Advances in Imaging and Electron Physics, Volume 153: Aberration-corrected microscopy (Advances in Imaging and Electron Physics)

Advances in IMAGING AND ELECTRON PHYSICS VOLUME 153 Aberration-Corrected Electron Microscopy EDITOR-IN-CHIEF PETER...

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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.

3

4

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

6

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

8

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.

12

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

14

Harald Rose

103 B U

V T

Frontlinse 4 2

103G2 mm2 Korrektiv

10 I

5

V III

50

0

100

z /mm 250

150

II

IV

25 mm

I1 I2 S1

10 1 0 ZG ZF Z21 f 210

S2

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.

16

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.

18

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

QPel&mag

QPel&mag

QPmag

Objective lens

2.5

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

20

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 z

zm 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.

28

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

Q1 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).

32

Harald Rose

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.

33

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.

34

Harald Rose

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

Å21

35

Aberrationcorrected EM TEAM Stuttgart (1.2 MV)

1

Haider et al. (200 kV)

Dietrich (200 kV)

1021 Zach, Haider (1 kV, SEM)

22

Borries and Ruska, Marton, Ardenne (100 kV)

Electron microscope

Resolution

10

1023

Light microscope Ruska (75 kV) Abbe Amici

1024 Ross 1025 1800

1850

1900

1950

2000

2050

Year

FIGURE 20 of time.

Increase of resolution in diffraction-limited microscopy as a function

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

44 44 46 46 48 55 61 69 70 70 72 80 91 101 101 103 105 112 114 116 117

* 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 20.4

Point resolution

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

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(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).

0.07 nm

0.10 nm

0.08 nm 0.10 nm 0.07 nm

0.09 nm

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

Present and Future Hexapole Aberration Correctors

(a)

65

(b)

50 pm

(c)

Ga 63 pm

(d)

63 pm

Intensity (a.u.)

Intensity (a.u.)

Au(660) 48 pm

GaN(2555) 49 pm

0.0

0.4 Distance (nm)

0.8

Scattering angle

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

U 300 kV, fOL 2.3 mm, C3,OL 1.3 mm, CC,tot 2.4 mm

120

120

100

100

Probe size d50 [pm]

140

Probe size d50 [pm]

140

80 60 40 20 0 10

dE 0.7eV, A5 0.7 mm dE 0.3eV, A5 0.7 mm dE 0.3eV, A5 0.15 mm dE 0.3eV, A5 0

20

30

80 60 40 20

40

Aperture semi-angle [mrad]

50

0 10

dE dE dE dE

0.7eV, 0.3eV, 0.3eV, 0.3eV,

20

A5 3.0 mm A5 3.0 mm A5 0.5 mm A5 0

30

40

50

Aperture semi-angle [mrad]

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.

Present and Future Hexapole Aberration Correctors

2 nm

69

2 nm

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

C3s/m0 [At/mm3]

0.015

0.01 L 0.005

0 230

220

210

0 z [mm]

10

20

30

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

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

|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.

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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

2

3iη 3 u uγ dz · uα + γ −

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

3

3

2

2

2

2

1

1

1

1

0

0

0

0

21

21 C3s u u /10 u { { /10 { { /10 u {{ /20 u

22

23 230 220 210

Aberration coeffcients [mm]

3

Hexapole excitation [1023/mm3]

First- and second-order rays [mm]

3

22

0 10 z [mm]

20

23 30

21

21

22

23 230 220 210

C3s C {{ C {{ C {{ /2 C {{ /2 C3 0

Hexapole excitation [1023/mm3]

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22

10

20

23 30

z [mm]

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
(11)

4f0

In Figure 25 the course of the fundamental rays and the second-order aberration rays calculated for a realistic shape of the hexapole ﬁeld are depicted with L = 30 mm and f0 = 2 mm. The results are in good agreement with the piecewise approximation in Eq. (11). The aberration rays are represented by linear combinations of the paraxial rays with two independent integral coefﬁcients. These coefﬁcients are function of z. In general, for each aberration ray uX in arbitrary order with

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index combination X, two coefﬁcient functions cX and CX exist

uX (z) = uα (z) cX (z) + uγ (z) CX (z), u X (z) = u α (z) cX (z) + u γ (z) CX (z),

(12)

which fully describe the course of the aberration ray and of its ﬁrst derivative. The coefﬁcient function CX can be interpreted as the projection of the aberration ray uX on the off-axial fundamental ray uγ and cX as that on the axial fundamental ray uα . Evaluated at some image plane z = zi the (zi ) is the image aberration referred back to the object value CX (zi ) = uuXγ (z i) plane z = zo , while the other coefﬁcient cX governs the aberrations at the diffraction image where uγ = 0. For this reason the coefﬁcients CX and cX are sometimes called image and slope coefﬁcients, respectively. The complex-valued beam parameters allow classiﬁcation of the aberration rays and coefﬁcients according to their behavior under rotation of the frame of reference about the optic axis. The transformation

α → αeiθ ,

γ → γeiθ

(13)

causes the aberration ﬁgure to rotate by an integer multiple (ν + 1) of the angle θ. Therefore, for a general index combination αNα αNα γ Nγ γ Nγ the azimuthal multiplicity is given by

ν = Nα − Nα + Nγ − Nγ − 1.

(14)

For an optical system with cylinder symmetry only index combinations with multiplicity ν = 0 are allowed; if in addition hexapole ﬁelds are present, the allowed multiplicities are ν = 3 · k, with integer k. Most aberration coefﬁcients are complex-valued, characterized by magnitude and orientation. The real part describes the isotropic and the imaginary part the anisotropic component. The most important intrinsic aberrations that can occur in a system with threefold symmetry are listed by the common names and the corresponding aberration coefﬁcients in traditional notation and index notation in Table I. The coefﬁcient Cαα in Eq. (10) evaluated at an image plane is the threefold astigmatism A2 . In SCOFF approximation we ﬁnd the stigmator strength

A2 = Cαα =

zi

zo

3η 3s u3α dz = 3η 3s Lf03 .

(15)

78

Important aberration coefficients of hexapole-type correctors.

Aberration ray

Aberration coefficient

Multiplicity ν

Order n

Common name

uαα uααα uααγ , uααγ uαγγ uαγγ uγγγ uαααα , uαααα uαααγ , uαααγ uαααγ uααααα uααααα

A2 , Cαα C3 , Cs , Cααα K3 , Cααγ , Cααγ FC3 , Cαγγ FA3 , Cαγγ D3 , Cγγγ D4 , Cαααα , Cαααα Cαααγ , Cαααγ Cαααγ C5 , Cααααα A5 , Cααααα

−3 0 0 0 0 0 ±3 ±3 −3 0 −6

2 3 3 3 3 3 4 4 4 5 5

Threefold astigmatism Third-order spherical aberration Third-order off-axial coma Third-order ﬁeld curvature Third-order ﬁeld astigmatism Third-order distortion Fourth-order three-lobe aberration Off-axial star aberration Off-axial fourfold astigmatism Fifth-order spherical aberration Fifth-order sixfold astigmatism

These only have rotational, threefold and sixfold symmetry ν.

Maximilian Haider et al.

TABLE I

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For the system depicted in Figure 25 with a nodal plane uγ = 0 at the center of the hexapole ﬁeld, second-order threefold distortion

Cγγ =

zi

zo

3η 3s u2γ uα dz =

η 3s 3 L 4f0

(16)

also would be visible at the image plane. The course of all four aberration coefﬁcients for this system is depicted in Figure 25.

4. Secondary Aberrations We now direct our attention to the third-order aberrations of a single hexapole ﬁeld. For the next step of the iteration procedure, we substitute u → u(1) + u(2) at the right-hand side of the integral equation [Eq. (6)]. From Eq. (9) and

2 2 u(1) + u(2) = αuα + γuγ + ααuαα + αγuαγ + γγuγγ we ﬁnd by collecting the third-order terms

u(3) = α2 αuααα + α2 γuααγ + ααγuααγ + αγγuαγγ + αγ 2 uαγγ + γ 2 γuγγγ .

(17)

According to relation (14) only aberrations with multiplicity ν = 0 occur. This is a very remarkable and important result. The axial and off-axial third-order aberrations caused by a hexapole ﬁeld have exactly the same structure as those of a round lens. The ﬁrst term on the right-hand side of Eq. (17) evaluated at the image corresponds to the spherical aberration C3 . From the integral Eq. (6) we ﬁnd

uααα (zi ) = C3 = uγ (zi )

zi

zo

6η 3s u2α uαα dz.

(18)

For the system from Figure 25 we now substitute Eq. (11) and perform the integration. For the coefﬁcient of the spherical aberration this results in

C3 = Cααα = −3 |η 3s |2 L3 f04 < 0,

(19)

while according to the Scherzer theorem for a round lens the coefﬁcient of the spherical aberration is always positive. It is actually this ﬁnding that makes hexapole correctors attractive for correcting for round lens aberrations.

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C. Hexapole Corrector Use of hexapole elements for the correction of the spherical aberration requires a setup for which the strong axial second-order aberrations of the hexapole elements can be avoided while the third-order effect on C3 is maintained. Two hexapole stages can be combined such that their threefold astigmatism cancels and the negative contributions of both stages to the third-order spherical aberration add up. This step was taken by (Beck, 1979) and (Crewe, 1980). Unfortunately, these early systems suffered from axial fourth-order three-lobe aberration D4 = Cαααα , as pointed out by (Rose, 1981). This aberration prevents a useful application, even in a probe-forming microscope (STEM). Rose proved that the shape of the axial bundle must be cylinder-symmetric at least in second order everywhere outside the corrector and not only at the ﬁnal image plane in order to obtain a real improvement in resolution with a hexapole Cs corrector. In other words, it is necessary to avoid both the image coefﬁcient Cαα and the slope coefﬁcient cαα of the axial second-order aberration ray uαα outside the corrector to eliminate D4 .

1. Basic Setup Rose (1981, 1990) proposed a minimum setup of two hexapoles and two round lenses to circumvents the aforementioned problem. Today, this system (depicted in Figure 26) forms the basis for all available hexapole correctors. It consists of two symmetric hexapole elements with a telescopic 4f round-lens doublet in between. The mid-plane S1 of the ﬁrst hexapole ﬁeld is conjugated to the mid-plane S2 of the second. This arrangement has the advantage that all second-order aberration rays [Eq. (10)] vanish identically outside the corrector, since the four conditions

η 3 u3α dz

=

η 3 u2α uγ dz

=

η 3 uα u2γ dz

=

η 3 u3γ dz = 0 (20)

are simultaneously fulﬁlled due to a local double-symmetric arrangement of the ﬁelds and fundamental rays. The hexapole ﬁelds are symmetric with respect to the planes S0 , S1 , and S2 while the lens ﬁelds are anti-symmetric with respect to plane S0 . The axial ray uα is symmetric with respect to S1 and S2 and anti-symmetric with respect to S0 . The ﬁeld ray uγ has the opposite symmetry. It is symmetric with respect to S0 and anti-symmetric with respect to S1 and S2 . The full symmetry correction of all second-order residual aberrations makes this design attractive for STEM and CTEM. The course of the second-order aberration rays and coefﬁcient functions through the corrector is depicted in Figure 26. A piecewise representation in SCOFF approximation can be obtained directly from Eq. (11) by

81

Present and Future Hexapole Aberration Correctors

Rays and coefficient functions [mixed units]

3 2 1 0 21 22 u u 10 -u - - 10 u - - 100 u

23 24 2100

280

260

240

220

0 z [mm]

20

40

60

80

100

80

100

Rays and coefficient functions [mixed units]

3 2 1 0 21 22 -C -C - - 3 C - - 10 c

23 24 2100

280

260

240

220

0 z [mm]

20

40

60

FIGURE 26 Hexapole doublet with local double symmetry with respect to the mid-plane between the transfer lenses S0 and the mid-planes S1 and S2 of the first and second hexapole fields respectively. The top diagram shows the fundamental rays and the second-order aberration rays; the bottom diagram shows the corresponding coefficient functions.

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symmetric continuation since the course of the second-order aberration rays uαα and uγγ is symmetric with respect to plane S0 , while that of uαγ is anti-symmetric. Additionally, the one-to-one imaging between the planes S1 and S2 by the anti-symmetric round lens doublet is free of Larmor rotation and third-order off-axial coma and geometric distortion. According to Hosokawa (2004) the conditions in Eq. (20) can still be fulﬁlled if the focal length of the ﬁrst and second transfer lens are different f1 = f2 . In this case, the transfer lenses also must be operated as a telescopic doublet with hexapole elements of length L1 and

2 f2 L2 = L1 f1 placed at the front and back focal planes, respectively. For this system the hexapole excitations must fulﬁll the condition (1) 3

5 f1 (2) = 3 . f2

The latter relation shows that the required hexapole excitations rapidly diverge if the ratio f1 /f2 is substantially different from 1.

2. Third-Order Aberrations We now derive the third-order aberration coefﬁcients for the hexapole corrector as depicted in Figure 26. At the asymptotic image plane z → ∞ behind the corrector the third-order path deviation [Eq. (17)] can be rearranged in the form

u(3)/uγ = α2 α Cααα + α2 γ Cααγ + ααγ Cααγ + αγγ Cαγγ + αγ 2 Cαγγ + γ 2 γ Cγγγ .

(21)

With the SCOFF approximation for the second-order aberration rays and the iteration formula [Eq. (6)] it is straightforward to calculate all thirdorder aberration coefﬁcients in Eq. (21). The spherical aberration of both hexapoles simply adds up to

C3 = Cααα = −6 |η 3s |2 L3 f04 .

(22)

The coefﬁcient of the off-axial coma vanishes due to symmetry

K3 = Cααγ

1 = Cααγ = 2

1 6η 3s uαα uα uγ dz = 2

6η 3s uαγ u2α dz = 0 (23)

Present and Future Hexapole Aberration Correctors

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since both integrands are anti-symmetric with respect to mid-plane S0 . The relation between the two coma-type coefﬁcients Cααγ and Cααγ can be derived by partial integration from the explicit representation of the second-order aberrations rays. The coefﬁcient of ﬁeld curvature is

FC3 = Cαγγ =

6η 3s uαγ uα uγ dz =

3 |η 3s |2 L5 , 5

(24)

and for ﬁeld astigmatism we ﬁnd

FA3 = Cαγγ =

3 6η 3s uγγ u2α dz = − |η 3s |2 L5 = −FC3 . 5

(25)

The coefﬁcient of distortion again vanishes due to symmetry

D3 = Cγγγ =

6η 3s uγγ uα uγ dz = 0.

(26)

Finally, the slope coefﬁcient of the distortion ray uγγγ , which has no effect on the image but causes spherical aberration at the diffraction image, evaluates to

3 L7 E3 = cγγγ = − 6η 3s uγγ u2γ dz = − |η 3s |2 4 . 56 f0

(27)

Formula (22) veriﬁes that the hexapole corrector has negative spherical aberration C3 < 0 that can be used to compensate for the always positive spherical aberration C3,OL > 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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85

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]

40

60

80

100

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.

Present and Future Hexapole Aberration Correctors

87

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

f0 = MTL fOL

(30)

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 =

3η 3s u2αα uα dz,

(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

3η 3s u2αα uγ dz.

cαααα = −

(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.

Present and Future Hexapole Aberration Correctors

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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.

Present and Future Hexapole Aberration Correctors

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Rays and coefficient functions [mixed units]

3 2 1 0 21 22 23 24 250

u u /10 Re CIm CC5/3 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 IHP ⫽ 144 mA, Cc ⫽ 1.42 mm, C3 ⫽ 0.51 mm, f ⫽ 1.81 mm 4

4 3

2

2

1

1

0

0

⫺1

⫺1 OL TL12 C5/mm B3c B3s |A5|/mm

⫺2 B3c⫺ free

⫺3

⫺4 ⫺9 ⫺8 ⫺7 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1

0

1

2

3

⫺2

Change of lens excitation dl/l [%]

Aberration coefficients

C5 ⫺ free 3

⫺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 · FA3,TL . A5 = − L +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 2π π 3 · 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

z uαα (z) = uα (z) ·

z 3iη u2α uγ 3 dz − uγ (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

z uαα (z) = −uγ (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

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105

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

6ηuα u2γ · Re A2 3 dz,

Im Cααγ = −

(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 the symmetry uα (zm + z) = symmetry uα (zm + z) = −uα (zm − z), and u+ 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.

3

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

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.

Maximilian Haider et al.

108

1

Ray, coefficient [mixed units]

ua/10 u /100 A2x A2y

0.5

0

20.5

21

2100

250

0 z-zm [mm]

50

3

100

Im(C) 10*Im(D4) Cs

Ray, coefficient [mixed units]

2

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).

Present and Future Hexapole Aberration Correctors

109

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 u + A A 6η u− α γ 2x 3c 2y 3s dz,

where both terms contribute equally.

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110

1

Ray, coefficient [mixed units]

ua/10 u /100 A2x 1000*A2y

0.5

0

20.5

21

2100

250

0 z-zm [mm]

50

100

3 Im(C ) 10*Im(D4) Cs

Ray, coefficient [mixed units]

2

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).

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111

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

150

200

3 Im(C ) 10*Im(D4) Cs

Ray, coefficient [mixed units]

2

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

z 3η u3α 3s dz ,

A2x (z) =

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

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

HP Cαααγ HP Cαααγ

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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) =

2π 2π 1 · 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

• • • •

115

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

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.

REFERENCES Beck, V. D. (1979). A hexapole spherical aberration corrector. Optik 53, 241–255. Crewe, A. V. (1980). Studies on sextupole correctors. Optik 57, 313–327. Crewe, A. V. (1982). A system for the correction of axial aperture aberrations in electron lenses. Optik 60, 271–281. Crewe, A. V. (2002). Some Chicago aberrations. Microsc. Microanal. 8, 4–5. Crewe, A. V., and Kopf D. (1980). Limitation of sextupole correctors. Optik 56, 391–399. Essers, E, Benner G., Orchowski, A., Kappel, R., and Trunz, M. (2002). New approach for ultra-stable TEM column support frame. Proceedings of 15th International Congress on Electron Microscopy, Durban, South Africa, pp. 355–356. Freitag, B., Kujawa, S., Mul, P. M., and Ringnalda, J. (2005). Breaking the spherical and chromatic aberration barrier in TEM. Ultramicroscopy 102, 209–214. Haider, M. (1990). Electron microscope development and STEM application. EMBL Research Reports, 142–146. 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 200 kV TEM by means of a hexapole corrector. Optik 99, 167–179. Haider, M., Müller, H., Uhlemann, S., Zach, J., Löbau, U., and Höschen, R. (2008). Prerequisites for a Cc /Cs -corrected ultrahigh-resolution TEM. Ultramicroscopy 108, 167–178. 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. Haider, M., Uhlemann, S., Schwan, E., Rose, H., Kabius, B., and Urban, K. (1998). Electron microscopy image enhanced. Nature 392, 768–769. Hartel, P., Müller, H., Uhlemann, S., and Haider, M. (2004). Residual aberrations of hexapoletype Cs -correctors. In Proceedings 13th European Microscopy Congress, vol. I (D. Van Dyck & P. Oostveldt, eds.), pp. 41–42. Antwerp: Belgian Society for Microscopy. Hartel, P., Müller, H., Uhlemann, S., and Haider, M. (2007). Experimental set-up of an advanced hexapole Cs -corrector. Micros. Microanal. 13, 1148–1149. Hartel, P., Müller, H., Uhlemann, S., Zach, J., Löbau, U., Höschen, R., and Haider, M. (2008). Demonstration of Cc /Cs correction in HRTEM. Proceeding of 14th European Microscopy Congress, Aachen, Germany. Hosokawa, F. (2004). Spherical aberration corrector for electron microscope. United States Patent, US 6836373B2. Hutchison, J. L., Titchmarsh, J. M., Cockayne, D. J. H., Doole, R. C., Hetherington, C. J. D., Kirkland, A. I., and Sawada, H. (2005). A versatile double aberration-corrected energy ﬁltered HREM/STEM for materials science. Ultramicroscopy 103, 7–15. Jia, C. L., Lentzen, M., and Urban, K. (2003). Atomic-resolution imaging of oxygen in perovskite ceracmics. 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. Kazumori, H., Honda, K., Matsuya, M., Date, M., and Nielsen, C. (2004). Field emission SEM with a spherical and chromatic aberration corrector. Microsc. Microanal. 10, 1370–1371.

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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,a

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C0,1,b

C1,0 C1,2,a

C2,1,a

C1,2,b

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

1/(n+1)

= (n + 1)λ 8 Fn,m Cn,m ,

αgeom

(5)

m

which leads to a probe size of

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

1/(n+1) Fn,m Cn,m

λn/(n+1) .

(6)

m

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)

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

dchrom = 0.5 (λCc δE/E0 )0.5 .

(11)

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)

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 2 Ip = Bπ2 dsource α2/4,

(14)

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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

0.5 dsource = 2 Ip /B /(πα),

(15a)

0.5 dsource = 2 Ip /Br Vo∗ /(πα),

(15b)

or alternately

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 :

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

(16a) (16b)

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

Ip > 1.4 × 10−18 Br .

(17)

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

Probe size (for daber ⫽ 50 pm)/pm

300

0 0

0.2

0.4 0.6 Probe current/nA

0.8

1

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)

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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 <110> 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

850 880 Energy (eV)

Ti

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 ,

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(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,

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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) .

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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 << dd . Note that when the probe size dp is made smaller, decreasing the probe current or the acquisition time as (dp2 + dd2 )2 keeps the SNR at each pixel unchanged. As the resolution improves, the pixels need to become more closely spaced, but even so the required dose per unit area decreases as (dp2 + dd2 ). This means that decreasing the probe size by using aberrationcorrected electron optics provides the same SNRs for detecting individual atoms at smaller incident doses per unit area. The improvement is a major one: decreasing the effective probe size from 1 nm to 0.2 nm decreases the required dose per unit area 25×, even though the spatial resolution in the elemental map will now be 5× better. It is also worth noting that when the electron probe size is ∼0.1 nm in diameter and yet intense enough to produce high-quality EELS data, and the delocalization is not much larger than 0.1 nm, detecting minority and trace elements in normal samples becomes possible on an atom-byatom basis. The concentrations corresponding to a single atom in a probed volume in a sample 5–30 nm thick are 1–10% when the delocalization is better than 0.1 nm, and 0.1–1 % when the delocalization is 0.3–0.5 nm. In other words, because the atoms are probed one at a time, even trace atoms present in real concentrations << 0.1 % should be detectable. Applying Eq. (22) to the La detection experiment, we used experimental cross-sections derived from the EELS atlas data (Ahn and Krivanek, 1993) as much as possible. To do this, we ﬁrst calculated the Hartree-Slater

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cross-section for the La M4,5 edge continuum (3.3 × 10−7 nm2 for an energy interval 50-eV wide starting 30 eV past the threshold), then scaled the experimental M4,5 white line intensities and pre-edge background to the edge continuum using the La2 O3 data in the atlas. We estimated the contribution of the adjacent columns in the CaTiO3 sample by using the measurements of Varela et al. (2005) of how much La signal was recorded when the probe was positioned over the neighboring atomic columns. Taking the sample thickness to be ∼20 nm then meant that when the probe was centered on the Ca column containing the substitutional La atom, the contributions to the spectrum came from an effective total of 125 atoms (some of which were located on neighboring columns illuminated by the dechanneled probe), with the relative proportions La 0.8, Ca 38.2, O 46, and Ti 15. We also accounted for the pre-edge background being weaker in this type of sample than in La2 O3 by scaling the edge background cross-section by a ratio determined by computing theoretical spectra appropriate to the mostly Ca-Ti-O sample under the probe in the present experiment and to the atlas sample (La2 O3 ). Assuming that the DQE of the CCD detector was close to 1 then gave a net signal strength of 5.2 × 103 electrons in the La M5 threshold peak that was recorded as ∼3 eV wide in the experiment, and a background height of 1.5 × 104 electrons per 1-eV wide channel just in front of the La edge. This led to SNR of 22 for the total M5 threshold peak intensity from the single La atom relative to the statistical noise of the background signal under the peak. Assuming that the detector point spread function was 1 eV wide then gave a ratio of the M5 net peak height to the peak-to-peak noise in the background of 4.3:1. Both these numbers are in good agreement with the experiment, showing that our understanding of the principal factors determining the strength of the recorded signal and of its background is sound. The numbers also show why the acquisition time had to be 30 s long: the La M4,5 edge would not have been readily visible in a spectrum acquired in a second or less. We also estimated the La white line SNR using the EELS Advisor simulation software package (Gatan Inc., Pleasanton, CA) developed recently (Menon and Krivanek, 2002; Menon, 2005; Hunt and Menon, 2005). Unfortunately, we found that the present version of the software does not properly account for collection efﬁciency when the collection aperture is smaller than the illumination aperture (as was the case in the present experiment), does not model threshold white lines and background intensities with sufﬁcient accuracy, and further underestimates the shot noise contribution to spectra. Nevertheless, we found that the errors approximately canceled out and that a reasonable agreement with the above SNR estimate as well as with the experimental data was obtained. In future La imaging experiments in an optimized instrument, the probe current will probably be ∼10× higher and the collection efﬁciency also ∼10× higher. The same SNR for identifying a single atom as was attained

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by Varela et al. will thus be obtainable in ∼0.3 s. Using optimized signalextraction techniques such as principal component analysis should make it possible to reduce the per-pixel dwell time to ∼0.01 s, at which point a 256 × 256 pixel map optimized for La single-atom detection would take 11 minutes to acquire. With similar or better SNR than the one obtained in the experiment described above, it should also be possible to use the details of the white line intensity and shape to map the valence state and other properties of the single atoms. La is a heavy atom, and detecting it in an EELS spectrum image is easier because it can be identiﬁed in a HAADF image before any spectra are taken. Reliable EELS imaging of single atoms that are similar to their matrix and therefore cannot be identiﬁed by HAADF imaging, as was done by Leapman (2003) for Ca in a very thin carbon sample, is more challenging. Figure 7 explores this situation with EELS spectra computed using the EELS Advisor, with a 100 keV, 0.15 nm diameter probe with 1 nA current, a collection semi-angle of 30 mr, and a DQE = 1 for the EELS detector. The visibility of EELS edges from single atoms is explored by simulating spectra recorded with the probe positioned over a single impurity atom for situations likely to be encountered on samples of practical interest. In a complete mapping experiment, spectra from other pixels near the single atom’s center will also contain the atom’s signature, and the SNR requirement for each contributing spectrum will therefore not be as strict as for a single spectrum. The left column in Figure 7 explores detecting phosphorus in a carbon matrix, a situation that may arise in biological samples. The P L2,3 edge lies in front of the strong carbon K-edge and is easily visible in a raw spectrum recorded in 0.01 s when the phosphorus is present in 1 at. % concentration (spectrum a). Because the P L2,3 delocalization is ∼ 0.4 nm at 100 keV primary energy, the L2,3 signal will be ∼10× weaker than if the delocalization had only been 0.1 nm. The middle spectrum (b) therefore simulates the strength of the signal when P is present in the probed volume at only 0.1 at. %. (Carbon density is ∼140 atoms/nm3 , and 0.1 at. % concentration therefore corresponds to 1 P atom in a probed volume 0.5 nm in diameter in a 36 nm thick sample.) The P L2,3 edge then becomes essentially invisible in the unprocessed spectrum. However, calculating a simple difference spectrum, which can be readily obtained in a spectrum-imaging experiment by subtracting a properly scaled version of the average spectrum from the carbon matrix at places free of phosphorus atoms, still shows a strong P signal, provided that the acquisition time per pixel is lengthened to 0.1 s. Shortening the time to 0.01 s decreases the edge SNR 3×, but the signal is still unmistakable (spectrum c). We therefore conclude that rapid EELS mapping of isolated P atoms in a carbon matrix should be possible with presently available instrumentation, provided that radiation damage is avoided (for instance, by decreasing the primary energy to 60 keV).

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FIGURE 7 Simulated EELS spectra. Left column: phosphorus in C. Middle column: calcium in C. Right column: phosphorus in Si. 100 keV, 1 nA probe current, 0.15 nm probe diameter, 30 mr collection semi-angle (see text for detailed discussion).

The middle column in Figure 7 simulates a less favorable case, also with biological signiﬁcance: Ca detection in a carbon matrix. The beam parameters are unchanged, but we are now dealing with the Ca L2,3 edge riding on top of the dominant matrix edge. Better localization of the inelastic scattering (dd = 0.22 nm at 100 keV) helps to prevent the spreading of the Ca signal due to delocalization, and the visibility of the Ca edge is aided by its strong threshold lines. Even so, the Ca edge is barely visible in an unprocessed spectrum recorded in 0.1 s for 1 at. % Ca concentration (spectrum d). (1 at. % concentration corresponds to one Ca atom in a probed volume 0.3 nm in diameter in a 10 nm thick sample.) The difference spectrum, however, shows the edge clearly for 1% concentration recorded in 0.1 s (spectrum e), and the edge is still recognizable in a spectrum recorded in 0.01 s (spectrum f). This suggests that rapid mapping of single atoms of

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Ca in a carbon matrix with single atom sensitivity should also be possible, provided that the sample is not much thicker than 10 nm. Note that in the above case, we are essentially exploring how the experiment performed by Leapman (2003) will work in an aberration-corrected CFEG STEM. The ﬁnding that decreasing the size of a 1 nA probe to 0.15 nm will make it possible to see single Ca atoms reliably in thicker samples and with shorter spectrum acquisition time is not surprising. The right column in Figure 7 deals with a still more difﬁcult case for single-atom mapping: P in Si, an example of the type of problems relevant in the semiconductor industry. Here we are dealing with an edge with no major threshold lines, riding on top of a very strong matrix edge, and with poor delocalization. Concentration of 1 at. % produces no readily identiﬁable features in the raw spectrum, and we have therefore shown the unprocessed spectrum for 10% concentration (spectrum g). Concentration of 1% is readily detectable in a difference spectrum acquired in 0.1 s (spectrum h). Silicon’s atomic density is 50 atoms/nm3 , and 1% concentration therefore corresponds to 1 atom of P in a probed volume 0.5 nm in diameter and 10 nm thick. For a more realistic 30 nm thick crystalline Si sample, we would be dealing with an effective concentration of 0.3 at. %. In the corresponding difference spectrum (spectrum i), the P edge is barely detectable, even though the acquisition time is again 0.1 s. The same SNR would be obtained for a concentration of 1% and 0.01 s acquisition time. This example may therefore be at the edge of the possibilities for fast elemental mapping with presently available instrumentation, especially if the sample thickness cannot be kept to 10 nm or less. To summarize, chemical mapping with single-atom sensitivity performed in a reasonable time (<10 min for a 256 × 256 map) is possible with present-day instrumentation, but practical success will very much depend on the situation at hand. Provided that radiation damage does not make the experiment impossible, mapping with single-atom sensitivity should be possible for most cases where a strong edge from the element of interest is available in the range of ∼100–1000-eV energy losses, especially if the edge of interest is not located on top of a strong feature in the matrix spectrum.

V. CONCLUSIONS Aberration correction has greatly improved the performance and the range of useful applications of STEMs. It is now possible to form atomicresolution images and spectra at primary energies as low as 60 keV. Moreover, electron probes can be produced that are smaller than the typical atom and yet contain enough current for rapid EELS analysis with

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sensitivity down to the single atom level. This is leading to a new range of applications for the STEM-EELS technique, which may well begin to rival the sensitivity of the atom probe.

ACKNOWLEDGMENTS We are greatly indebted to many of our colleagues, especially to Phil Batson, Andrew Bleloch, Mick Brown, Matt Chisholm, Christian Colliex, David Muller, Pete Nellist, Steve Pennycook, and John Silcox, for their unsparing support of our efforts in STEM aberration correction.

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PART 3

Results obtained with aberration-corrected instruments

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CHAPTER

4 First Results Using the Nion Third-Order Scanning Transmission Electron Microscope Corrector P. E. Batson*

Contents

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

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

I. INTRODUCTION During the 1990s it became clear that a scanning transmission electron microscope (STEM) resolution of ∼2 Å would soon be inadequate for analysis of semiconductor device structures. Today device gate stacks include atomic-level materials engineered for both high- and low-dielectric constant, for diffusion barriers against migration of electrically active minority atoms, for stabilization against electromigration in conductors, and even for ideal interface structures to enable desirable electrical properties, such as high carrier mobility, in nearby regions. These structures are also susceptible to radiation damage in typical medium-voltage electron microscopes; thus the need exists for instruments that operate at as * IBM Thomas J. Watson Research Center,Yorktown Heights, New York 10598, USA Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01004-5. Copyright © 2008 Elsevier Inc. All rights reserved.

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low an energy as possible. Correction of aberrations is the obvious answer to these difﬁculties. With this in mind, the second-generation Nion thirdorder corrector Nion Company, Kirkland, WA, USA) was installed at IBM location. The ﬁrst task was to answer a few simple questions: (1) would it improve the resolution of the microscope? (2) what unforseen instrumental difﬁculties might arise, and (3) what kinds of physical phenomena might be found using the new resolution? This account will summarize, in the Section II, the basic instrument, including some of the modiﬁcations made since its initial installation. Section III describes the ﬁrst results, including effort to characterize the optical quality of the system, with emphasis on ADF STEM imaging and some differences from previous work using larger probe sizes. Calculation of probe properties for an arbitrary set of aberration measurements is described in Section IV. Section V describes imaging of crystalline objects as a way to verify the calculated probe properties. Section VI describes some initial work on imaging in an example, AlN/GaN/AlN, where a weak N scattering column lies very close to a strongly scattering Al or Ga column, to point out the strengths and a few weaknesses of the new capability. I next discuss in Section VII, an example of imaging of a hi-k* HfO2 /Si interface which aptly illustrates the power and some of the pitfalls of aberration corrected imaging. In Section VIII, I show multislice calculations using assumptions appropriate for very thin sample areas, limited in lateral dimension. In Section IX, I will discuss the important problem of atomic movement that occurs in molecular sized systems that we seek to image using the new sub-Angstrom sized probe.

II. THE IBM HIGH-RESOLUTION STEM/EELS SYSTEM The IBM Vacuum Generators (VG) (model HB501) STEM (Vacuum Generators, Ltd., Hastings, Sussex, UK) has been heavily modiﬁed over the years to improve its operation for spatially resolved electron energy-loss spectroscopy (EELS). As shown in Figures 1 and 2, the most signiﬁcant visible difference between this instrument and others is the high-resolution Wien ﬁlter spectrometer built in 1986 (Batson, 1986, 1988). This device is mounted within a high-voltage electrode, to allow deceleration of high-energy scattered electrons to 100–150 eV energy for analysis. It has demonstrated sub-100 meV resolution under controlled conditions (Mook et al., 1999) and routinely obtains ∼50 meV accuracy for locating core edge threshold energies. During the 1980s, the instrumental spatial resolution was improved to 2 Å by addition of the VG high-resolution polepiece and by increasing the electron high tension (EHT) of the instrument to 120 KeV. The objective lens current also has been increased ∼10% over that needed for the higher voltage to improve the postspecimen compression and to reduce Cs to approximately 1.2 mm. Finally, in the late 1990s the ﬁeld emission source

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FIGURE 1 The IBM VG HB501 STEM. Major modifications have included (1) the high-resolution Wien filter spectrometer mounted within a high-voltage electrode at the top, (2) replacement of the diffusion pump with an ion pump to eliminate oil from the system, and (3) extensive modification of electron high-tension (EHT) components for the Wien filter and for addition of an electron monochromator in the gun. The microscope is located within a shielded area to reduce low-frequency AC interference.

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New EELS CCD

Spectrometer

Aberration Corrector CCD

EELS Multipole Correction

EELS Transfer Optics Retractable ADF Detector

120 KeV –– 1 Å Probe

Aberration Corrector

Monochromator New Tip Environment

FIGURE 2 Line drawing of the IBM VG STEM. The Nion aberration corrector replaces the VG scan coils, between the condensor and objective lenses. A charge-coupled device (CCD) TV camera obtains Ronchigram measurements of objective lens aberrations. Extensive changes to the spectrometer coupling optics were required by the positioning of the Ronchigram camera and by the need to preserve reasonable collection efficiency using the larger angles making up the sub-angstrom probe.

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and its associated power supply were modiﬁed to allow addition of an electron monochromator. This device demonstrated an energy resolution of better than 60 meV in a controlled experiment (Batson et al., 2000; Mook et al., 1999). Although the monochromator was removed before installation of the aberration corrector, the mechanical and electrical modiﬁcations, including an extra EHT tank for the monochromator supplies, remain in the gun system. Therefore the EHT system for the IBM STEM is about three times more complex than the typical VG STEM (including several cables and the two extra EHT tanks) and is operating at 120 kV rather than the typical 100 kV for the VG systems. These changes introduced several difﬁculties. First, the additional complexity signiﬁcantly increases the failure rate for the entire system, and the opportunities for introduced electrical interference, particularly given the novel nature of the electronics that reside within the added EHT tanks (Batson, 1986; Batson et al., 2000). Finally, running the objective lens at 120 KeV with an enhanced postspecimen ﬁeld results in an increased power dissipation to ∼800 W, signiﬁcantly increasing required cooling, which contributes to mechanical instability in the specimen area. Each subsystem previously described—the STEM, the spectrometer, the monochromator, and the aberration corrector—is controlled through various software levels. Hardware interaction is obtained using simple C routines. For instance, image acquisition uses an inexpensive National Instruments video acquisition board, with their supplied software libraries, to grab scanned frames controlled by the VG electronics. High-level control is achieved using an interpretor language, IBM APL2 (IBM Armonk, NY, USA). This software was invented during the late 1960s to allow very rapid and concise, vector-based programming that could then be applied to rapidly changing situations. Since that time it has evolved with operating system and hardware changes to embrace most current systems. Today APL2 handles complex arithmetic using primitive operations, running under Windows, Linux, Unix, and in the mainframe environment. Under Windows it handles many operating system calls to handle graphics, disk transfer, cursor functions, and network communication, among others. It is therefore an ideal language for handling prototype hardware and software environments. Additionally, its interpreter structure has preserved its usability across disparate platforms, allowing routines written for the mainframe in the 1980s to be used today on the personal computer. All hardware control, image acquisition, data storage, processing, model calculations for interpretation, and generation of ﬁgures for this chapter were implemented using APL2. A Hertzian active damping antivibration system supports the machine. This system uses springs to provide passive vibration isolation with a resonance near 0.5–0.8 Hz. Active damping eliminates the peak transmission at this frequency to suppress ﬂoor vibration above the resonance frequency while maintaining unit transmission at and below the resonance frequency.

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III. FIRST RESULTS First results for the Si [011] projection, shown in Figure 3, were obtained within a few weeks of the installation in 2001. Figure 3a shows the raw data, and Figure 3b shows a Fourier-ﬁltered section cut from the center of the image. These results were quite exciting at the time, because they

(a)

002 113 004

1 nm

(b)

022 111 002

113 004

0.5 nm

FIGURE 3 (a) First results for the Si [011] projection. Dumbbell contrast is just visible, but instabilities, particularly associated with AC mains frequencies, are prominent. (b) Fourier-filtered section cut from (a) suppresses the AC distortions to highlight the dumbbell contrast. Notice that bands of the dumbbell intensities are very weak due to the interference.

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unambiguously showed that the aberration-correction optics was capable of improving the resolution of the instrument from the 2 Å level to something of the order of 1.35 Å, the spacing of the Si dumbbell. However, much additional work on instrumental instabilities clearly remained in order to produce a practical instrument. Since the basic instrument was designed originally in the mid-1970s for operation at the 5–10 Å level, instabilities at the 1–1.5 Å level were tolerated. During the 1980s, when the resolution was improved to the 2 Å level by the introduction of the high excitation lens, some electrical instabilities were corrected so that it could begin imaging atom columns in thin crystals using the high-angle annular dark-ﬁeld (HAADF) detector (Howie, 1979; Pennycook and Jesson, 1990). At the 2 Å level, these improvements were modest, but improvement beyond 1.5 Å required much more aggressive work. The largest problem was at the line frequency, so an important VG instrumental mitigation was to synchronize line scans with the alternate current (AC) mains frequency, exchanging random interference for regular distortion in the ﬁnal image. This work introduced digital acquisition synchronized with the microscope scan to preserve this operation. A reduction of AC interference was also obtained by attention to ground loops and removal of some AC sources from within the microscope power supplies. Finally, isolation from ﬂoor vibration revealed that much of this source was driven by building air handlers, and so, it too had large mains frequency components. Mechanical instabilities also were a problem in the lower-resolution design. The largest contribution in this regard is probably due to turbulence in cooling water in the objective lens. Modiﬁcations to allow the higher acceleration voltage and to improve EELS collection resulted in a doubling of power dissipation and consequent cooling requirements. Thus vibration due to coolant ﬂow is higher than in the original instrument, and reducing this will likely require signiﬁcant design effort. Therefore this work uses a relatively fast image acquisition—5–10 Hz per frame—to convert random ﬂuctuations that reduce the resolution, into distortions that are distributed across the image. For a 1024 × 1024 pixel acquisition, 5 frames per second (fps) can be obtained using a 200 ns pixel acquisition time. Acoustic noise and AC magnetic ﬁelds were controlled by a shielded enclosure as shown in Figure 1. The rapid acquisition is made possible by the use of the VG microscope’s cold ﬁeld emission source, which has a brightness of β = 1.0 − 3.0 × 109 amps/Sr/cm2 , allowing 40–150 pA current to be obtained in the small probe. This current is sufﬁcient to obtain usable images in a single 200 ms frame, but it is not sufﬁcient to explore the optical limits of the system using 1024 × 1024 pixels, so some averaging within an image or over several images is required.

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Eventually it became possible to acquire larger areas, requiring little or no processing, for evaluation of the probe size as discussed in Batson (2003). Usually workers have used power spectra of crystal images, taking the smallest identiﬁable spatial frequency for an estimate of the instrumental resolution. In trying this, it was discovered that it was possible to obtain nonphysical spots at expected spatial frequencies, thus making this method unreliable. We suggested that this resulted from nonlinearities in signal detection (which in this case was black level), which we believe become more important as more Fourier coefﬁcients become needed to describe the newly deep minima between fringes in the HAADF image using the new probe size (Yu et al., 2002). However, it was not appreciated in that work that the nonphysical spatial frequencies could be removed by careful averaging of subregions of the original, larger image, suggesting that scan distortions may also play a role. Ultimately, we used this spatial averaging over subregions to produce a comparison of the performance of the system before and after aberration correction (Figure 4) (Batson et al., 2002). It is important to note that while we know that the black-level setting for this image is not 0, nonphysical spots in the power spectrum for the whole image, out to 0.48 Å, have been removed by the subregion averaging (Batson, 2003). Furthermore, the forbidden [002] type spots also have been suppressed. These can certainly be produced by scan distortions, so their disappearance under a spatial averaging is understandable. As summarized in Figure 4, the remaining lattice spots are visible out to 0.76 Å. Given the uncertainty in the use of power spectra for evaluation of resolution, this result is more important for the clarity with which it conveys the improvement that aberration correction brings to the existing instrument. Attempting to characterize the probe shape and size using lattice planes that are comparable in spacing to the probe size is very difﬁcult. A better approach would be to use a much smaller object, so that the object’s dimensions do not interfere with interpretation of the results. Single atoms ﬁt this criterion. These have been visible from time to time since Crewe’s original work in the 1960s (Crewe et al., 1970). Figure 5 shows results for a large Au island on an amorphous carbon substrate. This image has been been processed to reduce contrast by suppressing low spatial frequencies so that the strong scattering from the island does not overwhelm the signal from single Au atoms. The processing produces a result that has characteristics of both high-resolution bright-ﬁeld imaging, where the low spatial frequencies are suppressed by the contrast transfer function (CTF) at Scherzer defocus conditions, and HAADF imaging, where the atomic number contrast dominates. Patches of monolayers of Au atoms coexist with the large island. These are characterized by power spectra around the border of the image. In some areas, the Au atoms are well separated and can be imaged at very high magniﬁcation to evaluate the probe size.

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(a)

(b)

10 mR

(c)

10 mR

(d)

1.36 Å

1.36 Å

(f)

(e) 1.92 Å 3.13 Å 2.7 Å

0.76 Å 0.96 Å 1.04 Å 1.10 Å 1.63 Å 1.36 Å

FIGURE 4 Comparison of the results for Si[011] before and after aberration correction (Batson et al., 2002). The top (panels a and b) summarizes the Ronchigram plots, the middle (panels c and d) shows the images, and the bottom (e and f) shows the Fourier power spectrum for the images. The constant phase patch in the Ronchigram has been opened from 10 mR half angle to 25 mR. The probe size has been reduced to ∼0.8 A° from 2.0 A°, and spatial frequencies to 0.76 A° are visible in the power spectrum.

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1 nm

FIGURE 5 High-angle annular dark-field image of Au island on carbon, accompanied by single atoms and ordered monolayers of atoms. The image has been processed to reduce low spatial frequencies so that identification of the positions of atoms and atom columns can be obtained. Power spectra are shown for various regions in the image.

It was clear with the ﬁrst imaging that the smaller probe size produces a remarkable improvement in single-atom contrast. Figure 6 shows a single Au atom on a carbon support. This image was obtained in 200 ms and consisted of 1024 × 1024 pixels, with a scan step size of 0.23 Å. This image has been ﬁltered with a Gaussian-shaped Fourier ﬁlter with a width of 0.2–0.3 Å to reduce the measurement noise. Figure 6a compares the ﬁltered single-atom image with HAADF simulations for a 0.75 Å and 1.9 Å sized

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(a)

0.75 Å Simulation

2 Å Simulation

2Å

500

(b)

Counts

0.75 Å 400

300

200

100 0 28

26

24

22

0 2 Position (Å)

4

6

8

FIGURE 6 (a) HAADF image of a single Au atom on carbon. An image simulation using a 0.75 A° probe is at the upper left, while a simulation using a 2 A° probe is shown in the lower left. The interesting result is that the atom is not visible at 2 A° resolution here due to the background statistics. (b) Comparison of image data with a line plot for the calculated atom image.

probes. Clearly, this source size improvement achieves a new performance level where single atoms are always visible. Figure 6b shows that, while the image data are certainly consistent with 0.75 Å, there is enough scatter, even in the ﬁltered data, to make the precise width uncertain.

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30 25

Number

20

0.081 .01

15 10 5 0 0.00

0.02

0.04

0.06 0.08 0.10 Width (nm)

0.12

0.14

0.16

FIGURE 7 Histogram analysis of 100 single Au atoms obtained from 14 images used in Batson (2007). The cited error is the root mean square deviation of all the measurements from the mean of 0.81 A°.

Use of a ﬂexible scripting system such as APL2 allows construction of a consistent method to measure many atom images to produce a histogram plot. Brieﬂy, the algorithm uses cursor input to locate an atom. It estimates the atomic contrast above the background, restricting usable images to three times the background. It next ﬁts a two-dimensional (2D) Gaussian surface over the atom image and calculates a full-width half maximum (FWHM) from this ﬁtted surface. Finally, cuts in the surface are plotted with the data to help verify validity. Fourteen images were chosen on the basis of quality of focus from studies of interaction of small groups of single atoms (Batson, 2007). These images produced 100 measurements of atom image widths (summarized in Figure 7). This exercise produces a histogram with a mean of 0.81 Å with an root mean square (RMS) deviation of 0.11 Å, comparing with 0.78 Å measured in earlier work. (Batson, 2003) The histogram analysis is made necessary by the fact that the Au atoms move readily under the electron beam. It is therefore possible for an atom to be detected for only the part of the scan that it remains under the probe. This is probably the reason for the long tails to small widths that are evident in the ﬁgure.

IV. CALCULATION OF PROBE PROPERTIES In the preceding discussion, the reciprocal space Ronchigram and the realspace probe are critical characterizations of the performance of the optics of the microscope. As discussed elsewhere, the optics has many degrees of

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freedom and may be described by a set of aberration coefﬁcients in ways similar to those used in the past to describe defocus, astigmatism, coma, and other, more complicated defects. The ﬁrst-generation Nion corrector has the ability to measure up to ﬁfth-order aberrations (∼25 coefﬁcients) and accurately control up to third-order aberrations (12 coefﬁcients). Given the number of independent distortions, Krivanek et al. (2000) suggested a shorthand method to simplify the identiﬁcation of the coefﬁcients. We for an electron wavefront in a lens back focal plane, deﬁne a phase shift χ(k) with k = (θ, φ), where φ is an azimuthal coordinate and θ is a scattering angle for the probe electrons. We can then specify this phase shift in terms of aberration coefﬁcients:

χ(θ, φ) =

2π θ (n+1) λ n (n + 1)

[Cnma cos(mφ) + Cnmb sin(mφ)].

m
(1) In this expression, the scattering angle θ k/K0 , where K0 ≡ 2π/λ is the wave vector of the incident electrons. Aberration coefﬁcients are identiﬁed by subscripts; the ﬁrst is the exponent for the radial wave vector (or angle) for the aberration, the second its axial symmetry and the letter indicating the orientation of the axial symmetry. Thus C1 deﬁnes defocus, and C12a,b denotes astigmatism; coma, which is second order in angle, becomes C21a,b , and C3 identiﬁes spherical aberration. The probe wave function can then be written:

p (r) ∝

aperture

− ik · r] dk, exp[−iχ(k)

(2)

where the probe amplitude is given as a function of the real-space coordinates r. This expression is a generalization to nonaxial symmetry of the more typical results, which consider only defocus and spherical aberration (Kirkland, 1998). Equations 1 and 2 provide a full prescription for calculating the size and shape of the probe, including astigmatism and other nonaxial coefﬁcients. Chromatic aberration is included by summation of probe intensities for different defocus values weighted by the electron beam energy distribution, ρ(E).

| p (r)|2 =

| p (r, C1 (E))|2 ρ(E) dE

(3)

For the calculated chromatic aberration, Cc of 1.7 mm for the VG microscopes, Cs = 1.3 mm, high-resolution pole piece, this contribution is unimportant for probe sizes greater than ∼0.5 Å. Finally a source size

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contribution is included by convolution of the result with a Gaussian distribution having a FWHM of 0.3–0.5 Å. In earlier reports, Dellby et al. (2001), Krivanek et al. (2000), and Lupini (2001) discussed a method for obtaining these coefﬁcients based on evaluation of out-of-focus electron shadow maps, or Ronchigrams, using 5–10 nm gold islands on a carbon support as an object. Use of shadow maps for characterization of optical systems was extensively discussed by Ronchi (1964) and more recently by Lin and Cowley (1986). When the Ronchigram is taken at probe focus, a microdiffraction pattern is obtained that is strongly related to the integrand in Eq. 2. Thus (Lupini, 2001)

r0 ) ∝ R(k,

T (r) p (r − r0 ) exp(ik · r)dr,

(4)

r0 ), is obtained from the Fourier where the Ronchigram amplitude, R(k, transform of the specimen transmission function, T (r), multiplied by the r0 )|2 . To calprobe wave function. The observed intensity is, of course, |R(k, culate this for an amorphous thin ﬁlm, I use a phase object approximation for the specimen function:

T (r) = exp[−iσR(r)],

(5)

where R(r) is a random amplitude with a characteristic length scale of ∼0.25 nm and a maximum amplitude of 1. The scattering cross section, σ, is set to 0.1 for an easily observable contrast. For each microscope setup, aberration coefﬁcients can be measured and used to calculate the probe shape and in-focus Ronchigram for comparison with measurements. Figure 8 shows an example of this comparison reproduced from Batson (2006). In a semi-automatic manner, this comparison is made part of a daily setup of the corrector to verify that the measured aberration coefﬁcients are consistent with the observed Ronchigram and atom images. The coefﬁcients can then be used later in multislice calculations for image interpretation.

V. IMAGING OF CRYSTALLINE OBJECTS As noted, use of crystalline objects to assess image resolution is complicated by the many errors that can be introduced during the imaging process, including nonlinearities in the acquired image intensities, scan distortions, and background noise. These effects can be reduced by cross-correlating and averaging small areas over a large ﬁeld of view (see Figure 4).

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Figure 9a shows an example of this process for the Si [211] projection. The dumbbell spacing here is the [444] fringe at 0.78 Å. The line scans in Figure 9 show comparisons of measured data with multislice calculations for two different probes with different source size contributions, 0.8 Å and 0.5 Å. Clearly the dumbbell structure is visible in this example, placing upper limits on the source contribution. However, the question arises whether the [444] spacing has been resolved, because the image contrast is smaller than typical resolution criteria. On the other hand, the presence of the [444] fringe is clear in the averaged image. A better way to discuss the optical quality, therefore, is to use the Contrast Transfer Function (CTF), which was discussed extensively for HAADF imaging by Loane et al. (1992). This is just the Fourier transform of the probe intensity for an incoherent imaging model, where the image intensity is the convolution of the probe shape with a specimen object function. It is ﬁnite at zero spatial frequency and trends monotonically to zero for large spatial frequencies. Figure 10 shows a typical CTF for the 0.8 Å probe (Batson, 2006). There are two line curves, obtained from two sets of

(a)

(e)

(c)

Au

Probe

Au Image

.080

.088

.006

0.1 nm

20 mR

(b)

20.4 20.2

0.0

0.2

0.4

0.6

(nm) 1400

(d)

(f) 1300 1200 0.09 (6) nm

1100 1000

20 mR

0.1 nm

900 20.4

20.2

0.0

0.2

0.4

(nm)

FIGURE 8 (a) Calculated Ronchigram with dashed line indicating extent of 50 mR diameter objective aperture. (b) Measured Ronchigram through a 70 mR diameter aperture. (c) Calculated annular dark-field (ADF) gold atom image intensity. (d) Measured ADF image of a single Au atom. (e) Calculated line scans for the gold atom potential, the probe, and a gold atom ADF image. (f) Line scan measurements of the Au atom intensity along perpendicular directions indicated in (d). (Reproduced from Batson, 2006.)

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(a)

0.1 nm

Digitized Image Counts

(b) 18500

20 nm Multislice

0.5 Å Source

18000 0.8 Å Source 17500

0.78 Å

17000

16500 20.2

20.1

0.0 X (nm)

0.1

0.2

FIGURE 9 (a) Averaged image result for Si [211] projection. (Reproduced from Batson, 2006.) (b) Line scans show the fringe amplitude ratio for the [444] and [111] spacings, compared with multislice calculations using two different probe sizes. The [444] fringes would not be visible using a 0.8 A° source size.

aberration coefﬁcient measurements. Two sets of small data points show azimuthal averages of single-atom images. Large data points are obtained from ratios of fringe amplitudes taken from images of Si[011] and [211] crystal projections as described in Batson (2006). For a 0.8 Å probe size,

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179

1.0 Contrast Transfer Function

5.43 Å 0.8 3.14 Å CTF

0.6

0.4

1.36 Å 16–23% 0.78 Å 0.5–2.0 %

0.2

0.0

0

20

40

60

80 100 k (2p/nm)

Noise 0.4 %

120

140

160

FIGURE 10 The contrast transfer function (CTF) provides a means to estimate the relative amplitude of fringes having different spacings. The Si [444] spacing is visible at the 0.5%−2.0% level if the noise level can be reduced lower than this.

the [444] fringe contrast is ∼0.5%–2.0%. Thus, the image noise must be reduced below this level to allow the fringe to be visible. This discussion is a reminder that probe size and resolution are not interchangeable.

VI. ALN/GAN/ALN QUANTUM WELL STRUCTURE If this probe size analysis is correct, then the CTF suggests that a 1 Å column spacing should be visible with a contrast of about ∼8%–10% if the columns contain similar atoms. If one of the columns contain lower atomic number atoms, then scattering from that column will be weaker and it may not be visible. In order to assess the impact of aberration correction in this situation, we looked at an AlN/GaN/AlN quantum well structure grown by molecular beam epitaxy (MBE) on a sapphire substrate (Mkhoyan et al., 2006a). The question for which an answer was sought was whether the polarity of the nitride matrix can be directly imaged. This is a structural property, related to the direction of the Al–N bond, which can be responsible for large internal electric ﬁelds and thus is fundamental to piezoelectric effects near the quantum well. Figure 11 shows a low-magniﬁcation image of the quantum well structure. The sapphire substrate is out of the ﬁeld of view in the upper right. Multiple GaN quantum wells in the lower left become increasingly disordered as growth proceeds. Dislocations extend from right to left in the AlN

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AlN GaN

50 nm

FIGURE 11 Low-magnification image of the Al/GaN/AlN quantum well structure. The sapphire substrate is in the upper right out of the field of view. Many dislocations extend from the upper right toward the lower left, where they induce defects in the quantum well structure.

and introduce defects within the quantum wells. Small, darker patches in the AlN are due to radiation damage during imaging in the STEM. Figure 12 shows an image of the sapphire substrate in the [1010] projection. Oxygen is readily visible in this image, as shown by the model structure overlay, but the ∼ 0.45 Å Al-Al column spacing is not resolved. This image was obtained by cross correlation and averaging of 24 patches from a lower-magniﬁcation image. Although the Al-Al column spacing is not resolved, the spot corresponding to their position is elongated in the correct direction. Figure 13 shows a small area of the AlN substrate. As indicated by the structure overlay, each bright structure is located over an AlN double column. Since the Al scatters more strongly, the column is asymmetric, with a shoulder pointing in the direction of the N column, thus exposing the polarity of the structure. A multislice calculation for this situation is also included as an inset in the ﬁgure to show that the shoulder is expected and that it occurs on the correct side, toward the N. It is tempting to assign the intensity in the shoulder to a direct beam interaction with the N column, thus encouraging use of this intensity for analytical experiments—using

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181

O O

O

Al Al

Al

Al

O

O

Al Al

Al Al O

0.1 nm

FIGURE 12 An average of several patches in a high-angle annular dark-field (HAADF) image of the sapphire substrate. The graphical overlay indicates the atomic column identities. The Al-Al projected separation of ∼0.9 A° is not resolved in this image.

0.2 nm

FIGURE 13 Low-pass filtered image showing the AlN double columns. Each column is asymmetric with a shoulder extending in the direction of the N atom column. The two insets show a ball-and-stick model and a multislice calculation for this structure.

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EELS, perhaps. It can be shown by means of multislice calculations that the Al and N atom columns are close enough for the high-energy electron beam to couple to both columns at the same time. Thus, the electron beam channels strongly, ﬁrst on one column and then the other, with a periodicity of ∼60–70 Å. Therefore, the intensity of the shoulder cannot be expected to be predicted properly by a simple convolution of the probe intensity with the N column occupancy. Figure 14 shows a side view of the multislice calculation for a 0.8 Å– diameter 120-keV electron beam when it is positioned between the Al and N columns. It clearly shows that the beam quickly channels away from the center position onto both columns and eventually concentrates on the Al column. This ﬁgure is a superposition of the beam intensity as a function of depth in the sample, with appropriately scaled atomic potentials for the Al and N, to show the relationship of the electron beam to

Al

N

Al

N

2 nm

0.2 nm

FIGURE 14 Side view of a multislice calculation for a 0.8 A° 120-KeV electron beam incident between the Al and N columns in AlN. This image is a superimposition of the beam intensity with the atomic potentials. The beam almost immediately channels onto the two columns. It then couples more strongly to the Al column, eventually concentrating at the Al position.

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Unfaulted Calculation

Faulted Calculation

FIGURE 15 AlN/GaN/AlN quantum well. N positions cannot be seen near the heavy Ga, but the polarity can be related to the orientation of the AlN by comparing the registry of the Ga columns with the Al columns. Multislice calculations for faulted and unfaulted GaN are shown to illustrate this.

the column positions. The randomness of the atom column positions is a consequence of the shifts assigned by the frozen phonon approximation (Kirkland, 1998). In moving to the GaN well, a problem is posed by the large difference in atomic number between the two atomic species. Generally, it was not possible to identify the N side without reference to the AlN matrix. However, as shown in Figure 15, it was possible to decide the polarity of the well by comparing with multislice calculations that can be aligned with the AlN matrix. By comparing the positions of the strong Ga columns with positions predicted by the calculation, we could decide that the well continued the polarity of the underlying AlN matrix.

VII. HAFNIUM OXIDE GATE STACK An initial look at the very interesting HfO2 gate stack structure has resulted in many difﬁcult questions. As summarized in Figure 16, atomic resolution HAADF images provide a wealth of detail that is not obvious in bright-ﬁeld results. This structure was fabricated as an amorphous hafnium silicate, deposited over a thin nitrited silicon dioxide on the silicon substrate (Lee et al., 2006). A high-temperature anneal promoted phase separation of the silicate into a polycrystalline HfO2 layer overlying a thick layer of silicon oxide. In Figure 16a, the crystalline and amorphous nature of the

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P. E. Batson

Poly ⫺ Si

Poly ⫺ Si

HfO2

HfO2

SiOx SiOx SiOx⫹ Hf Si (a)

Si

1 nm

1 nm

(b)

FIGURE 16 (a) Bright-field and (b) high-angle annular dark-field (HAADF) lowmagnification images of HfO2 /SiO2 /Si gate stack. The irregular, bright contrast in the HAADF image is caused by mobile Hf atoms. The structure at the Si/SiO2 interface can be resolved and used to understand the behavior of the Hf atoms within the SiO2 matrix. (Reproduced from Batson, 2005.)

various structures are visible, but not much can be deduced about interface structure or atomic composition. In Figure 16b the HAADF results clearly identify the Hf-containing layers. In the silicon dioxide some heavy atoms, most likely Hf, remain at a 5% level. Identiﬁcation of these as Hf relies on their very strong scattering, as EELS was not operating with aberration correction at the time that this material was examined. Surprisingly, the heavy atoms within the silicon dioxide are mobile under the electron beam. This was the ﬁrst indication that HAADF imaging using aberration correction might present a new experimental situation that needed to be understood. Clearly, as the probe size is reduced, an opportunity to produce higher beam current densities is presented. Modiﬁcation of the specimen during observation may then be possible. In addition, when current densities are similar to those used in the past, the additional information that is accessible using the smaller beam may show modiﬁcations that have not been noticed previously. Conversely very small beams may modify only limited regions of the specimen, allowing healing mechanisms to limit permanent damage (Mkhoyan et al., 2006b). In the IBM instrument, using a beam current of 150 pA into a 0.8 Å probe, scanning 1024 × 1024 pixels in 200 ms, the image dose is approximately 5 × 106 e− /Å2 . This dose is modest by materials physics standards, where doses up to 1010 e− /Å2 are commonly assumed to produce results applicable to as-built structures. In this example, a histogram analysis of HAADF images of the oxide showed no mass loss for doses several times higher than used for Figure 16.

First Results Using the Nion Third-Order STEM Corrector

185

The atomic movement at thermal frequencies is very fast, so even 0.2-sec exposures do not show motion but are instead snapshots of quasi-stable atomic conﬁgurations. Occasionally, double exposures are present—where a single atom moves within a frame and is later scanned a second time—but more often an atom is simply located in a different place for each imaging scan. Atoms may move out of the ﬁeld of view in one motion so that they disappear between frames. They may also move to an out-of-focus depth in the specimen and thus are less recognizable. Figure 17 shows a sequence of exposures obtained at 1-sec intervals while the beam is scanning at 5 fps. Asperities in the Hf distribution in the oxide smooth out under the beam, and in general, the center of the distribution moves toward the Si substrate. There appears to be a narrow band within about 3 Å of the substrate, where the number of Hf atoms is limited. (a)

(b)

0.5 nm

(c)

0.5 nm

(d)

0.5 nm

0.5 nm

FIGURE 17 (a)–(d) Sequence of gate stack images acquired at 1-sec intervals. The overall HfO2 /SiO2 /Si structure is unchanged, but the Hf atoms migrate toward the Si under the influence of the electron beam. Arrows indicate areas where Hf atoms are moving. (Reproduced from Batson, 2005.)

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This behavior presents an opportunity to follow physical processes at the atomic scale. If the energy deposited by the electron beam is comparable to atomic binding energies, then atomic movement under the beam can be used to deduce binding energies and local forces within the solid. In this particular example, it is known that the bond mismatch at the Si/SiO2 interface is accompanied by a small amount of strain, tensile in the Si and compressive in the SiO2 (Bongiorno et al., 2003). Since hafnium silicate contains a higher density of SiO4− 4 units than SiO2 , it follows that diffusion of 4+ Hf within SiO2 from regions of low strain to regions of high compressive strain would be favored. The resulting Hf distribution therefore points out regions that are likely to be under this compressive strain in the original structure. Within 2–3 Å of the interface, this strain goes through 0 as the interface reconstructs to match bond densities, explaining why there appears to be a narrow region next to the Si where Hf does not occur. Why does the strain not have a singularity at the interface? Several proposals have been made for the detailed atomic structure of this interface. The most widely accepted today is due to Tu and Tersoff (2000), who suggest that oxygen bridging at the interface can produce a strain-free interface by matching the bond density. This bridging is then accompanied by a bond-free tunnel at the interface. I suggest here that the presence of the tunnel contributes to the apparent low density of anions (Hf or Si) within 3 Å of the interface. Figure 18a shows a high-magniﬁcation detail of a section of the interface. Atom column positions from three images have been overlaid to show reproducibility of the indicated atom locations. The structure consists of a single-atom column at the interface, accompanied by double columns on each side. A region of low atom density is immediately above the single column. Figure 18b shows the model proposed by Tu and Tersoff (2000) for this interface, with the oxygen-bridging column indicated by the (c)

(a) (b)

0.2 nm

0.2 nm

FIGURE 18 (a) The SiO2 /Si interface in the gate stack showing a “tunnel” structure surrounding a single column. (b) Model structure includes an O bridge. (Reproduced from Tu and Tersoff, 2000.) (c) Calculated high-angle annular dark-field (HAADF) intensity using the model structure shown in panel (b). The oxygen bridge is visible.

First Results Using the Nion Third-Order STEM Corrector

187

arrow. Figure 18c shows the result for a multislice calculation that incorporates the model structure into a crystalline/amorphous Si/SiO2 interface. This exercise indicates that the oxygen, as in the Al2 O3 , is readily visible and that the appearance of a “tunnel” is conﬁrmed for this structure. The calculation also conﬁrms that the relative magnitude of single-atom Hf scattering in the SiO2 matches the Si column scattering in the 10 nm thick Si. Apparently the Hf distribution actually tells us about the distribution of strain in the material and where details of the atomic structure can be obtained to compare with the multislice simulations.

VIII. MULTISLICE SIMULATIONS Multislice image simulations for the previous example were performed using standard techniques (Kirkland, 1998) modiﬁed for aberration correction and for quasi-random atomic structure. In these imaging experiments, the specimen thickness was kept very small—of order 10 nm. Thus, beam spreading to large lateral distances is not a big problem, and so the lateral extent of the calculated model structure can be kept small. At the same time, the scanned image area can be bigger than this minimum model size. This situation can be handled by constructing a model for the calculation that is anchored to the probe position. This limited model is derived from a much larger, nonperiodic model that summarizes the long-range behavior of the sample. Choosing an anchor point in this large model allows selection of the relevant nearby atoms for the multislice calculation for the beam propagation. After the atom coordinates are chosen, they are shifted according to the frozen phonon model rules, and appropriate atomic potentials are then centered on each coordinate. This process is done layer by layer so that random structures can be simulated. In these simulations, the initial probe wave function is calculated using Eq. 2, including defocus, C3 , and C5 , but excluding nonsymmetric terms such as coma and astigmatism. The nonperiodic nature of the model allows variance of the physical content of the strucure within the same imaging calculation to test various possibilities. Figure 19a shows a [110] projection of a model structure used for the Si/SiO2 simulation. First, there is a single oxygen bridge at the center of the model. On the right at the interface, a silicon pair was inserted for a comparison with the oxygen bridge. The coordinates for the pair were obtained from measurements at the free [001] 2 × 1 reconstructed silicon surface (Tromp et al., 1985). This pair is viewed end on, and so it could be confused with an oxygen bridge, but its position and scattering strength are different. In the center of the model, an interface step was constructed by rotating the model 90 degrees about the [001] interface normal. This illustrates (on the left) that “tunnels” will not be visible if they happen

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to be oriented perpendicular to the beam propagation direction. Within the SiO2 matrix, Hf atoms replace Si atoms with a 5% frequency. Placing these on a Si position is an arbitrary choice for convenience, although there is some theoretical support. (Kawamoto et al. 2001) If it is possible that a quasi-silicate persists, then these atoms should be positioned between SiO4 tetrahedra, but this was not attempted in this study. Figure 19b shows the result for a 10 nm thick model using 10 phonon conﬁgurations. The interface structure is well resolved. The interface step is visible and provides a qualitative comparison with the low-magniﬁcation

(a)

(b)

FIGURE 19 (a) Model structure for a more comprehensive consideration of the gate interface. This model includes O bridging, Si dimers, and a step that rotates the interface by 90 degrees about the interface norm. (b) ADF multislice image calculation. Small arrows indicate interface columns. Large arrow indicates out-of-focus Hf atom.

First Results Using the Nion Third-Order STEM Corrector

189

result in Figure 16. Both the oxygen bridge and the Si pair are visible (small arrows); the Si pairing is more intense and positioned slightly deeper into the SiO2 side of the interface. In brief, the observed single-column intensity and position lie somewhere between the two possible interface structures. This will be discussed in a future publication. Hf atoms in the SiO2 are about as bright as the Si columns, as seen in the experiment. In the upper right corner of the calculated image a ring structure is present, highlighted by the arrow. This is an out-of-focus image of a Hf atom embedded within the SiO2 . Simulations of the probe shape, using Eq. 2, show that a ring structure such as this can be obtained using a defocus of –60 to –100 Å. The intensity of this feature is about 10% of the in-focus peak. There is some evidence for this type of structure in the experimental images. The left side of Figure 20 shows two successive frames of the HfO2 stack (a)

(b)

000518

0.5 nm

(c)

(d)

000519

0.5 nm

FIGURE 20 (a, b) Successive frames of the HfO2 stack structure, showing a transformation of a single atomic column into a “ring” structure. The subsequent image shows that the column has been removed. This appears to show an example of specimen modification by the beam. (c, d) Magnified areas of a, b. (d) Shows a ring structure, characteristic of out-of-focus atom.

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structure with a tunnel structure framed by a small box. On the right side the boxed area is shown at increased magniﬁcation. In the upper image, the single column under the tunnel is clearly in focus. In the lower image, the single column appears as a partial ring. Two interpretations are possible: (1) the probe focus has changed by ∼5 nm between these two exposures, or (2) the column structure has shifted downward, away from the beam entrance surface by the appropriate amount. Since the probe focus stability is ∼0.3–1.5 nm in this instrument and the Si columns appear not to change in focus, it seems most likely that the single column moved in the vertical direction under the beam. Subsequent images show this column to be removed, suggesting that the entire column may have moved downward and out of the bottom of the specimen.

IX. ATOMIC MOVEMENT UNDER THE ELECTRON BEAM The previous descriptions clearly show that movement of individual atoms, resulting in structural modiﬁcation of the specimen, will be a common problem in using aberration-corrected electron beams. In the past, there has been extensive discussion about radiation damage, and much effort was expended on mitigation for, or elimination of, its effects (Hobbs, 1979). It appears that for nanometer-sized objects consisting of few atoms that atomic movement during measurement is unavoidable. This conclusion is consistent with previous work by others on very small metal islands (Iijima, 1985; Marks, 1994; Smith et al., 1986). Figure 21 shows a set of six frames of a 1.5 nm diameter gold cluster on an amorphous carbon substrate, located near a relatively large cluster. This particular sequence illustrates several characteristics. First, objects can remain relatively unchanged for signiﬁcant periods of time and then change rapidly. In this case, the small cluster remained separate from the large cluster for nearly 50 sec under the beam and then coalesced with the large cluster within 1 sec. Second, single Au atoms usually remain bonded to a small area of the carbon substrate. For instance, two atoms in frame 245 (indicated by arrows) remain within 1–2 Å of a single point during the ﬁve subsequent frames, suggesting that single Au atoms can be relatively strongly bonded to the substrate. Third, groups of Au atoms are less tightly bound to the substrate, forming small objects with high surface curvature. When these clusters are smaller than ∼1.5 nm, curvature at the edges can be very large, resulting in protuberances, or tendrils, that extend as much as 1× the diameter of the nanoparticle. Frame 89 is a good example of this behavior. Fourth, when Au clusters are greater than 1.5 nm in diameter, surface curvature is more moderate, possibly due to the development of surface tension, and the clusters become more

First Results Using the Nion Third-Order STEM Corrector

89 17.6 sec

191

245 48.8 sec

0.5 nm

246 49.0 sec

0.5 nm

247 49.2 sec

0.5 nm

248 49.4 sec

0.5 nm

249 49.6 sec

0.5 nm

0.5 nm

FIGURE 21 Coalescence of a 1.5-nm Au cluster with another, larger cluster. Some Au atoms remain bonded to the carbon substrate. The 1.5-nm cluster is probably just large enough to illustrate the transition from molecular bonding with the substrate to metallic bonding within the cluster. (Reproduced from Batson, 2008.)

compact. In this size range, clusters rapidly coalesce under the beam, illustrated by the sequence of frames 245–249, leaving a few atoms behind that were apparently tightly bonded to the substrate. It seems possible that electronic ﬂuctuations, either spontaneous or driven by the electron beam (surface plasmons), may produce an attractive force that pulls the two clusters together.

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This rich set of behaviors is controlled by interaction with the electron beam and by fundamental physical processes. For instance, the transition from the strong bonding of single atoms with the substrate to a compact nanocluster, which is weakly bound, is likely driven by the development of metallic bonding in the cluster with increasing size. The disappearance of spiky tendrils in favor of a compact surface indicates the appearance of surface tension for similar reasons. The question that merits in-depth exploration is whether the behavior viewed under the beam is representative of the behavior in the absence of the beam. For instance, can the sensitivity of structure to the beam be characterized in some simple way, perhaps as an equivalent local temperature? Experiments using temperature-controlled stage may address this issue.

X. CONCLUSIONS These initial studies using the Nion third-order aberration corrector in the STEM were intended to characterize the probe size performance and to explore some limits of the performance. We found that the new optics is capable of reducing the probe size by a factor of about 3× to 0.8 Å, with minimal loss of current. Single heavy atoms and columns of light atoms, including oxygen, are easy to image with this probe. It was found that multislice calculations are needed to fully understand the imaging results at this probe size, and that it is possible to use measured aberration coefﬁcients to accurately deﬁne the initial probe wave function for the multislice simulation. Finally, an efﬁcient multislice method was developed to deal with HAADF STEM imaging in thin nonperiodic structures. New instrumental capability often raises new questions. These preliminary studies are very good examples of this behavior. In the past, single-atom behavior was not normally visible, particularly within crystalline structure, and so the possibility of atomic movement during imaging was not considered in the absence of obvious extensive damage. These studies show that all angstrom-level imaging is accompanied by atom movement. The task now must be to understand this movement in the context of the as-built structure. How energetic is the beam-specimen interaction, relative to thermally driven processes? This depends strongly on details of the probe size, its impact position, and the electronic excitations available at its point of impact. How quickly do structural changes occur under the beam? Can structural changes be minimized by taking fast exposures or by lowering the specimen temperature? Extensive effort in these studies was expended just to set up the optics for particular spot size within a particular specimen area. As ﬁfth-order aberration correction becomes routine in new instruments, this process

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will become much easier. Rather than being limited by the microscope optics, the probe will be limited by the source size and brightness. Imaging of crystalline objects at atomic resolution will become routine, and large beam currents will allow the acquisition of atomic resolution electronic structure maps using EELS (Muller et al., 2008). It is also probable that tiltseries tomography will be enhanced by control over focus height, so that full three-dimensional structures will be obtainable at atomic resolution. Thus aberration correction ensures that the next few years of instrumental and technique development in electron microscopy will be very exciting.

ACKNOWLEDGMENTS I thank O. L. Krivanek, N. Dellby, and M. Murﬁtt for many hours of discussion and help in operation of the new corrector. Some of this work was done in collaboration with A. Mkhoyan and J. Silcox. I also acknowledge, discussion with D. A. Muller and E. J. Kirkland about the multislice technique.

REFERENCES Batson, P. E. (1986). High resolution electron energy loss spectrometer for the scanning transmission electron microscope. Rev. Sci. Inst. 57, 43–48. Batson, P. E. (1988). Parallel detection for high resolution electron energy loss studies in the scanning transmission electron microscope. Rev. Sci. Inst. 59, 1132–1138. Batson, P. E. (2003). Aberration correction results in the IBM STEM instrument. Ultramicroscopy 96, 239–249. Batson, P. E. (2005). Challenges and opportunities of angstrom-level analysis. In “Microscopy of Semiconducting Materials” (A. G. Cullis, ed.), pp. 836–837. Royal Microscopy Society, Oxford, UK. Batson, P. E. (2006). Characterizing probe performance in the aberration corrected STEM. Ultramicroscopy 106, 1104–1114. Batson, P. E. (2007). Au cluster behavior using aberration corrected ADF imaging. In “Microscopy and Microanalysis” (M. Marko, J.-H. Scott, E. Vicenzi, S. DeKanich, J. Frafjord, P. Kotula, S. McKernan, and J. Shields eds.), pp. 858–859. Cambridge University Press, Cambridge. Batson, P. E. (2008). Motion of gold atoms on carbon in the aberration corrected STEM. Micros. Microanal. 14, 89–97. Batson, P. E., Dellby, N., and Krivanek, O. L. (2002). Sub-angstrom resolution using aberration corrected electron optics. Nature 418, 617–620. Batson, P. E., Mook, H. W., and Kruit, P. (2000). High brightness monochromator for STEM. In “International Union of Microbeam Analysis 2000,” vol. 165 (D. B. Williams and R. Shimizu, R. eds.), pp. 213–214. Institute of Physics, Bristol, UK. Bongiorno, A., Pasquarello, A., Hybertsen, M. S., and Feldman, L. C. (2003). Transition structure of the Si(100)-SiO2 interface. Phys. Rev. Lett. 90, 186101. Crewe, A. V., Wall, J., and Langmore, J. (1970). Visibility of a single atom. Science 168, 1338–1340. Dellby, N., Krivanek, O. L., Nellist, P. D., Batson, P. E., and Lupini, A. R. (2001). Progress in aberration-corrected scanning transmission electron microscopy. J. Electr. Microsc. 50, 177–185.

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Hobbs, L. W. (1979). Application of transmission electron microscopy to radiation damage in ceramics. J. Am. Ceram. Soc. 62, 1151–2916. Howie, A. (1979). Image contrast and localized signal selection techniques. J. Micros. 117, 11–23. Iijima, S. (1985). Electron microscopy of small particles. J. Electr. Micros. 34, 249–265. Kawamoto, A., Jameson, J., Grifﬁn, P., Cho, K., and Dutton, R. (2001). Atomic scale effects of zirconium and hafnium incorporation at a model silicon/silicate interface by ﬁrst principles calculations. IEEE Elect. Dev. Lett. 22, 14–16. Kirkland, E. J. (1998). “Advanced Computing in Electron Microscopy.” Plenum Press, New York. Krivanek, O. L., Dellby, N., and Lupini, A.R. (2000). Method for measurement of optical aberrations, US patent application. Lee, K. L., Frank, M. M., Paruchuri, V., Cartier, E., Linder, B., Bojarczuk, N., Wang, X., Rubino, J., Steen, M., Kozlowski, P., Newbury, J., Sikorski, E., Flaitz, P., Gribelyuk, M., Jamison, P., Singco, G., Narayanan, V., Zafar, S., Guha, S., Oldiges, P., Jammy, R., and Ieong, M. (2006). Poly-Si/AlN/HfSiO stack for ideal threshold voltage and mobility in sub-100 nm MOSFETs. In 2006 Symposium on VLSI Technology, Digest of Technical Papers, pp. 160–161. Lin, J. A., and Cowley, J. M. (1986). Calibration of the operating parameters for an HB5 STEM instrument. Ultramicroscopy 19, 31–42. Loane, R. F., Xu, P., and Silcox, J. (1992). Incoherent imaging of zone axis crystals with the ADF STEM. Ultramicroscopy 40, 121–138. Lupini, A. R. (2001). Aberration Correction in STEM. Cambridge University, Cambridge, UK. Marks, L. D. (1994). Experimental studies of small particle structures. Rep. Prog. Phys. 57, 603–649. Mkhoyan, K. A., Batson, P. E., Cha, J., Schaff, W. J., and Silcox, J. (2006a). Direct determination of the local polarity in wurzite crystals. Science 312, 1354. Mkhoyan, K. A., Silcox, J., Ellison, A., Ast, D., and Dieckmann, R. (2006b). Full recovery of electron damage in glass at ambient temperatures. Phys. Rev. Lett. 96, 205506. Mook, H. W., Batson, P. E., and Kruit, P. (1999). Operation of the fringe ﬁeld monochromator in ﬁeld emission STEM. In “Electron Microscopy and Analysis 1999” vol. 161 (C. J. Kiely, ed.), pp. 223–226. Institute of Physics, Bristol, UK. Muller, D. A., Fitting-Kourkoutis, L., Murﬁtt, M., Song, J. H., Hwang, H. Y., Silcox, J., Dellby, N., and Krivanek, O. L. (2008). Atomic-scale chemical imaging of composition and bonding by aberration-corrected microscopy. Science 319, 1073–1076. Pennycook, S. J., and Jesson, D. E. (1990). High resolution incoherent imaging of crystals. Phys. Rev. Lett. 64, 938–941. Ronchi, V. (1964). Forty years of history of a grating interferometer. Appl. Optics 3, 437–450. Smith, D. J., Petford-Long, A. K., Wallenberg, L. R., and Bovin, J. O. (1986). Dynamic observations of atomic-level rearrangements in small gold particles. Science 233, 872–875. Tromp, R. M., Hamers, R. J., and Demuth, J. E. (1985). Si(001) dimer structure observed with scanning tunneling microscopy. Phys. Rev. Lett. 55, 1303–1306. Tu, Y., and Tersoff, J. (2000). Structure and energetics of the Si-SiO2 interface. Phys. Rev. Lett. 84, 4393–4396. Yu, Z., Batson, P. E., and Silcox, J. (2002). Effects of detector black level in ADF-STEM imaging. In “Microscopy and Microanalysis 2002” (E. Voelkl, D. Piston, R. Gauvin, A. J. Lockley, G. W. Bailey, and S. McKernan, eds.), pp. 474–475. Cambridge University Press, Cambridge.

CHAPTER

5 Scanning Transmission Electron Microscopy and Electron Energy Loss Spectroscopy: Mapping Materials Atom by Atom Andrew L. Bleloch*

Contents

I Introduction II Experimental Heuristics III Case Studies A Bulk Crystal B Interfaces/Planar Defects C Nanostructures D Biological Systems IV Discussion References

195 196 199 199 202 211 217 221 221

I. INTRODUCTION Knowing speciﬁc atoms’ location and arrangement and how they are bonded is one deﬁnition of a complete characterization of a material. Indeed, many materials characterization techniques are a quest for this, or a speciﬁc subset of this, information. Aberration-corrected microscopy has signiﬁcantly improved our ability to acquire these data from individual structures in mostly crystalline materials that can withstand the electron dose. However, we are still, unable to directly image the atomic structure of amorphous materials and this remains a challenge to be met (Van Dyck et al., 2003). Otherwise stated, transmission electron microscopy (TEM) has always been most productively used where translational symmetry is interrupted as at an interface, grain boundary, defect, or nanostructure. Aberration correction means the productivity has progressed to the point * SuperSTEM Laboratory, Daresbury and University of Liverpool, United Kingdom Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01005-7. Copyright © 2008 Elsevier Inc. All rights reserved.

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that the atomic structures can be routinely found; therefore, the majority of this chapter is concerned with defects and nanoparticles. Having made the pitch for emphasizing the study of defects, many early publications used a known perfect crystal structure (often silicon) as a standard against which the performance of the instrument could be measured. While this is not an end in itself, it does build conﬁdence in the technique and the technology.

II. EXPERIMENTAL HEURISTICS Although this is not a handbook, it is one of the early books in the ﬁeld and therefore some general hints and tips from our experience may be useful. An electron microscope (EM) experiment can be divided into four elements: the sample, the instrument, the site, and the operator. The site and operator will not be discussed here except for strongly advising anyone planning to construct a laboratory for an aberration-corrected instrument to consult those who have been through the process, in addition to talking to the instrument supplier. Good samples are essential to realize the beneﬁts of aberration correction. At the most mundane level, the largest number of wasted days can be blamed on excessive carbonaceous contamination. STEM is much more susceptible to this scourge than TEM because the only illuminated part of the sample is exactly the point of interest. For most samples, this problem can be avoided by “good hygiene”, —all users always use gloves (properly), avoid all solvents, and ensure that all holders/components are plasma cleaned (or cleaned according to a typical ultrahigh-vacuum component cleaning cycle of degreasing followed by alcohol and then deionized water, all in an ultrasonic bath). If solvents are essential, their cleanliness can be assessed by allowing a drop to dry on a glass slide. If there is any residue that allows you to indentify the drop’s location on the slide, then the solvent is not clean enough. In short, contamination can be eliminated and should not be acceptable. For samples that need to be thinned from a bulk material, the best specimens we have studied have been mechanically tripod or wedge polished followed by a brief (typically a few minutes) low-angle, low-energy argon ion–milling step. Acceptable specimens can be made by conventional ionmilling techniques, but this seems to frequently leave too many nanometers of damaged material at the surfaces. We have not yet seen a focused ion beam (FIB)-prepared sample that is suitable for both imaging and electron energy loss spectroscopy (EELS) at the highest spatial resolution with our 100-kV microscopes. However, a combination of FIB followed by lowenergy ion polishing with either Ar+ or Ga+ ions shows promise and may become the technique of choice in the future.

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Gatan Enfina Coupling module Projector lenses Objective lens Aberration corrector

Condensor lenses Gun

FIGURE 1 Images of the two dedicated STEM microscopes. The older microscope is on the left. All that is visible (except the Nion coupling module and Gatan spectrometer) on the microscope on the left was manufactured by Vacuum Generators (the Nion corrector is not visible externally). On the microscope on the right, the entire instrument that is visible (except the Gatan spectrometer) was manufactured by Nion.

The SuperSTEM laboratory has two dedicated STEM instruments; each is equipped with quadrupole-octupole aberration correctors designed and fabricated by Nion Co. (Figure 1). The older instrument is a VG HB501 retroﬁtted with a second-generation Nion corrector and accounts for almost all the results presented here. The pertinent components of both instruments are more completely and appropriately described elsewhere and in this volume (Hawkes, 2008; Krivanek, Dellby, and Lupini, 1999; Krivanek et al., 2003, 2008). The particular operating parameters relevant for the work reported here are shown in Table I. All images (unless otherwise stated) represent raw data (i.e., unﬁltered). In some early work, a geometric distortion caused by nonorthogonal scans was removed. This distortion was calibrated by reference to a known perfect lattice such as silicon [110]. The only other treatment to the images was to remove ﬂuctuations in the beam intensity. This was done either by assuming that each line in the image should have the same average intensity or, if that was a poor assumption, that a uniform region of the sample would be expected to have the same average intensity. Methods of aberration diagnosis also are discussed in other chapters (Hawkes, 2008). The only additional practical advice is to diagnose “early and often.” The high-order aberrations (third order and above) are

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TABLE I

Typical Operating Parameters of the SuperSTEM instrument

Accelerating voltage

100 kV

100 kV

Typical probe current

60–100 pA

2 modes (100 pA, 600 pA)

Probe forming aperture semi-angle 24 mrad

Typically 30 mrad

HAADF collection semi-angle

70–210 mrad Typically 65–200 mrad

EELS collection semi-angle

19 mrad

<60 mrad

usually remarkably stable for a given optical conﬁguration. In contrast, the extreme constraints on the ﬁrst- and second-order aberrations necessary to realize the advantages of correcting the third- (and higher-order) aberrations means that these low-order aberrations are, in practice, the active constraints while using the microscope and they require “tweaking” approximately every 30 minutes to maintain peak performance. This can be done either by hand (which requires considerable experience and skill) or by using one of the aberration diagnosis algorithms. An important ingredient in successfully operating the older HB501 instrument retroﬁtted with the Nion corrector is the Nion software used to diagnose the aberrations. This software allows the aberrations to be measured in less than 10 s. After acquiring the “perfect” set of data, it is usually desirable to simulate the images to conﬁrm one’s understanding of the experiment. We have used two software packages to simulate high-angle annular darkﬁeld (HAADF) images—one from Kirkland (Kirkland, 1998) and one from Koch (Koch, 2008). Each program has its advantages; the omission of other packages merely indicates that we have been content with the results from these. An important input to the simulations, in order to calculate quantitatively the interaction of the electrons with the sample, is the probe geometry. Given the drift and uncertainty in the low-order aberrations, this is a nontrivial task. Wang et al. (2008) have matched, to within a constant factor, the variation of the HAADF signal with position by simulating a good operator. This was done by assuming the measured second-order and higher aberrations are correct and that the operator optimally compensated for the residual consequent astigmatism and defocus. It is possible to match the shape of the HAADF signal, but the overall intensity is signiﬁcantly overestimated by the simulation. This is reminiscent of the so-called Stobbs factor in high-resolution electron microscopic (HREM) simulations and is believed to be due to the amorphous layer on the entrance surface (Mkhoyan et al., 2008), point defects, and so on. It is hoped that this effect can be understood and included in simulations in the near future. It has long been accepted that, by the principle of reciprocity, each pixel of a bright-ﬁeld scanning transmission electron microscope (BF-STEM) image is equivalent to that of a conventional transmission electron microscope (BF-CTEM) image taken under equivalent conditions.

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However, each geometry is particularly suited to speciﬁc modes. In the case of STEM, this is the combination of HAADF imaging and electron energy loss mapping. Despite the principle of reciprocity, most EM manufacturers have only recently begun to construct microscopes capable of realizing the same point resolution in STEM mode as in CTEM. Although commercial reasons undoubtedly existed for not doing so, in principle, there were no impediments to this happening decades earlier. Although the improvement of the probe forming ability of modern TEM/STEM instruments was progressing, it could be argued that the advent of successful aberration correction was the ﬁnal impetus required to bring STEM performance up to the standard of CTEM performance across most manufacturers. This impetus took the form of early successes with breaking resolution barriers using the STEM geometry rather than CTEM mode (Batson, Dellby, and Krivanek, 2002; Nellist et al., 2004). The reasons for this merit examination because I believe they illustrate a signiﬁcant and underappreciated beneﬁt of STEM. Pennycook and Jesson (1990) and many publications since showed that under suitable imaging conditions, the HAADF signal in a given orientation increases monotonically with Z and with the specimen thickness. This differs from the contrast reversals expected in high-resolution transmission electron microscopic (HRTEM) imaging. This difference does not necessarily render interpretation of HAADF images trivial, and it is still necessary to simulate to quantitatively understand the contrast but, crucially, it does make it easy to identify quickly whether a data set is good (or not) and to optimize the microscope parameters during the experiment to improve the data quality. The cycle of proposing hypotheses and testing them at least qualitatively can proceed much more easily. For HAADF (but not for EELS), the constraint on suitable sample thicknesses is also less severe. This rapid feedback and the relaxed sample constraint do give the STEM geometry an advantage in the early stages of an investigation.

III. CASE STUDIES A. Bulk Crystal Complex metal oxides such as perovskites are widely investigated because of the remarkable range of physical phenomena they exhibit. The material investigated in this study (Bosman et al., 2007) was Bi0.5 Sr0.5 MnO3 , which shows colossal magnetoresisitive behavior and was believed to show charge ordering at room temperature (Van Tendeloo et al., 2004). Despite the general notion that electron microscopy is most often applied to imperfect crystals, this study of a bulk crystal is important for

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several reasons. It was the ﬁrst publication in a refereed journal, although not the ﬁrst data presented at a conference. The motivation for the study was to look for direct evidence of charge-ordering domains, which can be thought of as an interruption of the crystalline periodicity, but such ordering was not found. Most important, it demonstrates both beneﬁts and pitfalls of column by column mapping. EELS of individual columns and planes has been demonstrated before (Browning et al., 1993; Varela et al., 2004), but this ability to routinely map the chemistry atom column by atom column has resulted in a ﬂurry of papers in high-proﬁle journals (Bosman et al., 2007; Kimoto et al., 2007; Muller et al., 2008). Chronologically, this mapping occurred about ﬁve years after the ﬁrst aberration-corrected instruments that were capable (in principle) of doing the experiment were installed because the demands on the instrumentation are the most stringent for this experiment. Before discussing the data, the method of spectrum image acquisition and data reduction deserves some attention. Bosman et al. (2007) used the SuperSTEM instrument to acquire spectra of remarkable quality by implementing a method to average away what is oxymoronically termed ﬁxed pattern noise. Although in principle, both dark current and pixel to pixel gain variation can be arbitrarily accurately compensated, in practice the pixel to pixel gain variation appears to be a complicated mixture of effects in the pixel itself and in the readout row of the charge-coupled device (CCD). By moving the spectrum along the CCD chip using the drift tube voltage and then realigning the spectra afterward (either by knowing the sequence of shifts or by cross-correlating the spectra), the ﬁxed pattern noise is made to behave like random noise and the signal to noise ratio (or more correctly, the useful information content) of the spectra is improved. The spectra were then subjected to a principal component analysis, which was used to compute a background to subtract from each raw spectrum that was much less susceptible to noise than traditional background subtraction methods. (Despite the careful acquisition, no evidence was found in the ﬁne structure of the Mn edge for charge ordering.) HAADF images and atomically resolved EELS spectrum images were acquired from three orientations but only the <110> is shown in Figure 2. The simulated images include both elastic and inelastic scattering and show remarkable agreement with the data. The most revealing is Figure 2b in which the oxygen columns that also contain Mn are almost invisible compared with the oxygen-only columns despite equal oxygen concentrations. This is a signiﬁcant but subtle effect of the interaction between elastic and inelastic scattering. In one of the orientations studied but not shown here, the peaks of intensity in the Mn EELS spectra do not occur on the Mn columns, an effect entirely predicted by the calculation. (The slight zigzag along the rows of oxygen columns is seen both in the simulation and the image and is due to the tilting of the MnO6 octahedra.)

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(a)

201

Bi, Sr Mn O

1 nm

(b)

515 605 695 Energy loss (eV)

(c)

515 605 695 Energy loss (eV)

FIGURE 2 Bi0.5 Sr0.5 MnO3 oriented along the 110 zone axis. (a) Z-contrast, (b) oxygen K-edge, and (c) manganese L2;3 -edge STEM images. The zigzag pattern of the oxygen columns is most noticeable in this specimen orientation and is caused by the tilted MnO6 octahedra. The simulations assume a 12-nm thick sample. (From Bosman et al., 2007.)

As carefully noted in Bosman et al. (2007), although one of the intrinsic advantages of HAADF and EELS mapping is that they often allow useful qualitative interpretation, they do not always allow this direct visual interpretation. This is clearly an example where the data cannot be interpreted without simulation. As a rough rule of thumb, large collection apertures, thin samples, and loss edges beyond 200 eV tend to provide more directly interpretable atomic EELS maps. In any case, quantitative agreement should be sought; therefore, theoretical modeling and calculation is highly desirable in all cases and essential in some. The two other more recent publications showing column by column mapping were also studies of perovskite structures. Kimoto et al. (2007) mapped one of the elements with two different edges, and Muller et al. (2008) were able to use a sufﬁciently large collection angle, a suitably thin sample, and core loss edges that were not too low in energy. The resulting map was therefore intuitively interpretable.

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B. Interfaces/Planar Defects In our experience, solving interface structures is most often hindered by sample growth and preparation. Once a sample is available that is known to have atomically sharp interfaces with ideally one (but not more than a few) atomic structures and the sample can be prepared so that the interface can be viewed from two directions, the heuristic for devising a candidate structure is straightforward. With aberration-corrected STEM and atomic resolution EELS, this candidate structure is not guaranteed to be unique but is very tightly constrained and is very likely to be the correct structure. Conﬁrmation by theoretical calculations then yields great conﬁdence that the proposed structure reﬂects nature. Given two atomically resolved images from two directions parallel to the plane of the interface, the ﬁrst step is to ascertain the relative position of the two known lattices far (i.e., a few unit cells) from the interface. With two views of the interface, relative orientation and positions are determined in three dimensions. The next constraint to apply is to determine from the images how many atomic planes are reconstructed at the interface. The two perfect crystals can then be grown toward each other in a ball-and-stick model (either physically or in a computer). The contrast in the HAADF images is then used to determine where atomic columns may or may not be placed. If an attempt is made to grow the model from the bulk of one crystal through the interface to the next, the constraints of linking bonds prejudices the builder against a number of possible structures that may be correct. This is the case in the interface that will now be used to illustrate this procedure. Characterizing the atomic structure of metal silicide/silicon interfaces has provided a series of challenges that have waxed and waned in perceived importance over recent decades. The EM has played an important role in attempts to solve these structures with mixed success. However, the advent of aberration correction has signiﬁcantly changed our expectations. This system serves well to illustrate the extent to which the qualitative information from this new generation of microscopes allows the structure to be solved directly and quickly. Coincidentally two early publications by two independent groups proving their aberration-corrected EMs were on the cobalt di-silicide/silicon (100) interface (Falke et al., 2004; Haider et al., 1998). The lattice mismatch is small (−1.2% for CoSi2 and −0.4% for NiSi2 ) with atomically smooth epitaxial interfaces between the conducting silicide and the semiconducting silicon. (CoSi2 has the higher conductivity and has been considered as a possible interconnect material.) Figure 3 shows two images of the interface from a sample grown at Chemnitz Technical University by molecular beam epitaxy using a process known as allotaxy (Mantl and Bay, 1992). The specimen was prepared by tripod polishing followed by low-angle ion milling at 3 kV in a Baltec

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2 nm

[001] [110]

[001] [110]

[110]

[110]

FIGURE 3 Two views of the nickel disilicide/silicon interface taken from different places on the same sample. The views have been aligned with the disilicide lattice contiguous. This necessarily means that the silicon lattice planes are not contiguous.

RES010 (Falke et al., 2004). Some analysis of the prevalence of the two variants together with the known information that the interface showed a 2 × 1 periodicity (e.g., (Bullelieuwma et al., 1991)) led to the hypothesis that these were orthogonal views of the same structure. The higher cubic symmetry of the metal silicide (space group Fm3m compared with Fd3m of silicon) means that its projection is identical in all <110> directions. The images are arranged so that the silicide lattice planes continue through from one image to the next so the change in the alignment of the silicon columns is then clear. In summary, there are two possible growth positions for the silicide that are at (0, 0, 0) and (1 4, 1 4, 1 4) with respect to the silicon unit cell. Given these two images and the procedure outlined above, a model was constructed. It comprises ﬁvefold coordinated silicon and sevenfold coordinated nickel sites at the interface (Figure 4). Building this ball-and-stick model of the interface forces one to realize that the manner in which the electrons are distributed between bonds at the interface is far from obvious. On the other hand, it does lead to the observation that the interface could have been predicted by taking the 2 × 1 dimer reconstructed Si [100] surface known to be the stable surface in vacuum and simply butting it up to the metal terminated unreconstructed [100] surface of the silicide without undue concern for assigning the electrons to bonds. (The relative alignment in (010) and (001) is easily obtained from TEM images.) The proposed structure had not been deduced from experiment before but had been recently predicted theoretically by Yu et al. (2001) to be energetically favored using density functional theory to distinguish between candidate structures. They also provide a concise summary of the state of knowledge of the interface prior to aberration-corrected STEM. This model was

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CoSi2

Si [110]

Si Co

FIGURE 4 Two views (along [110] and [1–10]) of the cobalt disilicide/silicon sevenfold R interface with simulations and ball-and-stick models overlaid. The HAADF images are 2 nm across. (Adapted from Falke et al., 2005.) (See color insert).

designated by Yu et al. as the sevenfold R model. The model is called sevenfold because the metal (Ni or Co) atom at the interface is sevenfold coordinated. The R denotes that the model has been allowed to relax to lower the energy. This relaxation leads to the formation of dimers of the silicon atoms on the terminating monolayer of Si and signiﬁcantly, they found that there is indeed signiﬁcant electron density in the ﬁfth bond of the ﬁvefold coordinated Si atoms. Studies of the cobalt disilicide interface were complicated by the fact that more than one structure was present. In addition to the predominant sevenfold R interface, a second, previously described (Werner et al., 1993), 1 × 1 interface structure comprising eightfold coordinated Co was present again with differing projections in two orthogonal directions (Figure 5). This was found to be the second lowest energy structure by Yu et al. (2001). Although the metal atoms are in a bulk-like eightfold coordination, the silicon is signiﬁcantly undersaturated. The calculated interface energies are given in Table II. Having solved the interface structure, the quality of information in the HAADF images has led to the identiﬁcation of several defects at the interface in addition to the complex structure brieﬂy discussed. For the predominant sevenfold R interface structure, if the silicide is grown on the silicon, a single monolayer step on the silicon substrate would cause two adjacent domains to have the two different growth positions and would have their lattices shifted by (2n + 1)/4 times [001] with respect to each other. Such domains cannot be contiguous and must be divided by stacking faults, interface edge dislocations, or highly strained at their domain

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CoSi2

Si [110] Si Co

FIGURE 5 Two views (along [110] and [1–10]) of the cobalt disilicide/silicon eightfold interface with simulations and ball-and-stick models overlaid. In this case, the HAADF images have been averaged along the interface and are 2 nm across. (Adapted from Falke et al., 2005.) (See color insert). TABLE II Interface energies (relative to the sevenfold R model) for the structural models of the MSi2 /Si(001) interfaces Energies [eV/(2 × 1)] Model

Sixfold Eightfold LGY BJV Sevenfold U Sevenfold R

Reconstruction

NiSi2 /Si (001)

CoSi2 /Si(001)

— — Dimer (unstable) Compositional — Dimer

0.42 1.74 — 2.76 1.52 0.00

0.87 0.27 — 1.92 1.01 0.00

FromYu et al. (2001).

boundary. Since the translation vector between the origins of the two different growth positions is ¼a<111>, the interface edge dislocations at those domain boundaries would have a Burgers vector of ¼a<111>. Possibly due to the growth method (so called allotaxy) but possibly also due to more general thermodynamic constraints, the two domains were always observed to be separated by {111} facets (Figure 6). At the cobalt disilicide eightfold interface a number of interface defects (or variant structures) were also observed. These are still the subject of investigation, but Figure 7 shows two such structures. In Figure 7a, the column of Co has been exchanged with a column of Si every ﬁfth (010) atomic plane along the interface. This usually occurred near a step in the interface and thus may be associated with a different stress distribution than that at the plane interface. In Figure 7b an additional pair of Si columns

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FIGURE 6 A five-plane (111) facet between the two orientations of the sevenfold R cobalt disilicide/silicon interface viewed down the silicon [110] zone axis. The 2× reconstruction can be seen on the left. Notice the two layers of so-called B-type twin at the (111) facet. This step incorporates two dislocations: a misfit dislocation and a dislocation with the necessary Burgers vector to accommodate the difference in the two orientations of the interface. This structure is reproduced in detail wherever the two domains about in the sample, although the number of dumbbells in the B-type twin is not always five. (The image is 7 nm across.)

along the [110] direction penetrates the silicide lattice approximately every 9 (1–10) planes. A second investigation of a perovskite material is presented here (Bangert et al., 2006). Instead of the three-dimensional (3D) solid, the chemistry at a two-dimensional (2D) planar defect, known as a Ruddleston-Popper (RP) defect, is studied. This example illustrates that both HAADF and EELS can be used to provide chemical information and that, in tandem, they provide a very convincing analysis. Complex metal oxides with the perovskite structure are candidate electrocatalysts for fuel cells and metal/air batteries (McEvoy, 2001). The subject of this study is La0.6 Ca0.4 CoO3 (LCCO). The operation of the catalyst is believed to proceed by hole-hopping from Co4+ to Co3+ and by oxygen vacancy diffusion (Tanaka and Misono, 2001). RP defects are prevalent in both sub-stoichiometric and electrochemically cycled ﬁlms. These are a planar defect a half unit cell thick of rock salt structure parallel to the {0 0 1} plane. They can be thought of as arising from the removal of a plane of cobalt oxide for this material. Although in this case RP defects were expected to detrimentally affect the performance of the material by disrupting oxygen vacancy diffusion, an entire class of materials exists which have a periodic repetition of these

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FIGURE 7 Two defects or reconstructions observed infrequently on the eightfold cobalt disilicide/silicon interface viewed down the silicon [110] zone axis. In the top, the Co column appears to be swapped with a silicon column every fifth unit cell. In the bottom panel, an extra column of silicon dumbbells penetrates into the silicide. The images are 9 nm and 11 nm across, respectively.

RP defects and are known as RPn phases, where n = 1, 2, 3 is the number of unit cells of the perovskite phase between each RP defect. These RP phases have unique and desirable properties (such as giant magnetoresistance). RP defects and phases have been previously studied (e.g., Battle et al., 1998; Gallon et al., 2002; Sturm et al., 2001). The presentation here is meant to illustrate the capabilities of the aberration-corrected STEM to assign stoichiometry at the atom column scale as well as the ability to study the near-edge EELS shape at the same scale. Figure 8a and b show HAADF STEM images of electrochemically cycled LCCO material imaged in the [1–10] projection. The RP defects can be seen as dark lines in Figure 8a. Figure 8b shows a ﬁltered image of a region close to the middle of Figure 8a showing a single defect. The contrast can be assigned to atomic columns as follows: La with the highest Z and constituting 60% of the atoms in the column, clearly gives the brightest contrast in the HAADF image. The Ca- and O-atoms in the same column contribute little. Co has a lower Z and produces the medium-contrast values between the brightest columns; again, the O-atoms in the same column contribute little to the contrast. The atomic model with the planar defect structure is overlaid on Figure 8b.

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FIGURE 8 HAADF images of LCCO at lower and higher magnifications. The left image is 25 nm across, and the right image has been Fourier-filtered and is 4 nm across. The overlaid atomic positions are indicated as follows: The black dots are the mixed La and Ca columns, the white dots are oxygen, and the white crosses are Co. (Adapted from Bangert, Falke, and Weidenkaff, 2006.)

Figure 9a shows an unprocessed HAADF lattice image, including a domain, similar to those in Figure 8. The horizontal streaklike variations result from instabilities in electron emission due to an old ﬁeld emission tip. The electron beam was again incident along [1–10]. The curves show the background subtracted integrated EELS intensity in energy windows set around the Ca, O, La, and Co absorption edges. There is an obvious excess of Ca in the vicinity of the defect compared with the bulk stoichiometry as well as the expected deﬁcit of Co. The line traces appear noisy, but a signiﬁcant contribution that appears to be noise correlates with the relevant atomic column (i.e., in the lattice to the right of the defect the La columns have the highest La EELS intensity). No changes in the ﬁne structure of the Co edge were observed that may have indicated a change in oxidation state near the defect. The slight dark contrast and this observed excess of Ca both point to the Ca/La ratio being different in the defect. Here again the EELS signal is an inelastic signal formed from a coherent atomic-scale probe; both elastic and inelastic scattering need to be taken into account. The approach used here was the simplest empirical assumption possible but does reproduce the experimental proﬁles and led to the same stoichiometry as using the HAADF signal (described next). The assumption was that the EELS signal was proportional to the intensity of the probe electrons close to the relevant atomic species. This was modeled by taking the product of a function that decayed rapidly away from the atom and the probe intensity as modeled in a multislice calculation and summing this modeled intensity through the thickness of the sample. Although this approach is not rigorous, it is quick and simple to calculate and, with testing, may become an appropriate simpliﬁcation under certain experimental conditions. Figure 9b shows proﬁles modeled

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with different Ca excesses; the best ﬁt is obtained with La0.3 Ca0.7 CoO3 (i.e., roughly half as much La and twice as much Ca as in the bulk). In addition to extracting the absorption intensity of the edges, the ﬁne structure of the oxygen edge was also instructive. Figure 10 shows the oxygen K-edge from the bulk material and that obtained at the defect. Clearly the defect spectrum closely resembles the reference bulk rock salt CaO spectrum (Gu and Ceh, 1999).

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To corroborate the EELS ﬁnding, multislice simulations were performed using Kirkland’s code (Kirkland, 1998) to attempt to reproduce the lower HAADF signal at the defect due to the substitution of the less strongly scattering Ca atoms for the brighter La atoms. Figure 11 shows the best agreement between the simulated and experimental relative intensities was obtained for between 70% and 80% of A sites occupied by Ca in agreement with the EELS analysis. In this section, two examples of 2D structures have been solved in the sense of assigning atom positions unambiguously to sites. Both chemical and positional information is needed to constrain the number of possible structures that could ﬁt the data. The uniqueness problem is highlighted with particular reference to earlier TEM studies on the silicide/silicon interface in which multislice simulation and comparison led to proposed structures that seem not to be borne out and did not discriminate between competing models (Yu et al., 2001). I believe there is a bright future in the interaction between aberration-corrected analytical microscopy and theory. The ability to constrain the number of conﬁgurations that need to be investigated theoretically is coinciding with ever-improving computational methods and power.

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C. Nanostructures The investigations of ﬁve nanostructures chosen to illustrate hard materials analysis in aberration-corrected STEM all beneﬁt as much from the improved signal to noise provided by aberration correction as from the improved spatial resolution. Three of the ﬁve examples revolve around imaging gold atoms. Clearly the large scattering potential of such a large Z material makes the imaging easier, but these examples were not selected as test samples to pander to the instruments strengths. Rather, each example was investigated in response to a materials research characterization experiment brought to SuperSTEM.

1. Buried InGaAs Quantum Dots Buried In(Ga)As/GaAs quantum dots (QDs) are very promising candidates for optoelectronic device applications. The relevant properties of these dots depend on their shape and composition and on the variation from QD to QD. (for example, can the dependence of the optical absorption or emission on light frequency be ascribed to the indium concentration

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within an average QD, or is it due to the variability from QD to QD?) The materials for this study were grown on a GaAs (001) substrate by adding excess InAs (nominally 2.43 monolayers per layer) during the growth of 10 of the layers of GaAs. The Stranski-Krastonov process results in buried QDs ∼30 nm across by 6 nm high separated by ∼30 nm in a continuous InAs wetting layer 3 nm thick. Electron microscopy is the only technique able to characterize a single buried QD at the atomic scale. The approach used a combination of EELS and HAADF imaging with the following steps (Wang et al., 2006): 1. Use two HAADF images from two different directions (or assume a symmetry for the dot) to obtain a proposed 3D shape for the dot. 2. Hence postulate an idealized shape for the dot. 3. Measure the thickness of the specimen containing the dot, and thus calculate the number of arsenic or (group III) sites in each atom column of the image. 4. Calculate for each column the number of indium sites available using steps 2 and 3. 5. Use quantitative EELS to measure the number of indium atoms in each column. 6. Calculate the concentration of indium in each column within the dot by subtracting the contribution of the wetting layer from the measurement and dividing the corrected measured atomic density by the number of sites available. The probe was deliberately defocused during the collection of the EELS data for several reasons. The most important is the risk of aliasing unless the spatial sampling frequency signiﬁcantly oversamples the atomic columns. For example, in a perfectly uniform sample, if the atomic columns are spaced at 0.1 nm and spectra are collected at 0.09-nm intervals, over ﬁve columns the probe will move from on the columns to between the columns (assuming perfect calibration and no drift or jitter in the probe). This changes the intensity of the EELS spectrum across those ﬁve columns even for a uniform sample. In practice, unless column-by-column analysis is warranted and the necessary care in acquisition is taken, it is safer to underfocus the probe to cover a unit cell at the sample entrance. Figure 12 shows the HAADF image of a single QD and outlines the process of ﬁnding the number of indium sites available. Figure 13 shows the indium concentration along a line across the center of the QD. These results were consistent with theoretical predictions.

2. Au309 Clusters Although the clusters studied here have a very speciﬁc size, information obtained from the images is applicable to a wide range of nanoparticles

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FIGURE 12 Progressing from the HAADF image of an isolated InGaAs QD (a) by subtracting the GaAs background (b) to leave the In signal (d) to which a model structure (c) can be compared. The resulting 3D dot geometry is shown in (e). (From Wang et al., 2006.) (See color insert).

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typically found, for example, in heterogeneous catalysis. The monotonic dependence of the HAADF signal on cluster size is exploited to infer information concerning the third dimension from 2D images. Clusters of 309 gold atoms (Au309 ) produced by a cluster source capable of size selection were soft landed on a carbon grid (Li et al., 2008) (309 is a so-called magic number that corresponds to exactly ﬁlling the fourth shell around a single atom). The motivation for this study was to try to relate the experimentally determined 3D shape to calculations (Curley et al., 2007). In this size range

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many shapes with the same number of atoms differ in energy by very small margins and indeed experimentally many shapes were observed. Figure 14 shows a few clusters that demonstrate different projected symmetries, as well as a comparison of one of the clusters with an inodecahedral model. The experiment and model match well except for the outermost atom column. This mismatch may have several origins: It could be because the atoms in the outermost shell are highly mobile and therefore the atoms spent only a fraction of their time scattering the beam, or a few atoms may have been shaken off the cluster when it landed. The challenge with clusters (particularly metal) that consist of a few hundred atoms is that roughly half of the atoms in the cluster are surface atoms and so have a relatively low binding energy. The cluster thus is much less stable than the bulk materials especially when being observed. Conventional tomography, which requires many images, is not capable in practice of atomic resolution of such unstable structures. Indeed, in the study reported here, only the ﬁrst-pass image was used in each case. If the focus or astigmatism, for example, were not correctly adjusted, then that image was discarded. One aspect that was not fully explored was using

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FIGURE 15 Thiolated clusters of 13 gold atoms, many of which have aggregated. Nevertheless, the average signal from single atoms (circled) can be used to assign a number of gold atoms to each cluster. (From Abad et al., 2007.)

the lowest dose possible. This was because the tuning of the microscope requires a certain current in the probe to produce sufﬁcient signal to noise to measure the aberrations accurately. This provides one operating constraint. The other constraint is that to discriminate optimally against main-borne interference, each line of the scan is constrained to take one period of the main signal (i.e., the scan is synchronized to 50 Hz). Neither of these two constrains is fundamental and so there is considerable scope (and need) for implementing low-dose techniques in aberration-corrected STEM.

3. Au13 Clusters From a single gold atom, the next “magic number” cluster is 13 atoms. Such thiolate-stabilized clusters have been studied by Abad et al. (2007). Attempts to show that Au13 clusters were present by other techniques such as mass spectroscopy failed because the clusters fragmented even though several methods for producing ionized clusters were attempted. Studying such small clusters in the EM succeeded for two reasons—the clusters were stabilized by the ligands, and the eluted fraction was sufﬁciently free from hydrocarbon contaminants. In this study, the 13 atoms were not clearly imaged and in most cases aggregation of several clusters took place. However, by using the HAADF signal from single atoms, it was shown that the integrated signals from 150 clusters were multiples of Au13 and that the smallest clusters produced a signal corresponding to 13 ± 2 Au

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atoms. (The integrated signal was corrected for the mass of the ligands and for the carbon ﬁlm background.) Figure 13 shows a small subset of the clusters examined. It shows aggregated clusters, single atoms, and a 13-atom cluster.

4. Silicon Nanowires As stated at the beginning of this chapter, microscopy comes into its own in cases in which translational symmetry is interrupted. In order of decreasing dimensions, this means line defects and point defects. The projection of a 3D object into a 2D image means that it is intrinsically difﬁcult to

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image point defects. The exception to this is imaging of heavy atoms in a light matrix. This has been exploited in imaging heavy dopants in silicon in a landmark study (Voyles et al., 2002). A subsequent study by (van Benthem et al., 2006) localized a single Hf atom in three dimensions in silica. Locating atoms in the bulk as opposed to the surface can be important in distinguishing artefacts related to the thin sample from characteristics inherent to the bulk of the sample. Two recent studies of vapor liquid solid grown silicon nanowires (Allen et al., 2008; Oh et al., 2008) have attempted to distinguish whether gold catalyst atoms are included in the bulk or on the surface of the wires. Although gold atoms were certainly visible at all positions along the wire, both studies were hindered signiﬁcantly by the fact that many of the gold atoms, were mobile and therefore neither a through-focal series nor a tilt series is capable of localizing the atoms. In the latter study, gold atoms decorating a twin in the silicon nanowire were used as datum lines to localize two gold atoms as being present in the bulk but not at the twin boundary. However, the presence of the gold in a wire containing a twin boundary may still be caused by the inﬂuence of the twin boundary but remains a signiﬁcant ﬁnding. In the former study, the 300-kV beam appeared to cause even more movement of the gold atoms; however, if an atom was imaged, these authors were able to distinguish between interstitial and substitutional atoms. It remains to be shown that gold occurs within as-grown single crystal (untwinned) silicon nanowires.

5. Carbon Nanostructures—Toward Low Voltage Carbon-based nanostructures have been studied extensively by electron microscopy. Figure 17 demonstrates that when the dose is large enough to perform EELS, even at 100 kV there is signiﬁcant damage to multiwall nanotubes but no discernible damage occurs when the energy is reduced to 80 keV. This provides a justiﬁcation for the burgeoning trend in TEM to lower voltages now that the penalty in resolution has been mitigated by aberration correction.

D. Biological Systems Biological Systems are considered separately because of the particular care required to avoid artefacts caused by electron beam damage. It may not be intuitive that aberration-corrected STEM, which has a current density that is increased by roughly an order of magnitude or more, would offer any advantage for biological samples. Two factors militate against the damage. The ﬁrst and most signiﬁcant is that the BF imaging collector aperture can be enlarged to the point that instead of the STEM’s BF collection efﬁciency being a factor of 1000 worse than the equivalent conventional TEM image, it is within a factor of 10. Second, for samples such as biominerals (which are more robust than proteins), the high signal-to-noise means that

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FIGURE 17 A section of multiwalled carbon nanotube containing an iron catalyst particle shows the damage done by attempting to obtain a spectrum image at 100 kV (top left before and top right after). At 80 kV there is no discernable damage (bottom left before and bottom right after, the box shows the region of additional exposure).

we have found, in practice, that useful images can be obtained and useful spectroscopy can be performed. The following examples illustrate two cases in which useful data were obtained from biological samples.

1. Ferritin Figure 18 is an HAADF image of a section from a diseased human liver with a signiﬁcant overload of iron resulting in an abnormally high density

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FIGURE 18 STEM HAADF images of ferritin in an unstained fixed liver biopsy section. The higher magnification image on the right shows only one subunit of the ferritin mineral core with aligned atomic columns. (From Sader et al., 2008.)

FIGURE 19 A three-dimensional reconstruction of a typical ferritin core by single-particle analysis (From Sader et al., 2008.)

of ferritin cores (Harrison and Arosio, 1996; Pan et al., 2006). The higherresolution image in Figure 18b shows unequivocally that the ferritin core is polycrystalline, which was not known before this study. Figure 19 shows a 3D reconstruction resulting from treating a selected subset (with equal HAADF intensity) of images of the ferritin cores as a set of identical particles and performing a single-particle analysis. This shows that the center of the particle is not ﬁlled and strongly suggests that the ferrihydrite grows from eight loci in a pseudo-cubic fashion (Sader et al., 2008). This not surprising as it reﬂects the known symmetry of the protein shell. The ferritin cores were damaged noticeably after minutes of imaging, but they did not show any discernible damage in one or two image passes. EELS analysis (not shown) was also performed and by sharing the dose over many particles, a damage sequence was observed with increasing dose. First, the iron coordination changed from octahedral to tetrahedral, and then

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the oxidation state was reduced from Fe3+ to Fe2+ , whereupon obvious structural changes occurred. The EELS also showed that a typical ferritin core contains 1200–1600 iron atoms.

2. Cells Macrophage cells in the lungs remove the relatively small number of dust particles that reach the alveoli in the lungs by ingesting them; eventually the dead macrophage—with its cargo—is transported out of the lung by the bronchial cilia. An important question that must be answered before we can understand the toxicity of nanotubes is how macrophages react when they ingest or attempt to ingest the nanotubes. Cell viability studies have not yielded consistent results. In addition, it is important to know where the nanotubes end up in the cell to understand the mechanism of ingestion. Figure 20a (from Porter et al., 2007) shows a small region of a section of an alveolar macrophage that has ingested multiwall nanotubes. Also shown is a map of plasmon peak position. The nanotube has a measurably different response in this region of the energy loss spectrum and is visible as a faint contrast. The high-resolution BF STEM image of that region then reveals the single nanotube in the cell in (Figure 20c). Although the structure of the nanotube is preserved (this itself is useful information), the dose would have been too large to expect any of the cell proteins to retain their structure. This and the previous example both illustrate important niche applications for the new experimental space that aberration correction has opened.

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FIGURE 20 Plasmon shift map showing the location of a nanotube. Right, high-resolution BF image of the nanotube. (From Porter et al., 2007.)

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IV. DISCUSSION The examples presented are a selection of those studied at the SuperSTEM laboratory and are therefore an accidental subset of the areas of science to which aberration-corrected microscopy (and STEM in particular) has contributed. Two of the nanostructure applications use the symmetry of the structures to infer information about the third dimension from information in the 2D image, and one of the biological applications also concerns itself with a 3D reconstruction. These approaches are possible in a limited number of systems, but they highlight the need to work toward an essential goal of electron microscopy—obtaining 3D information at the atomic scale in a generally applicable way. Many current technologies are advancing by engineering materials at the atomic scale, whether in interfaces in semiconductors or grain boundary segregants in structural alloys. Aberration-corrected STEM and EELS provides a technique uniquely able to characterize the important few atomic columns (and even single atoms in favorable situations) in buried structures. Materials science research can be divided into three components: synthesis, characterization, and modeling. Aberration-corrected STEM is capable of interacting with both growers and theorists in a remarkably productive triumvirate. As for the future, this decade-old technology will contribute to a vast array of problems in the near future. Incremental progress will be made in aberration diagnosis algorithms and in the rigor of making the experiments quantitative. For beam-sensitive materials, opportunities remain in the optimal extraction of information from the unavoidable dose. Incentives for instrumentation development lie in the directions of in situ experiments, in lower accelerating voltages, and in detectors.

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CHAPTER

6 Aberration Correction With the SACTEM-Toulouse: From Imaging to Diffraction Florent Houdellier*, Martin Hÿtch*, Florian Hüe*, and Etienne Snoeck*

Contents

I Introduction A Description of the SACTEM-Toulouse B Performance II Aberration Correction and Strain Mapping A Precision B Spatial Resolution C Thin-Film Relaxation D Strained Silicon Nanostructures and Devices III Aberration-Corrected Convergent-Beam Electron Holography A Introduction to Diffracted Beam Holography Methods IV Aberration-Corrected Electron Diffraction A Improved Diffraction Patterns Through LACDIF B Applications of LACDIF V Pseudo-Lorentz Mode for Medium-Resolution Electron Holography VI Conclusions Acknowledgments References

225 228 229 232 232 236 237 238 240 240 245 246 248 252 256 256 256

I. INTRODUCTION The CNRS-CEMES laboratory in Toulouse where a spherical aberrationcorrected transmission electron microscope (SACTEM) has been installed since 2004 has more than 50 years of history in electron microscopy. The laboratory was founded in 1957 as the “Laboratoire d’Optique Electronique” (LOE-CNRS) with the original goal of building very high-voltage electron * CEMES-CNRS, 29 rue Jeanne Marvig, 31055 Toulouse, France Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01006-9. Copyright © 2008 Elsevier Inc. All rights reserved.

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microscopes (EMS). The ﬁrst microscope ever to operate at an accelerating voltage above 1 million volts (1.5 MV) (Dupouy, Perrier, and Durrieu, 1960) was housed in the huge metal sphere known as “La Boule” (Figure 1). The excellent energy stability meant that the microscope was in continuous operation until 1990 and dedicated primarily to electron energy loss spectroscopy (EELS) experiments. Thanks to the experience acquired in high-voltage technology, a second microscope operating at 3 MV was successfully built and came into operation in 1969 (Dupouy et al., 1969a, b). A rectangular building 24 m high was constructed for the purpose, with a separate foundation for the microscope to ensure its stability. Until 1995, this was the only microscope in the world that operated effectively at voltages ≥ 3 MV. In parallel to the development of high-voltage microscopes the laboratory was investigating many other TEM techniques. A holography setup was in place two years after the early work of Möllenstedt (Faget and Fert, 1956), a new Cs corrector design was proposed based on an axial conducting wire located along the optical axis of the TEM (Marais, 1970), electromagnetic prisms were developed for electron energy loss spectroscopy (EELS) experiments for the 1.5 and 3 MV (Sevely et al., 1973, 1982) including a Omega ﬁlter for the 1.5 MV microscope (Zanchi et al., 1977, 1982), the ﬁrst 300 kV TEM with superconductor lenses was constructed [Balladore and Murillo, 1977], and a new Si(Li) detector for counting electrons at 3 MV was designed and built (Pinna et al., 1978). In addition, many original new samples holders for in situ experiments were built such as He-cooled double-tilt stages, heating and cooling straining holders, environmental cells, and holders ﬁtted with magnetic coils for in situ Lorentz microscopy (Figure 2). In 1989, the LOE became the CEMES, a pluridisciplinary laboratory for fundamental research in materials science. Apart from the development of

FIGURE 1

La Boule (left) housing the 1.5-MV generator (right).

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FIGURE 2

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The 3-MV column.

electron microscopy instruments and methods, the CEMES extended its activities toward the synthesis and growth of different types of materials and their structural characterization. In recent years, research has centered on nanomaterials and nanosciences. The processes that occur at very small scale are studied, such as plasticity and deformation under mechanical stress, early stages of epitaxial growth, and structural studies of nano objects and nanodevices. Various new methods are being developed in TEM, near-ﬁeld microscopy (atomic force microscopy (AFM), ultra high vacuum (UHV), scanning tunnelling microscope (STM)), X-ray scattering, and optics to study nanomaterials at the nanometer scale. Concerning the developments in TEM, the CEMES is focusing particularly on quantitative high-resolution electron microscopy (Q-HRTEM), quantitative electron diffraction, off-axis electron holography, EELS and in situ experiments. To further developments in quantitative electron microscopy, a spherical aberration-corrected transmission electron microscope (SACTEM) based on a 200-kV Tecnai F-20 (FEI) ﬁtted with a lower spherical aberration corrector from CEOS was installed at the CEMES laboratory in 2004. It has been placed where the original 3-MV column stood on the foundations

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with proven stability against mechanical ground vibrations. The room is large with no special additional shielding given that the electromagnetic conditions were within the speciﬁcations of FEI; the building itself acted as a fairly efﬁcient Faraday cage. Two weeks after installation, tests indicated the optimum point resolution of 0.12 nm was reached.

A. Description of the SACTEM-Toulouse The following signiﬁcant optical elements of the SACTEM-Toulouse make the conﬁguration unique. The ﬁeld emission gun (FEG) is a standard Schottky emitter with an electronic rack (extraction supply, gun lens) separate from the electronics of the microscope column. The FEG reference potential is then set to the high tension (−200 kV), which induces highly stable electron emission. The electronics of the microscope and the FEG are then linked by an optocoupling process. The objective lens is a super-Twin design (FEI) with a microprobe and a nanoprobe mode. The large pole-piece gap with respect to a conventional HRTEM allows relatively high specimen tilts of ±45 degrees and the possibility of introducing specialized holders for tomography and in-situ experiments. A DigiScan (Gatan) scanning unit with a bright-ﬁeld–darkﬁeld (BF–DF) detector is also present in nanoprobe mode to allow full STEM operation. In order to suppress aberrations in the image, the corrector is introduced after the objective lens. The design is based on a combination of two hexapoles connected by a transfer lens doublet. A ﬁrst doublet is also introduced to transfer the coma-free plane of the objective lens in the center of the ﬁrst hexapole. This design, ﬁrst imagined by Harald Rose (1990), has been realized by CEOS (Haider, Braunshausen, and Schwan, 1995) and mounted on our Tecnai by FEI. The design of the Super-Twin objective lens was modiﬁed to perform this operation correctly. Indeed, the ﬁrst transfer lens doublet of the corrector must be inserted directly inside the lower objective lens pole piece. An adaptative lens is also included to adjust the corrector image plane with respect to the object plane of the diffraction lens. A rotatable electrostatic biprism (FEI) can be inserted in the selected aperture plane. This optical element, combined with the FEG, allowed us to realize electron interferometry experiments including off-axis electron holography (Völkl, Allard, and Joy, 1999). An off-axis 2k Ultrascan charge-coupled camera (CCD) camera (Gatan) is mounted directly behind the viewing screen to allow images to be captured in front of the imaging ﬁlter. Finally, the latest generation of Gatan imaging ﬁlter (GIF), the Tridiem ﬁlter, is positioned after the Ultrascan camera. The GIF can be set to perform either electron spectroscopy or energy-ﬁltered imaging (EFTEM).

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Field emission gun

Spherical aberration corrector (CEOS) Rotatable biprism

STEM unit

GIF tridiem

FIGURE 3 The SACTEM-Toulouse (with covering removed).

The Tridiem ﬁlter is particularly suitable for EFTEM thanks to the correction of ﬁrst- and second-order isochromaticity performed by the new multipole optics. The combination of the STEM unit and the GIF allows line and spectrum imaging. Due to the presence of the corrector, the pumping system has been modiﬁed from a standard Tecnai F20 to increase the vacuum quality inside the column. The line tube is pumped by two ion getter pumps, one in front of the corrector in the standard position of the object chamber, and the second behind the corrector and below the object chamber. The pumping process of the sample holder is then performed by a turbomolecular pump. The particularity of the SACTEM-Toulouse is not only due to the Cs corrector, but to the combination of these different elements, which provides the opportunity to study and develop new techniques (presented later in this chapter).

B. Performance The microscope resolution of 0.12 nm was ﬁrst investigated using Young’s fringe method and conﬁrmed by the observation of the so-called dumbbells

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in [110] zone-axis Si and GaAs and in fringe spacings in Al for high-order zone axes. In Figure 4 (raw data), the two silicon atomic columns can be easily separated. Furthermore (and in our opinion, the major improvement), the contrast is uniform over the entire image with a very high signal to noise ratio (SNR) compared to HRTEM data recorded with conventional instruments. This fundamental property will have major repercussions for quantitative microscopy studies, especially using the geometrical phase analysis (Hÿtch, Snoeck, and Kilaas, 1998). As discussed in the previous section, the objective lens is based on the Super-Twin design, which allows working with high tilt without loss of resolution. The third-order spherical aberration coefﬁcient (C3 ) can easily be tuned with the corrector to useful values of the order of microns with a precision of about a micron. Aberrations up to fourth order are corrected during the computer-controlled alignment process. The remaining aberrations are ﬁfth-order spherical aberration (C5 = 8 mm) and chromatic aberration (Cc = 1.7 mm). The latter is particularly high compared to a conventional Tecnai F20 microscope because of the corrector. Indeed, the four transfer lenses and the adaptative lens add their own chromatic aberration, which increases the overall chromatic aberration coefﬁcient of the microscope. It is therefore essential to combine the corrector with a ﬁeld emission source with a minimum energy spread.

2 nm

FIGURE 4 Aberration-corrected HRTEM image of a silicon sample observed along the [110] direction.

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Different conditions can then be used for HRTEM observations, in particular concentration on the so-called amplitude contrast conditions with Cs zero and the new Scherzer settings for C3 (Haider, Braunshausen, and Schwan, 1995). These conditions can be understood using a twodimensional (2D) drawing of the electron phase introduced by the global electron optical system into the electron wave function (Uhlemann and Haider, 1998) (Figure 5). Thus, the phase presented in the left part of Figure 5 corresponds to the amplitude contrast condition where C3 and C1 are set to zero. In this mode, the phase χ is equal to zero in a range of 18 mrad, leading to an amplitude contrast transfer function cos(χ) of 1. Moreover, these conditions are particularly suitable for imaging nanoobjects, because the range of angles for which the phase is free from microscope aberrations is large compared to common Bragg angles, and the gradient is also small, leading to a minimization of image delocalization [Haider et al., 1998]. As in conventional HRTEM, Scherzer conditions can be set to maximize the phase-contrast transfer function sin(χ) with χ = π/2 in the widest possible range of spatial frequencies. These conditions can be calculated using the following Scherzer expression (Scherzer, 1949; Haider, Braunshausen, and Schwan, 1995):

3 C3 ≈ −3.2 3 λC52 and C1 = 2 λ2 C5

(1)

For our microscope, the calculated values are C3 = −17 mm and C1 = 7 nm. The phase is then equal to χ = π/2 up to 25 mrad (see Figure 5). The HREM image is now understood using the phase contrast theory with some improvement concerning the resolution compared to the amplitude C15 0, C3 5 0, C5 5 8 mm

C1 5 7 nm, C3 5 217 mm, C5 = 8 mm

d 18 mra

rad

25 m

0

2

FIGURE 5 Two-dimensional drawing of the electron phase for the amplitude-contrast condition (left) and the new Scherzer setup (right). (See color insert).

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is higher contrast conditions. On the other hand, the phase gradient ∇χ using the Scherzer expression; thus conditions slightly increase delocalization. The choice then depends on the material studied in the microscope and the image analysis technique that is used.

II. ABERRATION CORRECTION AND STRAIN MAPPING HRTEM is a powerful technique for measuring strain at the nanoscale. Strains are determined directly by comparing the high-resolution image of the crystalline lattice with respect to a reference region of unstrained crystal using image-processing techniques such as peak-ﬁnding (Bierwolf et al., 1993) or geometric phase analysis (GPA) (Hÿtch, Snoeck, and Kilaas, 1998). The technique has been applied to several systems ranging from strained multilayers to quantum dots and semiconductor nanowires. Indeed, strain can be measured with remarkable precision using conventional microscopes, so it is interesting to note that even here, aberration correction provides considerable advantages. Aberration correction addresses the main limitations in HRTEM strain mapping: precision, spatial resolution, and thin-ﬁlm relaxation.

A. Precision Strain mapping depends on the measurement of displacements, either by determining the positions of intensity maxima in the image (for peakﬁnding) or of the local phase of lattice fringes (for GPA). The accuracy with which this can be done depends on the signal-to-noise (SNR) of the HRTEM image. The SNR can be improved by a higher image contrast or a lower noise. Aberration correction can help with both of these factors.

1. Contrast From an experimental point of view, the SACTEM-Toulouse produces images of very high contrast, certainly much higher than for our CM30 (300 kV, LaB6 ﬁlament, Cs = 1.2 mm). Figure 6 shows a comparison of images taken with the two microscopes for the same specimen of [110] silicon and similar specimen area. Both microscopes were operated to obtain the maximum contrast. Various factors can explain this result, and not all directly related to aberration correction. Let us ﬁrst consider the contrast of an individual set of lattice fringes. Theoretically, the contrast of lattice fringes depends on the complex contrast transfer function of the lens, both for linear and nonlinear interference (see Spence, 2003). The maximum contrast obtainable for each of these terms individually depends ultimately on the envelope function and is

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CM30

Resolution 0.18 nm

SACTEM

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1.36 Å

Resolution 0.12 nm

FIGURE 6 Comparison of a conventional HRTEM (CM30 300-kV TEM, Cs = 1.2 mm, LaB6 gun) with an aberration-corrected TEM (SACTEM-Toulouse 200-kV FEG-TEM, Cs = 0). Images from the same specimen of [110] Si and similar thin region of specimen, approximately 10 nm thick.

independent of whether the specimen is thick or thin. For example, if only linear interference is considered, the defocus that maximizes the contrast will depend on the phase difference between the diffracted and transmitted beams. However, the maximum value will depend only on the linear transfer envelope function for the particular frequency of the lattice fringes. Aberration correction improves the envelope function by almost eliminating the damping function due to spatial incoherence. The improvement would be large for a thermionic source (LaB6 gun) but less so compared with a conventional microscope equipped with a FEG. This cannot explain, however, our observed contrast for {111} lattice fringes in silicon. The fringe spacing is only 0.313 nm, which is too large to expect much change in the envelope function, even compared with an LaB6 microscope. Nonlinear interference terms also contribute to the contrast of a particular set of lattice fringes. The contrast of each nonlinear interference term will be maximized for different values of defocus. Aberration correction provides a smoothly varying transfer function that probably optimizes the nonlinear contrast for a wider range of defocus values than for a conventional microscope. This is the case particularly for mono-atomic crystals of high symmetry such as silicon. Many beams have the same phase (assuming zone-axis orientations) and therefore similar requirements for optimized contrast. An additional, and maybe more important feature, is that the nonlinear contrast will probably be positive. Symmetric beams will have no phase difference and hence can be imaged in positive contrast. This effect is independent of defocus. By choosing conditions for positive linear contrast, the linear and nonlinear terms will add constructively. Aberration correction uniquely allows the choice of the sign of the spherical aberration coefﬁcient, thus allowing positive contrast for underfocused images of weak phase objects. All these reasons may explain the experimentally observed high fringe contrast in silicon, without elaborating on details about inelastic scattering contrast mechanisms.

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For the overall image contrast, which is a sum of the contrast of the individual lattice fringes, only general arguments can be used. Aberration correction, through the smoother transfer function, allows many spatial frequencies to have near-optimum contrast for the same defocus setting. If contrast is of the same sign, all interference will be constructive. Our experience suggests performing microscopy under amplitude contrast conditions. Linear and nonlinear interference terms will add. Aberration correction also allows high spatial frequencies through the improved envelope function. The overall contrast will therefore, in general, be higher.

2. Uniformity of contrast For effective strain mapping, the contrast of the HRTEM image needs to be uniform across the ﬁeld of view, without inversions of contrast (Hÿtch and Plamann, 2001). This might be more difﬁcult to achieve with aberration correction than for conventional imaging. It has been shown that aberration-corrected images can be very sensitive to defocus (Chen, Zandbergen, and Van Dyck, 2004), and since specimens are never perfectly ﬂat, the effective defocus varies across the ﬁeld of view. However, our work has not been adversely affected by this problem. Strain mapping ideally requires the preparation of specimens with relatively uniform thickness. This becomes a necessity for noncentrosymmetric crystals or for imaging away from a zone axis (Hÿtch and Plamann, 2001). Specimens so prepared therefore should have a relatively ﬂat exit surface, and provided signiﬁcant tilting was not used to align the crystal, the effective defocus should be constant. An example was previously shown of [110] silicon (see Figure 3) in which the contrast is extremely uniform across the ﬁeld of view. This is particularly remarkable given the high resolution of the image, including signiﬁcant contributions from the (004) reﬂections to produce the dumbbell contrast. Nevertheless, uniform thickness is not always a precondition. Figure 7 shows a study of gold decahedral particles. The geometry of the crystal means that the specimen thickness varies considerably from edge to center, and the exit surface is signiﬁcantly inclined. However, we were able to acquire an image with uniform contrast, except at the extreme edge (Figure 7c). Gold is monoatomic, and the structure is simple and centrosymetric. All these factors are beneﬁcial, although the crystal had to be very carefully aligned on the exact zone axis. Slight misalignment produced contrast inversions in one or all of the segments. One of the practical advantages of aberration correction is that the images are of such high quality, and optical alignments are largely computer assisted, so the operator has more time and facility to address the problem of crystal alignment (Lentzen

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(a)

235

(b)

(c)

5 nm

FIGURE 7 Aberration-corrected HRTEM image of decahedral Au nanoparticle. (a) The fivefold rotational symmetry is marked by twin boundaries that intersect at the center of the particle (enlarged). (b) Fourier transform. (c) Enlarged views of the edge of the particle.

et al., 2002). Despite the difﬁculties, changes in lattice parameters can be measured in relatively small nanoparticles (Carbone et al., 2007). The quality of the image in Figure 7a allows accurate strain mapping of the internal rigid-body rotations (Figure 8a) and shear strains (Figure 8b) required to accommodate the geometric constraints imposed by the decahedral geometry. The measurements conﬁrm, for the ﬁrst time, the existence of a disclination (Johnson et al., 2008). Furthermore, comparison of the results to ﬁnite-element analyses revealed that shear strains, which are not predicted by the commonly accepted strain models for decahedral particles, result from elastic anisotropy.

3. Noise Except for the noise inherent in counting statistics and detector characteristics, the majority of noise in experimental HRTEM images comes from the amorphous layers coating specimen surfaces. These can be native oxides or damage layers from thinning processes. In either case, the contrast from amorphous or badly crystallized material perturbs the contrast from the underlying crystal. These layers are thin generally and act like weak phase objects. Aberration correction could therefore reduce their associated contrast.

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(a)

(b)

10 nm

10 nm

FIGURE 8 Strain analysis of decahedral Au nanoparticle. (a) Internal rigid-body rotation of the crystallographic lattice (color scale 0 to 7 degrees). (b) Shear-strain distributions (color scale −5% to +5%). The lattice rotation combined with the shear strains, which result from elastic anisotropy, accommodate the unique geometry of the decahedral particle. Image analysis with GPA phase 2.0 (HREM Research Inc.), a plug-in for DigitalMicrograph (Gatan). (See color insert).

Figure 9 shows an image of Pt nanowire resting on an amorphous support ﬁlm. For a spherical aberration constant close to zero and Gaussian defocus, the contrast of the amorphous carbon is almost zero. This is indeed one of the striking aspects of using an aberration-corrected TEM. Strain measurements in such circumstances would be expected to have lower noise. However, surface layers are necessarily located at a different height to the crystal and therefore at different defocus conditions. The contrast of amorphous material returns progressively with defocus. Systematic experiments should be conducted to verify whether the noise is actually reduced. From our experience, nanocrystals suspended in amorphous material are more visible with aberration correction, but this is probably due more to the increase of crystalline contrast rather than a reduction in the amorphous contrast.

B. Spatial Resolution This section discusses the spatial resolution of the strain information as opposed to the spatial resolution of the microscope. In GPA, the spatial resolution is determined by the size of the mask used in Fourier space to isolate the particular frequency of interest. Reducing the mask size helps noise reduction but at the expense of spatial resolution. Precision and spatial resolution are therefore highly dependent parameters. In strain mapping, the precision should always be stated for a particular spatial resolution. Improving SNR therefore means that for the same precision,

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FIGURE 9 Aberration-corrected HRTEM image of Pt nanowire. Notice the lack of contrast of the carbon support film. (Courtesy of Marie-José Casanove.)

measurements can be performed at higher spatial resolution. This is the ﬁrst beneﬁt of aberration correction but not the only one. Image delocalization depends on the gradient of the transfer function (Lichte 1991; Coene and Jansen, 1992). For GPA, this simpliﬁes to the gradient at the spatial frequency of interest (Hÿtch and Plamann, 2001). Aberration correction provides a smoother transfer function for a wide range of defocus values and therefore reduces image artifacts. Simulations show that these effects are particularly important for spatial frequencies comparable to one or two lattice fringe spacings (Hÿtch and Plamann, 2001). Therefore, only strain measurements at relatively high spatial resolutions beneﬁt directly from aberration correction. In comparison, we often perform our measurements at spatial resolutions of a few nanometers to increase precision.

C. Thin-Film Relaxation Because microscopy specimens must be thinned to electron transparency, strains can relax at the two additional free surfaces with respect to the bulk sample (Treacy, Gibson, and Howie, 1985). This thin-ﬁlm effect is particularly important for HRTEM work in which specimens are even thinner than for conventional microscopy. Bending of lattice planes can also affect image contrast (Tillmann, Lentzen, and Rosenfeld, 2000), although the effects are not as signiﬁcant as for Convergent Beam Electron Diffraction (CBED) (Clement et al., 2004; Houdellier et al., 2006). Measurements need to be compared with elastic modelling of the thin ﬁlm and necessary corrections made (Cherkashin et al., 2006). This procedure allows reliable

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Si

CM30

Si0.6Ge0.4

Si

SACTEM

Si0.6Ge0.4

FIGURE 10 Comparison of a conventional HRTEM (CM30 300 kV TEM, Cs = 1.2 mm, LaB6 gun) with an aberration-corrected TEM (SACTEM-Toulouse 200-kV FEG-TEM, Cs = 0). Images are from the same specimen of [110] Si and a similarly thick region of the specimen, ∼30 nm thick.

determination of bulk strains in simple expitaxial systems, but the ability to observe thicker crystals is preferred. A comparison of images obtained with a conventional HRTEM and the SACTEM-Toulouse of the same specimen is shown in Figure 10. In both cases, the specimen thickness is ∼30 nm. Strain can be measured with reasonable accuracy from the aberration-corrected image but not the other. The higher contrast of aberration-corrected images therefore allows study of thicker crystals. The precision is not as good as for a thinner crystal, but the result is closer to the bulk crystal and the accuracy therefore higher without modeling. A compromise is necessary among spatial resolution, accuracy, and modeling effort.

D. Strained Silicon Nanostructures and Devices Strained silicon has gained great technological importance in electron devices because of the associated increase in carrier mobilities. The advantages of aberration-corrected HRTEM for strain mapping have been exploited in studies of strained SiGe layers (Cherkashin et al., 2006; Hÿtch and Houdellier, 2007). Indeed, aberration correction has been essential for tackling the more difﬁcult problem of mapping strains in devices. To study transistors and devices, specimens need to be prepared by focused ion beam (FIB) to select the region of interest. Unfortunately, FIB specimens are thicker than those obtained by conventional techniques and are covered with a signiﬁcant layer of damaged material. Both factors make HRTEM work difﬁcult. On the positive side, however, specimens have uniform thickness, which is ideal for strain mapping. Even for modern devices, the active areas over which strains must be measured are large and can be hundreds of nanometres square. Images therefore need to be acquired at relatively low magniﬁcation, which puts pressure on the CCD camera characteristics even for a 2k camera. The modulation contrast transfer function (MTF) decreases rapidly when only a limited number of pixels per lattice fringe period are available.

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Nevertheless, we have recently shown that HRTEM strain mapping is possible in devices using aberration correction (Hüe et al., 2008). An example of the study of a strained channel MOSFET is shown in Figure 11. The precision of the measurements is 0.3% at a spatial resolution of 4 nm. Higher accuracy (0.2% at 2–3 nm spatial resolution) has been demonstrated with a conventional microscope for the study of an isolated dislocation in silicon (Hÿtch, Putaux, and Pénisson, 2003; Hÿtch, Putaux, and Thibault, 2006). However, the specimen was very thin, with very little surface amorphous layers and the ﬁeld of view an order of magnitude smaller. Under similar circumstances aberration correction can attain 0.1% accuracy at 2-nm spatial resolution (Hÿtch and Houdellier, 2007). (a)

(b) Gate

Source

Channel

Drain

p-type MOSFET z

20 nm

x (c) SiGe

(d) SiGe

FEM simulation

22%

2%

FIGURE 11 Strain mapping of a strained silicon transistor. (a) Schematic diagram of p-type MOSFET with Si80 Ge20 source and drain-inducing compressive strain in Si channel. (b) Low-magnification aberration-corrected image of channel region filling 2 k CCD camera field of view, FIB specimen preparation, [110] orientation, enlargement showing lattice contrast. (c) Finite element modeling (FEM) of strain distribution εxx . (d) Measured strain distribution of εxx , precision of 0.3% at 4 nm spatial resolution. Image analysis with GPA phase 2.0 (HREM Research Inc.), a plug-in for DigitalMicrograph (Gatan). (See color insert).

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III. ABERRATION-CORRECTED CONVERGENT-BEAM ELECTRON HOLOGRAPHY Energy-ﬁltered convergent-beam electron holography (CHEF) is a relatively new technique that capitalizes on the performance and optical conﬁguration of the SACTEM-Toulouse (Houdellier and Hÿtch, 2008). The technique is a variant of diffracted-beam holography (Herring et al., 1995) and combines electron diffraction with off-axis electron holography. We show how this new holographic method beneﬁts from aberration correction and how residual aberrations can be investigated.

A. Introduction to Diffracted Beam Holography Methods The formation of interference fringes in the diffraction plane were ﬁrst investigated by coherent CBED in a dedicated scanning transmission electron microscope (STEM) using convergent angles large enough that diffracted disks overlap and interfere (for a detailed description, see Spence and Cowley, 1978). In these experiments, a highly coherent source is needed, such as that provided by a cold FEG. Electron diffraction interferometry also can be realized by interfering convergent-beam diffraction disks with the aid of an electron biprism (Herring et al., 1995). Optimum conditions require a FEG (although not necessarily a cold emitter), rotatable biprism, and imaging ﬁlter, which are rarely found on a single instrument. The technique, which we called CHEF, has a number of interesting uses—the measurement of the degree of coherence of electron beams (Verbeeck, 2006), the determination of aberrations, and the study of rocking-curve phases (Houdellier and Hÿtch, 2008). CHEF patterns can be obtained using large-angle convergent-beam electron diffraction (LACBED) geometry (Morniroli, 2002) as proposed originally, or the CBED geometry (Houdellier and Hÿtch, 2008). These two conﬁgurations are shown in Figure 12. In LACBED geometry, the interference fringes are superimposed on the shadow image of the specimen, whereas the CBED conﬁguration gives access to a fringe pattern characteristic of a small area on the sample. The theoretical treatment is fully detailed elsewhere (Houdellier and Hÿtch, 2008); here we recall only some of the fundamental equations. The CHEF pattern is obtained from the interference of a diffracted disk and the transmitted disk. Let the two wave functions be described in terms of their complex amplitude, respectively ψg (k) and ψ0 (k), as a function of position k within each disk

ψg (k) = ag (k) exp iφg (k) and ψ0 (k) = a0 (k) exp iφ0 (k) ,

(2)

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

Objective lens Intermediate lens

Condenser lens

241

Projector

Image plane

Sample

Source Biprism

Object plane

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Focal plane

Screen : conjugate with the objective focal plane

4 mrad

Projector Condenser lens Sample

Intermediate lens Image plane

Biprism

Condenser aperture

Object plane

Focal plane Screen : conjugate with the objective focal plane

FIGURE 12 CHEF configuration in LACBED mode (top) and CBED mode (bottom). The red and green rays correspond, respectively, to the transmitted and diffracted beams before applying voltage; black lines correspond to the same beams with the biprism activated. (From Houdellier and Hÿtch, 2008.) (See color insert).

where a and φ are the corresponding amplitudes and phases. The phase depends on the diffraction conditions and the position of the probe with respect to the specimen (Spence and Cowley, 1978). At the imaging plane, the resulting wave function is then the coherent sum of the two. The interference pattern observed, I chef (k), is given by the modulus squared of the above:

I chef (k) = a2g (k) + a20 (k) + 2ag (k)a0 (k) cos φg (k) − φ0 (k)

(3)

averaged over time. The local amplitude and phase, denoted by Achef (k) and chef (k), respectively, are therefore given by Achef (k) = ag (k)a0 (k) and chef (k) = φg (k) − φ0 (k),

(4)

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which can be measured using geometric phase analysis (Hÿtch, Snoeck, and Kilaas, 1998). (Here, local refers to the position within the disk, not on the specimen, although in LACBED conﬁguration the two are related.).

1. Influence of Optical Aberrations This subsection considers the case of an ideally thin specimen, for which the amplitude and phase across the diffracted and transmitted disks at the exit surface of the crystal are uniform. In order to understand the inﬂuence of the corrector, we begin working the corrector deactivated (i.e., with hexapoles and associated dipoles powered down).

Hexapoles deactivated. Figure 13 presents an experimental phase from the CHEF pattern obtained by interfering the (000) and (220) beams in silicon. The phase of the interference pattern is retrieved using a basic GPA procedure (Hÿtch, Snoeck, and Kilaas, 1998). The phase measured from the raw data is disjointed and distorted by geometric distortions from the CCD camera, imaging ﬁlter, and projector lenses. These distortions are then removed using the procedure described by Hüe et al. (2005). After correction, the phase variation is smooth with extremely high SNR (Figure 13b), which attests to the stability of the microscope. The reference area for the phase analysis was taken at the center of the disk. The experimental phase, fringe deformation (variation of the fringe spacing relative to the center of the disk), and amplitude are shown in the middle column of Figure 14. Simulations are realized using aberration theory, which was applied for CHEF simulation in the reference paper (Houdellier and Hÿtch, 2008). The (a)

4 mrad

(b)

4 mrad

FIGURE 13 Retrieval of the CHEF phase using geometric phase analysis. (a) Experimental (220)/(000) CHEF pattern recorded without aberration correction; Fourier transform shown in the inset. (b) Calculated phase after distortion correction. (See color insert).

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

FCHEF

4 mrad

FCHEF

(a)

´

F

CHEF

4 mrad 2 ´

(b)

4 mrad ´

(e) 210%

(c)

(d) ACHEF

243

(f)

10%

ACHEF

ACHEF

(g)

(h) 0

(i)

1

FIGURE 14 Comparison between experimental and calculated (220)/(000) CHEF results. (b), (e), and (h) show, respectively, the experimental phase, fringe deformation, and amplitude. (a), (d), and (g) show calculated values for exact alignment and no defocus spread. (c), (f), and (i) present the best-fitting simulations after translation of the coma-free axis, adjustment of Cs , and addition of defocus spread. (From Houdellier and Hÿtch, 2008.) (See color insert).

right-hand column of Figure 14 shows the results for the best ﬁt of the experimental phase and amplitude. We have shown that the major inﬂuence in this shape comes from the spherical aberration of the objective lens, which has been found to be Cs = 1.34 mm—this compares favorably with the 1.28 mm measured for our system by CEOS. The spherical aberration coefﬁcient is slightly higher than that of a standard Super-Twin lens (Cs = 1.2 mm) due to the modiﬁcations required to include the corrector (Houdellier and Hÿtch, 2008).

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The amplitude can be ﬁtted using theory usually performed in HRTEM (Spence, 2003) or lattice imaging in STEM (Spence and Cowley, 1978). Indeed, the local amplitude and phase are identical to linear lattice-fringe contrast between two beams. The quality of the resulting ﬁt suggests that the imaging theory is correct and that the thin-specimen approximation is valid.

Hexapoles activated. The Cs corrector was then activated and the different aberration coefﬁcients involved in the transfer function were determined in imaging mode and corrected. Figure 15 shows the difference between the (000)/(220) CHEF phase observed when the hexapoles are activated and deactivated (as in the previous section). With aberration correction, the CHEF phase becomes much more uniform. Nevertheless, a parabolic variation is present, which probably results from residual coma still present in this conﬁguration. Indeed, CHEF experiments are nominally realized in the nanoprobe mode of the objective lens, whereas the corrector is used to correct the objective lens set in the microprobe mode. This difference might explain the residual coma present in the nanoprobe mode where the minilens is now optically inactive, which is not the case in microprobe. The close relation between CHEF phase and optical aberrations means that such experiments might be suitable for determining aberrations (Houdellier and Hÿtch, 2008) in a manner similar to coherent CBED ronchigrams from periodic objects (Lin and Cowley, 1986). We are currently developing software to retrieve the aberrations of the microscope for CHEF patterns with high precision (Hÿtch et al., 2007).

Hexapoles deactivated

Hexapoles activated

2

4 mrad

4 mrad

FIGURE 15 Experimental (220)/(000) CHEF phase with (right) and without (left) Cs correction recorded under similar conditions. (From Houdellier and Hÿtch, 2008.) (See color insert).

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

10 rad

0

245

1

0

FIGURE 16 Determination of diffracted phase (left) and amplitude (right): experimental (220)/(000) CHEF pattern for a t = 150 nm thick sample. (See color insert).

2. Rocking-Curve Phase As the thickness of the specimen increases, variations of the transmitted and diffracted intensities across the disks are observed: the well-known rocking curves (Morniroli, 2002). The thin-specimen approximation used previously is no longer valid. With aberration correction, the reduced phase from aberrations allows easy separation of the rocking-curve phase due to the specimen diffraction. This experiment has been performed using the CHEF (000)/(220) of a 150 nm thick silicon sample (Houdellier and Hÿtch, 2008). The main result is presented in Figure 16. The phase change from the residual aberrations has been removed using a reference CHEF pattern taken in a very thin part of the sample where the effect of the rocking-curve phase can be neglected. The corresponding amplitudes are divided, whereas the phases are subtracted. This new technique makes it possible to quantitatively ﬁt the rockingcurve CHEF amplitude with structure factor in a manner similar to quantitative CBED. More important, the rocking-curve CHEF phase also can be determined and ﬁtted, which is not done in quantitative electron crystallography. Furthermore, we anticipate the relationship between phases of structure factors to be more direct. Phase variations also are more sensitive to thickness, and therefore CBED analysis could be performed for thinner crystal. However, aberration correction is vital; otherwise, the aberrations from the microscope become too large with respect to the rocking-curve phase.

IV. ABERRATION-CORRECTED ELECTRON DIFFRACTION This section describes a novel experimental method to obtain spot patterns that simultaneously display the integrated diffracted intensities for a large number of reﬂections based on the LACBED geometry and the use of a Cs -corrector (Morniroli et al., 2008).

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A. Improved Diffraction Patterns Through LACDIF In order to obtain a spot pattern (or a disk pattern made of disks with a very small diameter), the incident beam should be parallel (selected-area electron diffraction, SAED) or nearly parallel (microdiffraction). The conventional spot pattern is observed in the back-focal plane of the objective lens. If the specimen is a perfectly ﬂat single crystal with parallel entry and exit surfaces, and if the deviation from the exact Bragg condition for the considered (hkl) lattice planes is “s,” then the hkl diffracted intensity is connected with a well-deﬁned position on the rocking curve (Morniroli, 2002). A convergent incident beam focused on the specimen can be used to obtain an integrated intensity for the hkl spot. This is the conﬁguration used to obtain CBED and Kossel–Mollensted patterns (Morniroli, 2002). The incident beam can be considered as a set of rays impinging on the specimen covering the angles represented by the incident cone. Therefore, at the bottom face of the specimen, the transmitted and diffracted intensities are displayed as a function of the orientation of these incident rays with respect to the (hkl) lattice plane—that is, as a function of the deviation parameter s. The problem is that the resulting pattern, located in the back-focal plane, is not composed of spots but of two disks—the transmitted and the (hkl) diffracted disks. Obtaining a spot pattern with a convergent incident beam requires that the sample be mechanically defocused by an amount of h with respect to the object plane (Figure 17). This is, in fact, the illumination conﬁguration used to produce LACBED patterns (Morniroli, 2002). There is no change in the back focal plane compared with the focused convergent beam, but a spot pattern displaying the integrated intensity is now observed in the object plane and in the conjugate image plane. This explains why this type of diffraction pattern is recorded in image mode and not in diffraction mode, as is usually the case. With a convergent incident beam, each incident ray has a different orientation inside the incident beam, and the resulting pattern corresponds to a combination of all diffraction patterns. Each diffraction pattern is characterized by the intersection of the Laue circle with reciprocal space. All the reciprocal lattice nodes located inside the area deﬁned by the extreme Laue circles produce diffracted beams. A large convergence, α, is very useful for obtaining a large number of visible reﬂections because more reciprocal lattice nodes are included in the formation of the pattern. With our microscope, a beam convergence of up to 3–5 degrees is available. The above considerations show that, besides the electron precession technique (Vincent and Midgley, 1994), a reliable and realistic method of obtaining a spot pattern with a large number of reﬂections and displaying the integrated intensity consists of using a large incident and

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Defocused incident beam Specimen

Defocus distance Object plane

Objective lens

Back focal plane

Image plane Cs CORRECTOR

FIGURE 17 Description of the LACDIF configuration. (From Morniroli et al., 2008.)

defocused incident beam (i.e., the experimental illumination conditions used to produce LACBED patterns).

1. Advantages of Aberration Correction Figure 18a shows a LACDIF diffraction pattern (observed with the corrector deactivated) of a silicon specimen oriented along the [001] direction and illuminated by a large defocused incident beam with a convergence semi-angle of ∼3 degrees. Because of spherical aberrations connected with the large beam convergence each spot is actually surrounded by a large disk of confusion that greatly decreases the pattern quality. Using the current Cs corrector on the SACTEM-Toulouse, the disk of confusion is suppressed while maintaining a large convergence (Morniroli

Florent Houdellier et al.

248

(a)

(b)

FIGURE 18 Experimental patterns of a [001] LACDIF pattern observed in silicon without (a) and with (b) the Cs corrector.

et al., 2008). Figure 18b shows an example of a pattern obtained in this manner. The spots are now sharply deﬁned and the diffracted intensities uniform since each spot represents the integrated diffracted intensity over the large angular domain located inside the incident cone. The pattern quality is therefore much improved with Cs correction.

B. Applications of LACDIF 1. Space-Group Determination The possible space groups of a crystal can be identiﬁed from a few zoneaxis microdiffraction patterns (Morniroli and Steeds, 1992). The method is based on the observation of the shift and the periodicity differences between the reﬂections located in the ﬁrst-order Laue zone (FOLZ) and those located in the zero-order Laue zone (ZOLZ). The method requires the observation of the FOLZ and considers only the positions of the reﬂections on the pattern. Nevertheless, with high symmetry zone axes, or with a crystal with small lattice parameters, observations of the FOLZ are not easy and sometimes impossible. A solution consists in tilting the specimen (or the incident beam) along the mirror planes present in the ZOLZ until a more or less large part of the FOLZ appears. The entire FOLZ can be rebuilt from these slightly misoriented patterns. On most conventional microdiffraction patterns, only a small number of reﬂections are visible in the FOLZ, except for very thin crystals. The identiﬁcation of the shifts and periodicity differences therefore can be very difﬁcult. As a consequence, any experimental method that produces a larger number of reﬂections should be very useful. This is the case with the present method, as shown on the Si patterns in Figure 19. The slightly misoriented patterns on Figure 19 were obtained

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

a//(001)

or

cl or cF

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d//(001)

la.. or Fd..

m1 m2

m3

m4

FIGURE 19 Example of identification of the possible space groups of a crystal from LACDIF patterns. (a) [001] zone-axis pattern. Only the ZOLZ is visible. (b, c) Patterns obtained after a slight misorientation from the [001] zone axis along the m2 and m3 mirrors. Many FOLZ reflections are visible that allow certain identification of the FOLZ/ZOLZ shifts and periodicity differences. (d) Theoretical [001] pattern from Morniroli and Steeds (1992) and corresponding to the partial extinction symbols Ia.. or Fd.. are in perfect agreement with the experimental patterns.

from the silicon [001] zone-axis pattern by tilting the incident beam along the m2 and m3 mirrors. Many FOLZ reﬂections are then visible on these patterns. It is clear that some FOLZ reﬂections are located on the m2 mirror (reﬂections circled on Figure 19) and absent on the m3 mirror. The FOLZ/ZOLZ periodicity difference is easily identiﬁed by drawing a 2D “unit cell” that describes the ZOLZ and the FOLZ lattices. These patterns are in perfect agreement with the simulation on Figure 19 extracted from Morniroli and Steeds (1992), which corresponds to the extinction symbols Ia.. or Fd.. Only the position of the reﬂections is required to identify the possible space groups. However, the integrated intensity observed on the LACDIF patterns also may be used to identify the “ideal” symmetry, which is connected with the point group and the Laue class (Morniroli, 2002; Morniroli and Steeds, 1992).

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2. Electron-Sensitive Materials LACDIF patterns are obtained in LACBED geometry. Depending on the defocus value h, a wide specimen area is illuminated, indicating that the electron dose received by the specimen is considerably reduced with respect to the same focused incident beam. Therefore, it becomes possible to use the LACDIF technique to study beam-sensitive materials such as minerals or polymers. To prove the validity of this assumption we performed some experiments on the polydioxanone polymer (Figure 20a) and obtained good quality patterns like the one shown on Figure 20c (Morniroli, et al., 2008). The pattern was very intense so only a very short exposure time (∼0.1 s) was needed to record it. Furthermore, the polymer specimen exhibits much better resistance to electron beam due to the low electron dose. Indeed, the quality of the diffraction pattern remains correct after a 10-s time, which is large compared to the exposure time. This highquality pattern can be compared to the same diffraction pattern obtained in standard low-dose SAED mode (Figure 20b). (a)

(b)

(c)

FIGURE 20 Application of the LACDIF technique to polymers. (a) Electron micrograph of the polydioxanone polymer. (b) Experimental conventional selected-area electron diffraction pattern of the polydioxanone polymer. (c) Experimental LACDIF pattern of the polydioxanone polymer. (From Morniroli et al., 2008.)

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3. Electron Energy Loss Magnetic Chiral Dichroism EELS is widely used to investigate local chemical and electronic properties of materials. Magnetic information can also be obtained at the scale of the size of the electron probe. Knowledge of local magnetic properties is important for issues arising from the reduction of magnetic devices, for which interfaces and surfaces play a dominant role. Recently a new approach has been proposed for measuring circular dichroism that is suitable for the study of ferromagnetic materials (Schattschneider et al., 2006). This technique aims to give access to the energy loss magnetic chiral dichroism (EMCD) spectrum in a TEM, analogous to X-ray magnetic circular dichroism (XMCD) signal. The dichroic signal is measured at two precise positions located between the transmitted and diffracted spots (Figure 21a). The EMCD signal is particularly sensitive to the orientation conditions and dynamic scattering effects. Indeed, the signal can be “delocalized” on the inelastic diffraction pattern and should be less signiﬁcant at the two kinematical positions predicted in the initial paper (Hébert and Schattschneider, 2003). We have showed that the LACDIF condition strongly increases the EMCD intensity compare to the classic SAED diffraction method (Warot et al., 2008). This can be explained by the intensity integration inside the diffracted spots performed by the convergent illumination. The corrector is essential to suppress the blurring effect introduced by the spherical

q1 0

q

q’1 q’

q2

g

q’2

Intensity (arb. units)

Pos 1

100 80 60 40 20 0

(a)

2 0 22 24

Pos 2

(b)

26 700

705 710 715 720 725 730 735 Energy (eV)

FIGURE 21 Inelastic LACDIF pattern taken on the L3 edge of the iron sample oriented in (110) two-beam conditions. The two spectra taken in Pos 1 and Pos 2 of the LACDIF data cube and their difference (the so-called dichroic signal) are reported in (a) and (b), respectively.

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aberration. Indeed, this blurring also affects the uncorrected EMCD signal detected on the two positions of the LACDIF pattern. Iron can be used as a ferromagnetic test material. We have recorded a data cube of a LACDIF pattern oriented on the (110) two-beam conﬁguration using energy spectrum imaging (ESI) with energy windows centered around the L2,3 edge [705 eV, 729 eV] with 1-eV energy steps between images (Warot et al., 2008; Calmels et al., 2007). The main result is shown on Figure 21, where the inelastic LACDIF pattern taken on the L3 edge is presented in Figure 21a. High intensity is present in the transmitted and diffracted spots. Figure 21b presents the superimposition of the two spectra taken from the data cube at positions named “pos 1” and “pos 2,” where the difference is now visible. The subtracted signal is then reported on Figure 21b from which the orbital and spin magnetic moment could be extracted using recently determined sum rules (Calmels et al., 2007).

V. PSEUDO-LORENTZ MODE FOR MEDIUM-RESOLUTION ELECTRON HOLOGRAPHY As a deviation from the original purpose of the Cs corrector, we investigated the use of the ﬁrst transfer lens as a pseudo-Lorentz lens to perform Lorentz microscopy and holography experiments in medium-resolution mode. Such experiments are usually achieved in a dedicated TEM ﬁtted with an additional lens (Lorentz lens) located within the lower pole piece of the objective lens. The objective is switched off and the Lorentz lens is excited. For medium-resolution holographic experiments, the Lorentz lens should be set so that it images a plane located below the SA plane containing the electrostatic biprism. This is achieved with an optimized Lorentz-holography lens conﬁguration that focuses on a plane located 12 mm below the SA plane. The diffraction and projector lenses of the microscope are then set in such a way that they image this new image plane. The optical elements of the SACTEM corrector are partially located within the Super-Twin objective lens at the place where a Lorentz lens would normally sit. To compensate for the lack of a dedicated Lorentz lens, we created a “pseudo-Lorentz” mode (Snoeck et al., 2006). In this optical setting, the objective lens and the lenses of the Cs corrector are switched off, except for the ﬁrst transfer lens and the associated quadripoles, which are used to adjust the focus and the alignment. A drawing of the optic scheme is shown in Figure 22. The optical characteristics of this pseudo-Lorentz lens (TL11) are Cs = 1.0 m, Cc = 54 mm. The magniﬁcation can be increased by tuning in free lens mode the current

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Axial ray path

Spec plane

OL pre OL post

ffp bfp

}

Off in Lorentz mode

TL11 Normal Cs-corrected mode

“Lorentz” mode

TL12

HP1 TL21

Off in Lorentz mode

TL22 HP2 ADL

Axial ray Biprism SAD plane 12 mm

FIGURE 22 Cs Optical diagram of Cs corrector used in pseudo-Lorentz mode. (See color insert).

in the two projector lenses (practically, only the current in the projector P1 is adjusted). Unfortunately, the step size for the current adjustment of the transfer lens power supply is not small enough to allow perfect adjustment of focus.

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With such a conﬁguration of the lenses, a maximum magniﬁcation of ∼100, 000 times can be achieved on the camera mounted below the screen. Table I shows the interference fringes distances and the overlap width for different values of the voltages applied on the biprism. This conﬁguration has been used routinely to perform Lorentz microscopy and medium-resolution electron holography. Figure 23 shows a hologram obtained on 30-nm iron nanocubes with a biprism polarization of 120 V. The fringes spacing is 1.4 nm for a contrast of ∼15%. We separated the mean inner potential (MIP) contributions to the phase shift from the magnetic component following a now standard method (Dunin-Borkowski et al., 1998). Two holograms were recorded between which the cubes were magnetized in two directions by tilting the sample by ±42 degrees and switching the objective lens on fully to apply a large in-plane ﬁeld of ∼1.7 Tesla to the sample. This saturates the sample in two opposite remanent magnetic states. In our example, the in-plane projection of the applied magnetic ﬁeld is deviated of ∼10 degrees from the [100] direction of the body centered cubic iron nanocubes (see H1 and H2 in Figure 24a). The MIP and the magnetic contributions to the phase shift are obtained by addition and subtraction of the phase images obtained from the two holograms. The magnetic contribution to the phase shift is shown in Figure 24b. It indicates that the magnetic ﬂux is running in the [100] direction within the two cubes; the ﬂux closure then occurs in the vacuum—like around a magnet. The derivative of the magnetic phase image gives access to the components of the magnetic induction within TABLE I Hologram characteristics changes vs voltage biprism

Voltage (V) Fringe spacing (nm) Overlap width (nm)

25 5.5 80

50 3.3 135

75 2.2 209

100 1.7 300

125 1.35 383

150 1.12 460

Ha

20 nm

FIGURE 23 Electron hologram obtained for two interacting 30 nm iron nanocubes.

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

(a)

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(b)

H1 H2

[100]

[001] (c)

[010]

20-nm

(d)

FIGURE 24 Hologram analysis. (a) TEM micrograph of the two interacting iron nanocube. (b) Phase image corresponding to the magnetic contribution to the phase shift. (c) Vectorial map of the in-plane components of the magnetic induction. (d) Micromagnetic simulation of the in-plane induction. (See color insert).

and outside of magnetic nano objects. A vectorial map of the induction can then be obtained (Figure 24c). To simulate the magnetic conﬁguration of the iron nanocubes, we used the 3D version of the micromagnetic code object-oriented micromagnetic framework (OOMMF) (Donahue and Porters, 1999). The nanocube shapes and sizes introduced for the calculations are deduced from the TEM images. Each iron nanocube is discretized in 3D cells of 2 nm size in each direction. The simulated magnetic conﬁgurations calculations are veriﬁed by recalculating with a smaller cell size. Bulk magnetic parameters for iron were used: Ms = 1.72 × 106 A/m, A = 2.1 × 10−11 J/m, and K1 = 4.8 × 104 J/m3 . A damping parameter of α = 0.5 was included to allow the equilibrium remanent state to be reached rapidly. For the simulations, we apply magnetic ﬁelds of the same amplitude and direction as those experimentally. Among different calculation outputs, OOMMF provides vectorial maps of the magnetization and the total ﬁeld of the magnetic element. The simulated induction map is shown in Figure 24d. Simulated and experimental maps agree well both qualitatively and quantitatively. This example demonstrates the capability of the pseudo-Lorentz conﬁguration to perform medium-resolution electron holography on magnetic

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nanomaterials in the SACTEM. Others medium-resolution holography experiments have been performed with this mode, such as doping proﬁle analysis in semiconducting devices and dark-ﬁeld holography for large-area strain measurements.

VI. CONCLUSIONS Happily, the beneﬁts of aberration correction have been underestimated and largely unexpected. Even for HRTEM, we have discovered the surprising advantages of taking relatively low-magniﬁcation aberration-corrected images of silicon (Hüe et al., 2008). Aberration correction allows strain mapping to be performed on thick and damaged FIB-prepared specimens. The least expected beneﬁts have been for electron diffraction experiments. Aberration correction allows LACDIF (Morniroli et al., 2008) and CHEF (Houdellier and Hÿtch, 2008) conﬁgurations to be used effectively and efﬁciently. Finally, the extra lenses provided by the corrector allow original optical conﬁgurations to be explored (e.g. the pseudo-Lorentz mode; Snoeck et al., 2006). Indeed, the stability of the microscope and the computer-assisted alignment has allowed the SACTEM to be used continuously and routinely in a wide range of modes ever since its installation.

ACKNOWLEDGMENTS F. Hüe’s grant is co-funded by the CNRS and CEA-Leti. The authors thank the European Union for support through the IP3 project ESTEEM (Enabling Science and Technology through European Electron Microscopy, contract No. 0260019) and are grateful to Peter Hartel (CEOS) for many fruitful discussions.

REFERENCES Balladore, J., and Murillo, R. (1977). New type of magnetic lens for electron microscope. J. Micr. Spect. Elect. 2, 211–222. Bierwolf, R., Hohenstein, M., Phillipp, F., Brandt, O., Crook, G. E., and Ploog, K. (1993). Direct measurement of local lattice-distortions in strained layer structures by HREM. Ultramicroscopy 49, 273–285. Calmels, L., Houdellier, F., Warot, B., Hÿtch, M. J., Serin, V., Snoeck, E., and Schattschneider, P. (2007). Experimental application of sum rules for electron energy loss magnetic chiral dichroism. Phys. Rev. B 76, 060409. Carbone, L., Nobile, C., De Giorgi, M., Della Sala, F., Morello, G., Pompa, P., Hÿtch, M. J., Snoeck, E., Fiore, A., Franchini, I. R., Nadasan, M., Silvestre, A. F., Chiodo, L., Kudera, S., Cingolani, R., Krahne R., and Manna, L. (2007). Synthesis and micrometer-scale assembly of colloidal CdSe/CdS nanorods prepared by a seeded growth approach. Nano Lett. 7, 2942–2950.

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Hÿtch, M. J., and F. Houdellier, F. (2007). Mapping stress and strain in nanostructures by high-resolution transmission electron microscopy. Microelectr. Eng. 84, 460–463. Hÿtch, M. J., and Plamann, T. (2001). Imaging conditions for reliable measurement of displacement and strain from high-resolution electron microscope images. Ultramicroscopy 87, 199–161. Johnson, C. L., Snoeck, E., Ezcurdia, M., Rodríguez-González, B., Pastoriza-Santos, I., Liz-Marzán, L. M., and Hÿtch, M. J. (2008). Effects of elastic anisotropy on strain distributions in decahedral gold nanoparticles. Nat. Mater. 7, 120–124. Lentzen, M., Jahnen, B., Jia, C. L., Thust, A., Tillmann, K., and Urban, K. (2002). High-resolution imaging with an aberration-corrected transmission electron microscope. Ultramicroscopy 92, 233–242. Lichte, H. (1991). Optimum focus for taking electron holograms. Ultramicroscopy 38, 13–22. Lin, J. A., and Cowley, J. (1986). Calibration of the operating parameters for an HB5 STEM instrument. Ultramicroscopy 19, 31–42. Marais, B. (1970). Etude d’un dispositif corrigeant partiellement l’aberration de sphéricité des lentilles électroniques. J. Microsc. 9, 683–690. Morniroli, J. P. (2002). “Large Angle Convergent Beam Electron Diffraction: Application to Crystal Defects.” Société Française des Microscopies Edt , Paris. Morniroli, J. P., Houdellier, F., Roucau, C., Puiggali, J., Gesti, S., and Redjaïmia, A. (2008). LACDIF, a new electron diffraction technique obtained with the LACBED conﬁguration and a Cs corrector: comparison with electron precession. Ultramicroscopy 108, 100–115. Morniroli, J. P., and Steeds, J. W. (1992). Microdiffraction as a tool for crystal-structure identiﬁcation and determination. Ultramicroscopy 45, 219–239. Pinna, H., Kihn, Y., Sevely, J., and Jouffrey, B. (1978). Electron counting by means of a Si(Li) detector—adaption to EELS spectra recording in HVEM. J. Microsc. Spectrosc. Electron. 3, 1–14. Rose, H. (1990). Outline of a spherically corrected semiaplanatic medium-voltage transmission electron microscope. Optik 85, 19–24. Schattschneider, P., Rubino, S., Hébert, C., Rusz, J., Kunes, J., Novak, P., Carlino, E., Fabrizioli, M., Panaccione, G., and Rosse, G. Detection of magnetic circular dichroism using a transmission electron microscope. Nature 441, 486–488. Scherzer, O. (1949). The theoretical resolution limit of the electron microscope. J. Appl. Phys. 20, 20–29. Sevely, J., Perez, J. P., and Jouffrey, B. (1973). Design for high-power electron-energy loss analyser (200 to 1200 kV). CR Acad. Sci. 276, 515–521. Sevely, J., Perez., J. P., Zanchi, G., Kihn, Y., and Jouffrey, B. (1982). Electron-energy loss spectroscopy from 1MeV to 3MeV. Electr. Microsc. 1, 589–590. Snoeck, E., Hartel, P., Müller, H., Haider, M., and Tiemeijer, P. C. (2006). Using a CEOS objective lens corrector as a pseudo Lorentz lens in a Tecnai F20 TEM. In Proc. IMC16 International Microscopy Congress (Sapporo, 2006), vol. 2, p. 730. Spence, J. C. H. (2003). “High-Resolution Electron Microscopy,” third edition. Oxford University Press, Oxford. Spence, J. C. H., and Cowley, J. M. (1978). Lattice imaging in STEM. Optik 50, 129–142. Tillmann, K., Lentzen, M., and Rosenfeld, R. (2000). Impact of column bending in highresolution transmission electron microscopy on the strain evaluation of GaAs/InAs/GaAs heterostructures. Ultramicroscopy 83, 111–128. Treacy, M. M. J., Gibson, J. M., and Howie, A. (1985). On elastic relaxation and long wavelength microstructures in spinodally decomposed InxGa1−x AsyP1−y epitaxial layers. Philos. Mag. A 51, 389–417. Uhlemann, S., and Haider, M. (1998). Residual wave aberrations in the ﬁrst spherical aberration corrected transmission electron microscope. Ultramicroscopy 72, 109–119.

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Verbeeck, J. (2006). Interpretation of “Energy-ﬁltered electron-diffracted beam holography” by R. A. Herring. Ultramicroscopy 106, 461–465. Vincent, R., and Midgley, P. A. (1994). Double conical beam-rocking system for measurement of integrated electron-diffraction intensities. Ultramicroscopy 53, 271–282. Völkl, E., Allard, L. F., and Joy, D. C. (eds.). (1999). “Introduction to Electron Holography.” Plenum Publishers, New York. Warot, B., Houdellier, F., Hÿtch, M. J., Calmels, L., Serin, V., and Snoeck, E. (2008). Mapping inelastic intensities in diffraction patterns of magnetic samples using the energy spectrum imaging technique. Ultramicroscopy 108, 493–398. Zanchi, G., Sevely, J., and Jouffrey, B. J. (1997). Energy ﬁlter for high-voltage electron microscopy. Micr. Spectr. Electr. 2, 95–99. Zanchi, G., Kihn, Y., and Sevely, J. (1982). On aberration effects in the chromatic plane of the omega ﬁlter. Optik 4, 427–436.

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CHAPTER

7 Novel Aberration Correction Concepts Bernd Kabius* and Harald Rose†

Contents

I Introduction II Concepts for Improving Resolution for In Situ TEM III Cc Correction A Benefits of Cc Correction IV Summary Acknowledgments References

261 262 265 265 279 280 280

I. INTRODUCTION Aberration correction for transmission electron microscopes (TEM) has a long history starting with Scherzer’s theorem (1936) that spherical aberration is inevitable unless one of the three following conditions is met: rotational symmetry is broken, the electron path is reversed, or the electron beam is pulsed. Since then theoretical and experimental attempts have been made to implement aberration correction for electron optics using the exceptions (Scherzer, 1947) to Scherzer’s theorem. A summary of these efforts can be found in this volume (Rose, 2008). The ﬁrst successful design demonstrating Cs correction for imaging mode was suggested by Rose (1990) and demonstrated by Haider et al. (1998). In the following years this hexapole corrector was applied to various material science problems (Kabius, 2002) (Mantl, 1999) (Lentzen, 2002). A quadrupole/octopole corrector was developed and applied by Dellby et al. (2001) for STEM mode. Since then aberration correction has been used to help solve a wide variety of material science problems (Van Benthem, 2006) (Houdellier, 2007)

* Argonne National Laboratory, Argonne, Illinois 60439, USA † Technical University Darmstadt, 64289 Darmstadt, Germany

Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01007-0. Copyright © 2008 Elsevier Inc. All rights reserved.

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(Kirkland, 2008) (Jia, 2004). Up to date these studies have focused on the beneﬁts of Cs correction for high-resolution TEM (HRTEM). The instrumentation available for these investigations achieved a resolution limit of 0.8 Å, high signal to noise ratio (SNR), and elimination of image artifacts, such as contrast delocalization. The next step in aberration correction was suggested by Gibson in 2000 (2000a, b) and received a strong endorsement from the Department of Energy (DOE). Gibson proposed the simultaneous correction of chromatic aberration Cc and spherical aberration Cs to enable high-resolution in situ experiments. Based on these initial ideas the TEAM project (Transmission Electron Achromatic Microscope) was suggested. One of the limiting factors of in situ TEM is the small gap between the pole pieces of the objective lens available for the sample, the stage, and further experimental instrumentation such as electrical contacts for transport measurements or magnetic coils for the investigation of dynamic magnetic processes. Increasing the width of this gap causes a larger value for Cc, thus degrading the information limit. Correcting Cc removes this obstacle and allows an instrument with high-resolution capabilities as well as space for experimental equipment.

II. CONCEPTS FOR IMPROVING RESOLUTION FOR IN SITU TEM An alternate concept for improving spatial resolution for in situ experiments in TEM mode is Cs correction and monochromatization. In this case, the loss in resolution caused by a wide-gap objective lens is compensated by a smaller energy width of the electron beam. This concept appears to be less demanding because it is based on existing technology. In the past, Cc correction has been viewed as not feasible because of its high stability requirements. (Uhlemann, 1995). Recently, novel designs have been published that are suitable for correcting chromatic aberration as well as spherical, axial, and off-axial aberrations that are within reach of present technology (Rose, 2002). The advantages and capabilities of both concepts can be derived from their contrast-transfer behavior. The phase-contrast-transfer function includes the damping envelopes for spatial and temporal coherence, which ultimately limit information transfer (Ishizuka, 1979; equations 1, 4, and 5). The phase shift term W(q) [1] has been expanded up to the seventh order of coefﬁcients of spherical aberration (C1 . . . C7 ). Defocus (C1 ) and C3 (Cs ) can be adjusted, whereas C5 can be optimized. C7 is the next higher coefﬁcient of spherical aberration that cannot be inﬂuenced and therefore must be considered as a

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constant because it can be the dominant aberration for small values of C3 and C5 .

PCTF q = sin W q ∗ Et ∗ E3 ; (1)

(2π) 1 1 1 1 W q = C1 λ2 q2 + C3 λ4 q4 + C5 λ6 q6 + C7 λ8 q8 λ 2 4 6 8 where λ represents the wavelength of the electron beam, q the reciprocal lattice vector, Et the damping envelope of temporal coherence, and Es the damping envelope of spatial coherence. If C3 is corrected, Scherzer defocus (1949) does not provide the best choice for phase-contrast imaging. Therefore, a different approach was suggested by Scherzer (1947). Defocus C1 and C3 can be balanced against C5 to obtain a broad range of spatial frequencies with uniform contrasttransfer [Eqs. (2) and (3)].

C1,opt

λ =2 C5

−16 C3 = 5

2

3

C5

λ C5

(2)

1 3

C5

(3)

Recently other methods have been suggested (Lentzen et al., 2002; Lentzen and Urban, 2006) to calculate parameter sets for optimizing contrasttransfer. In this paper, the Scherzer approach is used consistently because some of the newer concepts require a control of C5 that is beyond the reach of present technology. Spherical aberration also inﬂuences the damping envelope of spatial coherence [Eq. (4)]:

2 − (πsc )2 2 3 4 5 6 7 C E5 = exp q + C λ q + C λ q + C λ q 1 3 5 7 2 1.0 + πλsc ds q2

(4) where ds is the defocus spread. The small values for C3 that are possible with the present correction systems have the effect that spatial coherence and therefore beam convergence does not limit resolution in TEM mode for defocus values close to the optimum as deﬁned by Eq. (2). We conclude from Eq. (4) that for a Cs corrected system the larger C3 values of lenses with large pole-piece gap are less stringent for the optical properties because of the decrease in spatial

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damping. Nevertheless, the coefﬁcient of chromatic aberration increases with increasing focal length and pole-piece gap. Therefore, the damping envelope of temporal coherence [Eq. (5)] will be the limiting factor for resolution of a C3 -corrected system:

2 − 0.5 π λ ds ∗ q4 Et = exp 2 1.0 + π λ sc ds

with : ds = Cc

Ee H

2

+

H H

2

(5)

+

2 I I

2 .

(6)

TEMs designed for in situ experiments are likely to have a large polepiece gap and from Eq. (5), four methods can be derived for improving the resolution for this type of instrument: 1. Smaller wavelength. This approach has been realized for TEMs with an acceleration voltage >1 MeV. The disadvantages of this concept are radiation damage inﬂicted by the very high incident energy electrons and the high costs for the instrument and for maintenance. Spherical aberration correction systems for 1-MeV electrons are not available at present and chromatic aberration correction appears out of reach because the electrical ﬁeld gradients necessary limit Cc -correction to a maximum 300 keV. 2. Improving the stabilities of the accelerating voltage H and of the power supply of the objective lens I. In the following text a stability of 0.1 ppm is assumed for both. Higher stabilities have a negligible inﬂuence on resolution. 3. Improving the energy width of the electron emitter Ee . The overall energy width E, which includes high-voltage instability, H, and the energy width of the ﬁeld emitter Ee , will be assumed to be 0.7 eV, which is reasonable for a Schottky ﬁeld emitter and for beam currents sufﬁcient for TEM mode imaging (∼10–100 nA). Presently available monochromators (Ringnalda et al., 2007; Thesen, Matijevec, and Benner, 2005) can deliver an energy width of 0.1 eV. This concept has been proven to extend the information limit experimentally (Ringnalda et al., 2007). Including the inﬂuence of high-voltage instabilities, it appears to be reasonable to assume a total energy width of 0.15 eV. 4. Cc correction. Cc correction also strongly affects the damping envelope of temporal coherence. A much larger fraction of the emitted electrons can be used for a Cc -corrected system compared with a TEM with electron beam monochromatization and Cs correction. This property can be used for either shorter exposure times or higher SNR. The latter

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increases sensitivity and improves quantiﬁcation of HRTEM images. Shorter exposure times are important for in situ experiments, which can require extended time resolution. Furthermore, mechanical instabilities are less important if the exposure time can be reduced.

III. CC CORRECTION A. Benefits of Cc Correction 1. High-Resolution TEM From the discussion above of parameters inﬂuencing resolution we can conclude that for HRTEM a smaller wavelength has severe disadvantages in terms of increased radiation damage and in designing electron optical components that achieve the required correction strength. High stabilities for acceleration voltage and objective lens current already have been achieved such that the resolution limiting parameters are now Cc and E. This leaves two competing concepts: Cc correction and energy monochromatization. Based on Eq. (1) and the optical parameters previously mentioned the phase-contrast-transfer function for a Cs -corrected TEM with a monochromator can be compared to the abilities of a Cc -corrected system (Figure 1). Equation (6) must be modiﬁed for a Cc -corrected system. The contribution of voltage and current instabilities to the overall defocus spread are negligible for small Cc values. Under these conditions, defocus spread is strongly inﬂuenced by the stability of the corrector lenses, which must be included: ⎛ ⎞

2 2 2 2 Q 2 IQ 2

EQ 2

H 2 I

E Q ⎠ (7) ds = ds + dE + ⎝Cc + + IQ EQ E H I

where IQ represents the instabilities of the electrical current of the magnetic quadrupoles of a Cc corrector, Q

ds is the proportionality constant between IQ and defocus,

EQ represents instabilities of the voltage of the electric quadrupoles of a Cc corrector, and Q dE is the proportionality constant between EQ and defocus. Q

In the following example, a value of 8 nm/ppm will be used for ds Q and for dE. . This value has been calculated for a Cc and Cs corrector of the superaplanator design, which consists of two quadrupole quintuplets with two combined electric/magnetic quadrupoles for Cc correction and can be regarded as a simpliﬁed version of the ultracorrector (Rose, 2002), is

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based on septuplets. This corrector for spherical and chromatic aberrations is addressed in the following that as Cc corrector. Stability values of 2∗ 10−8 for IQ and 4∗ 10−8 for EQ have been measured for the power supplies built for the Cc corrector (personal communication, Haider 2008). Figure 1 displays contrast-transfer functions (CTFs) for a 300-kV Schottky emitter TEM with Cs correction and without monochromatization. A general-purpose objective lens with a pole-piece gap of a ∼5 mm is assumed with a Cs value of 1.2 mm and a Cc value of 2.02 mm for 300kV electrons. This value has been calculated for this objective lens design, including relativistic correction for electrons accelerated by 300 kV. C5 for this type of objective lens has been measured to 4 mm using a Cs corrected TEM. The coefﬁcient C5 for such a lens has been measured using Zemlin tableaus to be ∼4 mm. The coefﬁcient C7 is estimated at 20 mm. Furthermore, it is assumed that an aberration correction system adds 0.2 mm to the total value for Cc , which is in agreement with present corrector technology. (Haider, 2008). A stability of 0.1 ppm for both acceleration voltage and objective lens current is assumed for the contrast-transfer calculations. C1 and C3 are chosen according to Eqs. (3) and (4) to be 5 nm and −10.1 μm, respectively. The phase contrast-transfer function in Figure 1 represents the electron optical behavior of a TEM 1.00 0.80 Spatial coherence 0.60 0.40

Temporal coherence

PCTF

0.20 0.00 20.20 20.40 20.60 20.80 21.00 5.0 3.0 2.0 1.5 1.2 1.0

0.8 0.7

0.6

0.5

0.4

g [Å]

FIGURE 1 Phase contrast-transfer function for a Cs -corrected 300 kV TEM with damping envelope of spatial coherence (dotted line) and damping envelope of temporal coherence (dashed line).

Novel Aberration Correction Concepts

267

as described above without monochromatization ( E = 0.7 eV) or Cc correction. The contrast-transfer extends out to 0.7−0.8 Å but is weak below 1 Å. As seen from the damping envelopes plotted in Figure 1, spatial damping is negligible up to 0.4 Å because of the inﬂuence of Cs correction on spatial coherence. Under these circumstances, information transfer is limited almost exclusively by temporal coherence. A monochromator capable of reducing E to 0.15 eV, while preserving sufﬁcient beam current for TEM imaging (>2 nA), extends contrast-transfer well below 0.5 Å (Figure 2). The ﬁrst zero of the CFT occurs at 0.4 Å for the C5 value chosen. Further improvement of contrast-transfer by monochromatization would require an even smaller value of E, which comes at the expense of electron beam intensity and requires longer exposure times. From this we can conclude that E can be reduced by a factor of about 5 before experimental conditions become impractical for material science applications. The advantage of Cc correction is that the coefﬁcient of chromatic aberration can be reduced by a much greater amount than can E. Two orders of magnitude reduction are possible, assuming that the measurement accuracy for Cc determines the lower limit for Cc correction. The brightness of the electron source and the electron beam current are not degraded by Cc correction, which allows a high SNR. 1.00 0.80 0.60 0.40

C s & C c corrected

PCTF

0.20 0.00 20.20

C s corrected & monochromated

20.40 20.60 20.80 21.00 5.0 3.0 2.0 1.5 1.2 1.0

0.8 0.7

0.6

0.5

0.4

g [Å]

FIGURE 2 Influence of monochromatization (dashed line) and Cc -correction (continuous line) on contrast-transfer.

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Contrast delocalization can be calculated for these conditions using Eq. (8) (Lichte, 1991) to be 0.47 Å for qmax = 0.5 Å, which is below the resolution goal of the TEAM project of 0.5 Å. This residual contrast delocalization is therefore acceptable. Further improvements in resolution will require a better control of C5 .

R = |C7 λ7 q7 + C5 λ5 q5 + C3 λ3 q3 + C1 λq|qmax

(8)

The gain in contrast by Cc correction compared with a Cs -corrected system can be demonstrated by image simulations of diamond in a [110] orientation (Figure 3). Line scans along the [100] direction are derived from these image simulations (Stadelmann, 1987) at the positions indicated by the lines in Figure 3. These line scans are positioned across close pairs of columns of carbon atoms separated by 0.89 Å. Assuming a Cs -corrected instrument, this distance is transferred with a very low contrast (8%) while a Cc -corrected TEM enables 60% contrast to be transferred. Therefore, a gain in sensitivity of one order of magnitude for lattice distances <1 Å can be expected for Cc -corrected instruments. This gain is important for quantitative evaluation of high-resolution images—for example, measurement of local lattice distortions or quantitative elemental analysis. The advantages of Cc correction increase at lower acceleration voltage as can be demonstrated by a CTF calculation for 80 kV (Figure 4). This behavior is expected because of the smaller inﬂuence of the temporal coherence damping envelope on resolution for a Cc -corrected system. At 80 kV C s - corrected

C c - corrected

0.0 0.2 Contrast

0.4 0.6 0.8 1.0 1.2 1.4 0

1

2

3

4

5

x [Å]

FIGURE 3 Comparison of image simulations of diamond in [110] orientation for a Cs corrected and a Cs and Cc -corrected 300 kV TEM.

Novel Aberration Correction Concepts

269

1.00 0.80 0.60

C s corrected & monochromated 0.40

PCTF

0.20 0.00 20.20 20.40 20.60

C s & C c corrected

20.80 21.00 5.0 3.0 2.0 1.5 1.2 1.0

FIGURE 4

0.8 0.7 g [Å]

0.6

0.5

0.4

Contrast-transfer at 80-kV acceleration voltage.

a signiﬁcant gain in the information limit is realized—from ∼0.91 Å to 0.52 Å, assuming a minimum contrast-transfer of 10%. The ﬁrst zero of the CTFs occurs at the same value (0.73 Å) for both the Cc -corrected and the Cs -corrected, monochromated systems, because it is determined by C5 . More important for scientiﬁc applications than the gain in information limit is the increase in contrast-transfer for spacings between 0.8 and 1.5 Å, which includes lattice distances in many materials such as the (400) lattice spacing in diamond or the (220) lattice spacings in most metals. Decreasing the acceleration voltage has two major advantages. (1) The knock-on damage is lower, which is particularly important for radiation-sensitive materials such as biological samples, macromolecules, carbon nanotubes, and fullerenes. (2) The cross sections for electron energy loss spectroscopy (EELS) transitions or X-ray emission for energy-dispersive X-ray analysis increase, allowing a better SNR for elemental and chemical analysis. Cc correction thus enables the combination of unprecedented resolution with low radiation damage and high analytical signal. Aberration-free imaging requires more than just correcting for spherical aberration. Further axial and non axial aberrations must be taken into account. The criterion for aberration-free imaging is that the total phase shift of all aberrations should not exceed π/4 (Haider, Uhlemann, and Zach, 2000) for a certain resolution and ﬁeld of view. Aberration-free imaging up to a resolution of 0.5 Å requires consideration of aberration coefﬁcients

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up to the seventh order. The phase shift term can be rewritten for two dimensions by substituting the one-dimensional (1D) lattice vector g by the complex scattering vector α, taking into account spherical and axial aberrations (Haider, Bernhardt, and Rose, 1982):

$

1 1 1 W (α, α) ¯ = C1 α α¯ + A1 α¯ 2 + B2 α2 α¯ + A2 α¯ 3 + C3 α2 α¯ 2 2 3 2 1 1 + S3 α3 α¯ + A3 α¯ 4 + B4 α3 α¯ 2 + D4 α4 α¯ + A4 α¯ 5 4 5 1 1 3 3 4 2 5 6 + C5 α α¯ + S5 α α¯ + R5 α α¯ + A5 α¯ + B6 α4 α¯ 3 3 6 1 1 + D6 α5 α¯ 2 + F6 α6 α¯ + A6 α¯ 7 + C7 α4 α¯ 4 7 4 % 1 5 3 6 2 7 + S7 α α¯ R7 α α¯ + G7 α α¯ + A7 α¯ 8 , 8

(9)

Where A 1...7 denotes astigmatism, B2...6 coma, S3,5. star aberration, R5,7 rosette aberration, D4,6 three-lobe aberration, F6 pentacle aberration, and G7 is not yet named. The 1D representation of the CTF [Eq. (1)] must be expanded to two dimensions. By using Eq. (9), we calculate a set of maximum values for each aberration coefﬁcient that induce a phase shift φ = 2πW/λ of less than π/4 for a certain high voltage and resolution. Ideally, the sum of the contributions to the phase shift of all aberration coefﬁcients should be <π/4. The phase shift criterion for individual aberrations (Table I) is important in determining the required measuring precision for these aberration coefﬁcients when aligning the corrector such that the conditions for aberration-free imaging are met. Aberration coefﬁcients can be calculated from Zemlin tableaus (Haider et al., 1998; Zemlin et al., 1978) by measuring C1 and A1 from the diffractograms of such a tableau. We concluded from Table I that the required measurement precision for C1 and A1 is 0.32 nm if enabling a resolution of 0.5 Å is to be achieved with negligible artifacts from lens aberrations. According to simulations of the superaplanator, a ﬁeld of view of 30002 independent image points can be imaged up to the resolution goal of 0.5 Å, which is achieved by correcting off-axial coma up to the ﬁfth order.

2. Elemental Mapping Calculating elemental maps from energy-ﬁltered images is a standard method in analytical electron microscopy. This method does not yield the spatial resolution that can be obtained by a focused beam (STEM mode), for which atomic column resolution can be achieved, but it has the advantage

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271

TABLE I Maximum aberration coefficient values according to the residual phase shift of π/4 allowed for aberration free imaging. Aberration symbol

Aberration

θ [mrad] C1 [nm] A1 [nm] A2 [nm] B2 [nm] C3 [nm]

Semi-angle at info-limit Focus Twofold astigmatism Threefold astigmatism Axial coma Third-order spherical aberration Fourfold astigmatism Star aberration Fivefold astigmatism Fourth-order axial coma Three-lobeaberration Fifth-order spherical aberration Sixfold astigmatism Rosette aberration Fifth-order star aberration

A3 [nm] S3 [nm] A4 [μm] B4 [μm] D4 [μm] C5 [μm] A5 [μm] R5 [μm] S5 [μm]

80 kV/ 0.08 nm

120 kV/ 200 kV/ 300 kV/ 0.065 nm 0.05 nm 0.05 nm

50 0.42 0.42 12.5 4.2 330

50 0.33 0.33 10 3.3 270

50 0.25 0.25 7.5 2.5 200

39 0.32 0.32 12.2 4.1 410

330 83 8.3 1.7 1.7 200

270 67 6.7 1.3 1.3 160

200 50 5 1 1 120

410 100 13 2.6 2.6 400

200 33 33

160 27 27

120 20 20

400 67 67

of large ﬁeld of view. A realistic assumption for the resolution that can be achieved for elemental mapping with Cc correction is a ∼1 Å. This limits results from the energy loss process itself because core-loss transitions with energies larger than a ∼1 keV extend over distances of ∼1 Å. This resolution can be achieved for a ﬁeld of view larger than 30002 independent image points. The resolution for uncorrected energy-ﬁltered images is limited because of the large values for energy width and electron beam semiconvergence that must be used to achieve an SNR sufﬁcient for elemental or chemical bonding mapping. Energy width and semiconvergence degrade resolution by inﬂuencing the temporal and spatial coherence envelopes [Eqs. (4) and (5)]. From these equations we can conclude that Cs correction reduces the inﬂuence of semiconvergence, while Cc correction reduces the inﬂuence of energy width. Figure 5 shows plots of damping envelopes that include both temporal and spatial coherence for parameters typical for acquiring elemental maps using EELS transitions >1-keV energy loss, for example, the Ge-L and the Si-L edges.

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C s & C c corrected

0.80 Uncorrected

PCTF

0.60

C s corrected

0.40

0.20

0.00 10.0 5.0

3.0

2.0

1.5 g [Å]

1.2

1.0

FIGURE 5 Damping envelopes calculated for an energy width of 50 eV and a semiconvergence angle of 3 mrad at 200-kV accelerating voltage.

We deduce from the damping envelopes shown in Figure 5 that the resolution of an uncorrected instrument is reduced to 1 nm for the set of parameters listed above. Cs correction improves contrast-transfer signiﬁcantly but has only a small inﬂuence on resolution, which can be explained by the large energy width. As expected, Cc correction has a very strong impact on Energy-ﬁltered transmission electron microscopy (EFTEM) and enables atomic-level resolution energy-selective imaging using core-loss transitions whose delocalization is small enough.

3. In Situ TEM, Tomography, and Biological Samples In situ TEM has been a valuable tool to study topics such as radiation damage (Song et al., 2005), growth of thin ﬁlms (Ross et al., 2004), and dynamic magnetic behavior (Petford-Long et al., 2006; Craig et al., 2007). In situ experiments combined with the resolution and analytical abilities of a TEM are a very powerful tool for observing and characterizing dynamic processes. Nevertheless, relatively few groups have developed the expertise for in situ TEM. This can be explained by the large effort necessary to perform experiments inside the objective lens. Several limiting parameters are discussed in the following sections. Magnetic lenses suitable for high-resolution imaging have a gap between the pole pieces that is between 2.5 and 5 mm wide, thus limiting the space for in situ instrumentation such as magnetic coils for Lorentz

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microscopy. Pole pieces with a larger gap decrease the resolution significantly (e.g., the point resolution in Scherzer defocus for an uncorrected 300-kV TEM decreases from 0.16 nm to 0.27 nm if the gap is widened from 2.5 to 20 mm) (Figure 6). This decrease in resolution can be attributed to the higher spherical and chromatic aberration coefﬁcients for lenses with larger gap and therefore longer focal length. The PCTF calculation plotted in Figure 6 is for a lens with a 20-mm gap, and the aberration coefﬁcients have been estimated to be Cs = Cc = 4 mm. Such a lens would signiﬁcantly increase the volume available around the sample for experimental equipment, such as larger detectors for analytical electron microscopy. Higher values for energy width and semiconvergence of 1 eV and 0.2 mrad, respectively, were assumed, compared with the calculations for HRTEM (Figure 2), in order to achieve a higher beam current. Focus values where calculated according to Scherzer (1949). Many in situ experiments require high beam current densities to allow better time resolution. This can be achieved by focusing the electron beam on a small area, which increases beam convergence. However, this compromises spatial coherence, leading to a loss in information limit by up to a factor of 2 and a signiﬁcant decrease in contrast-transfer for all spatial frequencies. Cs correction strongly affects the spatial coherence damping envelope and can compensate for large beam convergence. Therefore, Cs correction is efﬁcient in achieving high SNR and high time resolution. 1.00

0.80

Uncorrected

0.60 0.40 C s corrected

PCTF

0.20 0.00 20.20 20.40 20.60

C s & C c corrected

20.80 21.00 5.0 3.0

2.0

1.5

1.2

1.0 g [Å]

0.8

0.7

0.6

FIGURE 6 Contrast-transfer for an objective lens suitable for in situ experiments.

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Furthermore, Cs correction offers a much better interpretability for highresolution images up to the information limit of the instrument. This is especially important for interpreting movies of dynamic experiments, for which it is impractical to perform complex image simulation to understand the contrast in a large number of images. The range of interpretable resolution can be extended by Cs correction from 0.27 nm to ∼0.2 nm (Figure 6). We have already seen the inﬂuence of Cc correction on the temporal coherence damping envelope and its effect for HRTEM. As can be concluded from the CTF plots in Figure 6, the improvement in contrast-transfer and information limit is more evident for an objective lens dedicated to in situ experiments, which has relatively high aberrations compared with those for a high-resolution lens. The information limit of a Cs - and Cc corrected in situ TEM, as deﬁned above, extends well below 1 Å, and the contrast-transfer for most common lattice spacings of crystalline materials (between 1 and 3 Å) is above 30%, which is much higher than for a Cs -corrected system. A further beneﬁt of Cc correction is that the dependence of resolution on energy spread is smaller, as seen for energy-ﬁltered imaging (see Figure 5). For in situ experiments, high beam currents are required to achieve a high time resolution. This requirement conﬂicts with the high-resolution requirement of a small energy width. The energy width of ﬁeld emitters and thermal emitters increases with increasing beam current because of the Boersch effect (Boersch, 1953), which reduces the information limit according to Eq. (5). A Cc -corrected system can be operated with higher emission currents without compromising resolution and therefore enables a better time resolution for dynamic experiments. Many in situ experiments cannot be performed with samples that are thin enough for HRTEM—for example, certain magnetic samples— or they require the presence of a liquid or gaseous environment, which increases the effective sample thickness. Multiple scattering and inelastic scattering are the processes that limit resolution for thick samples. Cc correction cannot compensate multiple scattering and, therefore, cannot enable high resolution for samples that are hundreds of nanometers thick, but it can decrease the inﬂuence of inelastic scattering on resolution. For samples close to the transmissivity limit, an electron beam energy width of <500 eV can be assumed which reduces the information limit to <5 nm. Figure 7 shows temporal coherence–damping envelopes for a large energy width of 500 eV, which can be expected for a very thick sample. The inﬂuence of spatial coherence can be neglected. The resolution of an uncorrected system is degraded to 3 nm while Cc correction can signiﬁcantly reduce the effect of the large energy width caused by inelastic scattering, allowing high contrast-transfer up to 1 nm.

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When multiple scattering and large beam spreading are not the limiting factors, Cc correction can improve resolution for thick samples. The potential of Cc correction under these circumstances is to enable nanometer resolution close to maximum sample thickness for a certain electron beam energy. This gain in resolution is important for the characterization of dislocation networks during in situ mechanical experiments or for analysis of the three-dimensional 3D shape of macromolecules in cells, which can be used for identiﬁcation of these molecules. The 3D analysis depends on tomographic tilt series, which leads to large effective sample thicknesses for the high tilt angles required for reconstruction. The expected beneﬁt for these biological applications is greater than for the material science applications because acceleration voltages lower than 300 kV are used to reduce radiation damage. The resolution of an uncorrected instrument for acceleration voltages between 80 and 200 kV is reduced to 10 nm and 5 nm, respectively, for sample thicknesses as described above. Furthermore, cryo-microscopy would beneﬁt from Cc correction for investigating “thick sections” where, according to the damping envelopes displayed in Figure 7 chromatic aberration is the resolution-limiting parameter.

Contrast-transfer

0.80

0.60 Uncorrected

0.40

0.20

C s & C c corrected

0.00 50.0 30.0 20.0

10.0 g [Å]

50

FIGURE 7 Influence of Cc correction on damping envelopes of temporal coherence for thick samples. The electron optical parameters of the in situ TEM previously described also have been assumed for these calculations.

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4. Lorentz Microscopy Lorentz microscopy can provide essential information regarding dynamic magnetic processes such as domain reversal and movement of magnetic vortices. Understanding the physics of nano-scaled magnetic structures requires knowledge about the magnetic structure and the microstructure. Layer thicknesses are of the order of a few nanometers, and the deposition techniques used tend to provide polycrystalline ﬁlms, resulting in variations in properties across the structures. The behavior of these materials relies on the local magnetic domain structure and magnetization reversal mechanism. One of the techniques enabling micromagnetic studies at the sub-micron scale is Lorentz transmission electron microscopy, which allows the magnetic domain structure and magnetization reversal mechanism of a ferromagnetic material to be investigated dynamically in real-time with a resolution of a few nanometers. At present, imaging of magnetic ﬁelds is strongly hampered by the large aberration coefﬁcients of the low-ﬁeld magnetic lenses required for Lorentz microscopy. Standard objective lenses require the sample to be located between the pole pieces where the magnetic ﬁeld is strong enough to destroy the domain or vortex structure. A study of magnetic samples can yield reliable results only if the sample is located in a magnetic ﬁeld of a few Gauss or less. This can be achieved either by positioning the sample between two lenses with opposite ﬁeld direction (thus canceling the magnetic ﬁeld) or by positioning the sample outside the pole pieces of the objective lens. Both types of Lorentz lens have in common that their focal length is about one order of magnitude larger than that of a standard high-resolution lens, which degrades the coefﬁcient of spherical aberration to values of up to 8 m. The best available Lorentz lens for 200-kV TEMs has a Cs value of ∼100 mm and a Cc value of 20 mm. These lens parameters have been assumed for the contrast-transfer calculation presented in Figure 8. Such a lens provides an information limit slightly better than 1 nm. Cs correction is capable of a small improvement but does not offer a signiﬁcant gain for the information limit. Cs and Cc correction allow a large improvement of the optical properties by extending resolution to atomic level. From Figures 2, 6, and 8 we can see that the strongest improvement introduced by Cc correction can be observed either for low acceleration voltages for objective lenses with high aberration coefﬁcients. The beneﬁt of Cc correction for Lorentz microscopy is not that it provides a better resolution for magnetic ﬁelds but that it then also enables observation of the inﬂuence of crystal structure, defects, and chemical inhomogeneities at a length scale of up to 1 Å while observing the magnetic structure at the same time. Information about the relationship between microstructure and magnetic properties can then be concluded directly from the experiment, which

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1.00 Uncorrected

0.80 0.60

C s corrected

0.40

PCTF

0.20 0.00 20.20 20.40 20.60 20.80

C s & C c corrected

21.00 10.0

5.0

3.0

2.0 g [Å]

1.5

1.2

1.0

FIGURE 8 Benefit of aberration correction for contrast-transfer of an objective lens suitable for Lorentz microscopy.

allows improved modeling of magnetic structure and dynamic magnetic processes.

5. Ultra-Fast TEM A frontier in materials science research is the understanding of the physics and chemistry of fast processes. Present technology for in situ TEM makes it a suitable tool for observing and characterizing these processes on the nanometer scale, but its time resolution is limited to video rate with few exceptions. Pulsed electron emitters offer a solution to this problem, provided that electron beam currents are sufﬁciently high. In this case, high spatial and temporal resolution, as well as high repetition rates, can be achieved. Electron optical parameters such as energy width and beam convergence are compromised by this need for very high beam densities in short electron pulses. Unfortunately, high current densities result in a large Boersch effect, which reduces spatial and temporal coherence signiﬁcantly. We deduce from the spatial coherence–damping envelope of the CTF [Eqs. (4) and (5)] that correction of spherical aberration can compensate the inﬂuence of the large values of semi-convergence that are inevitable when high beam currents are required. Furthermore, these high beam

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currents increase the energy width of the electron emitter via the Boersch effect. Therefore, both spherical and chromatic aberration correction are necessary for an ultra-fast pulsed TEM to achieve optimum contrast and high resolution. Figure 9 shows CTFs for a medium acceleration voltage microscope (200 kV) with a lens suitable for in situ experiments (see Figure 6). The coefﬁcient of spherical aberration and of chromatic aberration was assumed to be 2 mm. Large values for semi-convergence (5 mrad) and energy width (3 eV) have been used to take the high beam current required into account. As seen in Figure 9, contrast-transfer of the uncorrected lens at Scherzer defocus is not sufﬁcient for high-resolution imaging. Only a very small range of spatial frequencies around 4 Å is transferred with more than 10% contrast. Defocusing shifts this small range of contrast-transfer to other spatial frequencies but does not give access to a broader range. Cs correction offers a signiﬁcant improvement by broadening this range so that it extends 10 Å up to 3 Å. High-resolution imaging is still very limited due to contrast damping by the large values for energy width and chromatic aberration mentioned above. Cs and Cc correction offer a solution by providing a signiﬁcantly higher contrast-transfer between 1 and 3 Å. A common feature of the uncorrected and corrected TEMs shown in Figure 1 is the weak contrast 1.00 0.80

Uncorrected

0.60 0.40

C s corrected

C s & C c corrected

PCTF

0.20 0.00 20.20 20.40 20.60 20.80 21.00 10.0 5.0

3.0

2.0

1.5 g [Å]

1.2

1.0

FIGURE 9 Contrast-transfer functions for an in situ TEM with a pulsed electron source.

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transfer at low spatial frequencies. Combining aberration correction with phase-contrast imaging either by holography or phase plates can solve this problem by achieving a high SNR for structures larger than 1 nm. It should be mentioned that this discussion of ultra-fast imaging is valid only for phase-contrast imaging. Information about larger structures such as particles, grains, and dislocations can be obtained by diffraction or absorption contrast. Ultra-fast pulsed TEM offers a wide range of time scales for in situ TEM experiments (from microseconds to femtoseconds), which is only partially accessible now (LaGrange et al., 2006). The contrast-transfer calculations discussed previously are not valid for the entire range. A reasonable assumption is that atomic-level resolution can be maintained up to a time resolution of microseconds—which will enable investigation of processes such as grain boundary and dislocation movement with a time resolution several orders of magnitude above the video rate. Lorentz microscopy of dynamic magnetic processes, such as domain reversal, requires a time resolution of ∼0.1 ns. The low-ﬁeld objective lens discussed earlier (see Figure 8) enables atomic-level resolution if Cs and Cc correction are applied, but parameters affecting temporal coherence will degrade further for electron pulses of this duration. Electron interaction within an electron pulse is stronger, which increases convergence angle and energy width of the electron beam. Under these circumstances, it can be estimated that spatial resolution <1 nm for this time scale will be very challenging. Nevertheless, a resolution of 1 nm is sufﬁcient for the observation of magnetic structures that require contrast-transfer on a length scale from a few nanometers up to micrometers.

IV. SUMMARY The motivation for the development projects for Cs correction in electron microscopy was the expected improvement in resolution. This improvement could be demonstrated for TEM and STEM. Surprisingly, the more important beneﬁt of Cs correction for many materials science investigations was interpretability. Especially, contrast delocalization effectively prevents structural interpretation at interfaces and defects for uncorrected TEMs. Cs correction can reduce contrast delocalization below the information limit. A similar development can be expected for Cc correction, which is probably more effective in improving resolution than Cs correction because it inﬂuences the damping envelope of temporal coherence, which deﬁnes the information limit. We conclude that Cc correction will have a strong impact on a large variety of topics, such as in situ TEM, analytical electron microscopy, and electron microscopy of biological objects.

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ACKNOWLEDGMENTS The authors thank Max Haider, Stefan Uhlemann, and Heiko Mueller for valuable discussions. This work was conducted as part of the TEAM project, a multilaboratory collaborative effort funded by the Department of Energy, Ofﬁce of Science, Basic Energy Sciences.

REFERENCES Basic Energy Sciences Advisory Committee (BESAC). (February 2000). Review of the electron beam microcharacterization centers. Boersch, H. (1953). Energy distribution of thermal electrons from electron beam sources. Die Naturwissenschaften 40, 267–268. Craig, B. R., McVitie, S., Chapman, J. N., O’Donnell, D. O., and Johnston, A. B. (2007). Magnetic reversal behavior of multiscale permalloy elements. J. Appl. Phys. 102, 13911/1–5. Dellby, N., Krivanek, O. L., Nellist, P. D., Batson, P. E., and Lupini, A. R. (2001). Progress in aberration-corrected scanning transmission electron microscopy. J. Electr. Micros. 50, 177–185. Haider, M. (2008). Personal communication. Haider, M., Bernhardt, W., and Rose, H. (1982). Design and test of an electric and magnetic dodecapole lens. Optik 63, 9–23. Haider, M., Rose, H., Uhlemann, H., Schwan, E., Kabius, B., and Urban, K. (1998). Electron microscopy image enhanced. Nature 392, 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. Houdellier, F., Warot-Fonrose, B., Hytch, M. J., Snoeck, E., Calmels, L., Serin, V., and Schattschneider, P. (2007). New electron energy-loss magnetic chiral dichroism (EMCD) conﬁguration using an aberration-corrected electron microscope. Microsc. Microanal. 13(Suppl 3), 48–49. Ishizuka, K. (1979). Contrast transfer of crystal images in TEM. Ultramicroscopy 5, 55–65. Jia, C. L., and Urban, K. (2004). Atomic-resolution measurement of oxygen concentration in oxide materials. Science 303, 200–2004. Kabius, B., Haider, M., Uhlemann, S., Schwan, E., Urban, K., and Rose, H. (2002). First application of a spherical-aberration corrected transmission electron microscope in materials science. J. Electr. Microsc. 51, 51–58. Kirkland, A., Chang, L. Y., Haigh, S., and Hetherington, C. (2008). TEM without aberrations: applications to materials science. Curr. Appl. Phys. 8, 425–428. LaGrange, T., Campbell, G. H., Colvin, J. D., Reed, B., and King, W. E. (2006). Nanosecond time-resolved electron diffraction studies of the α → β in pure Ti thin ﬁlms using the dynamic transmission electron microscope (DTEM). J. Mater. Sci. 41, 4440–4444. Lentzen, M., and Urban, K. (2006). Contrast transfer and resolution limits for sub-Ångström high-resolution transmission electron microscopy. Microsc. Microanal. 12(Suppl 2), 1456–1457. Lentzen, M., Jahnen, B., Jia, C. L., Thust, A., Tillmann, K., and Urban, K. (2002). High-resolution imaging with an aberration-corrected transmission electron microscope. Ultramicroscopy 92, 233–242. Lichte, H. (1991). Optimum focus for taking electron holograms. Ultramicroscopy 38, 13–22. Mantl, S., Zhao, Q. T., and Kabius, B. (1999). Self-assembled patterning of ultrathin silicides by local oxidation. MRS Bull. 24(8), 31–5. Petford-Long, A. K., Kohn, A., Bromwich, T., Jackson, V., Castaño, F., and Singh, L. J. (2006). Application of TEM to the development of information-storage materials. Thin Solid Films 505, 10–15.

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Ringnalda, J., Freitag, B., Hubert, D., and Stekelenburg, M. (2007). Practical electron microscopy with correctors and monochromators. Microsc. Microanal. 13(Suppl 2), 860–861. Rose, H. (1990). Outline of a spherically corrected semiaplanatic medium-voltage transmission electron microscope. Optik 85, 19–24. Rose, H. (2008). History of direct aberration correction. Adv. Imaging Electr. Phys. 53. Rose, R. (2002). Theory of electron-optical achromats and apochromats. Ultramicroscopy 93, 293–303. Ross, F. M., Kammler, M., Reuter, M. C., and Hull, R. (2004). In situ observations of selfassembled island nucleation on patterned substrates. Phil. Mag. 84, 2687–2697. Scherzer, O. (1936). Über einige Fehler von Elektronenlinsen Über einige Fehler von Elektronenlinsen. Z. Phys. 101, 593–603. Scherzer, O. (1947). Sphärische und chromatische Korrektur von Elektronen-Linsen. Optik 2, 114–132. Scherzer, O. (1949). The theoretical resolution limit of the electron microscope. J. Appl. Phys. 20, 20–29. Song, M., Mitsuishi, K., Furuya, K., Allen, C. W., Birtcher, R. C., and Donnelly, S. E. (2005). Structure variation of nanometre-sized Xe particles embedded in Al crystals. Appl. Surface Sci. 241, 96–101. Stadelmann, P. A. (1987). EMS: a software package for electron diffraction analysis and HREM simulation in materials science. Ultramicroscopy 21, 131–145. TEAM Workshop Report (July 2000). Aberration correction in electron microscopy: materials research in an aberration-free environment. Available at: http://ncem.lbl.gov/ team/TEAM%20Report%202000.pdf. Thesen, A. E., Matijevic, M., and Benner, G. (2005). Design features and ultimate performance of an ultra-high resolution, aberration-corrected, monochromatized 200 kV FEG-TEM. Microsc. Microanal. 11(Suppl 2), 2144–2145. Uhlemann, S. (1995). Aufbau und Leistungsgrenzen korrigierter analytischer TransmissionsElektronenmikroskope. Ph.D. thesis, University of Darmstadt. van Benthem, K., Lupini, A. R., Oxley, M. P., Findlay, S. D., Allen, L. J., and Pennycook, S. J. (2006). Three-dimensional ADF imaging of individual atoms by through-focal series scanning transmission electron microscopy. Ultramicroscopy 106, 1062–1068. 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

8 Aberration-Corrected Imaging in Conventional Transmission Electron Microscopy and Scanning Transmission Electron Microscopy Angus I. Kirkland*, Peter D. Nellist, Lan-Yun Chang, and Sarah J. Haigh

Contents

I Introduction II The Wave Aberration Function A Measurement of the Wave Aberration Function in CTEM B Measurement of the Wave Aberration Function in STEM III Coherence Effects in CTEM and STEM A Coherent Image Formation Under Partially Coherent Illumination B The STEM Effective Source/CTEM Effective Detector C Partial Coherence for Incoherent Imaging IV Aberration-Correction Imaging Conditions A CTEM Imaging Theory B High-Resolution STEM Imaging C Aberration-Corrected Imaging Conditions in CTEM D Optimal Conditions for Aberration-Corrected STEM V Conclusions Acknowledgments References

284 286 289 295 297 297 305 306 308 308 311 312 315 319 320 320

* University of Oxford, Department of Materials, Parks Road, Oxford, OX1 3PH, United Kingdom.

Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01008-2. Copyright © 2008 Elsevier Inc. All rights reserved.

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I. INTRODUCTION Since its invention in the early 1930s (Knoll and Ruska, 1932a,b,c; Ruska, 1934) the conventional transmission electron microscope (CTEM) in various forms has evolved into a sophisticated characterization tool capable of atomic-resolution imaging (Smith, 1997; Krakow, Smith, and Hobbs, 1984; Kirkland, Hutchison and Chang, 2008). The somewhat later development of the scanning transmission electron microscope (STEM) provided complementary imaging modes to those available in the CTEM together with spectroscopic data at the atomic level (Crewe et al., 1968a,b). Both types of instruments have made many contributions to key problems in materials science, physics, and chemistry (Spence, 2002) and instrument development for CTEM and STEM also supports a substantial commercial industry (Hall, 1966; Hawkes, 1985; Fujita, 1986). Numerous CTEM and STEM studies covering a wide range of materials systems, including bulk semiconductors (Smith and Lu, 1991; Smith, Glaisher, and Lu, 1989); defects (Olsen and Spence, 1981; Voyles et al., 2002) and interface structures; (Bourret, Rouviére, and Penisson, 1988; Gutekunst, Mayer, and Rühle, 1994; McGibbon, Pennycook, and Angelo, 1995; Wallis et al., 1997); metals and alloys (Penisson, Nowicki, and Biscondi, 1988; Krakow, 1990; Ishida et al., 1983; Amelinckx, Milat, and van Tendeloo, 1993; Thomas, 1962); and ceramics, particularly oxides (Lundberg, Hutchison, and Smith, 1989; Buseck et al., 1989; Nakagawa, Hwang, and Muller, 2006), have been reported in a vast literature (for additional general reviews, see Buseck, Cowley, and Eyring, 1989; Smith, 1997; van Tendeloo, 1998; Spence, 1999; Pennycook, 2002; Nellist, 2008; and Kirkland Hutchison, and Chang, 2008). More recently CTEM and STEM have become essential tools in the characterization and discovery of nanoscale carbon structures (for example, see Wang and Hui, 2003). The primary beneﬁt of CTEM and STEM in all of these applications is their ability to record real-space images of atomic conﬁgurations, particularly at localized structural irregularities and defects, that are largely inaccessible to broad-beam X-ray and neutron diffraction methods. Developments in instrumentation for CTEM and STEM over the same time scale have enabled this information to be recorded with increasing resolution and precision, leading to improvements in the quantiﬁcation of the data obtained (Van Aert et al., 2005, 2007). Historically, the resolving power of the electron microscope rapidly matched and then exceeded that of the optical microscope (Ruska, 1934). However, further resolution improvements in CTEM and STEM proved relatively slow because of the need to identify and overcome various instrumental limitations. In particular, at the turn of the last century, the unavoidable spherical aberration present in round electromagnetic objectives lenses (Rose, 1990; Scherzer, 1947; Rose, 1981; Crewe, 1980,

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1982) proved to be an insurmountable barrier to resolution improvement beyond the 0.1-nm level (O’Keefe et al., 2001; Phillip et al., 1994; Nellist and Pennycook, 1998). In 1997 initial results from electron optical elements designed for direct correction of spherical aberration were reported for both CTEM and STEM. Commercial versions of these prototype instruments using varying arrangements of multipole electron optical elements for CTEM (Uhlemann and Haider, 1998; Haider et al., 1998a,b, 1995) and STEM (Krivanek, Dellby, and Lupini 1999; Batson, Dellby, and Krivanek, 2002; Dellby et al., 2001) have since been constructed that are now capable of resolutions below 0.1 nm at intermediate voltages and as such have provided new insights into many materials problems (Urban, 2008). This current generation of corrected CTEM and STEM instruments is capable of direct correction of the coefﬁcients in the wave aberration function to third-order (see later text) with partial compensation of selected higher-order terms also possible (see, for example, Hartel et al., 2004, which describes this using a hexapole-based corrector). At the time this chapter was written, initial CTEM and STEM results from second-generation corrected electron optical systems capable of ﬁfth-order correction had been reported (Haider et al., 2007; Krivanek et al., 2007) targeting a resolution of 50 pm. In parallel with the development of electron optical correctors, absolute resolution limits also have been improved by reducing instabilities in the high voltage and objective lens supplies (O’Keefe et al., 2001) and through careful mechanical column design and detailed consideration of environmental conditions within the microscope room (Muller et al., 2006). Of particular note are developments in monochromators for the electron beam (Mook and Kruit, 1999; Rose, 1999; Batson, 1986), which provide a reduced energy spread, thus further improving CTEM resolution and enabling enhanced spectroscopy and spectroscopic imaging. Most recently, complex designs for chromatic aberration correctors have been proposed that promise further improvements in both image resolution and spectroscopic capability (Hawkes, 2004; Rose, 2004). Overall, with these developments image resolution is expected to extend to ∼50 pm with simultaneous sub–0.1-eV spectroscopic data. This chapter reviews and compares the operation of aberrationcorrected instruments in both the CTEM and STEM conﬁgurations from a theoretical perspective. In addition to correcting the spherical aberration, electron optical correctors provide control over many other aberrations and may actually introduce certain higher-order aberrations if not optimized. Hence, achieving the highest possible resolution requires careful adjustment of the corrector settings, which are detailed for the CTEM and STEM geometries. Understanding this optimization also requires knowledge of the effects of partial coherence on the imaging process and a comparison of coherence effects in CTEM and STEM is also provided.

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II. THE WAVE ABERRATION FUNCTION The key electron optical component affecting image formation in CTEM and STEM is the objective lens. This section reviews its inﬂuence in terms of the wave aberration function (see also Hawkes and Kasper, 1996). For the ideal case of a point object, P0 , imaged using a perfect lens, the wave at the back focal plane of the lens is spherical and converges to point in the image plane, PiG (Figure 1). However, all electron lenses are aberrated such that this spherical wavefront is perturbed and for real lenses, therefore, a ray intersects the image plane at a point Pi (Figure 1) from which the image aberration is deﬁned by δr = ri − riG . It should be noted that the image aberration for defocus is linear in angle and is hence described as a ﬁrst-order aberration, whereas the image aberration for spherical aberration is cubic in angle and is hence described as a third-order aberration. This image aberration is the differential of the wave aberration function, W(k), which is deﬁned as the distance between the aberrated and ideal wave surfaces in the diffraction plane. In general, the wave aberration function is a function of the complex reciprocal space vector k = kx + iky , position in the image plane r, and the energy of the electrons (E0 ). However, for a well-aligned ﬁeld emission source the energy-dependent terms can be ignored (the monochromatic approximation), and at high magniﬁcation where the ﬁeld of view is small, the dependence of W on r also can be neglected in the isoplanatic approximation. The dependence of W on r can equally be neglected in Ideal wave surface Real wave surface

P0 r0

riG

ri PiG Pi

FIGURE 1 Image formation in a real lens system showing the wave aberration and image aberration functions defined in the text. (Adapted from Typke and Dierksen, 1995.)

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STEM because of the small probe dimensions. Furthermore, in STEM the double deﬂecting coils used for scanning the probe usually are placed between the corrector and the objective lens, which means that for all probe positions, the beam is axial in the corrector and hence off-axial aberrations are not observed even at low magniﬁcations (large scan areas). Using both these approximations, a Taylor expansion to sixth-order in k about the origin of zero scattering angle gives the wave aberration function in terms of the coherent aberration coefﬁcients as follows:

1 1 W(k) = {C0,1 λk∗ + C1,2 λ2 k∗2 + C1,0 λ2 k∗ k 2 2 1 1 + C2,3 λ3 k∗3 + C2,1 λ3 k∗2 k 3 3 1 1 1 + C3,4 λ4 k∗4 + C3,2 λ4 k∗3 k + C3,0 λ4 k∗2 k2 4 4 4 1 1 1 + C4,5 λ5 k∗5 + C4,3 λ5 k∗4 k + C4,1 λ5 k∗3 k2 5 5 5 1 1 1 + C5,6 λ6 k∗6 + C5,2 λ6 k∗4 k2 + C5,0 λ6 k∗3 k3 6 6 6 1 + C5,4 λ6 k∗5 k} 6

(1)

The nomenclature used to described these aberration coefﬁcients, their order in k, and radial symmetry are given in Table I. This table also compares the notation used in Eq. 1 (Krivanek, Dellby, and Lupini, 1999) with a more traditional form due to Typke and Dierksen (1995).1 The overall effect of these coherent aberrations is to increase the distance traveled by the electron wave and hence lead to a phase change, χ(k)

χ(k) =

2π W(k). λ

(2)

The inﬂuence of lens aberrations in CTEM image formation therefore can be described by multiplying the electron wave in reciprocal space, (k), by a phase factor dependent on the wave aberration function exp[iχ(k)]. 1 The seemingly counterintuitive notation (e.g., C 1,0 for a second-order term in W(k)) stems from the

ray-optical theory of Seidel aberrations, which are described in terms of displacements of ray-path intersections with the image plane as a function of r and k. As these displacements are proportional to the gradient of the wave aberration function, an nth -order Seidel aberration corresponds to a term of order n + 1 in the wave aberration function.

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TABLE I

Aberration Coefficient Nomenclature

Krivanek Typke and Order in Azimuthal notation Dierksen notation k symmetry

C0,1 C1,2 C1,0 C2,3 C2,1 C3,4 C3,2 C3,0 C4,5 C4,1 C4,3 C5,6 C5,4 C5,2 C5,0

A0 A1 C1 A2 B2 A3 S3 C3 A4 B4 D4 A5 R5 S5 C5

1 2 2 3 3 4 4 4 5 5 5 6 6 6 6

1 2 inf 3 1 4 2 inf 5 1 3 6 4 2 inf

Nomenclature

Image shift Twofold astigmatism Defocus (overfocus positive) Threefold astigmatism Axial coma Fourfold astigmatism Axial star aberration Spherical aberration Fivefold astigmatism Fourth-order axial coma Three-lobe aberration Sixfold astigmatism Fifth-order rosette aberration Fifth-order axial star aberration Fifth-order spherical aberration

Coefﬁcients of the wave aberration function to ﬁfth order under the achromatic and isoplanatic approximations. For each coefﬁcient, both commonly used notations (Krivanek and Typke and Dierksen) are listed.

The wavefunction at the image plane is thus given by

ψi (r) = F −1 [ (k)A(k)exp(iχ(k))]

(3)

with the recorded image intensity, I(r) as

I(r) = ψi (r)ψi∗ (r).

(4)

Similarly, the STEM probe wavefunction can be readily calculated

P(r) = F −1 [A(k)exp(iχ(k))],

(5)

In Eqs. 3 and 5 A(k) is a circular function that describes the size and position of the objective aperture, such that A(k) has value 1 for λk < α and 0 for λk > α, where α is the semi angle subtended by the objective

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TABLE II Accuracy in Aberration Coefficients d (nm)

0.12

0.1

0.08

0.06

1.43 102.9 6557 391800 22480000

1.00 59.52 3162 157500 7528000

0.64 30.48 1295 51600 1973000

0.36 12.86 409.8 12240 351200

Aberration coefficients (nm)

C1,0 , C1,2 C2,1 , C2,3 C3,0 , C3,2 , C3,4 C4,1 , C4,3 , C4,5 C5,0 , C5,2 , C5,4 , C5,6

Accuracy required in measured values of the coherent aberration coefﬁcients (to ﬁfth order) for CTEM imaging with an accelerating voltage of 200 kV, a minimum resolvable distance, d, and a maximum allowable phase shift of π/4.

aperture. (Note that for the STEM geometry, we use the terms objective lens and objective aperture to refer to the main probe-forming lens and beam-limiting aperture, respectively, both of which are located before the specimen). For an uncorrected microscope only aberration coefﬁcients up to thirdorder in Eq. (1) need to be considered because higher-order terms are overwhelmed by the uncorrected positive, third-order spherical aberration. However, for corrected instruments higher-order terms in Eq. (1) are observable. For practical purposes a maximum tolerable value for the aberration coefﬁcients can be deﬁned by limiting the maximum allowable phase shift introduced by each coefﬁcient to π/4 (Table II).

A. Measurement of the Wave Aberration Function in CTEM As noted, only the spherical aberration coefﬁcients (third- and ﬁfth-order) and defocus (which is a free parameter, adjustable by changing the objective lens excitation) are present in a perfect, round electromagnetic lens (Scherzer, 1936). Therefore, all other coherent aberration coefﬁcients in Table I are caused by lens imperfections with higher-order aberration coefﬁcients becoming more important at higher resolutions due to the increasing order of their spatial frequency dependence. Determination of the aberration coefﬁcients of the wave aberration function in CTEM relies on the measurement of one or more possible observables from suitably constrained data sets of images recorded with known illumination tilts. Experimentally, either tilt-induced image displacements (Typke and Dierksen, 1995; Koster et al., 1987, 1989; Koster, 1989; Koster and de Ruijter, 1992; Saxton, 1995) or tilt-induced changes in defocus and twofold astigmatism (Thon, 1966; Krivanek, 1976; Zemlin

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et al., 1978; Zemlin, 1979; Typke and Dierksen, 1995; Saxton, 1995) can be easily observed and both methods have their own merits. Tilting the incident beam direction changes the aberration coefﬁcients through a change in the origin of the Taylor expansion of Eq. (1) (Thust et al., 1996). The wave aberration function at an angle, τ, can be expressed (to ﬁfth-order) in terms of a new set of “tilted aberration coefﬁcients” (denoted with a prime) as

W (k) = W(k + τ) − W(τ) 1 2 ∗2 1 2 ∗ = {C0,1 λk∗ + C1,2 λ k + C1,0 λ k k 2 2 1 3 ∗3 1 3 ∗2 + C2,3 λ k + C2,1 λ k k 3 3 1 4 ∗2 2 1 4 ∗4 1 4 ∗3 λ k k + C3,4 λ k + C3,2 λ k k + C3,0 4 4 4 1 5 ∗5 1 5 ∗4 1 5 ∗3 2 + C4,5 λ k + C4,3 λ k k + C4,1 λ k k 5 5 5 1 6 ∗6 1 6 ∗4 2 1 6 ∗3 3 + C5,6 λ k + C5,2 λ k k + C5,0 λ k k 6 6 6 1 6 ∗5 + C5,4 λ k k} 6

(6)

The relationship between these tilted aberration coefﬁcients and the axial aberration coefﬁcients is given by

1 ∗ 2 2 C0,1 = C0,1 + C1,2 τ ∗ + C1,0 τ + C2,3 τ ∗2 + C2,1 τ + C2,1 τ ∗ τ + C3,0 τ ∗ τ 2 3 3 3 1 3 ∗ 3 + C3,4 τ ∗3 + C3,2 τ ∗2 τ + C3,2 τ + C4,5 τ ∗4 + C4,1 τ 2 τ ∗2 4 4 5 2 ∗ ∗ 3 4 1 1 ∗ 4 + C4,1 τ τ + C4,3 ττ ∗3 + C4,3 τ + C5,6 τ ∗5 + C5,0 τ 3 τ ∗2 5 5 5 2 5 1 1 2 ∗ 5 ∗ ∗ 4 + C5,4 ττ ∗4 + C5,4 τ + C5,2 τ 2 τ ∗3 + C5,2 τ τ 6 6 3 3 2 2 C1,2 = C1,2 + 2C2,3 τ ∗ + C2,1 τ + C3,0 τ 2 + 3C3,4 τ ∗2 + C3,2 ττ ∗ 3 3 2 9 12 ∗ 3 + 4C4,5 τ ∗3 + C4,1 τ + C4,1 τ 2 τ ∗ + C4,3 ττ ∗2 + 5C5,6 τ ∗4 5 5 5 + 2C5,0 τ ∗ τ 3

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4 3 12 C1,0 = C1,0 + { C2,1 τ ∗ + C3,2 τ ∗2 + 2C3,0 τ ∗ τ + C4,1 ττ ∗2 3 2 5 8 + C4,3 τ ∗3 + 3C5,0 τ 2 τ ∗2 } 5 3 3 C2,3 = C2,3 + 4C3,4 τ ∗ + C3,2 τ + 6C4,5 τ ∗2 + C4,1 τ 2 + 10C5,6 τ ∗3 4 5 + C5,0 τ 3 9 18 9 ∗ 2 C2,1 = C2,1 + C3,2 τ ∗ + 3C3,0 τ + C4,1 ττ ∗ + C4,1 τ + 9C5,0 τ ∗ τ 2 4 5 5 4 10 2 C3,4 = C3,4 + 4C4,5 τ ∗ + C4,3 τ + 10C5,6 τ ∗2 + C5,4 ττ ∗ + C5,2 τ 2 5 3 3 8 16 20 16 C3,2 = C3,2 + C4,1 τ + C4,3 τ ∗ + C5,4 τ ∗2 + 4C5,0 τ 2 + C5,2 ττ ∗ 5 5 3 3 12 C3,0 = C3,0 + { C4,1 τ ∗ + 6C5,0 ττ ∗ } 5 5 C4,5 = C4,5 + 5C5,6 τ ∗ + C5,4 τ 6 10 C4,1 = C4,1 + 5C5,0 τ + C5,2 τ ∗ 3 5 C4,3 = C4,3 + C5,2 τ + 5C5,4 τ ∗ 3 C5,6 = C5,6 C5,2 = C5,2 C5,0 = C5,0 C5,4 = C5,4

(7)

Considering Eq. (7) in more detail, it is apparent that the displace − C , taken at beam tilts differing by τ, ment, between two images C0,1 0,1 depends on all the other aberration coefﬁcients. Hence, for a known set of beam tilts, all aberration coefﬁcients can be determined by minimizing the least-square difference between measured and expected image shifts. Experimental image shifts can be determined, even under low-dose conditions, by measuring the location of the peak in the cross-correlation function (XCF) between two images (Typke and Dierksen, 1995). However, since all aberration coefﬁcients change as the beam is tilted, nonlinear phase variations are introduced in addition to the linear phase variation predicted as a result of the tilt-induced change in C0,1 . This distorts the cross-correlation

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peak and is especially serious at high spatial frequencies. The peak distortion can be compensated but this requires certain a priori knowledge of the imaging conditions (Saxton, 1994; Kirkland et al., 1995). Image shift measurements also fail for clean periodic specimens, where positions differing by an integer number of lattice vectors cannot be distinguished. Most seriously for high-resolution application, the tilt-induced displacement required is indistinguishable from any specimen drift. For these reasons, therefore, this approach has generally been limited to microscope alignment at low magniﬁcation or for initial alignment at high magniﬁcation (Pan, 1998). A well-established alternative procedure relies on the measurement of − C , and/or two-fold astigmatism, tilt-induced changes in defocus, C1,0 1,0 C1,2 − C1,2 , from diffractograms (power spectra of the recorded image intensities). Diffractograms are easily calculated from experimental images of small regions of amorphous material, which in most high-resolution CTEM specimens is present either as a carbon support ﬁlm or as a contamination or damage layer (Saxton, 1995; Zemlin et al., 1978; Chand, Saxton, and Kirkland, 1995). However, because carbon is a weak scatterer, achieving the required signal-to-noise ratio (SNR) for accurate measurement is often difﬁcult. Measurement of defocus and two-fold astigmatism from diffractograms can be performed manually (Chand, 1997) or can be automated by matching experimental data to simulations (Baba, Oho, and Kanaya, 1987; Fan and Krivanek, 1990; Uhlemann and Haider, 1998; Zemlin et al., 1978). For high-resolution imaging the major beneﬁts of this method compared to image shift measurement are its insensitivity to specimen drift and high accuracy. For corrected microscopes automated aberration measurement is essential as the non-round lens elements introduce a multitude of high-order aberrations (up to six-fold astigmatism) that require correction in an elaborate alignment procedure. Uhlemann and Haider (1998) have implemented an optimized algorithm that is robust to disturbances from the presence of crystalline material and can evaluate C1,0 and C1,2 in less than 400 ms based on a comparison of experimental diffractograms with an extensive library of precalculated patterns. Given measurements of either tilt-induced image shifts or tilt-induced changes in C1,0 and C1,2 , a least squares ﬁt is generally used to determine the full set of aberration coefﬁcients from a set of suitably conditioned experimental measurements. Where the injected tilt is known, the dependence of the aberrations coefﬁcients on the observables is linear with a unique solution that can be easily found (Saxton, 1995). Alternatively, if the beam tilt coil magnitudes and orientations are unknown, these can be found through an additional nonlinear iterative minimization of the residual misﬁt (Saxton, 1995).

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1. Alternative Approaches to Aberration Measurement in CTEM Given the aforementioned difﬁculties with conventional measurements, a novel approach to aberration measurement has been developed (Meyer, Kirkland, and Saxton (2002); Meyer (2002); and Meyer, Kirkland, and Saxton (2004)) that is less reliant on the presence of amorphous material while retaining sufﬁcient accuracy for high-resolution application. In this approach, a phase correlation function (PCF) (Kuglin and Hines, 1975) is initially calculated (deﬁned by Eq. (8) as the conventional XCF with the modulus set to unity)

PCF(r) = F

−1

' I1∗ (k)I2 (k) , H(k) ∗ |I1 (k)I2 (k)|

&

(8)

where Ii (k) are the image Fourier transforms and H(k) is a rotationally symmetric weighting factor used to suppress high-frequency noise. Modulus normalization is essential as it suppresses the crystal reﬂections that make the conventional XCF periodic, thus allowing aberration measurement in the absence of signiﬁcant amounts of amorphous material or aperiodic features. The PCF calculated between two images recorded at different defocus levels consists of a centrosymmetric ring pattern, the exact form of which depends on the relative defocus between them (Meyer, Kirkland, and Saxton, 2002) (under the assumption that the other aberration coefﬁcients remain constant between the two images). The phase shifts that give rise to this ring pattern can be compensated by the application of a phase factor dependent on the defocus difference, which deﬁnes a phase-compensated PCF (PCFW (r)) denoted by

PCFW (r) = F

−1

cos(χδC1,0 (k))I1∗ (k)I2 (k)

H(k) cos(χδC (k))I ∗ (k)I2 (k) + h 1,0 1

(9)

in which the very small positive number, h, prevents a zero denominator and where χδC1,0 (k) = πδC1,0 λ|k|2 describes propagation from the ﬁrst to the second image in the presence of only a defocus difference δC1,0 . When the value of δC1,0 matches the actual focus difference, the phasecompensated PCF collapses to a sharp localized correlation peak (Figure 2). Hence, the relative defocus difference between two images can be determined by maximizing peak height in the phase-compensated PCF as a function of the compensated defocus difference; and with subsequent reﬁnement, this yields relative defoci to an accuracy better than 1 nm (Meyer, Kirkland, and Saxton, 2002).

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294

(a) 119nm

Correlation peak height

(b)

89nm

79nm

1.0

1.0

0.8

0.98

0.6

0.96

0.4

0.94

0.2

0.92

0

0.9 0

(c)

20

40 60 80 100 120 Defocus difference (nm)

88 nm

(d)

69nm

118 nm

128 nm

0.3 Phase contrast index

0.25 0.2 0.15 0.1 0.05

0 20.05 2300 2200 2100

0

100

200

300

Defocus difference (nm)

FIGURE 2 (a) Phase correlation functions (PCFs) calculated between two images of a crystalline material with varying phase compensation as a function of C1,0 marked. Without phase compensation (left column), the peak is widened into concentric rings due to the focus difference. When the phase compensation for the correct difference in C1,0 , (69 nm) is applied, the PCF collapses into a single sharp peak. (b) Peak height of the PCF between two images as a function of compensated C1,0 difference showing a sharp maximum at the correct compensating focus difference in the PCF. (c) Phase contrast index (PCI) function fPCI . For mismatched values of C1,0 , −88 and −118 nm, fPCI shows dark rings, whereas at the correct value of C1,0 , −128nm. fPCI is close to 1 (white) at all spatial frequencies. (d) The mean phase contrast index FPCI plotted as a function of C1,0 with a sharp maximum at the correct focus value of −128 nm.

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From the relative defocus values determined as above, an initial estimate of the wavefunction ψsi (k), in the plane of a reference image can be determined using standard methods (Meyer, Kirkland, and Saxton, 2002). In the second step of this approach, absolute values of C1,2 and C1,0 are determined from this image wavefunction using a phase contrast index function (PCI) fPCI deﬁned as

fPCI (k, C1,0 , C1,2 ) = − cos(arg(ψsi (k)) + arg(ψsi (−k)) + 2χ(k, C1,0 , C1,2 )),

(10)

where

χ(k, C1,0 , C1,2 ) =

2π 1 1 1 { C1,2 λ2 k∗2 + C1,0 λ2 k∗ k + C3,0 λ4 k∗2 k2 } (11) λ 2 2 4

is the symmetric part of the wave aberration function with trial parameters C1,0 , C1,2 , and ﬁxed spherical aberration C3,0 . When these trial parameters are correct, the fPCI tends to 1 for all spatial frequencies (Figure 2(c)). This general method is best implemented experimentally using a combined tilt-defocus data set geometry described elsewhere (Meyer, Kirkland, and Saxton, 2002; Meyer, 2002) that comprises a short focal series of images recorded at a number of tilt azimuths. This data set is overredundant and can be used to calculate values of C1,0 and C1,2 at each recorded tilt azimuth using the PCF/PCI approach followed by least-squares ﬁtting of these to determine the aberration coefﬁcients.

B. Measurement of the Wave Aberration Function in STEM By reciprocity (Cowley, 1969; Zeitler and Thomson, 1970), position in the STEM detector plane is equivalent to illumination tilt in CTEM. In principle, therefore, the tilt-series approach to aberration measurement used in CTEM could also be implemented in STEM by recording images on small detectors at various collection angles. An extension to this approach is to record the entire detector plane intensity as a function of probe position. The intensity observed in the STEM detector plane is usually referred to as the Ronchigram (Lin and Cowley, 1986) and has been used previously for measurement of spherical aberration (Cowley, 1979; Cowley and Disko, 1980; Cowley, 1981; James and Browning, 1999). Figure 3 shows how the apparent magniﬁcation in a Ronchigram depends on the distance between the beam crossover and the sample. In the presence of aberrations, that distance depends on the angle of the beams and the Ronchigram shows an apparent change in magniﬁcation as a function of angle. Ronchigrams therefore provide a sensitive method of measuring aberrations. The method developed by Nion Co. (Dellby et al., 2001) involves moving a probe with a known defocus over a known distance on the sample. The movement observed as a function of position in the Ronchigram

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L

(a)

C1,0 M⫽

Sample

L C1,0

Ronchigram

(b)

(c)

(d)

10 mrad

10 mrad

FIGURE 3 Schematic diagrams of (a) Ronchigram showing uniform magnification, M in the absence of aberrations, (b) Ronchigram in the presence of aberrations causing the magnification to change across the detector. Experimental Ronchigrams; (c) in the presence of C3,0 , (d) following aberration correction.

then provides a measurement of the local magniﬁcation in the Ronchigram. The magniﬁcation is in general not isotropic and is therefore described by a tensor, M that is dependent on the second derivatives of the wave aberration function (Dellby et al., 2001)

⎛

∂2 χ ⎜ ⎜ ∂kx2 M = Lλ⎜ ⎜ 2 ⎝ ∂ χ ∂ky ∂kx

⎞−1 ∂2 χ ⎟ ∂kx ∂ky ⎟ ⎟ . ⎟ ∂2 χ ⎠ ∂kx2

(12)

Hence, given local measurements of the Ronchigram magniﬁcation at a number of different points, Eq. (12) can be used to ﬁt the aberration coefﬁcients in the aberration function polynomial. Alternative aberration measurement methods for STEM also have been reported. One such approach is to tilt the illuminating beam through the corrector while using a relatively small beam-limiting aperture such that different illumination angles are explored. By analogy with the CTEM methods described earlier, the probe displays angle-dependent defocus

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and twofold astigmatism, which can be measured by forming an image of a sample containing small nanoparticles. Hence, from a tableau of such measurements, the coefﬁcients of the wave aberration function can be measured as previously described. Both this method and the Ronchigram method require speciﬁc samples that are nonperiodic, which may necessitate a sample change. To avoid this step, a method that allows measurements from a crystalline sample has been developed (Ramasse and Bleloch, 2005) that relies only on interpreting Ronchigram contrast along the achromatic lines. However, this has not yet been widely adopted.

III. COHERENCE EFFECTS IN CTEM AND STEM Before the optimum conditions for imaging using aberration-corrected CTEM and STEM instruments can be considered, it is necessary to understand the inﬂuence of partial coherence in both modes. The contrast in a CTEM image arises from the interference of electron waves that have been scattered by the specimen. In STEM, a small probe is formed by the interference of partial-plane waves in the cone of illumination generated by the objective lens. In both cases, the degree to which these waves interfere, or equivalently, the extent of their correlation, is referred to as the degree of coherence (Born and Wolf, 1980). This is therefore a key factor in determining CTEM and STEM image contrast. Using the principle of reciprocity (Cowley, 1969; Zeitler and Thomson, 1970), we can make many links between the effect of coherence on CTEM and STEM. There are three aspects to partial coherence that need to be considered in electron microscopy: partial spatial coherence, arising from the ﬁnite dimension of the electron source (Born and Wolf, 1980); partial spatial coherence, arising from the ﬁnite dimensions of the electron detector and partial temporal coherence, associated with a ﬁnite energy distribution of the source and ﬂuctuations in both the accelerating voltage and the objective lens current (Hanssen and Trepte, 1971). At ﬁrst sight it might seem surprising that the ﬁnite size of the detector needs to be considered as an aspect of partial coherence because it does not affect the degree to which the waves interfere. However, it affects the detection of the results of the interference, if those interference features are smaller than the detector, and hence is included here.

A. Coherent Image Formation Under Partially Coherent Illumination Conventional CTEM imaging is a coherent imaging mode, and although coherent imaging can also be performed in STEM, incoherent imaging modes are more commonly used (Nellist and Pennycook, 2000) (although aberration correction is making STEM coherent imaging more attractive

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as described later). We therefore describe the effect of partial coherence on coherent imaging from the CTEM perspective with references to STEM imaging as appropriate.

1. The Effective Source (CTEM)/Effective Detector (STEM) As for all practical radiation sources (Reimer, 1997), the source in an electron microscope is only partially coherent and the degree to which waves originating from the source interfere is less than ideal. The combined effect of partial coherence is to restrict the amount of information that can be extracted from high-resolution images (Frank, 1973; Wade and Frank, 1977). In an electron microscope, the rate of electron emission from the source is such that the probability of electrons interacting with each other is exceedingly small. Consequently, secondary interference (or antibunching) (Kiesel, Renz, and Hasselbach, 2002; Hanbury-Brown and Twiss, 1956) of the electrons can be ignored by analogy with light optics, where secondary interference (or bunching) of photons also can be ignored if the intensity is low. Hence, the statistical properties of a low-intensity electron beam can be treated within a framework similar to that appropriate for a low-intensity photon beam, justifying the application of the theory of partial coherence as applied to light to electron microscopy (Ferwerda, 1976; Hawkes, 1978). Various electron microscopes have been constructed with different optical designs for the illumination system (Reimer, 1997). However, all of these have in common an electron source, such as a thermionic ﬁlament (Broers, 1967; Reimer, 1997) or ﬁeld emitter (Otten and Coene, 1993; Honda et al., 1994), and a multiple-element condenser lens system. Hence, calculation of image intensities requires the propagation of the electron wave from the source through the condenser lens system to the specimen entrance plane. The concept of an effective source provides a convenient way to calculate image intensities (Hopkins, 1951, 1953) within the above model for the illumination system. It is assumed that the illumination system is equivalent to an extended quasi-monochromatic, incoherently ﬁlled source with the same size and intensity distribution as that of the ﬁnal exit pupil of the illumination system. This effective source is located at the plane of the ﬁnal condenser lens aperture, which is regarded as being coincident with the illuminating aperture (Spence, 1988). The validity of this approximation has been investigated previously and has been shown to provide an adequate description for CTEM under both Koehler and Critical illumination conditions (Ferwerda, 1976; Saxton, 1977). By reciprocity the STEM equivalent of the CTEM effective source size is the angle subtended by the detector (Nellist and Rodenburg, 1994). In this case, it is clear that electrons arriving at different locations on a detector are added in intensity and any coherence between them is hence lost.

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2. The Transmission Cross-Coefficient For CTEM and STEM coherent imaging, it is possible to describe the effects of partial spatial coherence and partial temporal coherence in terms of a transmission cross-coefﬁcient (TCC). Figure 4 shows a simpliﬁed schematic ray diagram of an electron microscope, in which the illumination system is represented by an extended quasi-monochromatic incoherent effective source, S (Hopkins, 1953) illuminating the object plane, O. The mutual coherence function, o (ro , r o , t1 ), describes the correlation of wave functions at two points, ro and r o , in this object plane. To obtain the intensity in the image plane, I, the mutual coherence function is transmitted through the object function O(ro , t1 ) and subsequently propagated through the optical system, K (in this case only the objective lens), to points ri and r i in the image plane. The image intensity at a point, ri , is therefore equal to the mutual coherence function at ri = r i . In general the mutual coherence function, (r, r , t1 ), is given by the time average of wavefunctions at two different points (r and r’) with a time difference t1 between them as

(r, r , t1 ) ≡ ψ(r, t)ψ∗ (r , t − t1 ).

(13)

Under the isoplanatic approximation (as used previously in the description of the wave aberration function), the mutual coherence function in the image plane, i (ri , r i ; ω), expressed in the frequency domain, can be written as

i (ri , r i ; ω) =

∗ − o (ro − ro ; ω)O(ro , ω)O (ro , ω)

× K(ri − ro ; ω)K ∗ (r i − r o ; ω)dro dr o .

(14)

where − 0 (r0 , r0 , ω) is the mutual coherence function at the entrance surface of the object also expressed in the frequency domain.

Optical System, K

rO

S r⬘O

Source plane, S

Object plane, O

ri r⬘i

Image plane

FIGURE 4 Simplified schematic ray diagram illustrating the propagation of the mutual coherence function where the illumination system is represented by an extended quasi-monochromatic incoherent effective source, S, illuminating the object plane, O.

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The intensity, I(ri )in the image plane, is then obtained by substituting r i = ri and integrating over frequencies ω. If it is further assumed that the object function is independent of the illuminating source temporal frequency (or the source energy through E = ω), the object can be separated from all the other functions describing the microscope. Under these conditions the Fourier transform of the image intensity, I(k), is given by,

I(k) =

O(k1 )O∗ (k2 ) ∗ × − o (τ, ω)K(k1 + τ; ω)K (k2 + τ; ω) dτdωdk,

(15)

with τ deﬁned in Section II.A. The above treatment demonstrates that for a microscope with a coherent source and a perfect optical system, the image intensity is the sum of contributions from all pairs of spatial frequencies in the object function. However, for real optical systems each contribution is further modiﬁed by a function, T, describing partially coherent illumination and an aberrated optical system. This function, T, is known as the transmission cross-coefﬁcient (Born and Wolf, 1980) and is given by

T(k1 , k2 ) =

∗ − o (τ, ω)K(k1 + τ; ω)K (k2 + τ; ω) dτdω.

(16)

If the dimensions of the source and the ﬁeld of view of the object are small compared to the distance between them, then according to the generalized van Cittert–Zernike theorem (Born and Wolf, 1980), the Fourier transform of the mutual coherence function in the object plane equals the spatial and temporal intensity distribution of the source. If the energy distribution of the source is independent of its position in the source plane, the spatial and temporal intensity distributions can be described as their product, whence the TCC reduces to2

T(k1 , k2 ) = =

I(τ, ω)K(k1 + τ; ω)K ∗ (k2 +τ; ω) dτdω

(17)

S(τ)F(ω)K(k1 +τ; ω)K ∗ (k2 + τ; ω) dτdω,

(18)

2 For most electron sources, this is a suitable approximation. However, for monochromated sources

formed using single Wien ﬁlters, this no longer holds; a suitable treatment for this case has been given elsewhere (Lentzen, 2008).

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where S(τ) is the spatial intensity distribution and F(ω) is the energy distribution. In the CTEM, the propagation function due to the optical system, K(ri − ro ), is the product of the Fourier transform of the aperture function and the wave aberration function of the objective lens under the isoplanatic approximation. The Fourier transform of the propagation function, K(k), due to the optical system can therefore be written as

K(k) = A(k) exp(iχ(k)),

(19)

where A(k) is the objective aperture function and χ(k) the phase change due to the wave aberration function of the objective lens deﬁned previously in Eq. 3. For a partially coherent source, the wave aberration function of the objective lens is dependent on both the direction of the incident beam and on the energy of the source. The variation in the energy of the effective source (due to the intrinsic energy distribution of the emitted electrons and instabilities in the accelerating voltage) and ﬂuctuations in the objective lens current result in a change of focus (Hanssen and Trepte, 1971). If the ﬂuctuations in these three factors are assumed to be independent of each other, then according to the central limit theorem the resultant defocus variation can be adequately described as a Gaussian distribution, F( f ),

1 f2 F( f ) = √ exp − 2 ,

π

(20)

in which f is the defocus spread and is the e−1 value of the focal spread distribution given by

1/2 √ σ 2 (V0 ) 4σ 2 (I0 ) σ 2 (E0 )

= 2Cc + + , V02 I02 E20

(21)

where σ 2 (V0 ), σ 2 (I0 ), and σ 2 (E0 ) are the variance in ﬂuctuations of the accelerating voltage, objective lens current, and electron energy, respectively. Similarly if the variation in the incident beam direction, S(τ) can also be described as a Gaussian distribution with an e−1 value, τ0 this yields

S(τ) =

1 πτ02

* exp

−τ 2 τ02

+ (22)

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For dynamical electron scattering, the object function can be represented by the specimen exit plane wave function, ψ. Hence, the image spectrum can be written as

I(k) =

ψ(k1 )ψ∗ (k2 )T(k1 , k2 )dk

(23)

with the TCC at a defocus C1,0 given by

T(k1 , k2 ) =

S(τ)F( f )K(k1 + τ; C1,0 + f ) K ∗ (k2 + τ; C1,0 + f )dτdf .

(24)

Nellist and Rodenburg (1994) have demonstrated that Eq. (24) also applies to coherent STEM imaging; the only difference being that S(τ) is a function describing the shape of the detector elements (usually chargecoupled device (CCD) pixels). This result arises from the equivalence of the CTEM effective source size and the STEM detector size through reciprocity.

3. Numerical Calculation of Image Intensity The terms in Eqs. (23) and (24) can be regrouped and the image intensity (in real space) written as

I(r) =

S(τ)F( f )Icoh (r, τ, C1,0 + f )dτdf .

(25)

This expression for the image intensity is weighted by values of the beam tilt S(τ) and focal spread F( f ) distributions. Importantly, Eq. (25) can be evaluated numerically by summation of suitably weighted image intensities, Icoh over an equally spaced mesh of beam tilts, τ and focal spread values, f . Formally, this can be expressed as

I(r) =

1 1

S(τ)F( f )Icoh (r, τ, C1,0 + f ), hτ2 hf τ

(26)

f

where hτ2 and hf are, respectively, the sampling intervals used for the beam divergence and focal spread distributions. The error in this calculation method is proportional to the sampling interval (Press, 1992) with ﬁner sampling leading to increased accuracy. However, even coarse sampling intervals result in a large total number of sampling points and unacceptably long computation times. Another

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drawback with this method is the introduction of aliasing artifacts in the image intensity due to the use of equally spaced sampling of beam tilts and defoci (Chang, 2004). An alternative numerical approach relies on importance sampling (Press, 1992), which invokes nonuniform rather than random sampling with uniform probability. This approach takes advantage of the fact that the distributions of S(τ) and F( f ) are peaked at certain values. Using importance sampling, the image intensity can be rewritten as (Chang, Meyer, and Kirkland, 2005)

I(r) ≈

1

|ψi (r, τ, C1,0 + f )|2 , Ns

(27)

τ,f

where the sampling points, τ and f , are chosen according to the beam divergence and focal spread distributions given by S(τ) and F( f ), respectively. In addition to its computational efﬁciency, an important advantage of this Monte Carlo integration over equally spaced integration is that due to random sampling any numerical artifacts introduced are random rather than systematic.

4. The TCC for Strongly Scattering Objects Image intensities also can be calculated analytically using Eq. (25). If the beam tilt and focal spread are sufﬁciently small, then Taylor expansion of the lens aberration function can be truncated to ﬁrst order in τ and f as

χ(k1 + τ; C1,0 + f ) − χ(k2 + τ; C1,0 + f ) ≈ χ(k1 ; C1,0 ) − χ(k2 ; C1,0 ) + τ · ∇[χ(k1 ; C1,0 ) − χ(k2 ; C1,0 )] + f ∂C1,0 [χ(k1 ; C1,0 ) − χ(k2 ; C1,0 )],

(28)

where ∇ is the gradient with respect to k and ∂C1,0 is the partial derivative with respect to defocus, C1,0 , only. Inserting Eq. (28) into Eq. (24) gives an expression for the TCC for strongly scattering objects (Ishizuka, 1980) as

T(k1 , k2 ) = exp{i[χ(k1 ; C1,0 ) − χ(k2 ; C1,0 )]} × S(∇χ(k1 ) − ∇χ(k2 ))F(∂C1,0 χ(k1 ) − ∂C1,0 χ(k2 )).

(29)

with the spatial (S) and temporal (F) coherence terms given by

S(k1 , k2 ) = exp{−π2 τ02 (∇χ(k1 ) − ∇χ(k2 ))2 },

(30)

F(k1 , k2 ) = exp{−π2 2 (∂C1,0 χ(k1 ) − ∂C1,0 χ(k2 ))2 }.

(31)

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Combining Eqs. (29–31), the image spectrum for strong objects can be written as

I(k) =

ψ(k1 )ψ∗ (k2 ) exp{i[χ(k1 ; C1,0 ) − χ(k2 ; C1,0 )]}

(32)

× S(k1 , k2 )F(k1 , k2 )dk . Note that Eq. (30) shows a dependence on the slope of the aberration phase shift, χ. In an aberration-corrected instrument this slope will be smaller, reducing sensitivity to the beam divergence caused by the effective source size and it may therefore be possible to increase the effective source size. In STEM, the equivalent conclusion is that coherent imaging can be maintained while using a larger detector, thereby increasing the efﬁciency of coherent imaging in STEM.

5. The TCC for Weakly Scattering Objects If the specimen is assumed to be weakly scattering such that the intensities of the diffracted beams are much weaker than that of the central beam, then interference between the diffracted beams can be ignored. Under these conditions and assuming a symmetric source the TCC, T(k1 , k2 ) is reduced to T(k, 0) and T ∗ (−k, 0) (T(0, k) = T ∗ (k, 0) by deﬁnition). The image spectrum can then be approximated as

I(k) ≈ ψ(k)T(k, 0) + ψ∗ (−k)T ∗ (−k, 0),

(33)

which has a linear relationship to the specimen exit wave function. The TCC at a defocus C1,0 under this linear imaging approximation, T(k, 0), is given as (Frank, 1973; Wade and Frank, 1977)

T(k, 0) = exp{−iχ(k; C1,0 )}S(k, 0)F(k, 0),

(34)

S(k) = exp{−π2 τ02 (C1,0 λk + C3,0 λ3 k3 )2 }

(35)

F(k) = exp{−π2 2 (λk2 /2)2 },

(36)

with

assuming that C1,0 and C3,0 are the only aberration coefﬁcients included. Under this approximation, S and F decrease with increasing spatial frequency and are commonly referred to as coherence envelope functions. Although this approximation to the image intensity is the most computationally efﬁcient approach, it is a poor approximation, particularly at high resolution and for high defocus values; in contrast, numerical

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2Scherzer

25 Scherzer

29 Scherzer

(a)

(b)

FIGURE 5 Comparison of moduli of the TCC calculated as (a) under the linear approximation (b) using numerical integration. TCCs were calculated at 300 kV with C3,0 = 0.6 mm and using e−1 values of the Gaussian beam divergence and focal spread distributions of 0.28 mrad and 4.2 nm, respectively. C1,0 values are indicated. Black and white levels correspond to values of 0 and 1.

integration has general validity (Figure 5). However, calculation of the image intensity using the latter approach involves summation of coherent image intensities over the entire range of applicable beam tilts and defocus values and is therefore substantially more computationally demanding.

B. The STEM Effective Source/CTEM Effective Detector Having treated the effects of the ﬁnite source size in CTEM, we now consider the effect of ﬁnite source size in STEM. In CTEM the objective and other post-specimen lenses magnify the exit-surface wavefunction to form an image, whereas in STEM the principal optics are pre-specimen and act to demagnify the electron source to form a small probe. It is possible to deﬁne an effective source size, SSTEM (r), for STEM as the intensity proﬁle

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of the source accounting for the demagniﬁcation effects of the pre-specimen optics. If SSTEM (r) is much smaller than the diffraction limited extent of the probe, then the probe forming illumination can be regarded as coherent. As SSTEM (r) gets larger, coherence is lost, resulting is a loss of spatial resolution. The effect of a ﬁnite source size can hence be described mathematically as a convolution of the image formed under perfect coherence conditions with the effective source proﬁle

I (r) = I(r) ⊗ SSTEM (r),

(37)

which can be rewritten in reciprocal space in terms of a coherence envelope accounting for the loss of higher spatial frequency transfer

I (k) = I(k)SSTEM (k).

(38)

The above analysis can equally be applied to CTEM images where SSTEM (r) now represents the point-spread function of the detector. A crucial aspect to achieving high resolutions in STEM, therefore, is to ensure that the effective source size is sufﬁciently small, which is equivalent to the condition that the illumination aperture is coherently ﬁlled. Aberration correction results in smaller diffraction-limited probe sizes (resulting from larger illuminating apertures) and hence requires smaller effective source sizes. Achieving sufﬁcient source demagniﬁcation is therefore a challenge in aberration-corrected instruments, particularly for those with Schottky rather than cold ﬁeld-emission sources.

C. Partial Coherence for Incoherent Imaging Section III.A.1 showed that increasing the detector size in STEM (which is equivalent to a larger effective source size in CTEM) leads to a reduction in coherence in the image-forming process. Generally, this reduced coherence might be expected to be detrimental for high-resolution imaging. However, as the detector size becomes much larger than the beam convergence angle in STEM, the image still contains high spatial resolution information but can be treated as being incoherent (Nellist and Pennycook, 2000; Jesson and Pennycook, 1993; Hillyard and Silcox, 1995). In this regime, the image intensity is given as a simple convolution between the intensity of the illuminating STEM probe, P(r) deﬁned in Eq. 5 and a real-positive object function as

I(r) = |P(r)|2 ⊗ O(r).

(39)

Incoherent imaging in STEM is almost always implemented by use of an annular dark-ﬁeld (ADF) detector that is signiﬁcantly larger than the

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probe convergence angle. ADF-STEM imaging often is conﬁgured such that only high-angle scattering is detected, which is dominated by thermal diffuse scattering further enhancing the coherence breaking effect. A particularly useful aspect of ADF-STEM imaging is the strong sensitivity to atomic number (Z-contrast) and the absence of contrast reversals as the thickness of the samples changes; the latter being a result of the incoherent imaging process. Despite its description as incoherent, this imaging process still requires coherence in the pre-specimen optics. The convergent beam forming the probe must still interfere to form a compact probe, and partial coherence in the source inﬂuences this. The effect of partial source spatial coherence in STEM was described in Section III.B and results in the recorded image intensity being convolved with the effective source proﬁle. However, partial chromatic coherence has a somewhat different effect on the probe. The effect of partial temporal coherence on incoherent imaging can be modeled by integrating the probe intensity over the defocus spread. Nellist and Pennycook (1998) have shown that the effect of chromatic defocus spread on the probe intensity proﬁle is to reduce intensity in the sharp central maximum and to redistribute it as a broad intensity skirt around the probe (Figure 6). Although the width of the central maximum is not signiﬁcantly broadened, and hence spatial resolution is not reduced, the loss of intensity to the skirt results in loss of image contrast. This effect on the probe can be expressed in reciprocal space in terms of the optical transfer function (OTF) (Nellist and Pennycook, 1998). Partial temporal coherence suppresses the mid-range spatial frequencies whilst allowing the very low and high spatial frequencies to pass unattenuated. Krivanek et al. (2003) have noted that the above redistribution of intensity is problematic for analytical STEM. The fraction of the probe intensity shifted from the probe maximum is given approximately by

fs = (1 − w)2 ,

(40)

w = 2dg2 E0 /(δECc λ) or w = 1, whichever is smaller,

(41)

where

and where dg2 is the theoretical probe size in the absence of chromatic aberration. Because the probe size is dependent on the maximum convergence semiangle, α, Eq. (41) provides a criterion for the maximum value of α that can be used in the presence of partial chromatic coherence. As α is increased, dg decreases and thus w also decreases and more intensity is

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(a) Intensity (arb. units)

300

200

100

0 0

0.1

0.2

0.3

nm (b) Intensity (arb. units)

300

200

100

0 0

0.1

0.2

0.3

nm

FIGURE 6 Probe intensity profile plots for a 100 kV microscope limited by C5,0 = 100 mm (a) without and (b) with defocus spread of 7.5 nm. The width of the main peak of the probe is not greatly affected, but intensity is lost from the central maximum into a diffuse skirt around the probe.

lost. As will be shown in Section IV.B, this restriction on the convergence angle can affect the balancing of aberrations for optimum performance in an aberration-corrected STEM.

IV. ABERRATION-CORRECTION IMAGING CONDITIONS This section reviews optimum imaging conditions for uncorrected CTEM and STEM. Subsequently, conditions for aberration-corrected imaging in both modes are described. Finally, an account of optimum aberrationcorrected imaging conditions, incorporating higher-order aberrations appropriate to the next generation of corrected instruments, is provided.

A. CTEM Imaging Theory In conventional phase-contrast CTEM imaging, image intensities include the effects of phase shifts arising from the aberrations of the objective

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lens. Consequently, high-resolution images often show contrast reversals and delocalization and can be difﬁcult to interpret except under speciﬁc imaging conditions. Therefore, to assist image interpretation or to minimise delocalization in CTEM images, several imaging conditions have been analyzed that rely on adjustment of the objective lens defocus to partially compensate phase shifts related to third-order spherical aberration.

1. The Scherzer Defocus For a thin specimen satisfying the weak phase object approximation (Spence, 2002), CTEM images can be interpreted using a phase-contrast transfer function (PCTF), sin χ(k) (Scherzer, 1949). The Scherzer defocus setting (Scherzer, 1949) maintains the widest band in this PCTF without phase reversals and hence gives a constant phase shift over the widest range of spatial frequencies. Considering only C1,0 and C3,0 , the Scherzer defocus is deﬁned as3

C1,0 = −(C3,0 λ)1/2 ,

(42)

with a corresponding point resolution deﬁned as the spatial frequency of the ﬁrst zero crossover in the PCTF at this defocus of

d = 0.707(C3,0 λ3 )1/4 .

(43)

2. Minimum Phase Contrast An alternative defocus setting to that in the previous section can also be deﬁned in which phase contrast is minimized with amplitude contrast maximized (Heinemann, 1971). For this condition the defocus value is given as

C1,0 = −0.5(C3,0 λ)1/2 ,

(44)

with a corresponding resolution deﬁned as the spatial frequency of the ﬁrst zero crossover of the amplitude contrast transfer function, cos χ(k) at this defocus of

d = (C3,0 λ3 )1/4 .

(45)

3 This original deﬁnition of Scherzer defocus is used in this chapter. A commonly used alternative deﬁni tion is C1,0,Scherzer = −1.2 C3,0 λ, in which the passband of the PCTF contains a weak local minimum deﬁning a slightly better point resolution, d = 0.65(C3,0 λ3 )1/4 .

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3. The Lichte Defocus Lichte (1991) deﬁned a third defocus setting that yields the maximum amount of retrievable object information in a high-resolution hologram provided that the lens aberrations are corrected. This condition also deﬁnes imaging conditions for minimum delocalization, given as the radius of the point spread function, R, as

∂χ R = max , ∂k

0 ≤ k ≤ kmax ,

(46)

where kmax is the information limit and where the lens aberration function χ includes only C1,0 and C3,0 . For high-resolution holography optimization of the lateral coherence width, the effective width of the hologram, and the precision of any numerical (aberration correction) phase plate applied are achieved for this condition by choosing C1,0 such that the contrast delocalization is minimized. These constraints deﬁne a defocus setting

C1,0 = −0.75C3,0 λ2 k2max ,

(47)

with the corresponding minimized delocalization given by

R = 0.25C3,0 λ3 k3max .

(48)

Table III lists defocus values and corresponding resolution limits for a 200 kV TEM with a C3,0 value of 1 mm and an information limit of 7.7 nm−1 . It is clear that the Scherzer defocus setting provides the best point resolution. However, for uncorrected instruments this point resolution is worse than the information limit set by partial spatial and temporal coherence. At the minimum phase-contrast setting, the phase shift due to C1,0 compensates C3,0 at high spatial frequencies, but the point resolution is worse than that deﬁned by the Scherzer condition. TABLE III Defocus Values and Corresponding Resolution Limits

Defocus (nm) Point Resolution (nm)

Scherzer defocus

Minimum contrast

Lichte defocus

50.08 0.25

25.04 0.35

279.66 1.8

Values of defocus and their corresponding resolution for the three commonly used defocus settings in a conventional TEM at 200 kV, with C3,0 = 1 mm and an information limit of 7.7 nm−1 .

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The Lichte defocus provides reduced delocalization, which is achieved at the expense of extensive phase-contrast reversals, particularly at higher spatial frequencies.

B. High-Resolution STEM Imaging Although it is possible for STEM to provide coherent imaging modes analogous to those of the CTEM through the principle of reciprocity (Cowley, 1969; Zeitler and Thomson, 1970), the commonly used ADF mode is best treated using an incoherent imaging model as previously described. In practice, these imaging modes are partially coherent (Oxley, Cosgriff, and Allen, 2005; Jesson and Pennycook, 1993), but even accounting for this it is still reasonable to assume that optimal STEM images are formed by focusing the electron beam into the smallest probe possible. If perfect source coherence is assumed, then the probe shape is easily calculated from the wave aberration function (Eq. (5)). The smallest probe is formed by minimizing the wave aberration function, χ(k), over a largest extent of the probe-forming aperture possible. This condition is similar to that for optimal amplitude contrast in CTEM. In the presence of aberrations, a useful criterion is to assert that the wave aberration function must not make excursions of more than π/2 peak to peak over the extent of the objective aperture, thereby ensuring that there is no degree of destructive interference in the partial plane waves forming the probe. As an example, in an uncorrected microscope with signiﬁcant positive C3,0 , defocus, C1,0 , can be used to partially compensate for the spherical aberration, yielding a defocus value of 1/2

C1,0 = −0.71λ1/2 C3,0 ,

(49)

which allows an objective aperture radius of −1/4

α = 1.3λ1/4 C3,0

(50)

and a probe full width at half-maximum (FWHM) of 1/4

d = 0.4λ3/4 C3,0 .

(51)

Note that these values have the same form as those previously described for CTEM imaging but with different constant pre-factors. Including the effect of a ﬁnite source size does not affect this optimization because Eq. (39) shows that it is still desirable to minimize the diffraction (and therefore aberration) limited extent of the probe. Thus, although the coherence width of the illuminating beam may not ﬁll the objective

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aperture, it is still important to use the maximum aperture size permitted by the limiting aberrations. The conclusion of the preceding paragraph does not apply to conditions of partial temporal coherence. Since high-resolution STEM instruments require a high-brightness ﬁeld emission source, the C3,0 limited aperture is usually too small for temporal coherence effects to become signiﬁcant. However, in aberration-corrected instruments this is not necessarily the case.

C. Aberration-Corrected Imaging Conditions in CTEM The information limit in aberration-corrected CTEM is restricted by the effects of partial coherence. Hexapole-based correctors (Haider, Braumshausen, and Schwan, 1995) contribute a small positive value to the Cc of the objective lens (due to the presence of round transfer lenses), and thus corrected instruments generally have a poorer information limit than equivalent uncorrected instruments. For a resolution limit less than 0.1 nm, the settings given in the previous sections are still applicable to an aberration-corrected microscope. However, when C3,0 is an adjustable parameter, these must be re-evaluated (O’Keefe, 1998; Lentzen et al., 2002; Lentzen, 2004; Chang et al., 2003, 2006; Scherzer, 1949). This creates several new imaging conditions speciﬁcally applicable to aberrationcorrected CTEM.

1. C5,0 -Limited Imaging For current corrected CTEM instruments the limiting aberration is C5,0 when the corrector operates as an aplanat (Hartel et al., 2004).4 In the presence of ﬁxed C5,0 , restricting the phase shift due to the aberration function to between 0 and π/2 requires

dχ(k) = 0, dk kξ

at

χ(kξ ) = ξ,

(52)

where ξ = π/2. Considering only the rotationally symmetric aberrations to ﬁfth order, the phase shift due to the aberration function can be approximated as

2π χ(k) = λ

1 1 1 2 3 4 5 6 C1,0 λk + C3,0 λ k + C5,0 λ k . 2 4 6

(53)

4 We note that C 5,0 can be reduced to zero or to negative values at the expense of introducing off-axial

coma. This is particularly useful in STEM where off-axial aberrations are not important

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Optimum choices for C1,0 and C3,0 if C5,0 > 0 are then given by 2 C3,0 = −2.88(C5,0 λ)1/3 ,

C1,0 = 1.56(C5,0 λ2 )1/3 ,

(54) (55)

with a corresponding point resolution

d = (C5,0 λ5 )1/6 /1.47.

(56)

If C5,0 is set to zero (Hartel et al., 2004) or if the information limit is such that the effects of C5,0 are negligible, the above condition reduces to the well-known Scherzer defocus setting (Scherzer, 1949):

C1,0 = −(C3,0 λ)1/2 ,

(57)

with a corresponding point resolution of

√ d = (C3,0 λ3 )1/4 / 2.

(58)

When C5,0 < 0, the signs of C1,0 and C3,0 are inverted.

2. Chromatic Aberration Limited Imaging Since C1,0 , C3,0 , (and C5,0 ) are freely adjustable in a corrected microscope, maximum values of these coefﬁcients can be evaluated such that the ﬁrst crossover of the PCTF coincides with the information limit set by partial coherence (Chang et al., 2006). Matching the ﬁrst zero of the PCTF to the information limit and constraining the phase shift due to the aberration function (considering only C1,0 , C3,0 , and C5,0 ) to between 0 and π/2 yields maximum values of the coefﬁcients for this condition as

C5,0 = 1.3

(π )3 , λ2

C3,0 = −3.4

(π )2 , and λ

C1,0 = 1.7π .

(59) (60) (61)

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1

Sin χ(k)

0 20.1

0

5

10

15

Spatial frequency

20

25

30

(nm21

)

FIGURE 7 Phase-contrast transfer functions of the C5,0 and Cc limited conditions for aberration-corrected CTEM at 200 kV, represented by the solid line and the dashed line, respectively. The dotted line shows the temporal coherence envelope function for a focal spread of 0.6 nm, corresponding to an information limit of 20.56 nm−1 . Aberration coefficients for the C5,0 limited conditions are C5,0 = 2 mm C3,0 = −6.2 μm and C1,0 = 4.14 nm, and for the Cc limited conditions are C5,0 = 1.34 mm, C3,0 = −4.8 μm, and C1,0 = 3.18 nm.

By analogy with the C5,0 -limited condition, if C5,0 is set to 0 or its effect is insigniﬁcant, the choice of C1,0 and C3,0 reduces to5

C3,0 =

(π )2 , λ

C1,0 = −π .

(62) (63)

Figure 7 compares PCTFs for C5,0 - and Cc -limited conditions at an information limit of 20 nm−1 . Both contrast transfer functions extend to the information limit, but the point resolution given by the C5,0 -limited condition is slightly higher than that of the Cc -limited condition.

3. Amplitude Contrast Imaging If both C3,0 and C5,0 are set to 0, there is no contrast in images of a pure phase object at Gaussian defocus. However, most materials are not pure phase objects, and amplitude contrast for a complex object is preserved 5 This condition is essentially the same as that given by O’Keefe (2000).

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under this zero-aberration condition, providing high-resolution amplitude contrast imaging in a mode that is not available in an uncorrected instrument. In practice, it may be difﬁcult to adjust C5,0 , C3,0 , and C1,0 to exactly 0. Hence, as a practical alternative for amplitude contrast imaging, the transfer function can be adjusted so that it remains close to unity, up to the information limit, kmax (Foschepoth and Kohl, 1998). Considering only C1,0 , C3,0 , and C5,0 in an aberration-corrected CTEM, for maximized amplitude contrast these are set to

C1,0 = (1/5)C5,0 (λkmax )4

(64)

C3,0 = −C5,0 (λkmax )2 .

(65)

D. Optimal Conditions for Aberration-Corrected STEM We have shown that uncorrected spherical aberration can be compensated using defocus. With the introduction of an aberration corrector it might be assumed that the third-order aberrations are effectively zero and that accordingly higher-order aberrations are limiting. As an example for the case of the second-generation Nion quadrupole-octupole corrector, the lack of optical coupling between the corrector and the objective lens leads to signiﬁcant ﬁfth-order aberrations C5,0 , C5,2 , and C5,4 (Dellby et al., 2001). Each aberration can be partially compensated by lower-order aberrations with the same rotational order. The round C5,0 aberration and the twofold ﬁfth-order aberration C5,2 can be compensated using a combination of third- and ﬁrst-order aberrations (Scherzer, 1970; Haider and Uhlemann, 2000). However, the four-fold rotational C5,4 can only be compensated with C3,4 because there is no ﬁrst-order aberration with four-fold rotational symmetry. It is now clear that uncorrected C5,4 limits aperture size and therefore resolution more severely than C5,0 and C5,2 . The coefﬁcient C5,6 is even more limiting because it cannot be compensated by any lower-order aberration. Using the criterion that the wave aberration function must not have an excursion greater than π/2 peak to peak allows the following compensations to be used for C5,0 . Identical compensations apply for C5,2 , and hence for n = 0, 2, 2 1/3 C3,n = −2.0(λC5,n )

(66)

C1,n = 0.65(λ2 C5,n )1/3 .

(67)

Figure 8 shows that this compensation process dramatically extends the range over which the wave aberration function, χ, has low excursions,

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Uncompensated Compensated

χ(k)

0

21

0

10 20 Convergence angle (mrad)

30

FIGURE 8 The aberration phase shift, χ(k), for 100-keV electrons and C5,0 of 100 mm with and without compensation by lower-order aberrations. The effect of fifth-order compensation is to allow much larger objective apertures to be used.

allowing a maximum aperture angle of

1/6 α = 1.5 λ1/6 /C5,n ,

(68)

yielding a resolution of 1/6

d = 0.41λ5/6 C5,n .

(69)

Recalling that C5,4 can only be compensated by a third-order aberration, we can show that 2 1/3 C3,4 = −1.1(λC5,4 ) ,

(70)

allowing a maximum aperture angle of 1/6

(71)

1/6

(72)

α = 1.4(λ1/6 /C5,4 ), thus yielding a resolution limit

d = 0.44λ5/6 C5,4 .

Because C5,6 cannot be compensated by any lower-order aberration, it leads directly to a maximum aperture angle and resolution limit of 1/6

α = 0.93(λ1/6 /C5,6 ) 1/6

d = 0.66λ5/6 C5,6 .

(73) (74)

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317

Therefore, since C5,6 cannot be compensated by any lower-aberration (due to its high rotational order), its effect on resolution is much more severe than the other ﬁfth-order aberrations. This feature is also of some importance in sextupole-based correctors in which this term is, in principle, the residual uncorrectable aberration. For the second-generation Nion quadrupole-octupole corrector, the residual aberrations lead to a typical resolution (at 100 kV) of 90 pm. The CEOS sextupole-based corrector is limited by C5,6 of approximately 2 mm, leading to a similar resolution limit in STEM. More recently, Nion has developed a third-generation corrector that can correct all ﬁfth-order aberration, and is limited by seventh-order aberrations with the limiting aberration projected to be C7,8 of 1.2 cm (Krivanek et al., 2007). Similarly, CEOS has worked to reduce the C5,6 value by optimizing sextupole geometries (Müller et al., 2006). It is therefore probable that for the next generation of instruments compensation of the seventh-order aberrations will be required. Using a general formula for the value of the aberration Cp,n required to compensate the seventh-order aberration C7,n , where n is the rotational symmetry of the seventh-order aberration and the compensating lowerorder aberration, shows that [(1+p)/8]

Cp,n = γλ[(7−p)/8] Cq,n

.

(75)

Table IV gives the γ coefﬁcients for the optimal seventh-order compensation. Using these values leads to an expression for the resolution achievable as 1/8

d = ελ7/8 C7,n ,

(76)

with ε equal to 0.35 for C7,0 , C7,2 ; 0.38 for C7,4 ; 0.45 for C7,6 ; and 0.61 for C7,8 . For the third-generation Nion corrector with a limiting C7,8 of 1.2 cm, this leads to a resolution limit of 28 pm at 200 kV, although in practice the resolution will be limited by considerations of coherence to lower values. TABLE IV Aberration Coefficients for 7th Order STEM Compensation

C5,n C3,n C1,n

C7,0 ; C7,2

C7,4

C7,6

−4.27 +4.80 −1.10

−3.0 +1.9

−1.3

Value of γ [Eq. (73)] used to calculate aberration coefﬁcients for optimal seventh-order compensation in STEM.

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1. STEM Aberration Compensation With a Restricted Aperture Size Section IV.D demonstrated that higher-order uncorrected aberrations can be compensated using lower-order ones. In practice, there often is one dominant uncorrected aberration, such as C5,0 in a Nion second-generation corrector or C5,6 in a CEOS sextupole corrector, that limits the objective aperture radius. In addition to this dominant aberration, there will be a range of other signiﬁcant aberrations, including fourth- and sixth-order aberrations, that can be difﬁcult to null. It may also be the case that the maximum objective aperture size is not limited by uncorrected geometric aberrations but is instead limited by partial coherence. As shown in Section III.C for incoherent STEM imaging, the spatial coherence does not change the criteria for corrector adjustment. However, partial temporal coherence and chromatic aberration may require a restricted beam convergence, and the requirement to maintain sufﬁcient intensity in the central probe maximum in the presence of chromatic aberration restricts the maximum size of the objective aperture. Therefore, either a limiting geometric aberration or the result of chromatic aberration deﬁnes the maximum objective aperture size. This alters the criteria for compensating the signiﬁcant higher-order aberrations. Rather than compensation to provide the maximum aperture size, it is now necessary to compensate the aberrations to give a minimum excursion of the wave aberration function within the restricted objective aperture. One approach is to scale the zeros of the aberration phase shift, χ, with the maximum aperture size, α, which can be achieved by scaling the lower-order compensation aberrations by α raised to the power of the difference between the uncorrected residual aberration and the compensating aberration. Formally, this yields

Cp,n = ξp,q,n αq−p Cq,n .

(77)

Table V lists values of ξ for compensating ﬁfth- and seventh-order residual aberrations. As an example of the use of Table V, consider an instrument with a 100-kV beam, an energy spread of 0.5 eV, and a chromatic aberration coefﬁcient of 1.5 mm. It is necessary to limit the probe size to 80 pm and the objective aperture to 28 mrad if we wish to retain more than 70% of the electron beam within the central maximum. Further assuming that there is signiﬁcant C4,3 present, then Eq. (77) can be used to ﬁnd a compensating value of C2,3 ; the resulting aberration phase shift, χ, is shown in Figure 9. However, if the beam energy spread is 1.0 eV, the maximum aperture size halves to 14 mrad, and in this case the compensation can be reduced to minimize the χ excursion.

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TABLE V Aperture Radius Dependent Compensation in STEM C4,3

C4,1

C5,n C3,n C2,n C1,n C0,1

−0.75

−0.52

+0.062

C5,0 ; C5,2

C5,4

C7,0 ; C7,2

C7,4

C7,6

−0.92

−0.60

−1.42 +0.54

−1.19 +0.30

−0.69

+0.14

−0.044

Value of ξ [Eq. (77)] for aperture radius–dependent compensation. Note that the aberration coefﬁcient C0,1 (image shift) is included here, but experimentally it appears automatically and is not explicitly controlled.

1

28 mrad 14 mrad

χ(k)

0

21

0

10 20 Convergence angle (mrad)

30

FIGURE 9 Compensation of C4,3 by C2,3 for aperture radii of 28 mrad and 14 mrad illustrating that when the maximum aperture radius, α, is restricted by chromatic aberrations, then a reduced degree of compensation reduces the excursions of χ(k).

V. CONCLUSIONS The current status of CTEM and STEM is such that instrumentation, theory, and computational tools are available for structure determination of a wide range of technologically important materials at atomic resolution. Enabling this relatively recent advance are multipole-based aberration correctors that currently can correct all third-order aberrations, with initial results from ﬁfth-order corrected CTEM and STEM instruments recently reported. In parallel with instrument development it also has been necessary to expand and reevaluate the theory describing imaging modes, particularly with regard to the effects of partial coherence, and to extend the algebraic framework that describes the optimum adjustment of these complex optical elements. This chapter has provided an overview of the effects of the wave aberration function for both CTEM and STEM imaging and the effects of partial coherence on these modes. In addition, optimum corrector settings for both imaging modes are provided.

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ACKNOWLEDGMENTS We acknowledge ﬁnancial support from the EPSRC (grant Nos. EP/F048009/1 and GR/S83968/01), the Leverhulme Trust, and the John Fell Fund.

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CHAPTER

9 Materials Applications of Aberration-Corrected Scanning Transmission Electron Microscopy 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*,†

Contents

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

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

I. INTRODUCTION For almost the entire history of the electron microscope, spherical aberration has been regarded as the practical limit for microscope resolution, starting with the classic proof by Scherzer (1936) that spherical aberration is unavoidable in a round lens with static ﬁelds and no charges on axis. Furthermore, for most of the entire history of the microscope the physical * Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA † Department of Physics and Astronomy,Vanderbilt University, Nashville, Tennessee 37235, USA ‡ Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge,

Tennessee 37831, USA Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01009-4. Copyright © 2008 Elsevier Inc. All rights reserved.

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principles for correcting spherical aberration have been known. In 1947 Scherzer proposed the ﬁrst design for an aberration corrector based on breaking the rotational symmetry of the system. However, only in the era of fast computers, efﬁcient charge-coupled device (CCD) detectors, and highly stable electronics has it become possible to measure and correct aberrations iteratively and to ﬁnally surpass the limit set by spherical aberration. As a result, in the past few years the ﬁeld of transmission electron microscopy (TEM) has seen a rate of instrumental advance unheard of since the invention of the TEM in the 1930s (Batson et al., 2002; Haider et al., 1998b). Resolution has more than doubled, crossing well into the sub-Ångstrom range (Nellist et al., 2004). This chapter presents ﬁrst a chronological account of the incorporation of Nion aberration correctors into the VG Microscopes HB501UX and HB603U (Vacuum Generators, Ltd., Hastings, Sussex, UK) scanning transmission electron microscopes (STEMS) at Oak Ridge National Laboratory (ORNL), the improvement in resolution, and our realization of the additional beneﬁts of the sharper probe. We show the improved capability for imaging individual heavy atoms on surfaces (Nellist and Pennycook, 1996; Sohlberg et al., 2004) or inside a bulk material (Lupini and Pennycook, 2003; Voyles et al., 2002), which enables spectroscopic identiﬁcation of a single atom (Varela et al., 2004). Subsequently, we present a number of case studies in various areas of materials research in which these instrumental advances have provided new insights or provided a new explanation for an old issue. In many cases, microscopy data are coupled with density functional theory, allowing structures suggested from the microscope to be veriﬁed and reﬁned. Furthermore, theory can quantitatively investigate potential impurity sites and determine segregation energies and activation barriers. The substantial advances in computational power over the past few years now mean that realistic systems requiring hundreds of atoms, such as grain boundaries or nanostructures, can be tackled. The combination of microscopy and theory has therefore become extremely powerful, able to resolve not only long-standing controversies but also the new issues of the nanomaterials era, as shown in the following text. Much of the material has been previously published, and the original papers should be consulted for more details and additional references. Some of the material also has appeared in other review articles (Chisholm and Pennycook, 2006; Lupini et al., 2007; Pennycook et al., 2006, 2007; Varela et al., 2005b).

II. KEY INSTRUMENTAL ADVANCES In the absence of aberrations, the spatial resolution of the microscope in the transverse plane is given by the Rayleigh criterion as

dt = 0.61λ/α

(1)

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where λ is the electron wavelength and α is the objective aperture semiangle. Of course, the maximum useable aperture is limited by aberrations, speciﬁcally, for an uncorrected system, the third-order spherical aberration. In a system corrected for third-order aberrations, the resolution is limited by the next higher aberration, the ﬁfth-order spherical aberration, and in a ﬁfth-order corrected system by the seventh-order aberrations, and so on. Each order of correction approximately doubles the useable aperture size; hence, resolution was expected to double if third-order aberrations could be successfully corrected, representing a historical advance. Figure 1 shows the anticipated improvement for the 100 kV and 300 kV VG Microscopes instruments at ORNL, assuming the limiting third- or ﬁfth-order aberrations are optimally balanced by

(a) FWHM 1.9 Å

100 kV

(b) FWHM 0.93 Å

(c) FWHM 1.1 Å

300 kV

(d) FWHM 0.39 Å

FIGURE 1 Comparison of anticipated reduction in probe sizes from the uncorrected case (left) and after correction of aberrations (right) for 100-kV and 300-kV microscopes. Simulations include no source size, energy spread, or non-round aberrations. Corrected and uncorrected probes are normalized to the same total intensity passing through the probe-forming aperture. Along with the reduction in full-width half maximum (FWHM) comes a concomitant increase in peak probe intensity.

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lower-order aberrations (defocus for the uncorrected situation, defocus and third-order spherical aberration for the corrected case, see Krivanek et al., 2003). Although no source size, incoherent instabilities, or chromatic aberrations are included, it is obvious that the full-width half maximum (FWHM) of the probe intensity could decrease signiﬁcantly on aberration correction, which is the key to improved Z-contrast image resolution. In addition, the peak probe intensity is greatly increased; spacings that were resolvable even with the uncorrected probe would show greatly improved contrast with the smaller probe. As shown later, in this chapter, the smaller, sharper probe allows light atoms to be seen in a Z-contrast image even when adjacent to heavy atoms because the probe tails no longer overlap the surrounding heavy atoms. In addition, the higher peak probe intensity means higher peak scattering from single atoms and consequently much improved signal to noise ratio (SNR). Hence, aberration correction brings much greater beneﬁts than merely improved transverse resolution, as illustrated below. In particular, the depth resolution improves as the square of the aperture angle (Born and Wolf, 1980),

dz = 2λ/α2 .

(2)

On correction of aberrations we move from a situation where the depth of focus is larger than a typical specimen thickness (i.e., the image is a projection of the specimen) to a situation where the depth of focus is below the typical specimen thickness, a new regime that allows information to be extracted in the third dimension simply by changing focus. Therefore, with much anticipation we installed our ﬁrst aberration corrector on the HB501UX in March 2001, (Figure 2a). The improvement in performance was dramatic, as shown in Figure 3. The ﬂat-phase region in the center of the Ronchigram more than doubled in size, indicating successful compensation of the third-order spherical aberration, and the smaller probe allowed the dumbbell in Si 110 to be directly resolved in a Z-contrast image. The 100-kV microscope now had the performance previously only achievable on the 300-kV HB603U (seen by comparing Figure 3f and 3g). The improvement of the coherent aberrations, of course, uncovered signiﬁcant incoherent instabilities and environmental factors, that are apparent in the images. These effects were gradually remediated over the next few years by improvements to the instruments themselves, and in particular, by the construction of the ORNL Advanced Microscopy Laboratory, incorporating a “room in a room” construction with separate instrument and control rooms and special attention to ensure low ﬁelds, high temperature stability, and a clean power system. One notable result from the corrected HB501UX was the imaging of individual Bi atoms within the Si lattice. Although single-atom detection

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FIGURE 2 (a) Installation of the first ORNL Nion aberration corrector on the VG Microscopes HB501UX on March 8, 2001, with the column stripped down ready to be rebuilt to incorporate the corrector. (b) The rebuilt HB501UX and (c) the HB603U incorporating Nion aberration correctors, after being moved to ORNL’s Advanced Microscopy Laboratory.

had been demonstrated in an uncorrected 200 kV microscope (Voyles et al., 2003) the SNR seen in Figure 4 is much higher, sufﬁcient to allow the Bi atom to be located in the left or right column of the Si dumbbell (Lupini et al., 2003; Pennycook et al., 2003b). The observed concentrations were in accord with the known dopant proﬁle. The thickness of the sample was estimated to be 300 Å, slightly greater than the 1 s Bloch state extinction distance, so that the intensity for a single substitutional Bi atom would be expected to be depth dependent (Nakamura et al., 1997) following the intensity buildup and decay on the Si columns, which was conﬁrmed by the histogram of intensities. In 2002, a Nion aberration corrector was installed in the 300-kV HB603U, and again the ﬂat-phase area on the Ronchigram increased by more than a factor of two, and sub-Ångstrom information transfer was seen in the Fourier transform of the image intensity (Figure 3l). Again, signiﬁcant instabilities needed to be remedied, and the information transfer slowly improved over the next few years. One of the spectacular gains in performance concerned the visibility of single atoms. Although single Pt atoms had been imaged on the surface of γ-alumina in the uncorrected microscope (Nellist et al., 1996), the images were very noisy. Figure 5 shows the muchimproved visibility after aberration correction. The atoms become smaller brighter spots, which stand out much more clearly over the background of the support. Thus the sensitivity to single atoms is greatly enhanced by

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100 kV Before (a)

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(311) 0.164 nm

(220) 0.192 nm

(400) 0.136 nm

(331) 0.125 nm

(400) 0.136 nm

(533) 0.083 nm

(600) 0.091 nm

FIGURE 3 Ronchigrams observed in far field with the probe stationary on the sample (a) for the 100-kV microscope before tuning the corrector, (b) after aberration correction, (c) for the uncorrected 300-kV microscope, and (d) after aberration correction. In both cases, the extent of uniform phase has increased ∼2.5 times. Images of Si 110 show the improvement in image resolution on the 100-kV microscope from (e) an estimated 1.9 A° before correction to (f) 1.36 A° after correction, shown by the resolution of the Si dumbbell. The resolution of the 300-kV microscope is sufficient before correction to resolve the Si 110 dumbbell (g) at 1.36 A°, and the improvement after correction is seen by improved image contrast (h). (i–l) are Fourier transforms of the image intensities (e–h), respectively, indicating the information transfer acheived in each case. (Reproduced from Pennycook et al., 2003b.)

reducing the probe size. Indications of a sub-Ångstrom probe are found by ﬁtting Gaussians to intensity proﬁles taken across individual atoms. Strictly speaking, it is possible to form a sharper central peak by using a non-optimum defocus, as had been done in the uncorrected microscope to show 0.93 Å resolution in a CdTe single crystal (Nellist and Pennycook, 1998), but it is rather unlikely that such precise conditions would be maintained for all the different atoms in different parts of the specimen. Hence, the histogram of the FWHM of the Gaussians is expected to provide a good indication of probe size, and Figure 5 shows that a probe FWHM ∼0.7–0.8 Å was achieved in 2003. This was soon conﬁrmed by the

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FIGURE 4 (a) Z-contrast image of Bi-doped Si 110 revealing the columns containing individual Bi atoms with (b) intensity profile along A–B showing a Bi atom in the right-hand column of a Si dumbbell and (c) an intensity profile along C–D showing a Bi atom in each of the two columns of a dumbbell. (d) Histogram showing the variability of the Bi atom peak intensity relative to that of a Si column due to Bi atoms located at different depths. (Reproduced from Pennycook et al., 2003b, adapted from Lupini et al., 2003.)

direct imaging of the dumbbell in Si 112 at 0.78 Å separation (Nellist et al., 2004), as discussed in more detail in the next section. The improved single-atom visibility from a sub-Ångstrom probe was found to be sufﬁcient to allow the imaging of single La atoms on the surface of a γ–Al2 O3 ﬂake aligned in the [100] direction, as seen in Figure 5d. Some of the Al-O columns are excessively bright, indicating the presence of a La atom on a substitutional site. Careful quantiﬁcation of intensities revealed two different levels, consistent with image simulations for La atoms at the entrance and exit surfaces, those at the exit surface showing bright due to the concentrating effects of columnar channeling (Wang et al., 2004). A small number of La atoms also were observed in interstitial sites. A focal series also indicated that the La atoms were located on the two surfaces. Conﬁrmation that the La atoms would not enter the γ–Al2 O3 lattice was obtained by density functional theory, which also showed that the reason for the enhanced stability of La-doped γ–Al2 O3 was the stronger binding energy of La to the γ–Al2 O3 surface compared to the α–Al2 O3 surface. Hence the γ−α transition is delayed to higher temperatures and the lifetime of the catalyst is extended. It also became apparent that the depth of focus of the aberrationcorrected probe was much smaller than before, and atoms would come in and out of focus depending on their height. Therefore, in a light

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0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 FWHM (Å)

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10 Å FIGURE 5 Images of La-doped γ-alumina obtained with a 300-kV STEM before aberration correction (a) showing faint fringes from the γ-alumina (arrow) and some individual La atoms. After correction (b) there is a significant improvement in contrast, resolution, and signal to noise ratio. Indications that the probe size is sub-A°ngstrom come from (c) a histogram of FWHM of intensity profiles. In (d) the γ-Al2 O3 lattice is viewed in the [100] direction and individual La atoms can be seen taking substitutional and interstitial sites. By comparison to image simulations (e), it was established that the brightest substitutional La atoms (A') are on the beam exit side of the flake, the less bright (A) are on the entrance side, and the faint ones (B) are in interstitial locations. (Panels (a)–(c) courtesy of A. Y. Borisevich; reproduced from Pennycook et al., 2003a; panels (d) and (e) from Wang et al., 2004.)

matrix a through-focal series of images becomes a depth sequence of images; information on the third dimension is available as shown in Figure 6. Single-atom sensitivity and sub-Ångstrom lateral resolution are maintained. The single atom seen in Figure 6d provides an excellent test of

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FIGURE 6 Four frames from a through-depth series of images of a Pt2 Ru4 catalyst supported on γ-alumina. The alumina is three dimensional with thin, raftlike Pt-Ru clusters on its surface. Different clusters are resolved at different depths, until, at −40nm defocus, the carbon support film is reached and a single Pt atom comes into focus (circled). Line profiles correspond to the rectangles shown in the images. (Images courtesy of A. Y. Borisevich; reproduced from Pennycook, et al., 2004.)

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Relative Incremental Intensity

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FIGURE 7 Variation in incremental intensity of the single atom circled in Figure 6d as a function of defocus. The atom remains visible over a wide range of focus due to the high signal to noise ratio. The depth resolution (FWHM of the Gaussian fit) is 12 nm, and the depth precision 0.2 nm with 95% confidence. (Reproduced from Borisevich et al., 2006a.)

the expected depth resolution [Eq. (2)], which is expected to be 7–8 nm for our conditions. Figure 7 shows the incremental intensity of that atom as a function of focus, which is closer to 12 nm in this case, perhaps due to beam broadening in the carbon support ﬁlm or residual aberrations. Because of the high SNR the peak intensity can be located to a precision of ±2 Å with 95% conﬁdence (Borisevich et al., 2006a). This is a relatively ideal case, and the presence of a larger background can signiﬁcantly reduce the range of visibility for single atoms. For example, Figure 8 shows a sequence of frames from a through-focal series of images of individual Hf atoms in a high dielectric constant semiconductor device structure, where the atoms are visible only over a range of about 1 nm or so. The reason for this is seen from Figure 9, in which a line proﬁle is extracted from one image in the series and shows a Hf atom with a large background intensity coming from out-of-focus contributions from the highly scattering HfO2 layer. As a result, Hf atoms are visible only close to exact focus when their contrast is near the maximum, and the incremental intensity in this case shows a much narrower FWHM with focus, ∼1 nm as shown in Figure 10. As a result, the precision with which the peak intensity can be determined (i.e., the precision of the depth determination is better than ±5 Å in this case), is little worse than in the case of the single Pt atom on the C ﬁlm with higher SNR.

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Df 5 120 Å

Df 5 115 Å

Df 5 110 Å

Df 5 15 Å

Df 5 0 Å

Df 5 25 Å

Df 5 210 Å

Df 5 215 Å

Df 5 220 Å

5Å

FIGURE 8 A series of frames from a through-focal series of images, recorded with 5 A° focal steps, showing the appearance and disappearance of an individual Hf atom (circled) in a thin layer of SiO2 between the HfO2 dielectric (left) and the Si gate region (right). (adapted from van Benthem et al., 2006.)

In a crystal aligned to a zone axis, optical sectioning—the ability to focus the probe at a speciﬁc depth inside the specimen—is opposed by the tendency of the columns to channel the electrons. This had ﬁrst been seen in a Bloch wave analysis of annular dark-ﬁeld (ADF) STEM imaging with large probe-forming apertures (Peng et al., 2004). With a 35 mrad probeforming aperture the peak intensity was found to move 40 Å into a GaAs 110 crystal for a defocus of 50 Å. Later calculations for Si 110 with Bi atoms at various depths are shown in Figure 11, also for a 35 mrad probeforming aperture (Borisevich et al., 2006b; see also Cosgriff and Nellist, 2007). Since Si is less strongly channeling than GaAs, the different Bi atoms could be selectively brought into focus at the defocus values corresponding

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(a)

1nm

(b)

HfO2

Hf

Si

Si

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Si

FIGURE 9 (a) A Z-contrast image taken from the through-focal series with a line profile (b) showing a large, nonuniform background under the Hf atom due to out-offocus contributions from the HfO2 . (Reproduced from van Benthem et al., 2006.)

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FIGURE 10 Excess intensities of five Hf atoms as a function of focus showing a FWHM of ∼1 nm and consequently a precision in the depth measurement of ∼ ± 5A°. (Reproduced from van Benthem et al., 2006.)

to their depth. With the introduction of ﬁfth-order aberration correctors, probe-forming apertures in this range are now available experimentally and this prediction can be experimentally tested. Another somewhat surprising result was the improvement in brightﬁeld (BF) STEM imaging. It has long been understood that STEM (BF) imaging can be equivalent optically to TEM BF imaging via the principle of reciprocity (Cowley, 1969; Zeitler and Thomson, 1970), with the STEM probe-forming (objective) aperture equivalent to the TEM objective aperture and the STEM collector aperture equivalent to the TEM condenser aperture. However, the high coherence conditions necessary for phasecontrast imaging meant that the STEM collector aperture had to be much smaller than the objective aperture and only a small fraction of the incident probe current could be used to form the image. As a result, STEM phase-contrast imaging was very noisy and not often used. The situation is quite different after aberration correction. By reciprocity, the BF image is now equivalent to an aberration-corrected BF TEM image, and because of

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FIGURE 11 (a) A model doped Si crystal with four Bi atoms at different depths (Bi indicated in black; Si indicated in grey). (b) Simulated HAADF image intensity scans for an aberration-free probe with 35 mrad semiangle, detector angle from 70 mrad to 300 mrad. Defocus varies from 0 A° (top) to 38.4 A°, 76.8 A°, 115.2 A°, and 153.6 A°, consecutively. (Reproduced from Borisevich et al., 2006b.)

the large ﬂat-phase region in the Ronchigram, the collector aperture can be enlarged an order of magnitude without signiﬁcant loss of coherence, giving two orders of magnitude increased signal. The STEM BF image is now of high quality and is available simultaneously with the annular dark-ﬁeld (ADF) image with pixel-to-pixel correlation. An example of the simultaneous phase-contrast and Z-contrast imaging of SrTiO3 110 is shown in Figure 12. Note that the optimum parameters are tuned for the Z-contrast image, but also result in a good phase-contrast image, although the optimum focus is slightly different in the two cases. The phase-contrast images contain quantitative information on the composition of light columns such as oxygen, just as in aberration-corrected TEM (Jia, 2004, 2005, 2006; Jia et al., 2003). One of the key advantages of STEM has always been the ability for simultaneous Z-contrast imaging and electron energy loss spectroscopy (EELS) (Crewe and Wall, 1970). The ﬁrst demonstration of atomicresolution spectroscopy was in 1993, with spectra taken plane by plane across an atomically abrupt interface, the probe being centered on each plane in turn by reference to the simultaneous Z-contrast image intensity (Browning et al., 1993). Later it became routine to perform column-bycolumn spectroscopy in this manner (Duscher et al., 1998). Now, aberration correction has enabled dramatic gains in sensitivity for EELS and for

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FIGURE 12 Comparison of coherent and incoherent imaging of SrTiO3 in the 110 projection using a 300-kV VG Microscopes HB603U STEM with Nion aberration corrector. (a) A Through-focal series with 4 nm defocus steps in which the phase contrast images show complex variations in contrast while the incoherent angular dark-field (ADF) image contrast slowly blurs. All images are raw data and show some instabilities. (b) Under optimum conditions, the ADF image shows the Sr columns brightest, the Ti columns less bright, and the O columns barely visible, while the phase contrast bright-field (BF) image shows the O columns with high contrast. Microscope parameters are optimized for the smallest probe, with, nominally, a probe-forming aperture of ∼22 mrad, Cs ∼ −0.037 mm, and C5 ∼ 100 mm. (Courtesy of M. F. Chisholm, A. R. Lupini, and A. Y. Borisevich; adapted from Pennycook, 2006.)

imaging. This has allowed comparison between experiment and theory (Allen et al., 2003) and the ﬁrst spectroscopic identiﬁcation of a single atom inside a bulk material, as shown in Figure 13 (Varela et al., 2004). More recently, the ﬁrst true two-dimensional (2D) spectroscopic maps showing atomic resolution have been obtained (Bosman et al., 2007). In

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FIGURE 13 Spectroscopic identification of an individual atom in its bulk environment by EELS. (a) Z-contrast image of CaTiO3 showing traces of the CaO and TiO2 {100} planes as solid and dashed lines, respectively. A single La dopant atom in column 3 causes this column to be slightly brighter than other Ca columns, and EELS from it shows a clear La M4,5 signal. (b) Moving the probe to adjacent columns yields reduced or undetectable signals. (Adapted from Varela et al., 2004.)

addition, just as with X-ray absorption spectroscopy, the ﬁne structure at an absorption edge gives an indication of the local electronic structure, as seen around the atom excited. We show examples from complex oxides that illustrate the remarkable insights available into electronic structure by atomic-resolution EELS and how different it can be from the atomic structure. In the future, it may even be possible to perform EELS studies with true three-dimensional (3D) resolution, just as in the case of imaging (D’Alfonso et al., 2007). For the past few years we have also participated in the Department of Energy TEAM project (Transmission Electron Aberration-corrected Microscope project; see http://ncem.lbl.gov/TEAM-project/). An FEI Titan 80-300 (FEI, Hillsboro, Oregon USA.) electron microscope equipped with a Schottkey ﬁeld emission gun and a CEOS CCOR probe corrector (Haider et al., 1998a,b) was delivered to ORNL in July 2006, and within a few weeks it was delivering sub-Ångstrom information transfer, even without optimization of instabilities, as shown by images of Ge along the [121] zone

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FIGURE 14 (a) Z-contrast image of Ge along the [121] zone axis taken with the probe-corrected ORNL FEI Titan 80-300 in August 2006 shortly after installation. The image resolves many of the dumbbells separated by 0.82 A° (b), with spots extending to an information limit of 0.76 A° (circled) (c). (Image courtesy of S. Lopatin and A. R. Lupini.)

axis (Figure 14). Many of the dumbbells showed distinct dips between the columns, although, due to the residual instabilities, many others did not, and this does not meet the resolution criterion we discuss below, which is for an average of 10% contrast (Peng et al., 2008). A major advantage of this instrument is that it has a standard SuperTwin objective lens that allows high specimen tilts to be achieved so that several zone axes can be imaged in the same specimen. Figure 15 shows the same Ge sample imaged along the [110], [231], and [121] zone axes taken by tilting. The microscope also has the ability to record simultaneous Z-contrast and BF phase-contrast images, as shown by Figure 16 showing PbS nanocrystals with numerous individual Pb atoms visible in the Z-contrast image. Over the next few years, many improvements were made to improve the instabilities, and the image quality gradually improved. In 2007 two major improvements were incorporated into the ORNL Titan column, the installation of a CEOS DCOR aberration corrector (Müller et al., 2006) and a prototype FEI high-brightness gun. The microscope in this conﬁguration is shown in Figure 17. With the DCOR full correction of all third-, fourth-, and ﬁfth-order aberrations is achievable to a level that is theoretically sufﬁcient to allow a probe size of 0.5 Å. The high-brightness

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(a) [110]

(b) [231]

(c) [121]

FIGURE 15 Z-contrast images of Ge along the [110], [231], and [121] zone axes taken from the same sample by tilting. Insets show the projected crystal structures. Images are low pass filtered. (Courtesy of S. Lopatin and A. R. Lupini.)

gun is still a thermally-assisted Schottky-type gun but has a brightness that is higher by approximately a factor of four to ﬁve over the standard gun. Figure 18 shows the improved resolution and information transfer obtained for Ge [121] with the reduced higher-order aberrations and increased source brightness. Now all the dumbbells show a dip between the two Ge columns, and the Fourier transform of the image intensity

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FIGURE 16 Simultaneously recorded, aberration-corrected Z-contrast ADF (a) and phase contrast bright-field (b) images of PbS nanocrystals supported on a thin carbon film. The small dots in panel (a) are individual Pb atoms with FWHM ∼1 A°. (Sample courtesy of S. J. Rosenthal and T. L. Watt; images recorded by A. R. Lupini on the FEI Titan 80-300 with CEOS CCOR.)

shows information transfer to 0.63 Å. Figure 19 shows images of GaAs recorded along four different zone axes, distinguishing the lattice polarity in the [110], [231], and [112] directions where the structure projects as individual Ga and As columns.

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FIGURE 17 The FEI Titan 80-300 with high-brightness gun and CEOS DCOR in ORNL’s Advanced Microscopy Laboratory, provided as part of the Department of Energy TEAM project. (Photo courtesy of Larry Hamill.)

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FIGURE 18 (a) Z-contrast image of Ge [121] taken after installation of the CEOS DCOR and FEI high-brightness gun on the ORNL Titan 80-300 in November 2007. The image resolves almost all the dumbbells. (b) Fourier transform of the modulus of the image intensity showing spots extending to an information limit of 0.63 A° (circled). (Image courtesy of S. Lopatin and A. R. Lupini.)

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FIGURE 19 Z-contrast images of GaAs in various projections in the same specimen by tilting. (a) [110] with intensity profile (b) distinguishing the less bright Ga columns from the brighter As columns, (c) [231], (d) [121], and (e) [111]. Images are raw data from the ORNL Titan 80-300 with CEOS DCOR and FEI high-brightness gun. (Courtesy of A. R. Lupini.)

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III. MEASUREMENT AND DEFINITION OF RESOLUTION As a result of the rapid improvement in image resolution, the issue of its measurement and deﬁnition assumed renewed importance. The ﬁrst direct image of a crystal lattice to show sub-Ångstrom resolution was Si 112, where a clear dip in intensity was visible in the center of every dumbbell, thus incontrovertibly resolving the 0.78 Å separation of the two columns (Nellist et al., 2004). The Fourier transform of the image intensity showed indications of information transfer to 0.61 Å. Care was taken to avoid clipping in the data collection, which can add spurious high-frequency information in the Fourier transform, and the scan was rotated to ensure the spots were not artifacts due to instabilities (see supporting online material for Nellist et al., 2004). A match to the experimental image was obtained by a simple convolution simulation using Lorentzians of 0.3 Å FWHM to represent the Si high-angle object function. The probe was simulated using the measured aberrations to form the coherent probe, then broadening it by a Gaussian of 0.31 Å FWHM, which is consistent with the measured 35-pA beam current. Subsequently a more detailed analysis of the image contrast was performed, including an analysis of the image noise and a match to the experimental parameters of thickness, defocus, and incoherent broadening through comparison to image simulations (Peng et al., 2008). Figure 20 shows the procedure used to evaluate the information cutoff. First, the image was corrected for the wavy distortion by linear interpolation, then unwarped to correct for non-orthogonality and cropped to be strictly periodic. Fourier transforming and comparing the crystal spot intensities to the noise background revealed the information limit. The 444¯ spot, which corresponds to the dumbbell separation of 0.784 Å, gave a conﬁdence level of 97%. Higher-order spots were also present, with decreasing conﬁdence ¯ 3¯ reﬂection at 0.707 Å had 92% conﬁdence, while the 084¯ spot levels; the 17 at 0.607 Å had a conﬁdence level of 71%. This is very close to the noise level and represents the effective information cutoff. Thus, in practice, information transfer is usually limited by noise rather than just the diameter of the objective aperture. With the spot intensities in the diffractogram quantiﬁed, their intensities were compared to those calculated from simulated images and a least-squares ﬁtting routine used to determine the experimental defocus, thickness, and incoherent broadening. The zero-order spot was used as an adjustable parameter to avoid possible issues with any background intensity or Stobbs factor. The defocus could be well ﬁtted to −45 ± 5 Å, the Gaussian blur to 0.6 ± 5 Å, but the thickness dependence of the Z-contrast image is weak, and thickness could only be estimated to be approximately

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FIGURE 20 Z-contrast image of Si taken along the 112 zone axis, resolving columns of atoms just 0.78 A° apart (Nellist et al., 2004). Image recorded with the ORNL 300-kV VG Microscopes HB603U STEM equipped with Nion aberration corrector. (a) Raw experimental image. (b) Image from (a) with the wavy distortion removed by linear interpolation. (c) Image from (b) with the nonorthogonality corrected and then cropped to be strictly periodic. (d) Diffractogram formed from (c) with the objective aperture indicated by the white circle. (Reproduced from Peng et al., 2008.)

in the range of 200–400 Å. Figure 21 shows a comparison between the experimental and best-ﬁt theoretical data. The coherent aberrations alone give a probe FWHM of 0.46 Å, which when broadened by the 0.6 Å Gaussian becomes a FWHM of 0.74 Å, resulting in a dip of 11% between the two columns of the dumbbell. A resolution criterion of 10% was proposed, since at this level all the dumbbells are seen to be resolved experimentally, and requiring the Rayleigh value of 27% seems unnecessarily severe.

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FIGURE 21 (a) Subsection of experimental ADF image of Si 112 using detector spanning 90−200 mrad. (b) Simulated image for t = 300 A°, f = −45 A°, Cs = −0.037 mm, C5 = 100 mm, and probe-forming aperture of 22 mrad. Temporal and spatial incoherence have been accounted for by convolving with a Gaussian with FWHM of 0.60 A°. (c) Line scans averaged over width of 10 pixels comparing experiment and theory. The experimental result has been averaged over a total of 96 dumbbells and normalized by the average of the data points shown. (Reproduced from Peng et al., 2008.) (See color insert).

The difference between resolution and information transfer can be well illustrated through theory. Figure 22a shows a simulated image for the aberration-corrected coherent probe (unbroadened) for a thinner crystal, which clearly resolves the dumbbells, and the Fourier transform shows spots ﬁlling the entire objective aperture. If we then simulate the image

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FIGURE 22 Simulated STEM ADF images and diffractograms for a 100 A° thick silicon crystal in 112 zone axis orientation and a 90–200 mrad detector. (a) Simulated ADF image formed using an aberration-corrected probe with f = 20 A°, Cs = −0.037 mm, C5 = 100 mm, and a probe forming aperture of 22 mrad. (b) Diffractogram corresponding to (a). (c) Simulated ADF image formed using uncorrected probe with

f = −572 A°, Cs = 1 mm with the same 22 mrad aperture. (d) Diffractogram corresponding to (c). White circles represent the cutoff pmax , and the maximum values of the diffractograms have been cropped to show the structure within the cutoff. (Reproduced from Peng et al., 2008.) (See color insert).

for an aberrated probe with the same objective aperture, the dumbbells are now unresolved, as expected, since the resolution is now 1.26 Å, but the Fourier transform still shows spots ﬁlling the entire aperture. Thus the information transfer remains sub-Ångstrom as before, but the phase of the wavefront within the objective aperture is no longer uniform and the probe is broadened. The high-frequency content in the image no longer coincides with the atom columns as in the aberration-corrected case.

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FIGURE 23 Z-contrast image of LaMnO3 in the pseudocubic [001] zone axis. The arrow indicates the [110] direction. (Reproduced from Oxley et al., 2007.)

Substantial new insight into the resolution of spectroscopic images also has been gained through the improved resolution available with the aberration-corrected microscope. Interpreting images formed by core loss inelastic scattering can be more complicated than those formed by the highly localized high-angle scattering due to the effective nonlocal nature of the inelastic excitation (Rose, 1976; for a recent review, see Spence, 2006). This can lead to peaks in the image that are not located on the atomic columns (Oxley et al., 2005). Complex oxides are convenient specimens for investigating EELS image contrast as they comprise elements with edges that span low- to high-energy losses even for the same element. An image of LaMnO3 in the pseudocubic [001] zone axis is shown in Figure 23. The arrow shows how a line scan in the [110] direction can cross La and MnO columns. La has both a high-energy M4,5 edge and a low-energy N4,5 edge; similarly, Mn has a high-energy L2,3 edge and a low-energy M4,5 edge. Furthermore, the scattering power of the columns differs signiﬁcantly, which allows the effect of dechanneling to be investigated. Figure 24 compares experimental EELS line scans obtained from the zero-loss peak and the integrated core loss intensities for different edges (Oxley et al., 2007). All experimental data show reduced contrast compared to simulations, as

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commonly found in high-resolution TEM images (Hytch and Stobbs, 1994). The zero-loss image shows generally inverted contrast with respect to the ADF image, as expected, since electrons cannot contribute to both detectors simultaneously. Moreover, the shape and width of the inelastic images does not show a simple dependence with inner shell binding energy as had often been assumed in the past. Many of the simulated images show volcanolike features on the atom sites that were conﬁrmed experimentally for the O K and La N4,5 edges. Compared with simulations for very thin crystals, it was found that the “volcano” in the La N4,5 edge is due to dechanneling and scattering of the probe as it propagates along the column (estimated to be 250 Å thick experimentally), whereas in the O case it is due to the nonlocal nature of the inelastic interaction. Furthermore, atomic resolution is maintained at the low-lying La N4,5 edge at 99 eV, although the contrast is reduced by delocalization from that obtained at the high-energy La M4,5 edge at ∼842 eV. These effects are more clearly appreciated from the two-dimensional (2D) EELS image simulations shown in Figure 25 for the various edges as a

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FIGURE 25 Two-dimensional EELS simulations for the O K edge, the La N4,5 edge, the La M4,5 edge, and the Mn L2,3 edge as a function of specimen thickness, showing the formation of volcano-like structures on the O and MnO columns (blue circles) even in a specimen one-unit cell thick. With increasing thickness a volcano develops on the MnO column in the center of the cell due to depletion of the beam by scattering from the Mn, and a similar feature develops in the Mn L2,3 edge. The low-energy La N4,5 edge shows atomic resolution, although a long-range tail exists due to delocalization. Black lines indicate the [110] directions corresponding to the line scans in Figure 24. (Adapted from Oxley et al., 2007.) (See color insert).

function of specimen thickness. The simulations for the O K edge show how the signal from the pure O column (at the edges of the unit cell) is maintained to much higher thickness than the O signal from the MnO column (in the center of the cell), which is rapidly depleted due to the high-angle scattering of the Mn. This effect also causes a volcano-like feature in the Mn L2,3 image. Comparing the two La edges reveals the effects of delocalization. The high-energy La M4,5 edge drops close to zero in the center of the unit cell, whereas in the same location the intensity of the low-energy La N4,5 edge is still signiﬁcant due to long-range inelastic excitations. For further discussion of the inﬂuence of probe-forming and EELS collection angles on spectroscopic images, see Oxley and Pennycook (2008).

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IV. COMPLEX OXIDES: MANGANITES Complex oxides such as manganites possess a vast range of physical properties ranging from ferromagnetism, high-temperature superconductivity, and colossal magnetoresistance to ferroelectricity. In addition, their properties can be further tuned through growth of superlattices or heterostructures. Key to the tuning of desired properties is the ability to maintain a mixed-valence state within the Mn sublattice through suitable doping. The structure is based on the perovskite lattice, Ax B1−x MnO3 , where A is a trivalent cation (La, Nd, Bi, Pr) and B a divalent cation (Sr, Ca, Ba), but typically will show reduced symmetry and the unit cell will be a multiple of the basic perovskite unit cell. For x = 0 the Mn would be in a +4 3 e0 ), while for formal oxidation state, with a 3d3 electronic conﬁguration (t2g g x = 1 the Mn would take a +3 formal oxidation state with a 3d4 conﬁgura3 e1 ). A fraction 1−x of Mn ions per unit cell take the +4 oxidation tion (t2g g state, while the rest take a +3 oxidation state. For some x values, these inequivalent Mn species can form ordered arrays at low temperatures. Evidence of this has come from the observation of superlattice reﬂections in diffraction that correlate with x, for example doubling or tripling of the Pnma unit cell of the parent CaMnO3 compound along the [100] or [101] directions for x = 0.5 or 0.33 in the Lax Ca1−x MnO3 system. Such observations imply a stable spatial ordering of the eg electrons, leading to the notion of “charge ordering” in these materials. However, the superstructures are accompanied by tilts and distortions of the oxygen octahedra (such as Jahn–Teller distortions) so that it was not at all obvious to what extent the superlattice spots represent structural changes, changes in electronic conﬁguration, or real charge disproportionation between inequivalent Mn sites. Figure 26a shows a Z-contrast image of a Bix Ca1−x MnO3 (BCMO) manganite along the pseudocubic projection with x = 0.37, which has an ordering temperature above room temperature. It is quite clear that the bright spots (representing Bi atoms within the Ca sublattice) are not showing any appreciable order. However, atomic-resolution EELS shows that the electronic structure is, in fact, ordered. EELS is a particularly powerful probe of the electronic conﬁguration of ﬁrst-row transition metal oxides because the ratio of the intensity of the L3 peak to the L2 peak, the L2,3 intensity ratio, correlates with the 3d-band occupation, i.e., the formal oxidation state (Krivanek and Paterson, 1990; Kurata and Colliex, 1993; Leapman and Grunes, 1980; Leapman et al., 1982; Paterson and Krivanek, 1990; Pearson et al., 1993; Rask et al., 1987). Figure 26b compares EELS spectra from the parent compounds CaMnO3 and LaMnO3 , with Mn in +4 and +3 oxidation states respectively, to spectra obtained from selected columns in the BCMO. Figure 26c–e shows line scans of the L2,3 intensity ratio along the 100 and

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FIGURE 26 Z-contrast image (a) and dependence of the L2,3 ratio on position within the BCMO sample for a few specific scan directions. (b) Electron energy-loss spectra typical of a Mn+3 O column (2) and a Mn+4 O column (3). Spectrum (4) comes from CaMnO3 , while (1) corresponds to bulk LaMnO3 . The oxygen K edge (∼530 eV) and the Mn L edge (∼644 eV) are shown. (c) L2,3 ratio along the direction marked with a green arrow in (a). Horizontal lines represent the L2,3 ratio expected for a pure Mn+3 oxidation state (top) and a pure Mn+4 oxidation state (bottom). (d) L2,3 ratio along the direction marked with a blue arrow in (a). (e) L2,3 ratio along the direction marked with a red arrow in (a). Also shown is the dependence of the L2,3 ratio with position in the sample obtained from high-magnification EELS linescans: (f) scanning along the pseudocubic (100) direction. The positions of Mn planes along the scan have been marked with red arrows. (g) Along the pseudocubic (110). Blue arrows mark the estimated width of the stripe, ∼4 A°. (Reproduced from Varela et al., 2005b.) (See color insert).

110 pseudocubic directions where the Mn oxidation state jumps from +4 to +3 along the directions of the green and red arrows, but little change along the blue direction. The higher-magniﬁcation scans in Figure 26f and g show that the jumps occur every three or four Mn planes, consistent

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with the nominal doping level, and that the separation of the stripes and the width √ of the stripes along the 110 direction are larger by a factor of about 2, which is consistent with the existence of one-dimensional (1D) stripes perpendicular to the green arrow (Varela et al., 2005b). Although the dopant is randomly distributed, the additional electrons have formed into an ordered stripe geometry of occupied eg orbitals on Mn sites, the periodicity of which is consistent with the macroscopically averaged periodicity observed in electron diffraction. EELS sees electronic conﬁguration but not total charge, so the issue of whether there is really any charge ordering cannot be addressed from these observations. We therefore turn to density functional calculations (Luo et al., 2007). The calculations used “generic doping”—that is, extra electrons were introduced into the parent CaMnO3 system along with a compensating uniform positive background and the structure relaxed. For 33% doping two stable phases were found, one by tripling the Pnma unit cell along the [100] direction, as observed in the La and Bi doped compounds, and another stable phase was found by tripling the cell in the [101] direction, which matches the striping seen in Figure 26 (note that the Pnma [101] direction is equivalent to the [100] direction of the pseudocubic unit cell). Examination of the orbital occupation and total charge densities revealed clearly that there is primarily just orbital occupation ordering. The projected density of states (PDOS) on the two nonequivalent Mn sites is different, as shown in Figure 27a, and the calculated Mn 3d majority spin occupation and magnetic moment was also different, comparable although not exactly identical to that found in the parent LaMnO3 and CaMnO3 compounds with formal oxidation states of +3 and +4 respectively. Figure 27b shows that the integrated charge density compared around Mn atoms in several oxides and in Mn metal is practically indistinguishable. The charge associated with the additional eg electron is compensated by the outward relaxation of the two neighboring oxygen atoms, the Jahn–Teller distortion, which ensures the system remains essentially neutral. Thus the charge ordering in this and most likely all perovskite-based manganites is really orbital-occupation ordering, the different patterns reﬂecting different 3D spatial arrangements of the Jahn–Teller distortions. Interfaces between complex oxides can show many more fascinating properties due to the interplay between different and sometimes mutually exclusive order parameters across the interface. A particularly intriguing example is the case of YBa2 Cu3 O7-x /La0.7 Ca0.3 MnO3 (YBCO/LCMO) ferromagnetic/superconducting oxide heterostructures where a superconductor material and a colossal magnetoresistance manganite are in contact (De Melo, 1997, 2000; Holden et al., 2004; Pena et al., 2004a, b; Przyslupski et al., 2004; Sefrioui et al., 2002, 2003; Vasko et al., 1997). These materials contain six different cations, and to understand the properties of these systems is an ideal application of aberration-corrected STEM and EELS. Indeed, it is

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FIGURE 27 (a) Projected density of states (PDOS) for the Mn 3d orbital of the two inequivalent Mn atoms [Mn(1) and Mn(2)] in 50%-doped CMO and the corresponding 3d PDOS in Mn metal. The Fermi energy is at EF = 0. (b) The integrated valence electron density within a sphere around Mn atoms in Mn metal and several Mn oxides as a function of the radius of the integration sphere. (Reproduced from Luo et al., 2007.) (See color insert).

not clear that the interface atomic stacking sequence could be unraveled in any other way. Furthermore, the ﬁne structure on the O K edge can be quantiﬁed to yield the hole concentrations in both the superconductor and the manganite, so that any charge transfer effects can be quantiﬁed on a scale below the coherence length. Therefore STEM/EELS is uniquely placed to unravel the underlying physics of such complex heterostructures, which in turn can enable new spintronics device applications.

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FIGURE 28 Z-contrast image obtained in the VG Microscopes HB603U of a YBa2 Cu3 O7 /La0.67 Ca0.33 MnO3 interface. The interface atomic plane stacking sequence is determined by comparing the image to EELS profiles as . . .CuO–BaO–CuO2 –Y–CuO2 –BaO–MnO2 –La/CaO–MnO2 . . . (Adapted from Varela et al., 2006.)

Figure 28 shows an image of one YBCO/LCMO interface in a superlattice composed of alternating layers of approximately 10 unit cells of YBCO and 15 unit cells of LCMO to a total thickness of 100 nm, which we denote as [YBCO10 u.c. /LCMO15 u.c. ]100nm (Varela et al., 2003, 2006). In YBCO, the CuO chain plane is the lightest in the unit cell and is further away from the adjacent BaO planes than the CuO2 layer. The chain plane therefore appears signiﬁcantly darker in the image, while BaO planes show up with the brightest contrast, so that the YBCO image shows a characteristic striped contrast due to the presence of the chain planes. The LCMO shows a typical pseudocubic perovskite structure, and the interface with the YBCO appears coherent and free of defects (Sefrioui et al., 2002). It is immediately clear from the absence of a dark stripe that the YBCO/LCMO interface does not contain a Cu chain layer. However, since Ba and La have very similar image contrast, as do Cu and Mn, the interface stacking sequence is not uniquely deﬁned from the image.

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FIGURE 29 Ba, Cu, and Mn elemental profiles extracted from integrating the EELS intensity under the Ba M edge (solid circles), the Cu L edge (open triangles) and the Mn L edge (open squares) in a linescan across the YBa2 Cu3 O7 /La0.67 Ca0.33 MnO3 interface shown in Figure 28, recorded in the VG Microscopes HB501UX. The image in the background is from Figure 28, reproduced to scale. EELS shows the interface to be located on a MnO2 plane, which has been marked with a vertical dotted line, and the stacking sequence across the interface from YBCO to LCMO is CuO–BaO–CuO2 –Y–CuO2 –BaO–MnO2 –La/CaO–MnO2 . (Reproduced from Varela et al., 2006.)

However, if we turn to EELS, as shown in Figure 29, we see by superimposing the EELS line traces onto the Z-contrast image (not obtained simultaneously) that the EELS edges do peak on the appropriate planes; speciﬁcally, the Ba signal peaks on the BaO planes in the YBCO and the Mn signal peaks on the MnO2 planes in the LCMO. The ambiguous interface plane, (marked with the dotted line), corresponds to a peak in the Mn signal but the Cu signal has already dropped signiﬁcantly. Note that the limited apparent spatial resolution is due to use of a relatively thick sample, which is necessary to avoid a major contribution from surface amorphous layers but leads to signiﬁcant beam broadening. Although the chemical signals of the different elements were not acquired simultaneously, which is preferred for deﬁnitive identiﬁcation of the atomic stacking, this analysis

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suggested the interface plane to be MnO2 , in which case the interface stacking sequence is CuO–BaO–CuO2 –Y–CuO2 –BaO–MnO2 –La/CaO–MnO2 . This stacking sequence was subsequently conﬁrmed. Signiﬁcant further insight into transfer of charge across the interface is available by analyzing the ﬁne structure on the O K edges (for a preliminary analysis, see Varela et al., 2005a). The charge transfer leads to a hole depletion zone in the YBCO of ∼3 nm at each interface, with a corresponding depletion of electrons from the LCMO layer. These charge transfers are fully in accord with the measured properties of these superlattices. Charge transfer also has been conﬁrmed recently by X-ray absorption spectroscopy measurements (Chakhalian et al., 2007), although only the integrated charge could be extracted and not the high-spatial resolution carrier concentration proﬁles that STEM/EELS provides.

V. HIGH-TEMPERATURE SUPERCONDUCTORS Soon after the discovery of high-temperature superconductivity in YBCO it was found that even a single grain boundary could lead to a dramatic reduction in critical current. Systematic studies established an exponential reduction with increasing grain boundary misorientation (Dimos et al., 1990; Gross and Mayer, 1991; Gurevich and Pashitskii, 1998; Heinig et al., 1996; Hilgenkamp and Mannhart, 2002; Mannhart et al., 1988). More recently, there have been numerous attempts to improve the critical current by doping, and doping the boundary with Ca has been effective (Hammerl et al., 2000; Schmehl et al., 1999). It was widely assumed that the effect was caused by Ca2+ substituting for Y3+ as it is known to do in the bulk, which would introduce excess holes into the system (Kucera and Bravman, 1995). Previous studies of grain boundaries in SrTiO3 , combining Z-contrast imaging with ﬁrst-principles calculations, showed the dislocation cores forming the boundary plane were intrinsically nonstoichiometric due to the extremely high local strains (Kim et al., 2001; Klie et al., 2003). Similar effects were obviously to be expected in the structurally similar but more complex YBCO. First-principles calculations were undertaken to explore the possibilities. Figure 30a shows the energy required to substitute an isolated Ca dopant atom on different lattice sites in YBCO (in an oxidizing environment) as a function of biaxial strain in the a-b plane. In unstrained YBCO, Ca substitutes for Y, as expected (Su and Welch, 2005). In regions under compression, however, substitution for Ba becomes energetically favorable at strains greater than ∼6%. Similarly, in regions under tension, substitution for Cu becomes increasingly favored. In contrast, strain has only a small effect on substitution for Y. The explanation is relatively straightforward.

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FIGURE 30 First-principle calculations of Ca and O vacancy formation-energies in bulk YBCO. (a) Energy required to substitute Ca on different lattice sites in YBCO as a function of biaxial strain. It is energetically more favorable for Ca to replace Y, Ba, or Cu, depending on the local strain. (b) Formation-energy of O vacancies in bulk YBCO (left panel), undoped YBCO under 5% tensile strain in the a-b plane (central panel), and Ca-doped YBCO under 5% tensile strain (right panel), simulating the expanded regions at grain boundaries. For doped grain boundaries where Ca substitutes for Cu, the O vacancy formation-energy increases significantly. (Reproduced from Klie et al., 2005.)

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It is relative ionic size that determines the preferred substitution. The ionic radii of Ca and Y are similar, and so Ca substitutes for Y in an unstrained lattice. However, the radius of Cu is ∼30% smaller and the radius of Ba is ∼30% larger, making substitution for Cu favorable in regions of expansion and substitution for Ba favorable in regions of compression. Hence, the site preference for Ca substitution is driven primarily by atomic size mismatch and lattice strain. Local strain relief has a major effect on the O vacancy formation energies. Figure 30b shows that in strained, undoped YBCO the formation energies for O vacancies are signiﬁcantly reduced, both in the CuO2 planes and CuO chains. O vacancies would therefore segregate to an undoped grain boundary, explaining the oxygen deﬁciency and hole depletion known from previous studies (Browning et al., 1998, 1992). However, when Ca doping is introduced, the local strains are relieved because the Ca chooses that site which results in strain reduction, and therefore as a result the O vacancy formation energies are restored toward, or even to slightly above, values in the unstrained bulk. Hence, compared to the undoped grain boundary, the doped grain boundary should show much reduced deﬁciency of O, and therefore reduced hole depletion. The role of Ca doping is to reduce the local strain, which in turn removes the intrinsic excess of O vacancies from undoped YBCO grain boundaries. Hence, high critical currents are restored through a cooperative doping mechanism between Ca and O vacancies on nearby sites. Full experimental conﬁrmation of this cooperative doping effect was obtained using aberration-corrected STEM. The dislocation core structures were quite different in the Ca-doped case compared with the undoped sample. As shown in Figure 31, the Ca-doped core shows an additional atomic column at its center. The column pairs in the undoped dislocation cores typically are seen in perovskite grain boundaries, but in the doped boundary there are three columns instead. These structures were seen consistently along the entire grain boundary. Column-by-column EELS conﬁrmed the scenario predicted by theory. Not only were the expected Ca substitutions conﬁrmed, but the expected improvement in carrier concentration (determined from the ﬁne structure on the O K edge; see Browning et al., 1998, 1992) was also seen (Klie et al., 2005).

VI. COMPLEX DISLOCATION CORE STRUCTURES Intermetallics are expected to be the key to building high-performance jet engines and other high-temperature, high-performance structures. However, the poor low-temperature ductility of these compounds has hindered their commercial development for decades. The improved resolution of

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FIGURE 31 Structural differences in pristine and Ca-doped YBCO. (a) Z-contrast image of pristine 4◦ [001] tilt grain-boundary dislocation core on the Cu-O sublattice, showing the typical pentagonal arrangement. (b) Ca-doped grain-boundary: the Y/Ba column pentagonal arrangement now encircles three columns: two on the Cu-O and one on the Y/Ba sublattice. Note the weaker intensity of the Y/Ba column on the left of the core, due to Ca substitution, and the extensive changes in the intensity of the CuO columns to the right of the core, due to substitution of Ba (brighter columns) and Ca (darker columns). EELS verified the compositional changes suggested from the image. (Reproduced from Klie et al., 2005.)

the aberration-corrected microscope is able to provide new insight into the atomic structure of dislocation cores in complex materials. Z-contrast imaging revealed that the synchroshear dislocation core in the C14 variant of Cr2 Hf contained “extra” Cr columns, as shown in Figure 32 (Chisholm et al., 2005). During dislocation motion the Cr atoms must move into and out of these columns. Diffusion is therefore necessary, which hinders their motion and contributes to the brittleness of this material at low temperatures. From geometric considerations dislocation motion would be expected to involve only glide, but by revealing the nonstoichiometric nature of the core the reason for the slow dislocation motion is clear. Realistic theoretical modeling of the slip process could now be undertaken to unravel the key atomic conﬁgurations that deﬁne the energy barrier, leading to a true atomistic understanding of the factors determining the energy barrier, and helping pave the way for the design of new alloys with optimized mechanical properties. Another example of a complex dislocation core structure is shown in Figure 33, in the structural ceramic α-Al2 O3 (Shibata et al., 2007). The nature of the terminating column has been controversial (Bilde-Sorensen et al., 1996; Kronberg, 1957). In this case, because of the low signal on the Z-contrast image from the light material, the BF phase-contrast image

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FIGURE 32 Z-contrast image of a synchroshear dislocation core in Cr2 Hf internetallic. A stacking fault comes in from the right (arrow). As shown in the schematic, the core contains additional Cr columns so that dislocation glide cannot occur simply by slip processes but must involve atomic diffusion, accounting for the poor low-temperature ductility of this material. Recorded with the aberration-corrected VG Microscopes HB603U. (Reproduced from Chisholm et al., 2005.)

showed better contrast. The images clearly show that the dislocation slip plane is located between the Al and O layers and unambiguously reveal that the individual dislocation cores are nonstoichiometric in this ionic compound. Again, from such knowledge, density functional calculations of the barrier to glide will become feasible.

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FIGURE 33 Atomic-resolution BF-STEM image of a basal dislocation core in α-Al2 O3 . (a) Typical core structure of a basal dislocation observed from the [1¯100] direction. The dislocation core is dissociated with two partials (b1 = 1/3[10¯10] and b2 = 1/3[01¯10]) connected by a {112¯ 0} stacking fault. (b) and (c) are magnified images of the upper and lower partial dislocation cores, respectively. It is clearly resolved that the upper partial core is terminated by an Al column, whereas the lower partial core is terminated by an O column. Both core terminations lie between the Al and O atomic planes. Each partial core is nonstoichiometric due to the excess of Al or O, but the total basal dislocation preserves Al2 O3 stoichiometry. (Reproduced from Shibata et al., 2007.)

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VII. UNDERSTANDING STRUCTURE-PROPERTY RELATIONS IN CERAMICS Si3 N4 ceramics often are doped with rare-earth elements because they promote the growth of needlelike grains that provide toughening mechanisms, including crack bridging, pullout, and crack deﬂection. Such microstructures result in superior high-temperature properties (Hoffmann and Petzow, 1994; Lange, 1973). In processed ceramic material the Si3 N4 crystals are separated by ∼1-nm thick amorphous ﬁlms or larger glass pockets, where the rare-earth atoms are located. Clearly, determination of the precise bonding arrangements of different rare-earth atoms to the Si3 N4 crystal grains is essential in understanding the fundamental origins of structure-property relations in ceramics. Z-contrast imaging is ideal for this application because of the high atomic number of the dopants. Figure 34 shows simultaneously recorded high-angle annular dark-ﬁeld (HAADF) and BF STEM images of a region centered on an intergranular ﬁlm in La-doped β-Si3 N4 (Shibata et al., 2004). The grain on the right side of each image appears in [0001] zone-axis orientation, and is seen to be terminated by the prismatic {10-10} plane of the Si3 N4 . The BF image shows some mistilt (due to misalignment of the STEM collector aperture), but the speckle contrast typical of an amorphous material is nevertheless clear. The Z-contrast image, however, shows numerous bright spots in the amorphous intergranular ﬁlm, especially along the interface (see the line trace in Figure 34c). These spots represent the location of the La atoms. As shown by the line trace, they are found in a plane that would be occupied by Si atoms if the Si3 N4 grain were continued, suggesting that they are attached to the nitrogen-terminated prismatic surface of Si3 N4 . Some of the Si atom positions in this plane can be identiﬁed in the image, see Figure 34d, but show reduced intensity. This indicates they no longer form straight columns but are somewhat disordered due to the high density of La atoms in this plane. Thus it can be seen that the transition from an ordered crystal to a disordered ﬁlm begins in the plane containing segregated La atoms. A higher-magniﬁcation view of the interface is shown in Figure 35, along with a Pixon reconstructed image (Pixon LLC, Stony Brook, NY, USA) that shows the bright La atom positions more clearly (Puetter et al., 2005). The Si positions in the interfacial plane are shown by open circles, which do not exactly correspond to the La atom positions (shown by arrows). This is due partly to the fact that the La− N bond is longer than the Si− N bond, by ∼0.34–0.48 Å. Figure 35b also shows theoretically predicted segregation sites of La on a bare Si3 N4 surface. These are much closer to the observed La positions. The deviations from theory are most likely due to the presence

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FIGURE 34 Simultaneously recorded Z-contrast (a) and bright-field (b) STEM images of an intergranular film in a La-doped β-Si3 N4 ceramic. (c) A vertically-averaged intensity profile across the Z-contrast image shows the La atoms to be segregated at the amorphous/crystal interfaces. The La atoms adjacent to the [0001]-oriented grain appear to be in a plane that would correspond to Si atoms if the crystal structure were continued. Many Si sites have instead been occupied by La atoms. (d) An intensity profile through selected Si atom positions in the crystal (shaded green) shows that some Si atom positions in the La-containing interfacial plane can be detected (indicated by the red arrow). Such sites show reduced intensity indicative of a more disordered structure. (Reproduced from Shibata et al., 2004.)

of Si, O, and N atoms in the amorphous ﬁlm, which are not included in the calculations. Further insight into the nature of the crystal/amorphous transition is obtained from the image of a triple junction in Lu-doped β-Si3 N4 , where there is an extended amorphous layer. In Figure 36, the ordered arrangement of Lu atoms extends for at least two atomic planes into the

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FIGURE 35 (a) Z-contrast image showing La atom positions (arrows) compared to the projected atomic structure of the Si3 N4 lattice. The La atoms are located slightly further from the N-termination plane of the predicted Si sites (open circles) if the crystal lattice was extended. (b) Pixon-reconstructed image indicating the theoretically calculated La atom positions for a bare Si3 N4 crystal surface, which are in fair agreement with the observed positions. (Reproduced from Shibata et al., 2004.)

“amorphous” region (Shibata et al., 2006). The transition from order to disorder is not instantaneous, which would not be reasonable to expect. At higher magniﬁcation, as shown in Figure 37, it is seen that the Lu sites are also in good agreement with theoretical predictions, and it is actually possible to superimpose small crystallites of LuN onto the image. This allows further insight into why the material becomes amorphous. In Figure 37b, Lu is bonded to the N-terminated Si3 N4 surface, accounting for the bright spots in the ﬁrst plane. If the structure is extended according to the LuN crystal structure, then the bright spots in the second Lu plane can be well explained (Figure 37c). Hence the LuN structure appears to be attempting to grow into the amorphous pocket as an epitaxial ﬁlament. However, sideways extension of the ﬁlament is incompatible with the spacing along the interfacial plane (Figure 37d). This appears to explain the lack of observed Lu between the clusters, where presumably the Si, O, and N atoms are located. Now imagine growing a third ﬁlamentary crystal of LuN onto the second plane. The location and orientation of the third layer of LuN clusters is surely going to be increasingly randomized due to the inﬂuence of the surrounding Si, O, and N atoms, so that the projected Lu positions

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FIGURE 36 (a) Z-contrast image and (b) vertically averaged intensity profile showing the existence of La ordering for at least two planes into an amorphous pocket at a triple junction. (Reproduced from Shibata et al., 2006.)

will show increasing discrepancy, and the structure eventually appears amorphous. Thus the combination of the Z-contrast image and ﬁrst-principles theory provides insight into how the transition to an amorphous structure occurs and also suggests that the amorphous structure itself may have mediumrange order due to the preferential formation of LuN-like clusters. Such order has been inferred from diffraction data in the past, but now might be amenable to direct imaging in a sufﬁciently thin sample. Several further studies on structural ceramics systems have been reported (Dwyer et al., 2006; Shibata, et al., 2005; Winkelman et al., 2004, 2005; Ziegler et al., 2004).

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FIGURE 37 (a) High-magnification Z-contrast image of the interface with distinct La sites visible in the first and second layers away from the crystal. Sites in the first layer agree well with theoretical predictions (shown as blue circles). Green circles represent vacant Si sites based on an extension of the β-Si3 N4 lattice structure. (b–c) Sites in the first and second Lu layer can be bridged by a fragment of LuN crystal, but sideways growth of such crystals is frustrated (d) because it results in mismatched atomic positions. (Reproduced from Shibata et al., 2006.)

VIII. SEMICONDUCTOR QUANTUM WIRES Controlled growth of semiconductor quantum nano-object arrays is important for reproducible fabrication of optoelectronic devices, and therefore understanding the nature of the nucleation sites during growth, and the role of strain, is of critical need. In this example, we show how the sublattice sensitivity of aberration-corrected Z-contrast imaging, combined with elasticity calculations, can provide new and quantitative insights into these phenomena. Figure 38 shows a cross-sectional view of an InAsx P1−x nanowire sandwiched between InP layers, grown by molecular beam epitaxy (Molina et al., 2007). In this 110 projection, the cubic zinc-blende structure projects as dumbbells with the In column toward the left of the

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FIGURE 38 (a) Z-contrast image of an InAsx P1−x nanowire between InP barrier layers. The wire is faceted with its base lying on the (001) plane. (Reproduced from Molina et al., 2007.)

ﬁgure and the P or Asx P1−x column toward the right. The nanowire is visible because of the extra intensity from the As. Close examination of the InP barrier layers does in fact show a weak intensity in the appropriate position that corresponds to the P column. The interface between the InP substrate and the nanowire can be located to atomic precision because of this contrast difference (Figure 39). Scanning tunneling microscopy examination of the nanowires deposited on an InP substrate showed the tendency for the nanowires to be located near step edges. Although many of the steps were wavy, some wires nucleated near straight segments of steps; these gave a clear, sharp image in cross-sectional Z-contrast images, as shown in Figure 39. The transition from the substrate InP layer to the nanowire is atomically abrupt. The location of the step is slightly offset from the center of the nanowire, indicating that the initial nucleation took place on the upper terrace of the step. Nucleation on the upper terrace is entirely in accord with expectations based on the 3.2% lattice mismatch between InAs and InP. Because of the existence of the step, a few unit cells next to the step on the upper terrace can relax slightly sideways, reducing the lattice mismatch, thus making this area the preferential nucleation site. Finite element elasticity calculations showed that the magnitude of the relaxation was indeed signiﬁcant. With the nucleation taking place on the upper terrace the surface of the growing ﬁlm will roughen, so this represents the onset of the Stranski–Krastanov transition from a 2D to a 3D growth mode.

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FIGURE 39 Z-contrast images of the InAsx P1−x nanowires with increasing magnification, and colorized to highlight the contrast differences between the P columns of the InP layers and the Asx P1−x columns in the nanowires. It is clear that the transition from InP to the InAsx P1−x nanowire is atomically abrupt, and an interface step is present under the nanowire. (Reproduced from Molina et al., 2007.)

Figures 38 and 39 also show a slight asymmetry in the As distribution within the wire; the region above the upper terrace appears brighter, and therefore contains more As. This is typical and was also conﬁrmed by quantitative EELS using the P edge intensity in the barrier layers as calibration. The resulting As concentration maps were used as input into a ﬁnite-element elasticity calculation (Molina et al., 2006). In the actual

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FIGURE 40 Low-magnification TEM diffraction contrast image showing the alignment of the nanowire stack is offset from the growth direction. (Adapted from Fuster et al., 2004.)

device structure, multiple layers of InP barrier layers and nanowire layers are grown, and cross-sectional TEM showed that they were not quite vertically aligned but canted by a few degrees, (Figure 40). This observation could be accounted for quantitatively by the ﬁnite-element calculations. Modeling a buried nanowire showed that, due to the microscopic asymmetry of the As distribution, an anisotropy also appeared in the stress on the surface of the subsequent InP barrier layer, with the peak stress emanating from the nanowire appearing slightly offset from the geometric center of the wire. Therefore, again the favorable nucleation site for the next nanowire is the position that minimizes the mismatch, which is slightly offset from the center of the underlying wire. Under carefully controlled conditions all nanowires line up at a speciﬁc angle to the growth direction and the observed angle is in quantitative agreement with the ﬁnite-element calculations.

IX. CATALYSTS Aberration correction has resulted in enormous gains in single-atom sensitivity, as shown in Figure 41 for the case of Pt atoms on γ-alumina. Before aberration correction, with a probe size of ∼1.3 Å, single atoms were just detectable on the support (Figure 41a), and the result in fact was the ﬁrst to show that very small clusters or even single atoms could be important for catalysis (Nellist et al., 1996). The improvement after aberration correction, when the probe size was reduced to ∼0.8 Å, is striking, (Figure 41b; Sohlberg et al., 2004). Now the SNR is greatly enhanced, sufﬁcient to allow accurate extraction of Pt atom positions. In the case of the Pt3 trimers,

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FIGURE 41 Z-contrast imaging of Pt atoms on γ-alumina with a 300-kV STEM (a) before and (b) after aberration correction. Some Pt trimers and dimers are just visible in (a), but individual atoms are much more visible in (b). Some faint contrast from the γ-alumina lattice is also present in (b) as it was not far from a zone axis. (Image (a) reproduced from Nellist et al., 1996; image (b) from Sohlberg et al., 2004.)

the spacings were too large for a metallic cluster, and the image could be matched only by density functional calculations by assuming the trimer to be capped by an OH group. Another application of combined aberration-corrected STEM and ﬁrstprinciples theory is the elucidation of the mechanism for the lowtemperature oxidation of CO to CO2 by Au nanoparticles (Rashkeev et al., 2007). Although Au is inactive in bulk form, at the nanoscale it becomes one of the most active catalysts known, catalyzing the reaction even below room temperature. The cause of the activity of nanosized Au has been a mystery for many years, with several explanations proposed (Bamwenda et al., 1997; Valden et al., 1998; Yoon et al., 2003). Figure 42 shows images of the catalyst prepared by deposition/precipitation onto nanocrystalline anatase (Zhu et al., 2004). In the as-deposited state, many individual Au atoms can be seen, but after reduction to the active form, the atoms are much less visible. It is known that the melting point of Au nanoparticles reduces with diameter and reaches room temperature at approximately the size range observed. Therefore, it seems likely that the smallest nanoparticles may be in a liquid-like state. Most of the reduced nanoparticles are 1–2 nm in diameter, and quantifying the thicknesses by comparing to image simulations revealed that a high fraction of the thicknesses is just one or two monolayers thick. First-principles calculations were performed on an entire sequence of nanoparticles with different sizes and atomic conﬁgurations on anatase.

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FIGURE 42 Z-contrast images of Au nanoparticles deposited on nanocrystalline anatase by deposition/precipitation. (a) as deposited, with Au atoms resolved, and (b) after reduction to the active form. The average particle size is ∼1 nm (c), and many particles are only one monolayer in thickness (d). (Reproduced from Rashkeev et al., 2007.)

The key feature for the activity of the nanoparticles relative to bulk Au is the presence of low-coordination sites. As shown in Figure 43, with reducing coordination number, the energy required to desorb O2 exceeds the activation barrier for the reaction for a coordination number less than 5. This means that an O2 molecule can reside on the surface long enough to react with an adsorbed CO molecule. At sites with coordination number ≥6, desorption occurs on average before reaction proceeds, and the surface of the Au nanoparticle becomes inactive for the reaction. In this situation, only perimeter sites can still participate signiﬁcantly in the reaction. This is the underlying reason for the vast change in catalytic behavior for nanoscale Au (see Rashkeev et al., 2007, for details and additional references). In the case of Pt nanoclusters, the increased binding energy also occurs with reducing size, but in this case it quenches the reaction because the molecules are pinned too strongly to the surface. Part of the reason for the low activation barrier on the Au nanoparticle is the “looseness” of

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FIGURE 43 (a) Calculated desorption energies Ed for O2 molecules bound to Au nanoparticle sites as a function of coordination number, and the activation barrier Er for conversion of CO to CO2 . The desorption energy exceeds the reaction barrier below a coordination number of 5, which allows the reaction to proceed on the nanoparticle surface. Above this coordination number only perimeter sites are important. (b) An example of the activation barrier for the CO to CO2 reaction in which the flexibility of the Au nanocluster lowers the activation barrier at the transition state marked by the red arrow, a feature critical to achieving high activity at room temperature. The schematic shows the nanocluster in its transition state, with Ti in grey, O in red, and Au in gold. (Reproduced from Rashkeev et al., 2007.)

the Au−Au bonds (Yoon et al., 2003), which allows the molecules to move during the reaction and ﬁnd a conﬁguration that minimizes the activation barrier (Figure 43b).

X. FUTURE DIRECTIONS With the present generation of aberration correctors, which correct all aberrations up to third order, it is possible to obtain sufﬁcient sensitivity to image single atoms and sufﬁcient depth sensitivity to locate individual atoms in three dimensions, with sub-Ångstrom resolution laterally, depth resolution of a few nanometers, and depth precision (for well-separated atoms) at the Ångstrom level. The next generation of aberration correctors is also beginning to appear, correcting all geometric aberrations up to ﬁfth order. Initial results with the ﬁfth-order CEOS DCOR aberration corrector (Müller et al., 2006) in the ORNL FEI Titan 80-300 with a prototype high-brightness Schottky source have been presented. Later this year a Nion UltraSTEM is scheduled for delivery to ORNL; this instrument is equipped with a Nion ﬁfth-order aberration corrector and cold-ﬁeld emission gun (Krivanek et al., 2008). These instruments should achieve another jump in resolution, to the level of 0.5 Å laterally and ∼1 nm in depth, with

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a concomitant increase in single-atom sensitivity and depth resolution. Similarly, it will become possible to perform 3D EELS (D’Alfonso et al., 2007) to probe single-impurity atoms at grain boundaries and dislocation cores to link their electronic environment to macroscopic mechanical and electronic properties in a rigorous manner. With these new “eyes” it will become possible to see the ultimate atomic origins of materials properties with new clarity—whether for structural materials or electronic materials, for nanoscience or catalysis. Perhaps there has never been a more exciting time to be involved in electron microscopy, since the human race crosses atomic resolution only once in its history, and that time is now. For ever more, we will be able to see atoms with clear vision.

ACKNOWLEDGMENTS The authors thank their collaborators in the work reviewed here: P. D. Nellist, O. L. Krivanek, N. Dellby, M. F. Murﬁtt, Z. S. Szilagyi, Y. Peng, S. Travaglini, H. M. Christen, W. Tian, R. Jin, B. Sales, D. G. Mandrus, V. Peña, Z. Sefrioui, J. Santamaria, I. Arslan, N. D. Browning, J. P. Buban, D. P. Norton, S. Kumar, P. S. Hazzledine, S. F. Findlay, A. J. D’Alfonso, E. C. Cosgriff, L. J. Allen, R. F. Klie, M. Beleggia, A. Franceschetti, C. Y. Jooss, Y. Zhu, N. Shibata, T. R. Gosnell, G. S. Painter, P. F. Becher, R. L. Satet, M. J. Hoffmann, A. Nakamura, T. Yamamoto, Y. Ikuhara, T. Ben, D. L. Sales, J. Pizarro, P. L. Galindo, D. Fuster, Y. Gonzalez, L. Gonzalez, K. Sohlberg, S. W. Wang, M. V. Glazoff, S. N. Rashkeev, S. H. Overbury, G. M. Veith, W. H. Sides, and J. T. Luck. This work was supported by the Division of Materials Sciences and Engineering, USDOE, in part by the Laboratory Directed Research and Development Program of ORNL, and by appointments (K. van Benthem, A. Y. Borisevich, and M. P. Oxley) to the ORNL Postdoctoral Research Program administered jointly by ORNL and ORISE. Some of the instrumentation used in this research was provided as part of the TEAM project, funded by the Division of Scientiﬁc User Facilities, Ofﬁce of Science, U.S. Department of Energy.

REFERENCES Allen, L. J., Findlay, S. D., Lupini, A. R., Oxley, M. P., and Pennycook, S. J. (2003). Atomicresolution electron energy loss spectroscopy imaging in aberration corrected scanning transmission electron microscopy. Phys. Rev. Lett. 91, 105503. Bamwenda, G. R., Tsubota, S., Nakamura, T., and Haruta, M. (1997). The inﬂuence of the preparation methods on the catalytic activity of platinum and gold supported on TiO2 for CO oxidation. Catal. Lett. 44, 83–87. Batson, P. E., Dellby, N., and Krivanek, O. L. (2002). Sub-Angstrom resolution using aberration corrected electron optics. Nature 418, 617–620. Bilde-Sorensen, J. B., Lawlor, B. F., Geipel, T., Pirouz, P., Heuer, A. H., and Lagerlof, K. P. D. (1996). On basal slip and basal twinning in sapphire (α-Al2 O3 )—I. Basal slip revisited. Acta Mater. 44, 2145–2152. Borisevich, A. Y., Lupini, A. R., and Pennycook, S. J. (2006a). Depth sectioning with the aberration-corrected scanning transmission electron microscope. Proc. Natl. Acad. Sci. USA 103, 3044–3048.

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CHAPTER

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Spherical Aberration-Corrected Transmission Electron Microscopy for Nanomaterials Nobuo Tanaka* Contents

I Introduction II Imaging Theories of HRTEM Using Aberration Correctors A Projection Approximation in Electron Microscope Images B Scherzer’s Linear Imaging Theory and Its Extension to Higher Order C Realization of Charge Density Approximation by Aberration Correction D Lentzen’s Optimum Defocus for Cs -Corrected TEM E Imaging of Thicker Crystals and Nonlinear Image Contrast F Imaging with Negative Aberration Constants G Subtraction Method of Nonlinear Contrast in Cs -Corrected TEM H Improved Image Deconvolution Method III Actual Advantages for Observation by Cs -Corrected TEM A Improvement of Point-to-Point Resolution B Sharp Interface Images Without Fresnel Fringes C Discrimination of Elements in Cs -Corrected TEM Images D Removal of Nonlinear Image Contrast by Subtraction Method E Z-Slice and Tomographic Imaging in Cs -Corrected TEM F New Development of Selected Area Nanodiffraction in Cs -Corrected TEM G Advantages of Cs -Correctors for Electron Holography IV Actual Advantages for Observation by Cs -Corrected STEM A Increased Point-to-Point Resolution

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* EcoTopia Science Institute, Nagoya University, Nagoya 464-8603, Japan Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01010-0. Copyright © 2008 Elsevier Inc. All rights reserved.

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B Reduction of Analysis Time of STEM-EDX and EELS Mapping C Other Advantages V Three-Dimensional Observation of Atomic Objects in Cs -Corrected STEM A Depth of Focus B STEM Experiments for 3D Observation VI Conclusion and Future Prospects Acknowledgments References

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I. INTRODUCTION Aberration correction in transmission electron microscopy (TEM) has been a long historical issue since its development (Knoll and Rusuka, 1932). The objective lens as an important element in image formation has a large spherical aberration, one of Seidel’s ﬁve aberrations, resulting from single convex lenses, which were theoretically studied by Scherzer (1936). Owing to a short wavelength of electron using in TEM, however, the point-to-point resolution had attained to sub-nanometer areas sufﬁcient to detect single atoms in the 1970s, even though Gabor (1949) tried to overcome the problem in another way using his “poterior” image processing as holography. The study of spherical aberration correction (Cs -correction) by using pole lenses or foil lenses started with a theoretical study by Scherzer (1947). Actual development of the correctors can be traced to Hawkes and Cosslett’s (1962) work on quadrupole correctors and Deltrap’s (1964) work on quadrupole-octupole correctors at Cambridge University, followed by the correctors of Beck and Crewe (1974) and Beck (1979) at the University of Chicago in their scanning transmission electron microscope (STEM). Rose (1970) established a scheme for actual TEM correctors and performed the basic experiments for actual usage with the help of his graduate students, including Haider, at Darmstadt University in the 1980s. During this period, some basic studies on Cs -correction were performed also in Japan. At Nagoya University Maruse et al. (1970) advanced the study of thin foil lenses to correct third order spherical aberration, particularly for use in high-voltage scanning transmission electron microscopy (HVSTEM). In Tukuba Okayama (1990) studied various conﬁgurations of the correctors and made test models. At Osaka University Taniguchi et al. (1992) developed another type of aberration correction using defocus-modulation by changing accelerating voltage. In the 1990s Haider et al. (1998a,b) developed a 200-kV Cs -corrected TEM using hexapole lenses in collaboration with researchers in Juelich. Krivanek et al. (1999) constructed a Cs -corrected STEM using quadrupole

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and octupole lenses at Cambridge University at the same time. Since then we have used them in actual observation of nanomaterials. This chapter presents mainly the work of the author in application of Cs -corrected HREM to nanomaterials. A short overview on Cs -corrected TEM/STEM by using hexapole and quadrupole-octupole correctors is provided and their advantages are shown primarily by experimental data obtained by the author and other groups in Japan and elsewhere for observation of nanomaterials. The related electron optics and detailed aberration correction methods in the present state of the art are described by Haider and Krivanek in other chapters of this monograph.

II. IMAGING THEORIES OF HRTEM USING ABERRATION CORRECTORS A. Projection Approximation in Electron Microscope Images The exit wave function below a sample is written as a complex function in a general form of waves in real-space coordinates, x, y, or two-dimensional (2D) positional vector x.

ψs (x) = ψs (x, y) = A(x, y) exp[iη(x, y)],

(1)

where A and η are the amplitude and phase of waves, respectively. Electron microscopes see potential distribution φ(x, y, z) of matters through the exit wave function. Establishment of the relationship between φ and A(x, y) or η(x, y) is a problem of electron-atom interaction (a real-space version of scattering theory). It forms the imaging theory of high-resolution transmission electron microscopes (HRTEM). When high-energy incident electrons >100 keV pass through thin samples of z in thickness, we can assume, to the √ ﬁrst approximation, that the electrons spread laterally only in order of λz, as the radius of Fresnel’s ﬁrst zone (Hirsch et al., 1977). For thin samples ≤ 10 nm, the spread is less than a few sub-nanometers. We can see approximately a projected structure of the matters from the exit wave function below the sample. Next, we establish a projected potential and its relationship with A(x, y) and η(x, y), as shown below.

φp (x) = φp (x, y) = φ(x, y, z)dz.

(2)

In a sample with interior electrostatic potential, most incident electrons are elastically reﬂected due to the potential ﬁeld without absorption. The

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reﬂection is determined by the reﬂective index n as the following formula in nonrelativistic approximation:

n(x, y, z) =

φ(x, y, z) E + φ(x, y, z) 1+ , E 2E

(3)

where E is an accelerating voltage. The phase shift at passing of the sample is

π 2π × (n − 1) × dz (geometrical pass) = η(x, y) = λ λE π = φp (x, y). λE

φ(x, y, z)dz (4)

In the formulation, we have neglected amplitude modulation and diffraction of periodical potential in crystalline samples. For the crystalline samples, the electrostatic potential is expressed in a form of Fourier series, where the ﬁrst coefﬁcient is the mean-inner potential φ0 relating to the reﬂection discussed above. This treatment is called the phase grating (object) approximation in optics, where samples cause only a phase modulation after interaction with samples. The formulas can be summarized as

ψs (x, y) = exp[iσφp (x, y)],

σ = π/λE,

(5)

where σ was called the interaction factor by Cowley (1981). The relativistic form of the factor (Reimer, 1984) is

2π , m0 2π , 2π E + m0 c2 2 = 1+ = 1+ 1−β , σ= λE E + 2m0 c2 λE m λE

(6)

where m0 and β are the rest mass and v/c, respectively. The function of an objective lens that connects the exit wave function with image wave function is expressed by a convolution operation of Fourier transform (FT) of a lens pupil function, that is, an exponential function of the wave aberration χ composed of spherical aberration (Cs ; third order aberration) and defocus ( f ; ﬁrst order aberration) as

ˆ ψi (x, y) = {exp[iσφp (x, y)]} ⊗ F[exp −iχ(u, v)] 2 χ = 0.5πCs λ3 u2 + v2 + π f λ u2 + v2

(7)

( f > 0; overfocus) (8)

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where u, v are reciprocal coordinates called spatial frequencies that have the dimension of 1/(length) (Reimer, 1984). The minus sign of χ in Eq. (7) was previously discussed by Saxton et al. (1983).

B. Scherzer’s Linear Imaging Theory and Its Extension to Higher Order Electrons are scattered (reﬂected in a picture of waves) by an electrostatic potential of atoms almost without absorption. It is well known that a phase shift of an incident electron wave in real space is the origin of scattering observed further away. The exit wave function is represented to a good approximation using the following phase grating as

. ψs (x, y) = ψs (x) = A(x) exp iσφp (x)

(9)

where amplitude modulation A(x) ∼ = 1, σ = π/λE, x and φp is a 2D positional vector and a potential projected along the incident electron (Cowley, 1981). The term A(x) is generated primarily from the so-called scattering absorption caused by a objective aperture. In HRTEM to observe single atoms, A(x) can be assumed to be unity. One of the important issues of HRTEM is how to make the image contrast (intensity difference) from the wave function with only a phase modulation. The weak-phase object approximation (WPOA) is a simple formulation in which the exit wave function and intensity are obtained by adopting the ﬁrst and second (linear) terms from expansion of the exponential function as

ψs (x) = 1 + iσφp (x) Is (x) =

1 + iσ(φp (x) − φp∗ (x)) + [σφp (x)]2 ,

(10) (11)

where the third nonlinear term in the intensity is ordinarily neglected. With perfect lenses and a spherical potential of atoms with φ = φ∗ , the second term as the phase contrast is also zero and the image intensity is unity. Here φ∗ is the complex conjugate. Including the effect of aberration and defocus of objective lens, the formula is modiﬁed as

Ii (x) = 1 + 2σφp (x) ⊗ s(x),

(12)

ˆ where s(x) = F[sin χ(u)] with χ of the wave aberration of the lens, and the symbols ⊗ and Fˆ represent a convolution operation and 2D FT, respectively (Buseck et al., 1988). The wave aberration in 1D form is

χ(u) = 0.5πCs λ3 u4 + π f λu2 ,

(13)

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where Cs and f are a third order spherical aberration constant and an amount of defocus ( f > 0, overfocus), respectively. The sin χ is called the phase-contrast transfer function (PCTF). The image intensity is then a direct representation of projected potential φ(x), although it is convoluted by a point-spread function (PSF) s(x). The Scherzer imaging condition is that the PCTF, sin χ is close to −1 for a wider spatial frequencies u(= (u, v)). For small values of u, χ(u) has 3 4 a minimum point and then increases rapidly by the term of 0.5πCs λ u . The minimum point is obtained by dχ/du = 0 at u = f /Cs λ2 . From this,

χ = −0.5π ( f )2 /Cs λ.

(14)

At χ = −2π/3, sin χ has a ﬂat area in relatively wider spatial frequencies. Then,

fS = − 4/3Cs λ = −1.15 Cs λ.

(15)

This is the Scherzer defocus condition (Scherzer, 1949; Cowley, 1981). Under this defocus condition, sin χ becomes again zero at uS = 1.5/ 4 Cs λ3 . When the resolution limit is deﬁned as the reciprocal number of the uS , the 4 3 Scherzer resolution is 0.66 Cs λ . The resolution is indeed different from the Abbe resolution as 0.61λ/α without aberration of an objective lens, where α is an acceptance angle of the objective lens. The defocus for obtaining image contrast inevitably introduces a blurring of images. The total resolution of HRTEM is determined by the size of exit wave function below samples, ordinary instrumental resolution determined by the Abbe theory, and the present lens aberration (spherical and chromatic) and defocus (the ﬁrst-order aberration) for obtaining image contrast. Figure 1 shows PCTFs of ordinary (Cs = 0.5 mm) and Cs -corrected TEM (Cs = 0 mm) around the Scherzer defocus, including chromatic damping in the linear imaging theory. The transfer characteristic of PCTF of Cs -corrected TEMs is extended up to the chromatic aberration limit (information limit) without oscillation around (0.1 nm)−1 . Correspondingly, the point spread function (PSF) in real space becomes sharper. This is also effective for recording electron holography, as described in Section III.G. In the linear imaging theory under WPOA, the effects of incident beam convergence and ﬂuctuations of accelerating voltage and objective current can be included by multiplication of damping envelope functions of E˜ j and E˜ D onto sin χ in Eq. (12), which are derived from Ej (h, g) and ED (h, g)

391

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1 (a) Information limit

u (nm21) 0 0

2

4

6

8

10

12

Zero-crossings

Cs 5 0.5 mm

f 5 242 nm

21 1 (b)

Information limit f 5 13 nm

u (nm21) 0 0

2

4

6

8

10

12

f 5 23 nm No zero-crossings

Cs 5 0 mm

21

FIGURE 1 Phase contrast transfer function (PCTF) in (a) a conventional TEM (Cs = 0.5 mm, f = − 42 nm) and (b) a spherical aberration corrected TEM (Cs = 0,

f = ± 3 nm, E = 200 kV, = 3 nm, β = 0.5 mrad).

functions [see Eq. (47)] in the second-order imaging theory (O’Keefe, 1979; Ishizuka, 1980) as,

E˜ j = exp [−π2 (β/λ)2 (Cs λ3 u3 + f λu)2 ], E˜ D = exp(−0.5π2 λ2 2 u4 ),

= Cc ( E/E)2 + (2 I/I)2 ,

(16) (17) (18)

where β, , E, Cc , and I are semi-angles of incident beam convergence, defocus spread, voltage ﬂuctuation, chromatic aberration constant, and objective current ﬂuctuation, respectively.

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The WPOA does hold only in very thin samples composed of light elements because the phase shift σφp should be small. Including strong scattering followed by dynamical scattering in crystals was an important issue with attempts to overcome it since. In the 1970s, strong phase object approximation (Fejes, 1977) and charge-density approximation (Lynch et al., 1975) were proposed to manage the problem, but were not helpful. Under these situations of imaging theory, trial-and-error matching between actual image intensity and simulated images of various kinds of structure models has been adopted for an extended time period. Recently, Lentzen (2008) reconsidered the resolution limit of Cs -corrected TEM in view of the size of the exit wave function below low-index-oriented crystals such as silicon and germanium, calculated using a dynamical diffraction theory. The latter part of this text section describes an extension of the imaging theory for thin samples to higher order, beginning with the phase grating approximation (Cowley, 1981). This approximation is a theory including dynamical diffraction effects between multibeams located in the zero-th order Laue zone, which means a projection approximation. The projection approximation is a better approximation than the WPOA, which holds in a wider range of experimental conditions (as explained in Section II.A). The exit wave function is

ψs (x) = exp[iσφp (x)] = cos σφp (x) + i sin σφp (x)

1 1 = 1 − σ 2 φp2 + · · · · · + i σφp − σ 3 φp3 + · · · · · , 2 6

(19)

where x is a 2D vector derived from r = (x, z) of 3D positional vector. The wave function in the image plane can be obtained by convolution of the FT of an exponential function of the wave aberration as

ψi (x) = ψs (x) ⊗ [c(x) − is(x)] ,

(20)

ˆ where c(x) = F[cos χ(u)]. Combining Eqs. (19) and (20) provides the image intensity, including the effect of object lens to the second-order terms, as

Ii (x) = 1 + 2σφp (x) ⊗ s(x) − σ 2 φp2 (x) ⊗ c(x) .2 .2 + σφp (x) ⊗ s(x) + σφp (x) ⊗ c(x) .

(21)

A previous paper (Yamasaki et al., 2005a) proposed a simple subtraction method for eliminating nonlinear image contrast such as half-spacing lattice fringes in structure images of crystalline objects. A theoretical consideration in their paper shows that dynamical diffraction effect

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between g and h still includes a small amount of contrast caused by nonlinear terms. Considering Eq. (21) and that c(x) is an even function against amount of defocus, and s(x), odd function leads to the following conclusion: When the phase grating approximation does hold, the imaging intensity obtained by subtraction of two images with positively and negatively equal amounts of defocus yields

Ii (x) = 4σφp (x) ⊗ s(x) + (3rd order terms)

(22)

It is noteworthy that this equation is derived without assuming the WPOA (Tanaka and Yamasaki, 2007). This expression is one step forward to formula including second-order terms and considering the variation of both ˆ of c(x)(= F{cos χ}) and s(x)(= F{sin χ}). The most important result from Eq. (22) is that the Cs -corrected TEM and the subtraction method provide a potential distribution in a form of linear term, without assuming WPOA and neglecting the second-order term as described in Eq. (22). A remaining problem is that low-resolution images of structures are difﬁcult to observe with high contrast because of the well-known small transmittance of sin χ in the spatial frequency region. Methods to overcome this problem are use of a phase plate inside the objective lens (Gamm et al., 2008) or the image deconvolution processing (discussed in Section II.H).

C. Realization of Charge Density Approximation by Aberration Correction With development of Cs -correctors, the charge density approximation becomes actually usable. The approximation was proposed in relation to the interpretation of N-beam lattice images by Lynch et al. (1975) and applied to very thin oxide samples, assuming phase grating approximation (high-voltage limit) and a perfect lens without Cs , chromatic damping and an objective aperture. Lens transfer function (LTF) is then represented at the in-focus condition in 1D form as

LTF(u) = exp −iπ f λu2 1 − iπ f λu2 (Cs = 0, f 0) 2 ˆ ψi (x) = Fˆ F(exp[iσφ p (x)]) 1 − iπ f λu ˆ iσφ(x)) = exp[iσφp (x)] − Fˆ iπ f λu2 F(exp 1 d2 = exp[iσφp (x)] − iπ f λ − 2 2 exp [iσφ(x)] 4π dx

(23)

(24)

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N. Tanaka

' & i f λ 2 λ f = exp [iσφ(x)] 1 − σφp (x) − σφp (x) 4π 4π . 2 d2 ] = − exp[iσφ ] σ φ − i σφ exp[iσφ (25) (∵) p p dx2 Ii = 1 −

λ f λ f λ f ∂2 φp (x) =1+ σρp (x) σφp (x) = 1 − σ 2 2π 2π 2πε0 ∂x (∵)

Poisson s eq.

∂2 φ (x) ρ(x) =− . 2 ε0 ∂x

(26) (27)

In the formulation, we use Cs = 0, f 0, and the linear approximation for image intensity calculation, but do not use the weak phase object approximation. It is useful to directly obtain the projected charge density in place of the projected electrostatic potential of samples from image intensity as described in Eq. (26). The effective area of the approximation was discussed in Figure 2 in Lynch’s paper. The paper shows a possibility of wider extension of the effective area than in the weak phase object approximation, which is an approach related to N-beam dynamical theory.

D. Lentzen’s Optimum Defocus for Cs -Corrected TEM Scherzer’s optimum defocus fS [Eq. (15)] is derived under constant values of third order spherical aberration coefﬁcient Cs (= C3 ). Development of aberration correctors provides one to have a freedom on Cs values and defoci for optimum HRTEM imaging. Lentzen et al. (2002) discussed criteria for the optimum defoci and Cs -values for these cases of (1) Scherzer’s method for obtaining phase contrast, (2) amplitude contrast imaging up to the information limit, and (3) Lichte’s method for least confusion (minimizing delocalization). They compromised delocalization and gain of phase contrast as follows: Lichte (1991) deﬁned the radius of the PSF using the following formula:

∂x R = max , ∂u

u ∈ [0, umax ] ,

(28)

where umax is the information limit and χ is the wave aberration. The minimum delocalization is obtained at a defocus as follows (Lichte’s focus)

R = 0.5πCs λ3 u3max if

3

fL = − Cs λ2 u2max 4

(29) (30)

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In this condition, however, phase contrast is not large. Increasing phase contrast is accomplished by going to the Scherzer defocus, − 43 Cs λ [Eq. (15)] and setting fS = fL , we have

Cs,opt =

1 64 3 27 λ u4max

fopt = −

16 1 . 9 λu2max

(31)

(32)

The Scherzer defocus, which gives a maximum phase contrast, and Lichte’s focus for a minimum delocalization are not matched in diagrams of Cs - f relation for the most optimized condition for each of them (Lentzen et al. (2002)). When cos χ is used for contrast transfer function (CTF) in HRTEM by using phase plates, both defoci are located in nearer conditions (Gamm et al., 2008). CTF in a form of cos χ is reasonable and natural for electron microscopy (Tanaka et al., 2004a). As another compromise, phase contrast can be maximized up to the information limit. Setting uS = umax , then

Cs,Sch =

16 1 3 λ3 u4max

(33)

8 1 3 λu2max

(34)

f = −

Such kind of discussion is concerned only with viewpoints of electron optics and an application of the Scherzer’s theory for imaging of single atoms. It does not concern the relationship between thicker samples and their exit wave function. It is also not related to imaging of crystals with dynamical diffraction effects and interference effects on lattice images (discussed in the following section).

E. Imaging of Thicker Crystals and Nonlinear Image Contrast Thanks to the recent development of spherical-aberration correction, the point-to-point resolution of 200 kV TEM has now been attained to ≤ 0.1 nm (Haider et al., 1998b; Tanaka et al., 2002). However, a fundamental problem still exists for image contrast in HRTEM: nonlinear image contrast in lattice images. It is well known that the contrast originates from interference between diffracted waves from crystals when they become strong in

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thicker crystals. It is a crucial problem that the image contrast does not correspond to actual projected structures of samples. Since 1960, nonlinear image contrast has been known as half-spacing lattice fringes, which were used for resolution testing of high-resolution electron microscopes (Komoda, 1966). In observation of thin gold ﬁlms, for example, the spacing of the half-spacing lattice fringes is 0.102 nm as half of (200) spacing, and 0.117 nm as half of (111) spacing, respectively. At that time, the resolution of instruments was limited to ∼ 0.2 nm and we did not recognize serious problems for image interpretation because the halfspacing fringes are observed near in-focus conditions, not the Scherzer defocus. This situation coincides with a good justiﬁcation of the linear image theory of HRTEM as described in Section II.B. The nonlinear image contrast can be explained by using multibeam lattice images as follows. When electrons are incident on a crystal along a high-symmetry zone axis (such as the [001] axis.) many diffracted waves are emerging from the exit surface of the crystal. Multibeam lattice images and their optimized “structure images” are formed by interference between a transmitted wave and diffracted waves. Let the wave ﬁeld on the image plane be denoted as follows, including the defocus of an ideal objective lens (Cs = 0) .

ψS (r) = ψ0 + ψg + ψ−g + ψ2g + ψ−2g + ψ3g + ψ−3g + · · · = | 0 | exp 2πik0 · r + i g exp 2πi k0 − kg + g · r exp iφg + i −g exp 2πi k0 − k−g − g · r exp iφ−g + · · · , (35) where g and φg are amplitude and phase of the kinematical structure factor and kg is the excitation error. The dynamical diffraction effect is omitted for simplicity in this explanation; it is discussed in Section II.G. The intensity of the lattice image is written as

IS (x) = |ψ0 |2 + |ψg |2 + |ψ−g |2 + |ψ2g |2 + |ψ−2g |2 + · · · ∗ + ψ0 ψg∗ + ψ0∗ ψg + ψ0 ψ−g + ψ0∗ ψ−g ∗ ∗ + ψ0 ψ2g + ψ0∗ ψ2g + ψ0 ψ−2g + ψ0∗ ψ−2g + · · · ∗ ∗ ∗ + ψg ψ−g + ψg∗ ψ−g + ψ2g ψ−2g + ψ2g ψ−2g ∗ ∗ + ψ3g ψ−3g + ψ3g ψ−3g + · · · + ψg ψh∗ + ψg∗ ψh + · · · ,

(36)

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where ψg is a plane wave with g-direction in real space. The terms in the ﬁrst row in Eq. (36) correspond to amplitude contrast, in which the spatial frequencies are ordinarily smaller than (1 nm)−1 , making background contrast in HRTEM images. The terms inside the parentheses in the second and third rows are three-wave lattice fringes formed by 0-wave and ±ng-wave (n = 1, 2, 3, .....) with respect to each of the parentheses with two terms. This is the Fourier image (Cowley and Moodie, 1957). The terms in the fourth row are the half-spacing lattice images. Those in the sixth row are another type of nonlinear term formed by g and h-waves, although their intensity is not as strong. The higher-order effect caused by aberration of the objective lens, including the size of electron source, can be described by the second order contrast transfer theory developed by O’Keefe (1979) and Ishizuka (1980) as follows. The intensity in the image plane is described in terms of transmission cross-coefﬁcient (TCC), T, as

Ii (x) =

T u , u F u F∗ u exp − 2πi u − u · xi du du

T(u , u , f ) =

(37)

J0 (u)D( )K(u + u , f + )

× K ∗ (u + u , f + )d( )du,

(38)

where F is a structure factor of samples, J0 , angular distributions of electrons from an electron source, D, distribution of defocus ﬂuctuation, , and K, phase component of wave aberration functions [see Eq. (8)] as

K(u, f ) = exp −iχ(u, f ).

(39)

This expression accurately describes multibeam lattice images including half-spacing lattice images as discussed above. We shall discuss decreasing the nonlinear image contrast caused by symmetrical waves of g and −g, 2g, −2g, and so on and nonsymmetrical ones by h and g as shown in the fourth/ﬁfth and sixth rows in Eq. (36). The imaging theory, including higher-order terms than those in Ishizuka’s paper, was reported by Mitome (1990). The formulation was recently further developed for Cs -corrected TEM (Mitome, 2007). The TEM imaging theory has two types of formulations; they are based on (1) Cowley’s phase grating and (2) N-beam lattice images, including structure images. In both formulations, ordinarily, only linear terms are used in the calculation of image intensity. When phase shift in real space

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or diffracted waves in reciprocal space are not small, second-order term such as nonlinear terms should be considered. The next section explains a proposal by the Juelich group for managing the nonlinear image contrast. Section II.G discribes a treatment for the N-beam lattice image formulation.

F. Imaging with Negative Aberration Constants The previous section discussed the non correspondence of the nonlinear image contrast in multibeam lattice image to atomic columns of crystals. This is related by strong diffracted waves and their interference effects. The problem can be described as a simple form in Eq. (21). Assuming c(x) = 0, which affects a background intensity in low spatial frequencies, the third term in the equation may be neglected. The second term is a linear image contrast, which varies in white/black contrast with Cs and the amounts of defoci by the convolution operation of FT of the wave aberration. The fourth nonlinear term overlaps onto the linear image contrast or appears between the linear contrast in some case. To manage this problem, Jia et al. (2003) found an effectiveness of negative Cs , judging from various kinds of image simulation of thin SrTiO3 crystals. Although they did not clarify the origin of the effectiveness, the essence of their treatment of nonlinear image contrast can be summarized as follows. They explained that for positive Cs values, the nonlinear term complicates the atom-corresponding linear image contrast, but for negative Cs , it reinforces the linear image contrast. The idea is simply explained in a different way as follows. Figure 2 shows so-called Thon’s diagrams (1966) for negative and positive Cs values. The Scherzer defocus appears in overfocus or underfocus d(nm)

(a)

Sch

0.8

W

B

Cs 5 20.5 mm 2300 2200 2100

(under)

d(nm)

(b)

Sch

W

0.6

B

W

0.8

B

0.6

0.4

0.4

0.2

0.2

0

1100 1200 1300

Df (nm)

(over)

2300 2200 2100

(under)

0

W

B

Cs 5 10.5 mm

1100 1200 1300

Df (nm)

(over)

FIGURE 2 Thon's diagrams show the relationship between amount of defocus (f ) and spacing imaged (d). (a) Cs < 0, (b) Cs > 0. W and B denote white contrast and black contrast, respectively; the Scherzer defocus is denoted by Sch.

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sides. Correspondingly, the underfocus condition gives black image contrast, but for the overfocused one white contrast is seen near the Scherzer defocus. This holds also for the smallest Cs values, although the spacing with the high-contrast versus defocus relationship is slightly deformed. Reconsidering Eq. (21), the fourth nonlinear term provides white contrast at any time (excess electrons against background). The key point of their idea is that where Cs < 0 and f > 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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N. Tanaka

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

Co

O

O

Li

Li 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) =

4| 0 || g | cos(2πg · r + φg )

g

× 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

0.3 nm

PCTF

1

0.3 nm

PSF

1

0 21

0

(c)

step function

0.3 nm

0.3 nm

(d)

PSF

envelope function

1

1

0

0

0.3 nm

0.3 nm

PSF

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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N. Tanaka

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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407

(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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N. Tanaka

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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N. Tanaka

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

0.20 nm A

C

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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Spherical Aberration-Corrected Transmission Electron Microscopy for Nanomaterials

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)

Cs-corrected TEM (c)

(b)

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 <5-nm thickness was prepared by vacuum deposition of MgO onto (011) faceted crystals of sodium chloride (NaCl) from an electron beam evaporation source. Furthermore, a centered area of the ﬁlm was sputtered by a focused electron beam inside the TEM. The thickness of the area is almost one or two unit-cell thickness of MgO

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N. Tanaka

Cs . 0, 200 kV

(a) (b) Mg

O O

0.21 nm

FIGURE 10

MgO(110)

See distribution of its potential

Visualization of oxygen (Z = 8) atoms in a [011] MgO crystal.

(110) planes. In any location Mg atomic columns can be identiﬁed as well as oxygen columns between them (Tanaka and Yamasaki, 2004). In this observation, the author and colleagues have chosen a positive Cs value in contrast with the data obtained by the Juelich group on SrTiO3 crystals (Jia et al., 2003). The present image shows a distribution of the image contrast, which may be connected with a distribution of projected potentials around the oxygen atomic columns under an appropriate imaging condition. Those obtained by the Juelich group are only sharp peaks emanating from a channeling phenomenon of electrons along the high symmetrical axis of the crystals.

D. Removal of Nonlinear Image Contrast by Subtraction Method Phase-contrast images in HRTEM are formed by interference of a direct wave and diffracted waves. Until now, the linear terms formed by the interference have been used most often. The point-to-point resolution of conventional TEM previously was ∼0.2 nm and the nonlinear interference fringes (such as half-spacing lattice fringes) formed by two diffracted waves did not disturb the linear image contrast due to their weaker contrast (as explained in Sections II.E and II.G). After the Cs correction, the half-spacing fringes (such as those of gold 400 spacing) appear appreciably in the image. It has become necessary to remove or decrease the nonlinear contrast for proper image interpretation.

Spherical Aberration-Corrected Transmission Electron Microscopy for Nanomaterials

(a)

413

Intensity

exact focus

(b) overfocus

(c) underfocus

only non-linear component

(a)

position linear component non-linear component linear component non-linear component

(d)

(b) O

Mg

Mg

O

(e)

0.2 nm

(c)

(f)

atom positions

(d) subtraction (b) – (c)

only linear component

FIGURE 11 A newly developed image-subtraction method for decreasing nonlinear image contrast in simulation. Right: (a) atomic arrangement in MgO, (b) 3 nm overfocused image, (c) 3 nm underfocused image, (d) substraction image, (e) substraction and deconvolution image, (f) projected potential.

The author and his colleagues are studying the image formation process in Cs -corrected HRTEM and have developed a simple method of subtraction of two images obtained by Cs -corrected TEM (Yamasaki, Kawai, and Tanaka, 2005a). Figure 11 shows an example of the images before and after the subtraction for MgO(011) crystals. Artiﬁcial image contrast not corresponding to the actual atomic columns is perfectly removed by the present method. Furthermore, we have developed a newly revised imagedeconvolution method to recover less transmission of information in lower spatial frequencies (see Section II.H). By combining the use of subtraction and deconvolution methods, we have succeeded in obtaining the TEM image that corresponds well to projected structures of crystals as shown in Figure 11e (Yamasaki et al., 2008). We also proved theoretically that the present subtraction method also can be applied to amorphous samples by assuming the phase object approximation (Tanaka and Yamasaki, 2007). Another way to solve the nonlinear image contrast is to adopt negative Cs values, as proposed by the Juelich group (Jia, Lentzen, and Urban, 2003). In this case, an ordinary linear term has a positive contrast (white contrast in the bright-ﬁeld image), which is in agreement with a positive (white) nonlinear image contrast. As a result the non-linear contrast may not complicate the linear contrast as much (see Section II.F). As previously described, the method cannot manage dynamical diffraction effects and nonlinear images between atomic columns.

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E. Z-Slice and Tomographic Imaging in Cs -Corrected TEM Application to Single-Wall Carbon Nanotubes, Fullerenes, and Pt-Rh Chain Molecules Three-dimensional EM including electron tomography is one of the cutting-edge topics of advanced EM, particularly toward atomic resolution. Two methods for data collection are reported on this purpose: (1) computed tomography by tilting samples and using the backprojection algorithm, and (2) z-slice imaging ﬁrst performed in annular dark-ﬁeld ADF-STEM (Borisevich et al., 2006). The author and colleagues have performed another trial by using Cs -corrected TEM. Figure 12 shows a HRTEM image of the side wall in a multiwall carbon nanotube, where lattice fringes of 0.24 and 0.14 nm are resolved due to image improvement by a Cs corrector (Tanaka et al., 2004b). Even using the Cs -corrected TEM, however, the image is not essentially different from those in previous reports, because the lattice fringes (a)

0.246 nm

(b)

0.142 nm

FIGURE 12 Visualization of smallest lattice spacing of graphene sheets in multiwall carbon nanotubes.

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415

are overlapped fringes or Moiré fringes formed from upper and lower multilayers of the nanotube. Visualizing detailed structures such as defect structures in graphene sheets requires obtaining images of the upper or lower sheets separately in a single-wall carbon nanotube (SWCNT). For this purpose, a new method of z-slicing in Cs -corrected TEM was developed (Hirahara et al., 2006). The principle is based simply on Fourier images studied by Cowley and Moodie (1957). The image contrast of a phase grating in TEM is expressed in the Fourier images, whose period of image intensity variation is written as

zF = d2 /2λ

(54)

where d is spacing of the phase grating. In the Fourier images the image contrast at in- focus is zero, and at overfocus condition by the above value of Eq. (54), where a wave ﬁeld beneath the sample is observed, the contrast is white, and in the opposite underfocus, the contrast is black. For resolving hexagons with a spacing of 0.14 nm in a graphene layer, the Fourier period becomes ∼3 nm at 120-kV accelerating voltage. Using a SWCNT of 3 nm diameter, the upper or lower graphene sheets can be focused individually with maximum black or white contrast. We tried to observe the SWCNT under those conditions. In Figure 13a and b with different defoci, the direction of arrays of hexagons differs. The image contrast agrees with simulation of a structure model in Figure 13c

Experimental images

Simulated images

(a)

(c)

(b)

(21,18) SWNT

FIGURE 13 Cs -corrected TEM visualization of hexagons of carbons in a single-wall carbon nanotube (SWNT). Separated imaging of the left upper (a) and lower (b) planes as shown in a SWCNT model (c). (See color insert).

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N. Tanaka

[nm] Cl Pt Pt

0.25959 0.25987

Rh Rh

0.27337

Pt

Pt

0.284500

Pt

Pt

0.30634 0.284500

Pt

Pt 0.273370

Pt

N

Cl

H

0.259870 Rh

Rh C

Pt

0.25959

O Pt Pt CI

FIGURE 14 Atomic arrangement of a 1D Pt-Rh chainlike molecule. The figures indicate the separation distance between framed atoms determined by X-ray diffraction. (See color insert).

determined in advance by using nano-beam electron diffraction (Hirahara et al., 2006). Based on the successful results, the trials have moved to observation of single molecular chains composed of platinum and rhodium atoms as shown in Figure 14. The molecules were dispersed on a vacuum-deposited single crystalline ﬁlm of magnesium oxide (MgO) by using a polar solvent (Tanaka et al., 2006). Figure 15 shows high-resolution images of two kinds of molecules on a MgO ﬁlm. By using the 200 lattice fringes of MgO ﬁlms, we calibrated the direct magniﬁcation of the image and measured the spacing between black dots in the molecule accurately. The ﬁgures were in good agreement with those determined by X-ray diffraction. Proceeding with this technique, we may be able to resolve 3D molecules tilted from normal to the direction of incident electron. Figure 16 shows a serial stack of simulated TEM images along z-direction (Hirahara et al., 2007). The depth resolution using a STEM probe is expressed as a formula similar to the intensity distribution at a focal point in optical microscopes (Born & Wolf, 1970), as

D = λ/α2 ,

(55)

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(a)

417

(b)

0.252 0.257 0.291 0.270 0.283 [nm] 1 nm

1 nm

Cs 5 6 mm, E 5 200 kV

FIGURE 15 Cs -corrected TEM images of two kinds of Pt-Rh molecules supported on a MgO thin film.

Over-focus

Under-focus

FIGURE 16 Series of simulation images of the Pt-Rh molecule stacked at vertical direction (amounts of defocus are shown on the scale at the left). (See color insert).

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N. Tanaka

where α is the convergence angle of incident electrons (see Section V.A). In the present TEM case, however, the situation is phase-contrast imaging of a phase grating. The D is that expressed in the formula of Eq. (54). Decreasing the spacing the depth resolution improves to 2 nm and results in resolving the carbon-carbon bond in the one layer of hexagons from SWCNT. The value is comparable with that attainable in Cs /Cc -corrected ADF-STEM, which was calculated by using the formula [Eq. (55)] by Borisevich et al. (2006; see Section V.B). For tomographic observation of fullerene-related materials, we can provide ultimate examples. Early in the 1990s, we tried to observe vacuumdeposited C60 on MgO ﬁlms in a plane-view mode and obtained a related contrast of the individual molecules in comparison with their image simulation by using a non–Cs -corrected HRTEM (Tanaka et al., 1993). Recently, Suenaga’s group in Tsukuba (Liu et al., 2007) observed chemically modiﬁed C60 molecules by using a 120-kV Cs -corrected TEM with 0.14-nm point-to-point resolution. Figures 17a–c show a set of HRTEM

(a)

(d)

(g)

(b)

(e)

(h)

(c)

(f)

(i)

1 nm

FIGURE 17 Cs -corrected HRTEM images of C60 anchored to a SWCNT through an organic molecule. (From Liu, Suenaga, and Iijima, 2007). (See color insert).

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images of C60 fullerenes chemically functionalized with pyrrolidine in which the functional groups act as anchors at an ouside surface of a SWCNT. Each ﬁgure shows one fullerene molecule located at the surface of the SWCNT in different orientations. Each fullerene molecule does not appear in a simple ring-shaped contrast but rather exhibits some intramolecular features. To corroborate the observed HRTEM images, the image simulations for a C60 fullerene molecule were performed systematically in more than 30 different orientations considering the Ih symmetry. Compared with the simulated images shown in Figures 17d–f, the molecular orientation can be unambiguously assigned for these experimental images. The molecule showing a six-membered ring contrast inside is identical to the simulated one (image d), which corresponds to the C60 molecule aligned parallel to the sixfold symmetry axis (shown in Figure 17g). Similarly, the image in Figure 17b corresponds to the simulation (image e), in which two pentagons are overlapped in projection (Figure 17h) and thus gives rise to a small circular contrast in the middle of the C60 fullerene. It should be noted that the C60 fullerenes could be moving during image recording since they are loosely anchored to the SWCNT.

F. New Development of Selected Area Nanodiffraction in Cs -Corrected TEM As is well known in textbooks on electron microscopy, selected area electron diffraction has positional errors for selections of investigating areas due to the method of area selection. In ordinary TEMs with the threelens system, a selected area aperture is inserted on an image enlarged by about 100 times using an objective lens, where diffracted beams with larger angles have lateral shifts on the image plane due to spherical aberration of the lens. The shift is dependent on diffracted (scattering) angles α as

x = Cs α3 ,

(56)

where Cs is the 3rd order spherical aberration constant. Most cases are concerned with diffraction spots as far as 10 nm−1 , which correspond to 222 spots for Al, and 440 or 333 spots for Si. According to Eq. (56), the area from which a 440 diffraction beam comes is displaced by 9 nm from the area responsible for the direct beam and by 18 nm for a ¯ beam. When an apertures is used to select areas of 100-nm diameter, 4¯ 40 approximately one-ﬁfth of the intensity of the 440 beam comes from other areas. Therefore, a selected area aperture <100 nm in diameter has not been used, which is not truly meaningful. When a Cs corrector is used to make

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Cs equal to 0, smaller apertures can be used to analyze nanometer-sized areas while keeping diffraction spots sharp. Yamasaki et al. (2005b) have ﬁrst applied a selected area nanodiffraction (SAND) to an interface between Si0.54 Ge0.46 and TiSiGe alloys a stacking fault in the crystals using a Cs -corrected TEM. Figure 18 shows a SAND experiment on the SiGe/TiSiGe semiconductor bilayer. In Figure 18a, areas near the interface are selected by a selected area aperture 20 nm in diameter. The distance from the interface is 2 nm. The corresponding diffraction pattern shows only a [110] pattern of diamond structure as far as 333 spots. Figure 18b shows a result with the aperture on the interface. The pattern is composed of spots of SiGe and several other spots from polycrystalline TiSiGe alloys. These results show success of electron diffraction from 20 nm areas with incident convergence angle less than 10−4 rad. They checked effects of C5 and defocus and concluded those are negligible for this selected area nanodiffraction. Yamasaki et al. (2006) prepared for much smaller selected area apertures of ∼0.5 μm by a fabrication process using ion etching. Figure 19a shows an image of a selected area aperture inserted on an image of silicon lattice

(a)

(b)

(c)

Ti-Si-Ge alloy

Interface

111 lattice fringes

Si12x Gex

SAD aperture

5 nm

333 222 111

FIGURE 18 An example of selected area nanodiffraction of TiSiGe/SiGe interfaces by using a 200-kV Cs -corrected TEM.

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(a)

421

(b) 220 111

Airy disk in electrons 0.31 nm

2 nm

FIGURE 19 5-nm selected area nanodiffraction from an Si crystal obtained in a 200-kV Cs -corrected TEM.

fringes of 0.31 nm. By using the small aperture in the Cs -corrected TEM, analysis of individual stacking faults in silicon crystals has become possible with more parallel incident beams than those of the nano-beam electron diffraction method (Zuo et al., 2003). The technique affords the advantage also of diffraction imaging using the oversampling technique. The technique must be used for areas in which the image intensity is known in advance. In previous studies, vacuum areas were used when the sample is a carbon nanotube (Zuo et al., (2003)). However, it is not applied to cross-sectional samples and thin ﬁlms. The present technique solves this problem. The operator can set freely the oversampling area where the image intensity is known by using the present selected area aperture. The image intensity outside the aperture is 0 (as shown in Figure 19a). This selected area diffraction method yields sharp diffraction spots (Fig 19(b)), which contribute to the feasibility of phase reconstruction in diffractive imaging algorithms. It is noteworthy that the method is usefully applied also to angleresolved electron energy loss spectroscopy (AREELS) from a nanometerselected area of samples with Cc - correction, because diffracted beams with inelastic scattering are more shaperly selected than those obtained by the nano-beam method. The present technique has the additional advantage for spatial coherence of incident beams compared with nano-beam electron diffraction where the electron beam is focused by condenser lenses and the preﬁeld of objective lens (Dwyer et al., 2007). Figure 20 shows a comparison of spatial coherence judging from intensity distribution of Airy disk between (b) nano-beam diffraction and (c) the present Cs -corrected selected area electron diffraction methods (Dwyer et al., 2007; Morishita et al., 2007). Because of the more parallel nature of the incident beam in the present method, the intensity variation is more pronounced.

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(a)

Airy disk intensity profiles

(b)

Cs-corrected STEM nano-beam diffraction (NBD)

1

100 Intensity (log scale)

Cs-corrected selected area nano-diffraction (SAND)

(c)

dotted: EXP

EXP

1021

0.1

1022

0.01

1023

0.001

1024 2100

250

More spatially coherent

0.0001 0

50

100

Pixels

FIGURE 20 Comparison of the intensity distribution of the Airy disk showing the coherence (a), from Dwyer et al., 2007 (b); from Morishita, Yamasaki, and Tanaka, (2007) (c). Cs -corrected STEM method (b) and Cs -corrected TEM method (c). (See color insert).

G. Advantages of Cs -Correctors for Electron Holography Advantages of Cs correctors for electron holography are not sufﬁciently understood by most electron microscopists. Lichte’s group (Lichte et al., 2004; Geiger et al., 2008) has been engaged in this issue theoretically since 1990 and experimentally since 2004 by using a 200-kV Cs -corrected TEM. The Cs corrector extends the spatial frequencies of transmitted waves less than π/4 in phase modulation through objective lens. The limit of the PCTF, sin χ(u) is greatly improved as described earlier. In electron holography, it is important that the other, cos χ(u) function be extended to higher spatial frequencies because a phase modulation of the exit wave ﬁeld below samples is transferred through the function into a phase modulation of image wave functions, which is expressed by the local fringe shifts in holograms. What does Cs -correction in TEM contribute to electron holography? Geiger et al. (2008) pointed out two advantages, such as decrease of PSF and increase of signal to noise ratio (SNR) in recording holograms. The improvement by Cs correctors is discussed by Geiger et al. (2008) in terms of the detection limit of phase difference of exit waves below samples. They derived a formula of the detectable phase difference (δφ) as

δφ =

4πSNR Cs , |μ| · Vinel · Vinst · VMTF −ln(|μ|) ekB2 εtDQE(qc )

(57)

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where μ is spatial coherence, and Vinel , Vinst and VMTF are the contrast factors due to inelastic scattering, instabilities, and modulation transfer function of the Charge Coupled Device camera, respectively. B is the brightness of the gun, ε, illumination of ellipticity, t, exposure time, k, wave number of electrons, and DQE, quantum efﬁciency of the CCD camera, respectively. Straightforwardly from the formula, decrease of Cs provides higher sensitivity of phase detection in holography. Another advantage is improvement of accuracy of holographic aberration measurement, which has been performed since 1990. In ordinary Cs -correction in TEM, Zemlin’s tableau (Zemlin et al., 1978) is used for this purpose, but electron holography yields directly detailed information of various kinds of aberration of the objective lens in the image wavefunction. The aberration information can be derived from the amplitude and phase distributions in the wave function, which are more accurately obtained by using Cs -corrected TEM holography. Geiger et al. (2008) described the improvement of signal to noise ratio (SNR) in career fringes in electron holograms by a factor of 1.7 after Cs -correction (as shown in Figure 9 in their paper). Lichte et al. (2004) summarized the advantages of Cs -correctors in electron holography as follows: • Information limit, which is restricted only by the chromatic envelope function, extends to higher spatial frequencies. • Axial coma disappears automatically with Cs . Therefore, only 6 instead of 9 microscope parameters must be corrected. • Aberration correction with the holography method is more accurate due to a considerably smoother dependence of χ-function [Eq. (8)] with spatial frequency, which allows for more accurtate numerical modeling of the wave aberration with fewer pixels. • Since there is no axial coma, the tilt of illumination connected with elliptical illumination does not violate isoplanaticity, which requires complicated correction. • PSF is virtually zero. Therefore, the hologram width can be reduced to the interesting area (e.g., a grain boundary and its surroundings). The dose can be increased (e.g., by a factor 5)., which reduces (improves) the phase detection limit. • Phase loss of conventional TEM imaging is overcome in electron holography. Phase information can be evaluated at atomic dimensions (e.g., for atomic materials analysis) and for large-area phase modulations from mechanical, electrical, and magnetic ﬁelds; for example. • The reconstructed wave is linearly transferred through the optical system, which is easier to interpret. • From analysis of Fourier space, local object tilt and thickness change can be determined, therefore, increasing the accuracy of interpretation.

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• Only zero-loss energy information is reconstructed, which further assists interpretation; • There is clear evidence that the Stobbs factor is reduced in holography; • All coherent aberrations, that may occur in the future, can be corrected (e.g., ﬁfth-order spherical aberration, star aberration), simply by incorporating those into the correcting phase plates.

IV. ACTUAL ADVANTAGES FOR OBSERVATION BY CS -CORRECTED STEM A. Increased Point-to-Point Resolution The advantages of Cs -corrected STEM are well understood from the viewpoint of the probe forming lens (Krivanek et al., 1999). The aberration correction allows the lens to be used in wider convergence angles, and then the probe size is decreased from the principle of FT and the probe current becomes larger compared with the conventional one. This improvement is easily observable in a Ronchigram of STEM as a shadow image, in which a type of diffraction pattern is obtained by a stationary incident probe without objective (condenser) apertures (Spence, 2003). The ﬂat area with an equal phase in the center is extended from ∼10 mrad into ∼40 mrad at 200 kV after the Cs correction. The decreased probe size allows HR-STEM image of ≤ 0.1 nm in point-to-point resolution, and the brighter probe minimizes analysis time for various kinds of advanced materials such as semiconductors. The optimum defocus condition is slightly different from that in TEM (Scherer defocus), which was derived for STEM imaging and analysis by Mory et al. (1987). Figure 21 shows a 200-kV ADF-STEM image of a (011) diamond ﬁlm obtained in our laboratory. Without a monochromator below an electron gun, the dumbbell structure of 89 pm in spacing is resolved clearly. This ensures higher electrical and mechanical stability of the double correctors TEM/STEM (JEM-2100FCS-TEM/STEM) (Saitoh and Tanaka, 2008b) Improvement of the resolution allows observation of lighter elements, such as aluminum (Al), which was not visible in non-Cs -corrected STEM. Abe’s group at Tokyo University showed Al atom distributions in Al64 Cu22 Co14 by using the Z-contrast mode. Figures 22a and b show the ADF-STEM images of the decagonal Al64 Cu22 Co14 , obtained before and after Cs correction, respectively. The brightest spots from the atomic columns of Cu or Co appear to be much sharper after the Cs correction. Second, by carefully inspection of the image after the Cs correction, it

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89 pm

FIGURE 21 Annular dark-field STEM image of a [011] diamond crystal, showing the dumbbell structure of 89-pm separation at 200 kV.

shows many weak spots between the brightest spots; these are reasonably attributed to the Al atoms. This is a remarkable effect of Cs correction with suppression of secondary maxima that exist in uncorrected probe tails. The incoherent imaging allows use of a simple deconvolution procedure, as well as the maximum entropy (ME) algorithm, which gives the safest, most possible structure model that ﬁts the experimental image (shown in Figure 22c). It is noteworthy that even with applying the ME deconvolution processing, the weak peaks were never seen on the non-Cs corrected STEM (Figure 9(a)). Only when sufﬁcient SNR is provided by the Cs -corrected sharper/brighter probe, does the ME deconvolution method work effectively to represent light atomic sites (Taniguchi and Abe, 2008). On the basis of these experimental data, Taniguchi and Abe (2008) concluded that atomistic disorders are rather signiﬁcant around the center of the ﬁvefold symmetry clusters of 2 nm across, which are the building units of the decagonal Al64 Cu22 Co14 . Such localized disorders are likely to be the result of weak phason ﬂuctuations that do not destroy the long-range quasiperiodic structure. Comparison between Cs -corrected STEM and TEM images of the quasicrystals was shown previously in Figure 9. In pursuit of ultrahigh-resolution images of Cs -corrected TEM and STEM, a government-support project is ongoing in Japan (JST-CREST)

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(a)

Before Cs correction

1 nm

(b)

After Cs correction

(c)

After ME deconvolution

FIGURE 22 ADF-STEM images of AlCuCo decagonal quasicrystals. (a) no Cs -correction, (b) Cs -correction and (c) after maximum entropy deconvolution image processing. (From Taniguchi and Abe, 2008.) (See color insert).

to realize 50 pm (= 0.05 nm) point-to-point resolution. A new 300-kV TEM/STEM instrument has been developed with a cold-ﬁeld emission gun and new nonsymmetrical image and probe correctors (JEM: R-005). The tentative resolution test in real space, not Fast FT patterns, is 63 pm of dumbbell image of [211]-oriented GaN crystals. The Japanese researchers and engineers independently developed the correctors and aberration measurement software using Ronchigram. Figure 23 shows ADF-STEM images from a GaN [211] fragmental crystal [image (a) show the raw image, (b) the low-pass-ﬁltered image, (c) the line scan, and (d) the Fourier spectrum] (Sawada et al., 2007). Much higher resolution images down to 0.05 nm are being obtained by using Ge [114] crystals. Development of Cs -corrected STEM using the nonsymmetrical corrector was performed in Tokyo Institute of Technology and Furuya’s group in Tsukuba (Mitsuishi et al., 2006).

Spherical Aberration-Corrected Transmission Electron Microscopy for Nanomaterials

(a)

427

(b)

0.5 nm 63 pm (c) 63 pm

(d)

Raw data Low pass filtered data

(e)

63 pm

(000) 377 pm (f)

94 63 pm pm

189 pm

FIGURE 23 Achieving 63-pm resolution by a new 300-kV ADF-STEM using GaN [011] crystals. (From Sawada et al., 2007.) (See color insert).

B. Reduction of Analysis Time of STEM-EDX and EELS Mapping Aberration correction provides smaller and brighter electron probes for STEM imaging, which are very beneﬁcial for elemental analysis and STEM dynamic observation. Figure 24 shows a high-contrast detection of small germanium (Ge) quantum dots grown on Si(100) crystals and an ultimate analysis of Ge/SiO2 /Si(001) interfaces by using Cs -corrected STEM-EELS (images b through e) (Tanaka et al., 2008). A Ge-rich thin area at the interface also is clearly detected as well as in the ADF image. Figure 25 shows another example of dynamic observation of movement of an antimony (Sb) dopant atom in silicon crystals (Yamasaki et al., 2007). This series of images was recorded in each 11 sec. The authors analyzed the jump process of the impurity atom from the series of images. The

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(f)

(a)

(b)

(c)

Si 92 –148 eV

(d)

O 521– 577 eV

(e)

Ge 1238 –1336 eV

ADF Image

FIGURE 24 Cs -corrected STEM image and STEM/EELS mapping of Ge quantum dots, (a) low magnification image, (b)–(d) STEM EELS mapping, and (f) high-resolution ADF-STEM image.

1

L

2

R

3

R

4

L

0.135 nm

FIGURE 25 Series of Cs -corrected STEM images showing the “jumping” process of an antimony (Sb) atom in a silicon crystal.

mechanism of the jump process is now studied by using various diffusion models. At present, they consider that thermal agitation excited by incident electrons is the primary cause of the jump.

C. Other Advantages Direct correspondence with the first-principle calculation Pennycook’s group (2002) pioneered high-resolution ADF-STEM observations on various kinds of ceramic boundaries. Ikuhara’s group at Tokyo

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University has now begun to perform similar ADF-STEM observations by using a Cs -corrected 200-kV STEM (JEM-2100F CS). Doping of Praseodymium (Pr) atoms into ZnO ceramics is a method to obtain highly nonlinear current voltage characteristics. The presence of the Pr atoms at the grain boundaries is considered a key factor. Figure 26 shows a high-angle ADF (HAADF)-STEM image of a Pr-doped ZnO [0001] 7 grain boundary (Sato et al., 2006). Because the HAADF-STEM image is an incoherent image, location of the atomic columns was directly observed as the bright spots. In addition, there are much brighter spots periodically along the grain boundary (indicated by white arrows). Because the intensity of the image increases as the average atomic number (Z) included in the atomic column, the brighter spots indicate the presence of Pr atoms (Z = 59). Comparison with the ﬁrst-principle calculations as shown in Figure 26b revealed that Pr-doping does not change the grain-boundary atomic structure (indicated by the set of open circles) signiﬁcantly, and Pr atoms substitute the speciﬁc Zn sites at the grain boundary. Based on the grain-boundary atomic structure and the location of Pr segregation, a role of Pr atoms on the nonlinear current voltage characteristics was clariﬁed. Interpretation of ADF-STEM images still is not straightforward. Detailed theoretical studies for such imaging have been performed by many researchers; studies include STEM elemental mapping images using inelastically scattered electrons (Allen, 2003). These types of study should explore a new possibility of Cs -corrected STEM.

(a)

1 nm (b)

Zn

O

Pr

FIGURE 26 Cs -corrected ADF-STEM images of ZnO interfaces including Pr atoms (a) and the analysis by the first-principle calculation (b). (From Sato et al., 2006.) (See color insert).

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V. THREE-DIMENSIONAL OBSERVATION OF ATOMIC OBJECTS IN CS -CORRECTED STEM A. Depth of Focus The dramatic effect of Cs correction in STEM by decrease of the probe size is well described. The size can be estimated using classical optics, which calculates a wave ﬁeld near a focal point along z and x, y directions (Born and Wolf, 1970). At the moment, one may assume no aberration of lens of the acceptance angle α, which is presently ∼40 mrad in 200-kV STEM. The beginning of this estimation is the Huygens-Fresnel diffraction integral, termed as the Debye integral.

i U(p) = − A λ

exp −ikq · Rd

(58)

Using polar coordinates, it is proved to be a convergent series of Bessel functions (Lommel functions). Introducing u and v as

2π a 2 z λ f

2π a 2π a 2 v= x + y2 , r= λ f λ f

u=

(59) (60)

where a and f are the pupil radius of the lens and focal length, respectively. The intensity distribution in reciprocal coordinates is written as

2 2 I(u, v) = [U12 (u, v) + U22 (u, v)]Io u ∞ u n+2s

(−1)s Jn+2s (v) Un (u, v) = v

(61) (62)

A=0

where, Jn (v) is nth-order Bessel function. From these equations, the intensity distribution near the focal plane along the optical axis and x, y-plane are obtained as

' 2J1 (v) 2 Io v

sin u/4 2 Io I(u, o) = u/4 &

I(o, v) =

(63) (64)

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Eq. (63) is the well-known Airy disk, which gives the probe size, and Eq. (64) is similar to the diffraction pattern of a rectangular aperture and is related to the depth focus in STEM imaging. The ﬁrst zero intensity is given by u/4 = πa2 z/2λf 2 = ±π, that is, a distance z = ±2f 2 λ/a2 from the focus. An effective depth with a loss of ∼20% of maximum intensity can be adopted, thus, we can choose the area (distance) from the focus (u = 0) to u ∼ 3.2. The distance z along the optical axis is

λ

z = ±3.2 2π

2

f 1 f 2 1 λ λ= , a 2 a 2 α2

(65)

where α is the acceptance angle of the objective (condenser) lens of STEM.

B. STEM Experiments for 3D Observation As discussed in Section III.E, 3D atomic observation is one of the next important targets in electron microscopy. The ordinary 3D electron tomography method is also applicable to STEM and TEM. Particularly, ADFSTEM images provide information about the distribution of atoms through their scattering contrast (Z-contrast). One can use the signal more directly connected to the 3D reconstruction of atomic objects. In the case of TEM, strong dynamical diffraction or crystalline diffraction effects present

Single Pt atom

0.226nm

FIGURE 27 Cs -corrected STEM image of single platinum atoms and clusters embedded in an amorphous carbon film.

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obstacles to direct correspondence between atomic density and image contrast. Pennycook’s group proposed a z-slice imaging in ADF-STEM by changing the focus of the STEM probe inside samples, based on Eq. (65) (Borisevich et al., 2006). This approach may work well when the sample consists of atomic clusters and impurity atoms embedded in/on amorphous ﬁlms, but it does not work for crystalline samples, including various kinds of defects and inclusions. Figure 27 shows the author’s data of observed platinum atoms embedded in an amorphous carbon ﬁlm. Due to the high sensitivity of the Z-contract STEM method, single platinum atoms are clearly imaged (Yamasaki and Tanaka, 2008). The sample is an ideal one for studying details of 3D observation of atomic objects in ADF-STEM. The depth resolution is expressed by Eq. (65). Due to Cs correction and the use of wider convergence angles, the depth resolution is drastically decreased. Figure 28 shows data for comparison of 300 kV STEM without Cs correction (a), Cs /C5 correction at 300 kV (b) and 200 kV Cs /Cc correction (c) in view of the elongation of the probe in z-direction (Borisevich

STEM Z-scan

DD 5 l/a2

Cs, C5 small, a, larger

1Å

Cs 51.3 mm, 100 kV

Cs 524mm, 300 kV

Cs /Cs corrected, 200 kV

FIGURE 28 Simulated intensity profile of electron probes along z-direction for z-slice imaging by ordinary and Cs -corrected STEMs. (From Borisevich et al., 2006.)

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et al., 2006). The blurring of the depth resolution for crystalline samples represents a new problem to be studied in detail, which should manage dynamical diffraction and channeling effects. The ease of elemental analysis by EDX and EELS in Cs -corrected STEM is greatly improved. The elemental analysis in silicon devices has become a half-day’s work shorter than previously.

VI. CONCLUSION AND FUTURE PROSPECTS This chapter, starting with a short summary of HRTEM theories with including the effects of aberration correction, has reviewed the current status of Cs -corrected TEM/STEM in the observation of nanomaterials. Characteristic features in TEM and STEM are described by using the experimental data from the author’s group and published reports by other research groups in Japan and elsewhere. The Cs correction up to third-order aberration has already been established and the interest of researchers is now focused on correction of ﬁfth-order spherical aberration and chromatic aberration of the objective lens in combination with development of monochromators below electron guns. In order to obtain ultra high-resolution images with <0.07 nm pointto-point resolution, mechanical and electrical instabilities as incoherent aberration making damping characteristic onto contrast transfer functions become the most crucial problems facing the next decade. One basic interest in Cs -corrected HREM is determining the limitation of resolution in TEM and STEM by quantum mechanics. The minimum size of an exit wave function for TEM imaging can be calculated as a scattering problem where multiple scattering by atomic columns has to be considered in various dynamical diffraction theories. The probe size in STEM is determined by electron opticals, but the interpretable image resolution depends on various effects that occur inside samples, which mandates further study. One future prospect of Cs -corrected TEM/STEM may be related to advantages not treated in this chapter: an extension of space around sample holders due to aberration correction of chromatic and spherical aberrations. In comparison with previous pole pieces of the lens, we can use wider gap while maintaining resolution. This wider space facilitates in situ experiments and combinations of TEM and other analysis methods. Furthermore, we need to develop different types of in situ observation tools by using the wider area. “Higher resolution,” “in situ and dynamic,” and “3D atomic observations” are the key words in the next decade of aberration-corrected high-resolution electron microscopy (Tanaka, 2008).

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ACKNOWLEDGMENTS The author thanks all of the researchers who joined the research project for Cs -corrected TEM/STEM, particularly Drs. J. Yamasaki, K. Saitoh, K. Yoshida, K. Hirahara and S.-P. Cho of Nagoya University. In the beginning of the project, Drs. M. Haider, P. Hartel, and other members of CEOS GmbH provided much assistance to the author. Prof. Y. Ikuhara of Tokyo University and Dr. T. Hirayama of JFCC are acknowledged for various support. The members of JEOL Co. Ltd., particularly Messers. M. Naruse, T. Tomita, and Drs. Y. Kondo, H. Sawada, and E. Okunishi, kindly supported the author in development of Cs -corrected TEM/STEM instruments at Nagoya University and provided assistance for related experiments. The contributed data from Prof. Y. Ikuhara, Drs. K. Suenaga, E. Abe, H. Sawada, Y. Tanishiro and M. Takeguchi, are greatly acknowledged. Prof. Y. Takai provided guidance and insights during the writing of this chapter.

REFERENCES Allen, L. J., Findlay, S. D., Oxley, M. P., and Rossouw, C. J. (2003). Lattice-resolution contrast from a focused coherent electron probe. Ultramicroscopy 96, 47–63. Beck, V., and Crewe, A. V. (1974). A quadrupole-octupole corrector for 100 keV STEM. In “Proceedings of the 32nd annual meeting of the Electron Microscopy Society of America” (C. J. Arceneaux, ed.), pp. 426–427. Claitors Publishing, Baton Rouge, LA. Beck, V. (1979). A hexapole spherical aberration corrector. Optik 53, 241–255. Borisevich, A. Y., Lupini, A. R., Travaglini, S., and Pennycook, S. J. (2006). Depth sectioning of aligned crystals with the aberration corrected scanning transmission electron microscope. J. Electr. Microsc. 55, 7–12. Born, M., and Wolf, E. (1970). “Principles of Optics,” 4th ed. (Section 8.8). Pergamon Press, Oxford. Buseck, R., Cowley, J. M., and Eyring, L.(1988). “High-Resolution Transmission Electron Microscopy and Associated Techniques.” Oxford University Press, New York. Cowley, J. M., and Moodie, A. F. (1957). Fourier images. Proc. Phys. Soc. B 70, 486–496. Cowley, J. M. (1981). “Diffraction Physics,” 2nd ed. North-Holland, Netherlands, 1981. Deltrap, J. H. M. (1964). Correction of spherical aberration with combined quadrupole and octopole units. Proceedings of the 3rd EUREM Congress (Czechslovak Academy of Science, Prague), vol. 1, 45–46. Dwyer, C., Kirkland, A., Hatel, P., Müller, H., and Haider, M. (2007). Electron nanodiffraction using sharply focused parallel probes. Appl. Phys. Lett. 90, 151104–151106. Fejes, P. L. (1977). Approximation for the calculation of high-resolution electron microscope images of thin ﬁlms. Acta Crystallogr. A33, 109–113. Gabor, D. (1949). Microscopy by reconstructed wave-fronts. Proc. Roy. Soc. A 197, 454–487. Gamm, B., Schultheiss, K., Gerthsen, D., and Schroeder, R. P. (2008). Effect of a physical phase plate on contrast transfer in an aberration-corrected transmission electron microscope. Ultramicroscopy 108, 878–884. Geiger, D., Lichte, H., Linck, M., and Lehmann, M. (2008). Electron holography with a Cs -corrected transmission electron microscope. Microsc. Microanal. 14, 68–81. Haider, M., Schwan, E., Rose, H., Kabius, B., and Urban, K. (1998a). Electron microscopy image enhanced. Nature 392, 768–769. Haider, M., Rose, H., Uhlemann, S., Kabius, B., and Urban, K. (1998b). Toward 0.1 nm resolution with the ﬁrst spherically corrected TEM. J. Electr. Microsc. 47, 395–405.

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Morishita, S., Yamasaki, J., Nakamura, K., Kato, T., and Tanaka, N. (2007). Diffractive imaging of the dumbbell structure in silicon by spherical-aberration-corrected electron diffraction, In print on Appl. Phys. lett. (2008). Mory, C., Colliex, C., and Cowley, J. M. (1987). Optimum defocus for STEM images and microanalysis. Ultramicroscopy 21, 171–177. Nomaguchi, T., Kawasaki, T., Kimura, Y., and Takai, Y. (2004). Separation of linear and nonlinear imaging components in high-resolution transmission electron microscope images. J. Electr. Microsc. 53, 403–406. Okayama, S. (1990). A new type of quadrupole correction lens for electron beam lithography. Nucl. Instr. Methods A 298, 488–495. O’Keefe, M. A. (1979). Resolution damping functions in non-linear images. In “Proceedings of the 37th annual meeting of the Electron Microscopy Society of America” (G. W. Bailey, ed.), pp. 556–557. Claitors Publishing, Baton Rouge, LA. O’Keefe, M. A. (2008). Seeing atoms with aberration corrected sub-Å electron microscopy. Ultramicroscopy 108, 196–209. Pennycook, S. J. (2002). Structure determination through Z-contrast microscopy. Adv. Imaging and Electr. Phys. (Elsevier Science) Vol. 123, 73–206. Reimer, L. (1984). “Transmission Electron Microscopy.” Springer, Berlin. Rose, H. (1970). Abbildungseigenschaften sphaerisch korrigieter electronenoptischer Achromate. Optik 33, 1–24. Saitoh, K., Tsai, P., and Tanaka, N. (2006). Spherical-aberration-corrected HRTEM of Al-Ni-Co decagonal quasicrystals. Proc. IMC 16, 1777. Saitoh, K., and Tanaka, N. (2008a). In preparation. Saitoh, K., and Tanaka, N. (2008b). In preparation. Saitoh, K. and Tanaka, N. (2008c). In preparation. Sato, Y., Buban J. P., Mizoguchi, T., Shibata, N., Yodogawa, M., Yamamoto, T., and Ikuhara, Y. (2006). Role of Pr segregation in acceptor-state formation at ZnO grain boundaries. Phys. Rev. Lett. 97, 1068021–1068024. Sawada, H., Hosokawa, F., Kaneyama, T., Ishizawa, T., Terao, M., Kawazoe, M., Sannomiya, T., Tomita, T., Kondo, Y., Tanaka, T., Oshima, Y., Tanaishiro, Y., and Takayanagi, K. (2007). Achieving 63 pm resolution in a scanning transmission electron microscope with spherical aberration corrector. Jpn J. Appl. Phys. 46, L568–570. Saxton, W. O., O’Keefe, M. A., Cockayne, D. J. H., and Wilkens, M. (1983). Sign conventions in electron diffraction and images. Ultramicroscopy 12, 75–78. Scherzer, O. (1936). Uber einige Fehler von Electronenlinsen. Z. fur Physik 101, 593–603. Scherzer, O. (1947). Spaeriche und chromatisch Korrektur von Electronmikroskop. Optik 2, 114–132. Scherzer, O. (1949). The theoretical resolution limit of the electron microscope. J. Appl. Phys., 20, 20–29. Spence, J. C. H. (2003). “High Resolution Electron Microscopy,” 3rd ed. Oxford University Press, Oxford. Tanaka, N. (2008). Present status and future prospects of spherical aberration corrected TEM/STEM for study of nanomaterials, Science and Technology of Advanced Materials, 9, 014111–01421. Tanaka, N., Cho, S. P., Shklyaev, A. A., Yamasaki, J., Okunishi, E., and Ichikawa, M. (2008). Spherical aberration corrected STEM studies of Ge nanodots grown on Si(001) surfaces with ultrathin SiO2 coverage, Appl. Surf. Sci. 254, 7569–7572. Tanaka, N., Hirayama, T., Ikuhara, Y., Hosokawa, F., and Naruse, M. (2002) Direct observations of nano-particles and interfaces by Cs -corrected TEM. Proc. IMC15 (Durban) 3, 37–38. Tanaka, N., and Hu, J. (1998). Theoretical basis of image deconvolution method for structure images of thin crystals. J. Electr. Microsc. 47, 217–221.

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CHAPTER

11

Atomic-Resolution Aberration-Corrected Transmission Electron Microscopy Knut Urban*, Lothar Houben, Chun-Lin Jia, Markus Lentzen, Shao-Bo Mi, Andreas Thust, and Karsten Tillmann Contents

I Introduction II Fundamentals of Atomic-Resolution Imaging in a Transmission Electron Microscope A The Optical Principles B Contrast III Atomic-Resolution Electron Microscopy and the Inversion of the Scattering and Imaging Problem A What is Atomic Resolution? B From Images to Structure: The Inversion of the Scattering and Imaging Problem C Aspects of Practical Work IV Selected Materials Science Applications A Measurement of the Occupancy of Oxygen-Atom Columns in a twin Boundary of BaTiO3 B Structure of the 90 degree [100] Tilt Grain Boundary in YBa2 Cu3 O7 C Structure and Polarization of Ferroelectric Domain Walls in PZT V Conclusions References

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* Institute of Solid State Research and Ernst Ruska Centre for Microscopy and Spectroscopy with Electrons, Helmholtz Research Centre, 52425 Jülich, Germany Advances in Imaging and Electron Physics, Volume 153, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01011-2. Copyright © 2008 Elsevier Inc. All rights reserved.

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I. INTRODUCTION Ten years ago the ﬁrst high-resolution images recorded with an aberrationcorrected electron microscope were published (Haider et al., 1998). Although taken under quite unspeciﬁc conditions, the images of a Si/CoSi2 interface demonstrated a substantial reduction in contrast delocalization, a feature that until now is considered a major advantage of aberrationcorrected electron optics for quantitative high-resolution studies. Further decisive steps on the way to modern aberration-corrected electron microscopy (EM) were the introduction of the negative spherical aberration imaging (NCSI) technique, yielding negative phase contrast of atomic objects (Jia, Lentzen, and Urban, 2003) and the demonstration of the advantages of combining this technique with exit-plane wave function (EPWF) reconstruction (Tillmann, Thust, and Urban, 2004). Today’s state of the art can be characterized by the fact that real atomic-resolution studies can be performed with these techniques and that individual lateral shifts of atoms can be measured at an accuracy of a few picometers (Houben, Thust, and Urban, 2006). On the other hand, work with aberration corrected optics also indicates the demands of these new techniques and that they are as yet by no means routine. Furthermore, because atomic-scale resolution means probing the quantum world, contrary to both intuition and frequently held views, understanding what the images actually show is generally not straightforward. As a consequence, in many cases the ultimate result of state-of-the-art microscopy is an atomic model on a computer rather than an image. The developments in high-resolution electron microscopy (HREM) with and without aberration-correction have been described in recent reviews (Smith, 1997, 2008; Urban, 2007). This chapter provides an overview of the current state of aberration-corrected transmission electron microscopy (TEM) with an emphasis on atomic-resolution studies in materials science. The examples are from research work performed in Jülich. A less-detailed account, also discussing aberration-corrected scanning transmission electron microscopy (STEM) was recently published (Urban, 2008).

II. FUNDAMENTALS OF ATOMIC-RESOLUTION IMAGING IN A TRANSMISSION ELECTRON MICROSCOPE A. The Optical Principles The basic problem of TEM is determining the structure of a specimen whose atomic arrangement may be written by an object function of the form

ρ(r) =

i

ρi (r) ∗ δ(r − r i ),

(1)

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where r is a position vector, and ρi (r) denotes any type of density function peaked at the position r i . For this we apply in the microscope an electron wave ﬁeld with the wave function ψ(r) transmitted through the specimen. In general, the specimen is assumed to be crystalline, and the propagation direction e0 of the electron wave ﬁeld is oriented so that it is parallel to a high-symmetry crystal axis a. The angular distribution of the electron wave vectors in this wave ﬁeld around e0 depends on the details of the illumination system of the microscope. The angular width, the illumination angle, is typically a few milliradians. Since the present treatment serves for demonstration of the principles only, we describe this ﬁeld, to simplify matters, as consisting of a single-plane wave (corresponding to parallel illumination) of normalized amplitude

ψ(r) = exp(2πik0 r),

(2)

where the wave vector k0 = e0 λ−1 is given by the direction variable and the electron wavelength λ. Inside the specimen the wave ﬁeld interacts with the interatomic potential V (r). The wave function inside the specimen may be obtained from the relativistic Dirac equation, which is solved in a small-angle approximation where it adopts a Schrödinger-type form. For the special case of a periodic potential, the wave function is given by the superposition of Bloch states j, that is

ψ(r) =

ε( j)

j

( j) Cg exp 2πi k( j) + g r ,

(3)

g ( j)

where k( j) is related to the eigenvalues and Cg to the eigenvectors of the Bloch states and g is a reciprocal-lattice vector. The amplitude factors ε(j) are determined by the boundary conditions at the entrance surface of the specimen. In EM ideally periodic objects are rarely investigated. Therefore in practice we are dealing with objects whose real structure gives rise to a more complex potential. This requires a numerical solution of the Schrödinger equation. The technique generally used in EM is the multislice approach (Cowley and Moodie, 1957; Williams and Carter, 1996). At the exit surface the electron wave ﬁeld is leaving the specimen and the electron waves propagate into the ﬁeld-free vacuum. Again to simplify matters, staying with the Bloch wave formalism, at the exit plane of the specimen, a distance t from the entrance plane, the EPWF can be written in a Fourier representation as a superposition of plane waves

ψ(r) =

g

ψ(g) exp 2πigr ,

(4)

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with the complex-valued Fourier coefﬁcients

ψ(g, t) =

(j) ( j) ε( j) Cg exp 2πi kz t .

(5)

j ( j)

kz denotes the component of the wave vector parallel to the electron propagation direction taken as the z-direction of a Cartesian coordinate system. In the following, g is called the spatial frequency. It is related to the frequency of the respective Fourier component. Again, in the realistic case of nonperiodic objects Eqs. (4) and (5) are to be replaced by numerical solutions of the Schrödinger equation. The EPWF contains all the information about the specimen that may be obtained using the electron wave ﬁeld as a probe. This information is encoded in the quantum-mechanical particularities of the electron wave function, which has been modiﬁed by the interaction with the interatomic potential. It is emphasized that the EPWF is corresponding to the “real” situation. In particular, if the specimen is tilted with respect to the direction of incidence, that is, e0 is not exactly parallel to a, the EPWF will be modiﬁed compared with the “ideal” case where the electrons are assumed exactly parallel to a high-symmetry crystal axis. Even rather small deviations—in the range of fractions of milliradians—of the specimen orientation from the ideal Laue orientation have an effect on atomic-resolution details. Furthermore, since the electron wave ﬁeld is interacting with the atoms in a quantum-mechanical way the effect of tilt will not affect all atomic features in the same way. This means that, for example, an oxygen-atom column will be affected with respect to relative position and contrast differently than a gallium column. In practice, such small specimen misorientations cannot be avoided and must be dealt with a posteriori. Since the details of the wave function are sensitively dependent on sample thickness, the result of an electron wave passing through specimens of identical structure but different thickness will be quite different. As a consequence, the EPWF will be changed locally by specimen tilt and thickness in a quite complex manner. Therefore, apart from certain special cases (discussed below), plots of the EPWF cannot (as unfortunately is often done) be directly interpreted in terms of the structure or a high-symmetry projection of the crystal potential. The EPWF (and not the specimen itself) is acting as the object of the following electron optics. The wave ﬁeld passes the electromagnetic lens and we obtain an intensity distribution I(r) in the image plane given by the Poynting vector which, again in a small-angle approximation, is proportional to the probability density (i.e., the absolute square of the wave function)

ih ψ ∇ψ∗ − ψ∗ ∇ψ ∝ ψ∗ ψ, 4πm where h denotes Planck’s constant and m is the electron mass. I(r) ∝

(6)

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So far we have assumed an ideal objective lens characterized by the property that its action is nothing else but compensation for the different phase shifts in the electron wave ﬁeld arising from the different geometrical path lengths of a divergent bundle of rays, leaving a point P at the exit plane of the specimen and converging at a corresponding point P i in the image plane. As a result, the optical path lengths are the same irrespective of the angle of an electron ray with the optical axis and imaging is phase preserving. We note that even for an ideal lens the image intensity distribution contains artifacts—features that do not occur in the original specimen structure. This may easily be recognized by inserting Eq. (4) in Eq. (6). The resulting expression will contain terms of the form

ψ∗ g i ψ g j + complex conjugate,

(7)

corresponding, in explicit form, to a spatial frequency g s = g i − g j not contained in the original EPWF. A special case are the well-known halfspacings for g j = −g i . This means that in principle the width of the Fourier representation of the image is always twice that of the EPWF. In the case of a real lens, the aberrations produce relative phase shifts of the electron waves associated with individual rays. As a consequence, the amplitudes of the individual components of the EPWF are multiplied by a phase factor

ψ(g) exp −2πiχ(g) ,

(8)

where the core χ(g) of the exponential is called the aberration function. Taking z as the electron propagation direction and x and y as the orthogonal directions in a Cartesian coordinate system, we may describe the phase shifts induced by the aberrations as being, in general, nonuniform perpendicular to z. The aberration function may be written in compact form if g is expressed as a complex number, g = gx + igy , and the related complex scattering angle, ω = λg, is used (Uhlemann and Haider, 1998). Expanding χ(g) to ﬁfth order in g and substituting g by ω yields

& 1 1 1 1 χ(ω) = Re C10 ωω¯ + C12 ω¯ 2 + C21 ωω¯ 2 λ 2 2 3 1 1 1 + C23 ω¯ 3 + C30 ω2 ω¯ 2 + C32 ωω¯ 3 3 4 4 1 1 1 + C34 ω¯ 4 + C41 ω2 ω¯ 3 + C43 ωω¯ 4 4 5 5 1 1 1 + C45 ω¯ 5 + C50 ω3 ω¯ 3 + C52 ω2 ω¯ 4 5 6 6 ' 1 1 + C54 ωω¯ 5 + C56 ω¯ 6 + . . . , 6 6

(9)

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with the lens aberration coefﬁcients defocus C10 , twofold astigmatism C12 , second-order coma C21 , threefold astigmatism C23 , third-order spherical aberration C30 , third-order star aberration C32 , fourfold astigmatism C34 , fourth-order coma C41 , three-lobe aberration C43 , ﬁvefold astigmatism C45 , ﬁfth-order spherical aberration C50 , ﬁfth-order star aberration C52 , rosette aberration C54 , and sixfold astigmatism C56 . The lens aberration coefﬁcients Cnk are complex numbers denoting the two Cartesian components, except for C10 , C30 , and C50 , which are real numbers, and ω¯ denotes the complex conjugate of ω. A simpliﬁed but frequently used form is thus obtained writing with the third-order spherical aberration parameter CS ≡ C30 and the objective lens defocus parameter Z ≡ C10

χ(g) =

1 1 CS λ3 g 4 + Zλg 2 + . . . . 4 2

(10)

Third-order spherical aberration is schematically illustrated in Figure 1. The high-angle rays in a bundle of rays originating from a point P 0 in the object and passing the lens at its periphery come to focus far in front of the Gaussian image plane, which is deﬁned by the paraxial rays leaving the object point at small scattering angles and traversing the lens in the center. This means that the image of the point in the object is not imaged into a point but into an aberration disk of radius RS given by the pointspread function, which is related to the maximum of the derivative of the aberration function with respect to the spatial frequency taken over the entire range of spatial frequencies contributing to the image (Coene and Jansen, 1992; Lichte, 1991)

RS =

max /012

0 ≤ g ≤ gmax

∂χ = max CS λ3 g3 + Zλg . ∂g

(11)

The extraordinary progress in TEM in recent years is founded on the fact that in the recent generation of instruments equipped with aberration correctors the aberrations of Eq. (9) can be measured and compensated or controlled by suitable multipole and stigmator elements (Haider et al., 1998; Uhlemann and Haider, 1998). This means that, in principle the radius of the point-spread function may be reduced to very low values. Therefore, the optical resolution may be enhanced to values that are limited by aberrations of orders higher than those listed in Eq. (9).

B. Contrast For imaging by an optical system, resolution is not sufﬁcient and contrast also is required. Atomic-resolution EM is based primarily on phase contrast. This means that the information on the specimen structure is largely

Atomic-Resolution Aberration-Corrected Transmission Electron Microscopy

(a)

(b) P

(c) P

445

(d) P

P

Aperture

CS 5 C0

R

CS 5 C0

CS 5 0

CS 52Cr

Image plane

FIGURE 1 Schematic illustration of third-order spherical aberration. The high-angle rays in a bundle of rays originating from a point P in the object and passing the lens at its periphery, come to a focus much in front of the Gaussian image plane which is defined by the paraxial rays leaving the object point at small scattering angles and traversing the lens in the centre. As a result the image of the point in the object is imaged into an aberration disk of radius R whose radius is given by the point-spread function. (a) Uncorrected conventional electron lens with the spherical aberration parameter Cs having the value C0 . R denotes the radius of the point-spread disk into which the image of the object point P is broadened as a result of spherical aberration. (b) An aperture is used to limit the angular range of rays contributing to the image. R is reduced but at the expense of reduced resolution, since the diffraction angle increases in proportion to the inverse of the extension of sample details. (c) Spherical aberration is corrected by means of a divergent lens. (d) Ray path for negative spherical aberration. The corrector lens is overexcited by a small amount to induce a negative residual value Cr . From Urban (2007).

provided by locally varying phase shifts of the electron waves. Because phases cannot be seen, atomic-resolution EM faces the same problem as phase-contrast light-optical microscopy—phase information must be converted into amplitude information. In light microscopy this is achieved by the Zernike λ/4 plate shifting the phase of the diffracted beams by π/2 to bring the beams into a π phase relationship to the incident beam. This is the basis of the positive phase contrast.

1. Scherzer Theory Despite recent progress in producing electron-optical elements corresponding to the Zernike λ/4 plate (Majorovits et al., 2007), this is not yet a general option for phase-contrast EM. Contrast-producing phase shifts therefore must be introduced by other means. As described for the ﬁrst time by Scherzer (1949), the fact that the phase shifts described by the aberration

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function depend on g may be exploited to produce the equivalent of a Zernike phase plate. On the other hand, the result is far from ideal since the phase changes depend in a highly nonlinear fashion on g. On the basis of Eq. (10), Scherzer (1949) derived the following relations for optimum phase contrast:

1

4 ZS = − C S λ 3

2

(12)

− 1 4 gS = 2 3CS λ3

(13)

dS = gS−1 ,

(14)

where ZS is the so-called Scherzer defocus and dS is the Scherzer or pointresolution limit. Figure 2a displays the phase contrast transfer function (CTF) − sin 2πχ(g) for a 200-kV electron microscope equipped with a ﬁeld emission gun (Lentzen, 2006). It is obvious that the modulus of the transfer function close to 1 (phase shift close to π/2) over a certain range (in this case, for 1.75 ≤ g ≤ 3.75 nm−1 ) is achieved at the price of rapid contrast oscillations for g > 4 nm−1 . An objective-lens aperture must be used to avoid Scherzer defocus, FEG

0

21

Negative phase contrast (b) 1

(a) 1

2

4

6

8

10 g [1/nm]

0

2

4

6

8

10

g [1/nm]

21

FIGURE 2 (a) Contrast transfer function (CTF) − sin 2πχ(g) for a 200-kV electron microscope equipped with a field-emission gun (Lentzen, 2006). It is obvious that a modulus of the transfer function close to 1 (phase shift close to π/2) over a certain range, in this case for 1.75 ≤ g ≤ 3.75 nm−1 , is achieved at the expense of rapid contrast oscillations at g > 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)

to be combined with a defocus value of

Z=−

16 2 −1 λgI . 9

(17)

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) =

ρi (r) ∗ δ(r − r i ).

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 <110> 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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1 nm

(b)

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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

1

Cu2, O2 O1 Cu1

p2

exp.

0.5

p1 gb

Domain II

155 155 pm pm

gb p1 p2 p...

0

20.5

234 184 pm pm

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0 x (nm)

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Phase (rad)

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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.

Knut Urban et al.

184 pm

55%

100%

0.2

Cu1 O4

0

exp.

0%

20.2 20.4 20.6

calc.

22

21

0 y (nm)

Domain I

Phase (red)

0.4

Domain II

Domain II

Cu2, Cu1 Domain I O2 O1

0.6

243 pm 166 pm

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Y Ba Cu O

1

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

Y

Ba

O4 9±4

Cu1

29 ± 20

38 ± 26

6±4

6±4

Cu1 O1

18 ± 10

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 <100> 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

O

c 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

Atomic-Resolution Aberration-Corrected Transmission Electron Microscopy

PS

473

PS

1 nm

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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PS

I

II

PS

1 nm

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.

Atomic-Resolution Aberration-Corrected Transmission Electron Microscopy

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0.04 0.02 dZr/Ti

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20.04 (e) Ps (mC/cm2)

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5 0 Distance (in units of c)

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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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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

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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

18:00

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

1.0m

LogMag, Gauss

100.0u

10.0u

1.0u

100.0n

10.0n 100.0

200.0

300.0

Hz

400.0

500.0

600.0

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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(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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(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)

(d)

(e)

f 5 nm

5 nm

0.0

0.5

1.0

5 nm

1.5

nm

(a.u) (a.u)

(b)

(h)

(g) FWHM 0.18 nm

(f)

FWHM 0.8 nm U⫺O4,5 C⫺K 100

200

300

0.0

0.5

1.0 0.0

0.5

1.0 0.0

0.5

1.0

(nm)

(nm)

(nm)

Energy loss (eV)

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).

1

2

p 2 nm

3

4

5

p

FIGURE 17 Sequential image acquisition of uranium single-atom motion. Single atoms move due to irradiation by the electron beam. During this study specimen is cooled to −160◦ C with Gatan 670 liquid nitrogen cooling stage (image size 256 × 256 pixel, 0.04 nm/pixel, 64 ms/pixel).

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

(a)

(b)

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 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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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

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

U

Z

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

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

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

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