Advances in Electronics and Electron Phisics. Vol. 70

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 70 EDITOR-IN-CHIEF PETER W. HAWKES Laboratoire d’0ptique Electro...

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 70

EDITOR-IN-CHIEF

PETER W. HAWKES Laboratoire d’0ptique Electronique du Centre National de la Recherche Scientijique Totilouse, France

ASSOCIATE EDITOR

BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California

Advances in

Electronics and Electron Physics EDITED BY PETER W. HAWKES Laboratoire d'Optique Electronique du Centre National de la Recherche Scientijique Toulouse, France

VOLUME 70

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston San Diego New York Berkeley London Sydney Tokyo Toronto

COPYRIGHT @ 1988 by ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL. INCLUDING PHOTOCOPY, RECORDING. OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC. 1250 Sixth Avenue. San Diego. CA 92101

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24-28 Oval Road, London NWI 7DX

LIBRARY OF CONGRESS CATALOG CARDNUMBER:49-7504 ISBN 0-12-014670-3 PRINTED IN THE UNITED STATES OF AMERICA

8XX99091

9 8 7 6 5 4 3 2 I

CONTENTS CONTRIBUTORS TO VOLUME 70 . . . . . . . . . . . . . . . . . . . . . . . PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Scanning Electron Microscopy at Very Low Temperatures . . . . . . . . . . R. P. HUEBENER

vii ix

1

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Low-Temperature Stage . . . . . . . . . . . . . . . . . . . . . . . I11. Principles and Electron Beam Parameters . . . . . . . . . . . . . . IV . Interaction between Electron Beam and Specimen . . . . . . . . . . V. Superconducting Tunnel Junctions: Pair Tunneling . . . . . . . . . . VI . Superconducting Tunnel Junctions: Quasiparticle Tunneling . . . . . VII . Arrays of Superconducting Tunnel Junctions . . . . . . . . . . . . . VIII . Hotspots in Superconducting Microbridges . . . . . . . . . . . . . . IX . Current Filaments and Turbulence in Semiconductors . . . . . . . . X . Ballistic Phonon Signal . . . . . . . . . . . . . . . . . . . . . . . XI . Phonon Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . XI1 . Imaging of Structural Defects with Ballistic Phonons . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4 8 10 18 26 39 41 49 58 64 71 75 15

. . . . . . . . . . . . . . . .

79

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Introduction and Overview . . . . . . . . . . . . . . . . . . . . . I1. AR and ARMA Models . . . . . . . . . . . . . . . . . . . . . . . 111. Robust Estimation in Causal Autoregressive Models . . . . . . . . . I V . Image Restoration with Robust Image Modelling Techniques . . . . . V. Composite Edge Detection . . . . . . . . . . . . . . . . . . . . . VI . Summary and Suggestions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80 80 84 109 121 139 155 155

Physical Limits in Information Processing . . . . . . . . . . . . . . . . . . ROBERTW . KEYES

159

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Representation of Information . . . . . . . . . . . . . . . . . . . . I11. Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 161 163

Robust Image Models and Their Applications

R . L . KASHYAP AND KE-BUM EOM

V

vi IV . V. VI . VII . VIII . IX.

CONTENTS

The Nature of Devices . . . . . . . . . . . . . . . . . . . . . . . Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wiring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dissipation of Energy . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164 175 198 203 207 213 213

Synthetic Aperture Ultrasonic Imagery . . . . . . . . . . . . . . . . . . . KEINOSUKE NAGAI

215

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imaging System and Aperture . . . . . . . . . . . . . . . . . . . . Theory and Application of Holography . . . . . . . . . . . . . . . Fundamentals of Digital Ultrasonic Imaging . . . . . . . . . . . . . Properties of a Transducer Array . . . . . . . . . . . . . . . . . . Actual Digital Imaging System . . . . . . . . . . . . . . . . . . . . Diffraction Tomography as the Inverse Problem . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215 216 223 253 267 282 290 313

1. I1. 111. IV. V. VI . VII .

Dual Complementary Variational Techniques for the Calculation of Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . J . PENMAN

315

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. A Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . 111. Complementary Variational Principles . . . . . . . . . . . . . . . . IV . The General Engineering Field Problem . . . . . . . . . . . . . . . V. Field Problems in Engineering . . . . . . . . . . . . . . . . . . . . VI . Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . The Electrostatic Field . . . . . . . . . . . . . . . . . . . . . . . VIII . The Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . IX . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

316 316 318 323 331 336 342 347 358 364 364

CONTRIBUTORS TO VOLUME 70 The numbers in parentheses indicate the pages on which the authors’ contributions begin. Kie- Bum Eom, School of Electrical Engineering, Purdue University, West Lafayette, Indiana 47907 (79) R. P. Huebener, Physikalisches Institut 11, Universitate Tubingen, D-7400 Tubingen, Federal Republic of Germany (1) R. L. Kashyap, School of Electrical Engineering, Purdue University, West Lafayette, Indiana 47907 (79) Robert W. Keyes, IBM T. J. Watson Research Center, P O Box 218, Yorktown Heights, New York 10598 (159) Keinosuke Nagai, Institute of Applied Physics, University of Tsukuba, Sakura, Ibaraki 305, Japan (21 5) J. Penman, Department of Engineering, University of Aberdeen, Aberdeen, Scotland (315)

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PREFACE The five chapters in this volume range over the traditional subjects of these Advances, with an aspect of scanning electron microscopy in first place. Very low temperature SEM is a relatively new development: it merits separate treatment, for not only does it enable us to study superconductors by the techniques well developed in the scanning field, but it also proves to generate information of a different kind, due essentially to the localized heating effect caused by the energy deposited by the scanning beam. Not much of the work described here has yet found its way into the textbooks of SEM and we are delighted to publish so complete and authoritative an account by H. P. Huebner in these pages. The second chapter, by R. L. Kashyap and K. B. Eom, reflects my efforts to increase the coverage of digital image processing in these pages. Image models are needed in various types of image processing, image restoration by statistical methods and image segmentation in particular. K. L. Kashyap has made important contributions to our understanding of image models, especially those that are robust in the sense that they remain reliable even if the assumptions on which they are based are not exactly satisfied. He and K. B. Eom cover the theoretical background of AR and A R M A models, robust estimation in the causal case, and restoration and edge detection based on these statistical foundations. The third chapter examines questions raised by the extraordinary rate at which miniaturization is progressing. There must be a limit to this progress, but what is it, what governs it, and is it likely to be reached? R. W. Keyes considers the numerous factors involved, which are of very different kinds. At one extreme, we have physical laws governing speeds of propagation, mathematical laws associated with topology, the slippery rules of uncertainty, and the constraints of thermodynamics. At the other, there are economic pressures, which tend to mean that large scale progress, as opposed to isolated achievement, can only be expected if it pays off. R. W. Keyes succeeds in keeping all these pressures in mind in his discussion of the various devices used to process information. In the fourth chapter we return to imagery, and, in particular, to synthetic aperture ultrasonic images. Ultrasound images are found in medicine, wherever non-destructive testing is vital and in the study of natural resources. Unfortunately, the wavelengths employed are frequently comparable with those of the structures of interest, and synthetic aperture techniques have hence been developed to provide good images despite this. K. Nagai examines ix

X

PREFACE

the whole field, from wave propagation to the properties of transducer arrays and the performance of modern systems. This authoritative account will surely be of use to the experienced and invaluable to newcomers to the subject. The final chapter is concerned with a problem that is encountered in all branches of electronics and electron physics: the calculation of electromagnetic fields. The dual complementary variational techniques presented here are not as well known as they deserve to be, despite the ubiquity of the finiteelement method. J. Penman first sets out the mathematical tools needed and then examines first the two static cases, with examples from electrostatics and magnetostatics. He subsequently turns to the electromagnetic field, devoting a section to eddy current problems. This clear account of the ideas will surely help many of us confronted with the problem of field calculation to appreciate these methods. As usual, we conclude with a list of forthcoming chapters. Peter W. Hawkes Parallel Image Processing Methodologies Image Processing with Signal-Dependent Noise Scanning Electron Acoustic Microscopy Electronic and Optical Properties of Two-Dimensional Semiconductor Heterostructures Inverse Problems Pattern Recognition and Line Drawings Magnetic Reconnection Sampling Theory Dimensional Analysis Electrons in a Periodic Lattice Potential The Artificial Visual System Concept Accelerator Physics High-Resolution Electron Beam Lithography Corrected Lenses for Charged Particles Environmental Scanning Electron Microscopy The Development of Electron Microscopy in Italy Energy-Loss Spectroscopy Amorphous Semiconductors

J. K. Aggarwal H. H. Arsenault L. J. Balk G. Bastard et al. M. Bertero H. Bley A. Bratenahl and P. J. Baum J. L. Brown J. F. CariEena and M. Santander J. M. Churchill and F. E. Holmstrom J. M. Coggins F. T. Cole and F. Mills H. G. Craighead R. L. Dalglish D. G. Danilatos G. Donelli J. Fink W. Fuhs

xi

PREFACE

Median Filters Bayesian Image Analysis Vector Quantization and Lattices Aberration Theory Ion Optics Systems Theory and Electromagnetic Waves Phosphor Materials for CRTs The Scanning Tunnelling Microscope Multi-Colour AC Electroluminescent Thin-Film Devices Spin-Polarized SEM Proton Micro probes Ferroelasticit y Active-Matrix TFT Liquid Crystal Displays Image Formation in STEM Electron Microscopy in Archaeology Low-Voltage SEM Languages for Vector Computers Electron Scattering and Nuclear Structure Electrostatic Lenses Historical Development of Electron Microscopy in the USA Atom-Probe FIM X-Ray Microscopy Applications of Mathematical Morphology Focus-Deflection Systems and Their Applications Electron Gun Optics Electron Beam Testing

N. C. Gallagher and E. Coyle S. and D. Geman J. D. Gibson and K. Sayood E. Hahn D. Ioanoviciu M. Kaiser K. Kano et al. H. Van Kempen H. Kobayashi and S. Tanaka K. Koike J. S. C. Mc Kee and C. R. Smith S. Meeks and B. A. Auld S. Morozumi C. Mory and C. Colliex S. L. Olsen J. Pawley R. H. Perrott G. A. Peterson F. H. Read and I. W. Drummond J. H. Reisner

T. Sakurai G. Schmahl J. Serra T. Soma et al. Y. Uchikawa K. Ura


ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS. V O L 70

Scanning Electron Microscopy at Very Low Temperatures R. P. HUEBENER Physikalisches lnstitut 11. Uniuersitat Tiibinyen D- 7400 Tiibingen, Federal Republic of Germany

1. Introduction . . . . . . . . . . 11. Low-Temperature Stage . . . . . . 111. Principles and Electron Beam Parameters IV. Interaction Between Electron Beam and Specimen . . . . A. Thermalization of the Beam Energy. . . . . . . . B. Localized Heating EKect: Thermal Healing Length and Thermal Relaxation Time. . . . . . . . . . V. Superconducting Tunnel Junctions: Pair Tunneling . . . VI. Superconducting Tunnel Junctions: Quasiparticle Tunneling VII. Arrays of Superconducting Tunnel Junctions. . . . . . VIII. Hotspots in Superconducting Microbridges . . . . . . IX. Current Filaments and Turbulence in Semiconductors . . X. Ballistic Phonon Signal . . . . . . . . . . . . . XI. Phonon Focusing . . . . . . . . . . . . . . . XI1. Imaging of Structural Defects with Ballistic Phonons . . . Acknowledgments . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

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I5

I. INTRODUCTION

Today scanning electron microscopy (SEM) is a widely used analytical tool providing structural information in many different fields such as materials science, solid state physics, microelectronics, biology, and the medical sciences (Reimer, 1985).The principle of SEM simply consists of scanning the surface of the specimen with a well focused electron beam and recording simultaneously a proper response signal generated by the interaction of the beam with the specimen. If this signal is displayed following the same geometric pattern as in the scanning process, a two-dimensional image of some specimen property can be generated. Usually, the response signal is utilized for modulating the brightness on the screen of a cathode-ray tube, which is operated synchronously with the electron-beam scanning process. In many I Copyright ic 1988 by Academic P r e s , Inc All rights of reproduction re\crved ISBN O-I?-O14670-3

2

R. P.HUEBENER

applications of SEM, the response signal consists of the emitted secondary electrons or the back-scattered electrons. In addition, the emission of Auger electrons and x-ray photons is often utilized for structural imaging. Of course, the interaction processes generating these signals take place only within the penetration depths of the primary beam electrons in the sample material. Hence, the information obtained in this way is restricted to a region close to the specimen surface. The spatial resolution of SEM is determined by the diameter of the region perturbed by the electron beam and acting as the signal source. Hence, the beam diameter represents the ultimate spatial resolution limit. However, the beam-induced perturbation of the specimen often extends appreciably beyond the beam diameter, resulting in a corresponding deterioration of the resolution limit, perhaps by several orders of magnitude. If the primary electron beam irradiating the sample is temporally structured, time dependent phenomena can also be investigated by SEM. Using the stroboscopic principle, strongly time-dependent structures can be observed with high temporal and spatial resolution. Of course, the principle of scanning microscopy can be extended to any other probe that is movable in a two-dimensional pattern (Ash, 1980). A moving laser beam, acoustic beam, or mechanical micro-contact represent some examples that have been used for two-dimensional imaging. However, due to the well developed technology for generating and manipulating a sharply focused electron beam, so far electron-beam scanning has found the widest application in scanning microscopy. Here the small value of the beam diameter and the long working distance between specimen and lower polepiece of the final lens, which can be achieved, represent distinct advantages of SEM. Although SEM is now widely used as an analytical tool, its extension to the regime of very low temperatures is still relatively rare. Here we have in mind the temperature range provided by liquid helium, i.e., temperatures around 4 K and below down to about 1.5 K. For experiments in this temperature range one needs a scanning electron microscope equipped with a well-functioning liquid-He stage. With such an apparatus, two types of studies can be performed. First, typical low temperature phenomena such as superconductivity and low-temperature devices used in cryoelectronics can be investigated. Second, experiments can be performed where the temperature range of liquid He is required by the measuring principle. The ballistic phonon signal represents an example for the second case. This signal requires a long phonon mean free path and a highly sensitive phonon detector, both being realized only in the temperature range of liquid He. In the following review, we will deal with both types of applications of low temperature scanning electron microscopy (LTSEM). Of course, in addition to electron beam scanning other scanning

SCANNING ELECTRON MICROSCOPY AT VERY LOW TEMPERATURES

3

techniques can also be extended to the liquid-He temperature range (see Huebener, 1984; Bosch et al., 1986). In the following we only discuss these other scanning experiments if they bear directly upon the results obtained by LTSEM. We do not present a critical evaluation and comparison of the different scanning techniques which can be performed at low temperatures. The signals mainly to be utilized in SEM performed in the temperature range of liquid helium are expected to be different from those usually used during room temperature operation and mentioned above. Generally, the latter signals do not provide any new information if the sample iscooled to low temperatures. We will see that it is the localized heating effect caused by the electron beam during the scanning process that generates the important response signal providing the structural information about the specimen. This prominent role of the electron beam as a localized heat source and the importance of the beam-induced thermal perturbation of the sample results from the fact that at low temperatures many material properties can depend sensitively upon temperature. Here superconductors represent a particularly striking example since their energy gap often corresponds to thermal energies of only a few K. Of course, in an ancillary and helpful way the “usual signals” discussed above are always used in LTSEM in addition to the new signals only obtained at low temperatures. In this review we summarize the results obtained recently by LTSEM. Following a brief discussion of the main features of the low-temperature stage in $11, we treat the important underlying principles of LTSEM in $111. In gIV we discuss the interaction between the electron beam and the specimen, concentrating only on the signal generating processes important for the low temperature experiments treated in the remainder of this article. In Sections V-VII we deal with spatial structures observed by LTSEM in superconducting tunnel junctions. In $VIII we discuss experiments relating to spatial temperature structures in current-carrying superconducting microbridges. Spatial structures generated in semiconductors during avalanche breakdown at low temperatures and observed by LTSEM are treated in $IX. The signals discussed in Sections VI-IX have some similarity to the concepts of the electron-beam induced current (EBIC) or electron-beam induced voltage (EBIV) utilized often in studies of semiconductors by means of SEM (see Reimer, 1985; Ehrenberg and Gibbons, 1981). In gX-XI1 we deal with a distinctly different signal for spatial imaging, namely the ballistic phonon signal. Here the region locally heated by the electron beam acts as a source of ballistic phonons (quanta of sound energy) in a similar way as the heated filament in a light bulb acts as a source of photons (quanta of electromagnetic energy).The ballistic phonons can serve for imaging the phonon focusing effect based on the elastic anisotropy in a single crystal. They can also be utilized for imaging structural defects in a crystal. These two applications of the ballistic phonon signal are treated in $XI and 4x11,respectively.

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R. P. HUEBENER

TI. LOW-TEMPERATURE STAGE The best experimental setup for performing scanning electron microscopy at liquid-He temperatures appears to be an arrangement where one side of the specimen is in direct contact with the liquid-He bath, whereas the opposite side of the sample can directly be scanned with the electron beam. Such an arrangement is shown schematically in Fig. 1. Further, it is highly advantageous if the temperature of the liquid-He bath can be reduced from 4.2 K down to about 1.5 K by pumping. The operation in the temperature range below 2.17 K is of particular interest, since here the cooling efficiency of liquid He is strongly increased due to its superfluid state. Because of these considerations a bath cryostat extending into the sample chamber of the scanning electron microscope appears to be the best possible choice. The typical features of a low-temperature stage based on the principles indicated above is shown schematically in Fig. 2 (Seifert, 1982).On the left side we see the lower part of a conventional 4He cryostat consisting of a cylindrical liquid-He tank surrounded by a liquid-nitrogen tank for precooling and thermal shielding. The cryostat extends horizontally into the sample chamber of the microscope. On the right side in Fig. 2 we see the lower part of the electron-beam column and the sample chamber of the microscope. For thermal shielding it is important that the part cooled to liquid-nitrogen temperatures extends into the sample chamber in addition to the liquid-He tank. Horizontal adjustment of the sample position is possible by mechanically shifting the base plate carrying the low-temperature stage. Flexible connections between the low-temperature stage in the sample chamber and the cryostat on the left serve for gaining the necessary mechanical freedom. Helium gas bubbles forming perhaps near the sample and impeding the cooling process for the specimen can be removed by a circulation pump within the 4He cryostat.

1

electron beam thin-film specimen substrate

m ‘

‘

v

‘

v

V

r

m m Y

liquid He

a

FIG.1. Sample configuration for LTSEM.

SCANNING ELECTRON MICROSCOPY AT VERY LOW TEMPERATURES CRY OSTAT

5

MICROSCOPE

I CRY0 STAGE

FIG. 2. Schematics of the low-temperature stage. 1, outer wall; 2, LN, reservoir; 3, LHe reservoir; 4, vacuum space; 5, position of LHe transfer line; 6, driving shaft; 7, circulation pump; 8, LHe transfer tubes with bellows; 9, LN, shield; 10, copper ribbon for thermal coupling; I I, micrometer screw for shifting micropositioning stage; 12, mechanical vacuum feedthrough; 13, push rod; 14, mounting plate; 15, x-y micropositioning stage; 16, LN, base plate; 17, LHe base plate; 18, spacers for thermal isolation; 19, LHe tank; 20, sample; 21, microscope chamber; 22, microscope column; 23, electron beam (reproduced from Seifert, H. CRYOGENICS, 1982, 22, 657-660, by permission of the publishers, Butterworth & Co g?). (Publishers) Ltd. @?).

The top plate of the He tank in the sample chamber can be used directly as sample holder. Such a sample mounting configuration is shown schematically in Fig. 3. Here the sample material separates the liquid He from the vacuum of the electron-beam column. The sample, shaped preferably as a round disk (of, say, 20 mm diameter and 2-3 mm thickness), is fixed mechanically by a clamping screw which also compresses the indium seal between sample and top plate. It is important to keep the sample position sufficiently low such that the liquid-He level is always higher. The top plate is fixed by a clamping ring and sealed also with indium. A shield above the sample with a hole for the electron beam provides protection against thermal radiation. Typical dimensions of the He tank are about 4 cm height and 5 cm diameter, corresponding to a volume of about 80 cm3. Of course, this sample mounting configuration is also very useful for investigating thin-film structures deposited on the top of a proper substrate, the substrate being again preferably shaped as a round disk. Due to its high heat conductivity, single-crystalline sapphire is well suited as subtrate

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R. P. HUEBENER

FIG.3. Sample mounting configuration. I , electron beam; 2, sample; 3, sample holder; 4, clamping screw; 5, copper ring for wire heat sinking; 6, thermal shield; 7, LHe tank;

8, clamping ring; 9, indium seal; 10, LHe tubes (from Seifert, 1982).

material for such thin-film structures. In the same way, other specimens which are unsuitable to act directly as the separating wall between the liquid He and the vacuum because of their small size or their mechanical weakness, can be mounted on such a substrate material with high heat conductivity. Sufficient thermal contact to the substrate can be attained by a proper medium such as stycast cement, vacuum grease, etc. A photograph of a low-temperature stage which has been used for several years in the laboratory of the author is shown in Fig. 4.The circular flanche (diameter = 6.8 cm) represents the top plate of the He tank, the sample being located below the opening in the middle. The whole stage is to be inserted into the sample chamber of the microscope. In the back, the end plate of the horizontal extension of the cryostat with the rubber O-ring seal can be seen. If electrical current and voltage leads are to be attached to the top side of the sample or of the substrate (electrical connections to a thin-film structure, etc.), it is important that these lead wires are thermally anchored to the liquid helium bath after a short distance, in order to minimize sample heating effects. In some applications of LTSEM, it is necessary to apply an external magnetic field to the sample. Such a field can be generated by a small superconducting coil placed in the liquid He surrounding the sample location. On the other hand, it can become important to carefully shield the sample against any ambient external magnetic field such as the earth’s magnetic field. An effective magnetic shield of the sample can be fabricated from a magnetically soft material such as cryoperm (obtained from Vakuumschmelze GmbH, Hanau, FRG). Figure 5 shows a cross-section of the complete sample mounting configuration with the magnetic shield in place. Such a magnetic shielding can


7

FIG.4. Low-temperature stage.

has turned out to be highly effective in LTSEM studies of superconducting tunnel junctions, as will be discussed in Sections V-VII. The ease with which the low-temperature stage can be attached to and removed from the scanning electron microscope represents an important consideration. A quick turnaround time for changing the sample is always attractive. Further, intermittent operation of the microscope at room temperature for conventional applications is often required. A lowtemperature stage of the type shown in Figs. 2 and 3 can be self-supporting and

Fici. 5. Cross-Section of the sample mounting stage including the magnetic shield. I , electron beam; 2, sample; 3, sample holder; 4, clamping screw; 5, magnetic shield.

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R. P. HUEBENER

provides this flexibility.On the other hand, scanning electron microscopes are commercially available today, where the low-temperature stage can directly be attached to the hinged door of the sample chamber without any further mechanical provisions. The fact, that the weight of some of the more elaborate sample tables available today is similar to the weight of the complete liquidHe stage including the cryostat, provides such a possibility. (As an example, in the laboratory of the author, a liquid-He stage is mounted in this way to the hinged door of a Camscan Model S4DV scanning electron microscope). It is important to note that we have concentrated so far on a form of the low-temperature stage which is most universal in terms of its applicability and most effective in terms of its cooling power. For special applications simpler and less expensive cold stages are often adequate and commercially available. Here the specimen is mounted on some cold finger extending into the sample chamber of the microscope, and no direct contact between the liquid He and the specimen is provided. At present the lowest temperature which can be reached with such simple cooling stages is often limited to 10-15 K. In this review we exclude such applications from our discussion. 111. PRINCIPLES AND ELECTRON BEAMPARAMETERS The principle of SEM is illustrated in Fig. 6. The primary electron beam is scanned over the specimen surface and produces a localized perturbation of the sample. As a consequence the sample generates a response signal which generally depends upon the coordinates of the electron beam focus. The electron beam of a separate cathode ray tube (CRT) is operated synchronously with the primary electron beam. If the beam intensity of the CRT is modulated by means of the sample response signal, a two-dimensional image of the specimen property corresponding to this response signal is obtained. In addition to this two-dimensional display by means of the brightness on the CRT, a linear scan with the signal amplitude plotted against the scanned coordinate is often useful for a quantitative analysis of the results. This latter operational mode is referred to as “y-modulation”. For SEM studies at low temperatures, the localized heating effect of the electron beam already results directly in a highly useful response signal (see Clem and Huebener, 1980). At liquid-He temperatures the electronic properties of superconductors as well as semiconductors respond sensitively to small changes in temperature. Therefore, in many applications of LTSEM the electron-bombardment induced conductivity plays a central role for imaging. (A similar effect is often used at room temperature in SEM studies of semiconductors or electrical insulators; see, e.g., Reimer, 1985; Ehrenberg and Gibbons, 1981 .)


1

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beam

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screen X X

FIG.6 . Principle of scanning electron microscopy.

At low temperatures the phonon density in a crystal is strongly reduced compared to the region locally heated by the electron beam. Therefore, this heated region acts as a strong source of phonons which propagate ballistically, i.e., without any scattering, over long distances through the remainder of the crystal. This ballistic phonon signal can also be utilized for structural imaging. In many applications of LTSEM the electron beam should represent a passive probe. Hence, the beam-induced perturbation of the sample and the beam power should be as small as possible. On the other hand, the beam power must be sufficiently large for generating a detectable signal. With a typical beam voltage and current of 20 kV and 10 PA, respectively, we have a beam power of 0.2 pW. As an example we consider a thin-film specimen deposited on a substrate with high thermal conductivity. We assume that the thickness d of the specimen film is larger than the range of penetration for the electrons of the primary beam into the solid. (This range of penetration is about 1 pm for a beam energy near 20 keV and for materials such as the noble metals). In this case, the beam power is totally dissipated within the specimen film. The resulting temperature rise T - Tb in the film is given by

10

R. P. HUEBENER

where T is the temperature in the irradiated film region and Tb is the bath temperature. Po is the beam power and CI the heat transfer coefficient describing the heat transfer between the film and the substrate. The quantity q is the thermal healing length given by (see Huebener, 1984; Clem and Huebener, 1980),

where K is the heat conductivity of the film. Assuming the value n * q 2 = 10 pm2 and taking for the heat transfer coefficient near 4.2 K the value a = 1 Wcm-'K-' (Skocpol et al., 1974; Schulze and Keck, 1984), we find for Po = 0.2 pW the temperature increment T - Tb = 2 K. The sensitivity for signal detection can be strongly increased if the electron beam is modulated and phase-sensitive detection is applied. At sufficiently high modulation frequencies the spatial spreading of the modulated sample perturbation is reduced because of thermal or electronic skin effects (Clem and Huebener, 1980). In this way, the spatial resolution of LTSEM can be increased considerably. A limitation of this increase in resolution results from the fact that the amplitude of the modulated signal decreases with increasing modulation frequency. We shall return to this question in the following section. Investigations of time-dependent effects can be performed with high time resolution if short electron beam pulses are used in combination with a boxcar technique.

rv. INTERACTION BETWEENELECTRON BEAMAND SPECIMEN Due to the electron beam irradiation, both charge and energy are transferred to the specimen. Whereas the charge transfer is not of primary importance, it is the energy transfer that is generally utilized for imaging. This energy transfer takes place via inelastic scattering processes (generation of phonons). As a result, the sample is locally heated by the electron beam. This localized heating effect represents the important sample perturbation which is utilized for imaging in LTSEM (Clem and Huebener, 1980). Therefore, in the following, we concentrate on the thermalization of the beam energy and the corresponding time and length scales. Further, we deal with the characteristic healing length and relaxation time associated with the beam-induced thermal perturbation of the specimen. We do not discuss at all secondary and backscattered electrons, nor Auger electrons and x-ray emission. Although these latter probes provide important signals for imaging by conventional SEM (Reimer, 1985), they are not characteristic for imaging in LTSEM.


11

A . Thermalization of the Beam Energy

The energy loss of the beam electrons in a stopping medium and the sequence of collisions can be described as a continuous process if the energy transfer in a Coulomb interaction between the incident and the target electrons is much smaller than the energy E , of the incident electrons. The energy loss can then be expressed per unit path length along the electron trajectory in the target, leading to the equation (see Reimer, 1985; Birkhoff, 1958) dE e4 - = -4nNZ-In Bi (3) dx mu Here E is the electron energy, and x is the coordinate along the electron path in the target. The coordinate x is taken to increase from zero at the target surface as the beam electrons proceed into the target interior. Due to the sequence of collisions the quantity x follows a complicated path and is not simply the distance of the beam electrons from the target surface. N is the number of the target atoms per unit volume, Z is the number of electrons per target atom, e is the elementary charge, m is the electron mass, and u is the velocity of the electrons. The quantity Bidenotes the ratio of the maximum to the minimum impact parameter and depends upon the electron energy. The derivative d E / d x is called the stopping power, and Eq. (3) is often referred to as the Bethe loss formula (Bethe. 1930). Because of its weak influence, in the following, we ignore the factor In Bi and rewrite Eq. (3) in the form dE 2nNZe4 -=-(4) dx E using E = m v 2 / 2 .With the boundary condition E = E , for x = 0, we obtain the solution 1 - ( E i - E z ) = (2nNZe4)x (5 ) 2 According to Eq. ( 5 ) the total path length L travelled by the beam electrons before reaching the thermal energy of the target atoms is L

=

Et(4nNZe4)

(6)

Here, in view of an energy E , as high as about 20 keV and of a target kept near liquid-He temperatures, we have set the final energy E corresponding to thermal equilibrium with the target equal to zero. The thermalization time zb is obtained from

12

R. P. HUEBENER

Inserting u ( x ) using Eq. ( 5 ) we find m2u: 4

q, = ---(3nNZe4u,)-'

where uo is defined by Eo = mu:/2. As an example, we consider lead as a target material and take E , = 10 keV as the incident beam energy. From Eq. (8) we then obtain zb

=

.

3 10-14s

(9)

This value of the thermalization time appears to be consistent with the experimental observation that the emission of secondary electrons reaches its final value in a time less than lO-'Os following the impact of the primary beam electrons (Oatley, 1972). From Eqs. (6) and (8) one obtains the relation

Except for the factor 3 / 4 , this is the path length travelled by the electrons along their trajectory in the target material during the time zb if they would have the constant velocity 0., For the value u, = 6 lo9 cm s - l corresponding to E , = 10 keV and the thermalization time of Eq. (9), we find from Eq. (10) the path length L = 1.4 pm. Since scattering in the forward direction is expected to dominate for the higher energy range of the beam electrons, this value of L is similar to the penetration depth of the electron beam within the target material. The thermalization time of Eq. (9) shows that the beam energy thermalizes quickly at the coordinate point of the beam focus on the specimen surface. The energy of the beam electrons is transferred into phonons resulting in a localized heating effect. On the other hand, Eq. (10) tells us that these thermalization processes take place only in a thin layer at the specimen surface with a typical depth in the pm range. Following these thermalization processes, the interaction between the electron beam and the specimen can simply be treated in terms of a localized heating effect (Clem and Huebener, 1980),as will be discussed in the next section.

-

B. Localized Heating Eflect: Thermal Healing Length and Thermal Relaxation Time

For treating the beam-induced localized heating effect, we apply the usual concepts of heat diffusion. We start with the geometry of a homogeneous halfspace irradiated with the electron beam. The beam power Po is assumed to be homogeneously dissipated within a hemisphere of radius r o . Because of the


13

spherical symmetry of the geometry, heat conduction is determined by the equations

dT Po = - ~ ( T ) - 2 n r * . dr

for ro < r < co

(12)

Here ti(T) is the temperature dependent heat conductivity of the target material, r is the radial coordinate, and T is the temperature. For simplicity, we ignore for the moment the temperature dependence of K . Then we find from Eqs. ( 1 1) and (12) by integration

(13) and for ro < r < co The total beam-induced temperature increment (at the center of the hemisphere) is

The temperature profile is seen to depend critically upon the value taken for the radius ro. From Eqs. (14) and (15), the radius r* at which the beam-induced temperature increment is reduced to 50% is given by 4

r*(50”/,) = -ro 3 i.e., it is only slightly larger than the radius ro. This radius is approximately given by the penetration depth of the beam electrons in the target material, which typically amounts to about 1 pm for a beam energy of 10-20 keV. Of course, in a more accurate treatment of the temperature profile, the temperature dependence of the heat conductivity K of the target material cannot be ignored and must be taken into account. We note further, that the temperature dependence of ti is particularly strong in the low temperature regime where LTSEM is applied. We can include this temperature dependence in a convenient way by approximating the function K ( T )in terms of a simple power series, say, of the form K ( T )= a T 2 + bT+

c‘

(1 7)

14

R. P. HUEBENER

where a, b, and c are constants. After inserting this function K ( T )into Eqs. ( 1 1) and (12), the solution T(r) can be found analytically by integration. As an example (Metzger, 1987), we show in Fig. 7 the temperature profile T ( r )for germanium and Tb = 2 K based on the approximation (17) and the experimental heat conductivity data of Geballe and Hull (1958). Whereas so far we have considered an unmodulated electron beam, next we turn to the case where the beam current is modulated at the angular frequency o.Now we are dealing with time dependent heat diffusion. The beam-induced temperature modulation of the specimen extends up to the distance qmfrom the coordinate point of the beam focus given by (Clem and Huebener, 1980)

Here D is the thermal diffusivity D =K/C-P

(19)

where C and p are the specific heat and the mass density of the target material, respectively. The length q,, is often referred to as the dynamic thermal healing length. The frequency dependence in expression (18) is known as the thermal skin effect. We see that the length qu decreases with increasing modulation frequency. This results in a corresponding increase of the spatial resolution obtained by LTSEM, if the modulated signal is detected. On the other hand, the signal amplitude decreases with increasing modulation frequency. In this way the gain in spatial resolution becomes limited at high modulation frequencies. As an example we take again germanium. If we ignore for the moment the local beam-induced increment of the time-averaged temperature, we find from Eq. (18) q m = 1 mm, taking the temperature value T = 4.2 K and the modulation frequency o / 2 n = 1 MHz. However, due to the rise of the timeaveraged temperature near the coordinate point of the beam focus, the dynamic thermal healing length qw is expected to be considerably smaller than such a value. This expectation has been confirmed by experiment (Huebener and Metzger, 1985).Of course, an exact treatment requires the consideration of the correct temperature dependence of the thermal diffusivity, and the accurate value of the healing length qw can only be obtained by numerical procedures. So far we have dealt with the geometry of a homogeneous half-space. Next we consider scanning a thin film deposited on a substrate, as we have briefly discussed already in Section 111. The film thickness d is assumed larger than the range of penetration of the beam electrons. The substrate is assumed to have

7 .OO

8.00

2

5.00

Y Y

aJ

L

3 3

4-

2 aJ

4.00

a

E, I-

3.00

2.00

1.00

~

'

0.00

5.00

10 .oo

15.00

20.00

25.00

30.00

Radius [ p m l FIG.7. Temperature profile calculated for germanium based on the approximation (17), for different values of the electron-beam power as indicated. Tb = 2 K. (From Metzger, 1987).

16

R. P. HUEBENER

high heat conductivity and to be closely coupled to the liquid-He bath, such that the substrate temperature is about equal to the bath temperature Tb.For an unmodulated beam the thermal perturbation of the specimen film extends up to the distance from the coordinate point of the beam focus given by the thermal healing length v] of Eq. (2). The temporal response to the beam irradiation is governed by the thermal relaxation time (Huebener, 1984; Clem and Huebener, 1980) z, =

C*p*d ~

ff

If the electron beam is turned on or off, the steady state behavior is attained after the time 7., The thermal healing length v] and the thermal relaxation time t,, given by Eqs. (2) and (20), respectively, determine the spatial and temporal resolution limits of the thermally induced response signal in LTSEM. For a lead film of 1 pm thickness deposited on a substrate, at 4.2 K one obtains the typical values v] = 100 pm and t, = lo-' s. Here we have used u = 1 Wcm-2K-' for the heat transfer coefficient. (The heat conductivity K was found using the Wiedemann-Franz law and assuming an electrical resistance ratio between room temperature and 4.2 K for the lead film of 200). It is important to note that the length v] of Eq. (2) refers to an unmodulated electron beam. If the beam current is modulated at the angular frequency w, the static healing length q is replaced by the frequency-dependent thermal healing length

where v] is taken from Eq. (2). In the limit wz, > 1, Eq. (21) approaches the expressions given in Eq. (2) and (18), respectively. Of course, the increase in spatial resolution at high modulation frequencies expressed in Eqs. (18) and (21) can only be obtained if the modulated signal is detected. The frequency dependence of Eq. (21) has recently been confirmed experimentally (see Pavlicek et al., 1984). In this study a superconducting thin film microbridge fabricated from lead-indium alloy has been used as a bolometer for recording the beam-induced voltage signal as a function of the distance between the microbridge and the coordinate point of the beam focus. The width and length of the microbridge was about 5 pm and 10 pm, respectively. The frequency of the beam modulation had been varied between 100 kHz and 18 MHz. Typical experimental results are shown in Fig. 8. Comparison with the theoretical curve calculated from Eq. (21) indicates


0 0

2

4

6

17

8

10 12 14 16 18 20 f [MHzl FIG.8. Experimental values of the length qo for a thin film of lead-indium alloy versus the frequency of the beam modulation (circles).The solid line is calculated from Eq. (21). 7’’ = 4.2 K (from Pavlicek el al., 1984).

excellent agreement between experiment and theory. If one applies Eq. (21) to superconducting films of Pb, Pb-In alloy, or Nb,Ge with 0.5 pm thickness as typical model cases, values of near or below 1 pm are obtained for modulation frequencies in the range 10-1000 MHz, depending upon the thinfilm material (Pavlicek et al., 1984). Of course, in any calculation of the beam-induced thermal response of the specimen only the beam power absorbed in the sample plays any role. Therefore, correction factors may have to be applied due to the emission of secondary and backscattered electrons, photons, etc. So far, in our discussion of the thermal sample response to the electron beam irradiation we have concentrated on purely diffusive heat propagation. This mode of energy transfer dominates if local equilibrium is established within the phonon and electron system and also between both systems by means of sufficient scattering. However, at liquid-He temperatures phononphonon and electron-phonon scattering is strongly reduced and energetic excitations can propagate over relatively long distances ballistically, i.e., without scattering. For the application of LTSEM, ballistic phonon propagation is particularly important, and we will discuss this point in more detail in Sections X-XII. Ballistic processes are expected to contribute eventually to the spatial become resolution limit of LTSEM if the thermal healing lengths q or extremely small. Of course, ultimately the resolution of LTSEM is always limited by about the range of penetration of the beam electrons in which the thermalization of the beam energy takes place, and which we have discussed in Section IVA.

R. P.HUEBENER

18

V. SUPERCONDUCTING TUNNEL JUNCTIONS: PAIRTUNNELING As the first example of a recent application of LTSEM we consider the observation of spatial structures in superconducting tunneljunctions. Here we deal with a typical low temperature phenomenon particularly well suited for LTSEM. Usually, superconducting tunnel junctions consist of two superconducting films separated from each other by a thin electrically insulating barrier. The thin-film structure is deposited on a proper substrate. During the LTSEM experiments, the film structure on top of the substrate is scanned directly with the electron beam, whereas the bottom side of the substrate is in direct contact with the liquid-He bath (see Fig. 1). In addition to the strong fundamental interest in the observation of spatial structures in superconducting tunnel junctions by means of LTSEM, such experiments are also highly important for the cryoelectronic applications. In Sections V-VII we deal separately with different aspects of tunnel junctions, namely pair tunneling, quasi-particle tunneling, and large junction arrays, respectively. The typical geometry of a superconducting tunnel junction is shown schematically in Fig. 9. The two superconducting films crossing each other are separated by a thin electrically insulating barrier. The barrier is formed usually by an oxide of one of the superconducting electrodes and has a thickness of only a few A. Current and voltage leads can easily be attached to the tunnel junction in the cross-line geometry as indicated in Fig. 9. Standard thin-film technology and microfabrication techniques are employed for preparing the junctions. As first predicted by Josephson (1962), Cooper pairs (the particles constituting the superconducting electronic system) can tunnel across the barrier resulting in an electric current flow at zero voltage, if the junction barrier is sufficiently thin. The maximum Josephson current density that can flow across the tunnel junction without electrical resistance is given by the equation

Here J1(x,y ) is the local critical current density and #(x,y) the local difference between the phase of the superconducting wave function in both electrodes.

current voltage

FIG.9. Cross-line geometry of a superconducting tunnel junction.


19

The coordinates on the junction are denoted by x and y. In the following we assume J , ( x ,y ) to be homogeneous. (Spatial variations of J I ( x ,y ) can result from inhomogeneities in the barrier and in the superconducting electrodes of the junction.) For two identical and homogeneous superconductors one finds (see, e.g., Ambegaokar and Baratoff, 1963)

where A(T)is the temperature dependent energy gap, e the elementary charge, R , the tunneling resistance per unit area of the junction in the normal state, and k , Boltzmann’s constant. The phase-difference function &x, y) depends upon the junction geometry. Further, it is influenced by a magnetic field applied parallel to the junction barrier. Geometrically, the important length scale is the Josephson penetration depth 1, given by

where h is Planck’s constant divided by 2 q and po the permeability of free space. The length d is

d = I‘,

+ I,, + t

(25)

where ILL, and I L 2 are the effective London penetration depths of the two superconducting electrodes and t the barrier thickness of the junction. The typical range of A, is about 1 mm. For junctions with small area, where both dimensions are much smaller than I,, and for zero applied magnetic field, the phase-difference function and, hence, the current density J ( x , y ) is a constant. If a magnetic field H is applied parallel to the barrier of a small-area junction, the local maximum Josephson current density J ( x , y ) is modulated as a function of the magnetic field, and the total maximum Josephson current Imaxshows the “Fraunhofer diffraction pattern”

Here w and L are the width and length of the junction, respectively. cp = H . L d is the total magnetic flux threading the junction (assuming the magnetic field to be oriented perpendicular to the direction with the junction dimension L). cpo = h/2e is the magnetic flux quantum. For junctions with dimensions equal to or larger than 1, the phasedifference function depends upon the spatial coordinates of the junction since the magnetic self-field of the tunneling current cannot be neglected any more.

-

R. P. HUEBENER

20

0

L

FIG. 10. Density of the maximum Josephson current J ( y ) of a one-dimensional tunnel junction versus the junction coordinate for the case L = 15 1,. L =Junction length; 1, = Josephson penetration depth.

First we consider a one-dimensional geometry, where one dimension, say, the width w of the junction (in x-direction), is still much smaller than A,, whereas the length L of the other dimension (in y-direction) is much larger than I , . The first analysis of such a geometry has been given by Owen and Scalapino (1967). In Fig. 10 we show the maximum Josephson current density along the direction of the dimension of length L for the case L = 15 I , and for zero applied magnetic field. The current density J ( y )is seen to reach a maximum at a distance of about 1, from both ends of the junction and to decrease sharply in the interior of the junction. This expulsion of the Josephson current from the junction interior is similar to the Meissner effect in a bulk superconductor, the London penetration depth AL being replaced by the Josephson penetration depth I , . If a magnetic field H parallel to the barrier and perpendicular to the direction of the dimension of length L is applied to the junction, Josephson vortices penetrate into the junction interior. In Fig. 11, we show schematically the case where four Josephson vortices exist within the junction due to the applied magnetic field. The current applied to the junction is assumed to be zero. In addition to the current density J ( y ) , the current vortices, as viewed parallel to the barrier, can be seen. It is the existence of the Josephson vortices within the junction and the oscillating pair current density similar to that shown in Fig. 11 that results in the “Fraunhofer diffraction pattern” of I,,, expressed in Eq. (26) for a small-area junction. Two one-dimensional junction geometries which are particularly interestingdue to their simplicity are shown schematically in Fig. 12: the in-line and the overlap geometry. In both cases, the current ca)nbe fed symmetrically to the junction.


21

t

J1y’

top electrode

barrier

@H

bottom electrode

FIG, I I . Top: Density of the Josephson current J ( y ) of a one-dimensional tunnel junction in the case where four Josephson vortices are generated due to a magnetic field applied parallel to the barrier. The current applied to the junction is zero. Bottom: Josephson vortices as seen parallel to the barrier.

Two-dimensional junction geometries, where both dimensions are larger than A J , require a complicated numerical analysis. The influence of the geometry of the current feeding lines must be explicitly taken into account in addition to the particular junction geometry. The brief summary we have given so far on the spatial distribution of the pair tunneling current has concentrated only on the most essential points. Further details can be found elsewhere (see, e.g., Tinkham, 1975; Barone and Paterno, 1982; Bosch, 1986). Next we turn to the signal generated by LTSEM for imaging the spatial distribution of the maximum Josephson current density J(x, y). The local perturbation of the junction by the electron beam is expected to change the critical current density J1(x,y)by the amount SJ, in the irradiated region. If the phase-difference function #(x, y) were to remain unaffected, the measured beam-induced change of the junction critical current as a function of the beam position yo would be given by m Y 0 ) = SJ,(YO) * 6Y * w sin #(Yo)

(27)

Here and in the following, for simplicity, we assume a one-dimensional geometry, restricting any spatial dependence to the y-direction and ignoring

in-line geometry

overlap geometry

FIG. 12. In-line and overlap geometry. The barrier is indicated by the hatched part.

22

R. P. HUEBENER

the dependence upon the x-coordinate. The results obtained can easily be extended to a two-dimensional geometry. In Eq. (27) 6y is the length of the junction element perturbed by the beam, and w is the junction width measured along the x-direction. From Eq. (27) we see that the change 61:(yo) is proportional to the current distribution in the unperturbed junction biased at its critical current value. According to Eq. (22) this current distribution is determined by the phase-difference functions 4(yo). As pointed out by Chang and coworkers (Chang and Scalapino, 1984; Chang and Ho, 1984; Chang, Ho, and Scalapino, 1985), in addition to the local effect expressed in Eq. (27), there also exists a nonlocal contribution to the beam-induced signal 61,(yo) due to the change 64 in the phase-difference function. This contribution results from the increase in the penetration depths AL and I,, by 6AL and 61,, respectively, and is given by 61f(Yo) = w

-

s

dY * J,(Y) * cos $(Y)

*

64(Y,Yo)

(28)

Clearly, the beam-induced change 6 4 results in a signal contribution also from the nonirradiated part of the junction in addition to the irradiated portion, as expressed by the integral over the y-coordinate in Eq. (28).In the limit of weak perturbation (6R,/RL EL, (Bosch et al., 1985). The recording (b) in Fig. 14 was obtained near the local maximum associated with the 4-5 vortex state in the magnetic interference pattern. The recordings (a) and (c)were obtained on the low- and high-field side of this maximum, respectively. The increase and decrease in the maximum amplitude of the signal - 6 1 , ( x , y ) along the y-coordinate shown by the recordings (a) and (c), respectively, result from the nonlocal response due to the influence of the phase-difference function qh and agrees well with the predictions by Chang and coworkers (1984, 1985). The maximum amplitudes of the beam-induced change - 6 1 , ( x , y ) in Fig. 14 correspond to about 30% of the maximum critical current of the unirradiated junction at the corresponding magnetic field values.

24

R. P. HUEBENER

B e -BEAM

d

H-

90 p m

+

FIG. 14. The signal -61,(x,y) showing the 4-5 vortex state and obtained by scanning longitudinally along the junction. The line scans were performed for several values of the transverse coordinate. The position of the junction and the scanning direction are indicated at the bottom. Recording(b) was obtained near the local maximum of the magnetic interference pattern, whereas recordings (a) and (c) were taken on the low and high-field side of this maximum, respectively. Tb = 4.2 K, beam voltage = 26 kV, beam current = 10-100 pA (from Bosch et al., 1985).

In addition to the different vortex states in an applied magnetic field, the restriction of the flow of the maximum Josephson current to the region near both ends of the junction at zero magnetic field as determined by the penetration depth ;1 could be confirmed by LTSEM imaging. Of course, from these latter experiments the value of the Josephson penetration depth ;1 can be obtained. The evolution of the different vortex states in an increasing magnetic field parallel to the junction barrier (Bosch, 1986) can be seen in Fig. 15. Here the sample geometry is similar to that of Figs. 13 and 14, with a sapphire substrate and Nb groundplane covered by a SiO insulating film. The base electrode is a PbIn film of 109 nm thickness and the top electrode a PbBi film of 250 nm

SCANNING ELECTRON MICROSCOPY AT VERY LOW TEMPERATURES 25

+

I

Magnetic Field

[mAl Generating Current IF

0 Be

f

100 JJA

3 50pA

91 pm __* Y FIG. 15. Signal -61,(y) recorded in an increasing magnetic field applied parallel to the barrier and showing the different vortex states. For each value of the magnetic field only a single line scan is presented. The number of vortices in the junction are indicated on the left. The magnetic field generating current (passing through the top electrode) are given in the second column on the left. From top to bottom the sensitivity of the signal detection increases as indicated by the scale marks on the right. Tb= 4.2 K, beam voltage = 26 kV, beam current = 10- 100 pA (from Bosch, 1986).

thickness. The area of the tunneling window is 97 pm x 19 pm. The signal -dIl(y), obtained by scanning linearly along the middle of the junction in the direction of the long dimension L, is shown. For this junction the ratio L/A, is 5.2. The different vortex states were recorded for the magnetic field values corresponding to the local maxima of the magnetic interference pattern of the maximum Josephson current. Therefore, the maximum amplitude of the signal -61,(y) is expected to remain approximately constant along the junction length. The magnetic interference pattern of this junction is shown in Fig. 16. As seen from Fig. 15, the different vortex states show reasonably regular behavior. The small shifting of the signal seen at the highest vortex states containing 12-14 vortices was accompanied by an increasing difficulty at higher magnetic fields to accurately reproduce the recorded signal during repeated line scans. It appears that these high vortex states are extremely sensitive to small perturbations.

26

R. P. HUEBENER

Ic

I[

[rnAl

JJAI

1.5

-

-

150

1.0

-

-

100

0.5

-

-

50

0.0

-

-

0

1 I I I I I I I -30 -20 -10 0 10 20 IF ImAl FIG. 16. Magnetic interference pattern (&-Hcharacteristic) for the same sample as that of Fig. 15. Two measured curves are shown at different sensitivity of the vertical current axis as indicated. & = 4.2 K (from Bosch, 1986). I

In summary, we note that the different vortex states generated by the pair tunneling current have clearly been observed by LTSEM. The results agree well with the theoretical expectation. In particular, the nonlocal effect due to the influence of the phase-difference function predicted theoretically has been confirmed experimentally. So far, most experiments have concentrated on one-dimensional junction geometries. In these experiments a spatial resolution of 1-2 pm has been reached. For further experiments studies of twodimensional junction geometries are highly interesting. Here particular attention is expected to be given to the influence of the geometry of the lines feeding electric current to the junction. Comparison of the two-dimensional images of the pair current density with numerical calculations will be an important task. During all of this section we have assumed that the tunneling barrier is spatially homogeneous. Accurate evaluation of the barrier homogeneity and a sensitive detection of inhomogeneities in the barrier is possible by means of the quasiparticle tunneling current, as we will see in the following section. VI. SUPERCONDUCTING TUNNEL JUNCTIONS: QUASIPARTICLE TUNNELING In addition to the pair tunneling current at zero voltage drop, the normal excitations or single quasiparticles can tunnel across the barrier of a superconducting tunnel junction. In the latter case, the flow of the tunneling


current is accompanied by a resistive voltage. The current-voltage characteristic (IVC) is highly nonlinear, its shape depending on the energy gap of the superconducting electrodes. The tunneling process of quasiparticles across a superconducting tunnel junction has first been demonstrated experimentally by Giaever (1 960). The variation of the quasiparticle tunneling current with the voltage V applied to the junction is shown schematically in Fig. 17, assuming two superconducting electrodes with the energy gaps A1 and A,, respectively. At T = 0 the tunneling current remains zero until a discontinuous jump occurs at the voltage V = (Al + A2)/e. Following this jump, the current gradually approaches the normal tunneling characteristic (curve I , ) obtained when both electrodes are in the normal state. At T > 0 thermally excited quasiparticles result in a tunneling current also at voltages smaller than (Al + A,)/e. Now the tunneling current peaks at V = [ A , - A,l/e and rises sharply again at V = [A,(T) + A,(T)]/e approaching the curve I , . This structure in the IVC provides a simple way to measure the energy gap of the superconducting electrodes. The quasiparticle tunneling current is given by (Tinkam, 1975; Barone and Paterno, 1982)

(30)

Here, A is the tunnel junction area and R , the normal tunneling resistance per unit area obtained when both electrodes are in the normal state. E is the quasiparticle energy and f ( E ) the Fermi distribution function at energy E. In the integral of Eq. (30),the energy ranges IEJ < ) A I [and IE + eVI < IA2( are excluded. The complete IVC is found from Eq. (30)by numerical integration. In some regimes approximate expressions for the quasiparticle tunneling current can be used. For a symmetric tunnel junction where both electrodes consist of the same material (Al = A, = A) one obtains in the voltage range V < 2A/e (Solymar, 1972)

exp( -A/kBT) (31) According to (31) the current depends only weakly upon the voltage, except near V = 0 where it decreases linearly with decreasing voltage. Assuming the voltage-dependent terms to be of the order of 1, one finds A I z --(27~Ak,T)'~~ exp( -A/k,T) x

Rn

I

V 0 FIG. 17. Quasiparticle tunneling current I versus the voltage V for two superconducting electrodes with the energy gaps A, and A * , respectively. The solid line refers to T = 0 and the dashed line to T > 0. The straight line marked I , represents the normal tunneling characteristic.

/ I

/

Load line

\

V

FIG. 18. Pair-tunneling current at zero voltage. If its maximum value Imax is exceeded, the junction switches along the load line to the voltage state on the quasiparticle tunneling characteristic. The straight line marked I , represents the normal tunneling characteristic.


29

It is the resistive nonlinear quasiparticle IVC which is attained by the junction if the total maximum Josephson current I,,, discussed in Section V is exceeded. At I > I,,,, the junction switches from the zero-voltage state to the corresponding point on the quasiparticle tunneling characteristic along the load line (see Fig. 18). In this state of non-zero voltage in addition to the quasiparticle DC current, the Josephson AC current appears. The latter current flows without dissipation and oscillates at the frequency 2e o=-v h

(33)

These pair current oscillations in a superconducting tunnel junction are described by the equation

where $ is the phase difference function introduced in Section V. Equations (22) and (34) constitute the two Josephson equations describing the flow of supercurrent across the barrier of a tunnel junction. In the imaging experiments on the quasiparticle current distribution using LTSEM and described in the following, the Josephson current is suppressed usually by means of a small magnetic field applied to the junction. The highly nonlinear quasiparticle IVC of a superconducting tunnel junction and the extremely rapid switching between the zero-voltage and finite voltage state (Fig. 18) are of particular interest for many cryoelectronic applications, Further details about the basic properties and the cryoelectronic applications of tunnel junctions can be found elsewhere (Tinkham, 1975; Barone and Paterno, 1982; Solymar, 1972; Matisoo, 1980). As we can see from Eqs. (30)-(32), spatial structures in the distribution of the quasiparticle current of a tunnel junction can be caused by inhomogeneities in the barrier and in the energy gap of the superconducting electrodes. In this case, a two-dimensional voltage image of the inhomogeneous junction properties is obtained by recording the beam-induced voltage change 6 V ( x ,y) of the current-biased junction as a function of the coordinates x and y of the beam focus. This voltage signal 6 V ( x ,y) is due to the localized heating effect of the electron beam and is the signature of the beam-induced conductivity change we have discussed in Section 111. We shall see below that the information obtained in this way depends critically upon the bias point on the IVC of the junction. Whereas the first LTSEM studies were performed using single junctions (Epperlein, Seifert and Huebener, 1982; Seifert, Huebener and Epperlein, 1983; Gross et al., 1984), subsequent experiments dealt with double junctions with the standard injector-detector configuration (Gross, Koyanagi,

30

R. P. HUEBENER

Seifert and Huebener, 1985; Gross, Schmid and Huebener, 1986). Earlier summaries of these results can be found in Huebener (1984), Bosch et al. (1986), Huebener and Seifert (1984), and Huebener (1985). The beam-induced voltage signal 6 V ( x ,y) is generated in the following way (Seifert, Huebener and Epperlein, 1983). We consider a planar tunnel junction of total area A . Focusing the electron beam on the coordinate point x, y on the junction surface, results in the thermal perturbation of the area nA2 around this point. Usually the radius A can be identified with the thermal healing length of Eq. (2) or (18). In this area the tunneling current increases by the amount 6 I ( x , y ) . Hence, in the unperturbed area A - nA2 of the current-biased junction, the current is reduced by the amount S l ( x , y). Assuming nA2 T,, the resistance is practically temperature independent for the small temperature excursions expected due to the beam irradiation. The sensitive region at each hotspot boundary extends over a distance of about the thermal healing length v] in both directions into the adjacent superconductor. Therefore, we expect a peak in the voltage signal 6 V ( x , y ) of width 21 to appear at the hotspot boundaries, if the sample film is scanned in the longitudinal direction. A more detailed treatment of the origin of the voltage signal S V ( x , y ) , based on a mathematical analysis of the heat balance equation, can be found elsewhere (see Eichele et al., 1983). Of course, such an accurate treatment must include the influence of the external electronic circuit attached to the specimen. As

- increasing bias voltagew

E

>

I

. -

-;-E-

FIG.29. Two-dimensional voltage image of the hotspot boundaries indicated by the bright regions for a superconducting tin film. The narrow strip with the wide section at both ends is indicated by the black lines. The image shown at the top is obtained by detecting the backscattered electrons. Results are presented for different sample voltages as indicated on the right. Tb = 2.5 K, beam voltage = 30 kV, beam current = 1-50 pA. (From Eichele et al., 1983).

shown by the theoretical analysis, the voltage signal 6 V ( x ,y) arises because of a small expansion of the hotspot in addition to a small shift in its location caused by the electron-beam irradiation. Typical experimental results (Eichele et al., 1983) are shown in Fig. 29. Here we see the two-dimensional voltage signal 6 V(x, y) for increasing and decreasing sample voltage. The signal is used for modulating the brightness on the oscilloscope screen. Bright regions correspond to large values of the signal 16V(x,y)J.At the top of the figure the signal generated by the backscattered electrons is indicated for identification of the sample location. The experiments were performed using a tin film of 1.9 mm length, 102 pm width and 0.5 pm thickness at a He bath temperature of 2.5 K. Figure 30 presents the voltage signal obtained under identical conditions as in Fig. 29 plotted against the longitudinal sample coordinate (y-modulation). As we expect, a localized voltage signal 6V(x,y) appears at the two hotspot boundaries. Figures 29 and 30 clearly show how the length of the hotspot changes with increasing and decreasing sample voltage and current.


detector signal

I

I

X

I

> co

sample coordinptt

I I

I

I

I

I I

I I

!I

I

I

I

I

10.7 12s9

0

o ' I

or

5,l

I

I

'

I

12

FIG.30. Voltage signal S V ( x ) (in arbitrary units) obtained under identical conditions as in Fig. 29 versus the longitudinal sample coordinate. The amplitude of the signals S V ( x ) shown is 100 - 200 nV. The top shows the signal generated by the backscattered electrons. The two hotspot boundaries are marked by the voltage peaks. The different sample voltages are indicated on the right. (From Eichele et a!., 1983).

Figure 31 shows a plot of the electric resistance of the film versus the distance L, between the two maxima of the voltage signal for eight different values of the length L,. The data can be fitted well by a straight line passing through the origin and through the resistance value obtained if the total sample length L is in the normal state (last point on the upper right). These results refer to a tin film doped with 0, of 1.9 mm length, 91 pm width and 0.5 pm thickness. They clearly demonstrate that the distance between the two peaks of the voltage signal indeed measures the geometric length of the hotspot. We have pointed out above, that the width of the signal 6 V ( x , y ) at the hotspot boundaries is approximately twice the thermal healing length q

46

R. P. HUEBENER

f

200

-C E

i

cz

I I I

100

I I I

0

0.5

1 L,(rnrn)

1.5

1

2

FIG.31. Electric resistance versus the distance L,, between the two maxima of the voltage signal S V ( x ) for an 0,-doped tin film. Tb = 3.7 K, beam voltage = 30 kV, beam current = 1-50 PA. (From Eichele et a[., 1983.)

expressed in Eq. (2). Through the heat conductivity K, the length q depends upon the purity of the superconducting film. This dependence is demonstrated in Fig. 32 which shows the signal 6 V ( x ,y) marking the two hotspot boundaries plotted against the longitudinal sample coordinate for three samples strongly differing in their heat conductivity. All samples had the same film thickness of 0.5 pm and were deposited on single-crystalline sapphire substrates. From top to bottom in Fig. 32 the sample material was pure tin, 0,-doped tin, and 02doped aluminum. For the last material the image 6 V ( x ,y) of a single hotspot boundary is also shown at higher resolution. The distance 2q calculated from Eq. (2) for the three samples is indicated in Fig. 32 and agrees reasonably well with the geometrical width of the voltage signals. It is interesting that in the 0,-doped aluminum sample the voltage signal is localized in a region as small as only a few pm. Further details on these experiments including the measuring electronics can be found in (Eichele et al., 1983). In a superconducting microbridge fabricated from a highly impure material such as 0,-doped aluminum many hotspots can be generated simultaneously because of their small size. Indeed, hotspots with a total length of 15-20 pm have been observed by LTSEM (Eichele et al., 1983). Of


47

/

0

/

\

Fic. 32. Voltage signal SV(x) marking the two hotspot boundaries versus the longitudinal sample coordinate for three samples as indicated in the text. The thermal healing length q decreases from the sample at the top to that at the bottom because of the decrease in heat conductivity. For sample H9 (bottom) (he voltage signal 6 V ( x )of a single boundary is shown also at higher spatial resolution. The width 2q of each voltage peak calculated from Eq. (2) is indicated for each sample. T, differed for the three samples and ranged between 2.25 and 3.7 K. Beam voltage = 30 k V , beam current = 1-50 PA. (From Eichele et al., 1983.)

course, the high spatial resolution provided by LTSEM is crucial for such observations. According to Eq. (2 I), high-frequency modulation of the electron beam results in a shrinking of the dynamic thermal healing length v], with increasing modulation frequency o. Correspondingly, the width of the modulated voltage signal marking the hotspot boundaries is reduced with increasing frequency. Experiments performed with an 0,-doped tin bridge up to beam modulation frequencies of 10 MHz confirmed these effects (Freytag et al., 1985). Typical results are shown in Fig. 33. Here the dynamic thermal healing length qw found experimentally from the width of the signal peaks 6 V ( x ,y ) at the hotspot boundaries is plotted versus the frequency v of the beam modulation. The solid line represents a theoretical curve calculated from Eq. (21)

R. P. HUEBENER

0.0

4.0 8.0 frequency V IllHz1

12.0

FIG.33. Dynamic thermal healing length qo versus the frequency v = 4271 of the beam modulation. Crosses: experimental values obtained from the width of the voltage peaks. Solid line: theoretical curve calculated from Eq. (21) using the experimentally determined values q = 86 pm and 7, = 280 ns. Tb= 2.5 K. (From Freytag et al., 1985.)

using the values q = 86 pm and z, = 280 ns. The latter two values were found from independent measurements. Hence, the theoretical curve contains no adjustable parameter. Simultaneously with the dynamic healing length q w , the peak value of the voltage signal 6V(x,y) at the hotspot boundaries decreases with increasing modulation frequency. This effect is shown in Fig. 34. Again, the solid line represents a theoretical curve obtained from the theoretical analysis of the voltage signal at high-frequency beam modulation (Freytag et al., 1985).Both Figs. 33 and 34 indicate satisfactory agreement between experiment and theory. For further details we refer to Freytag et al., 1985. In summary, we have seen that the beam-induced voltage signal 6 V ( x ,y) clearly images the boundaries of a hotspot in a thin-film superconductor. Theoretically the origin of the voltage signal is well understood, and good agreement is found between experiment and theory, including the effects due to high-frequency beam modulation. Noting that the development of a hotspot represents the final stage (before destructive burnout) in the resistive behavior of a thin-film superconductor reached at relatively high values of the applied electric current, the possible role of LTSEM imaging of the “early” stage remains an open and interesting question. Here we have in mind the formation of a phase-slip center in a nearly one-dimensional geometry


49

40

0

0.0

4.0 fraquency 3

8.0

12.0

UHz 1

FIG.34. Peak value of the voltage signal marking the hotspot boundaries versus the frequency v = u)/2nof the beam modulation. Crosses: experimental values. Solid line: theoretical curve obtained from a theoretical analysis. Tb = 2.5 K. (From Freytag er al., 1985.)

(Huebener, 1979; Skocpol et a!., 1974). In a two-dimensional geometry, correspondingly we deal with a phase-slip line (Volotskaya et al., 1981, 1984) and flux flow (Huebener, 1979). So far, imaging of these latter structures in thin-film superconductors have not been reported. However, the generation of a voltage image of, say, a phase-slip center by means of LTSEM does not seem unfeasible, in particular, if the phase-slip center can be pinned at some location within the thin-film superconductor.

1X. CURRENT FILAMENTS AND TURBULENCE I N SEMICONDUCTORS So far we have discussed applications of LTSEM dealing with superconductivity as a typical low temperature phenomenon. Next we turn to semiconductors. Here we will concentrate on the application of LTSEM for imaging of the current filaments generated by impurity impact ionization induced avalanche breakdown at low temperatures. At liquid-helium temperatures in a semiconductor all charge carriers are frozen out, and the material acts as an electric insulator, if the applied electric field is not too large. However, above a threshold field, avalanche breakdown occurs resulting in electric current flow. In a doped semiconductor this process

R. P.HUEBENER

50

is caused by impact ionization of the donors or acceptors (Seeger, 1982; Monch, 1969). Recently, the subject of avalanche breakdown in semiconductors has received strongly increasing attention because of the observation of spontaneous oscillations and chaotic behavior of the electric resistance during avalanche breakdown (Teitsworth et al., 1983, 1986; Held et al., 1984, 1986; Peinke et al., 1985). Usually during avalanche breakdown of a homogeneous semiconductor, spatial structures develop such as current filaments or high-field domains (Bonch-Bruevich et al., 1975).These structures evolve as a result of the highly nonlinear sample response to the applied electric field, and their formation is associated with a strongly nonlinear IVC of the semiconducting material. One of the major issues of strong current interest is the question to what extent the complex temporal resistance behavior of a semiconductor during avalanche breakdown (Teitsworth et al., 1983, 1986; Held et al., 1984, 1986; Peinke et al., 1985)is correlated with complex spatial structures of the electric current flow. Here imaging by means of LTSEM promises to provide an answer. In the following we summarize recent experiments performed with p-germanium using LTSEM for the two-dimensional imaging of electric current filaments (Mayer, 1986; Mayer et al., 1987). The experimental arrangement is shown schematically in Fig. 35. The semiconductor crystal is glued to a single-crystalline sapphire substrate with

electron beam

_

I

-

-

I

I

I

-

-

-

-

liquid He T=L.ZK FIG.35. Experimental arrangement for imaging current filaments in a semiconductor by means of LTSEM. The p-germanium sample is glued to the sapphire substrate using Stycast cement. The ohmic contacts are indicated by the hatched areas on the sample surface. (From Mayer, 1986)


51

1 mm thickness and 20 mm diameter using Stycast cement for good thermal contact. During the LTSEM experiments the bottom of the sapphire substrate is in direct contact with the liquid-He bath, whereas the top surface of the semiconducting crystal can be scanned with the electron beam. Electric leads attached to proper ohmic contacts on the specimen serve for applying the electric field. The response signal to be investigated as a function of the coordinate point (x, y) of the electron beam focus on the sample surface is the beam-induced conductivity change. The experiments described in the following were performed with voltage biased operation. Hence, the response signal consisted of the beam-induced electric current change 61(x, y). We note that the beam irradiation only represents a small perturbation of the object to be studied. Typically, the absorbed beam power was about 1 pW, whereas the Joule heat dissipated in the specimen during avalanche breakdown was in the mW range. For increasing the sensitivity, the beam current was modulated at 5 kHz and the signal 61(x,y) was detected using a lock-in technique. Qualitatively, the imaging process of the current filaments by means of the signal S l ( x , y ) can be understood as follows (see Fig. 36). We refer to a semiconducting sample doped homogeneously with flat donors or acceptors. Taking Ge as an example and assuming a typical beam energy of 26 keV, the beam injects about lo4 electron-hole pairs per incident electron in the generation volume of a few pm diameter. If the beam is directed to a nonconducting region, where no filamentary current flow takes place, the injected hot carriers can locally induce avalanche breakdown by impurity impact ionization, resulting in a significant current increment 61(x, y) in the voltage-biased sample. On the other hand, if the beam is focused on a highly conducting region with filamentary current flow, where most of the shallow

FIG.36. Origin of the signal 61(x,y) for imaging current filaments in a semiconductor. Top: part of a semiconductor sample as seen from the top with a current filament in vertical direction indicated by the hatched section in the center. The horizontal dashed line indicates a line scan of the electron beam. Bottom: beam-induced signal 6 I ( x )versus the beam coordinate for the line scan shown at the top.

1

0

=-

1.00

=~ ,

Q 1.10

1 ,

1.20

T=L':

1.30

V I'

FIG.37. Upper part: current-voltage characteristic without and with the beam irradiation. Inset shows the sample configuration. Lower part: beam-induced signal 61(x, y ) plotted vertically for the series of horizontal line scans. The triangular markers on the images correspond to those on the inset and specify the scanned portion of the sample. The images(a)-(f) refer to the different bias voltages indicated. T, = 4.2 K, beam voltage = 26 kV, beam current = 300 PA, material is p-doped Ge. (From Mayer, 1986.)


53

impurities are ionized, no significant current increment is expected. In this way the boundaries of a current filament can be detected. (This imaging principle is in some way similar to that for hotspots in current-carrying thin-film superconductors discussed in Section VIII). The sample material was single-crystalline p-doped Ge with an acceptor (indium) concentration of about I O l 4 ~ 3 1 3 The ~ ~ . samples had the typical dimensions 9 x 3 x 0.26 mm3 and were polished with diamond paste. They were provided with ohmic aluminum contacts. Figure 37 (top)shows the IVC without and with the beam irradiation for a Ge sample with the geometry indicated in the inset. The hatched areas represent the ohmic contacts. The voltage was applied to the inner two contacts, and the beam-induced signal 6 1 ( x ,y) was detected by means of the 1 ohm series resistor. The beam irradiation is seen to cause a small shift of the IVC. During this measurement of the IVC for the irradiated sample the beam was focused on a fixed point on the sample surface showing a relatively large response signal 61(x, y). In the lower part of Fig. 37, the signal Sl(x, y) is plotted vertically for a series of horizontal line scans (y-modulation) across the center part of the sample. The triangular markers on the two-dimensional images correspond to those on the inset and specify the scanned portion of the specimen. The images (a)-(f) were obtained for increasing bias voltage as shown. Images (a) and (b) were recorded in the pre-breakdown regime with about 10 times the signal amplification than that of images (c)-(f) obtained in the post-breakdown regime. The results of Fig. 37 clearly indicate the formation of a current filament, its width extending up to about 2 mm at the highest voltage shown. Of course, the shape of the filament in Fig. 37 is influenced by the dipole-like electric field pattern caused by the small area of the inner ohmic contacts. This field pattern is likely to play a role in the rapid decrease of the signal 61(x, y ) seen in the images (d)-(f) outside the filament region. For a proper recording of the signal 61(x,y) the beam modulation frequency must be sufficiently low such that the sample can respond properly. For the Ge samples studied the typical signal decay time was found to be about 10-20 ps. Therefore, the 5 kHz modulation frequency appears adequate. Figure 38 shows the evolution of a multifilamentary structure for increasing voltage bias. The top part indicates the geometry, and the results refer to the upper sample portion. The two small tips on each of the upper two ohmic contacts serve for promoting filament nucleation. The TVC is shown also at the top. It does not display any beam-induced shift because of the small beam current of about 20 pA used in this experiment. The two-dimensional images presented in the lower part were obtained at the different levels of the voltage bias as indicated. Here the signal 61(x, y) is used for modulating the brightness on the oscilloscope screen. Bright regions correspond to a large

T I

I

electron beom

1.18

M A , FiZK; 1.20

1.22

1.24

126

1.28

L30

1

1.32 V I V l

FIG.38. Evolution of multifilamentary current flow in p-doped Ge. Upper part: geometry and current-voltage characteristic. Lower part: two-dimensional images of the beam-induced signal N ( x , y) obtained at the different voltage levels indicated. Bright regions correspond to a large signal 61(x, y). Tb= 4.2 K, beam voltage = 26 kV, beam current = 20 PA. (From Mayer, 1986.)


55

response signal 61(x,y). Again, the triangular markers on the two-dimensional image on the upper left correspond to those on the inset and specify the scanned portion of the specimen. All other images refer to the identical scanned section of the sample. The image obtained at 1.210 V refers to the prebreakdown regime according to the IVC displayed at the top. Apparently, there does not yet exist a self-sustained current filament, and current flow only occurs in the presence of the beam irradiation. The bright vertical band seems to be caused by this beam-induced current. At 1.220 V the onset of a self-sustained current filament can be seen on the upper right. At the same voltage a non-zero slope appears in the IVC. At 1.230 V, the filament is more pronounced. However, the image of the filament is still relatively noisy. At 1.235 V the filament looks more stable and the noise level is strongly reduced. The following images at 1.240 and 1.245 V show growth of the filament width. A t 1.260 V the filament extends up to the right sample edge, and the filament continues to grow (1.270 V). The last three images (1.275 V, 1.277 V, 1.278 V) were obtained at relatively small voltage increments and refer to the onset of the region with the steep slope in the IVC. The rapid formation of new current filaments on the left can be seen. If the voltage was increased further, the current flow appeared to become more and more homogeneous. The sequence of events displayed in Fig. 38 can be seen distinctly also on linear line scans performed perpendicular to the electric field direction across the current filament structure. Figure 39 shows such a series of line scans performed on the same specimen as that in Fig. 38. These scans were taken across the middle of the sample for bias voltages in the range 1.200-1.300 V. The triangular markers indicate the location of the left and right specimen edge. Again, the self-sustained current filaments show up in form of a depression of the beam-induced signal H ( x ,y). In particular, we point out the nucleation (1.222 V) and growth (1.224-1.274 V) of a current filament on the right side of the sample, the noisy behavior (1.229 V), and the strong signal increase at 1.274 V, just before several new filaments appear abruptly in the left part of the sample (1.278 V). Again, the signal disappears at higher voltages. We have seen that LTSEM provides valuable information on the spatial structures developed during avalanche breakdown of a homogeneous semiconductor. In addition, the temporal structure of the sample response to a local beam-induced perturbation can be investigated as a function of the coordinates of the beam focus. The results of such an experiment are shown in Fig. 40. They indicate that the beam irradiation can stimulate current oscillations if the beam is focused on the boundary region of a current filament. The beam was turned on and off periodically as shown by the trace on the top. The three temporal recordings of the response signal bl(t)were obtained

500 pm, 2

v

-

1.300

* 1.290

W 0 C 0

c

1.280

1.279-

i

U

-

> -

C

W c L

3 U

V W ul

U J

2 E 0 W

n

0 n

n

FIG.39. Signal 61(x)for a series of line scans performed on the same sample as that in Fig. 38. The scans were taken across the middle of the sample for the different bias voltages indicated. The triangular markers indicate the left and right specimen edge. Tb = 4.2 K, beam voltage = 26 kV, beam current = 20 PA. (From Mayer, 1986.)


m,

electron beam oft

electron beam on

/

/

I

\

curren filament

I

\

\

I

\

'\

= I *

I

7

/-------

-

/

/ /

0

..- _ _ - -- -

/

1 J

time 100psec Idiv. FIG.40. Time dependence of the beam-induced signal &(t) in p-doped Ge. The trace on the top indicates the pulsed operation of the electron beam. The three signal recordings (a)-(c) were obtained when the beam was focused on the corresponding locations marked on the inset. Tb = 4.2 K , beam voltage = 26 kV, beam current = 3 nA. (From Mayer, 1986.)

by moving the beam focus from location (a) to location (c) across the filament boundary, as indicated on the inset. When the beam was focused on location (b) on the filament boundary, stable current oscillations were observed as shown on the temporal trace (b). Beam-induced oscillations of the response signal dI(t) appeared in the frequency range 10-120 kHz, depending upon the location of the beam focus along the filament boundary and upon the beam intensity. The measurements shown in Fig. 40 were performed using a standard boxcar technique. Recordings (a) and (c) were obtained for the location of the beam focus outside and inside the current filament region and show a relatively large and small signal amplitude, respectively. This behavior is consistent with the results presented above. The temporal structure of the sample response observed when the beam is focused on the boundary region of the current filament suggests that it is this boundary region where the spontaneous oscillations and the chaotic behavior of the electric resistance (Teitsworth er ul., 1983, 1986; Held et ul., 1984, 1986; Peinke et al., 1985) develop during avalanche breakdown.

58

R. P. HUEBENER

The experiments described above have shown that LTSEM yields valuable new information on the spatially structured current flow in a homogeneous semiconductor during avalanche breakdown at low temperatures. From the examples discussed we conclude that in the near future such experiments are expected to contribute significantly to a more detailed understanding of various low-temperature properties in a semiconductor. As an interesting example for further investigations by means of LTSEM, we mention the new concept of a magnetic field effect transistor discovered recently (Mannhart and Huebener, 1986). This concept is based on the magnetic control and switching of current filaments in a semiconductor and appears highly promising as the key element of a new cryoelectronic device family (Mannhart et al., 1986; Huebener et al., 1985). Clarification of the mechanisms, which determine the spatial resolution limit in these applications of LTSEM to low-temperature semiconductor physics, still represents an important task. So far, systematic experiments relating to this question have not been performed. Empirically, a resolution of better than 30 pm, as indicated by the traces of linear line scans, has been obtained (Mayer, 1986). Finally, we note that in addition to the LTSEM experiments described above some other attempts for two-dimensional imaging of spatially structured current flow in a semiconductor have been reported recently (see, e.g., Kerner and Sinkevich, 1982; Jager et al., 1986).

X. BALLISTIC PHONON SIGNAL Conceptually, the applications of LTSEM discussed in Sections VI-IX were all based on the same principle: the electron-beam induced local change of the electric conductivity of the specimen. Depending upon the bias condition, the sample response then consisted of the voltage signal 6 V ( x ,y) or the current signal &(x, y). In these applications the local perturbation of the sample effected by the electron beam (thermal and/or electronic excitations) was spreading by means of diffusive processes, and the latter processes played a central role in determining the spatial resolution limit. In the final three sections, we deal with a distinctly different signal concept for spatial imaging, namely the ballistic phonon signal. Here the region of the specimen locally heated by the electron beam acts as a source of phonons (quanta of sound energy) which propagate ballistically (i.e., without scattering) to the opposite side of the crystal where they can be detected with a suitable phonon detector. This arrangement is shown schematically in Fig. 41. There is a close similarity between this emission of phonons from the source


I

'I'

59

beam

\

'phonon detect or FIG.41. Imaging principle based on the ballistic phonon signal

region and the electromagnetic radiation emanating from a heated source (blackbody radiation) (Ashcroft and Mermin, 1976; Fjeldly et al., 1973). For the ballistic phonon propagation through the crystal over distances in the range of mm or cm, it is essential that the crystal is kept at low temperatures of only a few K (except for the hot region acting as phonon source). In this way phonon-phonon scattering is sufficiently suppressed. On the other hand, in metallic electrical conductors phonons cannot propagate ballistically over long distances because of their interaction with the electrons. Hence, the ballistic phonon signal can be utilized only in electric insulators and semiconductors (which act as insulators at low temperatures since all charge carriers are frozen out). Electric superconductors at temperatures far below their critical temperature T, also represent an interesting case with a sutficiently long phonon mean free path because of the strong suppression of the electron-phonon interaction. However, in all cases imaging by means of the ballistic phonon signal is restricted to nearly single-crystalline materials. The ballistic phonons propagating through the crystal can serve for imaging the elastic anisotropy of the material by means of the phonon focusing effect. On the other hand, they can image structural defects in the crystal due to their scattering at these defects. These two imaging concepts are discussed in Sections XI and XII, respectively. The imaging procedure by means of the ballistic phonon signal (Huebener and Metzger, 1985) shown schematically in Fig. 41 is straightforward. The ballistic phonons are generated locally at the top surface of the specimen by irradiation with the electron beam. They are detected at the bottom surface of the specimen. The bottom surface is in direct contact with the liquid-He bath (note the general scheme shown in Fig. 1). In this way effective cooling of the phonon detector and its operation at a well defined temperature is assured. By scanning the specimen surface with the electron beam, the ballistic phonons

60

R. P. HUEBENER

arriving from the different directions are recorded by the detector. In this way the detector signal images the angular variation of the phonon intensity. Clearly, the angular resolution of this imaging procedure is determined by the geometric size of the source and of the detector for the ballistic phonons. For eliminating charging effects on the top side of the specimen due to the electron beam irradiation, this side is usually covered with an electrically conducting metal film. Typically, a granular aluminum film of about 0.5 pm thickness prepared in the presence of oxygen has been used for this overlay (Eichele et ul., 1982). Turning first to the phonon source, we assume that the source region is heated to an effective temperature T*. According to Planck's radiation law the spectral energy density u(w, T * ) of the phonons emitted from this region is given by (Ashcroft and Mermin, 1976; Fjeldly et al., 1973)

Here the second term on the left takes into account the zero-point energy. is the angular phonon frequency. The factor lju," is the average of the inverse third power of the long-wavelength phase velocities of the three acoustic phonon modes. The frequency dependence of expression (36) for the temperature T* = 10 K is shown in Fig. 42. Here we have taken the value us = 4000 mjs for the sound velocity. The spectral energy density is w

v)

U 01

z

P ' c

01

+ a u l a v)

0

I

0 1 2 v[THzl FIG.42. Spectral energy density u(w, T )versus the phonon frequency v = w/2n. (The zeropoint energy is subtracted; T = 10 K ; us = 4000 m/s).


seen to reach a maximum at some frequency (omax.From Eq. (36) one obtains the relation hwmax= 2.82knT*

(37)

which can also be written in the form

A,,,, T* = 6.81 -

m eK

0

(38)

Here i,,,, is the phonon wavelength corresponding to the frequency w,,,. The optical analog of Eq. (38) is known as Wien's displacement law. As an important result, by looking at Fig. 42 we note that the dominant phonon frequency is typically about 600 GHz corresponding to an acoustic wavelength of about 7 nm (based on the values us = 4000m/s and T* = 10 K). A rough estimate of the effective temperature T* of the source region can be obtained from the balance between the power input of the electron beam and the power output of the emitted ballistic phonons (Huebener and Metzger, 1985). Here we assume that the electron-beam power dissipated in the sample is completely removed by the emission process of the ballistic phonons. The power emitted by the phonons from a region at temperature T* into the adjoining half-space is given per unit area by

where I ) , is the sound velocity (averaged over the different acoustic modes) and w(T*) the phonon energy density in this region. The energy density w(T*)is found from the spectral energy density U ( Q , T * )of Eq. (36) by integration over all phonon frequencies, yielding 7c2

W(T*)= 10

(knT*)4 ~

(iiL1,)3

The optical analog of the result expressed in Eqs. (39) and (40)is known as the Stefan-Boltzmann law. For determining the emitted phonon power, one must know the total effective area of the phonon source. This area can be found experimentally from the angular resolution limit of the images based on the ballistic phonon signal. In this way a typical value T* = 10 K has been obtained recently for germanium (Huebener and Metzger, 1985). In our discussion given above we have assumed that the source region is in thermal equilibrium at the elevated temperature T*. It is important to note that such a treatment of the phonon source region in terms of an effective temperature T* represents an approximation where all nonequilibrium aspects are contained only in the value of this temperature T*. The questions relating to the diameter of the source region have been discussed in Section IVB, and the important results are contained in Eqs. (16)-

62

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(19). A typical temperature profile T(r)of the region locally heated by the electron beam is shown in Fig. 7. Again, high-frequency beam modulation can be utilized for reducingconsiderably the diameter of the modulated part of the source region. In this way the angular resolution of the phonon imaging technique can be increased. Experimentally, diameter values of the source region in the range of 10-100pm have been observed (Metzger, 1987; Huebener and Metzger, 1985; Huebener et al., 1986). We will see below that the effective area of the phonon detector can be made much smaller than that of the phonon source. Therefore, the angular resolution limit of the phonon imaging technique is dominantly determined by the phonon source. Turning next to the phonon detector, we note that the detector operation in the liquid-helium temperature range represents a distinct advantage since it allows the application of extremely sensitive low-temperature devices. The simplest device appears to be a superconducting microbridge attached directly to the specimen surface at the bottom by thin-film evaporation. Such microbridges have been used extensively for phonon detection at low temperatures (Eichele et al., 1982).They consist of a thin-film superconductor connected at both ends to much wider film sections to which current and voltage leads are attached. Such a detector is shown schematically in Fig. 43. For maximum sensitivity the detector is operated close to the superconducting critical temperature T,, where the electric resistance of the microbridge shows an extremely strong temperature dependence. During the experiment the temperature of the liquid-helium bath is adjusted for operation at the maximum slope of the resistance-temperature curve of the detector. Usually, the microbridge is current biased, and the voltage change due to the arriving ballistic phonons is recorded. As detector material, superconducting films of granular aluminum prepared in the presence of oxygen with a typical thickness of about 30 nm have been used. Granular aluminum films have a high electric resistance at low temperatures in the normal state yielding a relatively strong detector signal. The critical temperature of such films is near 2.12 K. As a consequence, the temperature of the liquid-helium bath is below

FIG. 43. Typical geometry of a thin-film superconducting bolometer. The hatched part shows the effective bolometer area which can be as small as a few pm’.


63

the A-point, and the superfluid state of the liquid helium provides additional thermal stability of the cooling arrangement. Theeffective area of the thin-film detector can be made as small as only a few pmZ by means of standard microfabrication techniques. In this case, the contribution of the detector to the angular resolution limit of the imaging method is insignificant, as we have pointed out above. Further details including the electronic measuring procedures can be found elsewhere (see Metzger, 1987; Huebener and Metzger, 1985; Eichele et a!., 1982). It is important to note that in this application as a phonon detector a superconducting microbridge represents a bolometer integrating over all frequencies of the arriving phonons. Therefore, dispersive effects requiring frequency selective phonon detection cannot be investigated in this way. On the other hand, frequency selective phonon detection at low temperatures is possible by means of superconducting tunnel junctions (Eisenmenger, 1976). So far, tunnel junctions have been used mainly for the detection of ballistic phonons above a distinct threshold frequency (Renk, 1972; Anderson and Wolfe, 1986). For further details and references we refer to these authors. The ballistic phonon signal for the two-dimensional imaging of the phonon propagation in a crystal can also be generated by a laser scanning technique. Here the electron beam is just replaced by a laser beam and scanning is performed by means of a two-mirror system operated in the form of precision galvanometers (Northrop and Wolfe, 1980). Laser-beam scanning has been used extensively for investigating phonon focusing as will be discussed in Section XI. From a comparison of electron-beam scanning with laser-beam scanning at low temperatures it appears that the former has distinct advantages over the latter because of the relatively small electron-beam diameter available even at relatively long working distances between the final lense and the specimen surface. The ballistic phonon signal in LTSEM is distinctly different from the acoustic signal generated for imaging in scanning electron-acoustic (SEAM) and scanning photo-acoustic microscopy (SPAM) (Huebener and Metzger, 1985). In the two latter schemes the electron or laser beam is modulated at the angular frequency w, and coherent sound waves of frequency w are generated near the specimen surface. Typically, modulation frequencies in the range co = 100 kHz to 500 MHz are used. On the other hand, the ballistic phonons we have discussed in this section are emitted incoherenrly from the region locally heated by the beam, and their frequency distribution is given by Eq. (36) (see also Fig. 42). The dominant frequency of these incoherent phonons is typically v = 600 GHz and at least more than three orders of magnitude higher than the typical acoustic frequencies in SEAM and SPAM.

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XI. PHONONFOCUSING As the first application of the ballistic phonon signal for two-dimensional imaging, we discuss in the following the phonon focusing effect. This effect is caused by the elastic anisotropy of the crystal. As a result the intensity of the ballistic phonon flux through the crystal depends upon the crystallographic direction of the phonon propagation. Because of the elastic anisotropy in a crystal the surface of constant phonon energy in wavevector space is, in general, not spherical and displays more or less pronounced anisotropy depending upon the material. Therefore, the energy flux or group velocity of an acoustic plane wave is generally not parallel to the wave vector. This situation is shown schematically in Fig. 44. We see that the phonon energy propagates preferentially along distinct crystallographic directions. At low temperatures, where the phonon mean free path is long and where ballistic phonon propagation dominates over diffusive processes, the anisotropic channeling of the phonon energy can be highly 1010;

FIG.44. Principle of phonon focusing. For an anisotropic surface of constant phonon energy in wave vector space, the phonon energy flux (oriented perpendicular to the surface of constant energy) is generally not parallel to the phonon wave vector. This results in preferential propagation of phonon energy along distinct crystallographic directions, as indicated by the arrows at the top.


65

pronounced. The first experimental demonstration of this “phonon focusing” has been reported by Taylor et al. (1969, 1971). In principle, the phonon focusing effect can be measured using a fixed localized phonon source at the crystal surface and a two-dimensional array of phonon detectors attached to the opposite surface. The phonon intensity measured with the different detectors of the array (possibly corrected for the differences in distance between source and detector) immediately yields the anisotropy of the ballistic phonon propagation. However, this scheme can also be inverted by attaching a single localized detector to one crystal surface and by operating a two-dimensional array of individual phonon sources on the opposite surface. The two-dimensional phonon source array can simply be realized by means of electron beam scanning, and we arrive at the scheme shown in Fig. 41. By recording the signal of the phonon detector as a function of the coordinate point of the electron beam focus, a two-dimensional image of the anisotropic phonon energy flux is obtained. Recently such experiments have been performed with single-crystalline a-quartz (Eichele et al., 1982), sapphire (Eichele et al., 1982), germanium (Schulze and Keck, 1984; Huebener et ul., 1986), and silicon (Metzger, 1987; Huebener and Metzger, 1985; Huebener et al., 1986). The samples were typically disks of 20 mm diameter and 2 mm thickness. As an example we show in Fig. 45 the two-dimensional image of the

FIG. 45. Two-dimensional display of the ballistic phonon signal in single-crystalline q u a r t z ( y-orientation). Bright regions indicate high intensity of the phonon flux. Further details are given in the text. Tb = 2 K, beam voltage = 30 kV, beam current = 30 nA. (From Eichele ef al.,1982.)

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FIG.46. Monte Carlo image calculation of the phonon-focusing pattern of @-quartz in y-orientation. Bright regions indicate high phonon intensity. Further details are given in the text. (From Koos and Wolfe, 1984.)

anisotropic energy flux of the ballistic phonons in single-crystalline a-quartz (Eichele et al., 1982). The crystallographic y-axis of the sample (a disk of 20 mm diameter and 2 mm thickness) is oriented perpendicular to the plane of the figure, and the x-axis lies horizontally in the plane of the figure. Bright regions indicate high intensity of the phonon flux. The scanned area shown is 3.5 mm x 3.5 mm, corresponding to an angle of about f40" around the crystal y-axis. The image shown in Fig. 45 represents the time-integrated bolometer signal. This signal is proportional to the total phonon intensity and, hence, does not display separately the individual contributions from the different acoustic phonon branches. For comparison we show in Fig. 46 the results of a Monte Carlo image calculation of the phonon-focusing pattern of a-quartz viewed along the y-direction (Koos and Wolfe, 1984). The calculated image extends f56" horizontally around the crystal y-axis. Piezoelectric effects, which cause only insignificant changes, are included in the calculation. Again the bright regions indicate high phonon intensity. We see that these theoretical results agree well with the experimental image shown in Fig. 45. Theoretical calculations of the angular dependence of the total phonon energy flux for the sum of the three acoustic modes in the long-wavelength limit reported earlier (Rosch and Weis, 1976) also show good agreement with the experimental results of Fig. 45. In addition to the time-integrated bolometer signal, the time-resolved


67

FIG.47. Two-dimensional image of the anisotropic time-integrated phonon energy flux in single-crystalline [ 1 1I]-oriented silicon. Bright regions indicate high intensity of the phonon flux. Further details are given in the text. Tb = 2.01 K, beam voltage = 26 kV, beam current 2 1 /LA.

bolometer signal can be recorded. In this way the anisotropic ballistic propagation of the three acoustic phonon branches can be investigated separately because of their different phase velocities. Such experiments based on electron-beam scanning have been performed also with a-quartz confirming the theoretical predictions. Whereas the time-integrated measurements were carried out using a lock-in technique with a modulation frequency for the electron beam of typically 10-50 kHz, the time-resolved images were obtained by means of boxcar integration (Eichele et al., 1982). Figure 47 shows the anisotropic time-integrated phonon energy flux of [11 I]-oriented single-crystalline silicon (a disk of 20 mm diameter and 2 mm

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1 rnrn

FIG.48. Ballistic phonon signal obtained for a single line scan of single-crystalline [I I I]oriented silicon versus the horizontal line coordinate (same sample as in Fig. 47). The location of the line scan within the two-dimensional phonon focusing pattern is indicated by the dashed line in the inset. T, = 2.01 K, beam voltage = 26 kV, beam current x 1 PA.

thickness). Again, bright regions indicate high intensity of the phonon flux. The scanned area shown is 7 mm x 8 mm, corresponding to an angle of about & 60" around the [111] axis. For quantitative studies, individual line scans performed with the electron beam are more advantageous. The bolometer signal can then be plotted vertically versus the horizontal line coordinate (y(y-modulation). As an example we show in Fig. 48 the result of such a line scan performed on the same [111] oriented silicon crystal as in Fig. 47. The location of this line scan within the two-dimensional phonon focusing pattern is indicated in the inset. The time-integrated phonon energy flux of [00 11-oriented single-crystalline germanium (a disk of 20 mm diameter and 2 mm thickness) is presented in Fig. 49. Most of the experiments mentioned above were performed using superconducting thin-film bolometers for phonon detection with an effective area typically ranging from 10 pm x 10 pm down to 2 pm x 2 pm. As we have pointed out in Section X, the angular resolution of the acoustic imaging method is dominated by the effective diameter of the phonon source if such highly miniaturized devices are utilized for phonon detection. High-frequency modulation of the electron beam can serve for reducing the diameter of the modulated phonon source region and thereby increasing the angular resolution (see Eq. (18)). Experimentally, modulation frequencies up to about

S C A N N I N G E L E C T R O N M I C R O S C O P Y A T VERY L O W T E M P E R A T U R E S

H

69

2Opm

FIG.49. Image of the time-integrated phonon energy flux of [001]-oriented singlecrystalline germanium. Bright regions indicate high intensity of the phonon flux. Tb = 1.93 K, beam voltage = 26 kV, beam current z 2 nA.

20 MHz have been used so far for imaging by means of the ballistic phonon signal. The effective diameter of the phonon source and, hence, the angular resolution limit of this imaging method can be estimated experimentally from line scans such as shown in Fig. 48 and a comparison with theoretical curves. The result of such a procedure (Metzger, 1987; Huebener et al., 1986)is shown in Fig. 50. Here the anisotropic intensity of the ballistic phonons has been calculated for the case of [001]-oriented germanium, assuming a phonon source of variable diameter and a point-like phonon detector. A Gaussian distribution was assumed for the intensity of the phonon source as a function of the radial distance from its center. Twice the radius at which the intensity has decreased to l/e of the value at the center was taken as the source diameter. In Fig. 50 we show a series of theoretical curves with the diameter d of the phonon source as parameter. An experimental curve is also shown for comparison. All results given in Fig. 50 refer to the slow-transverse phonon mode and the location of the line scan indicated in the inset. By comparing the first peak on the left of the experimental curve with the theoretical results, we

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-

scan coord

U

100 p m

FIG. 50. Intensity of the ballistic slow-transverse phonons in [001]-oriented singlecrystalline germanium for the line scan shown on the inset. Left: theoretical curves obtained for the different values of the source diameter d indicated. Right: experimental curve. Crystal thickness = 2 mm, Tb = 1.96 K, detector area 10 pm x 10 pm, beam voltage = 26 kV, beam current = 1 p A (from Metzger, 1987).

find for the diameter of the phonon source d = 40 pm. We note that this value has been obtained without high-frequency beam modulation. Of course, the effective source diameter can be reduced by means of high-frequency beam modulation and by detection of the modulated signal (Metzger, 1987; Huebener and Metzger, 1985; Huebener et al., 1986). The influence of a thin-film overlay, placed on the specimen surface irradiated with the electron beam, upon the effective diameter of the phonon source has been investigated in a series of experiments (see Metzger, 1987; Huebener and Metzger, 1985). However, it appears that more work needs to be done before this question of an optimum overlay film for improving the angular resolution limit of this imaging method is settled. The experiments described above yielding information on the phonon propagation through the bulk crystal clearly require a highly homogeneous specimen surface for irradiation with the electron beam In this way, additional


71

two-dimensional structures in the ballistic phonon image possibly due to the phonon generation process at the crystal surface are avoided. In addition to electron-beam scanning, laser-beam scanning has also been used for studying phonon focusing in single crystals. Such experiments have been performed with germanium (Northrop and Wolfe, 1980, 1979); silicon (Hurley and Wolfe, 1985); a-quartz (Koos and Wolfe, 1984); sapphire (Every et al., 1984); lithium niobate (Koos and Wolfe, 1984); diamond (Hurley et ul., 1984);calcium fluoride (Hurley and Wolfe, 1985); lithium fluoride (Northrop ef al., 1982); tellurium dioxide (Hurley et al., 1986); and gallium arsenide (Northrop er al., 1985). In those cases where a comparison is possible because of the material and orientation of the crystal (a-quartz, sapphire, and germanium) electron-beam scanning and laser-beam scanning yielded highly similar results. In a series of experiments, laser-beam scanning has also been employed recently for investigating dispersive effects in the phonon focusing pattern appearing at large phonon wave vectors. (See, for example, Dietsche et ul., 1981; Northrop, 1982; Wolfe and Northrop, 1984; Hebboul and Wolfe, 1986; Schreiber et al., 1986.) XII.

STRUCTURAL DEFECTS WITH BALLISTIC PHONONS

IMAGING OF

Structural defects in a nearly single-crystalline specimen, impeding the ballistic phonon propagation by absorption or scattering, cause a reduction of the ballistic phonon signal recorded by the detector. Therefore, this signal can also be used for imaging these structural defects. As we have seen in Section X, the phonon frequencies contained in the ballistic phonon signal are typically about 600 GHz corresponding to an acoustic wavelength of only a few nm. Hence, this acoustic imaging of crystal defects can simply be discussed in terms of geometric optics. The phonon detector just observes the “shadow” generated by such an object. Clearly, during electron-beam scanning an object will be imaged by its shadow falling on the detector, if the straight line connecting source and detector of the ballistic phonons just passes through the object. Furthermore, if two phonon detectors are operated simultaneously, three-dimensional imaging of structural defects with ballistic phonons becomes possible (three-dimensional acoustic tomography; Huebener and Metzger, 1985; Huebener et al., 1986; Huebener, 1986). This imaging principle is shown schematically in Fig. 51. By scanning the electron beam over the specimen surface, two different two-dimensional images by means of the ballistic phonon signal are obtained with the two phonon detectors placed

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I‘ \

electron beam

\

small - area phonon detectors FIG.51. Three-dimensional tomography based on the ballistic phonon signal. The acoustic “shadow” of an object (indicated by the open dot) is independently registrated by the two phonon detectors at the bottom.

at different locations. From these two images the three-dimensional configuration of the structural inhomogeneities can be reconstructed. Of course, with a simple scheme such as shown in Fig. 51 (two detectors only), the unambiguous reconstruction of the three-dimensional configuration of the inhomogeneities is possible only if, for a given coordinate point of the beam focus on the specimen surface, each detector signal is affected only by a single object. Therefore, the density of the objects affecting the detector signal must be sufficiently low. As an example we show in Fig. 52 the ballistic phonon image of two holes, drilled sideways into a sapphire single crystal using a laser technique (crystal thickness = 2 mm). A superconducting thin-film bolometer fabricated from

FIG.52. Ballistic phonon image (left) and optical image (right) of two laser-drilled holes in z-oriented single-crystalline sapphire. Parameters of the ballistic phonon image: Tb = 2.065 K, beam voltage = 26 kV, beam current x 0.3 PA. The background features in the ballistic phonon image are due to phonon focusing. (From Huebener et al., 1986.)


73

FIG.S3. Optical image (top) and ballistic phonon image (bottom) of a laser-drilled hole in single-crystalline a-quartz. Parameters of the ballistic phonon image: Tb = 1.91 K, beam voltage = 26 kV, beam current 2 0.3 PA.

oxygen-doped aluminum with an effective area of 10 pm x 10 pm has been used for phonon detection. Comparison with the optical image, which is also shown, indicates that nearly all details are well reproduced by the acoustic image based on ballistic phonons. Figure 53 shows the acoustic image of a hole drilled horizontally into an uquartz single crystal (crystal thickness = 2 mm) together with the optical image. At the tip of the hole a round “shadow” can be seen in the acoustic image, which is not present in the optical image and which may be caused by mechanical strain. The fact that different images are obtained for two bolometers placed at different locations on the specimen surface is demonstrated by the results presented in Fig. 54. Here the sample is the same as that in Fig. 53, and the hole is detected separately by two bolometers placed about 200 pm apart. In both

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H

100 prn

FIG. 54. Ballistic phonon images of a laser-drilled hole in single-crystalline a-quartz. The two images were obtained with two bolometers placed about 200 pm apart. The sample and the experimental parameters are the same as in Fig. 53.

acoustic images the hole appears at different locations relative to the phonon focusing pattern of the background, as one would expect. Of course, from both images together with the exact locations of the two bolometers the exact placement of the hole can be obtained in all three dimensions. The background features seen in the acoustic images of Figs. 52-54 result from phonon focusing effects. It appears that in general the anisotropy resulting from the phonon focusing effect does not present severe problems for the acoustic imaging of structural defects. On the contrary, phonon focusing represents an advantage since an angular regime with relatively high ballistic phonon intensity can be selected (Metzger et al., 1985). The acoustic imaging method based on the ballistic phonon signal appears * particularly interesting for a detailed characterization of the materials used in semiconductor microelectronics. The method looks highly promising for three-dimensional imaging of doping structures, chemical precipitates, dislocations, etc. Recently, encouraging results on imaging of oxide precipitates in silicon have been reported (see Metzger et al., 1985).


The spatial resolution limit of this three-dimensional acoustic imaging principle is found from the same considerations discussed in Sections X and XI. The accessible depth range is limited, of course, by the phonon mean free path and is expected to reach at least several mm. In all experiments reported so far the phonon detectors were evaporated directly on the specimen surface opposite to that scanned by the beam. In this way, multiple use of a single detector for different samples has been impossible. The development of a suitable highly miniaturized detector configuration, which can be removed from the sample and used again for other specimens, represents an interesting task.

ACKNOWLEDGMENTS A large part of the work on scanning electron microscopy at very low temperatures performed in the group of the author and described in this article has been supported financially by grants of the Deutsche Forschungsgemeinschaft and of the Stiftung Volkswagenwerk. The developments summarized in this article were possible only because of the outstanding contributions of the many former and present coworkers of the author: J. Bosch, John R. Clem, R. Eichele, P. W. Epperlein, L. Freytag, R. Gross, H.-U. Habermeier, R. J. Haug, E. Held, W. Klein. M. Koyanagi, J. Mannhart, K. M. Mayer, W. Metzger, J. Niemeyer, H. Pavlicek, H. Seifert, and H.-G. Wener.

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L. Solymar, Superconductive Tunneling and Applications, Chapman and Hall, London, 1972. B. Taylor, H. J. Maris, and C. Elbaum, Phys. Rev. B3, 1462 (1971). B. Taylor, H. J. Maris, and C. Elbaum, Phys. Rev. Lett. 23,416 (1969). S . W. Teitsworth, R. M. Westervelt, and E. E. Haller, Phys. Rev. Lett. 51, 825 (1983); S. W. Teitsworth and R. M. Westervelt, Phys. Rev. Lett. 56, 516 (1986). M. Tinkham, Introduction to Superconductivity, McGraw-Hill, New York, 1975. A. M. S. Tremblay, Nonequifibrium Superconductivity, Phonons, and Kapitza Boundaries, K. E. Gray, ed., Plenum Press, New York (1981),p. 289,309. V. G. Volotskaya, 1. M. Dmitrenko, L. E. Musienko, and A. G . Sivakov, Fiz. Nizk. Temp. 7 , 3 8 3 (1981) [Sov. J. Low Temp. Phys. 7 , 188 (1981)l; V. G. Volotskaya, I. M. Dmitrenko, and A. G. Sivakov, Fiz. Nizk. Temp. 10, 347 (1984) [Sov. J. Low Temp. Phys. 10, 179 (1984)l. J. P. Wolfe and G. A. Northrop, in Proceed. Fourth Internat. Con5 on Phonon Scattering in Condensed Matter, ed., by W. Eisenmenger, K. Lassmann, and S. Dottinger, Springer, New York, 1984, p. 100.

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ADVANC’ES IN ELtCTRONICS AND ELECTRON PHYSICS VOL . 70

Robust Image Models and Their Applications R . L . KASHYAP AND KIE-BUM EOM* cf Electrical Engineering Purdue Uniiwrsity West LaJayette. lndiana

School

Abstract . . . . . . . . . . . . . . . . . . . I . Introduction and Overview . . . . . . . . . . . . A . Robust Statistical Procedures . . . . . . . . . . B. Image Models . . . . . . . . . . . . . . . . C. Applications . . . . . . . . . . . . . . . . 1. Image Restoration . . . . . . . . . . . . . 2 . Boundary Detection . . . . . . . . . . . . . I1. ARand ARMAModels . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . B. 2D AR Processes . . . . . . . . . . . . . . . C . Simultaneous and Recursive AR Models . . . . . . D . Generalized ARMA Models . . . . . . . . . . . E . Generative Interpretation of Models . . . . . . . . F. Approximations to The Image Models . . . . . . . G . Summary . . . . . . . . . . . . . . . . . 111. Robust Estimation in Causal Autoregressive Models . . . . A . Introduction . . . . . . . . . . . . . . . . B. Causal Autoregressive Model . . . . . . . . . . C . Robust Parameter Estimation . . . . . . . . . . I . Perfect Observation Case . . . . . . . . . . . 2. Noisy Observation Case . . . . . . . . . . . D . Experimental Results . . . . . . . . . . . . . E . Discussions and Conclusions . . . . . . . . . . . IV . Image Restoration with Robust Image Modelling Techniques . A . Introduction . . . . . . . . . . . . . . . . B. Previous Robust Filters . . . . . . . . . . . . I . The L Filter . . . . . . . . . . . . . . . 2. The M Filter . . . . . . . . . . . . . . . C. Intensity Representation for Restoration . . . . . . . D . Image Restoration Algorithm . . . . . . . . . . E . Experimental Results . . . . . . . . . . . . .

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Partially supported by the Ofice of Naval Research under the grant N00014-85K-0611and by the National Science Foundation under the grant IST 8405052. * Currently with the Department of Electrical and Computer Engineering. Syracuse University Syracuse. New York 13244-1240.

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Copyright il 1 9 R R by Academic Press Inc. All rights of reproduction In any form reserved.

ISBN 0-12-014670-3

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R. L. KASHYAP AND KIE-BUM EOM

F. Discussions and Conclusions. . . . . . . . V. Composite Edge Detection. . . . . . . . . . A. Introduction . . . . . . . . . . . . . B. Edge Hypothesis Generation (Algorithm 1 ) . . . C. Confirming The Presence of Edges (Algorithm 2 ) . 1. Confirming a Texture Edge . . . . . . . 2. Confirming an Intensity Edge . . . . . . D. Experimental Results . . . . . . . . . . E. Discussions and Conclusions. . . . . . . . VI. Summary and Suggestions. . . . . . . . . . References . . . . . . . . . . . . . . .

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ABSTRACT Various types of image models for representing images are considered, and robust image models are developed. The robust methods in image models are also applied to some important image processing problems such as image segmentation by texture property and image restoration in the presence of impulse noise. Robust estimation algorithms for two different outlier processes in causal autoregressive models are developed. These algorithms are based on robust M-estimators. Theoretical properties of the robust estimation algorithms are presented. The robustness of the estimators are also shown in the experiment. The robust estimation algorithm for causal autoregressive models is applied to the image restoration problem. Traditionally, median or a-trimmed mean filters are used, but these methods result in blurred images. The restoration method based on robust image model cleans out impulse noise without involving any blurring of the image. Experimental results show that the quality of images restored by the model-based method is much superior to the images restored by other traditional methods. We considered the detection of both intensity edges and texture edges. An intensity edge is defined by an abrupt change of intensities and a texture edge is defined as a boundary between different textures. Traditional edge detection algorithms cannot effectively detect texture edges. We developed an image model based edge detection algorithm which can detect both intensity and texture edges. The validity of the model based method is demonstrated by comparing with the result of conventional methods.

I. INTRODUCTIONAND OVERVIEW In the past decade, there has been remarkable progress in the research on statistical image models and their applications. Statistical image models (often called random field models or spatial interaction models) represent the image

ROBUST IMAGE MODELS A N D THEIR APPLICATIONS

81

intensity of a given picture by a small number of parameters. There are many applications of image models in image processing and analysis. For instance, they can be used for image synthesis (Kashyap, 1984b; Cross and Jain, 1983), image restoration (Chellappa and Kashyap, 1982; Geman and Geman, 1984), image coding (Delp et al., 1979), texture boundary detection (Kashyap and Eom, 1985a), and texture analysis (Kashyap and Khotanzad, 1984a). For the application of image models to such image processing tasks, we need to estimate the parameters in the image models. There are many different estimation algorithms for different image models, but most of these methods are based on the assumption of Gaussian image intensity distribution. However, the actual distribution of image intensity deviates from the Gaussian assumption, and traditional estimation methods are very sensitive to minor deviations from the Gaussian assumption. During the past decades, many estimators which are robust to the deviations from the Gaussian assumption have been proposed (Huber, 1981),but they are rarely applied to image modelling. Robust estimation procedures for several different image models are developed and applied to some important image processing problems such as image segmentation and image restoration in this study. A. Robust Statistical Procedures

There has been considerable interest in robust methods in statistics in recent years. It is because most statistical inference methods are based on rather restrictive assumptions about the observations and models, such as independence of observations, distribution of observations, etc. However, these assumptions do not always hold, and many statistical procedures are very sensitive to minor deviations from the given assumptions. For example, it is well known that least squares methods are excessively sensitive to a small number of outliers. The term robust was introduced by G.E.P. Box in 1953, and a procedure is called robust, if it is reasonably good (optimal or near optimal) if the assumption holds, and it is not sensitive to small deviations from the assumption. Primarily robustness implies distribution robustness, i.e., the robustness about the small deviations from the assumed distribution (usually Gaussian). The resistance to outliers is considered equivalent to the distribution robustness (Huber, 1981). There are several types of robust procedures: M-estimators, L-estimators, and R-estimators. Among these, M-estimators have an advantage over other procedures because they can be extended to the parameter estimation problems in image models. In contrast, either L-estimators or R-estimators are

82


difficult to generalize well beyond one parameter location or scale problems. The robust M-estimators are applied to the parameter estimation problem of causal autoregressive models. Two different outlier processes are considered, and iterative robust estimation algorithms for both of the outlier processes are developed. Theoretical properties of the proposed robust estimators are investigated.

B. Image Models Image models characterize image intensity surface with a small number of parameters. Image models can be divided into two groups, namely, descriptive and generative models. A descriptive model for an image summarizes the intensity distribution into a finite number of statistics. An example is the cooccurence matrix (Haralick, 1973) used in texture analysis. The generative model, on the other hand, allows one to synthesize an image obeying the given model by using the model description and a set of random numbers. We will restrict ourselves to generative models since they can be used for many variety of applications. We can further divide the generative models into two large classes. In the first class, the observed intensity function y(i,j ) is assumed to be the sum of a deterministic function-usually polynomial or sinusoid-and an additive noise. In the second class, the image intensity function is generated as the output of a transfer function whose input is a sequence of independent random variables. The transfer function represents the known structural information on the image surface; the independent random sequence accounts for the unknown part. Note that the neighboring pixels are highly correlated, unlike in the earlier case, and the transfer function accounts for the covariance. C . Applications

Image restoration and image segmentation are two important branches of image processing. Image restoration is needed to recover the original image from the image corrupted by noise (including impulse noise), and image segmentation procedure, especially edge detection or boundary detection, is involved in most high level image processing problems. Robust image models are developed and applied to the above image processing problems in this study. 1. Image Restoration

An image may be subject to noise and interference from many different sources, and image restoration is used to remove noise from the given image.

ROBUST IMAGE MODELS AND THEIR APPLICATIONS

83

Traditionally, noise distribution is assumed as Gaussian distribution, and many different restoration algorithms based on Gaussian assumption have been introduced (Pratt, 1978; Rosenfeld and Kak, 1982). Recently, image models are used in image restoration applications. For example, Chellappa and Kashyap (1982) used simultaneous autoregressive model and conditional Markov model, Wu (1985) used nonsymmetric half plane autoregressive model and two-dimensional Kalman filtering approach, and Geman and Geman ( I 984) used a family of Markov models. Even though the above examples show some successful applications of image models in image restoration problem, all of the above methods are designed to remove Gaussian noise, and are not very effective in removing impulse noise (Pratt, 1978). Traditionally, median filter and its generalizations (Kassam and Lee, 1985) are used to remove impulse noise (also called salt and pepper noise) from the noisy image. These methods are simple applications of robust location parameter estimators, such as median or a-trimmed mean, where image intensity is assumed constant over a small size window. However, the restored images by these methods are blurred (Pratt, 1978). Robust image model approaches are applied to the image restoration problem in our study. The original image intensity is assumed to follow an image model, and parameters are estimated by a robust estimation algorithm. The image is restored by applying data cleaning algorithm with the robustly estimated parameters. The robust model-based method perform better than any other traditional methods in the experiment.

2. Boundary Detection Edge detection or boundary detection is a fundamental step in scene analysis. Traditionally, an edge is defined as a boundary between two uniform regions, where the intensity of each region is uniform and the intensity difference between two regions is large. Most edge detection algorithms are based on the gradient operator or the laplacian operator (Davis, 1977) which is sensitive to change of intensity. Recently, some model based edge detection approaches are proposed (Haralick, 1984; Zhou and Chellappa, 1986), but they are also based on the derivatives methods using decision rules with estimated model parameters. For the higher level processing, the edges should be able to distinguish the shape of each object from the background of an image. However, intensity edges are sometimes not satisfactory to represent an object and distinguish it from the background, because the intensity of an object or a background is not uniform. For instance, grass lawn in an outdoor scene is homogeneous by its texture property, but it has many intensity edges within the region. The

84


above example suggests the necessity of detecting boundaries (or edges) by its texture property. Image models are already used in synthesizing textures which is very similar to real textures, and the estimated parameters which are obtained by fitting an image model to the given image can be used as texture features. The texture features derived from image model or from other methods can be used to segment an image by a statistical classification method, if the number and types of textures in the given image is known in advance. However, the above prior information is generally not available. A composite edge detection algorithm is developed in this study. The composite edge detection algorithm combines the model-based texture boundary detection method and a conventional intensity edge detection method. This algorithm detects all potential edges by a directional derivatives method, and final edges are confirmed whether they are texture edges or intensity edges. This algorithm is also compared with other conventional edge detection methods in the experiment. The composite edge detection algorithm performs better than other conventional methods which detect only intensity edges in the experiment.

11. AR

AND

ARMA MODELS

A . Introduction

It is claimed traditionally that a complete stochastic description of an M x M array of pixel intensities y(s) is given by the joint probability density of the M Z intensity variables y(.). Even writing down the expression is horrendous considering that the typical value of M is 128 or 256 or 512. As a consequence it was often conjectured that probabilistic models may not be of much use in solving interesting problems in image processing. The purpose of this chapter is to draw attention to the existence of a large class of image models which can be characterized completely in terms of the second order properties of the image sequence, i.e., the correlations E [ y ( s ) y ( s+ r)] or the corresponding spectral density. Consequently these models are relatively easy to analyze. It must be emphasized that the joint probability density of all the intensities is not assumed to be Gaussian. In the beginning, we will focus our attention on the two-dimensional generalization of the autoregressive (AR) models and autoregressive moving average ( A R M A ) models popular in the time series analysis. Basically all these two dimensional models can handle rational spectral densities i.e., ratio of two


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linear combination of sinusoids in the two frequency variables in the direction, just as in the one dimensional case. However, there are many differences between the 1D and 2D cases which will be highlighted in this chapter. For example, in the 1D case, the correlation function is an exponentially decaying function of the lag variable. But in the 2D case, one rarely encounters the exponential correlation function. Similarly in the 1D case, the driving input random sequence is both statistically independent and uncorrelated with the dependent variables in the past. In the general 2D case, the input sequence cannot possess both these properties simultaneously. Secondly, we will consider the various possible ways of defining the weak Markov property in the 2D case. By weak, we mean that the corresponding Markov property can be described completely in terms of the second order properties like correlation or spectral density. The traditional Markov property defined in terms of the probability densities is termed as the strong Markov property. A sequence cannot be strong Markov without being weak Markov. We will characterize the various subclasses of 2D AR and ARMA models which possess various types of weak Markov property. We recall that the general AR or ARMA models mentioned above are not recursive, in general. Still these models are generative in principle, i.e., it is possible to give an algorithm which generates a sequence which obeys a prespecified model. However, the amount of computation involved may be considerable. We will consider modifications or approximations of the AR or ARMA models so that it is relatively easy to synthesize an image obeying a given model. Preliminaries. We will consider a covariance stationary array of the real numbers { y ( i , j ) , -co, 0

for all

Izll = 1

and

Iz2( = 1

(2)

In addition, the input u ( - ) in (1) is assumed to have zero mean and be orthogonal to all y ( - ) ,i.e., E[u(s)y(s+ r)] = 0

for all r # 0

(3)

We also assume E [ u 2 ( s ) ]= 1. The parameter p in (1) can specify the relative power of the input term. We can also rewrite Eq. (1) compactly in terms of the polynomial A (4)

In defining (4), zi are interpreted as the unit lead operators in the two directions. Equation (3) defines the process u ( - ) only indirectly. The precise structure of the process o(.) is not obvious. We will derive later an expression for the spectral density of u ( - ) using (1)-(3). Equation ( 3 ) can be thought of as defining a u ( - ) process given a y(.) process. It is not obvious here to generate a y(.) and a u ( - ) sequence obeying


88

simultaneously (1)-(3). We will later show constructively that there do exist infinite sequences y ( - )and u(.) obeying (1)-(3). Structure of u(.) Process. The following theorem gives the spectral densities of the processes y(.) and u ( - ) which obey (1)-(3). Theorem 1: The spectral density of y and u obeying (1)-(3) are given below:

S""(4 = A l ( 4 where A,(E,) = A ( z , , z2),zi = e x p [ f i l i ] . Proof: We will obtain a difference equation for the covariance function of y. Note E(y(.)) = 0. Let R ( t ) = E[y(s)y(s + t)]. Multiply (1) by u(s), take expectation on both sides and use (3). E[y(s)u(s)l = &ECv2(s)l

=& Next, multiply (1) by y(s + t ) on both sides and take expectation

by using (3) and (7), where

dt,o = 1 =0

if

t =0

otherwise

Take Fourier transform of (8)

i.e., or S,,(l) = p / A l ( l ) . To prove (6),take spectral density of both sides of (4). PS""(4 = IlA(z1 = e x p ( J r 4 ) , '72 = exP(J-1~,))ll2Sy,(4 =

IIA , (4II 'S,,(4

(7)


89

Using (5) for S,,(A),the above equation yields the required expression for &"(A) in (6). The proof is given in some detail because it gives the difference equation for Ry(r).In addition, the above proof indicates the existence of a process y(.) obeying (1)-(3) by demonstrating its spectral density. The u(.) process is an unalog of a one-dimensional moving average process. Its covariance function is E[u(s)u(s

+ r)] = -0,

if

rEN

=1

if

=o

elsewhere

r=o]

(9)

However, one important distinction between l-D and 2-D cases lies in the fact that it cannot have 2D version of moving average representation, i.e., it cannot be represented as a finite linear combination of independent random variables. The reason is that the symmetric polynomial A(z,,z,) cannot be factored, ie., it cannot be expressed, in general, as a product of 2 finite polynomials, in general. Conuerse of Theorem 1 . This section started with the assumption (3) on u(.). What would be the structure of the process y(-) if u(-) is assumed to be

white? We will prove the converse of Theorem 1 and show that a process with inverse sinusoidal spectral density does not in general have any representation other than (1). The exceptions will be handled later. Theorem 2: Consider a zero mean stationary process y(-) having a spectral density as shown below

S,,(A) = p/[a positive linear combination of sinusoids in A,, A,] i.e.,

s,,(A)= p / ~ ( z ,z,), , and

A(z,,z,) = 1

zi= exp(J-lAi) -

C 8,z' reN

where N is symmetric, 0, = 0_, and A ( - )obeys (3). Then define u(-) as:

Then

+

E[u(s)y(s r)] = 0

for all r # 0


90

Proof: By definition =&)y(s)/h

Multiply both sides by y ( s + t ) and take expectation

JGy(- t ) = ' 4 ( Z ) W ) / & Take Fourier transform of both sides

suy(1)

zi

exp(J-lJi)

=

~ ( ~ ~ , ~ 2 ) s y y ( ~ = ) / h ~

=

h

by (10)

hence E [ v ( s ) y ( s + r)] = 0 if r # 0 Expression for the Correlation. In the one dimensional case, the correlation function is a linear combination of exponentially decaying function of the lag term given that the spectral density is a ratio of linear combinations of sinusoids. Such a result is not true in the 2D case. Exponential correlation functions are rare. We can evaluate the correlations from the spectral density by numerical integration. We will give one example below. Example: Consider the 4 member symmetric neighbor set.

Let

Y b ) = 0 1 Y(S + rsN

N

=

+h V ( 4

[ ( i , j ) , Iil = 1 or ljl = 1, not both]

The spectral density is S(1) = p / { 1

-

20(cos I ,

+ cos &)}

Here y(.) is isotropic. Full Plane Weak Markov Property. The system in (1)-(3) possesses a type of Markov property which is called weak since it can be described in terms of second order properties only. The weak Markov property will be defined in terms of the operator E*, the linear estimate of y ( s ) based on all its neighbors having the least mean square error. Dejnition of E*: E * ( y ( s )1 all y ( s + r), r # 0) is defined as the estimate of y(s) based on all linear functions of y ( s r), r # 0 having the least mean square error:

+

i.e.,

EC(y(s)- E*Cy(s) I all y ( s + r), r 5 EC(Yb) - d Y ( S

z 01121

+ 4 r # ON2]

where y is any linear function of one or more variables y ( s + r), r # 0.

(1 1)

91


Dejinition (Weak Full Plane Markov): A sequence y ( - )is said to have the weak full plane Markov property if ~ * [ y ( sI ail ) y(s

+ r), r z 01 = C ctry(s + r) rsN1

(12)

where N1 is a finite neighbor set and a,, r E N l are constants. The word ‘full plane’ appears because y(s) is conditioned on all other intensities in the entire plane. We will introduce later the half plane Markov property also. We will introduce below the intimate connection between the weak Markov property and the A R models discussed earlier. Theorem 3:

(i) The process y ( . ) defined in system in (1)-(3) possesses the weak full plane Markov property with the neighbor set N , equal to N and a, same as Or in (1). (ii) A stationary process which possesses the weak full plane Markov property in (12) must have its spectral density as in (10). Prooj:

Part (i): By adding and subtracting E * ( y ( s )I y ( s + r ) , r # 0) to y ( s ) - g on the R H S of Eq. (1 I), and simplifying it using the linearity of the function, one can easily show that (1 1) is equivalent to the following: E [ ( y ( s ) - E*(y(s)1 all y(s

+ r I ) ,r I z O))y(s+ r)] = 0

for all r

z 0. (13)

By the definition of the u ( - ) process in (3), we have E [ ( y ( s )-

C 8,y(s + t))y(s + r ) ] = 0,

for all r # 0.

(EN

Consequently, from ( 1 3), we have

4. Part (ii): In (12), let us replace simplicity. Define:

c(,

u(s) = y(s) -

by Or and N , by N for the sake of

C flry(s + r).

rEN

By the equivalence of (1 1) and (13), we have E[u(s)y(s + r)] = 0

which is precisely Eq. (3). By Theorem 1, the spectral density will have the required form.


92

Gaussian Markou Models. Since a Gaussian distribution is completely characterized by the second order properties, we will obtain an explicit expression for the joint probability density of all the M 2 pixel intensities in the given image in terms of the AR-parameters 6, and p. The expression is useful in estimating the appropriate value of the AR parameters so that the model fits the given image. After, even if the actual density is not Gaussian, we will still use the Gaussian expression for estimating the parameters. The parameter estimates obtained from this approach possess good properties like relatively low bias and variance, even though their variance may not be the least attainable value. Let us arrange the M 2 pixel intensities in the form of a column vector, row by row y = Col.[y(O,O),y ( 0 , l ) .. . , y(0, M - l ) , y ( l , O ) ,. . .,y ( M - 1 , M - l ) ] . By definition, E [ y ] = 0. Let R = E [ y y T ] . p ( y ) = The joint probability density of y = Gaussian(0, R )

- (2n)-MZ/2 IR I - l / 2

exPC - (1/2)YT R - l Y l (15) where (RI = determinant of R. The size of the matrix R is M 2 x M 2 . Even though every element of R is a correlation of the type E [ y ( s ) y ( s+ r)] and hence can be related to the parameters 6, in principle, the sheer size of R is a drawback in the manipulation of R . Hence, for all practical purposes, the above expression is not very useful. We will give here an approximation to the probability density when M is large which clearly displays the explicit dependence on the parameters 6,. According to a result of Brillinger, the elements of the discrete fourier transform (DFT) of y are approximately independently distributed for large M , with the following density. Let i, j = 0,... ,M - 1) be the DFT of y ,

{xj,

P(Yj)

'V

N ( 0 , Sij)

where 8 = Column[B,,, (k, 1 ) E N , ]

where

zij = Column[cos[(ik

+ j l ) 2 n / M ] ,( k , 1 ) E N ,

N , is the asymmetric half of N . We can write an expression for the density of y from the density of Xj. Note that the determinant of the matrix of transform to y is one. from {

xj}

ROBUST IMAGE MODELS A N D THEIR APPLICATIONS

93

Thus the density displays the effect of the parameters Or clearly. The error introduced by the use of the Brillinger's result is of the order 0( 1/M '). Since M is of the order 128-5/2 in real images, the error is negligible. The logarithm of the density function is not quadratic in 8. Hence maximizing this function is relatively difficult. C. Simultaneous und Recursive A R Models We will consider the AR models in which the input random sequence is white. These models are referred to as simultaneous AR models. As mentioned earlier, class of these models is a proper subset of the class of general AR models discussed earlier. Simultaneous A R Models. A simultaneous model is described as a difference equation

where N , can be any arbitrury neighbor set, and [ w ( . ) ] is a sequence of zero mean and uncorrelated random variables with E [ w 2 ( s ) ]= 1. E[w(s)w(r)]= 0

I.e.,

r#s

if

(17)

The above equation can be rewritten in the operator form also A , ( z ) y ( s )= h where A , ( z ) = 1 -

(18)

W ( 4

1 Przr.If r = ( i , j ) , z r 4 z i z ; .

rsNl

For the stationarity of the process, the coefficients following condition A,(z,,z,)A,(z;',z;')> 0

for all

fi must obey the

and

J z l (= 1

1z21 = 1

(19)

The spectral density of y(.) obeying (16) or (18) can be written down by in spect ion

This expression clearly indicates the connection between the SAR models and the general AR models. Clearly the SAR model is one in which the denominator of the spectral density possesses a factorization, i.e., if Syy(A)= p / A ( z , , z , ) , then the process y can possess a SAR representation in ( 1 6 ) if and only if the polynomial A can be factored as follows A ( Z i 1 z 2 = A 1 ( 2 i 1 z 2) A1 (zT

' 2;

)

(21)

94


where A, can be any finite polynomial. Unlike 1D polynomials, 2D polynomials do not possess in general a factorization as in (21). Equation (20) clearly indicates that a process y(.) obeying a SAR model in (16) possesses a full plane weak Markov property with a neighbor set N defined by the product A ( z l , z Z ) A ( z ; ' , z ~ ' ) i.e., A ( z l , z z ) A ( z ; ' , z ~ ' ) = 6, + 6#. Consequently

2

re N

The minimum mean square error of the estimate = p/8,. We will show that the noise w(-) is correlated with y(.). We will determine the cross spectral density between w(.)and y(.),

Thus E [ w ( s ) y ( s+ r)] # 0. If we d o not put any restrictions on the neighbor set N , then the cross correlation between w(*) and y(.) does not have any particular pattern. But if N is limited to special cases, as in the recursive models, then the cross correlations E[w(s)y(s + r)] will possess special properties. Gaussian SAR Models. Let us write an expression for the joint density of the M 2 intensity variables of a given image assuming that the density is Gaussian. One way is to begin with the joint density of the all w(9) variables which are independent and Gaussian. Then we can use (16) to obtain the density for y(-).However we need the Jacobian of the transformation between variables. This the M z vector of the y(.) variables and M Z vector of the w(-) determinant does not in general equal one and it is not easy to evaluate exactly. The second method is to use the approximation mentioned in the earlier i,j = 0,. . ., M - 1) be the D F T of the image sequence section. Let { {y(i,j),i,j = 0,. .., M - 1):

x,j,

Let

S,

= Syy(Al = 2ni/M,

= p/[i

-

LZ = 271j/M)

e ~ ~ , l[ ie

~ztl

0 = Column[&, ( k , 1 ) E N ] zij = Column[exp[fi(2n/M)(ik Z$

yj

= Column[exp[ --(2n/M)(ik

- N(0,

Sij)

+ j l ) ] , (k, I ) E N ] + j l ) , (k,I ) E N ]

95


p(y(i,j ) , i, j

= 0,. . . , M -

I) =

n-

i,k=O

Since S,' is quadratic in 0, the log likelihood function or the logarithm of the density function involve quadratic terms in I9 in addition to nonquadratic terms in 8 and p . Recursive A R Models. We will introduce here a subclass of the simultaneous AR models which behave very much like ID time series models and consequently possess properties like recursive generation. Let us define an asymmetric half plane R as follows:

R- = {(i,j ) : (i = 0 and j < 0) or (i < 0 and j is arbitrary)} One important property of R- is: if r E R- and s E R- then (s A recursive AR model can be written as

(25)

+ r) E R-.

where (i) the noise w ( . ) is zero mean, unit variance and uncorrelated, and (ii) the neighbor set N is a subset of the asymmetric half plane R-. Equation (26) can be rewritten in the usual operator form Al(Z)Y(S)= &w(s)

where

We will prove that w(s) is uncorrelated with y ( s + r) for all r belonging to R-. This property is similar to that of time series AR models. To prove this result, we begin with the expression (22) for the cross spectral density

&

A l ( z ; 1 > z i 1 ) ~ w .= "(4

(27)

We will express S,,,,(L) as a power series in zi

where Rwy(r)= E [ w ( s ) y ( s+ r)].Since the right hand side of (27) is independent of z 1 and z 2 , (27) is possible only if RWy(i,j ) = 0

i.e., E [ w ( s ) y ( s+ r)] = 0 for all r e R

for all (i,j ) E R

96

R. L. KASHYAP A N D KIE-BUM EOM

Similarly we can see that E[w(s)y(s)l =

h

Recursive Generation. We will describe a procedure which recursively generates a sequence y(.) obeying (26) given a set of values for w ( - ) and a set of boundary conditions. Recursive generation is possible only with the recursive AR models discussed in this section. Our intention is to recursively generate a finite image { y(i, j), 0 I i, j I M - I}. We need the intensities of the boundary pixels to be defined presently. Let i* = Min{i; (i, j ) E N , for some j } j * = Min{ j ; (i, j ) E N , for some i} 0,= {(i, j), (i = - 1,. .. ,i* and j = 0,1,. . ., M - 1 ) o r ( j = - l , ...,j * a n d i = O , l , ..., M - 1 ) ) We need the intensities of all the pixels in Q,. For example if N , = ((0,- l), (- 1, l), (- 2,0)), i* = - 2, j* = - 1, a, is displayed below. x x x QB

x x -

x x

x x

x x

x x t

X

* = (0,O) 0 row M=5

X X

X

t 0 Column

We are given {w(i,j), 0 I i, j 5 M - l}. We will recursively generate row by row to generate the zeroth row using the following equation:

+m)

In this case all the values y(r are the intensities of the pixels in 0,. w(0,j), j = 0,1,. . .,M - 1 are given, a set of independent pseudorandom numbers with zero mean and unit variance. Next we can generate all the intensities in the row-1, namely {y(l,j),j = 0,. . , ,M - 13, as before. All the quantities on the right hand side are available, either precomputed or given apriori. In general, to compute the intensities of the i I h row, given all the intensities of the previous rows, we use the following

j=O,l,,..,M

-

1


97

State Variable Models. All the models defined above involve recursion in both the variables (i, j ) . Here we will focus on a subset of the recursive models which can be written as a multiuariable model with recursion in only one index, say i. Then we can use Kalman type algorithms for filtering and smoothing an image in the presence of additive noise. We consider the recursive model given below

{ w(-))is a zero mean uncorrelated sequence with unit variance. The neighbor set N , is restricted as follows N , = {(i, j ) , i I - l }

We can also handle a similar neighbor set in which i is replaced by j, in a similar way. For illustration, consider the following set N2 =

{(-~,-l),(-~,O),(-~,~)}

Hence y ( k , j ) = P , y ( k - I , j - 1) +P,y(k - 1, j ) + P 3 y ( k - 1 , j + 1) + f i w ( k , j ) . Let yk = Column vector of dimension M = Col.(y(k,O),. , . ,y ( k , M - I)). Let wk = Col.(w(k,O),. . . ,w(k, M - 1)). Then equation can be rewritten as follows y(k)

= By(k-

1)

+

J&k)

where P2

B = I

P3

k g'

In the above equation, we generate one entire row as one step. Thus we have converted the two dimensional recursion into an M-variable one dimensional recursion. If N , has neighbors like (-2, j ) , then the corresponding vector . can convert this difference equation for y ' k )will involve y(k- " and y ( k - 2 ) We equation into a 2M-variable first order vector difference equation. D. Generalized ARMA Models

An ARMA model is characterized by two polynomials A(z) and B(z). A(z)= 1 -

c &z', or

reNl

=

/Ir

98


N , and N2 are symmetric neighbor sets. In addition the parameters S and satisfy the following equation. A(z) > 0 and

B(z) > 0

for all Izl( = I

and

1z21 = 1

4

(30)

A stationary A R M A model is defined by the following difference equation involving the coefficients 8 and 4 defined above.

C BrY(s + r ) + J;v(s) A(z)y(s) = Jm Y(S) =

(31)

rENl

or

(32)

where the polynomials A and B obey (30) and the input u(-) is zero mean, and correlated with spectral density S,,(Z)= A(z)B(z)

(33)

The condition (30) on A and B are necessary to ensure the existence of a stationary process y(.) obeying (31). As before, (31) is a descriptive model. It is not obvious how to generate a y(.) using (31). This aspect will be treated later. The spectral density of y(.) obeying (31) is

It is easy to see that given any spectral density which is a ratio of two positive linear combinations of sinusoids in Al and A 2 , there exists a corresponding A R M A model as in (31). Unlike the one dimensional A R M A models, u(-) in Eq. (31) cannot be replaced, in general, by a finite movin average representation operating on an uncorrelated sequence because A(z)B(z)is not, in general, a finite order polynomial in z. In view of (31),the sequence o(.) has nonzero correlations only over a finite number of lags, as displayed below.

P-

E[u(t)v(t

+ s)]

=

otherwise

where

N"

=

{ ( r + s): r E N ; , s E N ; }

N f= N i u (03, i = 1,2 We are defining Bo = - 1, q50 (31) since E [ e 2 ( S ) ] = 1.

=

1. Note that the parameter v is introduced in


99

Let us define the cross correlations between 2) and y. Multiply both sides of Eq. (31) by y ( s + r), take expectation and sum over t from -m to a: A ( 4 S Y , ( 4 = J;S,?,(4

Using the expression for S,, in (34), we get

S U Y ( 4= JI;W

(35)

Expanding S,,(-)in powers of z and equating the coefficient of z'on both sides, we get if r E N ; otherwise

E[o(*s)u(s + r)] =

Thus the sequence u(s) has nonzero correlation with y(s + r) only for a finite number of values of r. The A R M A model in (31) does not, in general, possess the full plane weak Markov property. This will be established by showing that the best linear estimate of y(s) given all other intensities explicitly depends on all of them.

Theorem 4: The linear least squares estimate of y(s) based on all y ( r ) ,s # r has the following expression where the sequence y ( - ) is stationary and obeys (31). yl(s) = E * [ y ( s )I all Y ( S

and

+ r), r z 01

EC(y(s) - Yl(.$)21 = v / K

(37)

where - K = the -K g , =

i.e., Proof:

-K

C yrz'

constant term in the expansion of A ( z ) / B ( z )

the coefficient of z' in the expansion of A ( z ) / B ( z )

= A ( z ) / B ( z ) go , =

-

1.

(38)

Let

44 = Y ( 4 - allr,1r 3 0 Y,Y(S + where yr obeys (38) u(s) = ( - C g , z ' ) y ( s ) ,

since go = - 1

Find the cross spectral density of u and y SUY(4 = (-Cg,z')S,,(4

(39)


100

Substitute for Sy,(A) from (34) and zg,z‘from (38) S,,(A.)

1 B vA K A B

= -- * =

v/K

Consequently

+

E[u(s)y(s r)] = 0

for all r # 0

(40)

Let y’(s) be any linear function of all y(s + r), r # 0.

+ r) + Cgry(s + r) - ~‘(s))’] = E [ ( Y ( s )- Cgry(s + r))’] + E[Cg,y(s + r ) - Y’(s))*],using (40)

E [ ( Y ( s )- Y‘(s))’I = E[(y(s) - Cgry(s 2 E [ ( y ( s )- CgrY(s +

This proves (36).To prove (37), let us find the spectral density of S,,,, from the definition of u in (39)

Hence E[u’(s)]

=v/K.

As a consequence of the theorem, only the generalized A R model possesses full plane weak Markov property among all processes with ‘rational’ spectral densities. There are several special subclasses of A R M A models which possess interesting properties. The detailed description of these models and their properties are well explained in Kashyap (1981).

E . Generative Interpretation of Models

While defining the basic AR process in Section A, Eq. (1) was used to define the process u ( * ) in terms of the y(.) process. The spectral density of u ( * ) in (6) was derived from the corresponding properties of y(.). In this section, we will view the basic Eq. (1) as generative, i.e., given a sequence u ( - ) with spectral density in (6), how can we generate a sequence y(.) with the corresponding spectral density in (5) or equivalently y(.) is the output of the system whose input is u(-). If Eq. (1) were recursive, the generation would


101

not pose any problem. However (1) is not recursive and y,(i, j ) depends on intensities of pixels all around it. Thus solution of the generative process is not an initial value problem as in recursive equations, but a boundary value problem. We will pose the problem in a precise way and solve it accordingly. We are interested in generating a sequence { y(s)), which belongs to the grid R.

R

= { ( i , j ) ,0 Ii , j IM

-

1)

Let the spectral density of the required process be [ p / A ( z ) ] , where A ( z ) is defined in terms of a symmetric neighbor set N

Note that the neighbors of the pixels as defined by N on the boundary of R will not belong to R. Let 0, be the smallest superset of R to include all the neighbors of a11 the pixels in R, i.e., let s E R, imply either s E R or there exists a s, E R so that s = s1 I , r E N . For instance, if N = ((0, l), (0, - I), (l,O), ( - l,O)}, then R, = { ( i ,j ) , - 1 I i, j 5 M } . Let R, = R, - R. Let us rewrite Eq. ( 1

+

13

The boundary value problem can be posed as follows: Given {u(s), s E R} and { y(.s), s E R,} determine [y(s), s E R) by solving the M 2 equation given by (41). Let y = {y(O,O), y(0,I),..., y ( M - 1, M - l)} and u = (u(O,O), ~ ( 0l),. , . ., u ( M - 1, M - 1)). We can rewrite the M 2 equations in (41) in the following vector matrix format Ay

=

hV + YE

(42)

where A is a M 2 x M 2 matrix involving (Or, r E N ) . y , is a M 2 vector whose components are the image intensities belonging to the border region R,. The matrix A is a block Toeplitz matrix. The existence of the solution for y obeying (42) depends on the regularity of matrix A . It can be shown that the condition (2) on the polynomial A ( z ) implies that the matrix A is invertible. Thus it is interesting that the conditions (2) are needed in both the generative and nongenerative interpretation of the model. The existence of the inverse for A is not the same thing as saying that the inverse is easily computable for any neighbor set N . Recall that a value 130 for M is common or even low, and M 2 x M 2 dimension means 16,900 x 16,900. The sheer volume of computation is stupendous. We will discuss the approximation methods in a later section.

R. L. KASHYAP A N D KIE-BUM EOM

102

Let us discuss the structure of A for the simple case of 4 member first order or nearest neighbor set N . The basic equation is

+ 1, j ) + y ( i - 1, j ) ] + e 2 [ y ( i , j - 1) + y(i,j + 111 + &W),( i , ~Q

Let

y(i,j ) = O1[y(i

E

We can write down the following M 2 simultaneous equations for y in terms of u and the vector of boundary pixels Ay = & U

A 2 A 1A2

+ b,

(43)

...

A=

A,

=

1

-02

o

-e2 1 -e2 o -e2 1

o

0

To find the eigenvectors and eigenvalues of the matrix A , we need to solve the equation A X = AX x

i.e.,

X l M I x 2 1 , . . . ,X M M l - 0 1 x i - 1 , j - e 2 x i , j ~+1x i j - e 2 x i , j + l- o , ~ ~ + , A, ~ ~=

= C O l . ~ X , 1 ,x 1 2 , .

9

1

,

~

(44) ,

Let us try for xij, a solution of the type xij = sin ia sin j r (45) Substituting (45) in (44) and simplifying we can find the eigenvalues of A to be

(A) (A)

li,= j 1 - 28, sin - - 28,sin

~

~


103

The corresponding eigenvector is x(i,J)

[ (

= Col. sin

~

M+1

)sin('"),

M+l

k,l

=

1 ,..., M

Note that the condition (2) on A ( z ) implies that all the eigenvalues in (46) are strictly positive. Hence we can easily invert the A matrix and solve Eq. (43)for y in terms of u and the boundary conditions. The above procedure is elegant, but it does not appear that one can generalize it to handle Eq. (1) with an arbitrary neighbor set. In the next section, we will give approximation methods which are computationally elegant and still can synthesize an image close to the ideal image given in Eq. (43).

F. Approximations to The Image Models The models presented earlier are very interesting from the theoretical point of view. But if we are interested in applying the models for interesting applications like image synthesis, segmentation etc., the allocated computational problems are not easy to handle. To illustrate the nature of the difficulty, we will consider four cases. First of all, to obtain an exact value for the correlation R(s), we need to numerically solve the corresponding fourier inversion problem involving the spectral density. There is no analytic method. We have already indicated that obtaining an exact expression for the joint probability density of all the intensities in a M x M image is almost impossible, we can get only an approximate expression for the density. Next, synthesizing an M x M image obeying a given model is computationally stupendous, as indicated in the earlier section. As indicated, we can think of separate approximation for each task like joint density of synthesis. The alternative method is to think of approximating the model itself, like the AR model in ( 1 ) or ARMA model in (28) etc. so that all the tasks mentioned earlier can be done with relative ease and modest computation. We will consider several approximations, but the most approximation is the toroidal model. These approximations are also called finite lattice models since they are defined only for a finite lattice, say M x M . The mc del has no meaning for the intensities outside of the grid. Toroidal Approximation. The toroidal approximation of the generalized AR model (1) defined for an M x M lattice is given below


104

All the symbols in (48) such as N , etc. have exactly the same meaning as in (1) except that the symbol @ stands for summation modular M in both components, i.e., (i, j ) 0 (k, I ) = ((i + k)mod M , ( j + l)mod M ) . Equation (48) is closed, i.e., it gives M Zlinear equations involving the M Zimage intensities [y(i, j ) , i, j = 0,. , .,M - 11. They do not involve any values of y(s) for s outside the boundary. The modulo operation makes the pixel (0, M - l), the left neighbor of (0,O) and ( M - 1,0), the top neighbor of (0,O). A similar statement is valid for the pixels on the edge of the grid. It is, as if, we have folded the M x M grid into a torus by folding the grid such that the O’h row and ( M - 1) row are neighbors and similarly the OIh column and ( M - l ) I h column are neighbors. Often the toroidal models are criticized as being unrealistic for “real” images in view of the folding property. This criticism misses the point that this toroidal AR model is an approximation to the spatial AR model in (1). We measure the quality of approximation by the quality of the inferences such as correlations, synthesized images, etc. The toroidal model is accepted only if the error between the inferences given by the toroidal models and the original models is sufficiently small. Properties of the Toroidal Model. We will consider here the properties of the model in (48), the toroidal approximation of the AR model in (1). Toroidal ARMA models possess similar properties. i, j = 0,. . .,M - I} be the discrete fourier transform of the finite Let { sequence {y(i,j ) , i , j = 0,. . . ,M - 13. Let us assume that y(-) is stationary and E[y(.)] = 0. Similarly, {vij} be the DFT of the sequence { ~ ( s ) } . Recall that

xj,

A(z,,z,) = 1 -

1 eijz;z’,

i,jsN

The S”,(i, j ) , spectral density of v(-) is defined as the mathematical expectation of uijn$,* indicating complex conjugate. S,(i, j ) = A(vi, v j ) &(i, j ) = A(vi, v j )

v = exp[-2n/~]

R,Jr) = Discrete inverse of the spectral density S,, =

-0,

rEN

= I

r=O

=o

elsewhere


105

By transforming (48),

xj

= &vij/A(vi, v')

(49)

Consequently Syy(irj ) = spectral density of y ( . ) = E[Y,,.Y$]

= p/A(v', v ' )

Thus both the toroidal AR model in (48) and the ordinary AR model (1) have the same expression for the spectral density. This statement is true for an ordinary ARMA model in (28)and the corresponding toroidal model. We can restate this result as follows. Theorem 5: The spectral density of an ordinary AR and ARMA model, say S,(i, j ) , defined over the infinite grid { i , j ; i, j = - 00,.. .,a} is identical to the spectral density of the corresponding toroidal model defined over the finite lattice M x M , say S F ( i , j )i.e.,

Consequently we can rewrite (49) as

(51) suggests a relatively easy method of generating a finite M 2 sequence y ( . ) obeying (48) in the following 3 steps:

(i) Obtain a sequence of M 2 uncorrelated pseudo-random numbers with zero mean unit variance and any required marginal distribution. Let { i, j = 0,. . . , M - I } be the DFT of { w ( - ) } . (ii) Compute x j , , i ,j = 0,1,. . . ,M - 1 from (51). (iii) A DFT of { yields the required finite image {y(i,j), i , j = 0,1, ..., M - I } .

wj,

xj}

Joint Probability Density. An exact expression for the probability density for the finite image { y ( i , j ) , i, j = 0,. . . ,M - I}, given that it is Gaussian since are Gaussian with zero mean, unit variance and independent, (51) implies that { i, j = 0,1,. . . ,M - 1 } are also zero mean, independent and has density

{w,)

xj,

x,'is Gaussian(0, p / A ( v ' , v'))

R L KASHYAI' A N D KI17-BIJM EOM

106

p(Y+ i , j = 0,1, . . . , h4 - 1)

Since the Jacobian of the transformation from [y(.s)i to [ write the joint density of y(.) in terms of y.j p ( y ( i , j ) , i , j = 0....,M 1 -

~

-

(2np)

1)

n

M-I

M~L ~

x j ) is one, we can

( ( ~ ( v 'v, . j ) ) " 2 e x p [ - ( ~ / 2 p ) ~ j ~ ~ ~ ( ~ ~ ' . \ ~ ' ) ]

1,/=0

Recdl that .

1

M

.

A(\jl, \ t J )

=

1

-

I (),,,\,'k'+J1)

h.l-0

Thus the exponent of the density function is linear in 0,. Also note that the above expression which is the exact joint density for the image obeying (48),is also the o p p r o x i m n r c expression for the density of the model in Eq. ( I ) . as discussed in Section (2). Thus the toroidal model throws light on the precise nature of thc approximation in the density expression of Section (2). The correlation function of y can be obtained by summation of S,

Thc correlations of the toroidal lattice model arc not exactly identical to [he correlations of the corresponding infinite lattice AR and ARMA model. But the dift'erence between them is small. Synr/w.si.s. Let i w(s,,j),i..j = 0,. . . , M - I ) be a sequence of zero mean uncorrrlatc~dvariables with unit variance. Let { W i j )be its DFT. We can relate v f to as follows:

wj

y j = J A ( l l ' , d )I q i Note that v i ) is real. Finally, i t is possible to arrange the M 2 equations for i j f i , j ) , i, j 0, I , . . . , M - I given (48) in the matrix form: A(\l',

=

u =Jpu

4'= Col.[y(0,0),..., y(0, M

-

I ) , y ( l , O ), . . . , 4'(M

-

I, M

-

I)]

u = Col.[v(0,0),. . . , \ ' ( M , M ) ]

where B is a doubly circulant matrix, i.e., B can be written as a block circulant matrix and each block is also a circiilant


1

Bo.0

B0.1

*

B0,M- 1

B0.M-I

Bo.0

Bo.1

B0.M-2

Bo.1

B0.2

Bo.0

107

where Eo.i, i = 0,1,. . ., M - 1 are also circulant For instance, if N is the 4 member nearest neighbor set then y(s) = 00,1(y(s

+ 8,,o(y(s

+ 0,1) + Y ( S + 0 7 ) ) + i,)+ + - 1,o))+ hw

Boo = circulant( 1, -do, 1

I,.

. . , - oo,l) -60.1

-00.1

1

-00.1

Po

-60.1

Bo.1 =

=0

1

-00.1

-00.1

- 4.0

1

1

for i # 0,1,M - 1

The eigenvectors of the matrix B are the fourier vectors fij.

B.. = C ~ ] . [ v ( ~ ~k,’ j ”=, 0, 1, . . . , A 4 ?I v

-

I]

= exp[J-127c/~l

The eigenvalue of B associated with the eigenvector fii is p i j .

.

pij = A(v’, V J )

For the 4 neighbor case pij = I =

+

v-j)

-

6,,o(vi + v-j)

27cj 27ci 1 - 2 0 0 ~ ~ ~ 0 ~ - - 2 0 ~ , ~ ~ 0 ~ - - , i , j = O ,Ml , . . 1. ,

M

M

we can compare these eigenvalues with the eigenvectors of the A matrix in Section 1I.E. These eigenvalues are identical to the approximate eigenvalues of the A matrix. This again throws light on the nature of approximation. Other Finite Lattice Models. As alluded earlier, the toroidal model of (48) is only one way of modifying the AR model in (1) so that the equation for y ( - ) are closed, i.e., they do not involve any y(i,j) where (i,j ) is outside the M x M


108

lattice. We can have two models in which the closure is achieved in different ways. We will give two examples. In both of them, we have the 4 member neighbor set.

N

=

{ ~1

(0,l), S; = (0, - l), ~2

= (1, 0),$ = ( - 1,O))

Example 1 (cosine vectors as eigenvectors): Let us modify the Eq. (1) as follows to achieve closure 2

y(s) =

1 8iyl(s + s i ) + h u b ) , s = (i,j), i, j = 0,1,. . .,M - 1

i=l

(52)

where Y,(S

+ s i ) E R, (s + g ) E R (s + s i ) E R, (s + 3)q! R (s + q) E R, (s + si) 4 R

+ si) = Y(S + si) + Y(S + Fi),

+ si) if = 2y(s + F), if = 2y(s

if

(S

(53) By definition M 2 equations for y(s) given by (52) are closed, i.e., we can write Eq. (52) as DY=&u

where

D,, and I are M x M matrices. To find the eigenvector of D , consider the Chebychev polynomials ci(xj), xj = cos ci(xj)= , , , ( A ) ,

( M j : ~

i, j

1) = 0,1,

.. . , M - 1

-

c j = (co(xj),cl(xj),.. . , c ~ - , ( x ~ ) M-vector )~

cij = column(c0xj)ci, c,(xj)ci,.. .,c M M,(xj)ci)


109

where 0 I i, j I m - 1. Then cij, 0 I i, j I ( M - 1) are the eigenvectors of D. The eigenvalue of D corresponding to the eigenvector cij is aij, aij = 1 + 2e,xj + 20,xj. Example 2: Consider again the 4 neighbor case; (52) is the finite lattice model with the following definition for y l ( . ) instead of ( 5 3 ) y,(s

+ si) = y(s + s i ) + y ( s + q), + si) = y ( s + Fj) = y(s

if

if

if

s + si E R, s + 5 E R

s + si E a, s

+ 5 .$ R s + < E R, s + si .$ n.

Then we can rewrite these equations

DY

= JpU

The eigenvectors of the matrix D are the so-called discrete sine vectors. G . Summary

We have defined the various types of AR and A R M A models and their toroidal and other approximations. We have concentrated on their second order properties. We have also discussed the various types of weak Markov properties which are completely characterized by the second order properties.

111. ROBUSTESTIMATION IN CAUSAL AUTOREGRESSIVE MODELS

A. Introduction

The importance of model based techniques for image processing tasks such as edge detection, image synthesis, image coding, image restoration, etc. has been well documented. However in all of these models, the image intensity array is assumed to be a multivariate Gaussian distribution. The Gaussian assumption is used primarily in estimating the parameters of the image model fitted to the image. The corresponding estimation procedure is relatively easy; for example, for the causal autoregressive model, the maximum likelihood method is the same as the least squares method. However in many applications, it is well known that the Gaussian assumption is not appropriate. A more realistic assumption is a contaminated Gaussian noise, il(i,j)=

w(i,j ) , u(i,j),

with probability 1 - p with probability p ’

(54)

110


where w ( i , j ) is a regular white Gaussian noise and u(i, j ) is an outlier process and the ratio of outlier is assumed small (less than 5%). Unfortunately, least squares estimators or maximum likelihood estimators under the Gaussian assumption are very sensitive to minor deviations from the Gaussian noise assumption. Even a single bad data (outlier) among 1000 observations can cause large error in the estimator. Because of this excessive sensitivity of least squares estimator, a robust estimator is needed in image models. The robust estimator should possess the following properties: (1) It should have a reasonably good (optimal or nearly optimal) efficiency at the assumed noise distribution. (2) It should be robust in the sense that a small number of outliers impair the performance only slightly. (3) Somewhat larger deviations from the assumed distribution should not cause a catastrophe. The resistance to outliers (e.g., impulse noise) is equivalent to the distribution robustness by Hampel’s theorem (Huber, 1981). Many different robust estimation algorithms have been developed in the last twenty years, mostly on the location parameter estimation. These robust estimation algorithms can be classified into three large types of estimators: M-estimator, L-estimator, and R-estimator. M-estimator is a maximum likelihood type estimator and it is obtained by solving a minimization problem. L-estimator is a linear combination of ordered statistics. R-estimator is derived from the rank tests. We are mostly interested in M-estimator for the application on the image models. M-estimator is easy to extend to the problems of image models, but other types of estimators are difficult to use in problems other than simple location parameter estimation. M-estimator is defined by the following minimization problem, Minimize

1p ( x i ;0)

or solve the following implicit function,

1+ ( x i ; 0) = 0 where p is a continuous and differentiable convex function possessing bounded and continuous derivative +(x) = a p ( x ) / d x , and p is symmetric about origin with p(0) = 0. The convexity of p function ensures the equivalence of (55) and (56). The boundedness and continuity of t,h function is essential in obtaining robustness of the M-estimator. If is not bounded, then a single gross outlier can completely upset the estimator. If I) is not continuous, then small changes in the observation x i may produce a large change in the estimator.

+


111

There are several different definitions of robustness of an estimator (Huber, I98 1). Qualitutiiv ruhttstrie.s.s is defined by weak continuity of the estimator. M-estimator is qualitatively robust if and only if the corresponding t+b is bounded and continuous. Mininiux rohusf estimator minimizes the maximum degradation over E deviations. The M-estimator of location is optimal in the sense of minimax robustness. Quantitutiue robustness is defined by the property of small change in asymptotic bias and asymptotic variance in the contaminated neighborhood. Even though a robust procedure is necessary in most image processing applications, very little research has been done on the use of robust procedure in image processing. In this section, we develop estimation algorithms for the causal autoregressive image model.

B. Ctrusul Autoreyressioe Model I t is well known that a large class of images can be effectively represented by various types of image models involving small number of parameters (Kachyap, 1981).Image models are already uscd in image coding (Delp et ul., 1979), image synthesis, texture analysis, edge detection (Kashyap and Eom, 198Sa).Of course, there are many different types of image models and these can be classified into two large classes of image models by their second order statistical structures: classical short correlation models and long correlation models. These different image models and their general properties are discussed by Kashyap (1981). The causal autoregressive model is a generalization of one dimensional autoregressive model. This model is simple but has good modelling performance as shown in previous studies. Consider the following m x n image (Fig. I ) .

t -1

11 1_1 I 1

11-I-i- t I - I - T + i -I -I -1-14 4-1-

c

n FIG. I .

An ni x

II

image and three causal neighbors

112


Assume that the image intensity in this image follows three neighbor causal autoregressive model. Let (i, j ) be an index for the coordinate location and y(i,j ) be the intensity at the coordinate (i,j ) . Then the causal three neighbors of this pixel are { y ( i - 1, j ) , y(i,j - l), y(i - 1, j - l)}. This causality is from the convention of raster scanning, and because of the causality, the resulting two dimensional model has all the convenience of one dimensional model. Suppose that { [ ( i , j ) } is a two dimensional white noise sequence with outliers as assumed in (54).The variance of the regular part of noise is IS’. Then the three neighbor causal autoregressive model is represented by the following equation:

+

y(i,j)= eTz(i,j) U , j )

(57)

where 0 is a parameter vector and z(i,j ) is a vector consists of intensities of three causal neighbors and unity. The last element of the vector z(i, j )is used to represent constant grey level in the image.

z(i, j ) =

It is assumed that every pixel has all of its neighbors, i.e., for each pixel at ( i , j ) , pixels at (i,j - l), (i - 1, j ) and ( i - 1, j - 1) are available. We consider the robust parameter estimation of the causal autoregressive model for two cases of outliers. First case, we assume that the process y ( i , j ) given in (57) can be perfectly observed. In this case, the outlier process is involved only in the noise process [ ( i , j ) to generate y(i,j ) . Second case, we assume that the observation x(i,j ) of the process y(i,j ) is corrupted by noise [ ( i , j ) . It is given by the following equation: x(i,j) = y(i,j)

+ t(i,j)

(59)

The noise process 5 is assumed to contain outliers. In this case, the outliers are not only involved in generating y(i,j ) but also involved in observation. In the next section, robust parameter estimation will be discussed for these two different cases of outliers. C . Robust Parameter Estimation 1 . Perfect Observation Case

The parameters of the image model given in (57) can be estimated by robust M-estimator. The M-estimator of the parameters in (57) is a gen-


113

eralization of location M-estimator. Define the following function Q(0,c):

where p is a continuous, differentiable and convex function possessing bounded derivative, and it is symmetric about origin with p(0) = 0. Then Mestimator of the causal autoregressive model is defined by the following minimization problem. Minimize Q(0,o)

(61)

The M-estimator can also be obtained by solving the following two equations simultaneously.

) x$(x) - p ( x ) ,function 1(1 is continuous and Where $(x) = dp(x)/dx and ~ ( x = bounded. The following p , $, and x functions satisfy the above conditions on these functions. In this section, it is assumed that the following functions are used in our robust estimation algorithm.

X I - c

x(4

= x$(x)

- P(X) = ~ / 2 [ $ ( x ) l 2

(66)

Asymptotic Property. The asymptotic property of the robust Mestimator for autoregression is investigated by Nasburg and Kashyap (1975). The asymptotic property of one dimensional autoregression is also applicable to two dimensional causal autoregressive model. First, the following conditions are assumed. (i) { y ( i , j ) }is a weakly stationary random sequence. (ii) is an odd, monotone increasing function satisfying a Lipshitz condition. (iii) The noise process [ has finite moments up to third order. (iv) E[$(> kT/q, so that nonlinearity in field-effect transistors is smaller than in bipolar transistors. Signals are larger in field-effect logic circuits than in bipolar circuits for this reason. The conclusion to be drawn from the preceding discussion is that voltage in electrical logic circuitry cannot be reduced indefinitely. Similar statements apply to logical operations on information represented in other physical forms.

PHYSICAL LIMITS IN INFORMATION PROCESSING

175

Large voltages are also used in logic to take account of the unavoidable differences among devices. Signal voltages must be large enough to switch all the devices of the system in spite of the differences. There are many sources of variation in the threshold voltages of insulated gate field-effect transistors: trapped charges, differences in insulator thicknesses, differences in doping concentrations. Bipolar transistors are less affected by physical differences, changes in voltages required vary only logarithmically with device parameters because of the exponential current-voltage characteristic. The additional voltages that take account of the difference among devices are part of the need for large noise margins. Voltages larger than necessary are sometimes used becausedevices must be adapted to existing power supply standards. A power supply may have several uses and is not easily changed to adapt to new device technologies. The demand for high voltages in solid-state electronics leads to two kinds of limits. One concerns the effects of high electric fields in devices. The other is the problem of removing the heat produced, the product of the voltage and the current supplied. These will be discussed in later sections.

V. TRANSISTORS

Continued miniaturization combined with a minimum voltage tends to increase the electric fields in electronic devices. Containment of high electric field effects as miniaturization is advanced led to a reduction in the voltages used in digital circuitry in the early stages of semiconductor device development. Reduction of voltage was also pursued as a means to reduce power dissipation. However, the intent to thwart high electric field effects by reducing voltage is eventually frustrated by the existence of the important voltage restrictions that have just been mentioned. High electric fields can produce undesirable or catastrophic effects, not encompassed in conventional transport theory. These include the production of hot electrons, reduction of mobilities, and dielectric breakdown. The nature of high-field effects is shown very schematically in Fig. 11, a plot of the current through a region of a semiconductor as a function of the electric field in the region. The current is controlled by mobility at low fields, the electron velocity and the current are proportional to the field. At around lo4 V/cm the electrons begin to acquire energy from the field at a faster rate than it is lost through the scattering processes that dominate at low field, the average electron energy increases, new scattering mechanisms come into play, and the velocity is no longer proportional to the field. The velocity eventually reaches a field-independent, saturated value. As the field approaches lo6 V/cm

176

ROBERT W. KEYES

'I02

lo4 FIELD (V/cm)

FIG. 11. Hot electron effects in a semiconductor. The current is proportional to electric field at small fields. The carrier velocity reaches a field-independent value at high fields. At still higher electric fields electrons acquire enough energy to excite more electrons across the energy gap and avalanche breakdown occurs.

some electrons acquire enough energy to excite a second electron from the valence band to the conduction band. The added electrons can also acquire energy from the field, excite more electrons, and an uncontrolled avalanche breakdown current rapidly develops. The exact fields at which these events take place are different for different semiconductors and also depend on doping concentrations and on the size of the region to which the field is applied. The voltage differences between different parts of a transistor are supported by layers depleted of electrons and holes. The thickness of such a depleted layer, x, is related to the potential difference that it supports, 4, and the doping level in the layer, N , by

Here E is the dielectric constant of the semiconductor. Equation 12 is the basis of various limitations on semiconductor devices. Elements of a device must be large enough to avoid being entirely depleted by applied voltages. The dimensions of the layers are shown in Fig. 12. The depleted layers occupy a large part of the volume of transistors, and miniaturization of devices must


0.1

0.2

0.5

I volt

2

5

177

10

FIG. 12. Widths of one-sided depletion layers as a function of voltage across the layer and doping concentration. The numbers apply to silicon and gallium arsenide.

include miniaturization of the depleted layers. Apparently the widths can be reduced by increasing N , Eq. (12); miniaturization is accompanied by higher doping levels. A . Bipolar Transistors

Practically all logic circuitry in large high-performance computers is based on bipolar transistors. As in the case of other solid-state electronic components, the evolution of bipolar transistors has been marked by rapid miniaturization. However, the current that these transistors are called upon to control in the fastest logic circuits has not decreased very much. Higher speeds have been attained by combining the reduction of capacitances produced by miniaturization with high currents. Therefore, the density of current has steadily increased. The trend of current density in bipolar transistors for highspeed logic in a sampling of the literature is shown in Fig. 13 (Keyes, 1987).The need to handle the high and steadily increasing current densities severely limits the design of transistors for use in logic circuits.

178

IIOBTKT W K l Y E S

1o6

I

I

I

I

I o5 0

0

NE

lo4

0

>

.-4-

0

8

0

2

0

o

I o3

0

ooo 00 oo

0

e

E L

al -0 4-

lo2

C

2L

3

10

1 10-1 1950

1960

1970

1980

1990

2000

l h e problenis arising from tlie voltage requirement of logic circuitry are less scvcrc i n bipolar devices than in field-efTect devices. 'The principle limit arising directly from the ~ipplicationo f voltage to ;I bipolar device i s that the 1 1 - 11 junctions should not break d o w n and that tlie base should n o t "punchthrough." T h e latter expression refers t o complete depletion of the base by the reverse bins applied to tlie collector. The essentiul compromise that must be m i d e is between heavy doping of the base, which reduces tlie penetration of t h e depleted layer i n t o it, and low doping to widen the collector depletion layer and increase i t s breakdown voltage (Hoeneisen and Mead, I972b). l-icavy doping of tlie base is also clesirable to decrcase the resistance in the path of the ha sc cii r re n t . The quantitative details depend on the doping profile ;it the junction. 'The doping i n ninny modern p1;inar bipolar transistors varies with distance from tlic stirfice ;IS shown in I-'ig. 14. T h e lightly doped collector :illows ;I wide deple[ion layer to forin atid the breakdown voltagc to be high. I t also reduces tlic capacitance of the collector junction. which is important i n determining tlic switching spced of ;I switching circuit.


179

102“

I

\

DISTANCE FIG. 14. Doping concentration as a function of distance from the surface in an npn bipolar transistor fabricated by the methods of planar technology. The transistor is formed in a lightly doped layer grown epitaxially on a heavily doped subcollector in the substrate.

However the use of the lightly doped collector in planar bipolar transistors introduces two new problems at high current densities. One is apparent from the physical form of the transistor and its contacts, Fig. 8. The collected current must flow from the region under the emitter to the contact through the substrate. The high resistance of a lightly doped collector in this current path would be inadmissible and it is avoided by the heavily doped “subcollector”, shown in Fig. 14, which provides an alternative, lower resistance current path to the contact. The other limitation of the lightly-doped collector is the phenomenon known as “base stretching”(Kirk, 1962). Base stretching occurs at high current levels, when the density of electrons required to carry the current is comparable to the density of the doping impurities. The charge of the currentcarrying electrons then drastically changes the potential distribution in the device. Quantitative analysis of the events in a structure such as that of Fig. 14 is complicated and must be done by numerical simulation (Poon et al., 1969; Knepper et al., 1985). An idealized model can show the nature and approximate magnitude of the effect, however. We consider the base-collector junction to consist of three uniformly doped regions: base, lightly-doped collector, and subcollector, separated by abrupt junctions. The sequence of events as the current is increased is shown in Fig. 15 in the form of graphs of potential, shown as the energy of the conduction band, as a function of position. (a) is the potential in equilibrium, with no applied voltage and no current flowing. (b)shows the effect of reverse bias on the collector; the regions depleted of carriers have widened. When current flows the electrons crossing the depleted region of the collector compensate the positively charged donor atoms there. The density of

180

ROBERT W. KEYES BASE

1

I

I

h

COLLECTOR

1I sc I I I

I

,

I (d)

- -I

FIG. 15. Potential in the base, collector, and subcollector of a bipolar transistor under various conditions. (a) No voltage on the collector, no current flowing. (b) Reverse bias on collector, no current flowing.The depleted regions are thicker than in (b).(c)The concentration of electrons needed to carry the current through the collector is equal to the collector doping. The collector is neutral and the field in it is constant. (d) The concentration of electrons carrying current through the collector is very high. Holes neutralize the electrons and the effective base width is greatly increased.

electrons with velocity u needed to carry the current j is

The electrons compensate the donors in the collector and effectively reduce the concentration, N,, to N , - n. When n = N, the collector is neutral, the field in it is constant, and the potential varies as in Fig. 15(c). Beyond this point the net charge in the collector becomes negative and the potential has the form of


181

Fig. 15(d). Holes enter the collector to compensate the electron charge and rapidly extend the netural base beyond the metallurgical junction. The extra charge stored in the expanded base slows the transistor action at current densities greater than .jL= NJUL (14) The electric fields in contemporary silicon transistors are large enough to make u close to the saturation velocity, uL = lo7 cm/s. For practical purposes j , is an upper limit to the current density in the transistor. The avoidance of base stretching interacts with the rest of transistor design. Thin bases are desired in bipolar transistors to minimize base transit times and the charge stored in the base. The widths of the depleted regions in uniformly doped base and collector are given by

Here = V, + Vbi,and V, is the reverse bias applied to the collector. However, uniformly doped bases and collectors are not used in the fastest logic transistors. Because of the limitations of the lightly doped collector that have been discussed, the collector doping is increased rapidly away from the junction to prevent the base from extending into the collector. The depletion layer is primarily in the base and extends into it to a distance given by Eq. (12), or with Nb >> N, in Eq. (15).

The condition that the base not “punch-through”, w b > x b , can be written in terms of the number of impurities per unit area of base, k f b = N b W b , (Mb = N b W b in a uniformly doped base) as

Here w b is the base thickness, &, is the punch-through voltage, and 5 is a constant that depends on the distribution of the dopant atoms in the base. From Eq. (17), 5 = 2 for a uniformly doped base, but is close to 1 for the type of doping profile shown in Fig. 14, and hereafter = 1 is assumed in Eq. (8). The sheet resistance of the base is another important parameter associated with M b . The sheet resistance is p _ = (‘b4p)-

(19)

182

ROBERT W. KEYES

Any base current must flow through this sheet resistance. The potential drop caused by the passage of base current through the sheet resistance decreases the forward bias at the emitter-base junction and decreases the density at which current is injected from the emitter. Because the current through a p - n junction depends on voltage as exp(qV/kT), if the voltage drop through resistance in the base is greater than (kT/q),little current flows across the emitter junction. If the emitter is a rectangle with dimensions W x L , then, very approximately, for the entire emitter to be active as a current source

Here i, is the base current, assumed to be injected along the long sides L. All current crossing the emitter junction is assumed to be captured by the collector in the ideal npn transistor; no base current is needed. The main source of deviation from this ideal is a component of current carried across the emitter junction by holes entering the emitter. The ratio of the current collected by the collector to that that must be supplied by the base is the current gain p. High values of p, p N SO, are needed in logic circuitry. All but a few percent of the current crossing the emitter junction reaches the collector. Nevertheless, the base current plays an important role in transistor design by limiting the distance from the base contact to which bipolar transistor action is effective. When ib = jWL/P is substituted in (20) the limit becomes

Many miniaturized logic transistors are square or circular, in which case j W 2 is approximately the transistor current. Current is injected along the entire periphery of the base, and a numerical factor appreciably larger than the 4 in Eq. (21) is applicable in this case. As an example of (21), if fl = 50, (kT/q) = O.O25V, and j W 2 = 1 mA, then p c I 5 x lo3Ohm. It is difficult to obtain high-speed switching of the required currents with prJ> lo4 Ohm. Equations (18) and (19) show the advantages of heavy doping of the base. However, the ratio of currents carried across the emitter junction by electrons and by holes is proportional to the ratio of the numbers of these entities in the emitter and the base. The large number of holes in the base associated with large M , leads to injection of holes into the emitter. This hole current is supplied through the base and decreases the current gain, 8. The number of dopant atoms in the base should be less than the number in the emitter to maintain high emitter efficiency. Heavy doping of the emitter helps to attain this end up to a point, but is not effective beyond about S x 1019cm-3 because of a decrease in the energy gap of silicon at higher doping levels (Slotboom and


183

deGraaf, 1976; Keyes, 1976; Selloni and Pantelides, 1982). The decrease in the energy gap lowers the barrier to the entrance of holes into the emitter. Therefore, keeping the number of dopants in the base below the number in the emitter means in practice that the average base doping must be below 10'8cm-3. The heavy doping in the emitter can also cause a decrease in lifetime by leading to Auger recombination, which increases injection of holes into the emitter. Solomon (1982) has discussed scaling and limits of bipolar transistors. In addition to the decreasing energy gap at high doping levels other limits to the scaling of bipolar transistors to higher current densities and smaller dimensions arise from effects that are more difficult to quantify. A further limit to the doping level in the base is the possible existence of a tunnel-assisted recombination current in the emitter-base junction (del Alamo and Swanson, 1986) at dopings above a few times 10'8/cm3. Another, less significant, effect of heavy doping in the base is the scattering of charge carriers that it introduces. The lowered hole mobility operates on p , _ and the lowered electron diffusivity increases the time taken for electrons to cross the base. In addition, neutrality is maintained in the base by holes that compensate the injected electrons. The number of holes in the base is equal to the sum of the number of acceptor atoms and the number of electrons injected. The latter number is determined by the current; the charge in the base is equal to the current times the time needed for an electron to traverse the base. The electrons cross the base by diffusion in an average time.

The number of electrons per unit area of base is thus

When j is small M , is small and the number of holes in the base is nearly equal to the number of acceptors. At high current M e can dominate the acceptors and the number of holes increases with current, thereby increasing the rate at which holes are injected into the emitter and reducing the current gain. The current gain begins to fall off when the current density becomes so large that M e exceeds the number of doping atoms in the base. To insure that the current gain is controlled by the base and does not fall off at high current density it can be required that the number of doping atoms in the base be greater than M e

184

ROBERT W. KEYES

Summarizing again, a high fl at large current densities, high punchthrough voltage, and low base resistance all call for a high M,. High emitter efficiency and high carrier mobilities conflict with these requirements. The various influences that confine the base doping on both the high and the low sides have prevented Mb from changing very much during the decades of miniaturization; it has remained in the range 10'2cm-2 to IOl3cm-*. Also, near-equality prevails in Eq. (24). The status of Eq. (24), which represents a limit on the base doping, is shown in Fig. 16, where the data of Fig. 13 is plotted against the base width (Keyes, 1987). Figure 16 shows a correlation of base width with current density. This figure includes points in addition to those in Fig. 13 derived from simulations of projected devices (Gaur, 1979; Kimura and Takahashi, 1982). No date was attached to such devices and no attempt to plot them in Fig. 13 was made. To the extent that Mb is a constant and (24) is an equality the relation in Fig. 16 is expected to be Wb =

(yy2

The line shown is calculated from (25) and deviates from proportionality to the - 112 power because of the dependence of D on doping. Increasing doping

lo2,

I

I

I

I

I

I

I

j(A/c~~) FIG. 16. Comparison of trends in npn bipolar transistors with Eq. (25), including some simulated transistors to which no date could be attached for inclusion in Fig. 13. The line is derived from Eq. ( 2 5 ) with Mb = loLz (Keyes, 1987).


185

levels reduce mobility and the electron diffusion constant in the base. The current density increases more slowly with decreasing base width than would be suggested by Eq. (25) with constant D. The line in Fig. 16 is calculated with Mb = 1012cm-3.The best value for M , is rather uncertain because of a variety of effects that are difficult to take into account quantitatively: the effects of fields associated with concentration gradients in the base, the effect of electron-hole scattering and compensation on mobility, averaging over the rapidly varying impurity concentration of Fig. 14, and the effects of degeneracy at high carrier concentrations. The limits described by Eqs. (18) and (24) and Nb < 3 x l O ' * ~ m -are ~ illustrated in Fig. 17, a plot in the M b - w b plane. As wb is decreased the value of M , becomes increasingly confined between the punch-through and heavy base doping limits. The window between these vanishes at the 106A/cm2 current density contour if the average acceptor concentration is kept below 3 x 10ls ~ m - The ~ . maximum current density transistor is close to that described by Solomon and Tang (1979). Certain other consequences of these considerations are worth noting. Eliminating pL and M , from Eqs. (19), (21) and (25) yields W2 By the Einstein relation the last part of (26a)is the ratio of the mobilites of the two carrier types, so w2 /Ix w; (26b)

-

There is a rough proportionality between the thickness of the base and the linear size of the emitter. A rapidly decreasing total number of impurities in the base is also implied. The decrease with time is shown in Fig. I8(a). It appears that in a decade or so it will be small enough that random fluctuations will affect yield. In terms of the above equations, the number of dopant atoms in a square base is hfbW2. The number can be written by using (19) and (21) as

The relation between M , W 2 and j is shown in Fig. 18(b), where the inverse proportionality described by (27) is illustrated by the line. There is an inevitability to the decrease in the number of dopant atoms as the current density is increased. According to Eq. (25) the increasing current densities that accompany miniaturization of bipolar transistors for high speed circuitry demand decreasing base widths. The fabrication of the ever-thinner bases is a difficult

186

ROBERT W. KEYES

10'

1 o6

lo5 A/crn2

lo4

I o3

10l6

?-

-E 0

W

v)

m

.-C

s

4-

0.

W

1015

0

8

1 o-2

10-I

1

10

Base width, w,, (pm) FIG. 17. Limits on bipolar transistors in the wb - M , space. Contours of constant current density (solid lines), and punch through voltage are shown. The average acceptor concentration in the base is indicated by the dotted lines. The limit set by a 3 V punch-through voltage and an average base doping level of 3 x lO''//cm3 is shown by the heavy line.

aspect of bipolar technology. The base width is defined by the difference in penetration beneath the surface of an acceptor and a donor (see Figs. 8 and 14). As neither of these can be controlled perfectly, controlling the base width within reasonable limits requires that both depths be scaled together; the depth of the emitter-base junction decreases with the base width. However, the contact of the emitter to a metal conductor at the surface has a high rctcombination velocity and forms an almost perfect sink for holes. The

1 Ol2

10I2

loll 1o1O 0, u)

m n .-c

E0

1 o9

a,

rn m n .-c

1 o8

?

0

4-

m

4-

4-

10’

m + c m

C

m

a

0

n

1 o6

Q

00

1 o5 1 o4 1o3

1950 1960

(a)

I

1970

1980

Year

1990 2000

I

I

I

I

I

( b) Current density (A/crn *) FIG. 18. (a)The decrease in the number of acceptor atoms in the base of npn transistors with time (Keyes, 1987).(b) The relation of the number of dopants in the base to the current density. The line illustrates Eq. (27).

188

ROBERT W. KEYES

gradient of the hole concentration in the emitter is increased by a shallow junction and the hole current flowing from the base is increased, decreasing the current gain. The passage of current through contacts of ever-decreasing cross section increases the loss in series resistance, which is another limitation on the miniaturization of bipolar devices. These limits on the base and emitter can be attacked by the heterojunction emitter and the polycrystalline emitter. The use of an emitter with an energy gap larger than the base creates an energy barrier to entry of holes into the emitter. The base can then be heavily doped without reducing emitter efficiency. The larger energy gap can be realized by forming the emitter from a different semiconductor than the base, making the emitter junction a heterojunction. A suitable larger energy gap semiconductor, one that can be heavily doped and grown epitaxially on the base, is not known for silicon. The requirement can be met by depositing a GaAs-A1As alloy on a GaAs base, and heterojunction emitter transistors are pursued in this system. The poly-emitter refers to the deposition of a layer of doped silicon on the surface of the emitter. The deposited layer is ordinarily not epitaxial to the base and is polycrystalline. Nevertheless, the interface between the single and polycrystals can be of such good quality that the deposit acts as an extension of the single crystal emitter, moving the metal contact from the metal contact away from the base-emitter junction and allowing the number of dopants in the emitter to be increased. The level at which these limitations on bipolar transistors are effective involves process details and qualitative judgements. The details depend on the exact doping profiles. The magnitude of current gain needed depends on circuit choice. The necessary control of base width depends on acceptable yield, among other things. B. Field-EfSect Transistors

Metal-oxide-semiconductor field-effect transistors are limited by phenomena associated with both the semiconductor and the oxide. The structure of a small n-channel MOSFET is shown in Fig. 19. The transistor is turned on by the application of a positive potential to the gate electrode, producing a variation of potential normal to the surface as shown in Fig. 20. Fp is the distance of the Fermi energy from the top of the valence band. The threshold voltage, V,, is the voltage that must be applied to the gate to induce a significant number of electrons at the semiconductor-oxide interface, conventionally defined as the point at which the energy of the conduction band is depressed by an amount EG - 2Fp at the interface. The electrons induced at the Si-SiO, interface, the “channel”, carry current when a positive voltage is applied to the drain.


189

OXIDE

SUBSTRATE DRAIN FIG. 19. The physical structure of a MOSFET, showing gate, oxide, source, drain, and substrate.

M

O

FIG.20. Form of the variation of potential perpendicular to the surface of a MOSFET at threshold. The potential in the semiconductor is represented by the energy of the conduction band. The Fermi energy is shown as the dashed line.

A brief description of the theory of the MOSFET current is given here to clarify the meaning of a few terms and introduce notation. The current carried by the electrons at the silicon surface, the “channel”, is (charge per unit area) x velocity x width. The charge density is equal to the excess of gate voltage minus threshold voltage over the channel voltage, V, acting through the capacitance of the insulator. The velocity is determined by the mobility and the electric field along the channel, dV/dx. Thus

190

ROBERT W. KEYES

is the dielectric constant of the insulator and ti is its thickness. The source and the substrate are regarded as the zero of potential. The current is constant along the channel. The voltage is therefore determined by integrating (28). E~

ix = ( + O V ,

-

V,)V-

"'I

2

V and X are measured from the source. At the drain V = VDand x length of the channel. The current depends on VDas

= L,

the

The current reaches a maximum as a function of VD when V, = VG - VT. When V, > VG - V, Eq. (30) no longer holds; the current remains constant at its maximum value

The current is said to be saturated at the value (31). The extent of the depleted regions in the miniaturized MOSFET is suggested in Fig. 18. A first limit on the design of the MOSFET is seen from Fig. 21(a): the depleted layers at the source and drain electrodes must not meet (Hoeneisen and Mead, 1972a). The widths are of the form of Eq. (12) 2E(VD+ 4%

xd=(

Ki)

l''

)

V, is the voltage applied to the drain and hi is the built-in voltage of the junctions. Thus it is required that the channel length satisfy

L > 2x,

(33) The limit (33) can be reduced by increasing N,, Eq. (32). However, heavy doping of the substrate increases the field normal to the surface at the interface

I - J l

__-i

I

I I

.

I---

\ \

/

--/

\

------

\

\ \

\---

(a1 (b) FIG.21. The dashed lines show the extent of the depleted region in a MOSFET. (a) Reverse biased drain below threshold. (b) With channel formed and current flowing.


191

with the oxide and the field in the oxide. For a uniformly doped substrate the field is (34) The potential supported by the depleted layer in the substrate, 4, approaches (E,/q), Fig. 20, in a heavily doped semiconductor at the threshold. The field in the oxide is ciFi = cF,, where subscript i refers to the insulator. Thus, the maximum field sustainable by the oxide sets a limit on N , and on L (Hoeneisen and Mead, 1972a). (VD

L> 1). Let the spot have diameter d, so that quanta of energy at least hcld are used in the exposure and an energy Mhcld is deposited in the spot. The rate at which spots are exposed must be proportional to l / d 2 to maintain a constant throughput, that is, to expose silicon area at a constant rate as d varies. Therefore, the rate at which energy must be delivered to a substrate varies as l/d3 in this limit. Stringent demands are placed upon radiation sources and optics of exposure tools. Some length parameters related to various exposure methods are shown in Fig. 26. A few general observations can be made. It becomes increasingly

j /cm

0.OOl

0.01

0.1

LENGTH

I

10

2

too

(pm)

FIG.26. Relations between length parameters relevant to lithographic exposure and energetic quantities. Line a is the energy of an electromagnetic quantum as a function of its wavelength. h is the range of electrons in silicon as a function of energy; its dashed extension suggests the range of secondary electrons produced by quanta of radiation. The shaded area encompasses the range-energy relation of various ions. The dotted lines are contours of the density of energy deposition when 100 quanta or particles are used to expose each spot of the diameter on the abscissa.

206

ROBERT W. KEYES

difficult to localize the effects of exposure as dimension is reduced because of the increasing range of electrons with increasing energy. For example, a one KeV quantum has a wavelength of only pm but can produce an electron with a range of 0.02 pm. The exposure produced by energetic particles is not confined to the area in which a beam is focussed, but occurs throughout the particle’s range. This effect is most prominent in exposure with electrons. Electrons experience many large-angle scattering events because of their small mass compared to atoms and produce exposure at an appreciable distance from their point of entry. Fortunately, a large part of this proximity effect can be compensated in computer-controlled electron beam tools by adjusting the exposure at each site to account for any exposure received from nearby sites. The sensitivity of resist materials in current use is to 1 j/cmz. The energy densities corresponding to 100 quanta per spot are also shown in Fig. 26. The quantum limit (M 100 quanta per spot) is not yet here. The energy deposited per unit area in the quantum limit increases rapidly with decreasing dimension. Further, less energy is deposited in exposure with photons than in exposure with electrons at a given dimension in the quantum limit. The long electron ranges shown cannot be regarded as a dimensional limit in the same way as the wavelength of radiation is a limit, however, a long electron range produces the proximity effect mentioned above, which can be corrected to a great extent. The radiation exposes some kind of resist material, changing its solubility in a developer, in conventional fabrication methods. The pattern created in the resist must be converted to some physical structure that forms part of a device, usually a pattern of metallic conductors or of dopants in a semiconductor. The conversion is effected by a sequence of processes involving material deposition and selective material removal by various kinds of etching. The accuracy with which the original pattern can be reproduced from one process step to another limits the fabrication of microminiaturized structures. A major improvement in the fidelity of pattern transfer must accompany any significant reduction of the dimensions of devices. Time-consuming development of new process methods limits the rate at which miniaturization can advance. Alignment is a further problem that limits the rate of advance of miniaturization. Device structures are made by several successive steps. For example, first a highly conductive region is formed in a semiconductor by introducing a dopant, then a metallic contact is deposited on the doped region. Alignment means locating the same place on a chip several times so that a sequence of steps are all properly placed, Increasing the accuracy of alignment is part of miniaturization and also requires the development of new methods as dimensions are reduced.

-


207

VIII. DISSIPATION OF ENERGY The dissipation of energy to heat in logical operations is one of the limits on the computational process that has attracted most attention. Removing the heat produced in logic operations is one of the limitations on the performance of large computing systems (Keyes, 1970; Vilkelis and Henle, 1979). A . Fundamental Limits I t is known that computation can be performed in principle without the dissipation of energy. Landauer (1986) and Bennett and Landauer (1985) have reviewed the subject. However, the idealized systems that demonstrate the possibility of dissipationless computation are far removed from the devices that have actually proved useful. Perhaps the concept closest to reality is the molecular Turing machine, whose operation closely resembles the transcription of genetic information in living systems (Bennett, 1982). A Turing machine consists of a black box or “head” that can assume any of a certain set of states and a movable tape on which a member of an alphabet of symbols can be written by the head. The machine operates step-by-step. At each step the head reads a symbol from the tape. Logic in the head combines the symbol read and the current state of the head to produce a new state in the head, write a new symbol on the tape, and move the tape to another position. The machine is then ready for the next step. A Turing machine can perform any possible computation. The tape of the molecular Turing machine would consist of a long molecule such as a strand of DNA that can bind various symbol molecules at a series of positions along its length. The state of the machine is represented by the attachment of one of a set of state molecules, a molecule selected to record the state of the machine, at the current position of the tape. The calculation is carried out by a collection of enzymes that can catalyze the reaction on the chain that places the next symbol molecule on the chain and replaces the previous state molecule with the appropriate one. Chemical reactions, such as those involved in the operation of the molecular Turing machines, are driven by imbalances in the concentration of reactants. At equilibrium a reaction proceeds equally rapidly in both directions. No progress is made in either direction and no energy is consumed; diffusion and Brownian motion cause reactants to come together or to dissociate occasionally. A reaction can be made to proceed preferentially in one direction by controlling the concentration of reactants, in accord with the law of mass action. The reaction in the case of the molecular Turing machine

208

ROBERT

w. K E w s

consumes the symbol monomers and can be caused to proceed in the direction of the computation by an excess of these monomers. However, the reaction is then no longer in equilibrium and dissipation occurs. The source of energy dissipation in logic can be identified as the discarding of information in an idealized system. The basic idea can be seen by thinking of a box partitioned into two equal parts. A bit of information can be stored by putting a gas of N particles in one or the other side of the box. If the partition separating the two chambers is removed, thereby erasing the stored information, the entropy of the system is increased by Nk In 2. A n amount of work NkT In 2 must be performed lo compress the gas and energy N k T In 2 dissipated to heat to restore ;in information-containing state. Since writing a bit can be regarded a s the simplest logic operation and the minimum value of N is one, kT In 2 represents a minimal dissipation per logic operation. This elementary system is still quite far from any real logic device. However, the result docs suggest that k7' is an appropriate unit in which to measure dissipation in logic and thus compare actual energy utilization with a crude limit. The dissipation in this cxnmple is essential because removing the partition between the two cells is thermodynamically irreversible; the cycle through which the system has been carried cannot be run in reverse. Common logic operations have a similar property. Consider the N O R in Table I. The inputs cannot bc determined in general from the output, the computation cannot be reversed. Pursuing the example, the dissipation could he avoided by eliminating the destruction of the original information. The new information could be written with an apparatus that could move the gas from one cell to the other at constant volume if the original location of the gas were known. Models of computational systems that would avoid dissipation in principle by storing all intermediate results to enable the system to be restorccl to its initial state without dissipation have been described (Bennett, 1973). Likharev (1 982) has proposed a scheme for doing this that does not seem too remote from reality. Stein (1977) has considered the effect of thermal fluctuations on information in clectrical form. If a bit is represented by a charge or absence of charge on a capacitor c', then the energy of storage is ( C V 2 / 2 ) .The mean thermal fluctuation energy in the capacitor ( k 7 / 2 ) , can produce an error in reading the bit. Stein shows, for example, that the probability of incorrect reading is reduced to lo-'" if ( C V 2 / 2 )> 165kT. Still another view of a fundamental limit to dissipation is based on the observation that according to the quantum mechanical uncertainty principle an event localized i n a time t must be associated with an energy-tilt, suggesting that this quantity is a lower limit to the dissipation of a logic operation performed i n time f . It does not seem possible to translate this rather general


209

thought into specific models involving particles, potentials, fields, or Hamiltonians, however. Its significance is, therefore, somewhat vague. A version, difficultto question, is that nothing can be done with aphoton of energy hv in a time less than l / v . B. Power Supply and Cooling

The energy dissipated in practical electrical logic is 10' kT or more and far exceeds any of these limits. Contemporary electrical logic uses large signals to influence many electrons at each step. We now turn to this topic, which constitutes an important limit. The need for large signals in logic means that large amounts of power must be supplied to logic chips and this power must be removed as heat. The ability of technology to meet these demands is limited by the properties of known materials. Power is supplied to chips through the same kind of connections that are used for logic signals. The connections on and off of the chip are subject to electromigration and are limited to current densities not much greater than lo5 A/cm2. Ohmic resistance may limit current densities to even lower values. For example, the resistive voltage loss in a conductor of resistivity ohm cm carrying current at a density lo5 A/cm2 is 1 V/cm, an amount that may not allow the tolerances on the power supply to be met on all chips and devices. The limited number of connections to the chip and their finite current-carrying capacity limit the power that can be supplied. The heat flow paths in a large system are shown schematically in Fig. 27. Most of the energy drawn from the power supply is converted to heat in the devices. Heat is conducted away from devices into the body of the chip, encountering the thermal resistance of the silicon in doing so. If the dimensions of the region under the emitter in a bipolar transistor are small compared to the thickness of the chip and the distance between devices the thermal resistance in the silicon can be regarded as a spreading resistance. The

FIG.27. Heat flow from devices to a cooling fluid. Heat spreads into the chip from the device where it is produced at high density. It is passed from the chip to cooling structures from which it is transferred at low density to a moving fluid that carries it out of the system.

210

ROBERT W. KEYES

I

2

3

4

5

10

L/W

FIG.28. Values of thermal resistance encountered in spreading of heat from a rectangular source region of dimensions L , W ( L > W ) into a semi-infinite substrate with thermal conductivity K.

spreading resistance of rectangular regions has been calculated and the results are presented in Fig. 28 (Loewen and Shaw, 1954). The relation is wellrepresented by the formula

Here K is the thermal conductivity of the substrate and L and W are the dimensions of the rectangle ( L is the larger of these dimensions in Eq. 66). Bipolar transistors projected for the future with 114 pm dimensions and currents around 0.5 ma (Solomon and Tang, 1979) will have thermal resistances to the substrate exceeding lo4 deg/W and rise 10°C above the substrate temperature. The time constants for a steady state heat flow to be achieved will be a few nanoseconds with 1 pm dimensions. As these times are longer than the anticipated delay times on the chips, the temperature of the device will depend to some extent on its switching history. Examples of calculated times and thermal resistances are presented in Fig. 29. The thermal resistance given by Eq. (66) actually refers to the relation of the maximum temperature of the rectangle, which is found at the center, to heat current uniformly distributed across the rectangle. The temperature increase will be only about half of this amount at the corners. The temperature gradient will cause a device to operate nonuniformly across its area. A 5 degree temperature difference changes the current density in a junction by a factor 213.


21 1

W (pm) FIG.29. Thermal resistances and thermal relaxation times for rectangular areas on a silicon surface. The solid lines represent the thermal resistance, left hand scale, and the dashed lines represent the relaxation times, right hand scale. The curves are labelled with the values of L/W to which they apply.

The heat spreads into the chip from the devices. It is eventually transferred to the atmosphere, a large heat sink. The density at which this is done depends on the investment in cooling hardware and to a less important degree on the permissible temperature difference between the chip and its environment. The allowable increase in temperature of a chip is limited by reliability; activated processes such as diffusion, creep, chemical reaction, and thermomigration alter device structures increasingly rapidly as temperature is increased. The activation energy of failure mechanisms in integrated circuitry is about 0.5 eV. At 70°C a temperature change of 15°C changes the mean life of components by a factor 2. Careful attention to the details of heat transfer is essential. It is desired to place the chips in Fig. 27 as closely together as possible in order to minimize the transit time of signals between chips. The spacing is limited by the density at which heat can be removed from the chip array. The historical trend of cooling in high performance systems is shown in Fig. 30. Early systems were cooled by free convection of the air. Fans were soon introduced to provide forced convection. At heat transfer rates above a few tenths of a watt per square centimeter, special structures must be provided to increase the area of solid-air interface across which heat can be moved. Heat

212

ROBERT W. KEYES

10010 -

1990 2000 YEAR FIG.30. Evolution of cooling technologies for large high-performance computers.

transfer to air becomes quite difficult around 1 W/cm2 and liquid cooling is used to take advantage of higher heat transfer coefficients at solid-liquid interfaces. There are several compromises involved in the design of heat transfer structures: thin fins introduce a thermal resistance into the path of heat flow into the fluid, thick fins reduce the number of fins that can be placed in a given volume; short fins increase the density at which heat must be transferred to the fluid, long fins increase the volume of fluid that must be provided; small separations between fins cause high viscous resistance to fluid flow, large separations mean that heat must flow a long distance through the thermal resistance of the fluid to reach the rapidly moving part and also reduce the number of fins. The optimum design must take account of many factors, including the constraints upon the system that is called upon to supply the fluid, notably, pressure and volume of fluid required. Tuckerman and Pease (1981) showed that there is a length parameter that characterizes a fluid defined by

Here pFis the viscosity of the fluid, KF is its thermal conductivity, and CFis its heat capacity per unit volume. P is the pressure driving the fluid through a channel of length L . r is a constant, approximately 100.The optimum channel width is close to b, depending to some extent on the constraints under which optimization is carried out (Tuckerman, 1984; Keyes, 1984).Tuckerman and Pease, (1981) recognized that if water is used as the cooling fluid, L is a typical chip dimension, 0.5 cm or 1 cm, and P is around one atmosphere, then b x 50 pm, a dimension that is easy to fabricate with techniques practiced in

PHYSICAL LIMITS I N INFORMATION PROCESSING

213

semiconductor technology. They demonstrated that hundreds of watts can be removed from a chip by this method, simply etching fins and channels into the back of a microelectronic chip.

IX. CONCLUDING REMARKS Many physical effects retard the advance of microelectronics towards smaller structures, higher levels of integration, higher speeds, lower powers, and greater reliability. Most of these depend on the properties of particular materials. In some cases, such as semiconductor materials, few materials are known and one can regard the consequences of the properties of these as limits. In other cases, for example, radiation-sensitive resists, there is substantial opportunity to devise new compositions and one anticipates a stream of improvements. Advances in the capabilities of microelectronics by and large stem from miniaturization and integration. Continuing these avenues of improvement depends on new device structures and new fabrication tools. The extent to which novel concepts can be utilized is frequently limited by economics, the ever-increasing cost of new instruments and facilities.

REFERENCES Abbas, S . A,, and Dockerty, R. C. (1975). Appl. Phys. Lerrers 27, 147-148. Attardo, M. J., and Rosenberg, R. (1970).J . Appl. Phvs. 41,2381-2386. Bennett, C. H. (1973).IBM J . Res. DCLX17, 525-532. Bennett, C. H. (1982).Inf. J . Theor. Phys. 21, 905-940. Bennett, C. H., and Landauer, R. (1985). Scient$c American 253 (I), 48-56. Blodgett. A. J., Jr. (1983). Scient$c American 129 ( I ) , 86-96. Brewer, G. R. (1980). In “Electron-Beam Technology in Microelectronic Fabrication” ( G . R. Brewer, ed.), 1-58. Academic Press, New York. Broers, A. N., Molzen, W. W., Cuomo, J. J., and Wittles, N. D. (1976). Appl. Phys. Lerter.s 29, 596-598. Cottrell, P. E.,Troutman, R. R., and Ning,T. H. (1979).IEEETrans. Elecrr. Deu. ED-26,520-533. DHeurle, F.. and Rosenberg, R. (1973).In“Physics of Thin Films”, 7(Academic Press, New York), 257- 310. del Alamo, J. A,, and Swanson, R. M. (1986). IEEE Elecrr. Deu. Lelters EDL-7,628-631. Donath, W. E. (1981).IBM Journal Res. and Deu. 25, 152-155. Drangeid, K. E., Sommerhalder, R., and Walter, W. (1970). Electronics Letters 6,228-229. Duvurry, C. (1986).IEEE Ckts., and Deu. Magazine 2 (6), 6-10. Eden, R. C., Welch, B. M., Zucca, R., and Long, S . L. (1979). IEEE Trans. Electr. Deu. ED-26, 299-317. El Gamal, A. A. (1981). IEEE Trans. Ckrs. Sys. CS-28, 127-135. Gabrielse. G., Dehmelt, H., and Kells, W. (1985). Phj,.s Rev Lerters 54, 537-539.

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Gaur, S. P. (1979).IEEE Trans. Electr. Deo. ED-26,415-421. Ghate, P. B. (1983).Solid State Tech. 26(3), 113-120. Gunn, J. B.(1968).J . Appl. Phys. 39,5357-5361. Heller, W. R., Mikhail, W. F., and Donath, W. E. (1977).Proc. 14th Design Automation Conference (New Orleans, 1977),32-42. Hoefflinger, B., Sibbert, H., and Zimmer, G. (1979). IEEE Trans. Electr. Deu. ED-26,513-520. Hoeneisen, B., and Mead, C. A. (1972a).Sol. SI. Elecfr. 15,819-829. Hoeneisen, B., and Mead, C. A. (1972b).Sol. Sf. Electr. 15,891-897. Howard, J. K., White, J. F., and Ho, P. S. (1978).J . Appl. Phys. 49,4083-4093. Keyes, R. W. (1970).Science 168,796-801. Keyes, R. W. (1976). Comments Sol. Sf. Phys. 7, 149-157. Keyes, R. W. (1981). In “Digital Technology - Status and Trends”, ed. H. Painke. Oldenburg, Munich. 253-271. Keyes, R. W. (1982a).IEEE J . Sol.-St. Ckts. SC-17, 1232-1233. Keyes, R. W. (1982b).Int. J . Theor. Phys. 21,263-273. Keyes, R. W. (1984). lEEE Trans. Elecfr. Den ED-31, 1218-1221. Keyes, R.W. (1987). IBM Research Report RC12843 (unpublished). Keyes, R. W. and Landauer, R. (1970).IBM J . Res. Deo. 14, 152-157. Kimura, K., and Takahashi, T. (1982). In “Large Scale Integrated Circuits Technology”, ed. L. Esaki and G . Soncini (Martinus Nijhoff, The Hague) 373-398. Kirk, C. T. (1962).IRE Trans. Electr. Deu. E9-9, 164-174. Knepper, R. W., Gaur, S. P., Chang, F.-Y., and Srinivasan, G. R. (1985). IBM J . Res. Deu. 29, 2 18-228. Kohonen, T. (1972).“Digital Circuits and Devices”. Prentice-Hall, Englewood Cliffs, N. J. Landauer, R. (1986). In “Der lnformationsbegriff im Technik und Wissenschaft”, eds. 0. G . Folberth, C. Hackl. Oldenbourg, Munchen. 139-158. Landman, B. S., and Russo, R. L. (1971).IEEE Trans. Computers, C-20, 1469-1479. Lo, A. W. (1961)./ R E Trans. Elect. Comput. EC-IO,416-425. Likharev, K. K. (1982).Int. J . Theor. Phys. 21, 311-326. Loewen, E. G.,and Shaw, M. C. (1954). Trans. ASME76,217. May, T. C. (1979).IEEE Trans. Components, Hybrids, Manu6 Tech., CHMT-2,377-387. Ning, T. H., Cook, P. W., Dennard, R. H., Osburn, C. M., Schuster, S. E., and Yu, H-N. (1979). IEEE Trans. Electr. Dev. ED-26, 346-352. Ogura, S., Tsang, P. J., Walker, W. W., Critchlow, D. L., and Shephard, J. F., (1980) IEEE Trans. Electr. Dev., ED-27, 1359-1367. Parrillo, L. C . (1983).In “VLSI Technology”, ed. S. M. Sze. McGraw-Hill, New York. 445-505. Poon, H. C., Gummel, H. K., and Scharfetter, D. L. (1969), IEEE Trans. Electr. Deu. ED-16, 455-457. Sedra, A. S. and Smith, K. C. (1982).“Microelectronic Circuits”. Holt, Rinehart and Winston. Selloni, A,, and Pantelides, S. T., (1982).Phys. Rev. Lett. 49, 586. Slotboom, J. W., and de Graaf, H. C. (1976).Solid-state Electron 19, 857. Solomon, P. M., and Tang, D. D. (1979).Inr. Sol. St. Ckts. ConjDigest, 86-87. Sze, S. M. (1969).“Physics of Semiconductor Devices”. Wiley-Interscience, New York. Stein, K.-U. (1977). IEEE J . Sol. SI. Ckts. SC-12, 527-530. Tuckerman, D. B., and Pease, R. F. W. (1981).IEEE Electron Device Lefters EDL-2, 126-129. Tuckerman, D. B. (1984). Thesis, Stranford, University. Vaidya, S., Sheng, T. T., and Sinha, A. K. (1980).Appl. Phys. Letters 36,464-6. Vaidya, S., and Sinha, A. K . (1981). Thin Solid Films 75,253-9. Vilkelis, W., and Henle, R. A. (1979). Spring Compcon 79 Digest, 285. Ziegler, J. F., and Lanford, W. A. (1979).Science 206,776-788.

ADVANCtS I N ELECTRONICS A N D ELECTRON PHYSICS . VOL 70

Synthetic Aperture Ultrasonic Imagery KEINOSUKE NAGAI Institute of Applied Physics Uniuersity of’ Tsukubu Sakura. lharaki. Japan

I . Introduction . . . . . . . . . . . . . . . . . . . . I1 . Imaging System and Aperture . . . . . . . . . . . . . A . A Problem in Ultrasonic Imaging Method . . . . . . . . B. Aperture . . . . . . . . . . . . . . . . . . . C. Resolution . . . . . . . . . . . . . . . . . . . 111. Theory and Application of Holography . . . . . . . . . . A . Holography and the Synthetic Aperture Imaging Method . . B. Principles of Holography . . . . . . . . . . . . . C. Application to Ultrasonic Imaging . . . . . . . . . . D . Synthetic Aperture Side-Looking Sonar . . . . . . . . . IV . Fundamentals of Digital Ultrasonic Imaging . . . . . . . . A . Propagation of an Ultrasonic Wave . . . . . . . . . . B. Numerical Reconstruction of an Image from a Hologram . . C. HolographywithaBroad-Band Pulse Wave . . . . . . . Appendix . Spatial Fourier Transform of a Spherical Wave . . . V . Properties of a Transducer Array . . . . . . . . . . . . A. Radiated Field from a Single Transducer . . . . . . . . B . Radiation Field from an Array of Transducers . . . . . . C . Combination of Transmitter-Array and Receiver-Array . . . VI . Actual Digital Imaging System . . . . . . . . . . . . . A . Speedy Processing System with a Transducer Array . . . . B. Synthetic Aperture Method Using a Broad-Band Pulse Wave . C . Ultrasonic Computerized Tomography Using the Time-of-Flight VII . Diffraction Tomography as the Inverse Problem . . . . . . . A . Wave in an lnhomogeneous Medium . . . . . . . . . B. Diffraction Tomography with Plane Wave Illumination . . . C. Diffraction Tomography with Fan-Beam Illumination . . . . D . DiffractionTomographywithBroad-Band Pulse Wave . . . References . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . 215 . . . . 216 . . . . 216

. . . . 217

. . . . . . 219 . . . . . . 223

. . . . . . 223 . . . . . . 224 . . . . . 231 . . . . . 244 . . . . . 253 . . . . . 254 . . . . . 260 . . . . . 265 . . . . . 266 . . . . . 267 . . . . . . 268 . . . . . . 272 . . . . . . 215 . . . . . . 282 . . . . . . 283 . . . . . . 287 Profiles . . . 288 . . . . . . 290 . . . . . . 291 . . . . . . 295 . . . . . . 300 . . . . . . 305 . . . . . 313

. . . . . . . .

I . INTRODUCTION The theory and the application of synthetic aperture ultrasonic imaging will be reviewed in the following article. Ultrasonic images of high quality can 215

.

Copyright ((5 1988 by Academic Press Inc. All rights of reproduction reserved. ISBN 0-12-014670-3

216

KEINOSUKE NAGAI

be obtained by the technique that removes diffraction effects and makes the images in focus. Ultrasonic waves play important roles in medical diagnosis, nondestructive evaluation (NDE), underwater observation and earth resource survey. This is because the waves have many advantageous properties: They are non-invasive. They can penetrate through solid and liquid. And, they convey information on spatial distribution of elastic parameters in a medium that is complementary to optical parameters. The disadvantage of ultrasonic waves to optical waves or X-rays is that they often have wavelengths comparable in size to object inhomogeneities. The observation values in these fields are often displayed in a form of images. The use of long wavelengths bring diffraction effects that degrade the images. Many researchers have investigated synthetic aperture ultrasonic imaging which adopts the concept of holography in order to remove the diffraction effects. Their results are reviewed in this article with emphasis on fundamental theory and concept.

TI. IMAGING SYSTEMAND APERTURE A. A Problem in Ultrasonic Imaging Method

Ultrasonic images at present are mainly produced by the pulse-echo method or the pulse-transmission method. The pulse-echo method, as an example, is shown in Fig. 1. The ultrasonic pulse wave is shot from the transducer. The pulse wave travels to the object from which it is reflected. Then the pulse is received by the transducer. The received signal is processed and displayed with a CRT (Cathode Ray Tube) display. The horizontal position and the vertical position of the spot in the CRT correspond to the round-trip time of the pulse wave and the position of the transducer, respectively. The brightness of the spot represents the intensity of the received wave. As the transducer is vertically scanned, the whole image is completed. The ultrasonic wave in the method is assumed to be a narrow beam. If the assumption is satisfied, a good image can be obtained. However, if the assumption is violated, the image is severely blurred. When one intends to obtain the ultrasonicimage of the inside of the object, the main problem is attenuation of the wave. Attenuation associated with wave propagation is represented by a = a, exp( - a x ) , where a is amplitude of the ultrasonic wave, x is the propagation distance, a, is a constant and a is named an attenuation constant. a is usually proportional to the square of the

SYNTHETIC APERTURE ULTRASONIC IMAGERY

217

1 0 ..

In

E!

object

> R

and approximating aR an

-=

1

the following equation is obtained, ag, ikexp(ikR) _ dn

27rR

Substitution of Eq. (66) into Eq. (65) yields the Rayleigh-Sommerfeld

260

KEINOSUKE NAGAI

formula of diffraction: exp(ikr)

2n

ds0

z

Equation (67) explains quantitatively the so-called Huygen’s principle: the spherical wave, exp(ikR)/(4nR), of which amplitude is - 2ikp,(ro) is generated at the point ro on the boundary. The spherical waves from each point are observed at the point r. b. Two-Dimensional Representation The process to obtain Eq. (67) from Eq. (61) is applied to Eq. (63) to yield the representation of the twodimensional Rayleigh-Sommerfeld formula of diffraction. The result is as follows,

“s

Pdr) = j

z

p , ( r , ) W ( k R ) dlo

(68)

B. Numerical Reconstruction of an Image from a Hologram 1. Recording Holograms

Figure 37 shows the geometry of recording a hologram using the piezoelectric transducers. An object is placed on the object plane (z = 0), which is insonified by the ultrasonic wave pi. The scattered waves are measured over the hologram plane ( z = z h ) by planar array of detectors or mechanical scanning of a detector if one has sufficient time for data acquisition. f(xo, yo) denotes the scattering coefficient of the object: pi(xo,yo) and po(xo,yo)are the illuminating wave and the object wave, respectively, on the object plane. Then, (69)

yo) = Pi(xo, Y o ) f (xo, yo)

~o(xo,

Ph(xh,

yh),the object wave on the hologram plane is derived from Eq. (67),

=

fZ: + (xh

- xo)2 + (Yh

- Yo)

2

1

1/2

The set of the measurements Ph(Xh,yh) is called the hologram. The illuminating wave pi(xo,yo) is usually known. Then f(xo,yo) is easily obtained from po(xo,yo)by the use of Eq. (69). po(xo,yo)as well as f(xo, yo)is often called ‘object’. Image reconstruction means reconstruction of the object, that is, f ( x o ,yo) or po(xo,yo) from the hologram.


26 1

Y

t

I

FIG.37. Geometry of recording ultrasonic hologram.

2. Numerical Reconstruction of Images The relationship between the hologram P h ( X h , y h ) and the object po(xo,yo) is presented by Eq. (70). Using this relationship, po(xo,yo)can be calculated from P h ( X h , y h ) . For example, Eq. (70) means that p h ( x h , y h ) is a linear combination of po(xo,yo), or for various P h ( X h , yh), Eq. (70) presents simultaneous linear equations which comprise unknowns po(xoryo). A numerical solution for po(xo,yo)is, of course, obtainable. However, the process takes too much time. The procedure which reconstructs images using a spatial Fourier transform is described here. Spatial Fourier transform P(u, u) of p(x, y) is defined by

-a

and its inverse, P(u, u) exp{i(ux + uy)) du du

(72)

-a.

Note that the sign of i, the imaginary unit, in the temporal Fourier transforms of Eqs. (56) and (57) is reversed in the spatial Fourier transforms of Eqs. (71) and (72).

262

KEINOSUKE NAGAI

It is known from Eq. (70) that the hologram p h is the two-dimensional convolution of the object po and the spherical wave, S(X,Y)

exp(ikR) 4nR

= ___

R

= (x’

+ y2 + z 2 ) l l 2

(73)

that is, y) = -2ikpo(x, y) * S(x, y)

(74) where * means the convolution integral. The property of the Fourier transform presents that the Fourier transform of the convolution of two functions is the multiplication of each transform: ph(x,

Ph(U,u ) = - 2ikP0(u,u)S(u, u)

(75)

where S(u, u) is the Fourier transform of the spherical wave and is derived in Appendix A attached at the end of this section. Now as z = z h ,

Ph(u,u), the spatial Fourier transform of the hologram, is numerically calculated by the use of the fast Fourier transform (FFT) from the numerical data of Ph(Xh, yh). Referring to Eqs. (75) and (76), Po(#, u), the spatial Fourier transform of the object, is obtained by Po(u,u) =

( k z - u z - u 2 ) l l 2 exp{ -izh(k2 k

- u2 - u

~ ) ’ /Ph(u, ~ } u)

(77)

The inverse Fourier transform of Po(u,u) can be again numerically calculated by the FFT, which results in the reconstructed image: m

-m

If the scattered coefficient f(x, y) is required, it can be obtained by

where pi(x,y) is the illuminating wave, which was already described in Eq. (69). The main calculation to reconstruct the image in the procedure is the two two-dimensional Fourier transforms executed by FFT. One is the transform of the hologram and the other is the inverse transform.


263

3. Imaging Procedure Based on Fresnel Approximation

The numerical reconstruction procedure presented in the preceding subsection imposes no particular condition to record the hologram. The procedure is applicable to almost all holograms. However, it requires two two-dimensional FFT. The calculation can be reduced to half, when the Fresnel approximation is valid. The procedure which is accomplished with only one two-dimensional FFT is described here. Consider Eq. (70) at first, m

If the lateral dimension is much smaller than the longitudinal distance from the object plane to the hologram plane, lxn19

Ixh19 IYol?

lyhl 0, the contour of the integral on the complex number plane of w can be taken as shown in Fig. 38. The integral of Eq. (A2) is evaluated by the residue as follows,

V. PROPERTIES OF A TRANSDUCER ARRAY A piezoelectric detector is usually used in ultrasonic imaging methods. As mentioned in Section 111, the piezoelectric detector is the most sensible of all detectors as it can detect a wave field as weak as lo-” W/cm2. However, the piezoelectric detector has the disadvantage that it cannot detect the entire spatial distribution of the wave field at one time. To alleviate this problem, either the single detector has to be scanned mechanically or many detectors have to be arrayed on the plane of

268

KEINOSUKE NAGAl

measurement. The mechanical scan is adopted in scanning acoustic microscopy (SAM) and the like mentioned in Section 111. The array of the detectors is discussed in this Section. The wave field produced by a single transducer is analyzed at first. Then, the result is used to derive the directivity of the field radiated from the array. In the following, a procedure, which combines the small array of the transmitter with that of the receiver to acquire the large number of hologram data needed to reconstruct images, is discussed. A. Radiated Field from a Single Transducer

The wave field produced by a single transducer is considered at first. The field in the unbounded medium is represented from Eq. (61) in Part IV.

where g,(r 1 r,) is the Green's function in the medium which was represented in Eq. (59) and is again cited as follows:

fa

expresses the wave source and appeared in the wave equation (58) as well,

v 2 P" + k2P" = -fo

(92)

The source of a vibrating plane is discussed. The region V, of the thickness 39 shows the region, of which the surface is denoted by So. Integrating Eq. (92)over V,,the second term of the left hand side vanishes as the limit t becomes zero. Then, t surrounding the plane is considered. Figure

the right = -

ss

tfmdS,

(94)

SO

Th integra ds of Eqs. (93) and (94) are equal, as both the equations are equal independently of the shape of V,,

where n, and n2 are unit vectors along the external normals of the plane.


269

FIG.39. Region V, and its surface So surrounding the planar source.

Let n = n, = -nn,,then

Substitution of Eq. (96) into Eq. (90) yields dp,, exp(ikR)

dS0

(97)

SO

The inverse temporal Fourier transform of Eq. (97) represents the wave field in the real domain,

(98) SO

where c is the acoustic velocity. If the plane source vibrates with the velocity u in the normal direction, Eq. (47) in Section IV becomes du

p-=-at

dp dn

(99)

Substituting Eq. (99) into Eq. (98),

where u’ represents the time derivative of u. Equation (100) expresses the wave field radiated from a single transducer.

270

KEINOSUKE NAGAI

When the whole surface moves uniformly, setting

SO

the wave field is represented by h(t) in the form p ( t , r) = h(t, r) * u’(t)

or alternately p(t, r) = h’(t,r) * u(t)

where

* denotes the convolution integral. Radiation Field from Rectangular Transducer

Now the radiation field from the rectangular transducer is discussed, which will be useful in later sections. The geometry is shown in Fig. 40(a). The width 6 of the transducer is, for the sake of simplicity, very small. If the observation point r is placed on the x - z plane, the angle of r from the z-axis is 0 and Irl = R, Eq. (101)becomes

h(t,r) = pb

dx.

2

t

FIG.40. Geometry. (a) Rectangular transducer (shaded area). (b) The x-z plane.


Referring to Fig. 40(b) d x is expressed in relation to d R by

dx

dR sin 9

=7

Using the identity 6

3

(

t --

=c6(R -ct)

and approximation for R >> a, the following equation is obtained: pcb [ R + ( a / Z ) s i n f J 6(R - c t ) d R h(t,r) = 27TR sinH R-(a/Z)sinfJ

Or R h(t,r) %

271R sin H

L

o

-

(:)sin9

if

C

sts

R

+ (:)sin9

otherwise,

schematically shown in Fig. 41.

h(t 1

Pcb 2rRs’s\B

0

FIG.41. h(t) and h‘(t)from the rectangular transducer.

C

272

KEINOSUKE NAGAI

Finally, differentiating both sides, the following equation is obtained:

h'(t,r) is also shown in Fig. 41. As an example, if the plane vibrates sinusoidally, that is, u(t) = U, exp( - iwt),

( 1 10)

the radiated field can be obtained by substituting Eqs. (109) and (110) into Eq. (103): pcb p(t, r) =

[

exi-ico[t

-

R - (I,"n0

)i

2nR sin 0

-exp

i( -io

R

+ (:)sin0)]]

t-

--ipcbUo exp [ - i o ( t n R sin 0

-

k R ) ) sin

where k is the wave number, k = w/c. Normalizing Eq. ( l l l ) , the directivity of the radiated field from the rectangular transducer is presented by sin

W )=

{(F)

sin o}

ka

sin 0

L

The numerical example for a

=

I is calculated and displayed in Fig. 42.

B. Radiation Field ,from an Array of Transducers

The directivity of the radiated field from a linear array of transducers is calculated here. The linear array is placed along the x-axis as shown in Fig. 43. The array consists of N transducers. Each transducer is the same as is discussed in the preceding section. These transducers are driven by a sinusoidal signal with a frequency w. However, the phase of the signal fed to each transducer is delayed by q,, from that fed to its left-hand neighbor. The phase of its right-hand neighbor is


273

FIG.42. The directivity of the radiated field from the rectangular transducer: lDs(0)l,a = 1.

Y

t

FIG.43. The radiated field from transducer-array.

further delayed by qo.Thus, the phase is delayed by qo one by one. The phase of the right most transducer, therefore, delayed by ( N - l)qofrom that of the left most transducer. The directivity of the elementary transducer is D,(d). qo is replaced with 0, by the relation qo = kdsin O,, where d is the interval of the transducer in the array. It will be found that 0, is the deflection angle of the radiated field. The

214

KEINOSUKE NAGAI

radiated field sufficiently distant from the array at the direction I3 from the zaxis is proportional to W(8)and is given as follows:

W ( 0 )= Ds(8) + Ds(13)exp{ikd(sin 8 - sin 0,))

+ . + Ds(13)exp{i(N - l)kd(sin I3 - sin do)} + .

(113)

The first term represents the field from the left most transducer. The second term expresses the field from the second transducer. In this manner, the last term corresponds to the field from the right most transducer. Equation ( 1 13) is easily calculated, as it is a geometric series,

(114)

Normalizing Eq. (114), the directivity of the whole array, D J 0 ) is obtained by,

D,(@ = DS(0)* Da(0)

(115)

where

45 O

goo

e

FIG.44. The directivity of the radiated field from an array of infinitesimally narrow transducers: lDa(0)(,0, = 0,d = 21.

SYNTHETIC APERTURE IJLTRASONIC IMAGERY

275

Equation ( 1 15) shows that the whole directivity is expressed by the multiplication of D,(d) and D,(H). D,(H) is the directivity of the single transducer, which was described in the preceding section. D,(O) represents the directivity of the array when Ds(d) = 1 ; that is, the width of the constituent transducers is infinitesimally small. Equation (116) shows that 6, is the maximum of D,(O), which is interpreted as the deflection angle of the radiated field. Figure 44 depicts the 1Da(8)1 for 8, = 0. Zero of the denominator of D,(O), that is, sin{(kd/2)(sinfl - sin Oo)} = 0, determines the position of the gratinglobes of the directivity. Figure 45 illustrates lD,(d)l for a = il, rl = 2A. As the deflection angle 0" is increased, the peak of the radiation pattern of the whole array, DJH), is relatively decreased by the directivity of the single transducer, D,(8). C. Comhinmtion of' Transmitter- Array and Receiver-Array 1. The Nurnher of Hologram Data

The relationships between the hologram Pl,(.q,, y h ) and the object po(xn,y,,) have been derived and expressed in Eqs. (70) and (86) in Part IV.

--x

Fic;.

45. The directivity of the radiated field from an array of transducers with finite width: = 10 , d = 2i,, N = i.

\Dw(0)\, O,,

276

KEINOSUKE NAGAI

po(xo, yo)

= i~

1

Ph(Xh? y h ) exp(-ikR)

’ Rdxh

dyh

(1 18)

-41

where R = { Z i (Xh - x,)2 -k ( y h - yo)2 } lj2 . These equations present that the hologram at a point is a linear combination of the object points and vice versa. Therefore, the required number of the hologram points is equal to that of the image points to be reconstructed. The number may be very large. If one datum of hologram is obtained by one element of an array, it might be too large to be practical. Methods called ‘super resolution’ reconstruct a large number of object points from a small number of hologram data. However, this method is not always successful. Figure 46 shows the structure of another method which combines a transmitter-array and a receiver-array to collect hologram data. It is plain that the hologram data at different points can be obtained by moving the receiver. However, the same data might be obtainable by moving the transmitter when the receiver is fixed. If this scheme is realized, N, N , hologram data can be collected by (N, + N,) elements of N, transmitter and N, receiver. A piezoelectric transducer can be used as the transmitter and as the receiver. This method has the advantage that the number of elements does not increase as rapidly as that of data. A large number of data could be collected, therefore, by the combination of small arrays.

+

.

FIG.46. Combination of a transmitter-array and a receiver-array to collect hologram data.

277


. . . ................ . . . . . . . .

.

transmitter

receiver 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

O o a o O o o o o o o o o o o o

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o a o o o o o

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(a) ( b) (C) FIG.47. Combination of small arrays equivalent to large array.

The geometrical structure of the array should be carefully examined to realize the method. There are two reasons for this. One is to avoid collecting redundant data. The other is for the numerical processing not to become especially troublesome. In other words, it is desired that the data collected by the structure of Fig. 47(a) can also be acquired by that of Figs. 47(b) or (c). The structure of Fig. 47(a) consists of one transmitter and N, N, receivers. Those of Figs. 47(b) and (c) show the combination of the array of N, transmitters and that of N, receivers. It is impossible to collect exactly the same data of Fig. 47(a) by using the structures of Figs. 47(b) or (c). However, it will be clear that these data are equivalent to the permitted limit of the Fresnel approximation. It will be presented that the data of Figs. 47(b) or (c) are processed as easily as the data of Fig. 47(a).

-

2. Structure of Arruys Let the illuminating wave be the spherical wave emanating from the transmitter at the point (xL,yL,z,),(refer to Fig. 46). Then, from Eq. (69) in Part IV, exp(ikR ,) Po(Xor Y o ) = f(XO,YO), 4nR,

278

KEINOSUKE NAGAl

where R, = {z: + (x, - x , ) ~ + (y, - yo)z}112,and f(x,,y,) is the scattering coefficient of the object. The scattered wave is received at the point (x,, y,, z,) as the hologram which is denoted by ph(xI,y,, xr, yr). Substitution of Eq. (119) into Eq. (117) yields

+

+

where R, = {zf (x, - x , ) ~ (yr - yo)2 112 Equation (120)expresses that the hologram data is the weighted sum of the object points. The weight, exp(ikR,) exp(ikR,) (RtRr) is symmetrically determined by the coordinates of both the transmitter and the receiver. For simplicity, notation related to the y-coordinate are omitted hereafter. Therefore, keep in mind that calculations with respect to x, or x, involve those with respect to y, or y,, respectively. Equation (120) is modified. The variables related to y-coordinate are omitted as just mentioned. The Fresnel approximation is made: R, and R, in the denominator are replaced with z, and z,, respectively.

and

are substituted into the argument of the exponent. Rearranging the result, the following equation is obtained,

where

and F is the operator of Fourier transform.


279

From Eq. (121), the spatial frequency u of the Fourier transform domain corresponds to,

(- + 2)

2n x, 1 z,

u =-

Now, let the intervals of the transmitters and the receivers be d, and d,, respectively: x, = n,d,,

(n, = 0, 1, 2,. . . ,N, - 1)

x, = n,d,,

(n, = 0, 1, 2,..., N , - 1)

( 1 23)

Substituting Eq. (123) into Eq. (122), the spatial frequency obtained is represented by

It is required that the hologram data should avoid redundancy. That is, the points indicated by Eq. (124) are regularly located and any two of them should not be coincident. Referring to Eq. (124), if either

z d, = d N , d , zr

or 2

d, = 2 N,d,

(126)

2,

holds, these points are regularly placed. Now, if Eq. (125) holds, Eq. (124) becomes

These points are regularly located and the data collected at them avoid redundancy. 3. Reconstruction of Images It was seen that the geometrical structure of the arrays should satisfiy Eq. (125) or Eq. (126). Now the image reconstruction procedure from the data ph(xl,y,)collected by the arrays is discussed. From the inverse Fourier

280

KEINOSUKE NAGAI

transform of Eq. (121),the image of the object is obtained as follows,

I Considering that u is expressed at sampling points, substitution of Eq. (122) into Eq. (128) yields

(’ +

x exp:{i

:)x}

where B is a constant. Let x be represented by sampling points

f,(x) and Ph(X,,x,) are rewritten by f ( m )and p,(n), respectively. Substitution of Eqs. ( 1 24), ( 1 27) and ( I 30) into Eq. (129) yields,

x exp

( ~

Equation (131) has the form of the two-dimensional discrete Fourier transform (2-D DFT) and can be calculated by executing 2-D FFT only once (note that the transform with respect to the y-axis is omitted here), which is similar to the case of ordinary data acquisition. That is, the combination of the arrays give rise to no extraneous calculations.

4. Point Spread Function The structure of the N, transmitter and N , receivers which satisfies Eq. (125) or Eq. (126) is equivalent to that of one transmitter and N,N, receivers from the viewpoint of not only the image reconstruction proce-


28 1

dure but also the resolution (point spread function). This is proved in this subsection. Consider a point as the object, at x,, f,(.w) = &x - x,) The hologram is given by the substitution of Eq. (132) into (121), Ph(-'Ct, X r )

= -

( 132)

~

Further substituting Eq. (133) into Eq. (129), the reconstruction image is obtained.

Using Eq. (124), Eq. (134) is calculated to obtain

(+I i ) ( x 2 - xi)}

f,(x) = Bexp{ - i E1 z,

n N d , ( x - x,) . nN,d,(x - x,) sin Jzt Azr sin nd,(x - x,) sin . nd,(x - x,) i.zt JZ,

sin X

(135)

Now, introducing the Fresnel approximation, x

.

X O

sin 8, = -,

sin H,, = -,

Z,

Z,

From Eq. ( 1 16), the approximated normalized field from the transmitter array D,(x) and that of the receiver array D,(x) are obtained by,

282

where k = 21-42 is used. Referring to Eqs. (137) and (138), Dc(x),the approximated normalized field of the combined arrays, is rewritten from Eq. (135), as follows, (139)

D C ( 4 = Dl(X) DS(4

From the relationship of Eq. (125) which characterizes the arrays, the following equation holds,

Then, the numerator of Eq. (138) is coincident with the denominator of Eq. ( 1 37). That is, in Eq. (135) or Eq. (1 39), the numerator cancels out the denominator. In other words, the grating-lobes of the transmitter-array are removed with the zeros of the receiver-array. Equation (135), therefore, is represented by sin nNtNrdr(x - xo)

{

1

/iZr

DJX) =

ndAx - xo) sin{ Azr

}

(141)

Equation (141) represents the reconstructed image of the point object given by Eq. (132), which is interpreted as a point spread function of the imaging system. This also expresses the directivity of the combination of arrays, which is coincident with that of the array with the interval of element, dr, and with N J , , the number of elements, from the Fresnel approximation of Eq. (116). VI. ACTUALDIGITAL IMAGINGSYSTEM The ultrasonic imaging system adopts piezoelectric transducers and digital techniques to attain high sensitivity and to speed up processing. The procedures which implement the system with electric circuits are discussed by Harmuth (1979). It is clear that the system becomes inevitably large and complex if one intends to obtain image in real-time. Simpler systems are discussed here, which use partially, for example, mechanical scanning and/or a general purpose computer.


283

A . Speedy Processing System with a Transducer Array

A speedy processing system can be realized by adopting the transducerarray to shorten the data-aquisition time. These systems are discussed first. Figure 48 shows the real-time imaging system proposed by Suarez et al. (1975).This system can detect the ultrasonic field as weak as lo-'' W/cm. The ultrasonic chirp signal (800 KHz bandwidth centered at 2 MHz) is collimated by a plastic lens, refer to Fig. 48(a), and the signal is scattered by the object in the auxiliary tank. The field is collected and focused by two polystyrene lenses onto the 192-element piezoelectric linear array. Thus, one line of the image is obtained. A pair of polystyrene prisms are counter rotated to sweep the ultrasonic field across the array. The image field is electrically scanned in the direction of the array, but it is mechanically scanned in the perpendicular direction by the prisms. The resultant image consists of approximately 400 lines. The image data is stored and displayed in a television screen. One of the images is shown in Fig. 48(b).This is the transmission image of the hand which shows the metacarpal-phalangeal and interphalangeal joint. The linear structures visualized between the bones are probably related to muscles, tendons, and neurovascular bundles. Figure 49 shows the ultrasonic imaging system using multibeam scanning proposed by Nitadori et ul. (1980). This system uses ultrasonic waves with about a 200 KHz operating frequency to view underwater of the near-shore bottom where a man is working. Figure 49(a) indicates the underwater unit consists of transducer-arrays. A 4 x 4-transmitter-array is seen outside and 32 x 32-receiver-array is inside. These relatively small arrays are combined to collect a large number of hologram data. The theory has been already described in Section C of Part V and it is applicable to the unit. These arrays are placed on the same plane; that is, as for the structure shown in Fig. 46 in Part V, 2, =

z,

( 142)

From Fig. 49(a), it is clear d , = N,d,

(143)

where d, and d, are the interval of the transmitters and that of the receivers, respectively, and the number of receivers, N , = 32. Referring to Eqs. (142) and (143), it can be seen that this structure of arrays satisfies Eq. (125) in Part V. Then, the hologram data collected by this system are equivalent to those collected by the combination of one transmitter and 128 x 128 receivers as discussed.

L

(b) FIG. 49. Ultrasonic imaging system using multibeam scanning (Nitadori et a!., 1980). (a) LJnderwater unit. 4 x 4-transducer-array is seen outside. 32 x 32-receiver-array is inside. (b) Ultrasonic image of a bicycle taken at a range of 3.6 m with the operating frequency of 200 kHz.

FIG.50. Underwater imaging system (Shibata and Koda, 1986). (a) Coaxial circular spherical arrays (CCS array). The outermost elements with horns are the transmitters and the inner three are the receiver-array. (b) Underwater vehicle equipped with CCS array. (c) The image with operating frequency of 95 kHz.


287

It was reported that the image could be reconstructed in 2 seconds by the computer. Figure 49(b) shows an example of the image. This is the image of a bicycle taken at a range of 3.6 m. Considering the long wavelength of 7.5 mm, the quality of the image is very high. Figure 50 shows the underwater imaging system too. This system was proposed by Shibata and Koda (1986). The structure of the transmitter-array and the receiver-array is indicated by Fig. 50(a). They are also set on the same plane. 8 transmitters with horns attached are placed in the outermost circle. Receivers form three inner coaxial circular arrays. Each circular array consists of 16 receivers. They are considerably sparse arrays. However, the radii of the four circles are so cleverly selected that the grating-lobes of an array may be canceled out by the zeros of another array. Thus, all the grating-lobes at small angles from the axis are removed. The canceling of the grating-lobes by the combination of linear arrays was discussed in Section C of Section V. The method extends to the circular arrays. The system has the distinctive feature of a small number of transducers. From the feature, the electric circuit could be implemented with a relatively simple set of delay-lines using charge coupled devices. Figure 50(b) shows the arrays with which the underwater vehicle is equipped. Figure 50(c)shows the image which is reconstructed by the system. The operating frequency is 95 KHz. The object is a character 'A' of 50 cm long and 5 cm wide which is made of aluminum. B. Synthetic Aperture Method Using a Broad-Band Pulse Wave

In order to attain the radial resolution, a broad-band pulse wave is used. The aperture is synthesized in real domain. An example of the technique was reported by Ishii and Sasaki (1983). It is shown in Fig. 51. The data in the example were collected by mechanical scanning of a transducer instead of using the arrays. The transducer is used not only as a transmitter but also as a receiver. Thus, R in Eq. (89)in Part IV, the distance from the object point to the receiver, should be replaced with 2R, the round-trip distance. A metal block is drilled to have 5 holes as shown in Fig. 5l(a). The transducer is scanned along a line on the upper surface of the block. At the spatial sampling points, the broad-band pulse wave is transmitted. Then, the scattered wave is received and recorded as time series. The image reconstruction is based on Eq. (89); that is, the data collected are delayed by the corresponding round-trip time from the transducer to the object point, and summed. The result of the two-dimensional cross-sectional image is shown in Fig. 5l(b).

288

KEJNOSUKE NAGAl

transducer

22

FIG.51. Synthetic aperture method using pulse waves (Ishii and Sasaki, 1983). (a) Specimen. (b) Image.

C. Ultrasonic Computerized Tomography Using the Time-of - Flight Profiles Computerized tomography (CT) which obtains clear x-ray images of the cross-section of human body has achieved great success. However, this technique may not be directly applicable to ultrasonic imaging. Diffraction effects cannot be neglected in ultrasonic waves but cause no problem in x-rays. The image of the ultrasonic CT can be reconstructed from the time-offlight profiles, where the time-of-flight means the propagation time from the transducer to the receiver through the specimen. The profiles are measured and then the two-dimensional distributions of the ultrasonic velocities are painted out. In this method, diffraction effects are minimized since the earliest arrival time is most probably that of the straightest ray path. The method is not a synthetic aperture method, but it is a promising new method. Figure 52(a) shows the geometry of ultrasonic transmission for algebraic reconstruction. The time of flight z j is measured by two opposing transducers on either side of the specimen in the fluid.

where Lijis length of ray path j in region i, uti is the velocity in the specimen in

,/” ” ”

S C A N LOCUS

( W a t e r - F i l l e d , 2 2 ”C )

I cm

FIG. 52. Ultrasonic computerized tomography using the time-of-flight (Greenleaf ct ul., 1975).(a) Geometry of ultrasound transmission for algebraic reconstruction. (b) Reconstruction of relative propagation delay within canine heart (left) compared to photographs of sections through corresponding levels (right).

290

KEINOSUKE NAGAI

region i, u w , the velocity in the fluid, and D is the distance from the transmitter to the receiver. Usually the differenceAzj between the arrival time zw through fluid and the arrival time z j through specimen is measured,

The measured data at sampling points are collected by scanning rectilinearly the transmitter and the receiver. Then, the object is rotated by an angle of AO. The measurements are repeated. A large number of AT, are obtained by the rotation and the rectilinear scanning. Using the Azj, ( l / u w - l/ut,), the unknowns, in Eq. (145) are found at each value of i by solving the simultaneous linear equations. As the number of the measurements increases, the quantity of the images becomes high since noises are removed by the measurements. Figure 52(b) shows the image reconstructed by the method. This is the profile of relative propagation delays within a canine heart for two separate transverse levels separated by 2 cm (left)compared to a photograph of sections through corresponding levels (right). Pulses were propagated through the tissue at each of 256 equally spaced points along the 12 cm scan and were digitized with a temporal resolution of + 10 nanoseconds. This is repeated at each step of 36 angles of view separated by 5”.

VII. DIFFRACTION TOMOGRAPHY AS THE

INVERSE PROBLEM

The term “ultrasonic imaging” has been used without clear definition in this article so far. Its meaning is as follows: An object is illuminated by a ultrasonic wave at first. The scattered and diffracted wave is then measured. The form of the object is evaluated by the measurements, and it is finally displayed in the visual fashion. The evaluation of the form of the object from the scattered wave has been studied for a long time as inverse scattering problems. Considerably exact solutions of some problems are obtained. An image of high quality might be obtained if such inverse solutions are used, because the scattering and diffraction effects are removed from the image. The evaluation requires the measurements of the wave over a certain region, which corresponds to the synthesized aperture. Diffraction effects degrade the image reconstructed by ultrasonic tomography. The ultrasonic tomography which takes diffraction into account


29 1

and compensates its effects is called diffraction tomography. The diffraction tomography which uses the inverse scattering solution to reconstruct the clear image is described here. Two-dimensional problems are discussed here. The configurations are assumed to be constant with respect to the y-axis (d/iiy = 0). Therefore, the two-dimensional image similar to the B-scan is obtained. The discussion could be simply extended to three dimensions, and a three-dimensional image could be reconstructed. The assumption of two dimensions is necessary, however, for the discussion to be simplified and easily developed. A. Wave in an Inhomoyeneous Medium

1. Wane Equafion

The object in which we are interested is like a human body. It exists in the homogeneous medium. Its elastic properties, the density p and the compressibility K, are slightly different from those of the surrounding medium. The wave equation in an inhomogeneous medium is derived at first. The coordinate system is shown in Fig. 53. The object, the inhomogeneous medium, is assumed to exist only inside the region S. Inside S, that is, p and K are functions of the position but outside S are they constants, po and K ~ .

Z

FIG.53. Coordinate system.

292

KEINOSUKE NAGAI

Where there is no wave source of the body force, substituting F = 0 in Eq. (53) of Part IV, the following equation holds, aU

p-

at

=

-vp

Equation (54) is cited here,

Equation (147) is differentiated with respect to t. Substituting Eq. (146) into the result, the following equation is obtained 1

at

va-vp P

Now, the following equations are defined using p and ti,

It is clear that I, and

fid

are zero outside S . From Eqs. (149) and (150), K = Ko(Bc

+ 1)

(151)

Substituting Eqs. (151) and (1 52) into Eq. (148) and rearranging the result,

i 2 the acoustic velocity in the homogeneous medium. where co = ( p o ~ o ) - L is Using the definitions of the Fourier transform pair, Eqs. (56) and (57) in Section IV, Eq. ( 1 53) becomes

where k = o / c o , wave number and

The solution of the wave equation in two dimensions is obtained from Eq. (63)


293

in Section 111 as follows,

where two-dimensional Green's function is

R = Ir - ro(

(158)

H t ' : zero order Hankel function of the first kind. f, in Eq. (58) of Section IV expresses the wave source of the body force but in Eq. (155) of this section, it represents the source of the second wave caused by the inhomogeneity of the medium. In other words, the incident wave is scattered by the inhomogeneity and fl,is the source of the scattered wave. When the boundary is taken at infinity, the line integral in Eq. (156) expresses the incident wave pi. Substituting Eq. (155) into Eq. (156), a term becomes

Since

td

vanishes on L , Eq. (156) becomes finally,

2. The Born Appraximation cind the Rytov Approximation The total field P , , ~is the sum of the incident wave pi and the scattered wave phdue to the inhomogeneity of the medium, Po = Pi

+ Ps

(161)

294

KEINOSUKE N A G A l

Being compared to Eq. ( 1 6 l), Eq. (1 60) becomes

The relationship between p s and fiC or fid is not clear, because Eq. (162) comprises p m in the integral. However, assuming (163)

IPiI >> Ips1 p , in Eq. (162) could be replaced with pi to obtain

+ l d v g o * vPi)

P s ( ~= ) l[{rck2pigo>

ds

( 164)

This approximation is called the Born approximation and is applicable to the case where scattering is weak (refer to Eq. (163))and multiple scattering can be neglected. The other approximation called the Rytov approximation is as follows. The wave field is represented in the exponential form p = eq (165) And q is represented by the sum of the component due to incident wave and the component due to the scattered wave, 4

=

qi

+ 4s

where qi is expressed by the incident wave of Eq. (161), Pi = exP(qi)

(167)

P = Pi exP(qs)

( 168)

that is, Assuming )q,1 = y 11 =

Sr

M 'w

+ u', = M'

I

-

E + c = h on ?R,

i n Q,

(30) E r = ij on JR,

We may now gather together all of the notions developed thus far and express thcm schematically a s shown in Fig. 8.

DUAL A N D COMPLEMENTARY VARIATIONAL TECHNIQUES

335

We can now clearly see the choices. It has been shown that each canonical set can be solved in one of two ways, giving in all, four possible solution schemes. They are: (i) Standard Primul Method

w = Mu, Tu + us = u in R, Bu = g on dR,, enforce: minimize: O(u), Taw = p in R, B'w = h on 80,. solves:

This is a direct equilibrium method, as it gives a weak solution to the equilibrium equation. (ii) Complementary Primal Method

enforce:

w

=

maximize: Z(w), solves:

1

Mu, Taw = p in 0, B'w = h on do2,

Tu + us = u in R and Bu = g on 130,.

(32)

This is an indirect compatibility method, since it gives a weak solution to equations which in turn satisfy the original compatibility equations. (iii) Stundard Dual Method

enforce:

w = Mu, Sr

+ w, = w in R, Er =

on do,,

minimize: O ( r ) , solves:

S"v = 4 in R and E'u

=

h on dR,.

A direct compatibility method. (iv) Complementary Dual Method

enforce:

w = Mu, S"v = 4 in R, E 'v

= hon

dR,,

maximize: E(u), solves:

Sr

+ w, = w in R and Er =

(34) on dR,.

An indirect equilibrium method. This concludes the general development and we shall now show how it can be directly related to problems in electromagnetism.

336

J. PENMAN

VI. MAGNETOSTATICS We begin by examining the magnetostatic field. The magnetostatic field is such that the following equations must be satisfied in the domain of the problem, denoted by Q.

-

(i) V B (ii) B

-compatibility

=0

= pH

(iii) V x H

-constitutive

=J

-equilibrium

Where B is the magnetic flux density, H is the magnetic field intensity, J is the current density, and p is the permeability. Relating the above operators to the general operators used in the previous section, it is easy to see that the following equivalences are implied,

S " = V . and

T"=Vx

-

Therefore, by considering the adjoints of the vector operators V and V x,

S=-V

and

T=Vx

Now, introducing A, the magnetic vector potential, defined by

B=VXA we have

V.B=V.(VXA)EO, and so the compatibility equation in Q is satisfied. Also, for two dimensional fields in which the current density can be expressed as, j = Oi

+ Oj + jk

it is sufficient also to define A by A

= Oi

+ Oj + Ak.

That is, the vector field has only one unknown, namely the z directed component of A. We also choose to introduce another potential $, termed the reduced magnetic scalar potential, which is defined by,

H=H,-V$ where

V x H = ( V x H , ) + ( V x (-V$) =J+O

DUAL AND COMPLEMENTARY VARIATIONAL TECHNIQUES

337

This means that the equilibrium condition is automatically satisfied in R. Following Penman and Fraser (1984), H, is called the reduced magnetic field intensity and represents the magnetic field produced by the current carrying elements of the system. The -V$ term represents the “deformation” of the H field, from H,, due to the effects of magnetic materials. Also, for the primal equations, we see that

M = v , the reluctivity and

us = 0.

For the dual equation we have, M = p, the permeability us = H,

and

.

Associated with the operators V and V x are their boundary operators Ef

=

-n.

Bt = - n x

and

therefore we may define their adjoints as

E=I

and

B = -nx,

noting that in two dimensions n = n,i

+ nyj

The manner in which these boundary conditions arise is discussed later in this section. Note though that the boundary operator B acts on the magnetic vector potential. That is, -nxA=g

If we remember that A has only a z-component and use the form of n given above then we have (-n,i

+ n,j)A

=g

Since A is the only unknown in this equation, it is best expressed as A=g

where g, although still a vector is now in the z direction. The boundary conditions for two dimensional magnetostatic problems are normally specified using the following sign convention for the boundary operators, n.B=h

$=g

nxH=h

A=g

338

J. PENMAN

Taking this into account and substituting the magnetostatic terms into the general functionals given by Eqs. (12) and (16), the following system of equations and functionals is obtained: A. The Primal Set

Primal Canonical Equation A = g on

B=AxA H

=

VB

an,

inn

(35) nx H

J=VXH

= h on

aR,

for which the appropriate functionals are,

Standard Primal Functional (V x A ) . v ( V x A ) & with A

=g

specified on

h * A d ( a R ) (36)

an,.

Complementary Primal Functional

with V x H = J in R and n x H = h on 8 0 , both satisfied. We can express this functional in a more useful form by introducing the magnetostatic scalar potential, t+b, defined by H = H, - V$. This gives, r

r

with II/ = 9 specified on 8QZ.

B. The Dual Set We can proceed in an identical way to that above, again using the general expression to produce a dual set.

Dual Canonical Equation

H

= H, - Vt+b

$=gonaR,

B = pH O=V.B

]

n .B = 6 on 8 0 ,


Standard Dual Functional @($) =

with $

-

ln

4

-

(H, - V$) p(Hs - V$) dR

+

Ian, h3/ d(dR)

339

(39)

specified on (3Q2,.

=

Complementary Dual Functional

-

=(B)=

B.vBdQ+

-L

In

-

H,.BdRIn

-

L2

@-Bd(dR)

with V B = 0 in R and n B = h on an, enforced. If we replace B by V x A in this expression we arrive at,

-

z(A)=-i

In

(VxA).v(VxA)dR+

In

H,.VxA-

L

Sn.Vd(d0) (40)

with A = g specified on dR,. The structure of the magnetostatic field can be summarized by Fig. 9 and certain points now emerge. When we wish to solve a magnetostatic field problem, several choices are open to us. First we may choose to solve either the Primal Equation, V x VV x A

V x VV x A

=

J

V . jc(H, FIG. 9.

= J,

~

V$) = 0

The magnetostatic system.

340

J. PENMAN

or we use the Dual Equation,

V * p(Hs = V$) = 0 Once this choice has been made we may then use the functionals associated with each equation. The choice is Eqs. (39) and (40) for the dual set. All four forms will provide an answer but a further benefit is that when considered in pairs, as given above, these pairs provide solutions which bound the exact solution. C. A Magnetostatic Example

We can illustrate this by means of an example. Only the primal set is considered, i.e. the functionals described by (36) and (37) will be extremized. This is done using the finite element method, and for a series of uniform meshes with an increasing degree of refinement. This course is chosen because it is generally the case that an increase in mesh refinement will improve solution accuracy, and hence the convergence from above and below to the exact value of field energy should be easily seen. The problem involves a simple magnetic circuit as shown in Fig. 10. The figure represents one quarter of the complete system and gives the appropriate boundary conditions. The four meshes with varying degrees of refinement are

air

. - . . . .. .. .. .. .. .

... ... . iron . . . . ' '. *

*

nxH=O FIG. 10. A magnetostatic problem.

DUAL AND COMPLEMENTARY VARlATlONAL TECHNIQUES

341

FIG. I 1. Meshes used for finite element solutions. (a) 64 nodes. (b) 225 nodes. (c) 484 nodes. (d) 841 nodes.

shown in Fig. 1 1, whilst the solution fields are illustrated in Fig. 12. It should be noted that in this problem we have used a total scalar potential function, defined by H = -Vx, in the iron region to avoid the cancellation errors discussed by Simkin and Trowbridge (1979, 1980). The convergence of the functional values for the standard and complementary forms is shown in Fig. 13. The system energy is clearly bounded, and as expected, the error reduces with mesh refinement. The advantages are immediately recognizable

342

J. PENMAN

FIG.12. Solution fields. (a) $-potential. (b) X-potential.(c) H field. (d) A-potential.

too, for it is now possible to solve a relatively simple problem twice, using both functional forms, and average the answers. This will give an accurate value that would otherwise have required a much greater computational effort. The additional benefit of being able to quantify error is obviously highly desirable. VII. THEELECTROSTATIC FIELD

We can now consider the electrostatic field by noting that the following equations must be satisfied in the domain of the problem.

I

I

I I I I

I

I I

I I I

I \

\\

\

J. PENMAN

344 (i) V x E (ii) D

=0

-compatibility

= EE

-constitutive

-

-equilibrium

(iii) V D

=p

(41)

where D is the electric flux density, E is the electric field intensity, p is the charge density, and E is the permitivity. If we introduce an electric scalar potential 4 by letting

E=-V$

(42)

then this automatically satisfies the compatibility condition, since - V x (V4) = 0. Similarly introducing an electric vector potential C, such that

D=D,+VxC

(43)

where

V-D=V-D,+V-VXC =p+o

automatically satisfies the equilibrium condition. For two dimensional problems C, like the magnetic vector potential, requires only the z component to model the system correctly. Following the detail presented earlier for magnetostatics, we see that the boundary conditions for the electrostatic problem are, n x E =h

-

n D

and

=h

which gives C = g, which gives

4 =g

Also, inserting the electrostatic quantities into the general equations and functionals gives the following electrostatic forms.

A . The Primal Set

Primal Canonical Equation -V$

E

=

D

= EE

p=V*D

4 = g on dR, (43)

.

n D = h on dR,

and this generates the following pair of complementary functionals:

D U A L A N D C O M P L E M E N T A R Y VARIATIONAL T E C H N I Q U E S

345

Standard Primal Functional

(V4) * &( - V4) dR with

&(an)

(44)

4 = y specified on ifR,.

Complementary Primal Functionul 1

(Ds + V x C)*-(D, + V x C)$R &

with C = g specified on dR,.

B. The Dual Set As with the magnetostatic set it is also possible to develop a dual system which is classified by the equations given below.

D u d Canonical Equation

D=D,+VxC

C

1

E=-D

=g

on dR,

in R

8

n x E=hondR,

O=VXE

for which the following functionals are appropriate: Standurd Dual Functional

O(C) = f with C

=g

I

1

(D, + V x C).-(D,

specified on dR2.

Complementary Dual Functional

z(+)= with

-

+

+ V x C)dR +

E

I

-

( - Vd) c( - v4)dR

4 = y specified on dR,.

+

lo -

Ds (v4)d~

-

L

6Cd(dR)

(47)

gn x ( - v4)$(do) (48)

346

J. PENMAN

FIG. 14. The electrostatic system.

The full structure of electrostatic systems together with its boundary conditions is illustrated in Fig. 14. Again we see that we have two choices of system equation. The standard form, v ( -EV4) = p and the dual equation,

-

1

V x -(Ds &

+ V x C) = 0

Both of these yield pairs of functionals which give error bounds. It also illustrates the striking symmetry inherent in electric and magnetic fields. We shall see later how it is possible to combine Figs. 9 and 14 to provide a complete structure for the electromagnetic field. First, we illustrate the provision of error bounds in the electrostatic field by means of an example. C . An Electrostatic Example

The problem considered here is that of a point to plane discharge. The system geometry, which is axi-symmetric is defined in Fig. 15, and as before a series of solutions were computed for a set of increasingly refined meshes. The


347

4=0

n.D=O

n.D=O

qJ=O FIG. 15. An electrostatic example.

solution fields produced using the two functionals generated by the standard form, given above, are illustrated in Fig. 16, whilst the energy convergence curves are given in Fig. 17. The provision of error bounds is once again clearly illustrated, and all of the conclusions drawn in the discussion of the magnetostatic problem solved in the previous section are obviously still appropriate.

FIELD VIII. THE ELECTROMAGNETIC A . The Structure

In this section, we will show how time varying systems can also be incorporated into our general scheme, and how our schematic representation can also be extended to accommodate this additional complexity. Much of the general structure proposed here first appeared in Penman and Fraser (1984) and was published in the Proc. I E E (Pt. A), 131, 1984. When sources of the electric and magnetic fields are subject to variation with time then the two fields became coupled. This is expressed in the time

348

J. PENMAN

FIG. 16. Solution fields. (a) $-potential. (b) C-potential. ( c ) Orthogonal nature of potentials in charge free regions. (d) E field.

varying forms of Maxwell's Equations,

VXE=-and

aB at

dD VXH=J+at

(49) (50)

I

I

I I I

I I I I I I I

I

I

I I

.

J. PENMAN

350

We can extend (49),for the sake of symmetry, to include magnetic current @. This may be regarded as the transport of free poles which will of course

always be zero. Its inclusion does however allow us to write,

VXE=@-or

l3B at

(51)

VXE=@+Bd

where

ad,the

magnetic displacement current

= - aB/dt.

Also, we define

@t=@+@d.

Equation (50) is often expressed in the form,

VxH

=J

+ Jd

(52)

with Jd, the electric displacement current = aD/at, and we may also let J, = J + Jd. It is now simple to include time variation into our diagrammatical representation. Consider for the moment the combination of the top loop in Fig. 9 combined with the lower half of Fig. 14. We can connect these diagrams through the mechanism of time differentiation by forming the three dimensional diagram shown in Fig. 18, which assumes for simplicity that no free

/1

FIG. 18. The time linkages in the electromagnetic field.


35 1

charges or poles are present. It may be seen that the links joining the ‘electric field plane’ to the ‘magnetic field plane’ are not space differentials but time differentials and that moving from the electric to the magnetic field implies the operation a/&, whilst moving in the reverse sense implies the adjoint of this operator, -a/&. We can extend our representation further by noting that we previously split the magnetic field H into two vectors thus,

and we may compare this with the usual expression for the non-conservative electric field, dA C7t

--

v+

(54)

Clearly we may write,

H

dC at

=-

(55)

and this confirms our earlier choice of the name electric vector potential for the quantity C. These additional relationships allow us to expand Fig. 18 to the fuller form shown in Fig. 19. For completeness, we have added the links between the electric current and field, and the corresponding magnetic variables. The result is a striking symmetry of structure in which our previously developed notions of equilibrium and compatibility, and primal, dual, and complementary still hold. It also shows that when we solve electromagnetic field problems we are effectively solving both compatibility, or both equilibrium equations, whereas for static fields only one need be considered.

B. The Primal and Dual Equations With reference to Fig. 19 we can identify the following principal relationships. (i) Compatibility Equutions

V-B=O V x E = @,

aB dt

= --

in the absence of free poles

FIG. 19. The electromagnetic system.


353

(ii) Constitutive Equations =p

B

H

D = EE

(iii) Equilibrium Equations V.D=/, V xH

= J, = J

aD +at

From this set of expressions we can now construct the primal and dual equations. The Primal form is developed in the following way: if B is defined by VxA=B,

V * B = V * V xAGO,

then

.

i.e. V B = 0 is satisfied.

Also,

V x E = V x (-V4)+V x

(

s)

--

= o - - c?B at

Thus both the compatibility equations are satisfied. The constitutive relationships may be written as,

and the equilibrium conditions as,

Equations ( 5 6 ) and ( 5 7 ) may now be combined with the expressions for B and E,

[-:I=[

v x

$

0

v][;l

J PFNMAN

3 54 to give the Primal equation,

Examination o f Eq. (59) shows that it has precisely the general Form we developed earlier, i.c.

7 " M ( R r + us) = p in which 0, = 0. This implies that the standard solution to the primal form in electromagnetic systems is in A and (I),and it is a direct equilibrium method since i t satisfies ( S X ) and directly solves Eq. (57). The complementary primal form is therefore in H and D and is an indirect comptability method, which solves Eq. (58). We shall develop complementary fuiictionals that achieve these two solutions later. First, for completeness, we develop the dual equation for the electromagnetic system. We begin by remembering that D has been defined as,

D=D,+VxC

(60)

I t is therefore simple to satisfy the condition

-

V I)

=

-

V D,

=

by direct iiitcgration.

p

Also since,

V x H

=

V x H,

=

v

x

+ V x (-V$)

(g).

then using (60) and (6 I)we have

and if the time variation of D, is such that (7 D,

-=-J

6/

then V x H = J + ?D/i'r is al\o satisfied. Thi\ mean\ i f we emure that V D, = p and (7D,/(7t= equilibrium conditions are automatically satisfied.

-

-

J , then the


The remaining conditions are,

L

J

and

[:]

=

a at

The above equations may be combined to yield the Dual equation,

Again we see that it has the general form given above, and is therefore amenable to our variational treatment. Note too, that posing the problem in (C, $) is a direct compatibility method giving the standard dual form, whilst using (B,E) is the indirect equilibrium approach, for the complementary dual form. The structure is thus preserved for electromagnetic systems. We can still develop a standard or a dual partial differential equation, and they can both provide two functionals which provide error bounds. In the specific cases of Eqs. (59) and (64), the extraction of appropriate functionals is a significant task. As previously, in the case of the static systems, we shall only consider the standard form, and limit our practical illustrations to two dimensional steady state eddy current problems. C . Complementary Function& for Eddy Current Systems

We begin by returning to the primal Eq. (59) for which it is easy to show that if there are no free charges and no displacement currents, it reduces to the usual form of the diffusion equation in A , dA V x v(V x A ) = -a-+ J (65) at together with the condition, V.A=O Furthermore, in the absence of eddy currents, that is, in a region of zero conductivity, it yields the usual form of the wave equation. And, in the absence

356

J. PENMAN

of time variation, it reduces to the standard magnetostatic and electrostatic partial differential equations. This highlights the fact that Eq. (59) is very general indeed, but we now limit our attention to the particular case where Eqs. (65) and (66) are applicable. Furthermore, we will restrict ourselves to situations where the vector potential and the impressed current may be defined by, A

=

Ak

J = Jk,

and

where k is the unit vector in the z-direction. For a steady-state sinusoidal analysis A(x, y, t ) = A(x, Y )cos(wt + O(x,Y ) )

and

JAx, Y , t ) = j,(x, Y )c o s ( 4

where .& and jsrepresent the peak values of A and J, respectively, o is the angular frequency of the current, and 0 is a phase angle. Equation (65) can now be more conveniently represented in complex number notation. That is, we use the identity cos(ut) = $(ejwt + e-j"' ) and substitute it in (65) to give

v + vv x AC+ j o o R c = 3, v x vv x Rc* = jwoAc* = 3, where AC= Ads is a complex function, and A'* denotes the complex If the following notation to denote real and imaginary conjugate of components is used:

A'.

and

Rc = Ack = u' + ju" Ac* = ;ic*k = u' - j u t ' 3, = 3,k = f' + jo,

then (66) may be rewritten as

This can of course be expressed in our operator form as

TaMTu = p(u)

DLlAl. A N D ~ ' O M f ' l . l ~ M t ~ N ' I ' AVKA~K I A ' I I O N A I _ f 7 : C ' H N l ( ) ~ l f ~ S

357

Here ii is a n ordered pair o f the rcal functions ( u ' , ~ " ) ,and they replace tile complex vcctor potential A'. Wi t h h oin oge ti o u s bo LI t i d a ry co t i d I t io t i s, t his opera I o r eq u a t i o n c;i t i be M ritten i n its canonical form as -

7" = F. _ -

M 1' = ,1.

1

I-"\(. = j ( G )

Ui

=0

on CQ,

in 0

(6%)

B

I(. =

0 011 ?Q2

We set' t h a t Eq. (66) has thcrcfore been transformed i n t o a form to which complementary wriational principles can be applied, since it is identical in form to I q . ( X j . The pliysiciil significance of I', 113, and p can again be highlighted, thus.

Standard and complementary functionals f o r (6X)can therefore be constructed i n the manner described in Section 4" and 4D. The resulting functionals are:

358

J. PENMAN

which becomes

s

E(H) = - 4 pH’ * H ‘ d R +

+-

‘s

UC7

V x

s

jiH” * H ” d R

Ur7

J * ( V x H ” ) d R (73)

The functional represented by expression (72) is the usual form for the 2dimensional eddy current problem, and together with expression (73) it provides a complementary pair of functionals which bound the exact value of the functionals from above and below. D. A n Electromagnetic Field Example

In this type of problem we have a further variable to consider, namely that of frequency. We have therefore solved the problems illustrated in Fig. 20 for the range of finite element discretizations also shown in this figure, and for a spread of frequencies. By splitting the functionals into real and imaginary parts it is possible to equate these components to the stored magnetic energy and the energy dissipated. This, in turn allows the inductance and resistance to be calculated for each condition. The results of how each of these parameters vary with mesh modification and frequency are shown in Figs. 21 and 22. It can be seen that bounded solutions are still provided, but the absolute accuracy also depends on how well the meshing models the skin depth. It is clearly noticeable that a small number of elements in the iron region results in an increased error for each individual solution as the frequency increases. Nevertheless the accuracy of the average value is still good under these conditions. This can obviously result in a much improved computer utilization, for it allows a greater degree of confidence to be placed on the results. For completeness, the computed real and imaginary components of the standard potential, A, are shown in Figs. 23 and 24, for each frequency, at the highest level of mesh discretization.

IX. CONCLUDING REMARKS In the preceding pages we have presented a more or less self-contained development of how complementary variational methods can be applied to problems in electromagnetism. A general structure has been introduced which


359

air A=O

t

FIG.20. An electromagnetic problem. (a) Problem definition. (b) 93 nodes. (c) 238 nodes. (d) 471 nodes.

leads to a powerful schematic representation of the interrelationships among the various field quantities. Examples of the application of the method via the finite element method have demonstrated its utility and power, and may hopefully provoke others to find yet wider areas of application for these techniques.

xlO-8

5

5

4

F.

E

\

I

" W

u Z < t-

u

3

n

5 S

W

I-

v) )v)

200

3

5

5

FIG.21. Bounds on system inductance as a function of mesh discretization and frequency.

V_ A R_ I A T I_ O N _OF_RESISTANCE WITH FREQUENCY ~ __~10-5 40

- 0 0

5

10

15

20

5

xlO' FREQUENCY

(

Hz )

364

J. PENMAN

ACKNOWLEDGMENTS The author would like t o warmly acknowledge the debt he is due, in the development of this work, to Dr. J. R. Fraser and Dr. M. D. Grieve. They both contributed massively t o the thoughts and substance contained in this article through their labours as graduate students in the Faculty of Engineering at the University of Aberdeen. The U.K. Science and Engineering Research Council are also due thanks for financially supporting a large amount of this work.

REFERENCES Arthurs A. M. and Anderson, N. (1970).“Bounds for capacities in a microwave filter problem”, 28, NO. 3, pp. 259-262. Fraeijs de Veubeke, B. M. (1964). “Upper and lower bounds in matrix structural analysis”, Agurdoyraph, Pergamon Press, 72, pp. 165-192. Hammond, P. (1981).“Energy methods in electromagnetism”, Pergamon Press. Hammond, P. (1976).“Physical basis of the variational method for the computation of magnetic field problems”, Compumag-76, Rutherford Laboratories, Oxford, pp. 28-34. Hammond, P. and Penman, J. (1978). “Calculation of eddy currents by dual energy methods”, Proc. IEE, 125, No. 7, pp. 701-708. Hammond, P. and Penman, J. (1976). “Calculation of inductance and capacitance by means of dual energy principles”, Proc. IEE, 123, No. 6, pp. 554-559. Lanczos, C. (1970).“The variational principles of mechanics”, Uniu. of Toronto Press, 4th Edition. Noble, B. (1966). “Complementary variational principles for boundary value problems”, Report 473, Mathematics Research Centre, Univ. of Winconsin. Oden, J. T. and Reddy, J. N. (1974). “On dual complementary variational principles in mathematical physics”, In/. J . of Engineering Science, 12, pp. 1-29. Oden, J. T. and Reddy, J. N. (1976). “Variational methods in theoretical mechanics”, SpringerVrrlay. Penman, J. and Fraser, J. R. (1982). “Complementary and dual energy finite element principles in magnetostatics”, Trans. IEEE, MAG-18, No. 2, pp. 319-324. Penman, J. and Fraser, J. R. (1984).“Unified approach to problems in electromagnetism”, Proc. IEE, 131, NO. I, pp. 55-61. Prager, W. and Synge, J. L. (1947).“Approximations in elasticity based on the concept of function space”, Quart. of App. Math., 5, No. 3, pp. 241-269. Simkin, J. and Trowbridge, C. W. (1979).“On the use of the total scalar potential in the numerical solution of field problems in electromagnetics”, h i . J . Num. Mefh. Eny., 14, pp. 423-440. J. Simkin and Trowbridge, C. W. (1980). “Three dimensional nonlinear electromagnetic field computations using scalar potentials” Proc. IEE, 127, No. 6, pp. 368-374. Tonti, E. (1972).“On the mathematical structure of a large class of physical theories”, Acad. Naz. Dai, Lincei., Lii, Series 111, pp. 48-56. Turner, M. J., Clough, R. W., Martin, H. C.,Topp, J. L. (1956).“Stiffness and deflection analysis of complex structures”, J . q / Aeronautical Sciences, 23, No. 9, pp. 805-824. Vainberg, M. M. (1973).“Variational methods and methods of monotone operators in the theory of non-linear equations”, John Wiley.

Index A

Broad-band pulse wave holography and, 265-266 synthetic aperture method using, 287-288

1-trimmed median filter, 125 Adjoint linear operators, 320-323 Aperture imaging system and, 216-223 azimuthal resolution, 219-221 radial resolution, 221-223 structure of, 219 real, side-looking sonar, 244-246 synthetic, broad-band pulse wave and, 287-288 side-looking sonar, 247-253 ultrasonic imagery and, 215-314 Approximations image models and, 103- 109 toroidal, 103-107 AR models. 84- 109 recursive, 95-97 simultaneous, 93-95 ARMA models, 84-109,97-100 Arrays image reconstruction and, 279-280 point spread function and, 280-282 structure of, 277-279 Autoregressive models, casual, robust estimation and, 109-120 Azimuthal resolution, 2 I9 - 22 1

C Casual autoregressive models, I 1 I - 1 12 robust estimation and, 109-120 Chip interconnection, 202-206 wire, 198-201 Complementary variational principles, 3 18-323 adjoint linear operators, 320-323 electromagnetic fields, 347-358 electrostatic field, 342-347 field problems, 33 1-336 general engineering, 323-331 magnetostatic, 336- 342 Composite edge detection robust models and, 139- 155 confirming edge presence, 144-148 edge hypothesis generation, 141 144 experimental results, 148-154 Computerized tomography, ultrasonic, 288-290 Current filaments and, semiconductors, turbulence in, 49-58 -

B D

Ballistic phonon imaging structural defects with, 71-75 signal, 58-63 Binary representation of information, I62 Bipolar transistors, 177-188 current density in, I78 doping concentration and, 179-182 limits on, 186 Boolean logic, I62 Born approximation, 293-295 Boundary detection, 83-84 Box, C.E.P.,81

Diffraction tomography, 290-313 born approximation, 293-295 broad-band pulse wave and, 305- 3 13 fan-beam illumination and, 300-305 plane wave illumination and, 295-300 Rytov approximation, 293-295 wave equation, 291 -293 Digital ultrasonic imaging, 253-267. 282-290 wave propagation, 254-260 equation, 254-256

365

366

INDEX

Digital ultrasonic imaging (Continued) Green's theorem, 257 two-dimensional problem, 258-260

Fresnel approximation, imaging procedure based on, 263-265 G

E Eddy current systems complementary functions, 355-358 Edge detection robust models and, 139- 155 confirming edge presence, 144-148 edge hypothesis generation, 141-144 Electromagnetic fields, 347- 358 calculation of, variational techniques for, 315-364 dual equations, 351-355 eddy current systems complementary functions, 355-358 example, 358 primal equations, 351-355 structure of, 347-351 Electromigration, 201-202 Electron beam energy loss of, 1 I- I2 heating effect of, 12-17 parameters, principles of, 8-10 specimen interaction with, 10-17 thermalization of, 11-12 Electrostatic fields, 342-347 dual set, 345-346 example, 346-347 primal set, 344-345 Energy dissipation of limits of, 207-213 power supply and cooling, 209-213 loss, electron beam, 1 1- 12

Gateaux differential, 329-331 General engineering field problem, 323-33 1 Germanium, 14 temperature profile, 15 Green's theorem, 257 complementary functional, 328-329 generalized form of, 325-326 standard functional, 328

H Heat conduction, 13-14 electron beam and, 12- 17 sapphire and, 5-6 Hologram defined, 224 Fresnel approximation and, 263 -265 geometry of recordings, 230 numerical reconstruction from a, 260-265 offset-reference, 226-228 recording, 260-261 Holography broad-band pulse wave and, 265-266 principles of, 224-231 reconstruction, 225 - 226,228 - 23 1 recording, 225 theory and application of, 223-253 ultrasonic imaging and, 231-244 liquid surface deformation method, 232-235 solid surface deformation method, 235-239 ultrasonic beam, 239-244

F Field effect transistors, 188-194 Field problems, 331-336 general engineering, 323-331 Filters a-trimmed, 125 L, 124-126 M, 126 robust, 123-126 Finite lattice models, 107-109 Focusing, phonon, 64-71

I Image reconstructed from, liquid surface, 234-235 recording, 225 restoration, 225-226,228-231 Image models, 82 robust, approximations to, 103-109 AR and ARMA models, 84-109 ARMA, 97-100

367

INDEX boundary detection and, 83-84 image restoration, 82-83 recursive, 95-97 simultaneous, 93-95 Image restoration, 82-83 algorithm for, 127- I29 robust models and, 121-139 experimental results, 129-137 intensive representation for, 126-127 robust filters and, 123-126 Imaging pulse-echo method, 216,217 system. aperture and, azimuthal resolution, 219-221 radial resolution, 221-223 holography and, 223-224 ultrasonic, aperture and. 216-223 diffraction, 290-3 13 holography and, 223-253 structure of, 219 synthetic aperture, 215-3 14 Information binary representation, 162 representation of, 161-163 digital, defined, 161 Information processing devices for, 164- 175 three terminal devices, 164-166 two terminal devices, 166-171 voltage and, 171- I75 dissipation of energy and, 207- 2 I3 physical limits of, 159-214 fabrication and, 203-207 systems, 163-164 transistors and, 175- 198 bipolar, 177- 188 FET. 188-194 MSFET and, 194-197 sofi errors and, 197-198 wiring and, 198-206 chip interconnection, 202-206 chip wiring and, 198-201 electromigration and, 201-202 L L-estimators, 81, I10 L filter, 124-126 Lateral resolution, 219--221

Liquid surface deformation method. 232-235 images reconstructed from, 234-235 transfer function of, 232-235 Longitudinal resolution, 221-223 Low temperature scanning electron microscope, see Scanning electron microscope LTSEM, see Scanning electron microscope

M M-estimators, 81, 82, 110, 111 computation of, 114-1 15 M filter, 126 Magnetostatics, 336- 342 dual set, 338-340 example, 340-342 primal set, 338 Median filter, 124-125 Medical diagnosis, mechanically scanning imaging for, 243-244 MESFET, 194-197 structure of, 195 Metal-oxide-semiconductor field-effect transistor, 189 theory of, 189-191 Microbridges, superconducting, hotspots in, 41 -49 Microscopy, mechanically scanning acoustic, 240-243 Models AR, 84-109 recursive, 95-97 simultaneous, 93-95 ARMA, 84-109.97-100 casual autoregressive, 1 11- I 12 finite lattice, 107-109 Gaussian Markov, 92-93 Gaussian noise, 109-1 10 image, 82 robust image, 79-157 boundary detection and, 83-84 composite edge detection and, 139- 155 generative interpretation of, 100- 103 image restoration with, 121-139 recursive, 95-97 simultaneous, 93-95 SAR Gaussian, 94-95 state variable. 97

368

INDEX

Monte Carlo image, 66 MOSFET, see Metal-oxide-semiconductor field-effect transistor

N NOR circuit, emitter-coupled, 173 0

Offset-reference hologram, 226- 228

P P-n junction in semiconductors, 171

Pair tunneling, zero voltage, 28 Phonon ballistic, imaging structural defects with, 71-75 signal, 58-63 focusing, 64-71 Power supply and cooling in dissipation of energy, 209-213

Q Qualitative robustness, I 1 1 Quasiparticle tunneling, 26- 39 current formula, 27

R R-estimators, 81, 110 Radial resolution, 221-223 Receiver, array, transmitter array and, 275-282 Rectangular transducer, radiation field from, 270- 272 Representation, intensive for image restoration, 126- 127 Resolution, 219-223 azimuthal, 2 19- 22 1 radial, 221-223 Robust estimation, casual autoregressive models and, 109-120 Robust filters, 123-126 Robust image models, see Models Rytov approximation, 293-295 S

Sapphire, heat conductivity of, 5-6 SAR models, Gaussian, 94-95

Scanning acoustic microscopy, 240-243 Scanning electron microscope low temperature stage, 4-8 cross-section of, 7 mounting configuration, 6 schematics of, 5 principle of, 1-2,X-10 spatial resolution of, 2 very low temperatures and, 1-78 Schottky-gate transistor, see MSFET Semiconductors hot electron effects in, 176 p-n junction in, 171 Side-looking sonar real aperture, 244-246 synthetic aperture, 247-253 technique, 246-247 Simultaneous AR models, 93-95 , Single transducer, radiated field from, 268-272 Soft errors, transistors and, 197- 198 Solid surface deformation method, 235-239 Sonar side-looking real aperture, 244-246 synthetic aperture, 247-253 Spatial fourier transform, spherical wave and, 266-267 Specimen electron beam interaction with, 10-17 energy loss and, 11-12 heating effect, 12-17 State variable models, 97 Statistical procedures robust, 81-82 types of, 8 1-82 Stefan-Boltzmann law, 61 Superconducting current filaments and, turbulence in, 49-58 microbridges, hotspots in, 41-49 tunnel junctions, 18-26 arrays of, 39-41 cross-line geometry, 18 quasiparticle tunneling, 26-39 Synthetic aperture broad-band pulse wave and, 287-288 holography and, 223-253 ultrasonic imagery and, 215-314

369

INDEX

U

T Texture edge, confirming, 145-147 Thermal healing length, 12-17 Thermal relaxation time, 12-1 7 Thermalization, electron beam, 1 I - 12 Throughput, defined, 204 Tomography computerized ultrasonic, 288-290 diffraction, 290-3 I 3 born approximation, 293-295 broad-band pulse wave and, 305-313 fan-beam illumination and, 300-305 plane wave illumination and, 295-300 Rytov approximation, 293-295 wave equation, 291-293 ultrasonic computerized, 288- 290 Toroidal approximation, 103- 104 Toroidal model, properties of, 104-107 Transducer array, processing system with a, 283-287 radiation field from, 272-275 ultrasonic imagery and, 267-282 rectangular, radiation field from, 270-272 single. radiated field from, 268-272 Transistors, 175-198 bipolar, 177-188 current density in, 178 doping concentration and, 179- 182 limits on, 186 field effect, 188-194 metal-oxide-semiconductor, 189- 194 MSFET, 194-197 structure of, 195 operation of, 173 soft errors and, 197-198 Transmitter, array, receiver array and, 275-282

Tunnel junctions superconducting. 18 -26 arrays of. 39-41 cross-line geometry, 18 quasiparticle tunneling, 26.- 39 Turbulence, superconducting, current filaments and, 49-58 Turning machine, 207-208

Ultrasonic beam, mechanical scanning of, 239 - 244

Ultrasonic computerized tomography, 288-290

Ultrasonic imagery holography and, 223-253 liquid surface deformation method, 232-233

ultrasonic beam, 239-244 problem in, 216-217 solid surface deformation method, 235-239 synthetic aperture, 215-3 14 transducer array and, 267-282 Ultrasonic imaging diffraction tomography and, 290-313 digital, 253-267,282-290 wave propagation, 254-260 real aperture, side-looking, 244-246 synthetic aperture, side-looking sonar, 247-253

V Variational principles complementary, 318-323 adjoint linear operators, 320- 323 electromagnetic fields, 347-358 electrostatic field, 342-347 field problems, 331-336 magnetostatics, 336-342 field problems, general engineering, 323- 331

historical perspective of, 316-318 Voltage information processing devices and, 171-175

scale of nonlinearity, 17 1 - 173 transistor operations and, 171 W

Wave propagation, 254-260 equation for, 254-256 two dimensional problem, 258- 260 Green’s theorem, 257


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