 eV per channel, and the pixel size typically 0.6 × 0.6 nm. SNR of 5 for single Ca atom detection was obtained. Similar methodology applied to atoms of iron on the same type of substrate yielded an SNR ∼10 for detecting a cluster of four Fe atoms–not quite high enough for single-atom detection. An experiment by Varela et al. (2004) on isolated La atoms substituting for Ca in CaTiO3 points to the possibilities opened by aberration correction (Figure 6). Spectra were collected from a Ca column containing a lanthanum atom (spectrum b, for which the bright spot in the HAADF image is more intense than at neighboring Ca-only columns), and from various points in the vicinity (spectra a, c, and d). The EELS SNR was sufficient to detect the La atom using its M4,5 edge at 832-eV energy loss. A weak La M4,5 signal was still detectable when the probe was centered on neighboring O and Ti-O columns (0.2 and 0.28 nm distant, respectively), but not when the probe was centered on a neighboring Ca-only column (0.4 nm distant). Simulation of the probe propagation through the sample showed that the signal was due to probe dechanneling rather than the delocalization of the inelastic scattering. A C3 -corrected VG HB501 with an

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0.5 nm d a

b

c

d

c b a

820 Ca

O

Ti

850 880 Energy (eV)

FIGURE 6 HAADF image of CaTiO3 sample with a La atom impurity, and EELS spectra recorded in 20–30 seconds from various points on and near the La atom. (Courtesy Maria Varela and Stephen J. Pennycook, with permission from Physical Review Letters.) (See color insert).

energy loss spectrometer equipped with a 2D CCD detector was used at 100 keV, with an illumination half-angle of 25 mr, a probe current of 100 pA in a 0.11 nm probe, and a collection half-angle of 7 mr. The nonoptimized EELS collection geometry meant that less than 8% of the available EELS signal was collected. This was partially compensated by a long total acquisition time: 30 s for the spectrum collected over the La atom and 20–30 s each for the other spectra. In order to understand the relative importance of the different factors contributing to the quality of the collected spectra, it is useful to derive the SNR that should have been reached in the above experiment. Neglecting multiple scattering, the signal from the atom of interest is given approximately by

   Sa = Ip σa εa t/ dp2 + dd2 e ,

(20)

(Leapman, 2003), where Ip is the probe current, σa the cross-section for the atom’s monitored signal, εa the detection efficiency for the atom’s signal, t the acquisition time, dp the probe size, dd the delocalization of the inelastic scattering [given by Eq. (19)], and e the charge carried by an electron. (d2p + dd2 )0.5 may be taken as an “effective probe size” due to the combined influence of the probe-forming optics and the scattering delocalization.

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The background under the signal Sa is given by

Sb = Ip σb ρb εb t/e,

(21)

where σb is the average cross-section for the matrix atoms’ contribution to the background, ρb the number of atoms per unit area of the sample, and εb the detection efficiency for the background signal (which may differ from εa if the angular distributions of the two signals are different). Assuming that the dominant noise term is the statistical variation in the background signal, and that the efficiencies εa and εb are both equal to ε, the SNR of the single atom’s signal is then

  0.5   √ SNR = Sa / Sb = σa / dp2 + dd2 Ip εt/ (σb ρ b e) .

(22)

Equation (22) shows that maximizing the signal cross-section σa improves the SNR linearly, whereas increasing the probe current, collection efficiency, or acquisition time or decreasing the background cross-section or atomic density per unit area (by decreasing the sample thickness) only improve it as the square root. Decreasing the probe diameter improves the SNR as 1/dp2 when dp >> dd , as ∼1/dp when dp ∼ dd , and does not change the SNR at all when dp 0 (overfocus), we have a white linear term, which is reinforced with similar white contrast by the nonlinear term. The treatment, however, does not correct for nonlinear images appearing between linear images. Thus another method to overcome the nonlinear image contrast and strong diffraction effects in thicker crystals is needed; that is the subtraction method of two Cs -corrected images as discussed in the next section.

G. Subtraction Method of Nonlinear Contrast in Cs -Corrected TEM Very few studies were made of the nonlinear terms except for a recent work by Nomaguchi et al.(2004), relating to an image-processing method for correcting spherical aberration of an objective lens. This text section describes a simple method to minimize the nonlinear terms in HRTEM images with spherical aberration correction (Yamasaki et al., 2005a) (see Section III.D).

1. Vanishing of a Nonlinear Term Formed by g and −g Waves

Consider a three beam lattice image formed with −g, 0, and g-waves picked up from Eq. (36), which is the same as the Fourier image (Cowley and Moodie, 1957). The image intensity has a simple form in 2D notation (x, z) from r(x, y, z) as

IS (x, z) = | 0 |2 + | g |2 +| −g |2 + 4| 0 || g | cos(2πgx + φg ) sin(2π kg z) + 2| g |2 cos(4πgx + 2φg ),

(40)

where the notations are the same as those in Eq. (35). The fourth term is an ordinary lattice fringe with spacing of 1/g and fifth term, half-spacing lattice fringes with a spacing of 1/2 g (Komoda, 1966). As shown in a previous paper (Tanaka and Hu, 1998), the sine function in the fourth term corresponds to sin χ(u) in Eq. (13), the PCTF, where χ(u) = πλ fu2 = 2 kg z with f = z ( f > 0; overfocus). Equation (40) yields two facts: that the contrast of lattice fringes is reversed while keeping the same intensity by a change from z to −z, and that half-spacing lattice fringe is independent of the amount of defocus.

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An important fact is that this is realized for any spacing (d) as shown in the Thon diagram with Cs = 0, where the imaging characteristic for any spacing is symmetric for the reversal of defocus from z to −z for Cs = 0 (Thon, 1966; Tanaka and Hu, 1998). When the terms in the last row in the Eq. (36) are negligible in regard to image intensity and the spherical aberration constant Cs is zero, the nonlinear terms as the half-spacing lattice fringes are vanished at any time by subtracting an image contrast taken at defocus f from that taken at defocus − f . For thicker crystalline samples, dynamical diffraction effect should be included by an amplitude modulation factor Ag and a phase-deviation factor from π/2, exp(iηg ) in Eq. (9). As discussed in the paper by Tanaka and Hu (1998), the phase factor makes the origin of the Fourier image shift along the z-direction by the following formula in a case of Cs = 0,

f ∗ = f + ηg /πλu2

(41)

When f ∗ = 0, the contrast of a lattice fringe of spacing of d(= 1/u) is zero despite the phase deviation from π/2. In this case, we should have two images taken at (−ηg /πλu2 + f ) and (−ηg /πλu2 − f ) and subtract them to double the ordinary (linear) phase-contrast term, although the nonlinear term as half-spacing fringes always vanishes because they are independent of defocus.

2. Vanishing Nonlinear Terms Formed by g and h Waves Next let us consider lattice fringes formed by g and h with different modulus in the last row in Eq. (36). In the first step, we neglect the phase-deviation term, exp(iηg ). The corresponding cross-term intensity is

IS (x, z) = (i) × (−i)| g || h | exp{2πi(k0 − kg + g) · r + iφg } × exp{−2πi(k0 − kh + h) · r − iφh } + (−i) × (i)| g || h | exp{−2πi(k0 − kg + g) · r − iφg } × exp{2πi(k0 − kh + h) · r + iφh } = | g || h | exp{2πi(g − h) · x} × exp{2πi(− kg + kh ) · z} × exp{i(φg − φh )} + | g || h | exp{−2πi(g − h) · x} × exp{−2πi(− kg + kh ) · z} × exp{i(φh − φg )},

(42)

where the vector r is rewritten as (x, z) using a 2D vector x. Similar formulations can be made on reflections of −g and −h, which are in symmetrical

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positions with respect to the origin in reciprocal space as follows,

IS′ (x, z) = | g || h | exp{−2πi(g − h) · x} × exp{2πi(− kg + kh ) · z} × exp{−i(φg − φh )} + | g || h | exp{2πi(g − h) · x}

× exp{−2πi(− kg + kh ) · z} × exp{−i(φh − φg )},

(43)

where we assume | g | = | h |, | kg | = | kh |, and so on. Combining the two formulas, we obtain

IS (r) = IS (x, z) = IS + IS′

= 4| g || h | cos{2π(g − h) · x + (φg − φh )} × cos{2π(− kg + kh )z}

(44)

Since the function cos{2π(− kg + kh )z} is an even function, when the images are taken at the defocus z and −z, the nonlinear term is not changed and vanished also by subtraction between the two images in this case. This result provides an important “recipe” for minimizing the nonlinear terms that disturb direct interpretation of image contrast in Cs -corrected HRTEM. When sample crystals are thin enough to be treated as a weak phase object without dynamical diffraction effect, it is the only thing that one should subtract two images taken at f and − f . In the second step, we consider the effect of dynamical diffraction in thicker crystals, phase-deviation from π/2, exp(iηg ). Including the terms into the formulation in Eqs. (42) and (43), the final form is written as follows in place of Eq. (44):

IS (x, z) = 4| g || h | cos{2π(g − h) · x + (φg − φh )} cos{2π(− kg + kh )z + α},

(45)

where α = ηg − ηh . Obviously, the additional phase α should shift the origin of defocus for the cancellation of the nonlinear term as discussed in Eq. (41). Unfortunately, the value α is dependent of the reflections such as g and h and the fitted shift of the defocus origin is varied with the corresponding reflections. The formulation considers only the wave aberration by usingχ(u). In the second-order imaging theory (Ishizuka, 1980), the effects of an electron source size and a chromatic aberration need to be included by using the concept of transmission cross coefficient (TCC). Using the notation of by Horiuchi (1994), the TCC between g and h reflections in axial multibeam lattice images is written similarly to

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Eq. (38) as

T(h, g; ε) =



J0 (u)D(δε)A(u + h)A(u + g)

× exp[−i{χ(u + h, ε + δε) − χ(u + g, ε + δε)}]d(δε)du, (46) where J0 is angular distribution of electrons from the source, D(δε), distribution of defocus fluctuation, A, aperture function, χ, the wave aberration, and ε and δε are the amount of defocus (ε > 0, overfocus) and its fluctuation, respectively. In HRTEM conditions, the above equation is proved to be reduced to the following equation in a multiple form of some envelope functions, Ej , ED , Ex , and P, as

T(h, g; ε) = A(h)A(g) exp[−i{χ(h, ε) − χ(g, ε)}] Ej (h, g; ε)ED (h, g)Ex (h, g; ε)P(h, g; ε).

(47)

The functions ED and Ej are damping functions due to chromatic aberration and beam divergence. The functions Ex and P as phase factors are negligible for HRTEM conditions (Ishizuka,1980; Horiuchi, 1994). ED is independent of the amount of defocus, which is the key parameter in this present subtraction method. The remaining function Ej , due to the effect of source size, can be reduced to the following simple form for Cs = 0.

  Ej (h, g; ε) = exp −(πu0 )2 ε2 λ2 (h − g)2 ,

(48)

where u0 is the reciprocal of the source size. The term is an even function for the defocus ε (= f ), so the values for ε and −ε are the same, which means that the present simple method still holds by including the TCC for second-order imaging theory, when the dynamical diffraction effect is neglected.

3. Simulation Study Figure 3 shows an example for LiCoO2 [100] crystals. Positions of Li atoms are well reproduced at t < 4 nm. The result shows that subtraction of Cs -corrected images with f = ±3 nm succeeds in purifying linear image contrast only when the thickness of samples is small in the crystal with medium atomic number elements such as cobalt (Co).

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(a)

(b)

(c)

(d) Co Li

Co

O

Li

O 0.1 nm

FIGURE 3 Image simulation of the subtraction method on LiCoO2 crystals ( f = ±3 nm, = 3 nm). (a) overfocused image, (b) under-focus image, (c) “subtraction” image, and (d) “subtraction + deconvolution” processing (see Section II.H).

In order to understand the result, we rewrite the formula for subtracted images from Eq. (45) as

IS (+z) − IS (−z) =

g

4| 0 || g | cos(2πg · r + φg )

× sin(2π kg z) cos(η0 − ηg )



+ 2| g || h | cos{2π(g − h) · x + φg − φh } g =0 h =0,±g

× sin(2π( kg − kh )z) sin(ηg − ηh ).

(49)

First and second summations in Eq. (49) show linear and nonlinear terms, respectively. The condition for minimization of the nonlinear terms is that at least one of | g || h |, sin(2π( kg − kh )z), and sin(ηg − ηh ) terms is very small. The second sine function has a considerable value only when differences between |g| and |h| are large because of kg = λg2 /2. For example, if |g| = 10 nm−1 , |h| = 6.7 nm−1 and z = 1 nm, sin(2π( kg − kh )z) is ∼ 0.42. The term | g | is attenuated roughly by a Gaussian function of |g| as opposed to saturating the above sine function up to 1.0. The term, sin(ηg − ηh ), cannot be managed since the dynamical phase shift of g-beam, ηg , depends generally on materials, thickness, index g, and so on. The term is also sin(ηg − ηh ) ≤ 1. Consequently, not so large nonlinear components may be expected in the second summation in Eq. (49). One should, however, consider another situation that the nonlinear terms are enhanced when | 0 | decreases and | g | and | h | increase strongly by dynamical diffraction effects. The situation is seen in the subtracted

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images of LiCoO2 at t > 4 nm. Owing to the large lattice constant of LiCoO2 , many reciprocal lattice points located in the zero-th-order Laue zone are sufficiently excited. This may cause remarkable dynamical diffraction effects in LiCoO2 crystals. It can therefore be concluded that the present image subtraction method is sufficient in that weak phase object approximation does hold. The next problem is accurate correction on the modified linear term even for Cs -corrected HRTEM.

H. Improved Image Deconvolution Method After subtraction of two images with positive and negative defoci, the image intensity can be obtained as a linear expression of the projected potential, modified by PCTF, sin χ(u) in the second-order approximation as shown in Eq. (22). This is perfectly different from the weak phase object approximation and pseudo–weak phase object approximation proposed by Han et al. (1986). In this stage, the image deconvolution for canceling the effect of an objective lens, ‘⊗s(x)′ , becomes important to the higher-order approximation (Yamasaki et al., 2008). Making the FT of Eq. (22), the image intensity in reciprocal space representation is approximately

ˆ p (x)} × sin χ(u) × ED × Ej , Ii (u) = δ(u) + 4σ F{φ

(50)

where ED and Ej are damping functions due to chromatic aberration and beam divergence. By dividing the equation by sin χ and chromatic and beam divergence envelope functions as ED and Ej , aberration-corrected images can be obtained to the second-order approximation. That is an ordinary image deconvolution method. However, three problems are involved with this technique. The first problem arise from many zero crossings in oscillating PCTF. Figure 1a shows a typical PCTF in a conventional TEM (E = 200 kV, Cs = 0.5 mm, f = −42 nm, defocus spread = 3 nm, convergence semi-angle α = 0.5 mrad). At the zero crossings (indicated by black arrows) the above filtering process is diverged. The solution is that one should carefully select processing images where Bragg spots are not located on the zero crossings. This tip is not available for unknown structures. Aberration-corrected TEM solves the problem because there is no oscillation in PCTF up to the information limit, as shown in Figure 1b. The aberration-corrected TEM makes it possible to perform the image deconvolution process properly over all spatial frequencies up to the information limit. The second problem involves nonlinear components. The deconvolution process without elimination of nonlinear image components induces improper increase of the nonlinear components with resultant artificial

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image contrast. This problem is solved by the subtraction method described previously. The third problem is artificial image contrast induced by the so-called Fourier termination effect. Use of PSF is helpful in understanding the issue. 2D expression of PCTF (E = 200 kV, Cs = 0.5 mm, f = −42 nm, defocus spread = 3 nm, convergence semi-angle α = 0.5 mrad) in Figure 1a is shown in Figure 4. The PSF is the inverse FT of the PCTF, which is convoluted onto the projected potential φp (x, y) as shown in Eq. (12) under the linear approximation. The shape of the PSF is not monotonically decreasing but includes oscillation. This causes artificial maxima in the linear image intensity. Moreover, nonlinear components are overlapped in the image. It may be rather difficult to eliminate these artifacts from recorded images and recover proper structure images. A Cs -corrector is an effective hardware to derive image contrast corresponding to atomic structures due to suppression of both of the PCTF oscillation and the nonlinear components in a combined use of the image subtraction method. As shown in Figure 4b, however, there is still some uncertainty for correspondence between atomic positions and image contrast maxima. Then, the image deconvolution method is applied on Figure 4b. This procedure results in making a flat filtered transfer function up to the information limit, which is a step function due to lack of information beyond the information limit (Figure 4c). In this case, the artificial image contrast comes from oscillation in the PSF as a kind of “sinc” function as the FT of the step function.

(a)

0.3 nm

PCTF

0.3 nm

(b)

PSF

1

0.3 nm

PCTF

1

0.3 nm

PSF

0 0

21

(c)

1 0

step function

0.3 nm

PSF

0.3 nm

(d)

envelope function 1

0.3 nm

0.3 nm

PSF

0

FIGURE 4 Filtering function with PCTF, PSFs, and simulated images of [110] silicon (t = 2 nm) (a) conventional TEM (Cs = 0.5 mm, f = −42 nm); (b) aberrationcorrected TEM (Cs = 0, f = +3 nm) after the subtraction method; (c) after ordinary deconvolution method; and (d) after the new deconvolution method. Profiles along the dotted lines.

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For this situation, a new deconvolution process was developed by the author’s group, in which FT of images are divided by sin χ itself in place of a product of sin χ and envelope functions. This procedure creates an improved transfer function gradually decreasing like actual envelope functions. The PSF in real space has an ideally damping nature as shown in Figure 4d. In this case, contrast maxima appear only at real atom positions. This new deconvolution procedure has another advantage: it eliminates the need to measure actual envelope functions by using Young’s fringe method etc. A Gaussian function may be ideal as a filtered transfer function. The FT is also Gaussian without inducing any artificial intensity maxima and minima. However, use of the Gaussian function is less practical than use of the envelope functions because of increased broadening of PSF and experimental measurement of the information limit. It should be noted that previous HRTEM images had given an apparent high resolution made by the artificial contrast that did not correspond to real atomic columns. In a later section, Figure 11e shows a result of the new deconvolution process applied to the subtraction image of magnesium oxide crystals. The extra bright spots (arrows) that slightly remain even after the subtractive operation are perfectly eliminated in the deconvolution image. Moreover, the elliptical form of image contrast at the position of oxygen atoms is corrected to a spherical shape. The shape corresponds to isotropic blurring of the projected potential in Figure 11f.

III. ACTUAL ADVANTAGES FOR OBSERVATION BY CS -CORRECTED TEM A. Improvement of Point-to-Point Resolution The point-point resolution of TEM images of a thin sample is expressed in terms of a PCTF, including chromatic damping functions, when the sample is assumed as a weak phase object (Spence, 2003). In ordinary cases for measuring the resolution, one can use thin amorphous carbon or germanium films and use their high-resolution granular images. The digital FT patterns give the Scherzer limit corresponding to the point-to point resolution and the information limits determined by the chromatic aberration of the instrument. For estimation of the latter resolution, the Young fringe method is useful: that is, a FT pattern from two overlapped images laterally a little shifted at the recording by using image shift knobs of TEM. The spherical aberration correctors improve the former Scherzer resolution, which is determined in the coherent imaging with aberration.

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(a) Diamond [110]

Cs-corrected TEM (200 kV)

d 5 89 pm

(b)

170 160 150 140 130 120 110 100

0

50

100

150

200

250

300

FIGURE 5 200-kV Cs -corrected TEM image of a diamond crystal along [011] direction (a) and scanned intensity along a bar (b).

The features of PCTFs of a conventional TEM and a Cs -corrected TEM are already shown in Figure 1a and b. The Scherzer resolution of the former is ∼ 0.2 nm, but the Cs -corrected TEM provides less than 0.1 nm in the resolution. Dumbbell structures of silicon (Si) crystals seen in observation along the [110] direction are easily resolved by the latter instrument. Figure 5 shows a high-resolution image of the dumbbell structure of a diamond crystal, where small black dots correspond to carbon atomic columns and white circular contrast indicates vacuum tunnels, which show a point-to point resolution of 89 pm (= 0.089 nm), even by using a 200 kV

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Cs -corrected TEM (JEM-2100F CS) without a monochromator and a Cc corrector (Saitoh and Tanaka, 2008a). The point-to-point resolution of the coherent imaging is expressed by the Scherzer formula as

 δ = 0.664 Cs λ3 ,

(51)

where Cs is the third order spherical aberration constant, and λ, is the wavelength of incident electrons (Scherzer, 1949; Spence, 2003). The resolution also determined by the chromatic damping function coming from the energy spread of incident electrons, energy loss in samples, and chromatic aberration of objective lens is written using the defocus spread as

δc = Cc α

  

2

E 2

E0 2

I

= + +4 , E E I

(52)

where Cc is the chromatic aberration constant, E is fluctuation of accelerating voltage, E0 energy loss in samples, I fluctuation of lens current, and α is a semi-angle for use of objective lens (Spence, 2003). This is a case of the linear imaging. The theory including nonlinear and cross terms between g and h reflections was developed by O’Keefe (1979) and Ishizuka (1980) using mutual intensity (Born and Wolf, 1970; Horiuchi, 1994). Improvement of the Scherzer resolution also gives a kind of insensitivity of image contrast in higher spatial frequencies with amount of defocus (Tanaka et al., 2004c). This fact is related to advantageous measurement of lattice strains by using a Cs -corrected TEM,—for example, with the help of the geometrical phase analysis method (Hytch et al., 1998). What is the ultimate point-to-point resolution after establishment of Cs and Cc -correctors, as well as monochromators and effective imageprocessing software? Lentzen and Urban (2006) and Lentzen (2008) calculated the size of exit wave functions below a silicon crystal as 0.06 nm at 200 kV on the basis of quantum mechanics. For imaging by electron lens, the resolution may be worse than the value. The point-to-point resolution of 0.078 nm was already obtained on Si (112) crystals using a non–Cs -corrected 300 kV TEM and advanced image-processing software (O’Keefe, 2008). Do we anticipate seeing experimental data over the above limitation by future use of Cs -corrected TEMs?

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B. Sharp Interface Images Without Fresnel Fringes The basis of HREM of atomic objects was established by Scherzer (1949). He used positive spherical aberration and defocus (first-order aberration) to obtain image contrast from almost pure-phase objects (such as single atoms). The phase object causing only phase modulation does not yield an image contrast in just-focus condition (Gaussian focus) as explained in Eq. (11). He had the idea of using the aberration of the objective lens to cancel the phase shift caused by scattering in a sample π/2 and to chang the sample into a kind of amplitude object, as performed by Zernike (1942) in optical microscopes (phase-contrast method). For this purpose, Scherzer derived the optimum defocus in an underfocus (weak lens) condition presently called the Scherzer defocus in the following form as:



fS = −1.15 Cs λ

(53)

Similar defocusing is adopted also for imaging of lattice fringes. In that case, one needs defocus corresponding to the period of Cowley and Moodie’s Fourier images (1957) and canceling the phase shift of scattering. The defocusing for obtaining image contrast necessarily blurs the image and makes white or black surrounding contrast, particularly observable  at interfaces and surfaces, which is called Fresnel fringes. The width is λ f (n − 0.25) from the consideration of Fresnel’s first zone (Horiuchi, 1994). The Fresnel fringe contrast mixes with contrast of lattice fringes. This should result in minor positional shifts of the lattice fringes. These shifts are serious for accurate determination of the position of atomic columns at interfaces and surfaces from lattice fringes. In Cs -corrected TEM, the Scherzer defocus is approaching zero [See Eq. (15)], which does not yield Fresnel fringes at all. When high-resolution images of interfaces of metals and ceramics are obtained by TEM, the images are surprising because the lattice fringes are smoothly connected across the interfaces without strong diffraction contrast. This is the true feature of the interfaces. Previous HRTEM images of various interfaces showed artificial strong contrast misleading into special interface structures and strain fields. Figure 6 shows a Cs -corrected high-resolution image of a SiO2 /Si(100) interface prepared by a dry etching process at 700◦ C without the Fresnel fringes (Tanaka et al., 2003). The second characteristic result obtained with Cs -corrected TEM is that on glassy metals such as Pd-Ni-P prepared by a melt-spun method. It is believed from previous TEM studies and neutron diffraction studies that there are medium-range order (MRO) structures floating in amorphous matrices. Without Fresnel fringes in Cs -corrected TEM images, the MRO structures have become clearly visible as shown in Figure 7 (Hirotsu et al., 2006).

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SiO2

0.135 nm Si

FIGURE 6 200-kV Cs -corrected TEM image of an interface between silicon and silicon oxide and the atomic model (inset). The image shows a clear image at the top surface without Fresnel fringes. (a)

(c)

(b) 123

B

D 107

0.23 nm

C

0.20 nm A 125

2 nm

FIGURE 7 200-kV Cs -corrected TEM image of a Pd-Ni-P metallic glass, showing a medium-rang order (MRO) state near in-focus image. A, B, C, and D in panel (b) indicate the MRO clusters. The angles of lattice fringes are also indicated.

C. Discrimination of Elements in Cs -Corrected TEM Images In Cs -corrected HRTEM images, scattering waves in sufficiently large angles are used for image formation. The minute difference of atomic scattering factors between elements can be visualized in the image intensity. Figure 8 shows a cross-sectional image of a thin edge of a GaAs crystal observed in the [110] direction. Black dots correspond to two kinds of atomic columns composing the crystal (dumbbell structures). A slight difference can be recognized in the size of the black dots in the image. This is caused by the difference of two atomic numbers between Ga and As atoms through the difference of their atomic scattering factors as fGa and fAs (Saitoh and Tanaka, 2008c). Figure 9 shows another example of AlCuCo and AlNiCo decagonal quasi-crystals observed along the fivefold axis by a 200-kV conventional

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Ga As

0.141 nm

FIGURE 8 200-kV Cs -corrected TEM image of a thin area of a GaAs crystal observed along the [011] direction. Ga (Z = 31) - As (Z = 33); A difference of two atomic numbers is discriminated. Cs-corrected STEM

200 kV-STEM (a)

(b)

Cs-corrected TEM (c)

Al

0.3 nm

Al

FIGURE 9 Comparison among images of AlCuCo and AlNiCo decagonal quasicrystals by 200-kV ordinary (a) Cs -corrected (b) STEMs, and Cs -corrected TEM (c). The Al atoms can be visualized by Cs correction in both samples. (From Taniguchi and Abe, 2008; Saitoh et al., 2006.)

STEM (image a), a Cs -corrected STEM (image b), and a Cs -corrected TEM (image c). The Cs -corrected STEM reveals the position of aluminum (Al) atomic columns of AlCuCo (Taniguchi and Abe, 2008), but the image contrast is not strong because of its Z-contrast (Z1.5∼1.7 ). On the contrary, the Cs -corrected TEM image of AlNiCo clearly shows the position of Al columns as indicated by white arrows. (Saitoh et al., 2006). Figure 10 shows the third example of discrimination of elements— detection of oxygen atomic columns in a [110]-oriented magnesium oxide crystal (MgO). The thin film of 4 nm−1 . To avoid the corresponding artefacts in the image an objective-lens aperture has to be employed to keep beams corresponding to spatial frequencies higher than 4 nm−1 from contributing to the image. This causes that the information on sample details corresponding to the high spatial frequency range between gS and gI , the information limit, is lost in spite of the fact that this information is transferred by the optical system. (b) The CTF for NCSI conditions in the aberration-corrected instrument. The modulus of the CTF is apart from the maximum always substantially lower than 1. However, no contrast oscillations occur and the whole range of g up to the information limit is contributing to the image. From Lentzen (2006).

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the associated artifacts in the image by preventing beams corresponding to spatial frequencies higher than 4 nm−1 from contributing to the image. Consequently, the information on sample details corresponding to the high spatial frequency range between gS and gI , the information limit, is lost despite the fact that this information is transferred by the optical system. The information limit is determined by the temporal coherence of the electron source. We also emphasize the existence of a transfer gap at low spatial frequencies; images obtained under Scherzer conditions unavoidably have substantial shortcomings in contrast or information transfer both at high and low spatial frequencies. Another problem inherent in the application of the classical Scherzer conditions has rarely been recognized and treated in sufficient detail (Lichte, 1991): the phenomenon of contrast delocalization. The origin of contrast delocalization is the imaging of an object point P into an aberration disk of radius R (Eq. 11) rather than in the desired ideal image point P i . With Scherzer conditions, the radius of the point-spread function adopts a rather high value of ∼ RS = 83 gS−1 . In an atomic-resolution image this means that the information on a given atom position is delocalized, that is, distributed over a disk of radius RS . As a result, intensity measurements performed in the image at a particular atom position generally may not be trusted since in this position intensity contributions also originate from neighboring atomic sites.

2. The New Contrast Theory for Aberration-Corrected TEM The optimum imaging conditions for aberration-corrected atomicresolution EM have been investigated in detail by Lentzen et al. (2002); Jia, Lentzen, and Urban (2004); and Lentzen (2004, 2006). The corresponding new contrast theory for aberration-corrected TEM takes advantage of the fact that the added freedom of variable spherical aberration may be used together with the variable defocus to create contrast conditions substantially improved with respect to Scherzer conditions. In the following text, we refer to the aberration function in the simplified form of Eq. (10). For an exact compensation of the aberration of the objective lens and for a vanishing defocus [according to Eq. (11)] contrast delocalization vanishes. The phase CTF − sin 2πχ(g) adopts the value 0. On the other hand, the amplitude CTF, given by cos 2πχ(g), adopts the maximum value 1. Under these conditions, EM is conducted under pure amplitude-contrast conditions. High-resolution imaging under amplitudecontrast conditions has rarely been discussed in the literature because this imaging mode is not available in the standard uncorrected electron microscope. The images exhibit a local intensity distribution originating from electron diffraction channeling. Details have been discussed by Lentzen et al. (2002).

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Taking advantage of the tuneable spherical aberration coefficient, according to Eq. (13), the point resolution at Scherzer defocus may be extended by reducing CS . However, it is not reasonable to increase gS beyond the value of gI . Another concern regarding an adequate contrast theory for aberration-corrected instruments is related to the problem that phase-contrast microscopy requires the application of a suitable value of defocus to convert phase into amplitude information. This automatically increases contrast delocalization, partially eliminating the gain in resolution by aberration correction. A compromise between large phase contrast and reduced delocalization may be evaluated by equating the Scherzer relations according to Eq. (12) with the Lichte defocus of least confusion (Lichte 1991)

3 ZL = − CS λ2 gI2 . 4

(15)

This yields the result for optimum imaging conditions in an aberrationcorrected transmission electron microscope (Lentzen et al., 2002; Lentzen, 2004). This consists of an adjusted value of

CS =

64  3 4 −1 λ gI 27

(16)

16  2 −1 λgI . 9

(17)

to be combined with a defocus value of

Z=−

This results in a value of the radius of the point-spread function of

R=

16 −1 g . 27 I

(18)

The corresponding amount of contrast delocalization is 9/2 times smaller compared with Scherzer conditions with about half of the resolution at the information limit negligible for most cases. For a Titan 80-300 instrument operated at an electron energy of 300 keV, we obtain the following values: λ = 1.97 pm; gI = 80 pm; CS = 12.7 µm; Z = 5.78 nm; R = 47 pm. Typical CS values of uncorrected objective lenses amount to 0.5–3 mm. This indicates that less than half a percent of the original CS value is required to obtain optimum contrast; the remaining contribution is compensated by the aberration corrector.

3. The Negative CS Imaging Technique One of the most striking discoveries in the early years of experimental research work with aberration-corrected instruments was the entirely

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unexpected dramatic gain in contrast und effective resolution when the residual CS value adjusted for contrast is adjusted to negative values (Jia, Lentzen, and Urban, 2003). Negative spherical aberration values are obtained by overexcitation of the correcting element. By application of the NCSI technique, the contrast of low nuclear-charge atoms becomes sufficiently strong and sharp so that oxygen, for example, could be imaged directly for the first time ever. Before this was accessible only by the technique of EPWF reconstruction (Coene et al., 1992). Although the latter technique has been used only rarely to study oxygen in oxygencontaining compounds (Jia and Thust, 1999), the NCSI technique has the potential to be widely used in studies where low-nuclear charge atoms (e.g., oxygen, nitrogen, and boron), are to be imaged close to heavy atoms. This has been demonstrated by a number of recent results (Jia and Urban, 2004; Urban, 2007; Jia et al., 2008). The theoretical foundation of the NCSI technique was investigated by Lentzen et al. (2002) and Lentzen (2004). The result is that Eqs. (16) to (18) are still valid but the signs of the equations defining CS and of Z are to be inverted. The traditional imaging mode with positive spherical aberration combined with underfocus and the new imaging mode with negative spherical aberration and overfocus produce within linear kinematic imaging theory an atom contrast symmetrical with respect to the mean intensity (Lentzen et al., 2002). If nonlinear contrast contributions (in a fully dynamical theory) are considered, however, a striking asymmetry occurs. The CTF for NCSI conditions is displayed in Figure 2b. Special emphasis is placed on the bell-shaped curve typical for contrast transfer in aberration-corrected instruments. Aside from the maximum area, the modulus of the CTF is always substantially lower than 1. However, no contrast oscillations occur and the entire range of g contributes to the image up to the information limit. The new imaging mode produces stronger contrast modulation at atom column sites than the traditional imaging mode. This asymmetry has been investigated by image simulations for a number of crystal structures (Jia, Lentzen, and Urban, 2004). An analytical treatment invoking the full imaging formalism using the transmission cross-coefficients for partially coherent illumination is rather complicated and provides no simple picture of the underlying contrast mechanism. A simple picture may be gained, however, if an ideal Zernike phase plate is assumed for both imaging modes (Jia, Lentzen, and Urban, 2004)—that is, transmission of the direct, unscattered wave ψ0 with a coefficient of 1 and transmission of the scattered wave

ψsc (r) = πiλU(r)t with a coefficient of +i for positive phase contrast and −i for negative phase contrast. Here U(r) denotes the projected crystal potential and t is

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again the specimen thickness. The exit wave in the object plane,

ψob (r) = ψ0 + πiλU(r)t,

(19)

is altered by the phase plate to the exit wave in the imaging plane,

ψim (g) = ψ0 ∓ πλU(r)t

(20)

and the resulting image intensity to second order in U(r) is

I(r) = ψ02 ∓ 2πψ0 λU(r)t + (πλU(r)t)2 .

(21)

The upper sign holds for positive phase contrast and the lower sign for negative phase contrast. A common phase of ψ0 and ψsc has been chosen to set ψ0 to a real value. The comparison of both cases shows that the linear contribution and the quadratic contribution have a different sign for positive phase contrast. As a consequence, the local intensity modulation at an atom column site is weak, because the linear modulation is partially canceled by the nonlinear modulation. On the other hand, the linear contribution and the quadratic contribution have the same sign for negative phase contrast. The local intensity modulation at an atom column site is comparatively stronger, because linear modulation and nonlinear modulation reinforce each other. In other words, using negative spherical aberration combined with an overfocus enhances contrast originating from thin specimens compared with a setting with positive spherical aberration and underfocus. The above considerations have treated the case of thin samples, where the scattered wave has only a small modulation relative to the direct, transmitted wave, and where single scattering by the projected crystal potential within the first Born approximation leads to a phase of the scattered wave of π/2 relative to the direct wave. The case of thicker samples has been treated by Lentzen (2004, 2006).

III. ATOMIC-RESOLUTION ELECTRON MICROSCOPY AND THE INVERSION OF THE SCATTERING AND IMAGING PROBLEM A. What is Atomic Resolution? The term high-resolution imaging has been used for decades to describe studies performed in conventional uncorrected instruments in which images were obtained showing a contrast distribution resembling an atomic pattern. In this context, the terms crystal structure imaging, lattice plane imaging and lattice fringe imaging also are frequently used. The characteristics of

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these techniques and the type of results obtained are discussed in textbooks (e.g., Carter and Williams, 1996; Reimer, 1984; Spence, 1988). None of these techniques provides genuine atomic resolution. In so-called crystal structure imaging, for example, the CTF is manipulated by proper adjustment of the objective lens defocus such that certain spatial frequencies, characteristic for the crystal lattice, are transferred at high amplitude in so-called pass bands. This means that the structure must be known a priori. Since in areas between these pass bands the CTF normally exhibits bands of low transfer, transfer gaps, and even oscillations, the information on other details is maltreated if it is available at all. In other cases, the patterns are created by crossed lattice fringes. In on-axis latticefringe imaging the effect of certain aberrations is canceled from symmetry reasons. As a result, the images may show apparent high-resolution qualities that they actually do not really exhibit. Crossed lattice-fringe patterns of this type therefore are not considered physical with respect to imaging an atomic structure. Quantum-mechanical and optical image calculations allow a crucial test with respect to the actual resolution of such images. For these purposes, a selected single-atom position in the structure model is shifted by a certain amount or even totally eliminated. This does not substantially alter the contrast of the related alleged “atomic” position in the image. The reason for this behavior is, that although the information on the basic Fourier components of the crystal structure is transferred, that of the higher spatial-frequency components required to describe deviations from the ideal undisturbed lattice structure is not. In addition, the contrast delocalization unavoidable in uncorrected instruments contributes to a nonlocality of the information. On-axis lattice-fringe images therefore are best used as a measure of local crystal structure, orientation, and symmetry, and in suitable cases even the local lattice parameter may be determined. These are collective features of the structure that do not require attention to the individual atomic site. How then should atomic resolution be defined? The answer is simple— the information must be entirely local on the atomic level. Any change in the position or occupancy of an atomic site in the sample must show up in the image as an individual signal localized only at the corresponding atomic position. In this strictest sense, the images obtainable in modern aberration-corrected instruments match the standards of atomic resolution. Figure 3 shows calculated images for oxygen in SrTiO3 demonstrating genuine atomic resolution in an aberration-corrected transmission electron microscope. The arrows mark oxygen columns for which the occupancy is set to 0 and to 0.5. Any changes are purely local. For completeness we also emphasize the necessary distinction between lateral and depth resolution. Currently, discussion about “atom positions” nearly always refers in reality to “atom columns” in a specimen aligned such that a high-symmetry crystal direction is parallel to the incident

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(a)

(b)

(c)

(f)

(g)

Sr Ti O (d)

(e)

FIGURE 3 Demonstration of atomic resolution by calculated images for oxygen in SrTiO3 : (a) Structure model. (b) Image under Scherzer conditions in an uncorrected conventional electron microscope. (c) Positive phase contrast under optimum conditions in a corrected instrument. (d) Negative phase contrast under NCSI conditions. (e) Oxygen column (arrow) for which the occupancy is set to 0. (f) Oxygen column (arrow) for which the occupancy is set to 0.5. (g) A local shift of the oxygen column by 50 pm is clearly seen and the effect of contrast does only affect this column, the neighbours remain unchanged. Any changes are purely local. From Jia, Lentzen, and Urban (2003). (See color insert).

electron wave. Thus, the results of atomic-resolution EM in general refer to lateral structural details. The term occupancy is defined as the fraction of atom sites in a column aligned parallel to the viewing direction. A fully occupied column has occupancy 1, whereas the value 0 appertains to a vacant column. The occupancy therefore should be considered the extreme case of a “concentration” where the range of the specimen for which this value holds is the atom column. Such occupancies may now be measured with single-atom resolution. Modern aberration-corrected TEM is on its way to improved depth resolution, exploiting tomographic techniques and the low depth of focus for highly convergent illumination (Bar Sadan et al., 2008; Midgley and Weyland, 2003; Nellist et al., 2008). However, presently the depth resolution generally does not match atomic-resolution standards.

B. From Images to Structure: The Inversion of the Scattering and Imaging Problem The availability of aberration-corrected instruments has opened a new era of TEM in which atomic-resolution studies may in principle be conducted in many materials. In fact, the first applications have demonstrated a wealth

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of information that can be obtained by studying the atomic structure (e.g., at crystal defects and interfaces). However, the world of atoms is that of quantum physics and the results obtained by TEM in the form of what we call “images” is usually not open for intuitive interpretation.1 These early studies also have demonstrated the need for a new view on TEM in atomic dimensions. This technique in general does not produce “images” in the conventional sense that these would directly display the sample structure. The human brain is trained to interpret black-white contrast automatically in terms of structure. But the brain’s reference is light optics and not the quantum mechanics of electrons interacting with an interatomic potential. It would be quite surprising if electron waves probing the Coulomb interaction potential would produce “images” that could be interpreted as if they would be produced by light and a type of light-matter interaction of a completely different nature. Unfortunately, electron microscopists for decades have nurtured oversimplified illusions about the real nature of their work. Understanding images produced by electron waves probing atomic structures is only possible by a “brain” that knows how to deal with quantum mechanics, and this “brain” is the computer. Therefore, atomicresolution TEM is based on a two-step process: (1) taking images in the microscope followed by (2) the subsequent evaluation of their contents by quantum-mechanical and optical image calculations. Therefore, strictly speaking atomic-resolution TEM does not provide “images” of a sample structure; it provides us with a computer model of the structure. Even when an “image” taken under certain conditions seems to show expected features (on the basis of preknowledge or some intuitive guess), the actual information contained in this “image” may only be elaborated when safely interpreted by the computer. This appears even more necessary when the imaging conditions under which the “image” was recorded are in general not known a priori. In particular, this holds for such basic parameters as the direction of incidence of the electrons and the sample thickness at the actual location of the sample investigated. We also note that a map of the phase part of the numerically retrieved complex electron EPWF usually does not show, as is often claimed, the specimen structure. This only holds, within certain limits, if the projected potential approximation is justified, but whether it is indeed justified (or not) must be proven by careful self-consistent computer treatment. Electron microscopists presenting their results to the public should clarify that the selection of a certain “image” (e.g., out of a focal series since it is suited for a sort of intuitive interpretation) is based extensively on computer 1 In this paragraph, the term “image” is set in inverted commas to emphasize the difference between

the everyday meaning of the word image and the intensity patterns produced in the image plane of an atomic-resolution transmission electron microscope.

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modeling. Without such modeling we cannot know for certain what the structure really is and what a given “image” is showing. The following text scrutinizes the inversion of the scattering and imaging process in more detail. As described in Section II.A., the basic problem of TEM is finding the structure of a specimen whose atomic arrangement is given by an object function of the form

ρ(r) =

i

ρi (r) ∗ δ(r − r i ).

We are interested not only in the set of atomic positions r i but also in the chemical nature of the atoms and the occupancy of certain atomic columns, which we hope to determine via the local variation of the atomic scattering potential. To solve this problem we use an electron wave field which, while it passes through the specimen, is modified by scattering—the interaction with the interatomic potential V(r). The EPWF is not directly accessible but it is the object of electron optics as it provides images (in the limited sense just discussed). Concluding backward from the images to the object structure entails the formidable task of inverting a highly nonlinear problem. The first step consists of deriving the EPWF on the basis of the electron optical images. A solution to this problem is provided by computer-based wave function retrieval techniques. There are a number of variants, but all have in common that the microscope is operated not as a conventional optical imaging system but as an interferometer, which exploits the fact that the objective lens introduces phase shifts into the electron waves. The technique most widely used is the focus-variation technique (Coene et al., 1992, Coene et al., 1996). Typically a series of 20 images is recorded by a charged-coupled device (CCD) camera while the objective lens focus is varied incrementally step by step. These images are transferred to the computer system, where the EPWF is calculated by least-square fitting to a set of simulated image series (Thust, 2006). In addition to defocus, all other lens aberrations also affect the image intensity distribution, which typically requires the handling of more than 10 parameters. These values must either be known from dedicated scattering experiments or compensated to negligible values before taking the image series (Barthel, 2007; Thust and Barthel, 2006; Uhlemann and Haider, 1998). Wave function retrieval allows compensation for the shortcomings of contrast transfer (Figure 2b), which deviates substantially from the “ideal” case of a uniform transfer that would be independent of spatial frequency. However, there is no means to retrieve the information on spatial frequencies that have not been transferred at all; that is, their effect on the interferograms is missing at all focus settings. This means that the EPWF is retrieved on

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the basis of a set of Fourier waves that is incomplete. In particular, this holds for the spatial-frequency range at very low g-values. Corresponding artifacts are indeed quite frequent. Despite some progress (Lentzen and Urban, 2000; Tang et al., 2006), no previous technique has allowed the calculation of the potential and the underlying structure “backward” from the EPWF. The solution to this problem is to do a “forward” calculation, in which the Schrödinger equation is solved numerically on the computer for a model structure based on a first guess and iteratively improved to obtain a best fit between the calculated and experimental EPWF. In addition to the formidable task of properly adjusting in the model the positions of a large number of atoms with atomic precision, this procedure is hampered by the fact that in atomic dimensions there is no direct access to such important imaging parameters as sample thickness and the precise direction of the incident electrons. There is no other solution but to treat these parameters as variables that also must be determined in the fitting procedure. The result generally is therefore not an image in the conventional sense but a computer model of the structure that gives the atomic species and coordinates.

C. Aspects of Practical Work The aforementioned procedures are facilitated if the structure is known in some areas of the imaged sample region because this information may be used as a reference. This is the case if defects in otherwise perfect structures are investigated. Also, the presence of various types of atoms with major differences in atomic scattering power accelerates the computer fit. In very thin samples, computer modeling benefits from using a tentative projected-potential approximation, according to which the maxima in maps of the phase part of the complex electron EPWF are considered as a projection of the atomic potential minima in the sample (Carter and Williams, 1996; Spence, 1988). This allows taking the precisely measured positions of the intensity minima (positive phase contrast) or maxima (negative phase contrast) in the images at a certain focus as a first approximation for the atomic positions in the model. However, as noted, whether the projected potential approximation is really valid must be explicitly shown a posteriori on the basis of a stable self-consistent solution of the scattering problem. Recall that the retrieved EPWF (apart from the possible occurrence of artifacts) reflects the “real” conditions. This introduces the impact of small deviations from the exact Laue orientation. These deviations arise from projection effects and changes in the electron wave field arising from the slightly modified projected potential. The local amplitude distribution of the EPWF also is sensitive to sample thickness. A self-consistent fit from

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the calculated to the experimental EPWF provides a value for both tilt and thickness. Another point with respect to the EPWF must be considered since it affects practical high-precision work. The precision of the reconstructed wave function cannot be better than an optimum determined by the precision to which the optical aberrations [see, for comparison, Eq. (9)] are measured. Measuring the aberrations by means of the technique of Zemlin tableaus of diffractograms (Uhlemann and Haider, 1998; Zemlin et al., 1978) requires an amorphous specimen on which the measurements are performed. In addition to the fact that such measurements are tainted with a limited accuracy, a priori errors may occur due to electronic or mechanical instabilities. Depending on the quality of the instrument, certain parameters (e.g., the defocus value) can vary at substantial rates with time. Even small shifts of the specimen position can change the aberration values appreciably. If the aberrations are measured with a special test sample, it is quite unlikely that after the actual specimen has been moved into place the aberrations at this very specimen position are identical with those measured when the test specimen was inserted. Even when an amorphous part of the specimen is used for aberration measurement, there is no guarantee that even the small specimen shift required to move the sample area of interest into the field of view will leave the aberrations unaffected. Even though time dependence of aberration-parameter values can be controlled by repeating the measurements in short intervals (e.g., before and after an image series is taken), effects of the actual position of the specimen on the values of the aberration parameters can be detected only by evaluating the real images taken from the specimen and scrutinizing them in terms of residual aberrations (Houben, Thust, and Urban, 2006). Practical work has shown that in the iterative procedure to match a wave function calculated on the basis of a structure model to the experimental EPWF, effects of twofold and threefold astigmatism, as well as axial coma, are detected, particularly if the structure is known for parts of the sample. These effects can be taken into account by using proper phase shifts to the calculated wave function in the framework of adopted numerical phase plates (Thust et al., 1996b) producing the artifacts observed in the experimental wave function. Finally, in some cases a complete focal series cannot be recorded because of dynamic specimen effects. Examples are electron beam illumination– induced atomic motion or radiation damage due to radiolysis irreproducibly changing the structure after a few images have been taken. Experience shows that in favorable cases the structure of the sample may be reconstructed in a forward calculation even for the extreme case where only a single image is available. This is feasible if an acquisition equivalent to the interferometric information can be obtained by exploiting the different effective extinction conditions pertaining to atom columns occupied

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by different types of atoms. Such an example involves perovskites where along high-symmetry orientations a number of differently occupied atomic columns occur from pure high-nuclear charge cation columns to columns exclusively occupied by oxygen. If heterostructures composed of different materials are investigated, the different structures and atomic species yield additional information, provided that the sample thickness is equal throughout the sample. We stress that in this special case the results can be considered reliable only after extensive cross-checks and tests of the set of data for self-consistency. An example of the different effective extinction distances of BaO and Ti atom columns in BaTiO3 imaged along a direction is shown in Figure 6 later.

IV. SELECTED MATERIALS SCIENCE APPLICATIONS A. Measurement of the Occupancy of Oxygen-Atom Columns in a twin Boundary of BaTiO3 Oxygen features eminently in materials that are of considerable technical importance. Such materials include the large group of electroceramic materials comprising dielectrics, ferroelectrics, piezoelectric materials, and ionic conductors (Dawber, Rabe, and Scott, 2005; Scott, 2007; Setter and Waser, 2000). The physical properties in all these materials, are sensitively controlled by the oxygen stoichiometry. In extended crystal defects (e.g., grain boundaries and dislocations) the oxygen occupancy of lattice sites can deviate substantially from the stoichiometric value either because of structural constraints or as a result of the interaction of charged mobile oxygen vacancies with the electrically active defects (Reˇcnik et al., 1994). Therefore a key issue is measuring the oxygen occupancy of individual lattice sites in or close to defects. The first and hitherto only technique that allows such measurements is based on the NCSI operation mode in aberration-corrected TEM (Jia, Lentzen, and Urban, 2003; Jia and Urban, 2004). This technique not only allows imaging of oxygen directly, but the intensity of the oxygen signal is, as shown below, linearly dependent on the oxygen occupancy over a wide range of imaging conditions. The BaTiO3 thin films investigated were grown by pulsed-laser deposition at 750◦ C on the platinum layer of Pt/Ti/SiO2 /Si heterostructure wafers in an oxygen atmosphere at 0.1 mbar. The instrument used for aberration-corrected imaging is the Jülich prototype, a Philips CM200 FEG microscope equipped with the Haider-Rose double-hexapole corrector and operated at an accelerating voltage of 200 kV. Figure 4a shows an image of a [110]-oriented BaTiO3 crystal.Embedded   in a “matrix” of crys¯ ¯ twin lamellae of tal orientation “I” are nanometer-scale 111 and 111 orientation “II” and “III,” respectively. A detailed analysis shows that the

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(a) [001] [110]

(b)

FIGURE 4 (a) Image of a [110]-oriented BaTiO3 crystal taken under NCSI conditions. Embedded in a “matrix” of crystal orientation I are nanometre-scale (¯111) and (1¯11) twin lamellae of orientation II and III, respectively. A detailed analysis shows that twins are of the type 3{111}. (b) The magnified image indicates that the oxygen-atom columns located between two Ti columns are imaged with strong contrast, both in the matrix and in the twin boundary plane (vertical black arrows). From Jia and Urban (2004).

twins are of the type 3{111}. The magnified image in Figure 4b indicates that the oxygen-atom columns (vertical arrows) located between two Ti columns are imaged with strong contrast, both in the matrix and in the twin boundary plane (diagonal arrows). The goal is to measure the atomic-column occupancy by quantitatively evaluating the local image intensity. To obtain a reliable result of the occupancy of a given atomic column, the image intensity measured at a given atomic position must be free of contributions arising from electron scattering at neighboring atomic columns. This condition is guaranteed only if the contrast delocalization defined by the radius of the point-spread function is sufficiently small. As described in Section II, one advantage of the NCSI mode is that contrast delocalization is rather small; under the conditions used in this work RS ≃ 80 pm. Therefore, the image intensity at

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the position of an oxygen site remains essentially unaffected by scattering at the neighboring Ti sites, 140 pm away, providing genuine atomic resolution. Another prerequisite for quantitative measurements is knowledge of how the image intensity of an individual column depends on oxygen occupancy. This relation may be obtained by numerical calculations solving the Schrödinger equation for different imaging conditions (Figure 5). Two arrows in the calculated image mark oxygen columns (B) for which the (a)

(b)

0.0

0.5 0.14 nm

Intensity (arb. units)

(c)

Z 5 11 nm t 5 2.5 nm t 5 3.1 nm t 5 3.7 nm

(d)

t 5 3.1 nm Z 5 9 nm Z 5 11 nm Z 5 13 nm

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Occupancy

FIGURE 5 (a) The structure of the 3{111} twin boundary in [110] projection. (b) Calculated image of the boundary (CS = 40 µm; specimen thickness 3.1 nm; defocus: +9 nm; semi-angle of beam convergence 0.2 mrad; defocus spread 5.6 nm). Two arrows mark oxygen columns with an occupancy of 0 and 0.5, respectively. The corresponding change in contrast is restricted to the column position itself, providing evidence for actual atomic resolution. (c) Plot of the image intensity of oxygen sites in the boundary plane versus occupancy calculated for thicknesses t of 2.5, 3.1, and 3.7 nm and for defocus Z = 11 nm. (d) Plot of the intensity versus oxygen-site occupancy calculated for a thickness of 3.1 nm and for the defocus values of 9, 11, and 13 nm, respectively. For an occupancy higher than 0.3, the intensity increases linearly with occupancy. (From Jia and Urban, 2004). (See color insert).

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occupancy is set to 0 and to 0.5. The corresponding change in contrast is restricted to the column position itself, providing evidence for actual atomic resolution. Figure 5c and d depict the calculated peak intensity of the oxygen columns as a function of occupancy for different sample thicknesses t and defocus values Z, respectively. The intensity depends both on thickness and defocus, and for occupancy values >0.3, increases linearly with occupancy. The use of this relation for oxygen-occupancy measurements presents at least two difficulties. First, to date it is barely possible to measure absolute intensities that could then be directly compared with calculated values. Therefore, only relative occupancy values can be determined, using the intensity of the oxygen signal in the matrix as a reference. Second, as described in Section II, neither the absolute value of the objective-lens defocus nor that of the sample thickness is known with sufficient accuracy. This problem is overcome by an iterative procedure, in which the intensities for a sample area far away from the boundaries are calculated and compared with the measured values. In particular, this method exploits the substantially different thickness dependences of the signal from the Ti- and Ba-containing atom columns for an accurate determination of the imaging parameters. The measurements are performed in three steps. For step I, the matrix area on the right-hand side of Figure 4a is studied. - . By integrating the ¯ direction over a disintensity values (at pixel resolution) along the 110 tance of ∼3.5 nm, one obtains an intensity function whose maxima are plotted (open symbols) in Figure 6 versus the distance from the top of Figure 4a (in units of the [001] lattice parameter of 0.4 nm). The intensity labeled “Ti-O2 ” in Figure 6 arises from both the Ti and the O2 columns. The Ti-O2 signal increases, whereas the BaO signal decreases with distance. In addition, corresponding steps in both intensity curves can be recognized. The solid symbols represent calculated values obtained for Z = 9 nm and the thickness values indicated. In these calculations the objective-lens defocus, the sample thickness, and the scaling factor (Hÿtch and Stobbs, 1994) required for a quantitative comparison of calculated and experimental image contrast were taken as variables and optimized iteratively until an optimum and stable fit to the entire range of experimental values was achieved. The results of Figure 6 can be understood if a wedge-shaped sample is assumed whose thickness increases roughly vertically in Figure 4a, from top to bottom. However, the thickness does not change continuously, but rather increases in steps of 0.3 and 0.6 nm in height. These values correspond to 2 and 4 , where is the thickness of a {110} atomic monolayer of BaTiO3 . The simultaneous measurement of both the Ti-O2 and the BaO signals permits an extremely accurate determination of thickness and defocus because, owing to the difference in effective extinction distance, the thickness dependence of the Ti

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(nuclear charge 22) intensity differs substantially from that of Ba (nuclear charge 56). In step II, in the area of the twin lamellae (containing the twin boundaries “1” to “6”) in Figure 4a, the individual intensity profiles of all Ti, BaO, and O2 sites are measured, with the exception of the oxygen sites in the boundary planes. From these data, an average peak value can be calculated for each of these three types of sites. Repeating the iterative computerbased fit further improves the overall fit to the data by the introduction of a deviation of 0.6 degrees from the exact Laue orientation. In this way, a simultaneous fit to all three peak values within an error of 2% is obtained. For the lamellae area, one obtains Z = 9.2 nm and t = 3.1 nm as parameters that need to be considered in the evaluation of the intensities measured for the oxygen sites in the boundary plane.

Thickness (nm) 1.9

2.5

Intensity (arb. units)

1.6

2.8

3.4

Ti-O2

BaO

0

10

20 Distance (unit cells)

30

FIGURE 6 Plot of an intensity function versus distance from top to bottom in Figure 4a (in units of the [001] lattice parameter of 0.4 nm). The function is obtained by integrating the intensity values - for . the matrix area on the right-hand side of Figure 4a along the horizontal 1¯10 direction over a horizontal distance of 3.5 nm. Black open squares, BaO; black open circles, Ti-O2 . The corresponding steps in the BaO and the Ti-O2 signals indicate that the sample thickness increases in steps of two and three atomic monolayers in height. The filled squares and filled circles depict values calculated for a defocus of 9 nm and the thicknesses indicated. The figure illustrates an example of the different effective extinction distances of Ba-O and Ti-O2 atom columns. While the Ti-O signal is increasing with increasing thickness, it is already decreasing for Ba-O. Such differences in contrast behavior are helpful for the accurate determination of the imaging conditions a posteriori. (From Jia and Urban 2004).

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This quantitative basis now permits, in step III, a determination of the occupancy of the individual oxygen sites in the boundary planes of the twins “1” to “6.” The data for boundary “4” are given as an example in Figure 7. The occupancy values calculated on the basis of the measured intensities range from 0.4 to 0.7. From the data obtained for all six boundaries, a mean value is obtained for the occupancy of 0.68 ± 0.02 at a standard deviation of 0.16. With respect to the measurement error, the scatter of the intensity values measured for the individual oxygen sites in the matrix area leads to essentially the same values for the deviation from the mean occupancy (assumed to be 1) and for the standard deviation. The measurements provide quantitative evidence for a substantial reduction of the oxygen occupancy, that is, the presence of oxygen vacancies in the 3{111} twin boundaries in BaTiO3 . On average, 68% of the boundary oxygen sites are occupied and the others, about one site out of three, are left vacant. The error in the mean occupancy can be accounted for by an atomic-scale specimen surface roughness. Also, despite the low sensitivity of BaTiO3 to electron radiation damage, a contribution of radiation-induced chemical disorder cannot be entirely ruled out. From the results of an electron energy loss spectroscopy (EELS) study of the near-edge structure of the L23 ionization edge of Ti in the 3{111} twin boundary, it was concluded that the Ti atoms adjacent to the boundary plane occur in an oxidation state

Occupancy

1.0

0.5

FIGURE 7 Oxygen occupancy in twin 4 (see Figure 4a for comparison) as a function of position. One of the oxygen positions is marked by an arrow. The occupancy values calculated on the basis of the measured intensities range from 0.4 to 0.7. From the data obtained for all six boundaries, a mean value for the occupancy of 0.68 ± 0.02 at a standard deviation of 0.16 is obtained. (From Jia and Urban, 2004.) (See color insert).

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lower than that, 4, in the stoichiometric compound (Reˇcnik et al., 1994). This suggests that the boundary plane is oxygen deficient. Since this is the first-ever measurement of its kind, it is interesting to compare the results with data obtained by other techniques. In the 3{111} twin boundary, the original TiO6 octahedra change from corner to face sharing, forming a Ti2 O9 group unit. This group is a genuine element of the BaTiO3 system because it is also the basic structural element of the hexagonal high-temperature phase formed from the cubic phase at 1460◦ C (Jia and Thust, 1999). In a study of the hexagonal phase by Grey et al. (1998) performed by X-ray scattering, tetravalent Ti+4 was replaced by trivalent Fe+3 . It was found that in this Ba(Ti1−x Fex )O3 phase, those sites [called O(1) sites] of the Ti2 O9 group unit that correspond to the oxygen sites in the 3{111} twin boundary plane are left partially vacant. When the iron content of the compound is varied, there is a maximum concentration of Fe+3 that may be accommodated. This corresponds to an occupancy in which one oxygen atom in three is missing. This corresponds favorably with the measured oxygen occupancy of 68% for the oxygen boundary sites. Thus, the incorporation of oxygen vacancies reduces the grain boundary energy and allows the system to react to oxygen-deficient conditions (e.g., during thin-film deposition) by the formation of a nanotwin lamella structure.

B. Structure of the 90 degree [100] Tilt Grain Boundary in YBa2 Cu3 O7 Whereas the previous example of an application of aberration-corrected TEM was selected for demonstration of the ability of this technique to measure occupancies of atomic columns, the present example serves as a demonstration of ultraprecise lateral atomic-distance measurements (Houben, Thust, and Urban, 2006). In the high-temperature oxide superconductor YBa2 Cu3 O7 the critical current density decreases by orders of magnitude if measured across a grain boundary compared with the bulk value of ∼107 Acm−2 at 4.2 K (Dimos et al., 1988; Gross and Mayer, 1991; Gross, 2005). Although the detailed mechanism responsible for this reduction is still under debate, it is generally accepted that the high anisotropy of the superconductivity in YBa2 Cu3 O7 confined to the two {001}-type crystallographic planes on both sides of the Y atom, together with the extremely short coherence length perpendicular to these planes, less than 1 nm, plays a crucial part in this. There have been a number of studies on the electronic properties of bicrystal and other grain-boundary Josephson junctions. Of the EM studies, none is actually atomically resolving and in particular oxygen could not be studied at all. Oxygen is particularly important since it is well known that oxygen is crucial for all oxide superconductors not only because of its electronic

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effects but also due to the fact that its high mobility allows oxygen to relax local stresses by changing its position in the lattice. The studied YBa2 Cu3 O7 films were deposited by high-pressure direct current sputtering on (001) NdGaO3 substrates in order to achieve [100]-axis growth. As a consequence, the c-axes of the two grains lie in the image plane. The instrument used for aberration-corrected imaging is again the Jülich prototype, a Philips CM200 FEG microscope equipped with the Haider–Rose double-hexapole corrector and operated at an accelerating voltage of 200 kV. Through-focus series for EPWF reconstruction were taken under NCSI conditions and recorded on a 1k × 1k CCD camera with a focal increment of 3.3 nm between successive exposures. A magnification yielding a sampling rate of 0.02 nm per pixel was chosen; that is an image discretization well below the Nyquist frequency of (2gI )−1 = 65 pm corresponding to twice the information limit gI−1 of 7 nm−1 . The focal range of each series included the special defocus setting of Z = +11.6 nm for optimized NCSI phase contrast [see Section 2, Eq. (17)]. The EPWF was retrieved for the frequency band between 1 and 8 nm−1 using the Philips/Brite–Euram software for focal-series reconstruction (Coene et al., 1996; Thust et al., 1996a). An a posteriori correction of residual aberrations was carried out on the basis of a phase plate derived in the fit between the calculated and the experimental EPWF for perfect YBa2 Cu3 O7 regions for which the structure is accurately known. Such a procedure is mandatory since practical issues (e.g., electronic instabilities and a frequently observed change in the twofold astigmatism upon change in the region of interest on the sample) leads to changes of the aberrations and deviations from previously adjusted values (as explained in Section III.C). Figure 8 displays the amplitude (a) and phase (b) of the complex EPWF. For noise reduction the images were averaged parallel to the boundary over five repeat cells. The image calculations yield a specimen thickness of 0.8–3.2 nm within which the simulated image patterns are fully compliant with the observance of brightest spots and darkest spots on the barium positions in the phase and the amplitude of the experimental EPWF, respectively. This means (as was checked by the calculations) that the weak-phase object approximation is fulfilled. Not all Cu atoms show identical contrast. A distinctively different local contrast pattern with a bright amplitude and a blurred phase peak is observed for the copper column at the Cu1 site (see Figure 12 for identification) in the grain boundary. For clarity, one of these sites is circled in Figure 8a. To quantify the atomic positions in the grain boundary, peak positions in the phase of the EPWF belonging to atomic columns are measured individually to subpixel accuracy by two different procedures. First, for the heavy-element columns (the Ba, Y, and Cu columns) a least-squares fit by single two-dimensional (2D) Gaussian profiles is applied. Second, the

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(a)

1 nm

(b)

465

1 nm

FIGURE 8 A 90 degree tilt grain boundary in YBa2 Cu3 O7 . Amplitude (a) and phase (b) of the complex exit-plane wave function. The structure is illustrated in a ball model (left insert). For noise reduction the images were averaged parallel to the boundary over five repeat cells. The image calculations (inserts lower right) yield a specimen thickness of 0.8 to 3.2 nm within which the simulated image patterns are fully compliant with the observance of brightest spots and darkest spots on the barium positions in the phase and the amplitude of the experimental exit-plane wave function, respectively. This means (as checked by the calculations) that the weak-phase-object approximation is fulfilled. One of the Cu atoms with particular contrast is marked by a circle in (a). (From Houben, Thust, and Urban, 2006.) (See color insert).

position of oxygen columns is evaluated by a simultaneous multiple Gaussian fit to line profiles extracted from the phase image. The second method is applied to account for the overlap between neighboring phase peaks. Peak overlap primarily affects the peak position of weaker scattering atoms (here, the oxygen) but is assured to be negligible for the stronger signals of the Ba, Y, and Cu columns. An example of a decomposition of a line profile into five Gaussian profiles is given in Figure 9. Error estimates for the position of a single phase peak were calculated by the Gaussian regression procedure. The standard deviation in the phase residuum image obtained after periodical averaging of the EPWF is taken as an estimate for the noise figure in the phase image. Knowledge of this noise figure allows for a reasonable calculation of the parameter confidence intervals in the nonlinear regression procedure. As a result, the position of a single column of cations is measured within a 2σ error radius of 2 pm demanding a statistical confidence of 95%. The distance between the phase peaks of two cation columns is measured within a 2σ error radius of 4 pm. The 2σ error radius for the oxygen positions is larger due to the smaller signal and the overlap with the tails of neighboring phase peaks of the heavier cation columns. The 2σ error radii for the oxygen positions are ∼6 pm in the bulk. Close to the grain boundary plane, the error radii become larger, up to 20 pm, due to a decrease in the distance to the neighboring cation columns.

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0.3

O2, Cu2 O1 Cu1 O1 Cu2, O2

Phase (rad)

0.2 0.1 0

20.1 G1 20.2

20.5

G2 G3 G4 0 x (nm)

G5

Experimental phase data Sum of Gaussians Residuum

0.5

FIGURE 9 An example of a nonlinear least-square regression to a phase profile along the grain boundary plane. Five Gaussians G1 ;. . .; G5, centered at the atomic positions, were fitted to the experimental phase data. The dashed curve represents the difference between the sum of the fitted Gaussians and the measured phase profile. The grey-shaded band superimposed to the graph shows the ±1σ-width for the noise in the phase image. (From Houben, Thust and Urban, 2006.)

The aforementioned error estimates reflect the statistical error correlated to the residual noise in the phase image. Additionally, in this case systematic errors are due to imaging artifacts. Although imaging artifacts due to residual aberrations are widely eliminated from the retrieved EPWF, the finite range of spatial frequencies, determined by the information limit of the microscope, causes a convolution of the wave function. This convolution may lead to an apparent shift of phase maxima with respect to the real atom positions (Jia and Thust, 1999). As a consequence, as column spacings get closer and closer to the information limit gI−1 , the measured distances of the intensity maxima deviate from the actual atomic separations. In this particular example, the error is negligible for the Ba, Y, and Cu columns with their relatively large spacings of ∼0.4 nm. More care is needed for the closely spaced oxygen and cation columns. For these cases, EPWFs were simulated for a structure model based on the measured atom positions with a resolution comparable to the experiment. The shift of phase maxima in the simulated wave functions induced by the finite resolution becomes significant only for the smallest oxygen–copper spacings of ∼160 pm in the grain boundary. A separate refinement of these spacings was performed by measuring the apparent column positions for simulated wave functions. The analysis shows that the spacings are underestimated by an artificial shift of the oxygen phase peak of up to 8 pm in the worst case. This systematic error is still significantly smaller than the 2σ radius for the statistical error for the position of an oxygen column in the grain boundary.

Atomic-Resolution Aberration-Corrected Transmission Electron Microscopy

Cu2, O2 O1 Cu1

p2

155 155 pm pm

exp. gb

0.5

p1 gb

Domain II

p1 p2 p...

0

20.5

234 184 pm pm

20.5

0 x (nm)

Domain I

Phase (rad)

1

467

calc. 0.5

1

Y Ba Cu O

FIGURE 10 Line profiles referring to the phase of the exit-plane wave function parallel to the grain boundary and parallel to the [001] direction through successive Cu1-O1-Cu2-O2 planes in the upper domain II. For details see text. (From Houben, Thust, and Urban, 2006.) (See color insert).

Figure 10 shows line profiles referring to the phase of the EPWF parallel to the grain boundary and parallel to the [001] direction through successive Cu1-O1-Cu2-O2 planes in the upper domain II. The atomic sites can be identified in the schematic drawing in Figure 12. Qualitatively, the grain boundary phase profile exhibits a smaller peak amplitude on the Cu1 site and a reduced Cu1-O1 distance compared with the periodic case approach of the neighboring oxygen on O1 sites toward this Cu1 site. Quantitative data for the bond lengths Cu1-O1 and Cu2-O1 can be obtained from the line profiles. The symmetry-related curves coincide within the statistical error. This symmetry conservation confirms that the removal of aberrations from the wave function was successful. In any plane but the grain boundary plane, the measured Cu2-O1 and Cu1-O1 phase peak separation of 234 ± 6 pm and 184 ± 6 pm is marginally different from the nominal bond lengths of 230 pm and 186 pm in the periodic structure. In the grain boundary plane, however, a significant shift of the O1 column toward the Cu1 column by ∼30 pm is measured and the Cu1-O1 phase peak separation reduces to 155 ± 20 pm. A contraction of the Cu1-O1 bond length is also measured perpendicular to the grain boundary plane. Figure 11 shows experimental and simulated phase profiles perpendicular to the grain boundary through the Cu1 site. A shift of the O1 column in domain I toward the Cu1 column in the grain boundary by ∼18 ± 10 pm is observed. The change in the Cu1−O1 bond lengths parallel and perpendicular to the grain boundary indicates the significant distortion of the square oxygen coordination around the Cu1 site.

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243 pm 166 pm

184 pm

55%

100%

0.2

Cu1 O4

0

exp.

0%

20.2 20.4 20.6

calc.

22

21

0 y (nm)

1

Domain I

Phase (red)

0.4

Domain II

Domain II

Cu2, Cu1 Domain I O2 O1

0.6

Y Ba Cu O

FIGURE 11 Experimental and simulated phase profiles perpendicular to the grain boundary through the Cu1 site. A shift of the O1 column in domain I toward the Cu1 column in the grain boundary by approximately 18 ± 10 pm is observed. The change in the Cu1-O1 bond lengths parallel and perpendicular to the grain boundary indicates the significant distortion of the square oxygen coordination around the Cu1 site. (From Houben, Thust, and Urban, 2006.) (See color insert).

In Figure 11, the phase maxima on oxygen columns in the bariumoxygen plane, (that is, on the O1 site) in domain I and on the oxygen columns in the copper-oxygen chain in domain II (that is, on the O4 site) are significantly different. An occupancy of 66 ± 11% is measured from the average amplitude ratio of the phase peaks of several 10 O4 and O1 oxygen columns. Figure 13 shows a geometrical model of the grain boundary in which all measured column displacements are summarized. Arrows indicate the direction and numbers specify the magnitude of a displacement relative to the position in the periodic structure of either. The adaptation of the spacing between copper-oxygen lattice planes in domains I and II results mainly in a distortion of the square pyramidal oxygen coordination of the Cu2 sites in and close to the grain boundary plane and the approximate square oxygen coordination of the Cu1 site in the grain boundary plane. (We point out the very high measurement accuracy of better than 5 pm.) To analyze the origin of the distinct amplitude and phase signal in the EPWF at the Cu1 site in the grain boundary, exit-plane waves simulated for the reconstructed grain boundary structure were compared with the experimental results. A close pattern match between experiment and simulation can be attained only by assuming displacement disorder on the Cu1 site. Both the amplitude and the phase pattern match when an

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(b)

(a)

0.415 nm

0.415 nm

0.338 nm

Ba

Grain boundary Y

Cu2

O3

O2 O1

Cu1

O4

O5 0.382 nm

FIGURE 12 (a) Model of the structure of the YBa2 Cu3 O7 unit cell. Cu atoms are labelled green, Ba orange, Y red, and O blue, respectively. Oxygen atoms are ordered occupying the O4 sites along the [010] axis. O5 shows schematically that along the viewing direction the two orientations (with a- or b-axis parallel to the viewing direction) occur. (b) Schematic of the grain-boundary area on the basis of the undistorted structure above and below the boundary. Upon joining the two domains, the shaded area disappears and the two Cu2 sites below the Y atom will coincide with the respective Cu1 sites above the central Ba atom. The resulting stresses are accommodated by atom rearrangements, depicted in Figure 11. From Houben, Thust, and Urban (2006). (See color insert).

effective Debye–Waller factor is assumed, which is increased by an order of magnitude, corresponding to an increase in the mean atom displacement by roughly a factor of 3 from 8 pm to 25 pm. Changes in the bond lengths of the plane copper Cu2 and the chain copper Cu1 to the apical oxygen O1 have been reported to reflect an internal charge redistribution between the plane copper and the chain copper and to affect the superconducting properties in bulk YBa2 Cu3 O7 (Cava et al., 1990).

C. Structure and Polarization of Ferroelectric Domain Walls in PZT The final example of work with aberration-corrected instruments is selected to demonstrate that accuracy in the picometer range allows for the high-precision measurement of atomic displacements, which in turn

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Cu Ba

O4 Cu1

Y

Ba

9±4 29 ± 20

38 ± 26

O1 18 ± 10

6±4

6±4

Cu1

Cu2 6±4

16 ± 12

Ba

11 ± 4

Ba

FIGURE 13 Geometrical model of the grain boundary, where all measured column displacements are summarised. Arrows indicate the direction and numbers specify the magnitude of a displacement relative to the position in the periodic structure. Note the very high measurement accuracy of better than 5 pm. From Houben, Thust, and Urban (2006). (See color insert).

permits the derivation of values for physical parameters on the atomic level (Jia et al., 2008). Ferroelectric thin films find potential applications in electronic and electro-optical devices, including, nonvolatile and high-density memories, thin-film capacitors, and piezoelectric and pyroelectric devices (Dawber, Rabe, and Scott, 2005; Setter and Waser, 2000; Scott, 2007). In bulk ferroelectrics, the paraelectric to ferroelectric phase transition is accompanied by the formation of polarization domains to minimize the system energy with respect to the depolarization field and mechanical strain. Because of the six equivalent cubic directions that can be chosen as directions for the polarization dipoles in PbTiO3 and Pb(Zrx Ti1−x )O3 , both 90 degree and 180 degree domain walls occur, where, on passing the wall from one domain to the other, the polarization vector changes direction by 90 degree and 180 degree, respectively. The 180 degree domain walls are particularly important for the understanding of polarization switching under an external electrical field (Gysel et al., 2006; Jung et al., 2002; Roelofs et al., 2001). A number of EM studies have been conducted on domain boundaries in ferroelectric materials (for references, see Jia et al., 2008). However, because of instrument-based limitations these investigations had to be performed at a resolution far from an atomic level. The first atomically resolving investigation of domain walls was done by Jia et al. (2008). In this work, the atomic details and dipole distortion were studied on the atomic scale near 180 degree inversion domain walls in thin epitactic films of PbZr0.2 Ti0.8 O3 (PZT) sandwiched between two SrTiO3 layers.

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The work was done using the Titan 80-300 instrument (equipped with an image corrector) of the Ernst Ruska Centre operated at an electron energy of 300 keV. The aberration measurement and the hardware correction was carried out directly before and directly after recording the images, demonstrating that these conditions remained unaltered during recording the images in NCSI mode. Because of PZT’s strong sensitivity to electron radiation damage, the adjustment of the imaging conditions had to be carried out close to the region of interest to shift to this region only for taking the images. Single-image analysis was performed because a focal series could not be taken without the risk of damaging the structure. Single-image analysis is facilitated by the fact that one is dealing with an epitactic twomaterial system: PZT with two types of polarization domains and SrTiO3 . The different effective extinction distances of three differently occupied atomic columns in PZT and in SrTiO3 and the different geometry of the corresponding atom columns can be exploited to obtain a self-consistent picture of the situation. All images were suitably Fourier-filtered to eliminate noise originating from amorphous surface layers and possible effects of radiation damage. The triple-layer films of SrTiO3 /PbZr0.2 Ti0.8 O3 /SrTiO3 investigated were grown on (001) SrTiO3 substrates by pulsed-laser deposition (Vrejoiu et al., 2006). Figure 14a shows the approximately 10-nm-thick PZT layer between the two SrTiO3 layers in a cross-sectional geometry. The view- . ¯ direction. The interfaces ing direction is along the crystallographic 110 are marked by two horizontal arrows. The two insets show magnifications overlying the respective areas in the upper left (domain I) and the lower right side of the figure (domain II). Yellow symbols denote PbO atom columns seen end-on. Red symbols denote Zr/Ti columns, and blue symbols represent oxygen. As illustrated by the schematic diagrams displayed in Figure 14b and c, the atom arrangement exhibits shifts of the atoms with respect to the cubic perovskite structure. In domain I (left inset), the Zr/Ti atom columns are shifted upward (along the [001] direction) toward the upper PbO positions and away from the respective lower ones. The oxygen atoms are also shifted upward but more strongly and thus are no longer collinear with the Zr/Ti atoms. This results in reduced symmetry of the atom arrangement from cubic to tetragonal with the fourfold axis parallel to [001]. The modified atom arrangement leads to a separation of the center of the anionic negative charge of oxygen from that of the cationic positive charge of the metal cations. The corresponding charge dipoles define the direction of the vector of spontaneous polarization P S (pointing - . from net negative to net positive charge) parallel to 001¯ . The atom shifts in domain II (right inset) are in the opposite direction compared with those in domain I. In fact Figure 14a displays two inversion polarization domains where, on changing from one domain to the other, the polarization vector changes by 180 degree.

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(a)

SrTiO3

I

Ps

PbZr0.2Ti0.8O3

II [001]

Ps

[110]

2 nm

SrTiO (b)

(c) ␦Zr/Ti

c

␦O

001

a

110

FIGURE 14 (a) Image of a SrTiO3 /PbZr0.2 Ti0.8 O3 /SrTiO3 thin-film heterostructure. The image - was . recorded under NCSI conditions with the incident electron beam parallel to the ¯110 direction. The atom columns appear bright on a dark background. The horizontal arrows denote the horizontal interfaces between the PbZr0.2 Ti0.8 O3 and the top and the bottom SrTiO3 film layers. The dotted line traces the 180◦ domain wall. The arrows denoted by ‘P S ’ show the directions of the polarisation in the 180 degree domains. The insets show magnifications of the dipoles formed by the displacements of ions in the unit cells (yellow: PbO, red: Zr/Ti, blue: O). (b) Schematic perspective view- of .the unit cell of ferroelectric PbZr0.2 Ti0.8 O3 . (c) Projection of the unit cell along the ¯110 direction. δZr/Ti and δO denote the shifts of the Zr/Ti atoms and the O atoms, respectively, from the centrosymmetric positions. From Jia et al. (2008). (See color insert).

With respect to the habit plane, three types of inversion domain wall morphology are observed. In the first (horizontal segments in Figure 14a), in the henceforth termed longitudinal domain wall (LDW), the normal of the domain-wall plane is parallel to the polarization vector in the two domains. Across this type of wall, the electric dipoles are head-to-head or tail-to-tail, resulting in a nominally charged domain wall. In the second type, termed transversal domain wall (TDW), the normal is perpendicular to the polarization vectors, so that the domain wall is head-tail coupled

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PS

473

PS

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FIGURE 15 Mixed-character domain wall (dotted vertical line) consisting of TDW segments, whose habit plane is parallel to the (110) plane. Their vertical extension is c, 2c, and 3c, where c denotes the c-axis lattice √ parameter. These are separated by short (001) LDW segments of horizontal width a 2/2, where - . a denotes the a-axis lattice parameter. The TDW segments run parallel to the ¯110 viewing direction and are seen edge-on. As indicated by the vertical shift of the oxygen atoms, down on the left-hand side and √ up on the right-hand side, the width of the wall is just a single projected unit cell (a 2/2). From Jia et al. (2008). (See color insert).

and not charged. Most frequently observed are planar and curved domain walls of mixed character where the habit plane normal makes an arbitrary angle with the polarization vectors. A closer inspection shows that this type of domain wall consists of LDW and TDW segments. Figure 15 shows a mixed character domain wall (dotted vertical line) consisting of TDW segments, whose habit plane is parallel to the (110) plane. Their vertical extension is c, 2c, and 3c, where c denotes the c-axis lattice parameter. √ These are separated by short (001) LDW segments of horizontal width a 2/2, where a denotes - . the a-axis lattice parameter. The ¯ TDW segments run parallel to the 110 viewing direction and are seen edge-on. As indicated by the vertical shift of the oxygen atoms, “down” on the left-hand side and “up” on the √ right-hand side, the width of the wall is just a single projected unit cell (a 2/2). This is corroborated by quantitative measurements of the vertical displacements δZr/Ti of the Zr/Ti and δO of the O atoms (defined in Figure 14c). Although no displacement can be measured in the wall center (dotted yellow line in Figure 15), the values approach the normal ones (far from the wall) of δZr/Ti ≃ 0.01 nm and δO ≃ 0.038 nm within the next projected unit cell (dotted bright line) on either side of the wall. The observation that the width of the TDW is of the order of one unit cell is in excellent agreement with results of first-principles totalenergy calculations ( Meyer and Vanderbilt, 2002; Pöykkö and Chadi, 1999) and with a calculation based on the scaling law for ferroic stripe domains (Catalan et al., 2007), indicating an abrupt change of polarization direction across the wall.

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FIGURE 16 Image of a LDW. The horizontal arrows indicate the geometrical centre of the wall. From Jia et al. (2008). (See color insert).

Figure 16 shows an LDW. The horizontal arrows indicate the geometric center of the wall. The following steps are taken for quantitative measurements of the lattice parameters and the atomic shifts as a function of vertical distance from the wall center. First, a least-square fit by 2D Gaussian profiles to each of the individual atomic intensity profiles in the figure is carried out (Houben, Thust, and Urban, 2006). The position of all the maxima of the Gaussians defines the experimental data set of atomic positions in the image. On this basis, all of the parameters defined in Figure 14c (the c- and a-axis lattice parameters, as well as the two shift parameters δZr/Ti of the Zr/Ti and δO of the O atoms) can be determined for each of the unit cells shown in Figure 16. If one is interested only in the behavior of these parameters as a function of distance from the domain wall center, one can calculate for a certain distance from the central plane a mean value by taking the average of the contrast-maximum position data parallel to the domain wall over the horizontal width of Figure 16. However, for ultrahigh-precision measurements, one considers that owing to residual objective-lens aberrations and an unavoidable small deviation of the crystallographic zone axis from the direction of the incident electron beam, the positions of the contrast maxima in the image may deviate slightly from the real atomic positions. To eliminate these artifacts, a model structure of the domain wall is constructed and the image expected from this structure is calculated using quantum mechanical and optical image simulations in which the imaging parameters are treated as variables. An iterative procedure is carried out in which the model and the imaging parameters are adjusted until a best fit between calculated and experimental data is reached.

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FIGURE 17 Quantities of the structural and electric behavior of the LDW as a function of the distance expressed in units of c from the central plane of the LDW shown in Figure 16. (a) shows the c-axis lattice parameter. Blue squares and red squares show the values measured from Pb to Pb atom positions and from Zr/Ti to Zr/Ti, respectively. (b) a-axis lattice parameter. Blue circles and red circles show the values measured from Pb to Pb atom positions and from Zr/Ti to Zr/Ti, respectively. Gaussian regression analysis indicates a measurement error of better than 5 pm. (c) Tetragonality c/a calculated from (a) and (b). (d) The displacements of the Zr/Ti atoms (δZr/Ti ) and the O atoms (δO ) across the LDW. Positive values denote upward shifts and negative values downward shifts. (e) The spontaneous polarisation PS calculated assuming the c-lattice parameters and the atomic displacements shown in (a) and (d). The positive values represent upward polarisation and the negative values downward polarisation. From Jia et al. (2008).

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The following shows the data of the structure determined in this manner. In Figure 17a, blue squares denote measurements of the c-axis lattice parameter carried out on PbO atomic positions. Red squares denote measurements on Zr/Ti positions. The distances are calibrated with reference to the lattice parameter (0.3905 nm) of the SrTiO3 layers. The minimum value of c occurs in the domain wall center with an accompanying increase of the a-axis lattice parameter (shown in Figure 17b). With increasing distance from the central area, c increases adopting a plateau value of ∼0.413 nm in domain I. In domain II this value is 1.3% higher. This results in a maximum tetragonality value of c/a = 1.059 in domain I and 1.072 in domain II (Figure 17c). The value for the Zr/Ti separation in the wall center adopts a local maximum (open square in Figure 17a). This is a geometrical effect because the two inner Zr/Ti atom rows belonging to different domains are shifted in opposite directions. In addition, in the central area of four to five unit cells in width, the c spacings between the Zr/Ti atom columns are larger than those between the PbO columns. This is due to the continuous increase of the Zi/Ti shift away from the center. In this area, the average value of c is ∼0.397 nm and that of a is ∼0.395 nm, leading to a value of 1.005 for c/a. Figure 17d shows the values of the vertical displacements δZr/Ti and δO as a function of the vertical separation from the domain wall plane. These displacements show essentially the same behavior as c—they follow the tetragonality, with the exception of the central area of four to five unit cells, where the increase of the displacements is not accompanied by an increase of the c value. Considering the variation of c, one infers a width of the LDW of 3c (domain I) plus 7c (domain II), in total about 10 lattice constants, considerably wider than the respective value for the TDW. Figure 17e shows the spontaneous polarization versus distance from the central plane of the domain wall. The values of P S are calculated on the basis of the c-axis lattice parameters and the atomic displacements shown in Figure 17a and d and the effective charge values of the ions for PbTiO3 given by Zhong, King-Smith, and Vanderbilt (1994). The maximum value of the modulus of P S is ∼75 µC cm−2 for domain I and ∼80 µC cm−2 for domain II. Inside the domain wall, the polarization changes direction. Interestingly, in the central area four to five unit cells wide, the polarization increases from zero to ∼40 µC cm−2 , whereas the tetragonality (c/a = 1.005) remains essentially constant. Irrespective of their lengths, all of the LDW segments observed in the studied PZT films are of the head-to-head type. The fact that these walls have a width of the order of 10 unit cells can be understood in terms of their particular charge status. The system energy can be reduced by distributing the polarization charge over an extended thickness of the wall. This is supported by the strong reduction of the c-axis lattice parameter and therefore the tetragonality in the wall area.

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As described in Section IV.B, a detailed Gaussian regression analysis by Houben, Thust, and Urban (2006) revealed that the accuracy of the measurements on the 90 degree tilt grain boundary in YBa2 Cu3 O7 is ∼4 pm for a 95% confidence level. This work was carried out in an aberration-corrected microscope with an information limit corresponding to dI = 0.12 nm. The present work was carried out in an instrument with dI < 0.080 nm. Therefore the accuracy should be even better, in the order of 2 pm. A crucial test of whether this accuracy is actually realized in the current work is the behavior of the a-axis lattice parameter shown in Figure 17b. The dashed horizontal line traces the position of the values calculated using a model for the domain wall in which the a-axis lattice parameter is assumed to be constant independent of the distance of the unit cells from the wall. The straight line is for the central points (close to the domain wall) located just at the limit of the 95% confidence level error bars. The slight expansion of the a-axis lattice parameter is seen not only in the measurements but also in the model structure in which this parameter was varied until an optimum fit to the experimental results was achieved. Regarding Figure 16 at a glancing angle from bottom to top this bowing out of the lattice planes along the horizontal [011] direction is detected even by the unaided eye.

V. CONCLUSIONS The introduction of aberration correction into electron optics by Haider et al. (1998) has set TEM on an entirely new track. Today’s state of the art can be characterized by the fact that real atomic-resolution studies can be performed using these techniques and that individual lateral shifts of atoms can be measured at an accuracy of a few picometers. This opens up new opportunities for materials science where precise values for physically relevant parameters can be determined on the atomic level by measuring atomic shifts, atomic column for atomic column. On the other hand, work with aberration-corrected optics also highlights the demands of these new techniques. The three examples selected from recent work in Jülich describe high-precision measurements of local occupancies (concentrations) atom column for atom column in delocalization-free images, of atomic shifts in a grain boundary, and of the polarization-induced atomic shifts in a ferroelectric material. These examples have been described in detail to illustrate the effort required to produce reliable results and the need for careful cross-checks in understanding the images. Atomic-resolution aberration-corrected EM provides with new insight into atomic structures at a time when nanophysics and technology urgently need quantitative information on physical phenomena and their structural background. In addition, nanotechnology will benefit from the ability to

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analyze structures in quantitative detail down to the picometer range. This is possible only if we take into account the genuine features of quantum mechanics. Scientists working with these formidable new machines should beware of the danger that the need to show “images” for evidence may lead to inadequate oversimplifications in the reception of the work in the general public. First, electrons are seeing the atomic world according to their own laws of interaction with matter. Understanding the information offered by the electron waves about the inner structure of the specimen is a highly nontrivial task. This will be underscored with greater numbers of such ultrahigh-resolution studies within the next few years. These studies also demonstrate the enormous potential of the new electron optics and the wonderful science waiting for us “at the bottom” (Feynman, 1959).

REFERENCES Bar Sadan, M., Houben, L., Wolf, S. G., Enyashin, A., Seifert, G., Tenne, R., and Urban, K. (2008). Toward atomic-scale bright-field electron tomography for the study of fullerenelike nanostructures. Nano Lett. 8, 891–896. Barthel, J. (2007). Ultra-precise measurement of optical aberrations for sub-Ångström transmission electron microscopy. Thesis, RWTH Aachen University. Carter, C. B., and Williams, D. B. (1996). Transmission Electron Micorscopy: A Textbook for Materials Science, Plenum Press, New York and London. Catalan, G., Scott, J. F., Schilling, A., and Gregg, J. M. (2007). Wall thickness dependence of the scaling law for ferroic stripe domains. J. Phys. Condens. Matter 19, 022201-1–022201-7. Cava, R. J., Hewat, A. W., Hewat, E. A., Batlogg, B., Marezio, M., Rabe, K. M., Krajewski, J. J., Peck W. F. Jr., and Rupp L. W. (1990). Structural anomalies, oxygen ordering and superconductivity in oxygen deficient Ba2 YCu3 Ox . Physica C 165, 419–433. Coene, W., and Jansen, A. J. E. M. (1992). Image delocalisation and high resolution transmission electron microscopic imaging with a field emission gun. Scan. Microsc. 6(Suppl), 379–403. Coene, W., Janssen, G., Op de Beeck, M., and Van Dyck, D. (1992). Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy. Phys Rev Lett. 69, 3743–3746. Coene, W. M. J., Thust, A., Op de Beeck, M., and Van Dyck, D. (1996). Maximum-likelihood method for focus-variation image reconstruction in high resolution transmission electron microscopy. Ultramicroscopy 64, 109–135. Cowley, J. M., and Moodie, A. F. (1957). The scattering of electrons by atoms and crystals, I. A new theoretical approach. Acta Crystallogr., 10, 609–619. Dawber, M., Rabe, K. M., and Scott, J. F. (2005). Physics of thin-film ferroelectric oxides. Rev. Mod. Phys. 77, 1083–1130. Dimos, D., Chaudhari, P., Mannhart, J., and LeGoues, F. K. (1988). Orientation dependence of grain-boundary critical currents in YBa2 Cu3 O7−δ bicrystals. Phys. Rev. Lett. 61, 219–222. Feynman, R. P. (1959). There’s plenty of room at the bottom. An invitation to enter a new field of physics [lecture at Caltech]; http://www.zyvex.com/nanotech/feynman.html. Grey, I. E., Li, C., Cranswick, L. M. D., Roth, R. S., and Vanderah, T. A. (1998). Structure analysis of the 6H–Ba(Ti, Fe+3 , Fe+4 )O3−δ solid solution. J. Solid State Chem. 135, 312–321. Gross, R. (2005). Grain boundaries in high temperature superconductors: a retrospective view. Physica C 432, 105–115.

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Gross, R., and Mayer, B. (1991). Transport processes and noise in YBa2 Cu3 O7−δ grain boundary junctions. Physica C 180, 235–242. Gysel, R., Stolichnov, I., Setter, N., and Pavius, M. (2006). Ferroelectric film switching via oblique domain growth observed by cross-sectional nanoscale imaging. Appl. Phys. Lett. 89, 082906-1–082906-3. Haider, M., Uhlemann, E., Rose, H., Kabius, B., and Urban, K. (1998). Electron microscopy image enhanced. Nature 392, 768–769. Houben, L., Thust, A., and Urban, K. (2006). Atomic-precision determination of the reconstruction of a 90° tilt boundary in YBa2 Cu3 O7 by aberration corrected HRTEM. Ultramicroscopy 106, 200–214. Hÿtch, M., and Stobbs, W. (1994). Quantitative comparison of high resolution TEM images with image simulations. Ultramicroscopy 53, 191–203. Jia, C. L., Mi, S. B., Urban, K., Vrejoiu, I., Alexe, M., and Hesse, D. (2008). Atomic-scale study of electric dipoles near charged and uncharged dopmain walls in ferroelectrics. Nat. Mater. 7, 57–61. Jia, C. L., and Thust, A. (1999). Investigation of atomic displacements at a 3{111} twin boundary in BaTiO3 by means of phase-retrieval electron microscopy. Phys. Rev. Lett. 82, 5052–5055. Jia, C. L., and Urban, K. (2004). Atomic-resolution measurement of oxygen concentration in oxide materials. Science 303, 2001–2004. Jia, C. L., Lentzen, M., and Urban, K. (2003). Atomic-resolution imaging of oxygen in perovskite ceramics. Science 299, 870–873. Jia, C. L., Lentzen, M., and Urban, K. (2004). High-resolution transmission electron microscopy using negative spherical aberration. Microsc. Microanal. 10, 174–184. Jung, D. J., Dawber, M., Scott, J. F., Sinnamon, L. J., and Gregg, J. M. (2002). Switching dynamics in ferroelectric thin films: an experimental survey. Integrat. Ferroelectr. 48, 59–68. Lentzen, M., and Urban, K. (2000). Reconstruction of the projected crystal potential in transmission electron microscopy by means of a maximum-likelihood refinement algorithm. Acta Cryst A 56, 235–247. Lentzen, M., Jahnen, B., Jia, C. L., Thust, A., Tillmann, K., and Urban, K. (2002). Highresolution imaging with an aberration-corrected transmission electron microscope. Ultramicroscopy 92, 233–242. Lentzen, M. (2004). The tuning of a Zernike phase plate with defocus and variable spherical aberration and its use in HRTEM imaging. Ultramicroscopy 99, 211–220. Lentzen, M. (2006). Progress in aberration-corrected high-resolution transmission electron microscopy using hardware aberration correction. Microsc. Microanal. 12, 191–205. Lichte, H. (1991). Optimum focus for taking electron holograms. Ultramicroscopy 38, 13–22. Majorovits, E., Barton, B., Schultheiß, K., Pérez-Willard, F., Gerthsen, D., Schröder, R. R. (2007). Optimizing phase contrast in transmission electron microscopy with an electrostatic (Boersch) phase plate. Ultramicroscopy 107, 213–226. Meyer, B., and Vanderbilt, D. (2002). Ab initio study of ferroelectric domain walls in PbTiO3 . Phys. Rev. B 65, 104111-1–104111-11. Midgley, P. A., and Weyland, M. (2003). 3D electron microscopy in the physical sciences: the development of Z-contrast and EFTEM tomography. Ultramicroscopy 96, 413–431. Nellist, P. D., Cosgriff, E. C., Behan, G., and Kirkland, A. I. (2008). Imaging modes for scanning confocal electron microscopy in a double aberration-corrected transmission electron microscope. Microsc. Microanal. 14, 82–88. Pöykkö, S., and Chadi, D. J. (1999). Ab initio study of 180° domain wall energy and structure in PbTiO3 . Appl. Phys. Lett. 75, 2830–2832. Reˇcnik, A., Bruley, J., Mader, W., Kolar, D., and Rühle, M. (1994). Structural and spectroscopic investigation of (111) twins in barium titanate. Phil. Mag. B 70, 1021–1034.

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Reimer, L. (1984). “Transmission Electron Microscopy.” Springer, Berlin. Roelofs, A., Pertsev N. A., Waser, R., Schlaphof, F., and Eng, L. M. (2002). Depolarizing-fieldmediated 180 degree switching in ferroelectric thin films with 90 degree domains. Appl. Phys. Lett. 80(8), 1424–1426. Scherzer, O. (1949). The theoretical resolution limit of the electron microscope. J. Appl. Phys. 20, 20–29. Scott, J. F. (2007). Applications of modern ferroelectrics. Science 315, 954–959. Setter, N., and Waser, R. (2000). Electroceramic materials. Acta Mater. 48, 151–178. Smith, D. J. (1997). The realization of atomic resolution with the electron microscope. Rep. Prog. Phys. 60, 1513–1580. Smith, D. J. (2008). Development of aberration-corrected electron microscopy. Microsc. Microanal. 14, 2–15. Spence, J. C. H. (2003). “High-Resolution Electron Microscopy.” Oxford University Press, New York. Tang, C. Y., Chen, J. H., Zandbergen, H. W., and Li, F. H. (2006). Image deconvolution in spherical aberration-corrected high-resolution transmission electron microscopy. Ultramicroscopy 106, 539–546. Thust, A., and Barthel, J. (2006). Ultra-precise measurement of residual aberrations for deepsub-Ångström HRTEM. Proc. International Microscopy Cong. (IMC16), Sapporo, Japan, p. 618. Thust, A. (2006). TrueImage [software package]. FEI Company, Eindhoven, Netherlands. Thust, A., Coene, W. M. J., Op de Beeck, M., and Van Dyck, D. (1996a). Focal-series reconstruction in HRTEM: Simulation studies on non-periodic objects. Ultramicroscopy 64, 211–230. Thust, A., Overwijk, M. H. F., Coene, W. M. J., and Lentzen, M. (1996b). Numerical correction of lens aberrations in phase-retrieval HRTEM. Ultramicroscopy 64, 249–264. Tillmann, K., Thust, A., and Urban, K. (2004). Spherical aberration correction in tandem with exit-plane wave function reconstruction: interlocking tools for the atomic scale imaging of lattice defects in GaAs. Microsc Microanal. 10, 185–198. Uhlemann, S., and Haider, M. (1998). Residual wave aberrations in the first spherical aberration corrected transmission electron microscope. Ultramicroscopy 72, 109–119. Urban, K. (2007). The new paradigm of transmission electron microscopy. MRS Bulletin 32, 946–952. Urban, K. (2008). Studying atomic structures by aberration-corrected transmission electron microscopy. Science 321, 506–510. Vrejoiu, I., Le Rhun, G., Pintilie, G., Hesse, D., and Gösele, U. (2006). Intrinsic ferroelectric properties of strained tetragonal PbZr0.2 Ti0.8 O3 obtained on layer-by-layer grown, defectfree single-crystalline films. Adv. Mater. 18, 1657–1661. Williams, D. B., and Carter, C. B. (1996). “Transmission Electron Microscopy.” Plenum, New York. Zemlin, F., Weiss, K., Schiske, P., Kunath, W., and Herrmann, K.-H. (1978). Coma-free alignment of high-resolution electron microscopes with the aid of optical diffractograms. Ultramicroscopy 3, 49–60. Zhong, W., King-Smith, R. D., and Vanderbilt, D. (1994). Giant LO-TO splittings in perovskite ferroelectrics. Phys. Rev. Lett. 72, 3618–3621.

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Aberration-Corrected Electron Microscopes at Brookhaven National Laboratory Yimei Zhu* and Joe Wall Contents

I Introduction II Environmental Requirements and Laboratory Design for Aberration-Corrected Electron Microscopes III The BNL Aberration-Corrected Instruments A The JEOL JEM2200FS and JEM2200MCO TEM/STEM B The Hitachi HD 2700C STEM C The FEI Titan 80-300 ETEM IV A Brief Comparison of the Three Instruments V Evaluation and Applications of STEM A Overview B Measurement of Probe Profile Using Single Atoms C Image Simulation D Quantification in Imaging and EELS E Position-Sensitive Coherent Electron Diffraction VI Outlook Acknowledgments References

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I. INTRODUCTION The past decade has witnessed the rapid development and implementation of aberration correction in electron optics, realizing a more than 70-year-old dream of aberration-free electron microscopy (EM) with a spatial resolution below 1 Å (Scherzer, 1936; Scherzer, 1947; Haider et al., 1998; Batson, Dellby, and Krivanek, 2002a,b; Jia, Lentzen, and Urban, 2003; Jia and Urban, 2004; Sawada et al., 2005; Borisevich, Lupini, and Pennycook, 2006). With sophisticated aberration correctors, modern electron microscopes * Brookhaven National Laboratory, Long Island, New York, 11973 USA Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01012-4. Copyright © 2008 Elsevier Inc. All rights reserved.

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now can reveal local structural information unavailable with neutrons and X-rays, such as the local arrangement of atoms, order/disorder, electronic inhomogeneity, bonding states, spin configuration, quantum confinement, and symmetry breaking (Zuo et al., 1999; Wu, Zhu, and Tafto, 2000; Volkov and Zhu, 2003; Klie et al., 2005; Jooss et al., 2007; Kimoto et al., 2007; Wu et al., 2007; Zhu et al., 2007). Aberration correction through multipole-based correctors, as well as the associated improved stability in accelerating voltage, lens supplies, and goniometers in electron microscopes, now enables medium-voltage (200- to 300-kV) microscopes to achieve image resolution at or below 0.1 nm. Aberration correction not only improves the instrument’s spatial resolution but, equally important, allows use of larger objective lens pole-piece gaps, thus realizing the potential of the instrument as a nanoscale property-measurement tool—that is, while retaining high spatial resolution, we can use various sample stages to observe the materials’ response under various temperature, electric and magnetic fields, and atmospheric environments. Such capabilities afford tremendous opportunities to tackle challenging science and technology issues in physics, chemistry, materials science, and biology. The research goal of the EM group at the Department of Condensed Matter Physics and Materials Science (CMPMS) and the Center for Functional Nanomaterials (CFN), as well as the Institute for Advanced Electron Microscopy, Brookhaven National Laboratory (BNL), is to elucidate the microscopic origin of the physical and chemical behavior of materials, and the role of individual, or groups of atoms, especially in their native functional environments. We plan to accomplish this by developing and implementing various quantitative EM techniques in strongly correlated electron systems and nanostructured materials. As a first step, with the support of the Materials Science Division, Office of Basic Energy Science, U.S. Department of Energy (DOE), and the New York State Office of Science, Technology, and Academic Research, we recently acquired three aberration-corrected electron microscopes from the three major microscope manufacturers (JEOL, Hitachi, and FEI). The Hitachi HD2700C is equipped with a probe corrector, the FEI Titan 80-300 has an imaging corrector, and the JEOL2200MCO has both. All the correctors are of the dual-hexapole type, designed and manufactured by CEOS GmbH based on the design of Rose and Haider (Rose, 1994; Haider et al., 1998). These three are one of a kind in the United States, designed for specialized capabilities in characterizing nanoscale structure. In this chapter, we review the performance of these state-of-the-art instruments and the new challenges associated with the improved spatial resolution, including the environmental requirements of the laboratories that hosts these instruments. Although each instrument described herein has its own strengths and drawbacks, it is not our

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intention to rank them in terms of their performance, especially their spatial resolution in imaging.

II. ENVIRONMENTAL REQUIREMENTS AND LABORATORY DESIGN FOR ABERRATION-CORRECTED ELECTRON MICROSCOPES An aberration-corrected electron microscope aims at a sub-angstrom spatial resolution or an improvement to a fraction of an angstrom in performance. This is not a trivial task. An addition to the requirement for a skillful operator who can tune the corrector and optimize the electron optics of the instrument, the instrument must be housed in an environmentally stable laboratory with a minimal amount of floor vibration, acoustic noise, airflow, and fluctuation in air pressure, temperature, and interference from electromagnetic fields. Environmental instabilities often are the limiting factors in achieving the expected performance of an aberration-corrected electron microscope. Since images are acquired serially in scanning transmission electron microscopy (STEM), any instabilities would appear as image distortions, while in the parallel recording of highresolution transmission electron microscopy (HRTEM) (e.g., through-focus series acquisition) would result in a loss of contrast and ultimately, of resolution. Indeed, an aberration corrector corrects only the electro-optical aberration of the microscope, not any of the instabilities. The doubleaberration corrected JEOL2200FS TEM/STEM located in the Department of CMPMS, Bldg. 480, at BNL is a good example (Figure 1). Before installing this microscope, we designed a new laboratory, completely renovating a 50-year-old building previously used as a gym. The changes were based on the required specifications with < 0.5 mG RMS (root-mean-square, which equals one-sixth of the peak-to-peak (p-p) measurement) electromagnetic fields at 60 Hz, a maximum airflow rate of 15 ft/min, and a temperature stability of 0.1◦ C/hr. Although active vibration-compensation systems were available, low-frequency vibrations (≤ 10 Hz) are best attenuated by large masses. To decouple the instrument from the vibrations emanating from surrounding laboratories, we cut the floor and poured a 2-foot thick concrete slab on which the instrument sits, isolated with de-coupling materials at the slab’s perimeter. To ensure the low airflow rate without jeopardizing the required temperature stability, we chose as an air-supply inlet a U-shaped tube covered with a small-pored “duct sox” that minimizes air movement. Since the electromagnetic field throughout the laboratory was on the borderline of the requirement (ranging from 0.2 to 0.5 mGauss), we installed an electromagnetic field-cancellation system (see Figure 1) to test the effectiveness of the system in compensating for

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Low Airflow Sound Isolation

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FIGURE 1 The JEOL JEM2200FS, equipped with an omega filter and a probe corrector, installed in December 2004 at the Department of Condensed Matter Physics and Materials Science (CMPMS), Brookhaven National Laboratory (BNL). The laboratory was renovated for the instrument with special features to improve the room environment. Note the six-coil magnetic-field cancellation system installed by IDE. The horizontal red arrow points to the “clamshell” to minimize the air pressure on the sample holder.

potential changes in the background alternating current (AC) and direct current (DC) fields. The compensation system did not work as well as expected, which we largely attribute to the small compensation we were seeking and because such a system can only cancel the field at one point in the room or on the microscope. Electromagnetic interference and stray magnetic fields, especially at the level above 0.5 mG, can induce considerable aberrations in HRTEM and scanning distortions in STEM imaging and electron energy loss spectroscopy (EELS). Initially, we encountered a problem of STEM image distortion (Klie, Johnson, and Zhu, 2008) due to 60-Hz noise that was associated with a ground loop in close proximity to the instrument (both in single- and three-phase circuits, the AC field generated by the grounding is inversely proportional to the grounding resistance, or proportional to the current lost to ground). After we eliminated the main source of the 60-Hz noise, the quality of the STEM images was greatly improved.

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The BNL JEM2200FS was equipped with an in-column energy filter that is better electromagnetically shielded compared to the post-column energy filter. The “clamshell” for the sample stage recently designed and provided by JEOL helps to reduce the fluctuation of air pressure. Nevertheless, the design of the instrument was based largely on the JEM2010F and JEM2010FEF with a small 25-cm diameter column. Our study suggests that the aberration-corrected JEM2200FS is more susceptible to airflow and electromagnetic fields than a conventional (i.e., uncorrected) JEOL instrument such as JEM2010F, which demonstrated an achievable STEM atomic resolution with airflow as high as 15 ft/min (Muller and Grazul, 2001). Significantly lower airflow rates are required for the JEM2200FS due to its much longer column. Since the stiffness of a microscope’s column improves roughly with the fourth power of its diameter but deteriorates by the third power of its length, increasing the length by adding correctors and energy filters therefore might dramatically lower the instrument’s performance. Recognizing the importance of the laboratory environment on the performance of the high-resolution instruments (Allard et al., 2005), we expended considerable effort on designing and constructing the EM suite at the CFN (Building 735, completed in May 2007, HDR Architecture, Inc.), one of the five nanocenters in the DOE’s nationwide complex. The CFN site was carefully selected within BNL’s 5,300-acre site for its few sources of vibration and electromagnetic interference. The entire building was constructed on compacted structural fill that was compressed to 98% maximum dry density using various vibration methods. The EM suite, located on the ground floor of the building, consists of six microscope laboratories and one sample preparation laboratory. Four microscope rooms originally were designed as high-accuracy laboratories for aberration-corrected microscopes, each consisting of an instrument room, an equipment room, and a control room (Figure 2); only two laboratories were constructed because of the unexpectedly high cost. The design criteria for the high-accuracy laboratories include floor vibrations below 0.25 mm/sec (RMS) in all directions and frequencies, acoustic noise below 40 dB, stray AC magnetic fields < 0.1 mG (p-p) at 60 Hz and at lower frequencies scaled by f/60, stray DC field below 1 mG vertical, below 0.01 mG horizontal above earth ambient field, airflow < 1 cm/min vertically, and no horizontal air current permitted. Temperature and humidity were set at 21.1 ± 0.1◦ C/hr (70 ± 0.18◦ F/hr) and 40–60%, respectively. The facility is located in the south side of the building, far away from vibrations induced by street traffic and from the elevator on the north side of the building. All the laboratories are adjacent to a 3-m (10-ft) galley that houses all vibrating equipment, such as vacuum pumps and water chillers. We placed additional shielding along the wall to limit magnetic emanations from the galley. Compared with the building slabs that are

B

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SERVICE TRENCH DUCT

PLENUM

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SCALE: 1/4"⫽1'–0"

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TYP. METAL INSULATED PANEL SYSTEM (MIP)

HIGH ACCURACY LAB FLOOR PLAN CONTROL RM 1L29 & LAB 1L30

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A 3" 4" 8'–7" 4" 12'–1" 1'–2" 3"

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INSULATED METAL PANELS TO UNDERSIDE OF SLAB

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RADIANT WALL PANELS ALL FOUR WALLS

HANGING FIXTURE ALL FOUR WALLS

INSULATED METAL PANEL SUSP PERFORATED RADIANT CEILING PANELS IN CHANNEL GRID SYSTEM

SUPPLY AIR PLENUM

RETURN AIR PLENUM

SECTION B-B ELEVATION

COMPACTED STRUCTURAL FILL

DAMPENING CONCRETE (CONCREDAMP)

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ISOLATION JOINT

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INSULATED METAL PANELS

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AIRFLOW

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J

GALLEY

486 Yimei Zhu and Joe Wall

FIGURE 2 Floor plan (above) and cross-section views (right) of the high-accuracy lab (1L30) in Brookhaven’s Center for Functional Nanomaterials (building 735).

15 cm (6 inches thick) (4 kips/in2 ), the floor slabs under the high-accuracy laboratories are 60 cm (24 inches) thick, reinforced with No. 5 reinforcing bars every 30 cm (12 inches) top and bottom, and are isolated with a halfinch isolation joint between the building columns and other slabs. The slab for the control room also is 60 cm (24 inches) thick and separated with a

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half-inch isolation joint. Furthermore, the top 6 inches of the slab contain a vibration-reducing agent (Concredamp, Durasol Corp.) that is reinforced with polypropylene fibers, which also was placed in the galley spaces. The instrument room (Figure 3), ∼ 3.7 m (W) × 5.3 m (D) × 4.6 m (H) (12 ft × 17.5 ft × 15 ft), of the high-accuracy laboratory was constructed with a room-in-room concept. An equipment area (equipment room, ∼ 2.6 m × 5.3 m × 4.6 m (8.5 ft × 17.5 ft × 15 ft)) was built outside the inner room (see Figure 2). We adopted this design for two reasons. The first was to minimize the heat load in the instrument room so that a minimum amount of cooling would be needed. The second was that the microscope’s power supplies added possible fluctuations in the noise and heat loads. Moving them to the space between the inner and outer rooms ensured that they were in a well-controlled environment but outside the most sensitive instrument area. A 15-cm (6-inch) space between the three inner and outer walls serves as a return air plenum. The walls and outer ceiling are constructed from prefabricated modular panels consisting of 100% 10-cm (4-inch) thick polyurethane foam insulation bonded by an adhesive to an aluminum outer skin. The inside of the instrument room offers a superb acoustically insulated wall surface and a support for the radiant wall panels. The equipment room also was constructed from prefabricated panels with gasketed, insulated doors, similar to freezers with good

acoustic ceiling & air filter

cooling panel thermistor HD2700C

air grill

FIGURE 3 The instrument room built with the room-in-room concept. It houses the Hitachi HD2700C, a dedicated STEM recently installed at CFN, BNL.

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FIGURE 4 The control room of the high-accuracy lab. Refrigerator-type doors were used to reduce the fluctuation of air pressure and acoustic noise. The lab has a thick double window to see through to the instrument room.

sealing. Double-glass panels, ∼ 6.8 m × 2.6 m (22.5 ft × 8.5 ft), between the instrument room and the control room (Figure 4) allow the operator to see inside the instrument room. Double doors with a total opening of 1.5 m × 2.4 m (5 ft × 8 ft) provide access to the instrument room. Although we laid down isolated slabs for the four high-precision laboratories, only two (1L30 and 1L24) were finished with the original room-in-room design; they currently house the Hitachi and FEI aberration-corrected microscopes, respectively. The other two laboratories will be developed and expanded in the future. Figure 5 shows the vibration measurement on one of the slabs in the microscope laboratory. We did not detect vibrations or an acoustic noise peak at 30 Hz (often associated with the ventilating motors and pumps running at 1800 rpm) in the instrument room, and very small (4- to 10-Hz) frequencies associated with belt-driven equipment. The equipment rooms (see Figure 2) are air-conditioned through a constant-volume VAV box from 100% outside air using an air handler. Silencers installed in the air handlers reduce acoustic noise. All air handlers in the building are direct-drive units specified for low vibration and noise. The air pressure in the equipment rooms is slightly positive at 2.5 pa (0.010in. wc) to prevent dust accumulation. The instrument rooms originally were meant to use laminar-flow air for 6 m/min (20 fpm) maximum. Later

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1000 Loc 3 - Rm 1L30 - Vert Loc 3 - Rm 1L30 - N-S Loc 3 - Rm 1L30 - E-W NIST-A RMS Velocity, muin/s

100

10

1

1

10 Frequency, Hz

100

FIGURE 5 Triaxial vibration measurement at the high-accuracy lab (1L30) in an evening in September 2007. For most of the frequencies the measured velocities are below 30 µin/s (RMS), or 0.75 µm/s. Note the NIST-A criterion, 25 nm (1 µin) between 1 and 20 Hz and 3.1 µm/s (125µin/s) between 20 and 100Hz, which maintains a constant RMS displacement amplitude at low frequencies, which was created to accommodate the ultrahigh-precision nanotechnology labs at NIST. Different strategies may be required to meet this criterion at sites with significant low-frequency vibration content.

we modified the design and placed radiant cooling panels (∼ 30 BTU/sq ft of panel area) in the ceiling and walls, thereby attaining almost zero airflow in the room. Currently, to ensure adequate ventilation for operators, we exhaust only 4.6 m/min (50 cfm) from the room through an exhaust grill located at floor level; the rate can be increased up to 27.9 m/min (300 cfm) during maintenance. Airflow measurements show 0 m/min horizontally in the space, and typically 0 to 0.3 m/min (1 fpm) vertically. In a normal day (a 24-hr period) without people entering and exiting the instrument room, a temperature fluctuation below 0.017◦ C (0.03◦ F) can be achieved (Figure 6). The radiant cooling system is isolated from the site’s 5.5◦ C (42◦ F) water with a heat exchanger, and the water temperature is held at 12.2◦ C (54◦ F). Piping was sized to keep the flow velocity below 0.9 m/sec (3 fps), and panels were sized at 0.6 m/sec (2 fps) to reduce noise. The variable-speed drives on the radiant cooling loops allow us to adjust flows to more closely match the system’s characteristics to actual cooling loads in the rooms. The aluminum acoustic ceiling panels used as radiant panels on the walls and the ceiling reduce noise in the room. The perforation sizes in the ceiling panels are just big enough to distribute the required airflow for cooling

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temperature (8F)

70.2 rm avg 4:04:00:00 03/09/2008 70.107

rm avg 13:52:00:00 03/09/2008 70.06251

70.0 rm avg 11:50:00:00 03/09/2008 69.94951

69.8 0:00 03/09/2008

6:00

12:00

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0:00 03/10/2008

time (hr)

FIGURE 6 A typical temperature measurement in the high-accuracy lab (1L30) with the instrument fully operational during a period of 24 hours on a Saturday. Using the radiant cooling panels, the fluctuation of the room temperature can be reduced to below ±0.03◦ F/day.

at reasonable velocities. Smaller holes in the panels increase their acoustic attenuation. In addition, we placed acoustic blankets enclosed in plastic above the microscope’s column to prevent airflow over the column and provide additional sound attenuation. Temperature sensors (thermistors) at multiple locations (see Figure 3) with high-resolution transmitters constantly monitor the temperature of the instrument room, and the entire system is remotely computer controlled. The room’s humidity is maintained between 30% and 50% relative humidity. The air into the space is cooled to 10◦ C (50◦ F) for dehumidification, while a humidifier is available if needed to increase humidity. We planned for the possibility of condensation formation on the cooling panels by installing a humidity sensor system to calculate dew point temperature. The controller resets the radiant cooling water temperature upward if the difference between the water’s temperature and room’s dew point is less than 1.1◦ C (2◦ F). The microscope laboratories have both fluorescent and incandescent lighting. The former lighting is used for maintenance, while the incandescent fixtures are dimmable, eliminating the radiofrequency interference that is generated from the solid-state electronic ballast of the fluorescent fixtures. The floor tiles are conductive and grounded to eliminate electrostatic charges. The area in the microscope suites and adjacent laboratories is considered a high-sensitivity area; accordingly, all circuits there >120 VAC are enclosed in rigid metal conduits and all conductors are twisted to mitigate the magnetic fields generated from the power circuits. The microscope equipment has a special dedicated equipment ground. All local electrical panels and transformers serving the high-accuracy rooms have aluminum and steel shielding to reduce electrical noise. Furthermore, the building’s main switchgear electrical room is totally shielded in two layers. The outer layer is a 6-mm (1/4-inch) thick aluminum plate with welded seams; the

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inner layer has two layers, 6 mm and 3 mm (1/4 inch and 1/8 inch), of lowcarbon steel plates to provide a total plate thickness of 9 mm (3/8 inch). The major feeder runs to the upstairs mechanical room are shielded to reduce stray magnetic fields that could affect the high-accuracy laboratories. The distribution transformers feeding the panels were isolated to establish a “clean” transformer for laboratories and sensitive equipment and a normal panel feeding nonsensitive equipment. Normal lighting and building equipment are fed from the “nonsensitive” distribution panels. The microscope’s manufacturer added an uninterrupted power supply (UPS) for more isolation, filtering, and safe orderly shutdown of the microscope and its subsystems. Electrical trenches within the floor slab accommodate the cables from the microscope equipment, thereby eliminating the hazards of them running across the floor. Both Hitachi and FEI room surveys before their instrument delivery showed very low magnetic fields in the high-accuracy labs. In general, in all x, y, and z directions at different heights in the instrument room, the measured AC fields are below 0.005 mG. Figure 7 shows the measurement of the Hitachi laboratory (IL30) in May 2007 when the building was just completed with no scientific equipment in place. Our own measurements in early 2008 using the Spicer Consulting Analysis System (SC11) suggest the average AC magnetic fields at 60 Hz of the room IL30 after the instrument is in operation are ∼ 0.15 mG in z-direction and 0.08 mG in x-y directions. An interesting observation is the dissimilar acoustic spectra measured in rooms 1L24 and 1L30, which were designed identically. The only difference between them when the measurement was taken was the

10.0m

G1, 1 X: 60.2 Y: 40.2u G1, 1 X: 180.5 Y: 3.3u dX: 120.3 dY: 36.9u

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100.0n

10.0n 100.0

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FIGURE 7 A typical measurement of AC field in IL30 (3 ft above the floor in-plane) during the day (4/18/07). For most frequencies, the electromagnetic interference is below 0.01 mG. Note the peak at 60 Hz and its multiples (excerpted from the Hitachi Room Survey Report).

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presence of the operational STEM in 1L30 (room 1L24 was in its as-built state). We recorded a source of acoustic noise from the mechanical fan used to cool the scan coils that were not in the lens water-cooling system. It increases the noise level from 38 to 44 dB.

III. THE BNL ABERRATION-CORRECTED INSTRUMENTS A. The JEOL JEM2200FS and JEM2200MCO TEM/STEM The Brookhaven JEM2200MCO TEM/STEM (Figure 8) is based on the JEM2200FEF TEM/STEM that features an in-column omega-type energy filter. The instrument consists of a probe corrector for STEM and an imaging corrector for TEM, as well as a double Wein filter monochromater (Mukai et al., 2005). We placed the purchase order in early 2002. When this chapter was written, the monochromater had gone through several iterations of redesign and had not passed its required specifications at the factory. The prototype version of the instrument, JEM2200FS (see Figure 1), which has a probe corrector, was loaned to BNL before the final instrument (JEM2200MCO—MCO refers to monochromated, aberrationcorrected, and omega-filtered instrument) could be built. The JEM2200FS was installed at BNL in December 2004 and the JEM2200CMO in October 2006 in the JEM2200FS room after the loaner instrument was sent from BNL to the University of Illinois at Urbana-Champaign. The monochromator is scheduled for delivery to BNL and retrofitting to the JEM2200MCO in the fall of 2008. Recent test results with the monochromator at the factory showed an energy resolution of 0.18 eV with a beam current of 10 pA at a probe size of 2 nm. The target for the probe size in STEM for the instrument is 0.07 nm. The performance details of the BNL JEM2200FS were reported previously (Zhu et al., 2005; Klie, Johnson, and Zhu, 2008). The Brookhaven JEM2200MCO microscope is a sister instrument to the one installed at Oxford University, U.K. also in October 2006 (the performance of the Oxford loaner instrument JEM2200FS can be found in Sawada et al., 2005). In addition to the omega filter and two aberration correctors, it has a 200-kV thermo-assisted Schottky field emission electron gun, an ultrahigh-resolution pole piece (URP) with a ±25-degree sample tilt, and a Gatan GAT894 2 k × 2 k Ultrascan charge-coupled device (CCD) camera (active area 28.7 mm × 28.7 mm). It has a Hamamastu camera instead of a fluorescent screen. It is also equipped with a JEOL STEM bright-field (BF) and dark-field (DF) detector and a Faraday cup. The JEM2200CMO has several important improvements over the JEM2200FS model. Because it must accommodate two aberration correctors and a monochromator, the total column height of the instrument can exceed 4 m (the measured height of BNL’s microscope is 3.68 m without the monochromator compared with

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FIGURE 8 The JEOL JEM2200CMO, equipped with an omega filter, a probe corrector, and an imaging corrector at CMPMS, BNL, May 2007. Note that a large metal frame has been added to improve the stability of the microscope column.

the 2.5-m height for the JEM2100F). A large rigid metal frame was added to the instrument to support the relatively tall and thin column (see Figure 8). Nevertheless, the new instrument still seems susceptible to perturbations in airflow. Figure 9 shows the quality of the STEM image resolution under various airflow conditions. Figures 9a and b show conditions before and after the air duct near the column was covered with aluminum foils. Figure 9c shows the effects of an additional heavy curtain halfway down below the sample stage, and part d shows the effect with the air conditioner off. Differences in the image distortion are clearly visible. Power-spectrum analysis of Young’s fringes of Au particles in TEM mode (Figures 9e–h)

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Yimei Zhu and Joe Wall

(b)

(a)

Before

Covered air duct with aluminum foils

(c)

(d)

Covered air duct with Air conditioner OFF aluminum foils and curtain (e)

(f)

Before (g)

Covered air duct with aluminum foils and curtain

Covered air duct with aluminum foils (h)

Air conditioner OFF

FIGURE 9 The influence of the STEM image quality (a–d) and information limit (e–h) of the JEM2200CMO due to the airflow at CMPMS, building 480. (See color insert).

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Aberration-Corrected Electron Microscopes at Brookhaven National Laboratory

indicates that there was a directional reduction of frequencies transmitted, in the case of Figures 9e and f but no remarkable differences in Figures 9g and h. The JEM2200MCO also has electro-optical design improvements: an additional intermediate lens, an additional projector lens, and a piezocontrolled stage in all x, y, and z directions. The ability to use the piezo control to adjust the focus, or z-height, of the sample without changing the objective lens current is particularly convenient for corrector alignment. The stability of the high tension of the instrument was improved from 1.0 × 10−6 to 3 × 10−7 , and the lens current stability was improved from 1 ppm to 0.5 ppm compared with the JEM2200FS. The geometrical aberrations of the objective lens and probe-forming lens in the instrument can be corrected up to third order. The aberration correctors can be tuned either manually or using CEOS’ autoalignment software (Haider, Uhlemann, and Zach, 2000). After properly tuning the corrector, either in TEM or STEM mode, the aberration coefficients A2 and B2 are typically smaller than 150 nm, and S3, A3, and C3 are smaller than 5 µm. Onsite acceptable tests demonstrated in both aberration-corrected TEM and STEM imaging a 0.1-nm p-p resolution in Fourier space. Measurement of a STEM image of a silicon single crystal in [110] orientation shows that a dip contrast at or better than 20% between the adjacent dumbbells can be achieved (see Figure 10a). The power spectra of the images show the (400) and the (511) diffraction spots, indicating the limit for attainable information is beyond 0.1 nm (Figure 10b). The Ronchigram analysis in STEM mode suggests (a)

(b)

400 511

FIGURE 10 (a) An ADF/STEM image of Si [110] showing Si atoms of the dumbbell structure with a contrast dip better than 20%. (b) The power spectrum of (a) showing (511) spot that corresponds to 1.05 A° of lattice spacing. (From Klie, Johnson, and Zhu, 2008.)

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Corrector Off: 14 mrad

Corrector On: 40 mrad

Amorphous Ge

FIGURE 11 Ronchigrams of Ge showing coherent convergence half-angle before (left) and after (right) aberration correction. The circle corresponds to 14 mrad and 40 mrad, respectively. (From Zhu et al., 2005.)

that the constant phase area is extended from 14 mrad before aberration correction to nearly 35 mrad afterward (Figure 11). This means that instead of using a 15-mrad convergence angle for an uncorrected STEM, we can routinely use a 30-mrad convergence angle, which allows a threefold increase in probe current. In STEM, the maximum probe current is 100 pA with the probe corrector off at beam size of 0.5 nm. JEOL’s in-column filter is advantageous for quantitative electron diffraction by filtering unwanted electrons, especially in convergent-beam electron diffraction (CBED). The filter’s acceptance angle is ∼ 60 mrad for a 4-eV slit and 170 mrad with no slit (Figure 12). The isochromaticity of the omega filter is ∼ 2 eV with an area of view of the full 2k × 2k CCD camera. The filter also can serve as an electron energy-loss spectrometer, particularly for spectroscopy imaging. However, for most spectroscopic applications it has an intrinsic drawback because it lacks an aberration-correction system to minimize the spectra distortion. Consequently, the smallest entrance aperture must be used for EELS to curtail distortion, which significantly reduces the flexibility of choosing different collection angles and beam intensities. Furthermore, the energy resolution for the omega system typically is 1.0–1.1 eV for the normal emission current (∼ 100 µA) and ∼ 0.7 eV for low-emission current (∼ 30 µA). Figure 13a shows an example of HREM imaging of a 3-degree tilt grain boundary in a SrTiO3 bicrystal using JEM2200MCO. Compared with conventional HREM images from an uncorrected microscope, there is a notable increase in image contrast due to the reduction of the contrast delocalization effects. Furthermore, corrections were made for the presence of the threefold aberration so that the images are better suited for quantitative analysis (Klie, Johnson, and Zhu, 2008). Figures 13b and 13c, respectively, are a reconstructed rotation map and a strain map from the same image in Figure 13a using geometric phase analysis (Hytch, Snoeck, and Kilaas, 1998; Johnson, Hÿtch, and Buseck, 2004). The color scheme shows the

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Aberration-Corrected Electron Microscopes at Brookhaven National Laboratory

(a)

(b)

FIGURE 12 Acceptance angle of the JEM2200CMO omega filter. Sample: Si 110 with camera length of 15 cm. (a) 170-mrad acceptance angle, marked by the circle, without energy slit. (b) Acceptance angle larger than 60 mrad, marked by the circle, with an energy slit of 2 eV (from the acceptance test of the JEM2200MCO). (b)

(a)

wxy 108

2108 (c) exy 5 nm

15%

215%

FIGURE 13 (a) An aberration-corrected HRTEM image from JEM2200MCO of a 3-degree tilt grain boundary in SrTiO3 . An edge dislocation array (indicated by arrows) oriented along the beam direction is clearly visible along the boundary. (b–c) Rotation and strain maps from the same area in (a). (b) The map of ωxy gives the rotation across the boundary. The bold 0-degree contour gives the trace of the boundary. The fine contours are every 0.75-degree to ±1.5-degree. (c) The map of the strain εxy perpendicular to the boundary plane shows the expansion (red to bright yellow) and compression (green to dark blue) of the lattice around the dislocation cores. The fine contours are every 1% to ±2% (From Klie, Johnson, and Zhu, 2008.) (See color insert).

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±1.5-degree rotation of each grain and a 15% expansion and compression of the lattice near the dislocation cores.

B. The Hitachi HD 2700C STEM The BNL Hitachi HD2700C is located in the newly established CFN. It is the first aberration-corrected electron microscope manufactured by Hitachi. The instrument is based on the HD2300 model (Nakamura et al., 2006), a dedicated STEM developed a few years ago as an alternative for the discontinued VG STEMs. The BNL instrument has a cold-field emission electron source with high brightness and small energy spread, ideal for atomically resolved STEM imaging and EELS. The microscope has two condenser lenses and an objective lens with a 3.8-mm gap compared with the 5-mm gap objective lens in HD2300, with the same ±30-degree sample tilt capability and various holders for heating and cooling (−170–1000◦ C). The projector system consists of two lenses that provide considerable flexibility in choosing various camera lengths and collection angles for imaging and spectroscopy. Table I summarizes the convergent angles and collection angles for various settings. The microscope has seven fixed and retractable detectors. Above the objective lens is the secondary electron detector for imaging the sample’s surface morphology. Below the lens are the Hitachi analog high-angle annular dark-field (HAADF) and BF detector for STEM and a Sony TV-rate (30 frame/sec) 8-bit CCD camera (480 × 480) for fast and low-magnification observations and alignment. The Gatan 2.6 k × 2.6 k 14-bit CCD camera located farther down is for diffraction (both convergent and parallel illumination) and Ronchigram analysis. The Gatan analog medium-angle annular dark-field (MAADF) detector and EELS spectrometer (a 16-bit 100 × 1340-pixel CCD) are sited at the bottom of the instrument. The spectrometer (Enfina ER) is a high-vacuum– compatible high-resolution device that Gatan designed particularly for BNL. The CEOS probe corrector, located between the condenser lens and the objective lens, has two hexapoles and five electromagnetic round lenses, seven dipoles for alignment, and one quadrupole and one hexapole for astigmatism correction. Other features of the instrument include remote operation, double shielding of the high-tension tank, and anti-vibration system for the field emission tank. The entire instrument is covered with a telephone booth–like metal box (see Figure 3) to reduce acoustic noise and thermal drift. The instrument was installed in July 2007. Within the first two weeks, we had achieved a 0.1-nm resolution of the HAADF-STEM image (Inada et al., 2008a). The instrument was accepted in November 2007. We recently added a residual gas analyzer (RGA-200, Stanford Research Systems, Sunnyvale, CA) to monitor vacuum quality, especially relative to specimen contamination. Figure 14 shows an HAADF-STEM image of BaTiO3 recorded with an inner collection angle of 53 mrad. The contrast between Ba and the

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TABLE I Presets for Different Collection Angles in the Hitachi HD2700C ADF collection angle ß Mode

Hitachi detector (mrad)

Gatan detector (mrad)

HAADF∗,† MAADF†,‡ CL1 CL2 CL3 CL4

114–608 53–280 64–341 45–242 24–126 16–85

46–104 21–49 26–59 18–41 10–22 7–15

∗ High-angle annular dark-field detector; † not compatible with spectrometer focus; ‡ medium-angle annular dark-field detector.

b

(b)

(a)

2 nm (c)

(d)

(e) a ~43mrad

FIGURE 14 (a) HAADF image of BaTiO3 (raw data) from Hitachi HD2700C. (b) Enlarged “b” area in upper-left corner of (a), ADF collection angle β = 53 mrad. (c) CBED pattern of Si (100) (convergence angle of α = 28 mrad), (d) Nano beam diffraction of Si (100) (convergence angle of α = 1.7 mrad. (e) Ronchigram of imaging condition, semiangle of flat area is 43 mrad. (From Inada et al., 2008a.)

background is 56%. Since the HD-2700C is equipped with a high-dynamic– range Gatan CCD camera, this dedicated STEM allows us to record CBED (Figure 14c, silicon (100) with convergence angle of 1.7 mrad and probe size < 10 nm) and nanobeam diffraction (Figure 14d, silicon (100) acquired with 28-mrad convergence angle). Figure 14e shows a Ronchigram of the area with a half-angle of flat region of 43 mrad with the aberration corrector on. In addition to preset modes, the microscope can operate in a variety of combinations of convergence angles and collection angles by using free lens control. Regarding EELS capability, Figure 15 illustrates a time trace of the zero-loss peak of the 200-kV electron beam during 130 sec; it indicates a stability of < 8 × 10−7 for this duration. The energy resolution derived

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, 8 3 1027 for 130 sec

For 10 sec 0.35 eV

130 (s)

FIGURE 15 The energy spread time trace for zero loss peak in EELS for a series of 1.0 s duration scans (left), indicating stability of less than 8 × 10−7 for 130 s acquisition and corresponding zero loss peak spectrum (right) FWHM of 0.35 eV for acquisition time of 10 s with cold-field emission source acceleration voltage of 200 kV. (Emission current of 1 µA, EELS collection angle of 10 mrad for 10 s duration time). (From Inada et al., 2008a.)

from this experiment is 0.35 eV (full width at half maximum, FWHM) for a 10-sec acquisition. Single heavy atoms on a thin (< 4 nm) carbon film represent a simple specimen from both a practical preparation and an analysis point of view. We selected uranium atoms for high atomic number Z, easy availability, and characteristic core-loss spectrum for atomic EELS and STEM (Figure 16). The sample is typical of negative staining used in biological studies except that the uranyl acetate is 100 × more dilute. Tobacco mosaic virus (TMV) was included to provide a thickness gradient with higher concentration of uranium atoms near the TMV, sometimes forming small clumps. The UO2 species observed on such a specimen has a nearestneighbor spacing of 0.34 nm, but the atom size “seen” by the electrons scattered onto the dark-field annular detectors is much smaller. The individual figure parts in Figure 17 are sequential images excerpted from a movie of uranium single-atom motion. Single atoms move due to irradiation by the electron beam. During this experiment the specimen was cooled to −160◦ C with a Gatan 670 liquid nitrogen cooling stage (Inada et al., 2008b). In an electron microscope, the detection of a single atom is accompanied by the passage of a high-energy electron within 0.5 Å of the atom (Batson, Dellby, and Krivanek, 2002a,b). Therefore, there is a significant probability that the atom will gain enough energy to remove it from its binding site; hence, a sequence of images will contain information about the movement of individual atoms, limited only by the time resolution of the image acquisition. This behavior is determined by the balance of several bonding energies, including Van der Waals forces, molecular orbital-valence and bonding-valence electrons states, surface energy, and electric attraction and repulsion. However, enough atoms remain stationary on sequential scan lines and from frame to frame to permit reliable measurement of probe diameter.

Aberration-Corrected Electron Microscopes at Brookhaven National Laboratory

(a) (a.u)

e

0.32 nm c

501

(c)

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FIGURE 16 (a) HAADF-STEM image of uranium atoms and clumps (probe convergence 28 mrad, collection angle of 53–280 mrad). (b) EELS spectrum showing U-O edge and C-K edge. (c) Intensity profile along line “c” in (a) showing 0.32-nm spacing. (d) HAADF intensity map of uranium clump (25 × 25 pixel, 0.4-nm pixel spacing). (e) Simultaneous spectrum imaging of area (d) using Gatan Enfina spectrometer set for 96 eV–150 eV loss with a spectrometer acceptance angle of 30 mrad. (f) Intensity profile of single uranium atom “f” in (a), FWHM of the peak represents 0.8 nm. (g) HAADF intensity profile of a small object (image not shown) 0.03 nm/pixel. (h) Simultaneous spectrum imaging of same area in (g). (From Inada et al., 2008b.) (See color insert).

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C. The FEI Titan 80-300 ETEM The FEI Titan 80-300 model is an instrument newly designed to fully realize the benefits of aberration correction in TEM/STEM. The Titan’s large column diameter (30 cm), compared with the JEOL2200MCO (25 cm) and

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the Hitachi HD2700C (24 cm), was specifically designed for mechanical and thermal stability with the added length of the probe, image correctors, and monochromator. The column assembly also is of the modular type wherein each module/lens section can be aligned accurately, and the aberration correctors and monochromator can be retrofitted with relative ease. The instrument sits on three active vibration dampers to further reduce mechanical vibration. Its unique design of proprietary constant power lenses and power supplies provide the needed thermal and electronic stability. The BNL FEI is an 80-kV to 300-kV TEM/STEM with a thermo-assisted Schottky field emission electron gun (Figure 18a). The advantages of higher acceleration voltages are improved spatial resolution and increased current in a small probe. The acceleration voltage of the instrument can be switched between 80 and 300 kV. The low-kilovolt setting is particularly attractive for radiation-sensitive materials, such as graphene and carbon nanotubes, since it is below the knock-on damage threshold of carbon and for EELS

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FIGURE 18 (a) FEI Titan 80-300, a dedicated environmental TEM/STEM, recently installed at CFN, BNL. The red circle marks the residual gas analyzer. (b) The newly installed plasma cleaner on the BNL FEI Titan 80-300 E-TEM. The plasma cleaner was manufactured by XEI Scientific, Inc.

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with its slightly improved energy resolution. The challenge is then to scale performance in line with the theoretical performance reduction due to the longer wavelength. This microscope, also located in the new CFN building, is a versatile instrument with its Super Twin objective lens and a CEOS imaging corrector mounted between the objective lens and the first intermediate lens (or diffraction lens) for high-resolution electron microscopy (HREM). It has a scanning unit with a Fischione HAADF detector for atomic-resolution STEM (Fischione Instruments). Its other components include a Lorentz lens (built into the CEOS corrector) for magnetic imaging and a biprism for electron holography; an EDX energy-dispersive X-ray system (EDS), a Gatan 2k × 2k top camera, and 863 Tridiem spectrometer (2k × 2k CCD) at the bottom. The large pole-piece gap (5.4 mm) of the objective lens and the five-axis motor-driven CompuStage support a ±40-degree tilt of the sample. The instrument also has several sample holders for heating, cooling, and tomography capabilities and various software for data acquisition and analysis, including HREM through-focus reconstruction, low-dose exposure, dynamic conical DF, and tomography reconstruction. The illumination system of the microscope consists of three condenser lenses. Together they form a double-zoom system (Koehler illumination), which allows increased flexibility in both the TEM and STEM mode. In the former mode, parallel illumination can be achieved over a wide field of view. In the latter mode, a large variety in probe angles and probe currents is available. The instrument has two intermediate lenses and a projector lens for diffraction and imaging at different camera lengths and magnifications. The Lorentz lens is a separate small lens located at the bottom of the objective lens’ lower pole piece (similar to the minicondenser lens; its main difference is a separate water-cooling circuit). The Lorentz lens occupies an intermediate position between objective lens on and off. Typically attainable maximum magnification values for the Lorentz lens are ∼ 60 kx as opposed to ∼3 kx for the low-magnification mode; the resolution lies between 1.5 and 2 nm. Configuration of the BNL instrument was finalized in late 2005. The main feature was its environmental cell (E-cell). FEI has built E-cells in the past (for the CM300 at Haldor Topsøe in Demark, in 1999, and for the Tecnai F20 at Arizona State University in 2002); nevertheless, BNL was the first to request an aberration-corrected E-TEM to study chemical reactions and catalysis. The E-cell has a gas-inlet system with four inlets connected to the objective lens’ octagon. A needle valve that can regulate gas flows in small increments is sited at the gas-inlet system to maintain a stable pressure in the E-cell. The special vacuum system allows researchers to observe a specimen under a gas pressure up to 20 mbar while maintaining a workable high vacuum in the rest of the TEM column and field emission gun (FEG).

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Various gases (e.g., He, N2 , CO2 , and H2 O) can be used, but some gases must be handled with utmost care because of fire or explosion hazards (e.g., CH4 , H2 ), toxic properties (e.g., CO), or strong contamination effects (e.g., sulfur-based gases). The instrument is equipped with a residual gas analyzer (PGA Pfeiffer Prisma QMS 200). The E-TEMs that FEI previously built had a vacuum ∼10−5 torr in the sample area. Nevertheless, we realize that imaging and spectroscopy quality, especially in STEM mode, is sensitive to vacuum quality. This is particularly important for E-TEM, where a clean environment at the sample before any reaction has taken place is crucial for interpretation of experiment data. Particular attention was paid to the in situ vacuum system of BNL’s E-TEM. There are five ion getter pumps (IGPs) (two for the gun, two for the column, and one for the E-cell), five Turbo Molecular Pumps (one for the column and sample airlock, one for the projector chamber, and three for the E-cell), along with two scroll pumps as backup. In the in situ mode, the IGP is switched off and the valve toward this IGP is closed. Other valves are open to allow pumping of the column by the in situ system. Gas is introduced to the E-cell via another valve. A BaroCell measures the gas pressure. Two fixed differential pumping apertures inside the tip of the upper and lower objective lens pole pieces restrict the gas flow from the E-cell into the rest of the column. Special pumping measures were taken to evacuate a large portion of this gas flow via the in situ tubing. Some additional bypasses were implemented in the column of the TEM to obtain the required pressure ratio, allowing a change in the vacuum level of 10 orders of magnitude within a distance < 0.5 m between the E-cell and the FEG’s emission chamber. Even though the E-TEM system entails significant modifications to the microscope, the system still will support normal TEM operation, and its resolution and performance are uncompromised compared with a standard Super-Twin system. Although the system is not yet suited to reach true ultrahigh-vacuum (UHV) pressure levels, in high-vacuum mode attainable pressure is 5 × 10−6 Pa. The primary limitation is the lack of differential pumping in the moving bearings, or O-rings, of the sample stage, or goniometer. Based on our agreement with FEI, if the piezo-controlled stage that does not use O-rings materializes (currently under development within the U.S. DOE TEAM project), then FEI will make the stage available and we will be able to significantly improve the vacuum of the system. When the system has been running in the E-TEM mode (with gas) after switching to standard mode, the vacuum quality often is too low to allow HRSTEM. This problem reflects strong sample contamination or etching. We unsuccessfully requested a sample pretreatment chamber and a heating lamp in the sample loading dock. With the help of FEI, a decontaminator, or radiofrequency (RF) plasma-cleaning system (Evactron model 45,

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manufactured by XEI Scientific, Inc., Redwood City, CA) has been installed at the octagon of the objective lens or E-cell (Figure 18b). The RF plasma ignites and produces oxygen radicals that clean the sample and the chamber. The source of the oxygen radicals is a down-flowing remote plasma device that does not directly expose the chamber or sample to plasma. The general underlying principle is that the plasma generates active species that flow into the chamber and react with the contaminants to produce volatile compounds that are removed by the vacuum pumps. Special care must be taken in gas-reaction experiments. For example, electrons scattered by gas atoms reduce image contrast, increase background noise, and hasten the aging of the detector. Infrared radiation of the heating holders also increases the background signal. It is good practice to retract the EDS detector when the equipment is in E-TEM mode or when the specimen is heated, using EELS for compositional analysis in the E-TEM mode. The beam’s opening angle and field of view at the specimen is limited because of the differential pumping apertures in the objective lens pole pieces, as is the maximum detector angle in STEM mode. The Brookhaven FEI E-TEM was installed in October 2007. One week after the start of the installation, the instrument in TEM mode showed an information limit of 0.7 Å using Young’s fringes of Au particles, demonstrating the stability of the instrument. Figure 19 shows the information limit test at different operation modes with E-cell’s 3 turbo pumps running mode without gas (Figure 19a) and with gas (Figure 19b), both operated at 300 kV. Figure 19c shows the E-cell 3 turbo pumps off mode without gas at 80 kV. The instrument can achieve energy resolution of 0.66 eV and 0.54 eV at 300 kV and 80 kV, respectively (Figure 20) at low emission (extraction voltage at 3000 V, compared with the normal operation condition of 4000 V for TEM and 4500 V for STEM). Figure 21 shows an image of twinning of an Au particle taken at 300 kV immediately after the E-cell (a)

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FIGURE 19 Information limit test at different operation modes. (a) 300 kV at E-cell 3 turbo pumps running mode (without gas), indicating information transfer better than 0.7 A°. (b) 300 kV at E-cell 3 turbo pumps running mode (with gas), indicating an information limit of 1.2 A° can be achieved. (c) 80 kV at E-cell 3 turbo pumps off mode (without gas) indicating information limit of 1.0 A°. (See color insert).

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FIGURE 21 An HRTEM image taken at 300 kV showing multiply twinned nanocrystal Au after E-cell test with the microscope system back to the high-vacuum condition.

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test with the turbo pumps running. A [112] silicon dumbbell image with a separation of 0.78 Å is shown in Figure 22 with a contrast dip of ∼ 20%, using the through-focus series and exit wave reconstruction method.

IV. A BRIEF COMPARISON OF THE THREE INSTRUMENTS Each of the three aberration-corrected electron microscopes at Brookhaven has its own strengths and weakness. Although all are capable of atomic imaging and spectroscopy, subtle differences in their electro-optics design accommodate other capabilities. They are sufficiently different to preclude comparisons. Table II is a summary of the major features of the three instruments. Because they all have STEM capabilities that are very sensitive to the instrument’s stabilities, we briefly discuss their performance within this context. All three aberration-corrected instruments have a 1 Å specification on STEM imaging resolution. They all achieve visible separation of Si 110 dumbbell images with a contrast dip, defined as (Imax − Idip )/(Imax + Imin ), of 25% and better. All three manufacturers used the high-order diffraction spot (say 511) of the fast fourier transform (FFT) from Si dumbbell images to signify an achievement of 1 Å and beyond in spatial resolution. In fact, this is not a convincing method since an instrument’s spatial resolution is defined by its ability to separate two objects in real space. The intensity or visibility of high-order reflections in FFT depends on many

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Primary features of the three aberration-corrected instruments at BNL Instrument

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80–200 Thermal FEG TEM/STEM Probe and imaging JEOL omega filter

factors, including the test samples, not just the spatial resolution of the instrument. The STEM operation of the Hitachi HD2700C is probably the “simplest” since it is a dedicated STEM and does not have other accessories to interfere with the STEM operation and performance. The cold FEG of Hitachi is extremely beneficial as it offers high brightness and a small energy spread. Compared with a monochromator, it has much higher beam current, is much easier to use, and can reach an energy resolution at 0.35 eV under normal operating conditions. The Hitachi instrument appears to give consistently reasonable STEM image resolution showing separation of Si 110 dumbbells at any given time. The aberration corrector improves the beam current by a factor of 10; our test results reveal that this probe current is ∼200 pA at 0.14-nm probe size and 400 pA at 0.2 nm. The JEM2200CMO STEM also demonstrated well-separated Si 110 dumbbells (see Figure 10), although we usually see scanning distortion, which likely can be attributed to the imperfection of the laboratory environment. Nevertheless, with the aberration corrector on, the specification on probe current (50 pA at 0.15 nm) still was not demonstrated when this chapter was written (March 2008). The energy resolution of the instrument during operation is ∼1 eV at normal emission current. The unique advantage of the JEM2200CMO is its in-column omega filter, which is invaluable for quantitative electron diffraction. However, this advantage becomes a significant drawback in EELS performance because the filter does not have multipole lenses to correct its second-order aberration of the omega filter, as does Gatan’s post-column filter. For most EELS applications, the smallest entrance aperture must be used to minimize the aberration and distortion in spectroscopy and spectroscopy imaging. When such an energy filter is used as a spectrometer, the value of the monochromator for the instrument in EELS is increasingly questionable. The FEI 80-300 ETEM probably shows the best instrumental stability among the three models because of its large column diameter. An

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FIGURE 23 High-resolution annular dark field STEM images recorded at 300 kV at E-cell 3 turbo pumps off mode. (a) Si dumbbells along the [110] direction (raw data) showing 1.34-A° separation and little distortion of the image.

information limit around 0.07–0.08 nm is routinely achievable using gold particles on carbon films in the TEM mode. Good Si 110 dumbbell STEM images without image distortion (90-degree lattice planes in images) have been frequently recorded (Figure 23). The energy resolution of the instrument using the Gatan 863 Tridiem spectrometer seems better than the JEM2200CMO under normal operational conditions (probably by 10–15%). Because the instrument does not have a probe corrector, comparison on beam current was not made. A major issue observed in the aberration-corrected STEM is the contrast reversal in annular dark-field (ADF) Z-contrast imaging. Because the condenser aperture can be opened for a large convergent angle without inducing spherical aberration, the ratio of the convergence and collection angles falls sharply. Consequently, an ADF image often is no longer a

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true amplitude image, but is mixed with phase information. Our recent study on thermalelectric Ca3 Co4 O9 , which has a layered structure of CoO2 and Ca2 CoO3 with a distinct difference in Co atomic density in adjacent atomic columns, revealed that contrast reversal might occur when the ratio between the collection angles and the convergent angle is smaller than 3. Hence, care must be taken in interpreting the Z-contrast images. Details on contrast reversal in STEM ADF imaging are reported elsewhere.

V. EVALUATION AND APPLICATIONS OF STEM A. Overview The challenge in STEM performance is simultaneous acquisition of atomically resolved ADF imaging and EELS. Because spectroscopy imaging necessitates a lengthy acquisition period (a 100 × 100-pixel map requires 104 times longer acquisition compared with a point acquisition), the quality of the data largely depends on the stability of the instrument that may be perturbed by various factors, including mechanical vibration, electrical noise, electromagnetic field, thermal drift, fluctuations in lens and emission current, and specimen charging. A monitoring protocol must be established for each category to detect any deterioration in performance, trace it to its source, and then correct it; otherwise, the instrument will not perform as expected. For any experiments involving small probes and long dwell times, contamination of the specimens is a severe issue and is more acute for high-end instruments due to their high beam current and better detection ability. Contamination not only reduces considerably the signal-to-noise ratio (SNR) but also makes interpreting quantitative data difficult; indeed, in the environment TEM (E-TEM), it can obscure in situ chemical reactions. We have eliminated the use of oil-based diffusion pumps for all our microscopes to minimize contamination. We developed an analytical method to quantify the rate of contamination and installed a gas analyzer to determine the source and ingredients. Generally, the problem is lessened by resorting to extended (overnight) pumping, pre-irradiation of a large area before data acquisition, or cooling (down to −160◦ C, which prevents surface migration and buildup of the contaminants). Extended pumping or pre-irradiation times are undesirable because they limit productivity. The unique plasma-cleaning apparatus for treating the sample inside our FEI microscope and various protocols we developed to maintain cleanliness, including transferring the sample from the cleaning chamber to the instrument, will help to circumvent the contamination problem. A major emphasis of the BNL effort is development of imaging methods to permit quantitative comparison to theory compared with “pretty

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pictures” with qualitative interpretation. Most commercial instruments automatically adjust contrast and brightness for optimum viewing (qualitative images) rather than treating the signals as quantitative data (numbers), thereby often leading to misinterpretation. We have developed a routine to retrieve the absolute values of all detector signals to eliminate any ambiguity in interpretation. Furthermore, we develop and use image simulation software to predict image signals (in percent of incident electron counts) for simple, well-characterized specimens and then collect data in various instruments for comparison; we also exploit our own cross-correlation algorithms to correct for sample drift. The effects of the delocalization of inelastic scattering also must be addressed to retrieve the true structural information from 2D chemical images. As we refine relevant parameters, we plan to move to more complex known cases and eventually to unknowns for which we hope to be able to interpret images by quantitative comparison to models with a high degree of confidence. There is a general assumption that spherical aberration correction removes all artifacts and makes image interpretation foolproof. Our efforts to date show that this is not the case. Aberration-corrected instruments usually are operated at the largest aperture angle to maximize resolution. This has several side effects: (1) chromatic aberration increases in proportion to aperture size, (2) depth of focus also decreases proportionately, (3) coherent effects such as channeling become even more important, (4) more of the elastically scattered electrons remain within the cone of the incident beam, and (5) a higher dose rate may accelerate specimen damage. The effects of chromatic aberration and depth of focus are shown in Figure 24. For reference, the diffraction-limited probe size decreases to below 1 Å at 15 mrad as shown. For a cold FEG with 0.25-eV energy spread and lens plus corrector chromatic aberration coefficient of 0.7 mm, the chromatic spread does not become an issue until the aperture reaches 40 mrad, whereas for a Schottky gun with 0.6-eV spread and no monochrometer, it can be a serious limitation at 25 mrad. For an amorphous specimen, the depth of focus is only ∼2 nm. At that point the blurring becomes equal to the diffraction-limited probe diameter, increasing the effective probe size by a factor of 1.4 (effects added in quadrature). This is the basis of the depth sectioning, as described by Pennycook (2006). An obvious example of this effect is seen in viewing Ronchigrams during alignment of the aberration corrector. Theoretically, the pattern should be flat and featureless when the probe is focused exactly on the specimen. In practice, this happens only with specimens thinner than 2 nm because Ronchigrams from all specimen planes are mixed coherently in the final pattern (Figure 25). In a crystalline specimen the defocus effect may be overcome by channeling, permitting atomic resolution on much thicker specimens. However,

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an awareness that the beam amplitude trajectory within the specimen is not a simple transit down an atom column is necessary and minor imperfections (or thermal vibrations) can have significant ramifications. In order to understand this we have adopted a two-pronged approach: start with the simplest possible specimens and use quantitative image simulation to bootstrap to increasing levels of complexity. The intensity distribution at any plane within the specimen can be calculated with image simulation.

B. Measurement of Probe Profile Using Single Atoms Uranium atom images (Wall, Hainfeld, and Bittner, 1978; Inada et al., 2008b) show clear bright spots with high SNR and quantized intensity (see Figures 16 and 17). Roughly two-thirds of atoms move less than 2 Å on subsequent scans and have symmetrical profiles suitable for probe profile measurement. Occasionally a spot has a flat side or a gap, indicating that the atom moved between successive scan lines. Uranyl ions have a tendency to form chains or clumps with 3.4-Å spacing, as well as vertical

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stacks that look like single atoms but with higher signal values. The atoms in the clumps appear to stack nearly on top of each other so the substrate signal level is visible between bright columns and profile measurements of columns give nearly the same full width half maximum (FWHM) as single atoms. From a practical point of view, use of this specimen as a “delta function” to measure beam profile is sensitive to contamination and atom motion. Atom motion comprises two domains: large jumps with unpredictable frequency and small movements related to thermal vibration. The large jumps are infrequent enough that many atoms can be traced from scan to scan and measured (both intensity and profile). Contamination can be eliminated by baking the specimen in the column overnight at 100◦ C and pre-irradiating a large area to polymerize the contaminant molecules. Alternatively, the specimen can be cooled below −40◦ C to stop surface migration. Cooling also could help with atom motion, but it appears that the gross motion is beam induced and not sensitive to temperature. Figure 26 shows the simulated probe profile for a uranium atom on a 40-Å thick amorphous carbon substrate as would be measured with various detectors. For comparison, we include a scan over a point atom that gives the beam profile. Several points are worth noting: (1) the increase in FWHM due to the finite size of the atom is minimal, (2) the signal on the large angle (LA, 60–200 mrad) detector is roughly 3× less than on the small angle (SA, 30–60 mrad), (3) a small shoulder on the probe profile is less pronounced on the detector signals, and (4) the contribution of the

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substrate is essentially negligible. Assuming the effective diameter of the atom adds to the probe profile in quadrature, the effective atom size would be 0.20 Å as seen by the SA detector and 0.1 Å as seen by the LA detector. Therefore, a specimen consisting of single heavy atoms on thin carbon should be a good test of STEM performance. Figures 26 and 27 illustrate tests performed with the Hitachi HD2700C. A similar specimen could be made using gold atoms; we find that the gold-island specimen used to align the corrector has numerous single gold atoms between large clumps. Uranium has the advantage of higher atomic number, giving 1.2 X higher peak signal and a distinctive energy loss peak at 100 eV, which is suitable for EELS and chemical identification. Gold atoms do not have a similarly distinctive spectrum. In addition, gold atoms in clumps tend to stack in an fcc pattern with atoms over gaps in the lower layer, so atom columns are less evident in clumps.

C. Image Simulation Image simulation provides a convenient method for sorting the importance of various factors in image formation. In a practical instrument the factors usually are interrelated in a complex way that makes it difficult to

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understand their individual importance. Simulations allow use of the same sample in a variety of circumstances. Our long-term goal is to develop quantitative imaging to the point where simulations agree with images within a few percent and significant differences indicate that the physics of the specimen is not as postulated. The software package provided by Kirkland (1998) is particularly useful in that it provides absolute intensities (100% of incident electrons are accounted for at the detector or image plane) in both TEM and STEM. We have added our own computer codes and software to generate atom coordinates of up to 107 atoms (floating point coordinates) in a single “unit cell” so that both crystalline and amorphous specimens can be treated in the same way. We developed a subroutine to generate “amorphous” films with specified nearest-neighbor spacing. We also added display and linkage subroutines to expedite studies of imaging and nano diffraction in various scenarios. The Kirkland subroutines provide wave-optical propagation through optical elements and multislice coherent propagation through the specimen using parameterized atomic scattering factors. An additional contribution to apparent spot diameter occurs as a result of thermal vibration of the atoms. This is documented for many crystals but is only a guess for atoms free to move on a surface. Using the crystal value for silicon of 0.075 Å, we expect the thermal contribution to not be

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measurable when added in quadrature. This is calculated by the “frozen phonon” method where atoms are given random displacements with RMS equal to the expected value and the results for roughly 10 runs averaged. In this case, the expected probe diameter would increase from 0.473 Å to 0.479 Å.

D. Quantification in Imaging and EELS As described previously, our long-term goal is to develop quantitative imaging. We have developed a procedure to calibrate the detector signals in terms of percent scattering. This involves reducing the annular detector gain by a factor of 10 or 100, removing the phase-contrast aperture ahead of the BF detector, positioning the direct beam (out of its normal position in annular detector hole) fully onto the annular detector and recording the signal relative to the BF signal obtained with the beam aligned in the detector hole. We then restore the detector gain to high sensitivity, realign the beam, and record AD and BF signals simultaneously during scanning. The normalization is done off-line on the recorded data. At each step care is taken to ensure that signals are never greater than one-half the saturation value of the recording system. With single-atom specimens on a thin amorphous substrate the signals are linear. The simulation software allows us to ask what happens as we stack up atoms or pack atoms into a crystal. The small size of the probe suggests that an atom might block the beam from reaching atoms at lower levels. The actual situation is much more complicated because of the coherence of the beam, giving constructive and destructive interference effects. This is accounted for in the multislice calculation. In the case of stacked uranium atoms, the linearity breaks down after only a few atoms. However, thermal motions disrupt the exact alignment of the columns, giving nearly linear signal versus column height up to four atoms. A slight tilt of the specimen has the same effect. A diffraction detector located in the detector plane permits direct visualization of the intensity of the scattered wavefront. The various annular and solid detectors integrate over portions of the pattern to give fast-response signals suitable for STEM imaging but discard much valuable information. Image simulation provides a convenient means to assess the information contained in the detector plane. Ideally, we would like to record the entire pattern at each scanned pixel and dynamically configure a set of virtual detectors to extract the maximum amount of information. In the extreme case, we would use an algorithm to reconstruct the phases at each point in the detector plane (lost when the electron amplitude is converted to intensity by the detector). With the amplitude (square root of intensity) and computed phase, we could transform back to retrieve the exit wave at the bottom of the specimen. This form of “diffraction imaging” is being pursued by several groups in addition to our own (see Section V.E).

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Presently available detectors are not fast enough to record such a 4D data cube, but progress is rapid in that direction. However, we use the diffraction detector to view the Ronchigram to check the performance of the aberration corrector. In addition, the central portion of the unscattered disc gives a phase-contrast image similar to that observed in TEM. Usually this is detected in STEM imaging by placing a small (3-mrad) aperture ahead of the BF detector. This is comparable to TEM phase contrast with a 3-mrad illumination. Viewing the entire beam disc gives an in-line hologram. The region around the central disc gives a CBED pattern. Both are characteristic of the very local area probed by the beam so they can be used for nanodiffraction studies of defects, grain boundaries, and so on. Modern STEM instruments have several intermediate condenser and projector lenses, as well as apertures of different sizes that can be chosen to give a diffractionlimited probe size commensurate with illumination angle and a wide range of camera lengths that can be optimized for any nanodiffraction experiment (see Table I). EELS is complementary to the diffraction. Both methods rely on keeping the beam stationary with respect to the specimen to within a small fraction of a beam diameter. For diffraction, that can be a matter of a fraction of a second but for EELS it can be many minutes. Aberration correction places proportionately greater demands on specimen drift, scan noise and drift, high voltage, and lens current stability. In addition, the larger illumination angle requires a correspondingly larger entrance aperture to collect most of the energy loss electrons. If the specimen is more than a few atomic layers in thickness, careful attention must be given to depth of focus and channeling. The ability to obtain an image showing atom columns does not guarantee that the beam intensity is probing a single-atom column all the way through the specimen. This question can be addressed by simulation and comparison to standard specimens of known structure. There is also the possibility that an atom column that is “invisible” in the DF image (due to low atomic number) could still give an EELS signal. From a practical point of view, a parallel EELS system (best for large entrance aperture) is relatively slow for spectrum imaging compared with a simpler system with a slit, since a spectrum must be recorded at each scan point. However, the results can be more accurate since the full spectrum is available for background subtraction. Based on these considerations, we have designated several generic types of study to assess the importance of various features and the factors in deciding which instrument to use: (1) isolated catalyst particles (core/shell) on thin substrate: crystal structure, size and shape distribution, surface and bulk composition, and tomography; (2) interface between two crystal phases, sectioned perpendicular to interface: strain due to mismatch, impurities at interfaces; (3) doped crystals, dopant concentration, and segregation; and (4) thin filaments, supported only at ends. In addition to the obvious criteria of resolution and stability, key factors in

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selecting an instrument for a specific application are speed of searching, ease of tilt alignment, eucentricity of tilt stage, ease of selection of operating parameters, and automatic documentation database. In addition, specimen contamination and means to minimize it may be critical to some experiments, as well as specimen exchange time before stable operation can be obtained. We are attempting to arrive at generalizations to assist in allocating projects to specific instruments.

E. Position-Sensitive Coherent Electron Diffraction For many applications, a fraction of an Angstrom improvement in spatial resolution of an instrument is not critical. A real value of a TEM and STEM corrector seems to lie in the large pole-piece gap that facilitates various in situ experiments and greatly increases probe current to achieve column-by-column spectroscopy imaging. A promising and exciting application in aberration-corrected STEM is position-sensitive coherent electron diffraction (PSCED) (Zhu, Wall, and Rehak, 2002), similar to the method recently commercialized by Gatan (2008). In conventional STEMs, analog image intensity is recorded by integrating signals due to electrons passing through each scanning point of the sample over large annular ranges. Although the interpretation of the incoherent angular-integrated images is straightforward, a wealth of structural information is discarded in this angular integration, including the electronic and magnetic signals from the specimen. In PSCED, an electron diffraction pattern is acquired for each scanning position; that is, an intensity distribution as a function of q in reciprocal space is recorded. In other words, we acquire signals of an object in reciprocal space (u, v, w) as a function of each scanning point in real space (x, y, z). For a 32 × 32 area detector, this gives ∼1000 times more structural information for each scanning point than with a conventional STEM. A 1k × 1k detector generates 106 times more data points. The signals of each scanning point would come from three parts: (1) attenuation of the direct transmitted beam, (2) diffracted beams or scattered electrons striking the detector off axis, and (3) deflection of the incident beam off the optical axis due to magnetic or electrical potentials and fields in the specimen. All of these signals can be recorded digitally for quantitative analysis. In general, electrons scattered at small angles are sensitive to charge and spin, whereas those scattered at large angles are sensitive to phonons and atomic positions. The information is paramount in nanoscale structure characterization. A major challenge to this development is processing the large quantity of data generated for each scanning point, as well as the need for an ultrafast detector suitable for PSCED. A novel position-sensitive detector, based on high-resistivity silicon active matrix pixel sensors (AMPS) was recently designed, manufactured, and tested for PSCED at BNL (Chen et al., 2003). Unlike the conventional

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CCD cameras used in electron microscopes, which convert electrons to photons and then back to electrons, the AMPS uses a direct-exposure format, thus eliminating the need for a scintillator and other electron-photon coupling components in the detector; hence, this design significantly improves the quantum efficiency of detection. The solid-state detector also gives fast parallel readout of the number of electrons striking each of the pixels in an area array. Our calculations and preliminary test results suggest that the detection speed for a 32 × 32 array is ∼10, 000 frames per second with detection quantum efficiency better than 99.9%. The electronics noise of the readout corresponds to only about 2000 electron hole pairs. Compared with the 60,000 electron hole pairs generated by a single 200-kV incident electron in the silicon of the detector, at the anticipated low count levels per pixel, the AMPS detector counts the true number of incident electrons in the STEM. With such a detector, we not only have the sensitivity to detect single electrons but also can distinguish pixels with, say, 30 counts from those with 31 counts. To interpret the wealth of new information that will emerge from position-sensitive electron diffraction, we carried out simulations of such an experiment (Zhu, Wall, and Rehak, 2002). Figure 28a depicts the calculated CBED patterns from an undecagold cluster (11 gold atoms) on a 2-nm thick amorphous carbon substrate. The image is composed from 14 × 11 diffraction patterns (one corresponding to each scanning position). (a)

(b)

(c) Au

C

FIGURE 28 (a) Simulated diffraction patterns for 200-kV electrons, based on the proposed position-sensitive coherent electron diffraction technique, from an undecagold cluster (11 gold atoms) on a 2-nm thick amorphous carbon substrate. Parameters used: Cs = 0.6 mm, defocus = 26 nm, and aperture radius = 10 mrad. The beam increment between patterns is 0.05 nm. (b) Enlarged diffraction patterns from (a) showing the large variation between adjacent patterns. (c) Reconstructed image obtained by integrating scattering between 10 and 20 mrad. The structural model that is slightly tilted away from the projected image of the gold cluster is shown on the right.

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Four enlarged patterns are shown in Figure 28b. We used a probe size of 0.15 nm with an increment of 0.05 nm. The black circle in each pattern represents the disk of the transmitted beam, and the intensity within the disk was attenuated by a factor of 100 for better visibility. We note the drastic change in intensity of the diffuse scattering surrounding the center disks when the probe scans across the area at a pace less than the interatomic distance of the specimen. Figure 28c is a reconstructed image from the diffuse scattering of the aperiodic gold cluster on amorphous carbon, together with a structural model of the Au cluster on its right. The integrated intensity of the center disks also can be used to form a BF image that maps the mean inner potential of the scanned area. Simulations based on 106 or more atoms of specified atomic number and spatial coordinates can be used to evaluate feasibility of proposed experiments and optimum imaging conditions.

VI. OUTLOOK Aberration correction provides superior spatial resolution in microscopy and ample opportunities for materials research; however, it does not automatically make interpretation of experimental data easier. The real challenge in electron beam–based structural characterization is quantitative EM, which in our view is the future direction of the field. EM traditionally was used to visualize microstructure to understand mechanical properties; with the advancement in instrumentation, EM now becomes an indispensable tool, not only for materials science and nanotechnology, but also for physics, in parallel with neutron and synchrotron X-ray, to understand electronic and magnetic structure in complex functional materials. Although quantitative data analysis and interpretation are still rare, with our continued efforts, we will be able to dispel the perception that EM is only for imaging and we microscopists only take pictures. As we adapt to the new capabilities at hand, we begin a new wish list. One key consideration is specimen preparation. As depth of focus decreases and resolution improves, we are increasingly concerned about the few layers of contaminant coating the top and bottom of the specimen. The atomic force microscopy (AFM) and scanning tunneling microscope (STM) communities have already faced this issue and moved to UHV systems with integrated specimen preparation hardware. A related issue is the need to examine specimens in more than one instrument to obtain complementary information. Exposure to air and poor load-lock vacuum limits this. A UHV transit system standard is needed for all analytical instruments. Unfortunately, for EM, that probably means stage designs not dependent on specimen rods.

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Another area sure to develop rapidly is direct electron detectors with smart electronics in the detector chip. These devices could be configured to approximate the bandwidth of present detectors but with many additional channels giving relevant specimen parameters. Similar detectors for EELS could enable spectrum imaging at the same rate as normal STEM imaging. For EELS detector, the ability of simultaneous acquisition of zero-loss and core-loss spectra (with attenuation for zero-loss) is desirable for fine structure and chemical shift analysis. Faster and more sensitive electron detectors are urgently needed not only for real-time aberration corrector alignment but for in situ experiments to observe structural dynamics and transient states in materials. With better detectors and more coherent beams, PSCED could become a reality, providing separate amplitude and phase maps (Volkov, Wall and Zhu, 2008) in reciprocal space of the specimen for studies of electronic and magnetic properties at nanometer scale.

ACKNOWLEDGMENTS We thank T. Nomura, H. Inada, and G. Hom for installing and testing the JEM2200CMO, Hitachi HD2700C, and FEI Titan 80-300, respectively, at BNL. We also thank C. Channing and O. Dyling for their useful discussions and contribution to Section II of this chapter, and R. F. Klie and C. Johnson for testing the JEM2200FS and 2200MCO. We acknowledge and appreciate, the assistance of various people in recording the data used in figures: Nomura (Figure 9), H. Inada (Figures 16 and 17), Y. C. Wang and L. Fu (Figures 21, 22, and 23). The work at BNL was supported by the U.S. Department of Energy, Division of Materials Science, Office of Basic Energy Science, under contract no. DE-AC02-98CH10886.

REFERENCES Allard, L. F., et al. (2005). Design and performance characteristics of the ORNL advanced microscopy laboratory and JEOLFS-AC aberration-corrected STEM/TEM. Microsc. Microanal. 11, 2136–2137. Batson, P. E., Dellby, N., and Krivanek, O. L. (2002a). Sub-angstrom resolution using aberration corrected electron optics. Nature 419(6902), 94. Batson, P. E., Dellby, N., and Krivanek, O. L. (2002b). Sub-angstrom resolution using aberration corrected electron optics. Nature 418(6898), 617–620. Borisevich, A. Y., Lupini, A. R., and Pennycook, S. J. (2006). Depth sectioning with the aberration-corrected scanning transmission electron microscope. Proc. Natl. Acad. Sci. USA 103(9), 3044–3048. Chen, W., et al. (2003). High resistivity silicon active pixel sensors for recording data from STEM. Nucl. Instrum. Meth. Phys. Res. A 512, 368–377. Haider, M., et al. (1998). Electron microscopy image enhanced. Nature 392(6678), 768–769. Haider, M., Uhlemann, S., and Zach, J. (2000). Upper limits for the residual aberrations of a high-resolution aberration-corrected STEM. Ultramicroscopy 81, 163–175. Hytch, M. J., Snoeck, E., and Kilaas, R. (1998). Quantitative measurement of displacement and strain fields from HREM micrographs. Ultramicroscopy 74(3), 131–146.

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Inada, H., et al. (2008a). The newly installed aberration corrected and dedicated STEM (Hitachi HD2700C) at Brookhaven National Laboratory. Proceedings of Microscopy and Microanalysis Conference, August 3–7, 2008, Albuquerque, New Mexico. Inada, H., et al. (2008b). Uranium single atom imaging and EELS mapping using aberration corrected scanning transmission electron microscope and LN2 cold stage. Proceedings of Microscopy and Microanalysis Conference, August 3–7, 2008, Albuquerque, New Mexico. Jia, C. L., and Urban, K. (2004). Atomic-resolution measurement of oxygen concentration in oxide materials. Science 303(5666), 2001–2004. Jia, C. L., Lentzen, M., and Urban, K. (2003). Atomic-resolution imaging of oxygen in perovskite ceramics. Science 299(5608), 870–873. Johnson, C. L., Hÿtch, M. J., and Buseck, P. R. (2004). Nanoscale waviness of low-angle grain boundaries. Proc. Natl. Acad. Sci. 52, 17936–17939. Jooss, C., et al. (2007). Polaron melting and ordering as key mechanisms for colossal resistance effects in manganites. Proc. Natl. Acad. Sci. USA 104(34), 13597–13602. Kimoto, K., et al. (2007). Element-selective imaging of atomic columns in a crystal using STEM and EELS. Nature 450, 702–704. Kirkland, E. J. (1998). “Advanced Computing in Electron Microscopy.” Plenum Press, New York. Klie, R. F., et al. (2005). Enhanced current transport at grain boundaries in high-T-c superconductors. Nature 435(7041), 475–478. Klie, R. F., Johnson, C., and Zhu, Y. (2008). Atomic-resolution STEM in the aberrationcorrected JEOL JEM2200FS. Microsc. Microanal. 14, 104–112. Mukai, M., et al. (2005). Performance of a new monochromator for a 200 kV analytical electron microscope. Microsc. Microanal. 11, 2134–2135. Muller, D. A., and Grazul, J. (2001). Optimizing the environment for sub-0.2 nm scanning transmission electron microscopy. J. Electr. Microsc. 50, 219–226. Nakamura, K., et al. (2006). Development of a Cs-corrected dedicated STEM. Proceedings of 16th International Microscopy Congress, 2006, Sapporo, Japan, pp. 633–634. Rose, H. (1994). Correction of aberrations, a promising means for improving the spatial and energy resolution of energy-filtering electron microscopes. Ultramicroscopy 55, 11–25. Sawada, H., et al. (2005). Experimental evaluation of a spherical aberration-corrected TEM and STEM. J. Electr. Microsc. 54, 119–121. Scherzer, O. (1936). Uber einige Fehler von Elektronenlinsen [Some defects of electron lenses]. Optik 101, 593–603. Scherzer, O. (1947). Sparische and chromatische korrektur vonelektronen-linsen. Optik 2, 114–132. Twesten, R. D., et al. (2998). Mixing real and reciprocal space. Proceedings of 14 European Microscopy Congress, September 1–5, 2008, Aachen, Germany. Volkov, V. V., and Zhu, Y. (2003). Phase imaging and nanoscale currents in phase objects imaged with fast electrons. Phys. Rev. Lett. 91(4), 043904. Volkov, V. V., Wall, J., and Zhu, Y. (2008). “Position-sensitive diffractive imaging in STEM by an automated chaining diffraction algorithm”, Ultramicroscopy 108, 741–749. Wall, J. S., Hainfeld, J. F., and Bittner, J. W. (1978). Preliminary measurements of uranium atom motion on carbon films at low temperatures. Ultramicroscopy 3, 81–86. Wu, L. J., Zhu, Y. M., and Tafto, J. (2000). Picometer accuracy in measuring lattice displacements across planar faults by interferometry in coherent electron diffraction. Phys. Rev. Lett. 85(24), 5126–5129. Wu, L., et al. (2007). Experimental confirmation of Zener-polaron-type charge and orbital ordering in Pr1-xCaxMnO3. Phys. Rev. B 76(17), 174210. Zhu, Y., et al. (2005). The aberration corrected JEM-2200FS at Brookhaven. Proceedings of Microscopy and Microanalysis Conference, July 31-August 4, 2005, Honolulu, Hawaii 11(Suppl 2), 1428 CD.

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Zhu, Y., et al. (2007). Nanoscale disorder in CaCu3 Ti4 O12 : a new route to the enhanced dielectric response. Phys. Rev. Lett. 99(3), 037602. Zhu, Y., Wall, J., and Rehak, P. (2002). Understanding aperiodic structure and non-spherical charge density, electron orbital and spin of functional materials. US DOE Transmission Electron Aberration-free Microscopy (TEAM) project R&D proposal. Zuo, J. M., et al. (1999). Direct observation of d-orbital holes and Cu-Cu bonding in Cu2 O. Nature 401(6748), 49–52.

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CONTENTS OF VOLUMES 151 AND 152

VOLUME 151 C. Bontus and T. Köhler, Reconstruction algorithms for computed tomography L. Busin, N. Vandenbroucke, and L. Macaire, Color spaces and image segmentation G. R. Easley and F. Colonna, Generalized discrete Radon transforms and applications to image processing T. Radlicka, Lie agebraic methods in charged particle optics ˇ V. Randle, Recent developments in electron backscatter diffraction

VOLUME 152 Nina S. T. Hirata, Stack filters: From definition to design algorithms Sameen Ahmed Khan, The Foldy–Wouthuysen transformation technique in optics Saverio Morfu, Patrick Marquié, Brice Nofiélé, and Dominique Ginhac, Nonlinear systems for image processing Tohru Nitta, Complex-valued neural network and complex-valued backpropagation learning algorithm Jerome Bobin, Jean-Luc Starck, Yassir Moudden, and Mohamed Jalal Fadili, Blind source separation: The sparsity revolution Ray L. Withers, “Disorder”: Structured diffuse scattering and local crystal chemistry

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INDEX

A Aberration coefficients, 44, 76, 175, 270, 291, 317, 444 axial, 290 coherent, 287, 289 determination of, 289, 292, 295 fifth-order aperture, 15 of hexapole-type correctors, 78 nomenclature, 288 values, 271 weighing factors for, 131 Aberration-corrected convergent-beam electron holography, 240–245 Aberration-corrected electron diffraction, 245–252 Aberration-corrected electron microscopes, 482. See also FEI Titan 80-300; Hitachi HD2700C; JEOL JEM2200MCO applications of, 140–154 comparison of, 507–510 environmental requirements and laboratory design for, 483–492 features of, 508 high-accuracy laboratories for, 485 control room, 488 design criteria, 485 equipment room, 487–488 instrument room, 487 performance of, 129–140 Aberration-corrected probes, 139 Aberration correction, 123, 130, 308–319 advantages of, 247–248 birth of, 5–9 combining, 279 complications resulting from, 138–140 concepts, 261–279 in CTEM, 312–315 amplitude contrast imaging, 314–315 chromatic aberration limited imaging, 313–314 C5,0 -limited imaging, 312–313 imaging theory, 308–311 failure to achieve, 18 by non-round lenses, 124–129 optimal conditions, 315–319 in high-resolution STEM, 311–312 and strain mapping, 232–239

Aberration correctors CEOS DCOR, 343, 346 future generation of, 377–378 on HB501UX, 330 Nion, 166, 328, 331 present generation of, 377 for TEMs, designers of, 126 Aberration-free imaging, criterion for, 269 Aberration function, 124–125, 130, 443–444 Aberration ray fourth-order, 92 second-order, 75–76, 80–81, 83, 104 of threefold astigmatism, 104 Aberrations axial, 44–45, 51, 100–101 axial misalignment, 8 chromatic, 10, 14–15, 32, 47, 69, 89–90 combination, 126–128 correction of systems with curved axis, 25–28 fifth-order, 93 fourth-order, 91–93 fourth-order off-axial, 93 geometric, 129, 132–133, 136 gun, 138 of hexapole fields, 72–79 higher-order, 91, 131 and information transfer, 44–46 integral kernels, 106 lower-order, 129 measuring, 456 misalignment, 126, 128 negative sign of spherical, 20 off-axial, 44–45, 101, 103 optical, 44 parasitic, 126–127 precise control of, 126 proportions of, 128 residual, 63 second-order, 15, 198 elimination of, 20, 26 size of, 127 spherical. See Spherical aberration third-order, 82–87 third-order spherical, 20, 45, 53, 68, 85, 87, 330, 444–445 three-lobe, 91–92 Achromatic aplanatic system, 32

527

528

Index

Active matrix pixel sensors (AMPS), 518–519 Air handlers, 488 Airy disk, 421–422, 431 Allotaxy process, 202 AlN double column, 180–181 AlN/GaN/AlN quantum well structure, 179–183 low-magnification image of, 180 multislice calculation, 182–183 on sapphire substrate, 180–181 Amorphous films, 367, 432, 515 Ampere’s law, 72 Anamorphotic image, 31–32 Angle resolved electron energy loss spectroscopy (AREELS), 421 Annular dark-field (ADF) detector, 306 of gold atom, 177 Aperture angle, 67, 511 probe diameter as function of, 512 APL2, 167, 174 Aplanatic system, 70 Aplanator, 16–17 Aplanats, hexapole. See Hexapole aplanats AREELS, 421 Astigmatic line images, 10 Atom columns, 451 Atomic objects, three-dimensional observation of, 430 Atomic-resolution imaging, 140–142 Atomic-resolution microscope (ARM) project, 47 Atomic-resolution spectroscopy, 142–147 Atomic scattering, 455 Atomic shifts, 472–473 in domain, 471 measurements of, 474 Atom motion, 513 Atom spacings, 466, 477 Auger electron, 142 Axial aberrations, 44–45, 51, 100–101 Axial coma, parasitic second-order, 14 Axial fourth-order aberrations, hexapole aplanats without, 105–112

B Ball-and-stick model, 202–204 Biological systems, 217–221 BNL, 482 aberration-corrected instruments comparison of, 507–510 features of, 508 FEI Titan 80-300 ETEM, 501–507

Hitachi HD 2700C STEM, 498–501 JEOL JEM2200FS and JEM2200MCO TEM/STEM, 492–498 Bright-field–dark-field (BF–DF) detector, 139, 228 Brookhaven National Laboratory. See BNL Bulk crystal, 199–201 colossal magnetoresistive behavior of, 199 orientation of, 201 reconstruction, 205–206 Burgers vector, 205

C Ca doping, 363 Carbon nanostructures. See Carbon nanotubes Carbon nanotubes damage to multiwall, 218 high-resolution BF image of, 220 toward low voltage, 217 Castaing–Henry filter, 26 Catalysts, 374–377 Cation columns, 465 CBED, 237, 240, 245, 496 CHEF configuration in, 241 Cc correction, benefits of, 265–279 biological samples, 275 elemental mapping, 270–272 high-resolution TEM, 265–270 Lorentz microscopy, 276–277 in situ TEM, 270–274 tomography, 275 ultra-fast TEM, 277–279 CCD cameras, 122, 228, 423 CCD detectors, 148–149, 328 C3 -corrected system, 264 CdTe single crystal, 332 CEMES, 225–227 Center for Functional Nanomaterials. See CFN CEOS, 55 CEOS CCOR probe corrector, 342 CEOS DCOR aberration corrector, 343, 346, 377 CEOS monochromator, dispersion-free, 26 Ceramics, structure-property relations in, 367–371 CESCOR, 57, 61 graphical user interface for control and alignment of, 63 for Hitachi HD2700, 59

Index

CETCOR, 57, 61 CFEGs STEMs, theoretical probe-forming performance of, 135, 137 CFN, 482, 485 high-accuracy lab (1L30) in, 486 AC field measurement at, 491 temperature measurement at, 490 triaxial vibration measurement at, 489 C60 fullerene molecule, 419 Charge density approximation, by aberration correction, 393 Charged particle optics instrument, properties, 44 Charge-ordering domains, 200 CHEF, 24, 240 aberration-corrected, 240–245 configuration in LACBED mode and CBED mode, 241 Chemical mapping, 142–147, 154 Chromatic aberrations, 10–11, 47, 69, 89–90, 133, 136, 138, 141, 144, 175 coefficient, 278 correction of, 14–15, 19, 32 effect, 132 Chromatic focus spread, 45, 58, 60, 65, 100 CM200F, unmodified objective lens of, 49 Coefficient function, 77, 85–87 Coherence effects, in CTEM and STEM, 297–308 Coherence envelope functions, 304 Coherent aberration coefficients, 287, 289 Coherent imaging, effect of partial coherence on, 298–305 effective source (CTEM)/effective detector (STEM), 298 numerical calculation of image intensity, 302–303 under partially coherent illumination, 297–305 transmission cross-coefficient, 299–302 for strongly scattering objects, 303–304 for weakly scattering objects, 304–305 Coherent imaging mode, 297 Cold field emission guns. See CFEGs Colossal magnetoresistive behavior, of bulk crystal, 199 Coma-free point, 21 of objective lens, 23 optically matching, 23

529

Coma-free transfer doublet, 22–23 Coma integral, 109 Combination aberrations, 126–128 Complex dislocation core structures, 363–366 example of, 364, 366 Complex oxides, 355–361 Condensed Matter Physics and Materials Science (CMPMS), 482 Contrast delocalization, 268, 447 defined, 458 Contrast theory. See also Scherzer theory for aberration-corrected TEM, 447–448 Contrast transfer function. See CTF Conventional transmission electron microscope. See CTEM Convergent-beam electron diffraction. See CBED Convergent-beam electron holography. See CHEF Corrected electron optical systems. See CEOS Cowley’s phase grating, 397 Crystalline objects, imaging of, 176–179 Crystal orientation, of BaTiO3 , 458 Crystal structure imaging, 451 Cs -corrected CM200F, information limit of, 52 Cs -corrected CTEM, 47, 53, 56 Cs -corrected HRTEM, 410, 418 Cs -corrected STEM, 424 advantages for observation, 424 atomic objects in, 430 depth of focus, 430 and EELS mapping, 427 experiments for 3D observation, 431–433 first-principle calculation, 428 future prospect of, 433 optimum defocus condition, 424 point-to-point resolution, 424–427 Cs -corrected TEM, 392 advantages for observation, 406–424 discrimination of elements in, 410–412 for electron holography, 422–424 future prospect of, 433 information limit of, 46 Lentzen’s optimum defocus for, 394–395 point-to-point resolution, 406–408 of Pt-Rh chain molecules, 417 selected area nanodiffraction in, 419–422 subtraction method of nonlinear contrast in, 399–404

530

Index

Cs -corrected TEM (continued) without Fresnel fringes, 409 z-slice and tomographic imaging in, 414–419 CTEM, 44–45 aberration-corrected imaging conditions in, 312–315 aberration measurement in, alternative approaches to, 293–295 bright-field imaging in, 45 coherence effects in, 297–308 Cs -corrected, 47, 53, 56 hexapole corrector, 59–60, 88, 103–112 transfer lens, 57, 88 imaging theory, 308–311 with objective lens, 88 off-axial coma, 70, 90 PCTF of uncorrected, 48 phase-contrast imaging in, 45 point resolution of, 47 spherical aberration in, 47 wave aberration function in, measurement of, 289–295 CTF, 53, 170, 177, 179, 266, 278, 395, 446 modulus of, 449 for NCSI conditions, 446 phase and amplitude, 447 Cubic zinc-blende structure projects, 371

D Darmstadt correction project, 12–17 D-COR-type corrector, 66 multipole elements, 62 Debye–Waller factor, 469 Delocalization, 25, 34 phase contrast, 447 Density functional theory, 203 Detective quantum efficiency (DQE), 148 Detector annular dark-field (ADF), 306 bright-field-dark-field (BF-DF), 139, 228 charge-coupled device (CCD), 148–149, 328 diffraction, 516–517 high-angle annular dark-field (HAADF), 169 medium-angle annular dark-field (MAADF), 498 Diamond crystal, 407 Dichroic signal, 251 Diffracted beam holography methods, 240–245

Diffraction detector, 516–517 Diffraction-limited resolution, 35, 130 Diffraction patterns, through LACDIF, 246–248 Diffraction plane, anamorphotic image of, 32 Diffractograms, 51, 53, 292 Diffractogram tableau method, 24 DigiScan (Gatan) scanning, 228 Dipoles, 126 Dislocation, misfit, 206 Divalent cation, 355 Dodecapole elements, 17–18 producing the hexapole fields, 24 Domain boundaries, 205 Dumbbells, 229

E Edge dislocations, 205 EDXS (Energy-dispersive X-ray spectroscopy), 123 EELS, 67, 142, 147, 164, 195–221, 251, 340, 355 column-by-column, 363 elemental mapping, 143, 226 energy spread time trace for zero loss peak in, 500 mapping, 210, 427 monochromation approach, 142 quantification in, 516–518 at RP defect, 209 simulations, 354 SNR of, 143, 148 spectrum-imaging approach, 144 transitions, 269 visibility of, 152 EFTEM, 143–144 Elastic scattering, 139, 143, 200. See also Inelastic scattering Electrocatalysts, complex metal oxides as, 206 Electromagnetic field-cancellation system, 483 Electromagnetic interference, 484 Electron-atom interaction, 387 Electron beam, atomic movement under, 190–192 Electron diffraction, aberration-corrected, 245–252 Electron diffraction interferometry, 240 Electron energy loss magnetic chiral dichroism, 251–252 Electron energy loss spectroscopy. See EELS

Index

Electron guns, Schottky, 135 Electron high tension (EHT), of IBM VG HB501 STEM, 164, 167 Electron holography, 422 advantages of Cs -correctors in, 422–424 Electron lenses, properties of, 121 Electron microscopes (EMs), 44, 126 Darmstadt corrected, 17 elemental mapping in, 142 projection approximation in, 387–389 resolution limit of, 5, 15 failure for improving, 7 improvement of, 8, 19, 25 in situ, sub-angstrom and sub-electron volt, 30 types of, 122 Electron microscopy (EM) suite, design and construction, 485 Electron monochromator, 142, 167 Electron-optical aberrations, of round lenses, 4 Electron optical correctors, 285 with curved axis, 25–28 Electron optical elements, 285 Electron optical parameters, 277 Electron precession technique, 246 Electron probes, 133, 154 Electron-sensitive materials, 250 Electron wave field, 441–442 Electron wavefront, 124, 132, 175 Electrostatic corrector. See Scherzer corrector Electrostatic-magnetic quadrupole, 12 Elemental mapping, 142–147, 270–272 EMBL hexapole corrector, 48–55 alignment of, 63 design of, 48–49 transfer lens, 48–49 twelve-pole design, 50–51 project, 46–48 quadrupole-octupole corrector, 18–19 Energyfiltered TEM. See EFTEM Energy loss magnetic chiral dichroism (EMCD), 251 Energy monochromatization, 265 Energy spectrum imaging (ESI), 252 Environmental cell (E-cell), 503 Environment TEM (E-TEM), 503, 510 EPWF, 441–442, 455 amplitude and phase of complex, 465 line profiles referring to phase of, 467

531

European Molecular Biology Laboratory. See EMBL Exit-plane wave function. See EPWF

F Fast Fourier transformation (FFT), 50 FEI Titan 80-300, 346, 377 FEI Titan 80-300 ETEM, 501–507 energy resolution test, 506 information limit test, 505 in situ vacuum system of, 504 Ferritin, 218–220 iron atoms in, 220 STEM HAADF images of, 219 three-dimensional reconstruction of, 219 Ferroelectric domain walls atom shifts in, 471 inversion, 472 LDW and TDW segments of, 473 mixed-character, 473 structure and polarization in PZT, 469–477 Ferroelectric thin films applications of, 470 heterostructure, 472 unit cell of, 472 Field emission gun (FEG), 228 Fifth-order aberrations, 93, 329 Fifth-order spherical aberration, 93 transfer lenses adjusted for, 95 tuning, 94–96 First-order Laue zone (FOLZ), 248–249 First-order Wien filter, 10–11 First-principle calculations, 361–362, 375, 428 First-principles theory, 375 Z-contrast image and, 370 Focused ion beam (FIB), 198, 238 Fourier termination effect, 405 Fourier transform (FT), 300–301, 348, 388 Fourth-order aberration rays, 92 Fourth-order aberrations, 91–93 Fourth-order off-axial aberrations, 93 Fresnel fringes, sharp interface images without, 409 Fresnel’s first zone, 387 Frozen phonon model, 187, 516 Fullerene molecule, 419 Full width at half maximum (FWHM), 130, 133, 311, 329, 500, 513 Fundamental rays, 31, 33 doubly symmetric course of, 28 second-order, 20–21 FWHM, 130, 133, 311, 329, 500, 513

532

Index

G Gatan imaging filter (GIF), 228 Gaussian distribution, 301 Gaussian function, 406 Gaussian image plane, defined, 444–445 Gaussian regression analysis, 475 Gd atoms, 148 Generic doping, 357 Geometric aberrations, 129, 132–133, 136 Geometric phase analysis (GPA), 232 Gold atom on amorphous carbon substrate, 190–191 annular dark-field (ADF) of, 177 histogram analysis of, 174 probe profile for, 514 Gold clusters, 212–216 HAADF images of, 214 magic number of, 213, 215 size of, 213 thiolated clusters of, 215 Gold nanoparticles, 375–376

H HAADF detector, 169 HAADF images, 139–140, 198–200, 205 of Au island on carbon, 172 of BaTiO3 , 498–499 of gold atom clusters, 214 of HfO2 gate stack, 183–187 of isolated InGaAs QD, 213 of LCCO materials, 208 multislice simulated, 211 of single Au atom on carbon, 173 of uranium atoms and clumps, 501 HAADF-STEM, of BaTiO3 , 498–499 Half-spacing lattice fringes, 396 HB501UX, 330 Hertzian active damping antivibration system, 167 Hexapole aplanats, 101–114 integral formulation of essential requirements for, 104–105 versus semi-aplanat, 101–102 without axial fourth-order aberrations, 105–112 minimum of four hexapoles, 109–111 three hexapoles only, 106–108 three hexapoles plus two weak hexapole pairs, 109 Hexapole correctors, 70, 80, 312 aberration coefficients of, 78 arrangement of elements of, 21

axial fifth-order coefficient functions for, 97 basic setup, 80–82 chromatic aberration, 89–90 commercialization of, 55–61 correction strength, 87–89 for CTEM, 59–60, 88 development of, 46 EMBL. See EMBL evolution of, 19 field aberrations, 90 future, 69–70 magnetic field, 71 progress of, 61–69 properties of, feasibility and prediction of, 112–114 semi-aplanatic condition, 90 third-order aberration coefficients for, 82–87 Hexapole doublet, 81, 83, 101 Hexapole elements, 70–72, 82, 87 magnetic, 71, 73 Hexapole fields aberrations of, 72–79 equation of motion, 72–73 primary aberrations, 75–79 secondary aberrations, 79 successive approximation, 74–75 dodecapole elements producing, 24 Hexapoles, 70, 127 activation, 244 deactivation, 242–244 Hexapole strength, 72–73, 87 Hf atoms, 336, 339 HfO2 gate stack structure, 183–187, 189 bright-field image, 184 HAADF image, 183–187 SiO2 /Si interface in, 186 High-angle annular dark-field. See HAADF High-resolution electron microscopy (HREM), 503 High-resolution STEM imaging, 311–312 High-resolution transmission electron microscopes. See HRTEM High-resolution Wien filter spectrometer, 164–165 High-temperature superconductors, 361–363 High-voltage scanning transmission electron microscopy (HV-STEM), 386 Hitachi HD2700, CESCOR for, 59

Index

Hitachi HD2700C STEM, 487, 498–501 collection angles in, 499 features of, 498 HAADF image of BaTiO3 , 498–499 of uranium atoms and clumps, 501 Hole-hopping, 206 Holograms, 254–255 HRTEM, 231–232, 274, 387, 483 Cc correction, benefits of, 265–270 conventional, comparison of, 233, 238 Cs -corrected, 410, 418 CTF in, 395 imaging theories of, 387–406 issues of, 389 nonlinear terms in, 399 phase-contrast images in, 412 resolution of, 390 HRTEM strain mapping, limitations in, 232–239 precision, 232–236 contrast, 232–234 noise, 235–236 uniformity of contrast, 234–235 spatial resolution, 236–237 strained silicon nanostructures and devices, 238–239 thin-film relaxation, 237–238 HV-STEM. See High-voltage scanning transmission electron microscopy Hysteresis effect, 18–19

I IBM high-resolution STEM/EELS system, 164–167 IBM Vacuum Generators (VG) HB501 STEM, 164–167 electron high tension (EHT) of, 164, 167 modifications of, 165 Nion aberration corrector in, 166 Image aberration, 286 Image deconvolution method, 404 Image-processing techniques, 232 Image resolution, measurement and definition, 348–354 Image simulation, 402–404 subtraction method on LiCoO2 crystals, 403 Incoherent imaging, partial coherence for, 306–308 Incoherent imaging modes, 297 In-column filter, 496

533

Inelastic scattering, 153, 200, 274 delocalization of, 143, 148–149 localization of, 153 Information limit, 25, 32, 44, 46–47, 52, 446, 494, 505. See also Resolution limit Inodecahedral model, 216 InP barrier layers, 372, 374 In situ electron microscope, sub-angstrom and sub-electron volt, 30 In situ TEM Cc correction, benefits of, 270–274 improving resolution for, 262–265 Instrumental resolution, 45 Intensity distribution, 442 Intensity function, 461 Interaction factor, 388 Interface energies, 205 Interface structures, 202–211 cobalt disilicide/silicon eightfold, 205 defects or reconstructions on, 207 cobalt disilicide/silicon sevenfold, 204, 206 nickel disilicide/silicon, 203 Interferometer, 454 Isoplanatic approximation, 286 Isotropic coma, 22

J Jahn–Teller distortion, 357 JEM2010F, 485 JEM2010FEF, 485, 492 JEM2200MCO TEM/STEM, 492–498 electro-optical design improvements, 495 HREM imaging of SrTiO3, 496 image quality under airflow condition, 494 information limit of, 494 with omega filter, 493–494, 497, 508 Ronchigram analysis, 495 of silicon single crystal, 495 JEOL JEM2200FS TEM/STEM, 483–484, 492–498 with in-column energy filter, 485 with omega filter and probe corrector, 484

L La atoms, 333 “Laboratoire d’Optique Electronique” (LOE-CNRS), 225

534

Index

LACBED, 240 CHEF configuration in, 241 LACDIF applications of, 248–252 electron energy loss magnetic chiral dichroism, 251–252 electron-sensitive materials, 250 space-group determination, 248–249 configuration, 247 diffraction patterns through, 246–248 improved diffraction patterns through, 246–248 Large-angle convergent-beam electron diffraction. See LACBED Larmor rotation, 20, 70, 82, 90 Lattice parameter, 461, 473, 475–476 Laue orientation, 442, 455, 461 LDW, 472–473. See also Transversal domain wall (TDW) domain walls, 474 quantities of structural and electric behaviour of, 475 Least-square fitting, 455 Lens transfer function (LTF), 393 Lentzen’s optimum defocus, 394–395 Lichte defocus, 310–311 Lichte focus, 395 LiCoO2 crystals, subtraction method on, 403 Linear imaging theory, 389–393 Linear kinematic imaging theory, 449 Local intensity modulation, 450 Longitudinal domain wall. See LDW Lorentz lens, 252, 276, 503 Lorentz microscopy Cc correction, benefits of, 276–277 of dynamic magnetic processes, 279 Low-energy electron microscope (LEEM), 27 Low-voltage scanning electron microscope. See LVSEM LVSEM arrangements of elements of, 19 Cc - and Cs -corrected, 46, 52 chromatic and spherically corrected, 18

M Macrophage cells, 220 Manganites, 355–361 Maximum entropy (ME) algorithm, 425 Mean inner potential (MIP) contributions, 254

Medium-angle annular dark-field (MAADF) detector, 498 Medium-range order (MRO) structures, 409 Minimum phase contrast, 309 Misfit dislocation, 206 Mn oxidation, 356 Modulation contrast transfer function (MTF), 238 Moiré fringes, 415 Molecular beam epitaxy (MBE), 179 Monochromatic approximation, 286 Monochromatization energy, 265 Monochromator, 492 Monte Carlo integration, 303 MOSFET, 239 Multiple scattering, 149, 274–275 Multipole quintuplet, 33 Multipoles, 126 Multislice image simulations, 187–190 Multiwall carbon nanotubes, damage to, 218 Mutual coherence function, 299–300

N Nanostructures, 211–217 Nanowires layers, 372 N-beam dynamical theory, 394 N-beam lattice images, 397–398 Negative aberration constants, imaging of, 398–399 Negative spherical aberration imaging (NCSI) technique, 448–450 Nion aberration correctors, 328, 331 in IBM VG HB501 STEM, 166 Nion quadrupole-octupole corrector, 315, 317 Nion third-order STEM corrector, results using, 168–174 Nion UltraSTEM, 136, 377 Nodal points, 21, 23 Nodal rays, 10 course of x- and y-components of, 32 Nonlinear image contrast, 395–398 removal of, 412–413 solution, 413 Nonlinear least-square regression, 466 Nonlinear terms formation by g and –g waves, 399–400 by g and h waves, 400–402 Non-round lenses, aberration correction by, 124–129 Nucleation, 372

Index

O Oak Ridge National Laboratory (ORNL), 328 Objective lens, 8, 58, 70 anisotropic off-axial coma of, 70, 101 of CM200F, 49 coma-free aperture plane of, 90, 94 coma-free point of, 23 cross section of, 17 CTEM with, 88 field distribution of, 14 spherical aberration of, 45, 70, 83 Object-oriented micromagnetic framework (OOMMF), 255 Occupancy, 454 defined, 452 mean value, 462 oxygen, 457–463 Octupoles, 126–127 Off-axial aberrations, coma type, 44–45, 101, 103 delocalization area of, 102 fourth-order, 93 for semi-aplanatic system, 90 Omega filter, 26–27 isochromaticity of, 496 JEM2200FS, 484 JEM2200MCO, 493–494, 508 acceptance angle of, 497 Optical aberrations, 44 influence of, 242–245 Optical elements, 126–128 types of, 126 Optical transfer function (OTF), 307 Optimum defocus condition, 424 Oxidation state, 220 Oxygen-atom columns, 452 calculated peak intensity of, 459 evaluating position of, 465 in magnesium oxide crystal (MgO), 411 occupancy in twin boundary of BaTiO3 , 457–463 phase maxima on, 468 Oxygen K-edge spectra, from RP defect and bulk LCCO, 210 Oxygen vacancies, 208, 363, 463

P Parasitic aberrations, 126–127 Parasitic axial astigmatism, 6 Partial coherence aspects to, 297

535

effect on coherent imaging. See Coherent imaging for incoherent imaging, 306–308 Partial spatial coherence, 297, 299, 307 Partial temporal coherence, 297, 299, 307, 318 PbS nanocrystals, 343 Peak-finding analysis, 232 Phase contrast delocalization, 447–448 negative, 450, 452 positive, 445, 450, 452 Scherzer theory for, 446 of TEM, 444–450 Phase contrast index function (PCI), 295 Phase contrast transfer function (PCTF), 47, 262, 266, 273, 309, 390 transfer characteristic of, 390 of uncorrected 200-kV CTEM, 48 Phase correlation function (PCF), 293–294 Phase grating approximation, 388, 392–393 Phase shifts, 262, 270, 443, 446 Photo-emission electron microscope (PEEM), 27 Pixon reconstructed image, 367, 369 Planar defects, 202–211 Plasmon shift map, 220 Pnma unit cell, 355, 357 Point spread function (PSF), 390, 444, 448 Point-to-point resolution, 406–408, 424 Pole faces, of deflection magnet, 27 Position-sensitive coherent electron diffraction (PSCED), 518–520 Post-column filter, 27 Praseodymium (Pr) atoms, 429 Principal rays, 10 course of x- and y-components of, 32 Probe-forming performance, of CFEG STEMs, 137 Probe intensity, FWHM of, 330 Probe profile for gold atoms, 514 for single uranium atoms, 512–514 Probe properties, calculation of, 174–176 Probes aberration-corrected, 139 electron, 133, 154 Probe size, 66–67, 129–138, 170, 520 Probe wave function, 175 Projected density of states (PDOS), 357 Projection approximation, in electron microscope images, 387–389

536

Index

Pseudocubic zone axis, LaMnO3 image in, 352 Pseudo-Lorentz mode, for medium-resolution electron holography, 252–256 Pt-Rh chain molecules, 414 atomic arrangement of, 416 Cs -corrected TEM images of, 417 series of simulation images of, 417 Pulsed electron emitters, 277

Q QDs applications of, 211 buried InGaAs, 211–212 characterizing, 212 HAADF image of isolated, 213 3D geometry of, 213 properties of, 211 Quadrupole fields, 9 excitation of, 17, 32–33 symmetric and antisymmetric, 31 Quadrupole-octupole corrector, 46, 52, 128–129, 132 Chicago and EMBL, 18–19 revival of, 29–33 Quadrupole-octupole quintuplet, telescopic, 32 Quadrupole quadruplet, telescopic, 10 Quadrupoles, 31, 126, 128 Quadrupole triplet, electric-magnetic, 15 Quantitative high-resolution electron microscopy (Q-HRTEM), 227 Quantitative imaging, 516–518 Quantum dots. See QDs Quantum wires, semiconductor, 371–374

R Radiant cooling panels, 489 Rayleigh criterion, 328 Reduction factor, 131 Residual aberrations, 63 Resolution limit reduction of, 24 Scherzer, 23–25 of STEM, 29 sub-angstrom, 30 Rocking-curve phase, 245 Ruddleston-Popper (RP) defect, 206–207 EELS at, 209

S SACTEM-Toulouse, 225, 227 description of, 228–229

optical elements of, 228, 252 performance, 229–232 Scan coma, elimination of, 100 Scanning transmission electron microscopy. See STEM Scattering elastic, 139, 143 and imaging problem, inversion of, 452–455 inelastic. See Inelastic scattering multiple, 149, 274–275 Scherzer conditions, 231, 310 Scherzer corrector, 6 Scherzer defocus, 394, 409 Scherzer focus, 47 Scherzer resolution limit, 23–25 Scherzer theory, 4, 389–393, 445–447 for phase contrast, 446 Schottky electron guns, 135 Schottky field emission gun, 65–66, 342 Second-order aberration, 15 eliminating, 20 Second-order aberration rays, 75–76, 80–81, 83, 104 Seeliger corrector, 7. See also Scherzer corrector Seidel order, 44 Selected-area electron diffraction (SAED), 246, 419 Selected area nanodiffraction (SAND), 420 Semi-aplanat, 23, 70, 90 versus aplanat, 101–102 off-axial coma for, 90 Semiconductor quantum wires, 371–374 Sevenfold R model, 206 Sextupoles, 20, 126–128. See also Hexapoles Sharp cut-off fringing field (SCOFF) approximation, 72, 76–77, 80, 82, 92 Siemens 102 TEM, 13, 17 Signal-to-noise ratio. See SNR Silicon nanowire, 216–217 VLS grown twinned, 216 Si3 N4 crystals, 367 Single atom spectroscopy and mapping, 147–154 Single carbon nanotubes (SCNTs), 67 Single-image analysis, 471 Single-particle analysis, 219 Single-wall carbon nanotubes (SWCNT), 414 Sixfold astigmatism, 93 minimizing, 96–100

Index

SMART beam separator, 27–28 SNR, 142, 230, 330, 422, 510 Space-group determination, 248–249 Spatial frequency, 389, 442, 444 Speckles, 25 Spectro-microscope for all relevant techniques (SMART), 27–28 Spherical aberration, 6, 49, 57, 62, 79, 82, 126–128, 130, 261, 263, 327, 386 coefficients, 233, 243, 278, 289 correction of, 14, 25, 29 in CTEM, 47 of objective lens, 45, 70, 83 Spherical aberration-corrected transmission electron microscope-Toulouse. See SACTEM-Toulouse Spicer Consulting Analysis System (SC11), 491 STEM aberration compensation with restricted aperture size, 318–319 aberration-corrrected, complication for, 138–140 ADF, 337, 340 ADF Z-contrast imaging, 509 advantages of, 340 applications for, 140–154 brightfield (BF), 339–340 coherence effects in, 297–308 Cs -corrected. See Cs -corrected STEM depth sectioning, 68 of dumbbell structure of GaN, 65 evaluation and applications of, 510–521 experiment, 196–199 of Ge, 64 high-resolution, 311–312 image distortion, 483–484 image simulation, 514–516 at low primary energies, 140–142 optimal conditions for aberration-corrected, 315–319 performance of, 129–140 probe current, 129–138 probe size, 66–67, 129–138 wave aberration function in, measurement of, 295–297 STEM effective source/CTEM effective detector, 305–306 Stobbs factor, 198 Stoichiometry, 207–208 Strained silicon nanostructures and devices, 238–239 Strain mapping, 232–239

537

Stranski-Krastonov process, 212 Stray magnetic fields, 484 Sub-Ångstrom probe, 332 Sub-Ångstrom range, 328 Superconductors, high-temperature, 361–363 SuperSTEM instrument, operating parameters of, 198 Super-Twin objective lens, 228, 252, 343 Synchroshear dislocation core, 364 Z-contrast image of, 365

T TCC, 299–302, 397, 401 for strongly scattering objects, 303–304 for weakly scattering objects, 304–305 TEAM, 32–33, 61, 64, 342 TEM aberration correction in, 126, 386 atomic resolution, 450–457 axial chromatic aberration of, 45 contrast theory for aberration-corrected, 447–448 Cs -corrected. See Cs -corrected TEM fundamentals of atomic resolution imaging in, 440–450 high-resolution, 265–270 imaging theory, 397–398 optical principles, 440–444 phase contrast of, 444–450 point-to-point resolution of, 406–408 resolution for in situ, 262–265 methods for, 264–265 ultra-fast, 277–279 TEM Cs correctors, commercial availability of, 56 Tetrapoles, 127 Thicker crystals, imaging of, 395–398 Third-order aberrations, 82–87, 329 Third-order spherical aberrations, 20, 45, 53, 68, 85, 87, 330, 444–445 Thon diagrams, 398 Threefold astigmatism, 14, 72, 80, 85, 109, 126 aberration ray of, 104 Three-lobe aberration, 91–92 Tilted aberration coefficients, 290 Tilt grain boundary accuracy of measurements on, 477 geometrical model of, 470 in YBa2 Cu3 O7 crystal, 463–469 Tobacco mosaic virus (TMV), 500

538

Index

Tomographic imaging, 414 Transfer lens, 82, 85, 90 CTEM hexapole corrector, 57, 88 EMBL hexapole corrector, 48–49 for fifth-order spherical aberration, 95 Transmission cross-coefficient. See TCC Transmission electron aberration-corrected microscope (TEAM), 32–33, 61, 64, 342 Transmission electron microscopy. See TEM Transversal domain wall (TDW), 472–473 Tridiem filter, 228–229 Trivalent cation, 355 Tunnel structure, 190 Twin boundary, 217 oxygen-atom columns in BaTiO3 , 457–463 structure of, 459 Twin lamellae, 458

U Ultracorrector, 31 Ultra-fast TEM, 277–279 Cc correction, benefits of, 277–279 Unit cell, 473, 476 of ferroelectric thin films, 472 of YBa2 Cu3 O7 structure, 469 Uranium atoms, probe profiles for, 512–514 Uranium single-atom motion, 500–501

V Valence electrons, 139 Van Cittert–Zernike theorem, 300 VG HB501, 136, 164, 166–167 VG HB603, 136 VG microscopes instruments, 328 anticipated improvement, 329 Z-contrast image in, 359

W Wave aberration function, 286–297

in CTEM, 289–295 in STEM, 295–297 Wavefront distortions, 124, 132 Wave function, 441–442 Wave scattering, 449–450 Weak-phase object approximation (WPOA), 389 Wien filter spectrometer, high-resolution, 164–165

X X-ray magnetic circular dichroism (XMCD), 251

Y YBCO, 359 high-temperature superconductivity in, 361 Young fringe pattern, 60, 229

Z Z-contrast image, 330, 367 of Au nanoparticles, 375–376 of Bix Ca1−x MnO3 (BCMO), 355 and first-principles theory, 370 of GaAs, 347 of Ge, 343–344, 347 high-magnification of, 371 of InAsx P1−x nanowires, 372–373 of LaMnO3 , 352 of Pt atoms, 375 of Si, 349 of SrTiO3 , 340–341 of synchroshear dislocation core, 365 in VG Microscopes, 359 Zeiss TEM, 13 Zemlin tableau technique, 50, 61 of uncorrected and corrected CTEM, 54 Zernike phase plate, 446, 449 Zero-loss image, 353 Zero-order Laue zone (ZOLZ), 248–249 Z-slice imaging, 414