Advances in Imaging and Electron Physics, Volume 102 (Advances in Imaging and Electron Physics)

ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 102

EDITOR-IN-CHIEF

PETER W. HAWKES CEMESILuboratoire d 'Optique Electmnique du Centre National de la Recherche Scientifque Toulouse, France

ASSOCIATE EDITORS

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

TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom

Advances in

Imaging and Electron Physics EDITEDBY PETER W. HAWKES CEMESILaboratoire d 'Optique Electronique du Centre National de la Recherche Scientifique

Toulouse, France

VOLUME 102

ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto

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CONTENTS CONTRIBUTORS .. F%EFACE . . . .

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

vii ix

Finite Element Methods for the Solution of 3D Eddy Current Problems R. ALBANESE AND G. RU~INACCI 1.Introduction . . . . . . . . . . . . . . . . . . . . . 2 4 11. Field Equations and Material Properties . . . . . . . . . . 9 111. Fields. Potentials. and Gauges . . . . . . . . . . . . . . 15 IV. Edge Elements for 3D Field Problems and Vector Potentials . . . 30 V. Integral Formulations for Linear and Nonlinear Eddy Currents . . 50 VI. Differential Formulations and Constitutive Error Approach . . . 81 VII. Discussion and Conclusions . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . 82

Nanofabrication for Electronics

w. CHEN AND H. m

I. Introduction . . . . . . . . . . I1. Nanofabrication Methods . . . . 111. Pattern Transfer . . . . . . . IV. Resolution Limit of Organic Resists V. Applications of Nanostructures . . VI. Conclusion . . . . . . . . . . References . . . . . . . . . .

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E D

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

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87 90 125 138 158 174 176

Miniature Electron Optics A. D. FEINERMANAND D.A. CREW

I. Introduction ............. IT. Scaling Laws for Electrostatic Lenses . . . 111. Fabrication of Miniature Electrostatic Lenses IV. Fabrication of Miniature Magnetostatic Lenses V. Electron Source . . . . . . . . . . .

V

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

. . . .

187 189 189 211 212

vi

CONTENTS

VI . Detector . . . . . . . . . . . . VII . Electron Optical Calculations . . . VIII . Performance of a Stacked Einzel Lens IX. Summary and Future Prospects . . References . . . . . . . . . . .

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217 220 225 232 233

Optical Interconnection Networks

KHAN M

. IFTEKHARUDDIN AND MOHAMMAD A . KARIM

. . . . . . . . . . . . . . . . . . 111. Architectures . . . . . . . . . . . . IV. Applications . . . . . . . . . . . . V. Packaging of Optical Interconnects . . . VI . Problems and Possibilities . . . . . . VII. Conclusions . . . . . . . . . . . . References . . . . . . . . . . . . . 1. Introduction

I1. Optical Interconnect Qpes

. . . . . . . .

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

Aspects of Mirror Electron Microscopy S . A . NEPIJKO AND N . N . SEDOV I. Resolution of the Mirror Electron Microscope . . . . . . . . I1. Distortion of Details of Object Image under Observation in a Mirror Electron Microscope . . . . . . . . . . . . . . . 111. Limiting Sensitivity of a Mirror Electron Microscope for Observation of Steps on an Object . . . . . . . . . VI. Image of Islands on an Object Surface in Mirror Electron Microscopy . . . . . . . . . . . . . . . . . . V. Calculation of Image Contrast in a Mirror Electron Microscope in . . . . . . . . . . . the Focused Operation Mode References . . . . . . . . . . . . . . . . . . . . . INDEX

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

235 236 249 253 263 264 267 269

274 288 293 300 311 323 325

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributions begin.

H. AHMED(87), Microelectronics Research Centre, Cavendish Laboratory, University of Cambridge, Cambridge, CB3 OHE, United Kingdom R. ALBANESE (2), DIEMA, Universith degli Studi di Reggio Calabria, Reggio Calabria, Italy W. CHEN(87), Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794-3800,USA; and MicroelectronicsResearch Centre, Cavendish Laboratory, University of Cambridge, Cambridge, CB3 OHE, United Kingdom

D. A. CREWE(187), Microfabrication Applications Laboratory, University of Illinois at Chicago, Electrical Engineering and Computer Science Department, Chicago, Illinois 60607-7053, USA A. D. FEINERMAN (187), Microfabrication Applications Laboratory, University of Illinois at Chicago, Electrical Engineering and Computer Science Department, Chicago, Illinois 60607-7053, USA KHANM. IFTEKHARUDDIN (235), BDM International,Kettering, Ohio 45420, USA MOHAMMAD A. KARIM(235), Department of Electrical and Computer Engineering, University of Dayton, Dayton, Ohio 45469-0227, USA S. NEPIJKO(274), Fritz-Haber Institut der Max-Planck-Gesellschaft, D-14195 Berlin (Dahlem),FRG; Institute of Physics, Technical University Clausthal, 38678 Clausthal-Zellerfeld,FRG; and (permanentaddress)Instituteof Physics, Ukrainian Academy of Sciences, Kiev, Ukraine G. RUBINACCI(2), Dipartimento di Ingegneria Industriale, Universid degli Studi di Cassino, Cassino, Italy N. N. SEDOV(274), Institute of Physics, Technical University Clausthal, 38678 Clausthal-Zellerfeld,FRG; and (permanent address) The Moscow Higher Military Command School, 109380 Moscow, Russia

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PREFACE The calculation of eddy currents, nanofabrication, miniature electron optical elements and instruments, optical interconnections and mirror electron microscopy, these are the themes of this volume. We begin with a discussion by Raffaele Albanese and Guglielmo Rubinacci of the use of finite-element methods for modelling eddy current distributions. Anyone familiar with the proceedings of the annual COMPUMAG conferences will know that great progress has been made with this difficult problem and the authors of this chapter have made important contributions. Here, they explain why the problem is so difficult and then show in detail how solutions can be obtained. The method is explained fully, with many examples, and the remaining difficulties are clearly indicated. This account will surely be much used by those confronted with this class of problems. It is a particular pleasure to introduce the second chapter, in which Wei Chen and Haroon Ahmed discuss nanofabrication for electronics, for it may be seen as a successor to a landmark paper published in these Advances in 1965 on the scanning electron microscope. The Departmentof Engineering and the Cavendish Laboratory of the University of Cambridgehave often contributed to this series and this latest survey is concerned with a subject of the highest interest: nanofabrication itself and its applications in the fields of quantum interference,superlattices,single electronics,quantum dots, and magnetic nanostructures. I believe that this account will be as heavily used in this new area as was its predecessor in an area that was new in 1965. It is entirely suitable that the third chapter should be concerned with miniature electron optics, a fascinating application of nanofabrication. Alan Feinerman and David Crewe have pioneered the design and construction of miniature electron optical elements and miniature microscopes composed of these elements but the information about them is scattered in a host of journals and conference publications. This connected account of their achievements and related work will, I am sure, draw the attention of a wide audience to the progress that is being made in this area. If anyone needs to be convinced, it is sufficient to glance at the “future prospects”, in which a subcentimeter scanning electron microscope, a ten-centimeter time-of-flight mass spectrometer, a ten-centimeter NMR instrument and a five-meter linear accelerator capable of producing hard x-rays are mentioned. In the past, these Advances have contained few contributions that could be classed as “photonic”, for electrons were placed at the heart of the series. But

ix

X

PREFACE

the distinction between these forms of radiation has become blurred and I find it entirely natural-and very welcome-that a chapter on optical interconnection networks should appear here. Khan Iftekharuddin and Mohammed Karim describe these in detail and show beyond any doubt how important they will be for the next generation of high-speed devices. They describe architectures, applications, packaging and future developments, I am delighted that these authors have agreed to publish their survey in this series. The volume concludes with a more traditional contribution on mirror electron microscopy. This is not a full survey in that it does not include any account of related developments world-wide; but it is a survey of developments that have not, in the past been accorded as much importance as they deserved, namely, those described in the Russian-languagejournals. The work of Sergei Nepijko and N. N. Sedov may be known from their conference abstracts and occasional papers in western European serials but a connected account of their research was urgently needed. It is with great pleasure that such a survey is presented here. As usual, I conclude this introduction by thanking all the authors for the time and trouble that they given to the preparation of their contributions and in particular, for their efforts to make their material accessible to non-initiates. Material to appear in future volumes is listed below. Peter W. Hawkes

FORTHCOMING CONTRIBUTIONS Mathematical models for natural images Use of the hypermatrix Image processing with signal-dependent noise The Wigner distribution Hexagon-based image processing Microscopic imaging with mass-selected secondary ions Modern map methods for particle optics ODE methods Microwave tubes in space Fuzzy morphology

L. Alvarez Leon and J.-M. Morel D. Antzoulatos H. H. Arsenault M. J. Bastiaans S. B. M. Bell M. T. Bernius M. Berz and colleagues J. C. Butcher J. A. Dayton E. R. Dougherty and D. Sinha

xi

PREFACE

The study of dynamic phenomena in solids using field emission* Gabor filters and texture analysis Liquid metal ion sources X-ray optics The critical-voltage effect Stack filtering Median filters The development of electron microscopy in Spain Space-timerepresentation of ultra-wideband signals Contrast transfer and crystal images Numerical methods in particle optics Surface relief Spin-polarized SEM Sideband imaging Vector transformation SEM image processing Electronic tools in parapsychology Z-contrast in the STEM and its applications Phase-space treatment of photon beams Image processing and the scanning electron microscope Representation of image operators Fractional Fourier transforms HDTV Scattering and recoil imaging and spectrometry The wave-particle dualism Digital analysis of lattice images (DALI) Electron holography X-ray microscopy Accelerator mass spectroscopy Applications of mathematical morphology Set-theoretic methods in image processing Focus-deflection systems and their applications Electron gun system for color cathode-ray tubes

M. Drechsler

J. M. H. Du Buf R. G. Forbes E. Forster and F. N. Chukhovslq A. Fox M. Gabbouj N. C. Gallagher and E. Coyle M. J. Herrera and L. Bni E. Heyman and T. Melamed K. Ishizuka E. Kasper J. J. Koenderink and A. J. van Doorn K. Koike w. Krakow W. Li N. C. MacDonald R. L. Morris P. D. Nellist and S. J. Pennycook G. Nemes E. Oho B. Olstad H. M. Ozaktas E. Petajan J. W. Rabalais H. Rauch A. Rosenauer D. Saldin G. Schmahl J. P. F. Sellschop J. Serra M. I. Sezan T. Soma H. Suzuki

xii

PREFACE

Study of complex fluids by transmission electron microscopy New developments in ferroelectrics Electron gun optics Very high resolution electron microscopy Morphology on graphs Analytical perturbation methods in charged-particle optics

I. Talmon J. Toulouse Y. Uchikawa D. van Dyck L. Vincent M. I. Yavor (vol. 103)

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ADVANCES IN IMAGING AND ELECTRON PHYSICS VOL . 102

Finite Element Methods for the Solution of 3D Eddy Current Problems R . ALBANESE

.

DIEMA. Universitd degli Studi di Reggio Calabria. Via E Cuzzocrea 48. 1-89128 Reggio Calabria Italy

.

G. RUBINACCI Dipartirnento di Ingegneria Indusrriale. Universitd degli Studi di Cassino. via Di Biasio 43. 1-03403 Cassino. Italy

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . I1. Field Equations and Material Properties . . . . . . . . . . . . . . . . . A . Electromagnetic Problem . . . . . . . . . . . . . . . . . . . . B . Stationary Fields . . . . . . . . . . . . . . . . . . . . . . . C . The Eddy Current Problem . . . . . . . . . . . . . . . . . . . Ill . Fields. Potentials. and Gauges . . . . . . . . . . . . . . . . . . . . A . Eddy Current Formulations in Terms of Fields . . . . . . . . . . . . . B . Vector Potentials . . . . . . . . . . . . . . . . . . . . . . . C . Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Edge Elements for 3D Field Problems andVectorPotentials A . Edge Element Shape Functions . . . . . . . . . . . . . . . . . . B . Tree-Cotree Decomposition . . . . . . . . . . . . . . . . . . . C . Boundary Conditions for Simply and Multiply Connected Regions . . . . . . V. Integral Formulations for Linear and Nonlinear Eddy Currents . . . . . . . . . A . Formulation of the Linear Eddy Current Problem . . . . . . . . . . . . B . Finite Element Approach to the Solution of the Linear Eddy . . . . . . . . . . . . . . . . . . . . . . . Current Problem . . . . . . . . . . . . . C . Nonlinear Eddy Currents and Magnetostatics D . Iterative Procedures forthe Nonlinear Eddy Current Problem . . . . . . . . E. Finite Element Approach to the Solution of Nonlinear Magnetostatic and Eddy Current Problems . . . . . . . . . . . . . . . . . . . . . . . VI . Differential Formulations andConstitutiveError Approach . . . . . . . . . . A . The Magnetostatic Problem . . . . . . . . . . . . . . . . . . . B . The Eddy Current Problem . . . . . . . . . . . . . . . . . . . . C . The Electromagnetic Problem . . . . . . . . . . . . . . . . . . . D . Analysis of Resonant Cavities . . . . . . . . . . . . . . . . . . . E. Open Boundary Problems . . . . . . . . . . . . . . . . . . . . VII . Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

2 4 4

5 6

9 9 11 12 15 17 20 25

30 30 31 38 41 44 50 51 54 63 69

78 81 82

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Copyright @ 1998 by Academic Press Inc . All rights of reproduction in any form reserved 1076-567W97 $25.00

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I. INTRODUCTION The rapid escalation of the performance of the computational tools in these last decadeshas given a big impulse to the developmentof numerical methods and associated codes in all fields of engineering and physics. Now the numerical techniques are so widespread that there is no need to demonstrate the paramount importance that computational methods have in modem analysis and design activities, which routinely involve the solution of nonlinear, inhomogeneous, anisotropic, or timedependent problems that cannot be solved by using analytical techniques. Thus, computational electromagnetics has also developed rapidly in the last few years and now modern electromagnetic codes are able to obtain sufficiently accurate solutions for a wide class of stationary, steady-state, and time-dependentproblems of electrical engineering. It is not our intention to provide an exhaustive review of the state of the art of computational electromagneticanalysis, for which the reader can refer to the specialists conferences like COMPUMAG (see, for instance, Biro, 1996) and CEFC (see, for instance, Sabonnadiere, 1995). We present instead a comprehensive treatment of a confined sector of computational electromagnetics. The distinctive points of such a sector, aimed primarily at the solution of the eddy current problems, include the edge elements (Nedelec, 1980; Bossavit, 1988), the error-based approach (Rikabi et al., 1988a,b), the two-component vector potentials, and the tree-cotree decomposition (Albanese and Rubinacci, 1988a,b). We try to present the matter in a systematic way, giving due prominence to the primary sources and to the scientists who have used and further developed the various ideas. We hope that our effort can be useful not only to the readers who have an expertise in computationalelectromagnetics,but also to those who work in different fields of computational engineering and physics as well as to the physicists and engineers who are not familiar at all with computational analysis. As far as possible, this article is self-consistent in the sense that it is not strictly necessary to have previous knowledge of computational methods. In addition, we take great care to show which real problems can be analyzed and with which approximation, i.e., what accuracy can be achieved. The assessment of the performance of the numerical methods and the associated codes is so important that the scientific community has decided to hold the Testing of Electromagnetic Analysis Methods (TEAM)workshops (see, for instance, Nakata, 1990), mainly devoted to magnetostatic and eddy current problems, and the Applied Computational Electromagnetic Society ACES workshops (see, for instance, Sabbagh et al., 1990), essentially devoted to high-frequency electromagnetics. The aim of these series of workshops is to show the effectiveness of the numerical techniques and the associated numerical codes to solve electromagnetic problems and gain confidence in their predictions. These symposia allow

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

3

one to compare the numerical results obtained for well-defined benchmark problems for which analytical solutions or experimentalresults are available. In more than ten years of activity, these workshops have been and still are a very useful guide to the development, assessment, and improvement of the electromagnetic codes. The aim of this paper is therefore to present a critical survey on our approach to the solution of the general three-dimensional eddy current problem, which originally started from the necessity of studying the transient electromagnetic phenomena in tokamaks. These thermonuclear fusion devices present very complex geometries in which massive structures and thin shells coexist and ferromagnetic effects are usually negligible. For this reason, the starting point for our activity was the finite element eddy current integral formulation for nonmagnetic shells proposed by Kameari (1981), Bossavit (1981), and Blum et al. (1983). The main obstacle to the treatment of fully three-dimensionalmassive conductors was the lack of a general method for generating a complete set of independent, solenoidal shape functions for the current density. We tried to solve this problem by introducing the discrete analogue of a two-component current density vector potential, which, however, in the form first proposed by Carpenter (1977) and Brown (1982) was not directly applicable because of the inconsistent treatment of electric interfaces (Albanese et al., 1985). The solution was definitely found by means of the edge elements, first proposed by Nedelec (1980) and applied in the field of computational electromagneticby Bossavit and V6rit6 (1982), in conjunction with the tree-cotree decomposition of the mesh (Albanese and Rubinacci, 1988a). We also showed that this framework, including edge elements, two-component vector potentials, and tree-cotree decomposition, was the natural one for the application of the differential formulations given by the constitutive error approach, a method that provides a useful means for establishing numerical techniques characterized by symmetric and positive matrices, upper and lower bounds for energy-related functionals, and straightforwardestimates of the numerical error to be used for mesh refinement and that are essential to reliably use the numerical results for design purposes. The fundamentals of the constitutive error approach date back to several decades ago (see, for instance, Moreau, 1966), but were first proposed for computational electromagnetics by Rikabi, Bryant, and Freeman in 1988 (Rikabi et al., 1988a,b), and shown to effectively work using the edge elements by Albanese and Rubinacci ( 1990b). The necessity of using edge elementsfor the convergenceof the procedure was then demonstrated by Bossavit (1992). These works gave a big impulse to the use of the edge elements, which, practically ignored since their first application by Bossavit and Vtritt? (1982) and Bossavit and VCritt (1983), are now of common use all over the world. Edge elements are also used for high-frequency problems, where their main application is for the resonant cavities, where the problem of the spurious modes that appear using the conventional nodal elements is avoided, as

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R. ALBANESE AND G . RUBINACCI

explained by Bossavit (1990), Cendes (1991), Bardi et al. (1992), and Albanese and Rubinacci (1993a). The paper is organized as follows. Section I1 describes the mathematicalmodels, including the assumptions made on the field equations and the material properties. An excursion from full Maxwell equations to the stationary and quasistationary fields is briefly presented, with attention focused on the magneto-quasistationary model, i.e., the eddy current problem. Section I11 introduces vector and scalar potentials along with the corresponding gauge conditions. The emphasis is put on the benefits offered by the use of the vector potentials with the classic Coulomb and Lorentz gauges in dealing with material interfaces and the unified treatment of magnetostatic and eddy current problem. Other vector potentials are also inuoduced, including the two-component vector potential, which may have a jump on the normal component at the discontinuity surfaces. Section IV summarizes the main features of the edge elements, whose degrees of freedom are associated with the tangential components (or the line integrals) of the vector field along the edges. They give rise to a set of vector shape functions, for which the continuity of the tangentialcomponents is preserved, allowing for the discontinuity of the normal component between adjacent elements. In Section IV we also describe the tree-cotree decomposition, a numerical method we showed to be essential to obtain the discrete analogue of the two-componentvector potentials. Section V is dedicated to the edge-element-basedintegral formulationsfor linear and nonlinear eddy current problems in terms of a two-component vector potential. Section VI introduces the constitutive error approach with its application to magnetostatic, eddy current, and electromagnetic problems. We show that dual electric and magnetic formulations can be regarded as by-products of the constitutive error approach, discuss the possibility of defining upper and lower bounds for global and local quantities, and explain why spurious modes do not appear using the edge-element-basedapproach for the study of resonant cavities. Finally, Section VII reports the main conclusions.

11. FIELDEQUATIONS AND MATERIAL PROPERTIES

A. Electromagnetic Problem In the volume of interest V, we assume the classic electromagnetic model, defined by the field equations

aD

VXH=J+VxE=--

at

aB at

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

5

the material properties

B = B(H)

D = D(E)

J = J(E) and initial conditions for B and D, verifying V B = 0 and V . D = p , interface conditions, and suitable boundary conditions like E x n = e on aV, 5 a V and H x n = h on a Vh = a V - a V, (or regularity conditions at infinity if the domain V is unbounded). The discussion on initial, boundary, and interface conditions is reported in detail in Section IIC for the eddy current case. Here t is the time, E is the electric field, H is the magnetic field, D is the displacement vector, B is the flux density, J is the current density, p is the charge density, and t is the time. The constitutive operators (3)-(5) are supposed to be single-valued, continuous, symmetric, and monotonic. So, the magnetic property (3) can also be replaced by H = X(B) with X = B-’.Therefore, in this survey, we neglect hysteresis. In most cases, the constitutive equations refer to linear media, for which (3)-(5) take the particular form

B=pH

+

J = a E J, D = EE

(6) (7) (8)

where the permeability p, the source term Js, the electric conductivity a (nonzero only in the conducting region Vc), and the permittivity E are specified as functions of the time and of the spatial coordinates.

B. Stationary Fields The stationary assumption (a/& +- 0) leads to simplified models of Maxwell equations. In particular, Faraday’s and Ampere’s laws become

VXH=J

(9)

VXE=O

(10)

Depending on the sources and the field quantities we are interested in, we have the three particular cases of magnetostatics, electrostatics, and direct current. In the magnetostatic case the source term is represented by the prescribed current density J = J,. The electric field is zero, so the problem is described by the field equations V x H = J,, in V (1 1) V*B=0, inV (12)

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R. ALBANESE AND G . RUBINACCI

the constitutive equation

B=B(H),

inV

(13)

and suitable boundary conditions like B .n = b on a Vb 5 a V and H x n = h on avh= av - avb. In the direct current problem, the source term is represented by the impressed electric field E, = J,/a (or given by the boundary conditions). The magnetic field is not zero, but it does not appear in the field equations that determine electric field and current density:

VxE=0, V.J=O,

inV inV

(14)

(15)

the constitutive equation

J=aE+J,,

inV

(16)

and suitable boundary conditions like J n = j on a Vj E a V and E x n = e on aVe = a V - aVj, along with additional integral constraints on currents or voltages if aVe is not a connected surface (i.e., a surface consisting of two or more disconnected pieces). In the electrostatic case the source term is represented by the prescribed charge density p = ps. The magnetic field is zero, and the problem is described by the field equations

VxE=O,

inV

V.D=p,,

inV

the constitutive equation

D=EE,

in V

(19)

and suitable boundary conditions like D . n = d on a Vd a V and E x n = e on aVe = a V - aVd, along with additional integral constraints if aV, is not a connected surface.

C. The Eddy Current Problem

This model is obtainedfrom the full Maxwell equationswhenever the displacement current effects can be neglected. Actually, inside a passive metallic conductor, the displacementcurrent a(&E)/atis always negligible with respect to the conduction current aE as far as the time scale of interest, e.g., the period T in steady-state problems, is longer than the relaxation time of the electric charge r, = &/a (for instance, in copper tris of the order of magnitude of lo-’’ s). To neglect the inAuence of the displacement current in the nonconducting region, the characteristic

THE SOLUTION OF 3D EDDY CURRENTPROBLEMS

7

length of the problem has to be smaller than the wavelength and the energy stored in the electric field must be a negligible fraction of the global energy. Therefore, in the conducting region, the mathematical model is defined by the field equations V x H = J, aB VxE=---,

in V,

B=B(H),

inV

at

inV,

(20) (21)

and the material properties

J = a E + Js,

in V,

In the external nonconducting region V-V, the field equations are those of the magnetostatic problem: V x H = Jsr V . B = 0,

in V-V, in V-V,

and the constitutive equation is

B = B(H),

in V-V,

(26)

The whole problem in V is closed by suitable boundary, initial, and interface conditions. Boundary conditions that ensure uniqueness of the solution are those on the tangential component of the electric or the magnetic field:

Exn=e, Hxn=h,

onaV&aV onaVhEaV

If V is an unbounded domain, e.g., the three-dimensional space !R3, conditions (27) and (28) are replaced by regularity conditions at infinity. It is worth noticing how the eddy current problem is intimately linked to the magnetostatic problem, as the statement in V-V, is the same as (11) and (12). For this reason, the initial conditions for the flux density B have to satisfy (25) and (28). Therefore, it is customary to prescribe the initial current density (everywhere, including the current density in the conducting region) and determine the initial conditions for the flux density by solving the corresponding magnetostatic problem. The interface conditions:

[HI X n = 0, [El X n = 0,

On

&

On

&j

where n is the outward normal and [.I stands for the jump of the quantity at the interface, have to be imposed at any surface where the material properties are

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R. ALBANESE AND G. RUBINACCI

subject to discontinuities. At the interface between conducting and nonconducting region, (29) and (30) become

[HIx n = 0, [B] . n = 0,

on av, n a(v-V,)

(31)

on aV, n a(V-V,)

(32)

Notice that (29) and (31) are based on the assumption of no surface currents. Let us now briefly discuss the consequences of neglecting the displacement current density aD/at in Ampere’s law. Firstly, the calculation of the electric field in the external region is not required to determine the eddy currents in V,. However, the electric field can be determined in the nonconducting region S3-V, after the solution of (20)-(30), once we know its curl (i.e., the computed value of -aB/ar), its divergence (specified by the charge distribution outside, which we suppose to be zero), its tangential components on aV,, which, according to (30), coincide with those computed from the V, side, and the regularity conditions at infinity. Secondly, it can be observed that a surface charge will appear on a V,, owing to the discontinuity of the normal component of the electric field. In fact, its value on the outer side of a V, is determined by the solution of the electric field problem outside V,; on the inner side, taking into account (23), (24), and (31), it is given by o-’J,.n, which, in particular, is zero if the source currents do not enter the region V,. The presence of this surface charge is not surprising, even when J . n = 0. In fact, the charge is, in general, not conserved in the eddy current model, since, As a matter of fact, neglecting the displacement current, we get V . J # +/at. (21) implies solenoidality of aB/at, hence of B, provided that the initial values of B are solenoidal, whereas (20) forces solenoidality of J. Thirdly, the characterof the system of equations, parabolic inside the conducting region and elliptic in the nonconductingdomain, is radically different with respect to the full Maxwell equations (1) and (2), which are hyperbolic. This has an influence on the boundary conditions. In the eddy current problem, even in the frequency domain, there is no radiation toward infinity. The regularity condition at infinity forces the fields to approach zero at least like l / r 2 , where r is the distance from an arbitrarily fixed point of the space. Therefore, the global energy stored in the magnetic field is always finite even at steady state. Finally, notice that there is the dual case, the electro-quasistatic problem, in which the aB/at term is neglected in Faraday’s law. This approximation is acceptable in cases like the analysis of the electric field in semiconductors or ionized gases. The main difference with respect to the eddy current problems is that the suppression of the aB/at term in Faraday’s law implies irrotationality of E, which allows for the formulation in terms of the scalar electric potential. This is not possible in the eddy current case, since the elimination of the displacement current from Ampere’s law is not sufficient to imply irrotationality of the magnetic field due to the presence of the conduction current.

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

9

In. FIELDS, POTENTIALS, AND GAUGES Let us consider the eddy current problem describedby the quasi-stationary Maxwell equations (20)-(30). The main differences between the various formulations that have been proposed and applied in the literature reside in the choice of the basic unknowns and in the character of the spatial operators. According to the spatial operators used, we may have integral and differential formulations, whose associated numerical methods are treated in Sections V and VI, respectively. Integral formulations are characterized by a lower discretization cost, automatically take account of the regularity conditions for open boundary problems, and usually yield flux density distributions that are smooth and solenoidal. However, they require a cumbersome treatment of inhomogeneities and nonlinearities. In addition, the associated numerical methods imply the calculation of full matrices defined by double-volume integrals, which may yield an escalation of the computational time and memory. For the selection of the primary unknowns there exist two classic families of numerical methods. In the magnetic formulations, the basic unknown is the magnetic field H or the electric vector potential T,which is described in Section IIIB. With these formulations Ampere’s law and constitutive equations are exactly satisfied, whereas Faraday’s law is imposed only in a weak form. These formulations yield solutions having a solenoidal current density J, whereas the solenoidality of the flux density B is fulfilled only in an average sense. The dual family is that of the electric formulations, in which the basic unknown is the electric field E or the magnetic vector potential A described in Section IIIB. With these formulations the roles of Ampere’s and Faraday’s laws are exchanged with respect to the magnetic formulations. Thus, Faraday’s law is exactly verified, hence B is rigorously solenoidal,whereas Ampere’s equation is fulfilled only in a weak form, which may yield V J f: 0 inside the conductors and [J] . n f: 0 at the electrical interfaces. Both electric and magnetic formulations are briefly illustrated in this section. However, in Section VI we show how dual electric and magnetic formulations can be considered results and complementary parts of an alternative approach, firstly proposed by Rikabi et al. (1988a,b) in the framework of computational electromagnetics. This method, known as the constitutiveerror approach, exactly enforces both Faraday’s and Ampere’s laws and minimizes error in the constitutive equations.

-

A. Eddy Current Formulations in Terms of Fields

In the conducting domain V,, using (20) and (23), we can express the electric field as a function of the curl of the magnetic field:

E = a-’(V x H - Js),

in V,

(33)

10

R. ALBANESE AND G. RUBINACCI

Substituting this expression into (21). we get the magnetic field diffusion equation in V,. Of course, a similar equation cannot be derived in the nonconducting region where CT = 0. Therefore, in the external region, we simply rewrite (24) and (25) in terms of H. Thus, the formulation in terms of the magnetic field can be stated as follows:

V x H = Js, V . B(H) = 0,

in V-V, in V-V,

along with suitable initial, boundary, and interface conditions like those described in SectionIIC, obviously expressed in terms of the magnetic field H unknown. For instance, the initial conditions for H must verify (35) and (36). Notice that with these formulations, (29) and (30) must also be applied on the interfaces where the source current density J, is discontinuous, where (34) cannot be imposed as it is. However, the treatment of the interfaces is not critical when the weak forms are used, provided that the unknown variables do not have jumps. The formulation in terms of the magnetic field (34)-(36) automatically takes account of the discontinuities of the conductivity, such as the boundary of the conducting domain a V,, because there is no jump of any component of H.On the contrary, on magnetic discontinuities, i.e., where the magnetic properties have a jump, the continuity of B . n calls for a discontinuity of H . n, which implies the introduction of double-valued functions. Therefore, only in the case of nonmagnetic conductors, this formulation can be used with the unknown vector field H expanded on a set of continuous basis functions. However, in the presence of magnetic interfaces, formulation (34)-(36) can make use of a particular set of vector shape functions, for which the continuity of the tangential components is preserved, allowing for the discontinuity of the normal component between adjacent elements. Such features can be obtained by using the edge elements, introduced by Nedelec (1980) and Bossavit (1988) and generalized by van Welij (1985), Mur and de Hoop (1985), Ren and VCritC (1989), Kameari (1990a), and Dular et al. (1994), whose degrees of freedom are associated with the tangential components (or the line integrals) of the vector field along the edges. These shape functions are illustrated in detail in Section IV. Inside the conducting domain V,, a dual formulation can be obtained in terms of the electric field or related quantities,leading to the electric field diffusion equation. The discussion about the continuity properties for the solution is similar to the case of the magnetic field formulation, obviously exchanging the roles of magnetic and electric discontinuities. The main difference with respect to the magnetic field formulation is that the electric field unknown is limited to the conducting domain

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

11

Vc; however, since the external region V-V, plays a role in the problem, a different unknown, related either to the magnetic field H or to the flux density B, has to be added in V-V,. B. Vector Potentials

It is possible to formulate the problem without introducing double-valued unknowns at the discontinuity layers. This approach is based on the introduction of vector and scalar potentials. Solenoidality of the current density suggests the introduction of the electric vector potential T, sometimes referred to as the current density vector potential, defined by

J=VxT

(37)

From this definition and the quasi stationary form of Ampere’s law (20),it follows that the difference between the magnetic field and the electric vector potential is irrotational. Therefore, their differencecan be expressed as the gradient of a scalar magnetic potential 51:

H=T-V51 In particular, in simply connected current-free regions, it is possible to set T = 0 and the magnetic field intensity can be simply expressed as H = -V51. In this way, at the magnetic interfaces, the possiblejump of the normal component of H is totally absorbedby a discontinuity of a S2/an, and both 51 and the three components of T can be kept continuous. In principle, the vector potential T could also be used for the solution of the electromagnetic problem, provided that in its definition (37) the left-hand side is modified into J aD/at. Likewise, solenoidality of the flux density suggests the introduction of the magnetic vector potential A, defined by

+

B=VxA

(39)

This definition and Faraday’s law (2) imply the irrotationality of the sum of the electric field and the time derivative of the magnetic vector potential. Therefore, this sum can be expressed as the gradient of a scalar electric potential q. However, to get symmetric matrices in the numerical formulations, it is customary to refer to a scalar potential 9 having the physical dimensions of a magnetic flux, which is actually the time integral of the scalar electric potential q. With this choice we can express the electric field as aA

a9

E=-at-V-a t

12

R. ALBANESE AND G. RUBINACCI

At the electric interfaces, the possible jump of the normal component of E is absorbed by a discontinuity of the time derivative of a$/an. So, both 4 and the three components of A can be kept continuous everywhere. C. Gauges

The introduction of the vector potentials allows us to deal with discontinuity surfaces without making use of discontinuous basis functions. The price to be paid for using a vector potential is the introduction of an additional scalar unknown (the scalar potential), which unavoidably gives rise to another equation. If we rewrite the magnetic field formulation in terms of the vector potential T and the scalar potential S2, we still have the uniqueness of the magnetic field H = T - V52 (Albanese and Rubinacci, 1990a; Fernandes, 1995). However, ,he solution in terms of T and S2, separately, is not unique. If (TI, S2l) is a possible solution, then (T2= TI - V@, 522 = 52, - +) is also a solution, with being any scalar function. Therefore, to assure the unicity of T and S2 separately, an additional condition must be imposed. In the classical approaches, this additional relationship, called the gauge condition, involves the divergence of T.The Coulomb gauge prescribes the divergence of T to be zero, whereas the Lorentz gauge links it to the time derivative of the scalar potential 52. In both cases, the uniqueness of the potentials is guaranteed by imposing the interface conditions and suitable boundary conditions (Albanese and Rubinacci, 1990a; Fernandes, 1995). Similar comments can be made for the electric field formulation and the magnetic vector potential A, for which both Coulomb and Lorentz gauges can be applied (Polak et al., 1983; Rodger, 1983; Biro and Preis, 1990; Biro and Richter, 1991; Bryant et al., 1990). A particular magnetic vector potential is that defined by

+

A* = - L E d t

(41)

+

which obviously satisfies the condition E = -aA*/at and B = B, V x A*. B, is the prescribed distribution of the flux density at the initial time t = 0. In this case the gauge condition is implicitly imposed as A* is uniquely defined by the solution in terms of fields. In other words, the scalar potential is forced to be zero. The vector potential A* allows one to write the diffusion equation in the conductor

V

x

H(V x A*) = -aaA*/at

+ J,

in V,

(42)

which can be regarded as the dual form of the magnetic diffusion equation (34). This potential can also be used for the electromagnetic problem defined by the full Maxwell equations, in which the dual potential F* is uniquely defined as:

F* = L H d r

(43)

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

13

+

satisfying the conditions H = aF*/at and DT = DO V x F*, with Do being the initial condition for the displacement vector and

DT = Do

+

l ( J+

aD/at) d t .

It is worth noticing that these vector potential A* and F* defined by (41) and (43), imply the use of only three unknowns, but unavoidably suffer of the same problems at the discontinuity surfaces as E and H. In our work, we have mainly referred to a different choice of the gauge condition, which prescribes the component of the vector potential along the director determined by an arbitrarily chosen non-vanishing vector field w which does not possess closed field lines (Carpenter, 1977; Brown, 1982; Albanese and Rubinacci, 1988a, 1990b). We prescribe this component to be zero:

with w selected as an arbitrary vector field without closed field lines. Then one of the three components is eliminated and the total number of scalar unknowns is reduced from four to three (two components of the vector potential along with the scalar potential). In some cases, the number of scalar unknowns can be further reduced (see Section V). The uniqueness of the vector potential for its corresponding vector field has been reported in Albanese and Rubinacci (1990b). Here we briefly recall the main points in the case of the electric vector potential; with obvious substitutions, the same proof holds for the magnetic vector potential. Let us suppose that there exist two different vector potentials TI and TZfor the same current density distribution J and that both TI and TZsatisfy gauge condition (44):

VxT*=VxT*=J

T I . w = T ~ . w = O , inV

(46)

If V is a simply connected region, then the difference can be expressed as the gradient of a scalar function

+:

TI - TZ= V+,

in V

(47)

Let us fix a reference point Po and take an arbitrary point P in V. If there always exists a field line y of w that connects P to the reference point Po, we get + ( P ) - *(Po) =

J

v+

*

tdl =0

(48)

povp

-

since t, the unit tangent vector to y. is parallel tow and V J , w = (TI - Tz).w = 0. Therefore, is everywhere equal to its arbitrarily fixed value in Po, hence TI - Tz = V+ = 0, which proves the uniqueness of T.

+

14

R. ALBANESE AND G. RUBINACCI

This proof can be extended to the cases in which the current density J and the vector field w are discontinuous, provided that the continuity of the tangential components of T is imposed on the discontinuity surfaces. What is strictly needed is that (1) the region V is simply connected, otherwise (47) cannot be imposed; (2) the field lines of w passing through the reference point Po span the whole region V, otherwise there exists a point P for which (48) cannot be applied; (3) the vector field w must not possess closed field lines, otherwise the circulation of T along a closed field line of w would be zero, which, due to Stokes’ theorem, would arbitrarily force the linked total current to be zero. A simple example of an admissible w is the radial field defined by w(P) = P - Po,i.e., the vector connecting P to the reference point Po. A convenient selection of the vector field w,well suited for the numerical calculation is widely illustrated in Section IVB, whereas the treatment of multiply connected regions is illustrated in Section IVC. The two-component electric vector potential was first used in conjunction with isoparametricelements, postulating the continuity of the two nonzero components of the vector potentials (Carpenter, 1977; Brown, 1982; Albanese et al., 1985). However, it was difficult to confine the electric vector potential inside the conducting regions even in the case of simply connected regions, because the vector potential could not be zero at the boundary of the conducting region. According to Carpenter (1977), the T vector appeared on a downstream side of the conductor as defined by w.In addition, large numerical errors arose in the magnetic regions (where H is small when compared to T and VQ separately). Finally, the discontinuity interfaces, including the boundary of the conducting regions, were not taken into account adequately. To understand why problems may appear at the electric (respectively, magnetic) interfaces when using the two-component vector potential T (respectively, A), let us consider the simple case in which w is uniform and parallel to the z axis of a Cartesian coordinate system, e.g., w = Vz, yielding

a TX

a TY , J y = -, w = VZ, Tz = 0, J, = -az

az

aT, aT, J, = - - - (49) ax ay

In this case the discontinuitiesof J, and J y across surfaces at constant z (where J, is continuous) can totally be absorbed by the jumps of the normal derivatives of T, and T, without forcing T to be discontinuous. However, there is a problem if w is not perpendicular to the discontinuity surface. For instance, an electrical interface at constant x calls for the continuity of J,, E,, and E,, hence the discontinuity of J, and J,. The continuity of J, = -aTy/az is properly taken into account by the continuity of aT,/az, the tangential derivative of T,. The discontinuity of J, can be absorbed by the jump of the normal derivative of T,. However, the continuity of J y implied by the continuity of aT,/az, the tangential derivative of T,, is in contradiction with the requirement. To allow both components of the current density to be discontinuous,it is necessary to relax the continuity requirement to the normal component of the vector potential. As shown in Albanese and

15

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

Rubinacci (1988a,b, 1990b), to avoid these problems, it is essential to refer to numerical schemes based on the edge elements, whose details are also illustrated in Section IV. At the end of this discussion on the vector potentials, we would like to summarize the reasons why they are so widespread in computational electromagnetics. One is probably historical, since classical solutions of electromagneticfields in the free space are based on the introduction of vector potentials. However, there are other motivations that are more related to the numerical solution of the problem. As we have seen above, the use of the vector potentials in conjunction with the Coulomb or Lorentz gauges allows for a good handling of the discontinuity surfaces. However, in our opinion, the most important thrust toward the use of the vector potentials lies in that they allow for a unified treatment of conducting and nonconducting regions, and of magnetostatic, direct current, and eddy current problems (see, for instance, Rodger, 1983; Biro and Preis, 1990; Kameari, 1990a; Nakata et al., 1990; Biro and Richter, 1991; Albanese et al., 1991; Albanese and Rubinacci, 1993b).

Iv. EDGEELEMENTS FOR 3D FIELD PROBLEMS AND VECTOR POTENTIALS When a variational formulation is introduced,the field problem reduces to give the minimum (or the stationary point) of a functional in a proper functional space. The fields E, H,B, D,and J should belong to L2(V), the space of the square integrable vector fields over V, because the associated energy and power are finite. Moreover, according to the previous discussion, the continuity of the tangential component of E, H, T,and A, as well as the normal component of B, D,and J, is required at any material interface. The normal components of E,H,T,and A are free to jump as well as the tangential component of B, D, and J. On the other hand, the scalar potentials S2 and 9 should be continuous. These properties can be formally stated as follows:

E, H, A, T E L&(V) = (W E L2(V), V x W E L2(V)} B, D, J E L&(V) = (W E L2(V), V * W E L 2 ( V ) ] n,9 E LLad(V) = ( w E L2(V), v w E LZ(V)}

(50)

(51) (52)

Additional constraints to the various vector fields are related to the essential boundary conditions (i.e., constraints like the Dirichlet boundary conditions that can easily be enforced for fields or potentials at the discretized level). These have been omitted in the functional space definitions (50)-(52) and therefore have to be explicitly enforced when required. The particular structureof the vector fields involved in the formulationsrequires us to proceed with care in the discretizationprocess involved in the finite element approach. The adoption of the usual nodal shape functions forces the fields to be continuous all over the mesh. For this reason, the nodal elements are “too

16

R. ALBANESE AND G. RUBlNACCl

rigid in this respect, and do not lead to convergent numerical schemes” (Bossavit and Mayergoyz, 1989; Bossavit, 1990). A better choice is to adopt edge elements (Nedelec, 1980; Bossavit, 1988) for representing vector potentials and fields, and nodal elements for the scalar potentials. Edge elements were first introduced in the electromagnetic community in 1982 by Bossavit and VCritC (1982), who proposed a mixed FEM-BEM (Finite Element Method-Boundary Element Method) eddy current formulation based on the H variable. In 1985, van Welij (1985) proposed first-order hexahedral edge elements for weak formulations in terms of H and E. In 1987, Albanese and Rubinacci proposed an eddy current integral formulation in terms of T (Albanese and Rubinacci, 1988a,b)and a T, S2 differential formulation (Albanese and Rubinacci, 1988a), both based on a gauge condition that takes advantage of a tree-cotree decompositionof the edges of the finite element mesh. The extension to magnetostatics and an A, 4 eddy current differential formulation were discussed by Albanese and Rubinacci (1988c, 1990b). The tree-cotree decomposition is illustrated in detail in Section IVB. In 1988, Bossavit (1988) provided a rationale for the introduction of the edge elements in the theory and practice of magnetic field computations. In this period, it is also appropriate to mention the work of Mur and de Hoop (1985). Since then, there has been an increasing interest inside the electromagnetic community, and the number of papers related to the use of the edge elements for electromagnetic computation has significantly grown. This behavior is illustrated in Fig. 1, which reports the number of papers dealing with

20

[

15

n Ic

9

s

10

5

0 1982 1983 1985 1988 1990 1992 1994 1996

Year FIGURE1. Number of papers dealing with edge elements presented at the COMPUMAG conferences and published in the IEEE Trunsucrionson Magnerics in the years 1982-1996.

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

17

edge elements presented at the COMPUMAG Conferences and published in the ZEEE Transactions on Magnetics in the years 1982-1996. A. Edge Element Shape Functions

Here, we briefly recall the fundamental points of the theory of the edge elements, mainly following Bossavit (1988, 1996). The degrees of freedom of the edge elements are associated with the tangential components along the edges of the elements. In this way, the adoption of the edge elements for the representation of vector fields E and H (or vector potentials A and T)automatically guarantees the continuity of the tangential components. For a tetrahedral element t, the shape function associated with the edge e = { i , j),connecting ith and jth nodes is N;j = N j V N j - N j V N ;

(53)

Here Nk represents the scalar shape function associated with the kth node. Nk is continuous and, in the case of four-node tetrahedra, piecewise linear, with the following properties:

where x , is the location of mth node, vd is the finite element discretization of the domain P, and P is the number of nodes of the finite element mesh. In the notation of the rest of the paper, we will neglect the possible difference between Vd and V. We notice that The line integral of N;j along the edge e = {i, j } is unity:

in fact, along the edge e of the tetrahedral r = { i , j ,k,Z], according to (54)-(56), Nk = Nl = 0, N; = 1 - N j , V N j = - V N ; , and N;j = V N j ; thereN;j dl = N j ( x j ) - N , ( x ; ) = 1. fore, The line integral of N;j is zero along any other edge where either N; or N , or both are zero:

s{i,j) -

s

1k.U

N;, . dZ = -

l,kl

Nij * dZ = 0,

{i, j } # [k,l } and {i, j ) # {Z, k} (58)

The tangential components of N i j are continuous across the faces of adjacent elements, since the scalar functions Nk are continuous.

R.ALBANESE AND G.RUBINACCI

18

7

8

N9

N1 FIGURE 2. The gradient of the nodal shape function N I in a hexahedral element is given by a linear combinationof the edge shape function Nl,N4,N9,i.e., V N I = N I+ N4 + N9.

The normal components of Ni, are not necessarily continuous. N i j and V Nk belong to the same functional space; in fact, taking into account property (56), it is easy to show that the linear combination of edge shape functions can reproduce the gradient of a scalar shape function (Fig. 2):

e= 1 where E is the number of edges of the mesh, and { C )is an E x P incidence matrix defined by Gem= 1, Gem= -1, Gem = 0,

e = {i, j ) andm

=j

(60) (61)

e = {i, j ] andm = i e = {i, j ] , m #

i andm # j

(62)

Property (59) drastically reduces cancellationproblems, such as those experienced in the T,!J formulation in magnetic materials (Simkin and Trowbridge, 1979). Similarly, a set of shape functions S f associated with the faces of the elements can be introduced. For each oriented face f = {i, j , k] = (k,i, j ) = { j , k, i) having three vertices, the nodes i , j, k, we have

s f = 2(NiVNj X VNk + NjVNk x V N , + N k v N i

x VNj)

(63)

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

19

It can be shown that the flux of S f is unity across the face f and zero across any other face. The normal component of Sf is continuous across the faces of adjacent elements,whereas the tangential componentis not necessarilycontinuous. Moreover, it can be easily verified that the curl of an edge shape function is a linear combination of facet shape functions: F

V x N, = C C , , S f

(64)

f= I

where F is the number of faces of the mesh, and (C) is an F x E incidence matrix defined by

Cf,=l, C,, = -1,

f =(m,n,I}ande=(m,n) f = ( m , n, I) and e = (n,m )

(65)

e$f

(67)

Cf, = O ,

(66)

In the language of differential geometry, the edge elements are 1-Whitney elements (or Whitney forms) (Whitney, 1957; Bossavit, 1988). They generate the finite dimensional vector space W' of dimension E, if E is the number of edges of a given finite element mesh obtained by subdividing the bounded domain V in a finite number of tetrahedra. Analogously, the facet elements span the finite dimensional vector space W 2of dimension F, the number of faces. For the sake of completeness,we notice that nodal elements with node shape functions N; generate the space W oof dimension P (P is the number of nodes), the 0-Whitney element. Each Whitney element has a different behavior across the element faces. Nodal functions Ni are continuous,edge vector functions have only their tangential component continuous across faces, while face vector functions guarantee only the normal continuity. This property can be stated in a formal way by saying that W" c Lirad, W' c L:o,, and W 2c Liiv.In this way, it is possible to obtain a discrete approximation of the electromagnetic fields that preserves their continuity properties. Remember in the following that the gradient of a 0-Whitney element is a linear combination of 1-Whitney elements; similarly the curl of a 1-Whitney element is a linear combination of 2-Whitney elements:

VW" c w' v x w' c w2

(68) (69)

In addition to the first-order tetrahedra, fist-order prisms (Dular et al., 1994), first-order hexahedra (van Welij, 1985; Dular et al., 1994) and second-order hexahedra (Kameari, 1990a) have also been proposed and utilized. Although the expression of the associated shape functions vary, these elements retain the set of

20

R. ALBANESE AND G . RUBINACCI

properties described in this section. Figures 3-5 show nodal, edge, and facet shape functions, respectively, in a hexahedral element.

B. Tree-Cotree Decomposition According to the discussion of Section IIIC, to guarantee the uniqueness of the vector potential, a gauge condition has to be imposed. As shown by (47), the difference of two admissible vector potentials of the same field is irrotational. Therefore, the role played by the gauge condition is to eliminate all the irrotational fields from the functional space in which we search for the vector potentials. In other words, the gauge condition should constrain the null space of the discrete curl operator to be void. The edge elements are well suited for dealing with a discrete analogue of the gauge condition T . w = 0 (or A w = 0) with w selected as an arbitrary vector field without closed field lines. We introduce our approach refemng to the T, 52 formulation, where J = V x T and H = T - V52. We consider a simply connected region V,,postponing the treatment of multiply connected regions to Section VC. We express T and J using the element introduced in Section IVA, namely 1-Whitney edge elements for T and 2-Whitney facet elements for J: E

T=

T,N, e= 1

-1

-0.5

0.5

0

1

csi at <

FIGURE3. Nodal shape function Nl in a hexahedral element as a function of 6 and q coordinates = 1.

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

5

21

fq

1 ’

FIGURE4. An edge shape function and its curl in a hexahedral element: (a) NI along the arc ( a + I)’ + (5 + = I in the plane 6 = 0; (b) N1 along the segment 5‘ = -1 in the plane 6 = 0; (c) N Ialong the segment < = 9 in the plane 6 = 0; (d) N I along the segment t) = - 1 in the plane 6 = 0; (e) V x N I along the arc ( 9 + I)’ + (< + 1)’ = I in the plane 6 = 0; (0 V x N1 along the segment < = - 1 in the plane 6 = 0: (g) V x N Ialong the segment 5‘ = t) in the plane 6 = 0; (h) V x N Ialong the segment q = - I in the plane 6 = 0.

R.ALBANESE AND G.RUBINACCI

22

5@ 1

/JU

--3

7 ,

5

I

/

1

/’

V

+

FIGURE5 . A facet shape function in a hexahedral element: (a) SI along the arc (t) I)’+ ( ~ + 1 )= ~ lintheplanet = O;(b)Sl alongthesegmentconnectingpoints(6 = 0.I) = O,( = -1)and (t = 0, t) = 1 5‘ = - 1); (c) S I along the segment connecting points (6 = 0.q = - 1, ( = - I ) and (6 = 0, t) = - I , ( = 0); (d) S I along the segment connecting points (6 = 0, t ) = 0, 5 = - 1 ) and (6 = 0.1 = 0, ( = I). I

and

f=I

where Ne’s are edge element shape functions, and S f ’ s are facet element shape functions. Thus, T, and .Ifare the degrees of freedom associated with the edge e and the face f. respectively. Notice that expanding J in terms of facet elements does not automatically assure its solenoidality. In fact, the flux of S f is zero across any face of the mesh except face f,and, for the tetrahedral elements, we have V S f # 0 inside the element where (63) holds. Solenoidality is instead assured by expressing J as V x T. In this way, (70) yields

-

E -

J=

C T,V x N, e=

I

and, using (a), E

F

e=l f = l

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

23

Equation (72) shows that the coefficients of expansions (70) and (71) are linearly related:

[JI

= ICm-1

(74)

where [J] and [TI stand for the numerical vectors of coefficients J f and Te, respectively. This relationship can also be derived as follows. Due to (57) and (58), the coefficient T, provides the line integral of T along the oriented edge e. On the other hand, the degree of freedom J f represents the flux through the oriented face f. This flux can be obtained, via Stokes' theorem, as the circulation of T along the edges of the faces. Therefore, according to the definition of incidence matrix (C) given by (64)-(66), we get E

Jf=CCfeT,,

f =l,...,F

(75)

e= 1

Relationships (74) and (75) can be regarded as a linear system:

ICWI = [JI

(76)

where [TI and [J]play the roles of unknowns and right-hand side, respectively. System (76) does not possess a unique solution. The existence of a solution is subject to the compatibility of the system, which subsists if and only if [J] corresponds to a solenoidal current density distribution. If the system is compatible, the solution is not unique, as clearly demonstrated by (59). Adding any linear combination of the columns of {G}to [TI does not modify [ J ] = ( C } [ T ]because , the columns of {G}yield irrotational vector potentials and zero current densities: {CHG}= {OI

(77)

here {0)is the F x P null matrix. Thus, in our discrete approximation, imposing the gauge condition is equivalent to the problem of assuring a uniqueness condition to system (76). The rank of the E x P (G) matrix is P - 1, as the sum of the coefficients of each row is zero. Therefore, (77) shows that the rank of the F x E (C]matrix is E - P 1. Let us refer to the graph formed by nodes and edges of the edge elements mesh. As shown by (79, the flux of J = V x T across any elementary face is related to the circulation of the vector potential along the edges identifying the face. We decompose the graph in an arbitrary tree, formed by P - 1 edges, and the cotree, formed by the residual E - P + 1 edges. According to the fundamentals of the circuit theory, each edge of the cotree closes a single independent loop with the edges of the tree, and the circulation along any other loop can be obtained by linear combination of the circulations along the E - P 1 independent loops associated with the edges of the cotree. Henceforth, and degrees of freedom corresponding

+

+

24

R. ALBANESE AND G. RUBINACCI

to the edges of the tree can be eliminated, leaving a reduced set of coefficients [TICassociated with the E - P 1 cotree edges, and a reduced F x (E - P 1) matrix (C)' with rank E - P 1. In this way, for any compatible [ J], the solution of system

+

+

+

ICIC[TlC= [JI

(78)

is unique. This is the discrete analogue of the gauge T w = 0, where the field lines of w are given by the edges of the tree, connecting all the nodes without forming closed loops. The number of unknowns is decreasedby the number of tree edges to remove the arbitrary choice of a gradient field. The degrees of freedom corresponding to the edges of the tree are eliminated, and the remaining degrees of freedom are thus related to the fluxes of J = V x T linked with the set of independent loops closed by adding each of the residual edges (i.e., each edge of the cotree) to the tree (Albanese and Rubinacci, 1988a,b, 1990b). Another way of presenting this theory is given in Kettunen and Forsman (1996). According to (59), a gradient field can be represented as a linear combination of edge shape functions. All the edge degrees of freedom describing this gradient field can be calculated from the values of the coefficients corresponding to the line integrals along the tree edges. In fact, the algebraic sum of these degrees of freedom has to be zero for any closed loop. Hence, considering the oriented closed loops associated with the cotree edges and corresponding set of tree edges, we can define an E x ( E - P 1) incidence matrix Q defined by

-

+

Qe, = 1, if the edge e belongs to the loop I , with the same orientation

(79) Qe/ = - 1, if the edge e belongs to the loop I, with different orientation (80) Qc/ = 0, if the edge e does not belong to the loop 1

(81)

All the degrees of freedom [ F] for a gradient field F can be calculated from the subset of coefficients [F]' corresponding to the tree edges, since the remaining coefficients corresponding to the cotree edges are given by

[FIC= -IQ)[Fl'

(82)

As a consequence, any gradient field can be eliminated by zeroing the degrees of freedom [F]'. Notice that the tree-cotree decomposition allows for an easy decomposition of the vector space L2 into the two complementary parts

W, = {F = '752,52 E Lirad}

(83)

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

25

and

W: = (J = V x T,TE L:ot, J - n = 0} which is in strict correlation with the T,2 ! formulation. The tree-cotree decomposition was first proposed in Albanese and Rubinacci (1988a). It was successfully applied in a linear and a nonlinear integral formulation of the eddy current problem by the authors and by Kettunen and ’hrner (1992), Kettunen et al. (1995), Kettunen and Forsman (1996), and Forsman et al. (1996). These formulations are discussed in Section V. The extension of the analysis to the differential formulations leads to very efficient magnetostatic and eddy current T, R formulations that are discussed in Section VI. The dual formulations in terms of A, instead, presented a poor conditioning and thus a high number of ICCG iterations (Kameari, 1990b; Fujiwara, 1992; Biro et al., 1996). In the high-frequency problem, the tree-cotree decomposition is useful from a theoretical point of view, for a better understanding of the question of spurious modes. The equivalence of the spaces spanned by the tree edges and by the gradient of a scalar field, implies an analogous equivalence between the field formulations (either E or H)in terms of edge elements and the potential formulations (either A, qj or T,R) when adopting the gauge based on the tree-cotree decomposition. These topics are also briefly discussed in Section VI. Nevertheless, quoting Menges and Cendes, who analyzed this approach in 1995, “it remains to refine and evaluate the method as a possibility for more efficient solution of high frequency problems.” C. Boundary Conditions for Simply and Multiply Connected Regions

In the formulations in terms of H (respectively, E),essential boundary conditions on H x n (respectively,E x n) can be imposed directly on the degrees of freedom associated with the corresponding boundary edges. In the A, 4 formulation, the boundary condition B . n = 0, implied by A x n = 0, can easily be imposed, in simply connected domains. We form the tree by first connecting the boundary nodes with boundary edges only. In this way, any edge of the cotree closes a loop lying on the boundary a V. The corresponding degree of freedom is the magnetic flux across a surface lying on a V, which must be zero. Therefore, the boundary condition can be imposed by eliminating the edge of the cotree lying on a V. In this way, the active edges, i.e., the edges to which nonzero degrees of freedom are associated, are the n,; edges of the cotree that are not on the boundary. Figure 6 shows the mesh, the tree, the cotree, and the active edges, with reference to a simply connected domain and the boundary condition B .n = 0. In the T,R formulation, T is usually defined only in the conducting domain V, and in the source volume V, where the current density source J, is supposed to

26

R. ALBANESE AND G. RUBINACCI

\

TREE COMPLEMENT

ACTIVE EDGES

(4

FIGURE6. A simply connected domain with the boundary condition B . n = 0.

be known. Obviously, to have 52 continuous at the interfaces between V,, V, and V - V, - V,, we should impose T x n = 0 on the boundary of V - V, - V,. This is straightforward if V - V, - V, is simply connected. In fact, in this case, we can use the same technique utilized to impose the boundary condition B n = 0 in the A, (p formulation. Namely, we form the tree in such a way that a subset of it, the boundary tree, is a tree for the graph formed by nodes and edges lying on the boundary of V - V, - V,. In this way the degree of freedom associated with each boundary edge of the cotree is the total current crossing a surface lying on the boundary (related to the loop closed by the tree and that boundary edge), which must be zero. Henceforth, the condition is met by simply eliminating the degrees of freedom associated with the boundary edges of the cotree. In this way,

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

27

the basis functions Te for the gauged potential T are given by the edge element shape functions N, associated with the n,i active edges. If V, or V, are multiply connected, the same conclusioncannot be drawn because a closed line lying on the boundary is not necessarily the contour of a surface entirely lying on the boundary (consider, for instance, the cross section of a wire). However, the T, 52 method can be used in these cases, avoiding the definition of multivalued scalar potentials. We discuss separately the cases of V, and V,. If V, is multiply connected, we simply extend the region, including an additional volume V,' (not necessarily nonconducting) in such a way that V, U V , is simply connected. At this point we obtain the vector potential T, associated with the source current J,, defined by V x T, = J,, T, . w = 0,

T, x n = 0,

in V, U V, in V, U V, on a(V, U V,')

and use the technique described above. If V, is the multiply connected volume, the same procedure cannot be applied because J is not known in advance. An efficient way to cope with this difficulty is the following. Again, we form the tree in such a way that a subset of it, the boundary tree, is a tree for the graph formed by nodes and edges lying on the boundary of V,. Then we consider the Euler's formula (Griffiths and Hilton, 1970; Albanese and Rubinacci, 1990c)for the polyhedral surface formed by the faces of the finite element mesh lying on aV,: Pb

- Eb -k

Fb

=2 -2p

(88)

where p is the genus of the polyhedron, & is the number of boundary nodes, Eb the number of boundary edges, and F b the number of boundary faces of the mesh. The number of degrees of freedom Nd belonging to the boundary i3 V, are Nd = Eb - ( P b - 1)

(89)

as pb - 1 is the number of edges of the boundary tree. On the other hand, we have to impose zero flux across each elementary face of the boundary. Thus, we have a homogeneous system of Ne independent linear equations: N, = F b - 1

(90)

since Fbth is implied by the first F b - 1equations, because J = V x T is solenoidal. If the region is simply connected then p = 0, Nd = N,, and the only active degrees of freedom are those related to the internal cotree edges. On the other hand, if p > 0 there are 2 p independent distribution TAdd on the boundary that give V x TAdd n = 0. However, only p distributions are related to net fluxes of V x TAdd across surfaces defined inside the region V,. The other p values, related

28

R. ALBANESE AND G. RUBINACCI

to net fluxes of V x T~ddlinked with cross sections of R3 - V,, give rise to nonzero V x T ~ d distribution d inside V,, which can also be obtained as linear combinations of the other (internal and boundary) degrees of freedom. A numerical procedure can be designed leading to the automatic preprocessing of multiply connected domains (Albanese and Rubinacci, 1990~).It consists in the minimization of the quadratic form:

where T is approximated, as usual, by a linear combination of shape functions. If the result of the minimization gives @h = 0, then TAdd is redundant. If @ h > 0, the distribution related to the additional degrees of freedom cannot be reproduced by internal edges and should therefore be included among the shape functions. The procedure goes on until the p independent additional basis functions T ~ d are d found. The mesh, the tree, the cotree, and the active edges are shown in Fig. 7, with reference to a multiply connected domain V,. In this simple case, there are four additional independentdistributionsT A d d on the boundary, but only the first two are active additional degrees of freedom, being related to currents circulating inside V,. Once the p independent additional degrees of freedom T ~ d dhave been determined on aV,, we select an additional volume V,' such that V, U Vi is simply connected. Of course, the line integrals of TAdd along the internal edges of V, are arbitrary and can be set to zero. Instead, in V,' we must impose J = V x T ~ d d= 0. Henceforth, we extend the computation of T ~ d din Vl, by solving the following magnetostatic problem for each of the p additional degrees of freedom (Albanese and Rubinacci, 1994): V x

T ~ d d= 0,

T ~ d d. w T ~ d dx

= 0,

n = K x n,

in VL in V,' on aV,l

where K is the known value of T ~ d at d the interface between V, and VL, and K = 0 at the boundary of V, U V,'. Notice that in this way there is no need to define a set of cuts in V - V, - V,, and all that is needed is readily and automatically computed by a suitable preprocessor. In conclusion, for a simply or multiply connected region, the basis functions T, for the gauged potential T are n = n,i p , namely the n,i edge element shape functions N, associated with the active edges, and, for p > 0, the p additional degrees of freedom T A d d .

+

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

MESH

29

w

u- n n

TREE COMPLEMENT (C) (4 FIGURE7. Multiply connected domain and additional basis functions Tadd: (a) mesh; (b) tree; (c) cotree; (d) boundary distributions; (e) active additional degrees of freedom.

30

R. ALBANESE AND G . RUBINACCI

(e) FIGURE 7. Continued.

v.

INTEGRAL FORMULATIONS FOR LINEAR AND

NONLINEAR EDDYCURRENTS

A. Formulation of the Linear Eddy Current Problem Let us consider linear nonmagnetic conductors in the free space !Ti3. The relevant constitutive properties are

B=~oH,

E = qJ, J = Js,

in%j

(95)

in V,

(96)

in !R3 - V,

(97)

where q = o-’is the resistivity and POis the vacuum permeability. Faraday’s law is automatically satisfied by assuming

E = -aA/at - vq

(98)

The magnetic vector potential is defined by (39) with the Coulomb gauge:

V*A=O

(99)

with regularity conditions at infinity. In this way, Ampere’s law and constitutive equations (95) and (97) allow us to express A in terms of the unknown solenoidal

31

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

current density:

where A, is the magnetic vector potential given by the sources located in 8’- V,:

Constitutive equation (96) can be verified in an average sense by adopting the weighted residual approach:

1

W*(qJ-E)dV = O ,

VW E S

(102)

JES

(103)

where S is the subspace of L&( V,) defined by S = { J E Lji,( Vc),V J = 0 at regular points of V,, J .n = 0 on aVc} (104) Substituting (98) and (100) in (102), we finally obtain the weak form:

J C . W . { ~ , JatJ -4n ~ [ Ev, /w Ix-x’l dV’]+%}dV=O,

VWES

JES In (105), the term related to the scalar potential cp disappears because of the solenoidality of the weighting function W and the boundary condition W n = 0 on aV,. B. Finite Element Approach to the Solution of the Linear Eddy Current Problem To obtain a numerical solution, J is approximated as a linear combination of n basis functions Jk E S : n

h 0 )Jk

J(x, t ) =

( 107)

k=l

which satisfy (106), whereas (105) is imposed by taking n independent weighting functions wk E S. Condition Jk E S can be satisfied by introducing the electric vector potential T and adopting edge element shape functions for T.The uniqueness of the vector potential is assured by the gauge T w = 0. This gauge is conveniently imposed directly on the basis functions, introducing the tree-cotree

.

32

R. ALBANESE AND G. RUBINACCI

decomposition of the mesh and eliminating the degrees of freedom associated to tree edges, as described in Section IVB. Condition Jk. n = 0 on a V, can also directly be imposed on the shape functions, using the approach explained in Section IVC. The shape functions for J are therefore derived from the n basis functions for the gauged vector potential defined in Section I V

Using Galerkin method, i.e., assuming Wk = Jk, Eq. (105) reduces to the following linear system of ordinary differential equations, which can be solved using time stepping:

where V x T;(x) * V x Tj (x’) dVdV’ Ix - X’I

(1 10)

A great deal of the computational burden resides in the numerical calculation of the ( L ) matrix, due to the double volume integral of (1 10). In addition, particular attention should be paid to the numerical computation, due to the presence of the singular contribution of l/lx - x’l. A method to tackle this difficulty has been discussed by Albanese and Rubinacci (1988b). The basic idea is as follows. A direct application of the Gaussian cubature formulas leads to interpret the L i j term as the mutual energy between the sets of the vector point charges (w(xk)Ji(xk)) and {w(xk)Jj(xk)),located at the Gauss points x k , where w(xk) are the associated Gauss weights. To keep the energy finite, the point charges arising in the discretization process are replaced by uniformly charged spheres of an adequate radius, for which an analytic expression of the mutual energy is available. If the spheres do not overlap, this expression of the mutual energy coincides with the energy of interaction between the set of point charges. The radius of each sphere is computed by imposing the condition that for each finite element the selfenergy of the uniformly charged spheres is equal to the double volume integral of p0/4nlx - x’l, computed with an inner analytical (Collie, 1976; Urankar, 1990) and an outer numerical integration. To obtain (109) from (105). we implicitly assumed that the conducting region V, is not moving in our coordinate system. The more general form of (1 09) in the

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

33

presence of the motion or deformation is (Albanese et al., 1996b) d

+

-({LHII) IRHII = [UI (1 13) dt where the calculation of the matrices may be required at each time step. On the basis of this numerical formulation, we developed CARIDDI, a finite element code for the analysis of the eddy current induced in nonmagnetic conductor (Albanese and Rubinacci, 1988b, 1990d). The code, which employs first-order hexahedral edge elements, has been successfully applied to the solution of several test cases proposed for the TEAM series of workshops (see, for instance, Nakata, 1990). In particular, results were obtained concerning the hollow cylinder (Davey, 1988),the hollow sphere (Emson, 1988),the Felix brick (Kameari, 1988), and the Bath plate (Rodger, 1988). Here, we report some numerical results concerning TEAM Problem 8, Coil Above a Crack: A Problem in Non-Destructive Testing (VCritt, 1984). It deals with the calculation of the differential flux linked with two small receptive solenoids due to the eddy currents induced by an exciting solenoid, ’ of austenitic all placed 8 mm above the surface of a 330 x 285 x 3 0 - m mblock stainless steel (a = 1.4 MWm) containing a 40 x 10 x 0.5-mm3rectangular slot. The probe is shown in Fig. 8, and the conducting block with the flaw is shown in Fig. 9. The objective is to calculate the two output signals and their difference for a motion parallel and perpendicular to the flaw, as defined in Fig. 10. The frequency of the source is 500 Hz. The main difficulty of the problem is that the differential flux is the difference of two values very close to each other. The application of the reciprocity theorem can conveniently be exploited (Albanese et al., 1990). Let us state of the theorem in a form suitable for the case under study. To this purpose, consider two current sources J: and J! localized in the volumes V, and vh, respectively, giving rise in all the space to the electric fields Ea and Eh,respectively. Then, from linearity and symmetry of the 0 and p tensors, it follows that the reciprocity can be expressed as

We rewrite each integral of (1 14) with reference to the solenoids of TEAM Problem 8:

Here is the flux linked with the differential receptive coil due to the magnetic field produced by the current laflowing in the exciting solenoid. On the other hand, represents the magnetic field linked with the exciting solenoid, that would be produced by a current I’ flowing in the receptive coil. Taking I’ = Za, Eq.( I 15) provides @: = @: at sinusoidal steady state. As a consequence, the differential flux due to the current flowing in the active coil

34

R. ALBANESE AND G. RUBINACCI

R coil

FIGURE8. TEAM workshop Problem 8. The probe (units are millimeters).

can be computed as the flux linked with the active probe induced by a suitable pair of opposite currents flowing in the receptive coils. In this way, the unknown of the problem is directly the current density distribution induced in the block by a differential probe, and the numerical superposition of almost equal signals is avoided (Albanese et al., 1990). Only one-fourth of the block has been discretized. For each position of the inducing solenoid, the solution has been computed as the superposition of the

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

35

FIGURE9. TEAM workshop Problem 8. The conducting block with the flaw (units are millimeters).

fields generated by symmetric and antisymmetric sources. The crack has been modeled with zero thickness, by imposing J . n = 0 on a number of faces. The relevant results obtained using reciprocity with a rather coarse mesh (448 elements, 765 nodes, 670 unknowns) are shown in Fig. 11 in comparison with the experimental data (Takagi et al., 1989). In Fig. 12 we also report the results obtained without exploiting reciprocity, using the same mesh and a finer mesh characterized by 798 elements, 1320 nodes and 1193 unknowns. These results show that reciprocity allows us to considerably reduce the size of the system matrix to achieve a given accuracy for a problem in Non-Destructive Testing. The CARIDDI code is widely used for the electromagneticanalysis of toroidal fusion devices. This integral formulation seems to be the most adequate to analyze the eddy currentsinduced in both massive structuresand thin shells, which typically coexist in toroidal fusion devices. As a matter of fact, in thin plates, the unknowns (the internal edges directed along the normal to the shell of thickness A) exactly correspond to the values of the surface current stream function @ = TA, where the surface current density K, = JA has been defined as

K, = V x T A n = V+ x n

(1 16)

In this way the integral formulation embraces the thin shell formulation studied by Bossavit (1981) and Blum et al. (1983). Here, we show an application related to the study of the eddy currents induced in the passive structuresof the InternationalThermonuclear Experimental Reactor (ITER)tokamak, an experimental fusion device designed in cooperation among the European Union, Japan, the Russian Federation, and the United States (Aymar,

R. ALBANESE AND G. RUBINACCI

36

Y

@

&

Motion parallel to the flaw

FIGURE 10.

TEAM workshop Problem 8. Trajectories of the probe.

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

37

(b) FIGURE I I. TEAM workshop Problem 8. Numerical results obtained by the integral formulation exploiting reciprocity with a coarse mesh characterized by 670 unknowns (dash-dot), in comparison with the experimental data (solid): (a) motion parallel to the flaw; (b) motion perpendicularto the flaw.

1996). In Fig. 13, we show the solid model of 1/40 of the conducting structure, including the first wall, the vacuum vessel, the toroidal field (TF) coil case and a TF coil plate, the intercoil structure, the poloidal field coils, the upper and lower crowns, and the cryostat (2084 elements, 4494 nodes, 2834 unknowns). Figure 14 shows the patterns of the eddy currents induced in the upper crown, in the TF case, and around a port of the vacuum vessel during the start-up phase of the plasma discharge.

38

R. ALBANESE AND G. RUBINACCI

Pewndicular:CR (-1 C(-.)F[-)

Exper(.)

0.8

0.2

FIGURE 12. TEAM workshop Problem 8. Numerical results obtained by the integral formulation with coarse (670 unknowns) and refined mesh (1193 unknowns). The computed curves are solid (coarse mesh, using reciprocity), dash-dot (coarse mesh, no reciprocity), and dashed (fine mesh, no reciprocity), whereas the dotted line corresponds to the experimental data: (a) motion parallel to the flaw; (b) motion perpendicular to the flaw.

C. Nonlinear Eddy Currents and Magnetostatics In the presence of magnetic materials, the formulation presented in Section VA cannot be applied as it is, because constitutive equation (95) must be replaced by its general form (3). We rewrite it in terms of B, H,and the magnetization

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

39

FIGURE13. The ITER tokamak. Solid model of 1/40 of the conducting structure, including first wall, vacuum vessel, toroidal field coil case and plates, intercoil structure, poloidal field coils, upper and lower crowns, and cryostat (2084 elements, 4494 nodes, 2834 unknowns).

vector M:

where Vf is the ferromagnetic region and the magnetization vector M is given by the constitutive relation:

- Vf, in which B = p0H. where B(B) = p;'B - X(B), whereas M = 0 in Therefore, in the presence of magnetic media, the expressionof the vector potential

40

R. ALBANESE AND G. RUBINACCI

(b)

(C)

FIGURE 14. Eddy currents induced in the passive structure of the ITER tokamak in different phases of a plasma discharge: (a) in the upper crown during start-up: (b) around a port of the vacuum vessel during start-up; (c) in the TF case during a plasma disruption.

becomes

which differs from (100) for the contribution of V x M in Vf.

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

41

The flux density B can be expressed in terms of J and M using the following form of Biot-Savart law: J(x’, t ) x (X- x’) d V’ B(x, t ) = Bs(xrt )

+

+ p O M - E L r V ’ . M ( x ‘ , t ) -(x - x’) dV‘ Ix - x’l-7 (x - x’) dS‘ M(x’, t ) . n’lx - xi13

+ where

and particular care must be taken to calculate the contribution of V M since M $ Lii,(Vf). Constitutive equations (96) and (118) can be verified in an average sense by adopting the weighted residual approach: +

Jvc

W . (qJ - E ) d V = 0,

WM. [M - O(B)]dV = 0,

VW E S V W M E L2(Vf)

( 123)

JES M E L2(Vf) in which the fields E and B can be expressed in terms of J and M using (98). (119), and ( 120). In this way, (122)-(125) provide an eddy current integral formulation whose unknowns are J and M. This formulation represents the natural extension of (105) and (106), to which reduces in case Vf is void. On the other hand, it provides a magnetostatic formulation in case V, is void. D. Iterative Proceduresfor the Nonlinear Eddy Current Problem For the linear problem described in Section VB, the current density at each time step can be determined directly by inversion of a linear system of algebraic equations. The presence of a magnetic material requires the use of iterative methods. Equation (122) can be used to determine J, assuming an arbitrary value of M. Equation (124), i.e., Biot-Savart law, can then be used to provide the flux density B. Finally, the magnetization vector M can be updated using the constitutive equation (1 18). The procedure is then repeated until convergence. Similar approaches have been studied by Reichert (1970), Holzinger (1970), Karmaker and Robertson (1973), Hantila (1979, Bloomberg and Castelli (1989, Kettunen and ’hmer (1992), Koizumi and Higuchi (1993, Urata and Kameari

42

R. ALBANESE AND G. RUBINACCI

(1995), as far as the static case is concerned and by Koizumi and Onizawa (1991). Albanese et al. (1992), Ruatto (1992), Gimignani et al. (1992), Albanese et al., (1996c), Kettunen and Forsman (1996), among the others, with reference to the time-dependent problem. More recently, Forsman et al. (1996) have discussed the numerical properties of these formulations. This section is dedicated to the convergence properties of the iterative procedures, and is mainly based on the papers by Hantila (1975). Bloomberg and Castelli (1983, Albanese et al. (1992, 1994e. 1996~). The iterative procedure (Hantila, 1975) consists of the following steps, starting from k = 0: 1. Select an arbitrary B'"). 2. Compute

M ( k )= (j'(B(k))= - &'(B(k)) ( 126) 3. Solve the linear problem described by (122) and (124) where M = M(k). 4. Compute B(k+')from the sources J(k+') and V x M(k)using Biot-Savart law (120). 5 . Go to step 2 until convergence is achieved, i.e., if E ( k + l ) = M(k+l)- M(k) (127)

II

II@,,

is smaller than a prescribed tolezance. Here, we adopt the notation

(u, v)a = (u,a v )

11u11 = (u, Ilullu = (u,au)'/2 If the constitutivecharacteristic (3) verifies suitable general conditions, the iterative method is a convergent Picard-Banach procedure. Here we report the main points of the proof given in Albanese et al. (1992). For a given M, let (B, H, E, J) be the electromagnetic field obtained by linear equations (96), (117), (120). (122), and (124). First, we show that the associated linear mapping

B =I(M) (132) is nonexpansive. Let (AB, AH, AE, AJ) be the difference of the electromagnetic fields obtained for two different magnetizations M' and M".Poynting theorem for the eddy current problem yields

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

43

and, using constitutive relation (1 17) we get

J,.L

A B . (uoAB - AM)dVds = IIABlli - (AB, PO AM)^ 5 0

(134)

where AM = M’ - M” and uo = p i ’ . Since IlABIl~ I (AB, POAM), I IIABll~~IIAMllpo~ Eq. (134) yields

IlABlla IIIAMIIPO

(135)

which proves that the linear mapping 7 is nonexpansive, i.e.,

IIT(M’) - 7 W ” ) l I ” o IIIM’ - M”llpo

( 136)

Second, we show that the nonlinear mapping (7 defined by constitutive relation (118) is contractive if the mapping ‘H = 23-’ verifies Lipschitz condition and is uniformly monotonic:

VB’, B” ll’H(B’) - ‘H(B”)llpo5 KIIB’ - B’’Ilm, VB’, B” (‘H(B’)- X(B”), B’ - B”) 5 kllB’ - B’’I&,

( 137)

(138)

with K and k positive constants that verify the condition 2k (139) Po K* AM), we can write an alternative expression of -

Since AB = po(A‘H(B) I1AMll;n:

+

’

m

I I A M I I= ; ~ I I A G ( B ) I=I ~I I, A B I I-;~ P ~ ( A ’ H F I ( B ) , + P ~ I I A ~ ( B ) I I ~ , , (140) and, taking into account (137) and (138), we get the following inequality

which demonstrates that (7 is nonexpansive under condition (139). For an isotropic material with a regular magnetic curve, k-’ and K-’represent upper and lower bounds for the differential permeability a B / a H , respectively. In this case, a sufficient condition for (139) is

which is satisfied by all materials of interest. Notice that if the magnetization vector had been updated as Mck) = p;’ 23(H(k)) - H(k),then convergence would not have been assured, because condition (142) would have been replaced by aB

gH

.c 2Po

(143)

44

R. ALBANESE AND G. RUBINACCI

Third, we observe that the solution of the nonlinear eddy current problem is given by the fixed point of the mapping E[I(M)]. This mapping is contractive, because it is the composition of contractive and nonexpansive mapping. Therefore, the convergence of the Picard-Banach procedure is assured and its convergence rate can be accelerated by overrelaxation methods.

E. Finite Element Approach to the Solution of Nonlinear Magnetostatic and Eddy Current Problems The iterative procedure illustrated in Section VD allows us to extend the integral method described in Section VB. Of course, to calculate the eddy currents in the presence of magnetic materials, the contribution of the magnetization M(k)has to be added to the integral equation for the solenoidal current density J'"+'). Expressing J and W in terms of the edge shape functions Tk, n

J(x, t ) =

Ik(t)V x Tk(X)

(144)

k= I

Wi(x)=VxTi(x),

i = 1, ..., n

(145)

and assuming that V, and Vf are fixed, we obtain the following linear system of ordinary differential equations: d (146) IL)#I {R"I= [UI [UYI

+

+

where the vector [U]and the matrices {L) and ( R ) are defined by (1 10)-(112) and

As M E L2(Vf), in the numerical approximation, we can use the simple zeroorder expansion of the magnetization vector. Its three components are taken to be uniform within each of the Efelements in which the ferromagnetic domain V, is discretized: Ilm

M(x, t ) =

Mk(f)Pk(x)

(148)

k=l

where Pk's are n, = 3Er discontinuous shape functions obtained by multiplying the unit vectors along the coordinate axes by the Efscalar pulse functions pi's: Pj(X)

1, x E jth element of Vf = 0, x 6 jth element of Vf

{

(149)

In this way, condition (125) is satisfied by our numerical approximation,and (147) can be rewritten as

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

45

where

The constitutive equation for M is imposed by applying the Galerkin procedure, obtaining the following form of (123):

Pi.[M-G(B)]dV=O,

i = 1, ...,n,

(152)

The flux density B is given by (120) as a function of J, M,and J,. Thus, [I]and [MI can be obtained by the nonlinear system of n n, equations given by (146) and (152). However, we prefer to replace (152) by the alternative weak form of

+

(118):

.I

Pi . [G-’(M) - B]dV = 0,

i = 1,. . ., n,

(153)

According to (148), M has been assumed to be piecewise constant, hence Q-’(M) is also uniform in each element. Therefore, the adoption of (153) implies that in each element we update M as the magnetization corresponding to the average value of B, instead of taking the average value of Q(B). Because the average is a nonexpansiveprocedure, the iterative method described in Section IVD preserves its convergenceproperties (Albanese et al., 1992). Notice that (153) presupposes the existence of Q-I, which is not implied by the existence of 7 i - I . However, Q is a one-to-one mapping for all materials of interest. Taking into account (120), (144), (148), and (151), Eq. (153) can be put in the following form:

IDIG-’([Ml) - IE)[MI - IFlT[I1- [WI = 101 where

Di, =

lf-

Pi Pj dV

Pi (x) - J~(x’,r ) x (X - x’) IX- xi13

(154)

(155)

dVdV’

(157)

a Vfi is the surface bounding the element where the shape function Pi is located and G, defined by (158), is the global relation generated by the local constitutive equation 6 defined in (1 18).

46

R. ALBANESE AND G . RUBINACCI

The resulting nonlinear system of equations provided by (146) and (153) is then solved using implicit time stepping. The solution [M]kN,+1',[I]kN,+1',corresponding to the (k 1)th iteration at the time instant t ~ + 1= f N Ar, is obtained by

+

+

+

+

[M]kN,+I'= C ( [ D ) - ' [ E ) [ M ] ~ + ' [ F ) T [ Z ] r f ' [W],"") (L)([I]kN++I' -[ZIN)

(159)

+ [F}([M]kN++1'- [ M I N )+ [R)[Il,N,+,'At= [ V I N + ' A t ( 160)

As a by-product, assuming [I]:" = 0, Eq. (159) provides a numerical formulation for nonlinear magnetostatics. The numerical approximation of the linear mapping (132) used in the iterative procedure (159) and (160) is given by

+

[BIkN++1'= [DJ-'([El- [F)[A)-'(F)T)[Ml,N+' [SI

(161)

+

where [ll]kN++1'= G-'([[M];$']), [ A ) = [ L ) A t { R ) ,and [ S ] is a term that does not depend on [MI:+'. As a consequence, we get vo [ ABI = ( T 1[ AM1

(162)

where [ T ) = V O [ D ) - ' ( ( E} { F ) [ A ) - ' { F ) T )whereas , [ A M ] and [ A l l ] are the differences between two possible choices of [MI:" and associated values of [B]:;', respectively. The ( T ) matrix is the sum of two terms, namely UO( D)-'(E) and UO[ D)-' [ F ) [A)-' [ F)T. The first contribution represents the average magnetic flux density associated with the magnetization. The second term corresponds to the average flux density induced by the time-varying magnetization in the conducting domain. The condition corresponding to (136) for the convergence of the procedure is that all eigenvalues AT of [ T ) verify the condition IAT I 5 1. In the numerical calculation of relevant volume integrals, we guarantee symmetry and positive definiteness of (D), ( p o ( D )- [ E ) ) ,and [ F ) [ A ) - ' ( F ) T ) There. fore, matrix (T) can be put in the form

IT) = ( 1 )- ~ o [ D ) - ' ( P o [-D[ )E l

+ IF)[A)-'[FJT)

(163)

where {I)is the identity matrix. Equation (163) shows that the eigenvalues of [ T ) are real and less than unity. From physical arguments (Bloomberg and Castelli, 1985), we expect eigenvalues in the range 0 5 AT I1. Therefore, the approximationintroduced in the numerical integration is not a critical problem. In fact, numerical error cannot give rise to eigenvalues AT 2 1, and we can get eigenvalues AT 5 - 1 only with very large numerical errors in quadrature formulas. The procedure has been applied (Albanese et al., 1996c) to analyze TEAM Workshop Problem 10 (Nakata et al., 1991). It deals with the calculation of the electromagnetic field produced by a coil of rectangular cross section carrying

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

47 0.0 5 x 5

P1 (

h h

1.6

1.6.6.25. 0.0)

( 41.8.6.25.632)

(125.3.6.25, 0.0)

FIGURE 15. TEAM workshop Problem 10. Calculation of the transient electromagnetic field in thin conducting ferromagneticplates with narrow gaps placed around a coil (units are millimeters).

FIGURE 16. TEAM workshop Problem 10. The finite element mesh for the nonlinear integral formulation, referring to 1/8 of the ferromagnetic plates around the coil. (0 1996 IEEE. Reprinted with permission from IEEE Trans. Magn. 32, 784787, May 1996, R. Albanese et a].).

a current that increases with time, in the presence of a magnetic circuit made up of thin iron plates and small air gaps (Fig. 15). The finite element mesh, which refers to one-eighth of the domain, is shown in Fig. 16. It is made of 459 elements giving 831 degrees of freedom for [I]and 1377 degrees of freedom for [MI.

R. ALBANESE AND G.RUBINACCI

48 0.5

0.4

0.3

i*z

/a

0.2 0.1 t

1 Time [s]

1.5

E A 9

1

0.5

0 0

0.05

0.1

0.15

Time [s]

FIGURE17. TEAM workshop Problem 10. Numerical results obtained by the nonlinear integral formulation compared to the experimental data. The experimental current densities are solid ( P I ) , dashed ( P z ) , and dash-dot (4).The experimental average flux densities are solid (SI), dashed ( S2) . and dash-dot (S3).(0 1996 IEEE. Reprinted with permission from lEEE Trans. Magn. 32,784-787, May 1996, R. Albanese et al.).

The relevant numerical and experimental results are reported in Fig. 17. They refer to the current densities evaluated at positions P I , P2, and P3 and to the average magnetic flux density normal to the surface S1, S2,and S, as defined in Fig. 15. Figure 18 shows the current density in the iron plates for t = 25 ms, whereas in Fig. 19 the magnetic flux density distribution is shown at the end of the transient. Figure 20 shows the distributionof the current density along the thickness of the iron plate in correspondenceof point P2 at several time instants. The edge element shape functions for J are piecewise constant along the thickness, giving a step-like behavior of current density. In this way, the current density at the surface of the plates is generally undervalued and sometime overvalued (Dular et al., 1993). On

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

49

t 5 IME

=

2.5EI@E-[32

FIGURE18. TEAM workshop Problem 10. Current density in the iron plates as computed by the 1996 IEEE. Reprinted with permission from nonlinear integral fonnulation at the time t = 25 ms. (0 lEEE Trans. Magn. 32,784-787, May 1996, R. Albanese et al.).

TIME =

1.5B@E-B1

FIGURE 19. TEAM workshop Problem 10. Magnetic flux density in the iron plates as computed by 1996 IEEE. Reprinted with permission the nonlinear integral formulation at the time I = 150 ms. (0 from lEEE Trans. M q n . 32, 784-787, May 1996, R. Albanese et d.).

the other hand, the numerical results for the average values of B are much closer to the experimental values. As pointed out in Dular et al. (1993), the introduction of a suitable smoothing, provides a better agreement with the experimental measwments of the current density. As for the convergence of the Picard-Banach iteration, we found that, during the early transient phase, 7000 iterations were required to obtain a relative emor E ( ~ 2 ) 10-4.

50

R. ALBANESE AND G . RUBINACCI

z[mI FIGURE 20. TEAM workshop Problem 10. Current density distribution along the thickness of the iron plate in correspondence of Pz as computed by the nonlinear integral formulation at the times t = 10 ms (solid), r = 25 ms (dashed), and t = 55 ms (dash-dot). (0 1996 IEEE. Reprinted with permission from IEEE Truns. Magn. 32,784787.May 1996, R. Albanese et a].)

VI. DIFFERENTIAL FORMULATIONS AND CONSTITUTIVE ERRORAPPROACH Conventional numerical formulations adopted for FEM can be classified into magnetic or electric methods, depending on the basic unknown. With these standard procedures, constitutive relationships and one of the field equations are explicitly enforced. The solution is then obtained by imposing the other field equation on the unknown variables. Magnetic methods are those in which the unknown is basically the magnetic field H (or strictly correlated variables like F or T,Q) and Ampere's equationis exactly satisfied as well as the constitutiveequations, whereas Faraday's equation is satisfied only in an average sense. Conversely, electric methods, which assume the electric field E or the magnetic vector potential A as the primary unknown, enforce exactly Faraday's equation and material properties, whereas Ampere's equation is imposed in a weak form. Here, we adopt a different approach known as the minimization of the constitutive error (Rikabi et al., 1988a,b), which is based on the exact enforcement of the field equations and on the subsequent minimization of the constitutive equation error inside the domain. In this way, complementary solutions can be obtained, providing in the frame of computational magnetostatics upper and lower bound for energy-related parameters and an estimate of the numerical errors of the approximate solutions. The possible extension of these results to the transientand steady-state eddy current problems has been discussed by several authors (Hammond and Penman, 1978; Penman, 1988; Hammond, 1989). However, in these cases the possibility of establishing upper and lower bounds for power and

51

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

energy related functionals has not been demonstrated(Rikabi et al., 1988b;Bossavit, 1992, 1994; Li et al., 1994). A. The Magnetostatic Problem Numerical formulations of the magnetostatic problem can be efficiently obtained by exploiting the concept of complementary solutions. Complementary solutions can be found by enforcing the canonical equations

V.B=O, V x H = Js,

inV

in V

and minimizing a functional A(B, H), which is related to the error on the constitutive equation B = B(H) and is zero if evaluated on the correct solution B = Bo and H = Ho (Rikabi et al., 1988a,b; Albanese and Rubinacci, 1990b, 1993b, 1994). The choice B = V x A automatically satisfies (I), while the choice H = T,- VQ enforces (165), provided that

V

x

T, = J,,

in V

(166)

The uniqueness of the solution for A and T, is obtained when a suitable gauge condition is applied. Thus, the problem of determining B and H is reduced to find the best estimates of A and n. Here we briefly recall the main points of the error-based approach to complementary formulations of magnetostatic field solutions, presented by Rikabi et al. (1988a) and applied with the two-component vector potential by Albanese and Rubinacci (1990b). As shown in (Rikabi et al., 1988a), the starting point is the definition of the local error density: rB

which is zero only if the field estimatesB and H satisfy the constitutiverelationship (3). Notice that, for linear media, i.e., if the constitutiveequation is given by Eq. (6), then the expression of the local error density becomes

This allows for the definition of a global error functional h(V x A, T,

- VS2)dV = Ev(A) + Ov(L?)+ r(A, L?) 2 0 (169)

52

R. ALBANESE AND G. RUBINACCI

where

r ( A , 52) = -

s,

V x A * (T, - V Q ) d V

If A and 52 are selected such as to verify the essential boundary conditions V x A . n = b on aVh and (T, - VQ) x n = h on aV,, then r ( A , 52) can be written as r(A,S2)=-

s,

A.J,dV-/

A.hdS+/

52bdS a v, and the error functional can be split into two separate contributions

(173)

iJ v h

h ( A , 52) = E(A)

+ O(52) 2 0

( 174)

with

Minimization of 3(A) yields

whereas from the minimization of @(a) we have vrpk .B(T, - V52)dV =

rpkbds,

Vrpk

(178)

i ) Vh

Here the two-component vector potentials T, and A are approximatedby means of edge element-based shape functions Tk and Ak , respectively, which automatically satisfy the gauge conditions, whereas the scalar potential 52 is approximated by nodal shape functions (pk. Notice that Ak x n = 0 on aV,, while Vk = 0 on aVh. Approximating T, as Ck ckTk, (166) can be enforced by determining the values of the coefficients Ck by integratingthe flux of J, linked with the loop closed by the tree with each single internal edge of the cotree. An alternative numerical procedure is the minimization of the error functional: I V x T, - J ~ I ~ ~ V

(179)

THE SOLUTION OF 3D EDDY CURRENTPROBLEMS

53

This minimization leads to the following weak formulation of the problem: V x Tk . V x T, dV =

s,

V x Tk J, d V , a

VTk

( 180)

Since both B and H are available on the same mesh, we can calculate both local error I and global error A. A compact visualizationof the local error density distribution can be obtained by reportingthe cloud of pairs (B = V x A, H = T, - VS2) in a B-H plane in comparison with the B-H curve representing the constitutive equation. However, to obtain a spatial localization of the error density in the domain of interest, one can plot surfaces at constant I or arrows representing the error fields B - B(H)or H - 'FI(B). Of course, the local error distributions offer a clear indication for a mesh refinement, because I = 0 is a necessary condition to =I &,although a proof of a direct relationship between have both B = Bo and € local error and mesh refinement is not available. In addition, there is the possibility of having energy bounds. If we call A0 and S2o the exact solutions for A and a,respectively, then we have well-defined upper and lower bounds for the energy-related quantity @(no)= - E (Ao) according to the inequalities O(S2) 2 @(no) = -E(Ao) 2 -=(A),

VS2, A

(181)

In particular, if a V = a Vh (or b = 0 on a vb), then O(520)= - 3 (Ao) is the coenergy of the system. Notice that the enforcement of the canonical equations is numerically obtained in a straightforward way, taking full benefit from the adoption of the gauge based on the tree-cotree decomposition. In other cases, use of (164) would have implied the introduction of an additional equation such as V A = 0 (Biro and Preis, 1990). The gauge associated with the tree-cotree decomposition has been successfully applied in the case of the T,S2 formulation. In fact, it reduces to the well-known scalar potential formulation, without suffering from the cancellation errors and allowing for a more efficient treatment of multiply connectedferromagneticregions. The magnetic vector potential formulation has been widely used in recent years. The main problems that have been discussed are related to the convergence of the ICCG (Incomplete Cholesky Conjugate Gradient) iterative solver. In principle, the choice of the tree does not affect the solution of the problem in terms of fields. This means that different choices of the tree lead to different solutions in terms of A, but these solutions provide exactly the same fields B and H.However, from the numerical point of view, it has been found that the choice of the tree seriously affects the convergence rate of the ICCG method (Kameari, 1990b; Fujiwara, 1992; Biro et al., 1996). In particular, if the tree is chosen at random, the convergencerate could be rather slow. This does not happen if the tree is formed in a direction close to the direction of B (Golias and Tsiboukis, 1994)

54

R. ALBANESE AND G . RUBINACCI

or starting from the trees of the subgraphs of the regions with uniform material properties or geometrical characteristics (Albanese and Rubinacci, 1992). On the other hand, the ungauged vector potential formulationmay show very fast ICCG convergence rates (Barton and Cendes, 1987; Kameari, 1990b; Fujiwara, 1992; Biro et al., 1996; Ren, 1996) even if the linear system is singular. In this case, the convergence is influenced by the discretization of the R.H.S. term of Eq. (177), which should guarantee the numerical compatibility of the system. Compatibility requires that the vector [ J ] defined by Ji = Ni .J, d V , should be orthogonal to the kernel of the discretized curl-curl operator and hence to the nullspace of the matrix (S)defined by

sv

Sij =

s,

V x Ni . p - ' V x N j d V

(182)

Adopting edge elements, it can be shown that (S) and the matrix (C), defined by Eqs. (65)-(67), share the same nullspace (Menges and Cendes, 1995). Since, according to Eq. (77), ( C } ( G }= (0),with the matrix ( G )defined by Eqs. (60)-(62). it results that [ y ] = ( G ) [ x ]belongs to the nullspace of IS),V [ x ] (i.e., ( S ) [ y ] = [O]). Hence, [ylT[J1= [JITIG)[xl= [ X I ~ { G ) ~=[ 0, J]

V[xl

(183)

because [ J ] = ( S ) [ A ]and

Therefore, the compatibility constraint

which requires that the divergence of J,, should be zero also in the discrete approximation, being (G ) T ,the adjoint of ( G ) ,the discrete form of the div operator (Bossavit, 1988). The convergence is greatly influenced by the way the compatibility condition is numerically satisfied. A convenient way (Ren, 1996) to assure the numerical solenoidality of J, is to express [ J ] as Ji

=

s,

V x Ni . T, dV

where T, is a vector potential for J,, which can, for instance, be computed using (180). A series of numerical tests showed that the convergencerate is very fast once the compatibility condition is correctly imposed. If not, the conjugate gradient fails to converge (Fujiwara, 1992; Biro et al., 1996; Ren, 1996). B. The Eddy Current Problem

To guarantee that the error is concentratedin the constitutiveequations (3)and (3, it is necessary to enforce the canonical equations (1) and (2) exactly. This can be

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

55

done by assuming

aA - V-” E=__ at

at

H = T - VS2 + T,

(and therefore B = V x A) (and therefore J = V x T

+ J,)

(188)

where J, is approximated by V x T, using, for instance, (166). Notice again that we introduced &$/at as electric scalar potential to obtain symmetric matrices in the numerical formulation. In analogy with the magnetostatic case, it is possible to define local error densities related to the constitutive equations (3) and ( 5 ) . The possibility of establishing upper and lower bounds for energy- or powerrelated functionals in the frequency domain has been also discussed in Rikabi et al. (1988b), Bossavit (1992, 1994), and Li et al. (1994), without a positive conclusion being reached. This possibility can be connected, as in the static case, to that of splitting a global error functional in two contributions, one depending only on H,and the other only on E.This hypothesis has been deeply investigated, but with the constitutive error-based approach it is impossible to extend the results obtained in the static case to the analysis of quasistationary electromagnetic field problems both in the time and in the frequency domain. A successful attempt was made by introducing a restriction of the Laplace transform to the real axis (Albanese and Rubinacci, 1993b). This allowed a real functional to be defined in the linear case; minimization of this functional gives equations corresponding to the weak formulations (obtained by applying the Galerkin method). In this case, two local errors can be defined, with referenceto the two constitutive equations (3) and (5). A direct approach based on the local and time-varying constitution errors was proposed in Rikabi et al. (1988b), who show that it is impossible to split the error functional into the sum of two contributionsdepending on E and H separately. However, the splitting can be achieved by using Laplace transforms. Assuming V = V, and zero initial conditionsfor the sake of simplicity, we can define

where p is the complex variable and 2 denotes the Laplace transform of X ( t ) . Both complex quantities A, and A, are zero if and only if the field estimates verify the constitutive properties, and assume real and positive value if p is real and positive.

56

R. ALBANESE AND G. RUBINACCl

From now on we shall assume p to be a real positive quantity. In this case we can also define the global error functional

h(A + V d , $ - V n , p) =

1

{ a m ( vx

A, T - VA + Ts, p )

V

+ L ( V x f + js,- p A - p V 4 , p ) }dV = z v ( A + v+,p ) + ev(T - v A , p ) + r(A + v&T - v A , p ) 2 0, V real and positive p

(191)

where

+ ('

@"(T- vil, p ) = r(A+V$,T-VA,p)=-

s,

TI2} dV

(193)

2UP

V-(A+V3) x (T-VA+Ts)dV

(194)

If T - V R and A + V @ are selected such as to verify the essential boundary conditions (A Vq5) x n = - J e d t on aV, and (T- V R T,)x n = h on a Vh, then r can be written as

+

+

r(A + v+,T - vA, p ) = - l v h ( A + V b ) .

hdS

and the error functional can be split into two separate contributions

h(A + V 4 , T - VA, p ) = =(A

+ V 3 , p ) + O(T - Vfi, p ) 2 0, V real and positive p

with

E(A

1 1,

+ V$, p ) = Bv(A + V$, p ) -

(A + V3)

hdS

(196)

(197)

vh

O(T - VA,p ) = O"(T - vfi, p ) -

(T - vn + fs)6 d S P

(198)

The stationary point of E (as well as its minimization for real and positive values

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

of p ) is given by

Similarly, the stationary point (minimumfor p 2 0) of 0 is given by

Equations (199) and (200) and (201) and (202) correspond to the Galerkin formulations

and

.-

in the time domain.

R. ALBANESE AND G.RUBINACCI

58

Finally, with reference to the general nonlinear case, we report the two weak formulations, the one in terms of the electric field E = -aA/at - Va$/at:

and the other in term of the magnetic field H = T - VS2

+ T,:

in the time domain. As in the magnetostatic case, both local and global errors can also be calculated, because both (E, B) and (J, H)are available in the same mesh. A compact visualization of the local error density distribution can be obtained by reporting the cloudofpairs (B = V x A, H = T - VS2+TS) and (J = V x (T+ T,), E = -aA/at - Va#/at) in comparison with the curves representing the constitutive equations. For a better localization of the spatial error, the surfaces at constant h or the arrows representing B - B(H)and J - J, - aE can be plotted in the domain of interest. Also, in the eddy current case, the local error distributions offer a clear indication for mesh refinement. In addition, there is the possibility of having bounds for functionals (197) and (198), according to the inequalities (holding for p real and positive)

E(A+

vfj,p) 2 E

(-$,

p ) = -O(H,, p )

-@(T - VR, p ) ,

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

59

Notice that if the fields are computed in the frequency domain, the splitting gives rise to the two following classic weak formulations in A, 4 and T, 52:

A

{V x Ak . p-'(V x A) =

Vvk . joa (A

A

Ak. & d V

+ V4) dV =

1 1 +

a Vh

Ak . j a a ( A

+ V#)} dV

-

Ak h d S ,

Vcpk .J, dV

+

(212)

VAk

1,

v(Pk *

hdS,

v (Pk (213)

P

r

Notice also that both formulationslead to a unique solution also in the static limit. As was also pointed out by Webb and Forghani (1993) for the T, 52 case, this leads to a better conditioning of the matrix at low frequency, compared to the H or E method. It is interesting to see how these 3D eddy current formulations in terms of twocomponent vector potentials using edge elements can directly and efficiently be applied for the analysis of 2D plane or axisymmetricproblems. For instance, a 2D problem defined in the x y plane with B, = 0 can readily be studied by means of the 3D A formulation. We use a mesh made of a single layer of hexahedral elements of length h along z (the nodes on the plane z = h/2 are obtained by translation of the nodes on the plane z = -h/2). Then we simply impose A x n = 0 at the boundaries. This implies A, = A, = 0; henceforth, all the active edges are directed along z and their number coincides with the number of nodes lying on each plane z = f h / 2 . We report (Albanese and Rubinacci, 1994) the results of the analysis of TEAM Workshop Problem 10, which was described in Section V.We analyzed two cases (Albanese and Rubinacci, 1993c, 1994). In the first case, we used the coarse mesh shown in Fig. 21. The results given by the A, 4 formulation (Fig. 22) were in good agreement with the experiment, whereas the T, 52 formulation led to large errors, especially in the transient phase (Fig. 23). Of course, analysis of the constitutive error cannot by itself tell us when at least one solution (in the present case the A, 4 solution) is still acceptable. However, looking at the constitutive

60

R. ALBANESE AND G. RUBINACCI TABLE 1 TEAM WORKSHOP PROBLEM 10. 0

Coarse mesh Refined mesh

_ I

Relative error

(d)

(mu

(%)

31.4 27.56

25.6 26.64

10.2 I .70

Note: Estimation of the coenergy for the magnetostatic field configuration achieved after termination of transients. The relative error is given by (0+ 6 ) / ( 0-

e,.

Source: Albanese and Rubinacci (1994).

FIGURE21. TEAM workshop Problem 10 (see Fig. 15). Coarse mesh (1936 elements) for the differential formulations. The discretization of the vacuum is not shown. (From Albanese and Rubinacci ( 1994).)

error distribution we see an acceptable agreement between the two dual solutions at steady state but large disagreement in the transient phase (Fig. 24). We see also that the error is mainly concentrated in the air gap along the thickness and in the corners, giving us an useful information for carrying out the mesh refinement. We then refined the mesh accordingly (Fig. 25). Notice that a single refinement of the mesh as suggested in Albanese and Rubinacci (1991) did not substantially improve the numerical results (Albanese and Rubinacci, 1993c, 1994). This is additional evidence that the error estimation is quite helpful, as demonstrated by the good results illustrated in Figs. 26 to 29 and in Table 1.

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

.0

.2

.4

.6 t i m e Csl

.e

.6 t i m e Csl

.e

1.0

1.2

61

1.4

*E-1

Ln

Y 61 u)

? P

n nJ

E

a

-61

)7xN

?

.0

.2

.4

1.0

1.2

1.4

*E-1

FIGURE 22. TEAM workshop Problem 10. Results given by the A, Q formulation using the coarse mesh. Comparison with the experimental values of average flux densities (cross sections SI, S2, and S3) and current densities (points P I , Pz. and P3). (Albanese and Rubinacci, 1994.)

62

R. ALBANESE AND G . RUBINACCI

UI

.0

.2

.4

.6

.e

1.0

1.2

t i m e Csl

?

.0

.2

.4

.6

1.4

*E-1

.e

1.0

1.2

1.4

*E-1 FIGURE 23. TEAM workshop Problem 10. Results given by the T, 52 formulation using the coarse mesh. Comparison with the experimental values of average flux densities (cross sections SI, S2,and S1)and current densities (points PI,9,and 4).(Albanese and Rubinacci, 1994.) time CSI

63

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

. . . ._.-.. . . . . . . . .?'

.

. . . . . . ..

,'

-. ..,.2 ..............

..

FWURI:24. TEAM workshop Pnhlcm 10. RCSU~LS obtained by the diITmtiol rormulations with the conrsc mcsh at the limes I = 30 ms and I = 140 ms. Here R ( A - V ) is the flux density field H = V x A: R ( T - 0 ) is the flux density B(H)= A T V n +T,): EWB) is the R U(H): J ( T - 0 ) is the current density J = V x (T+T,):J ( A - V ) is the c u m t density J, nE =J, cr(i)A/i)t V;)@/;tl): EmJ) is Ihc e m field J J, nE. (Albanesc and Kuhinnwi. 1994.)

-

+

-

-

+

-

-

C. The Electromagnetic Problem

The model, including the full set of Maxwell equations (1) and (2). can bc stated in the form

V X H = -ah at

VxE=--

aB ar

64

R. ALBANESE AND G . RUBINACCI

FIGURE25. TEAM workshop Problem 10 (See Fig. 15). Fine mesh (9765 elements) for the differential formulations. The discretization of the vacuum is not shown. (Albanese and Rubinacci, 1994.)

where

DT = D

+

I'

Jdt

with the constitutive properties and the initial, boundary and interface conditions discussed in Section 11. Using the vector potentials defined in Section 111

A*=we have that the fields

I'

E=--

Ed?

F=lHdt

aA*

aF H=-

at

at

automatically satisfy Maxwell equations (216) and (217) and the initial conditions B(x, 0) = Bo(x) and D(x, 0) = Do(x) for arbitrary values of F and A*. We can therefore proceed to define a local error functional A(x, t) 2 0, with strict inequality for field estimates that do not satisfy the constitutive equations, and a global error functional A whose minimization yields the best approximation in terms of potentials and fields. Here, for the sake of simplicity, we will focus our attention on the model of Maxwell equations in nonconducting media, for which & reduces to D. In any case, the magnitudes of the two terms J and aD/at are comparable at a

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

.0

.2

.4

.6 tlfre

.E

.2

.4

.B

.6 tlme

1.B

1.2

[sl

1.4

*E-1

.E

[sl

65

1.E

1.2

1.4

XE-1

FIGURE26. TEAM workshop Problem 10. Results given by the A, 4 formulation using the fine mesh. Comparison with the experimental values of average flux densities (cross sections SI, Sz,and &) and current densities (points P I , 4 ,and 4).(Albanese and Rubinacci, 1994.)

fixed position only in particular cases (e.g., lightning events or highly dissipative media). So, with the assumption of nonconducting bodies, we define the local error as (Albanese et al., 1994a)

rE

\

where the local values of the fields are given by Q.(220); 3.t and & are the inverse B*) and (E*,D*) are two pairs that mappings of B and D,respectively; (H*, satisfy the constitutive equations; a~ and ( Y E are weighting factors. In the linear case (B = pH,D = EE,H*= 0, B* = 0, E* = 0, and D* = 0), the local error

66

R. ALBANESE AND G . RUBINACCI

EJ

.E

.2

.4

.6 tlma

Csl

.B

1.0

1.2

1.4 UE-I

In

Y 0 UI

EJ t

? .0

.2

.4

.6 tlme

Csl

,E

l.E

1.2

1.4

NE-1

FIGURE 27. TEAM workshop Problem 10. Results given by the T, 52 formulation using the fine mesh. Comparison with the experimental values of average flux densities (cross sections S1, S2,and S3) and current densities (points PI,9, and 91). (Albanese and Rubinacci, 1994.)

density defined by (221) becomes

According to the discussion of Section III, edge elements are used to approximate the potentials A* and F, and linear approximation is assumed in each of the time steps in which the global interval (0,T) is partitioned. In this way the problem reduces to the solution of a number of subproblems defined by the time steps in which the unknown values of the potentials at the final time will be taken as initial conditions for the next step (since A* and F are continuous with the time by definition). The solution at each time step can thus be obtained via minimization

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

Err I J I

- TIK

-

67

3.8BBE-l

FIGURE28. TEAM workshop Problem 10. Results obtained by the differential formulations with the fine mesh at the times I = 30 ms and t = 140 ms. Here B ( A - V ) is the flux density B = V x A; B ( T - 0 ) is the flux density B(H) = B(T - V n +T,); Err(B) is the error field B - B(H); J ( T - 0 ) is the current density J = V x (T Ts);J ( A - V ) is the current density J, + U E = J, - u(aA/ar Va4/at); Err(J) is the error field J - Js - oE. (Albanese and Rubinacci, 1994.)

+

+

of a global error functional in V x ( t k , rk+l), related to h(x,t): A=

I+'

A(x, t ) dV d t

where V is the domain of integration, (fk, & + I ) is the kth time step. Notice that the boundary conditions must be taken into account properly in the minimization process. Both unknown potentials A* and F have to be found among the elements of L,2,!(V)that verify the relevant boundary conditions. Let us assume that the domain is bounded by a perfect electric wall a V, and a perfect magnetic wall a Vh, where (27) and (28) particularizeto E x n = 0 and H x n = 0,

68

R. ALBANESE AND G. RUBINACCI TfRllE K5H-1936 t im

-. 14EtBE

TEWlEl K5H-9765 t Ime

-. 14E+E0

P

m

m .B

.2

.4 .6 IHI C W m l

.E

.El

*E 4

.2

.4

.6

IHI C W m l

.B XE 4

FIGURE 29. TEAM workshop Problem 10. Results obtained in the iron by the differential formulation with coarse and fine meshes at the time t = 140 ms. B-H plots (pairs at the element centroids) with B = IV x A1 and H = IT - VR +T,I.(Albanese and Rubinacci, 1994.)

respectively. These boundary conditions can directly be formulated as constraints for the vector potentials

A* x n = 0, Fxn=O,

on aV, onaVh

From the numerical point of view, we then get a system of 2E equations in 2E unknowns, most of them given by the minimization process (i.e., imposing the partial derivatives of A with respect to each unconstrained degree of freedom to be zero), and those remaining given explicitly by the boundary conditions. However, for a suitable choice of the nondimensional coefficients (a" = a E ) the , two unknown potentials A*(tk+l)and F(tk+l) can be obtained as the solutions of two independent systems of equations of E equations in E unknowns each. For instance, in the linear case, approximating the time integral 1;' I h(x, r ) dt with the theta method, as h(x, ro)Ar with to = 8rk (1 - B ) r k + l , Ar = rk+l - tk, the choice a~ = OIE yields

+

8

x A*(tk) - -V At

x

Fi * A*(tk)

(226)

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

(1 - 6 ) + -A; At

e

. V x F(tk) + -V

At

69

V Ai

x A; * F(tk)

This splitting, however, is different from solving two separate problems for A* and F in (0, T), because in our case each potential at tk+l depends on both potentials at t k . A similar approach has been proposed for analysis in the frequency domain in terms of the complex field quantities E = -jwA* and H = j o F (Pichon and Bossavit, 1993) searching for the stationary point of the complex functional

NE, H) = j w

S, {

x H - jwEE)2

('

jWE

+ jwpH)2 }dV + (V x Ejw

(228)

which can be split as the sum of the two complex functionals

@,(E) = j w @h(H) = j w

JV

+

(229)

+

(230)

(jwEE2 (jwp)-l(V x E)2]dV

s,

(jwpH2 (jwE)-'(V x H)2] dV

These expressions are slightly modified by the inclusion of surface integrals in the case of nonhomogeneous boundary conditions. The stationary points of ',229) and (230) determine the values of the complex fields E and H, which are given by the two independent problems:

Jv V x N; . p-'V x EdV = w2 Jv Ni * EEdV, Jv V x Ni * E-'V x HdV = w2 J, N; * pHdV,

VN;

(231)

VN;

(232)

which can be regarded as weak forms of the field equations:

v x p-Iv x E = E ~ v x E-IV x H = ~ W

~ E ~ H

The error-based approach has been successfully applied to the analysis of nonlinear dielectric slabs (Albanese et al., 1994b,d), magnetic shields (Albaneseet al., 1994c),and resonant cavities (Albanese et al., 1995b). D. Analysis of Resonant Cavities In the last decade, many finite element formulationshave been proposed for solving closed cavity problems. The major difficultyencounteredin this kind of application is the appearance of the so-called spurious modes. The origin of these modes is

70

R.ALBANESE AND G. RUBINACCI

related to the incorrect numerical representation of the kernel (i.e., the null space) of the curl curl operator, which can give rise to different kinds of numerical eigenmodes. In particular, nonphysical solutions may appear that are not divergence free. The scientific community devoted a particular attention to the elimination of spurious modes in the numerical solution of electromagnetic problems. A number of methods have been proposed to overcome these difficulties: penalty function formulations (see, for instance, Webb, 1988), edge element formulations (see, for instance, Bossavit, 1990), or nodal element approximation of vector and scalar potentials (see, for instance, Bardi et al., 1992). The procedure illustrated here, based on the minimization of the constitutive error, rigorously satisfies the Maxwell equations and hence is a good candidate to avoid spurious modes. Pichon and Bossavit (1993) noted that the constitutive error approach yields in any case four fields that satisfy Maxwell equations and hence solenoidal D and B fields, even using nodal elements. However, the fields EE and p H properly converge to the solenoidal solution only using edge elements (Bossavit, 1990). In Albanese and Rubinacci (1993a) we presented a finite element formulation based on the introduction of vector and scalar potentials as primary unknowns (suitably gauged with the tree-cotree decomposition) and the use of edge elements. No spectral pollution is allowed, since at any nonzero frequency the numerical structure of the problem is the same as the edge element field formulation. However, in our formulation it is suitably constrained so that also at zero frequency the solution in terms of vector potential is unique, and the exact number of zero eigenmodes is given by the number of scalar potential unknowns. This formulation is for a better theoretical understanding of spurious modes. The equivalence of the spaces spanned by the tree edges and by the gradient of a scalar field implies an analogous equivalence between the field formulations (either E or H) in terms of edge elements and the potential formulations (either A, 4 or T,52) when adopting the gauge based on the tree-cotree decomposition. Here, the stability conditions to avoid spurious solutions are briefly recalled following the approach of Gruber and Rappaz (1985). The problem can be formulated as the determination of the angular frequencies for which

C[E]= 02R[E]

(235)

which reduces to the determination of the eigenvalues of the operator R-'C[E], where the operators C and R are defined according to the left- and right-hand sides of (233), respectively, along with the boundary conditions. The same representation can be obtained starting from the weak form (231). The discussion is limited to the E formulation, being obviously a similar form also valid for the H formulation. The spectrum a(R-'L)of R - ' C is the set of w2 such that (L- 02R)is

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

71

not continuously invertible (Gruber and Rappaz, 1985). Notice that, since C is singular, the value w2 = 0 belongs to a(R-'C), but it is an infinitely degenerate eigenvalue. The numerical approximation of this behavior is a critical point and can produce the so-called spectrum pollution (Gruber and Rapaz, 1985). In fact, let us consider the numerical approximation of (235) given by the Galerkin method and the standard finite element approach. Call v h a finite element discretization of the integration domain, and Lh and R h the corresponding discrete from of L and R (h tends to zero with the size of the mesh). Thus, the numerical approximation of (235) consists of the following problem: Find w l , Eh # 0 such that

Ch[Ehl= w;Rh[Ehl (236) The critical point in this approximation is to guarantee that for the sequence w i E a(RI,'Ch) we have lim w 2h = w2

h+O

(237)

with w2 E a(R-'L).If this condition is not assured, a set of spurious eigenvalues that do not belong to the physical spectrum of the original operator R - ' C are generated. Mathematical conditions to avoid this instability leading to the choice of convenient shape functions are discussedin Gruber andRappaz (1985). More practically, we see, using nodal shape functions, that for w2 = 0 the continuum spectrum (the kernel of R-IC), described by curl-free eigenmodes Eh = -V@h, is not consistently represented. For instance, using piecewise linear functions for E and tetrahedral elements, nonzero curl-free fields satisfying the boundary conditions can hardly be obtained. Hence, the related eigenmodes spread their presence on the whole spectrum, being characterizedby divergent eigenmodes. On the other hand, it has already been shown by Bossavit (1990) and in Section IV that using edge elements, the -V@h eigenmode is represented in the same functional space of Eh. Therefore, the kernels of C and R-'L are consistently approximated and in addition all the other eigenvalues different from zero are associated with eigenmodes that are divergence-free. This condition is weakly assured by the numerical formulation in terms of edge elements (Bossavit, 1990). We can use the two-component magnetic vector potential A and the electric scalar potential 4 defined by (187) to express the electric field

E = -jw(A + V#) (238) adopting edge elements and the tree-cotree decomposition to impose the gauge condition (45). In this way, for a given finite element discretization v h of the domain, we can approximate A and 4 as

72

R. ALBANESE AND G. RUBINACCI

The field equations are

+

V x p - ' V x A = &02(A V#)

A*w=O and the weak formulation (231) becomes

s,

s,

VXN,*~-'VXA~V=U N j~. & ( A + V # ) d V , 0 = w2

s,

+

Vp, * &(A V#) d V ,

Vp;

VNi

(242) (243)

Therefore, using matrix notation, the following eigenvalue problem is obtained: {Ch - o2%l[z1 = P I

(24)

where

Matrix C is singular and the multiplicity of the zero eigenvalue is equal to the number of scalar potential unknowns, i.e., to the number of the unconstrained nodal values. We now turn to Eq. (243). which is obtained weighting Eq. (233) by Vpi. However, apart the factor j w , it coincides with the weak form of

V.D=O

(248)

+

assuming D = EE = -j m ( A V#) and taking pj as weighting function. Therefore, we recognize that the relationship

[#I

= -R;,'R;"[AI

(249)

which is implied by (245)at nonzero frequency, is also true when w2 = 0 because it is related to the solenoidality of D. Hence, the eigenvalue problem can be restated in terms of the A variable only: [Lou

- W 2 ( R u u - Rau~;;,lR;u>] =0

(250)

This form is probably less useful from the computational point of view because -R , , R ; ~ R ~ matrix u ) is full. However, it is important from a theoretical the (Roo

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

13

point of view, because it clearly identifies the exact number of nonzero eigenvalues associated with the discrete problem. The key feature of this formulation can be better understood by examining again the structure of the problem in terms of E. We recall that, after discretization, it can be described by the following eigenvalue equation in matrix form (Bossavit, 1990; Bardi et al., 1992):

[Lee- w 2 ~ e e l= o

(25 1)

with E approximated as

Both Leeand L,, are singular, but the tree-cotree decomposition of the unknowns (the vector and scalar potentials) allowed for a clear identification of the subspace associated with the kernel of the curl curl operator in Eq. (233). Moreover, notice that in the formulation of Albanese and Rubinacci (1993a) the kernel of the curl curl operator in Eq. (240) (and its numerical approximation Laa)is empty since now, thanks to the gauge condition (241), La, is no longer singular. Finally, Proposition 1 in Bossavit (1990) (the property related to the edge elements approximation of giving, when the mesh is refined, by passing to the limit, a divergence free field) is proved by Eq. (243), which, apart the factor j w , corresponds to the weak form of V . D = 0. Of course, the success of this approach relies always on property (68). The introduction of two-component vector potentials and the use of edge elements in conjunction with the tree-cotree decomposition of the mesh allowed for a consistent representation of the solution of Maxwell equations in closed cavities, avoiding the problems related to the appearance of spurious modes and clearly identifying the number of curl-free eigenmodes at zero frequency in the discrete approximation (Albanese and Rubinacci, 1993a; Bardi et al., 1992, Menges and Cendes, 1995). Notice also that it yields a unique solution also in the static limit. As was also pointed out by Webb and Forghani (1993) for the T,S2 case, this leads to a better conditioning of the matrix at low frequency, if compared to the H or E method. For these reasons the two-component vector potential approach is very interesting from the theoretical viewpoint. In practice, the approach in terms of field is equivalent and in most cases more convenient from the numerical point of view. Three main procedures have been proposed to calculatethe resonant frequencies of a closed cavity. The most straightforward procedure is the standard solution of the generalized matrix eigenvalue problem (251) taking advantage of sparsity and symmetry of matrices Leeand Ree.Two additionalprocedures have been proposed; the Fourier analysis of the dynamics of suitable local quantities (Albanese et al., 1994b, 1995b) and the analysis of the frequency response of the system when forced by a sinusoidal input, locating the resonant frequencies at the peaks of the electromagnetic energy (Gomes and Ida, 1993; Albanese et al., 1995b).

74

R. ALBANESE AND G. RUBINACCI

=z ---

FIGURE30. Geometry of the rectangular cavity loaded with a dielectric slab. In the numerical examplea= I m , b = 0 . 3 m , c = 0 . 4 m , d = 0 . 4 m , r = 0 . 2 m , ~ / ~ 0 = 1 6 .

In the first procedure, the problem is solved in the time domain starting from the initial conditions D(x, 0) = Do(x) = 0 and B(x, 0) = Bo(x) = V x A0 at time c = 0. The vector potential A0 is selected equal to a linear combination (with random coefficients) of all edge element-based shape functions. The resonant frequencies are then obtained by performing a Fourier analysis of the numerical results. The alternative procedure based on the analysis of the frequency response refers to the solution of Maxwell equations in the frequency domain. The frequency response can be obtained (Gomes and Ida, 1993; Albanese et al., 1995b)by applying a sinusoidal input; the eigenfrequencies correspond to the peaks of the reactive power that, according to Foster’s reactance theorem, is an increasing function of the angular frequency. Both methods have been used to determine the resonant frequencies of the rectangular cavity loaded with a dielectric slab as shown in Fig. 30. One-eighth of the domain has been discretized by imposing n x H = 0 at x = a / 2 and z = c/2, and n x E = 0 at y = b/2. Tivo meshes have been considered: (1) a coarse mesh with 10 x 4 x 4 equally spaced hexahedral elements, and (2) a fine mesh with 20 x 8 x 8 equally spaced hexahedral elements. The analysis has been carried out in the time domain (8192 time steps of 50 ps). Figure 31 shows the spectra of E , obtained by a Fourier analysis of the transient numerical results (Fig. 32) calculated inside the dielectric at the point P2 (x = 0.050 m, y = 0.075 m, z = 0.1 m) for coarse and fine mesh. The resonance frequencies of the main modes are reported in Table 2 in comparison with the analytic solutions. The electromagnetic field distribution for a given mode can also be reconstructed. The eigenfunction of an arbitrary quantity q corresponding to the resonant frequency fr can be computed by q ; W = A(x, f i ) cos @(x9f i )

(253)

where A(x, f ) and @(x,f ) are amplitude and phase given by the Fourier analysis of q ( x , t) at frequency f . For example, Fig. 33 shows the E, distributions in

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

75

TABLE 2 RECTANGULARCAVITY LOADED WITH A DIELECTRIC SLAB. Numerical Mode ~~

101

121 103 30 1

Analytical f(MHz)

Coarse mesh f (MHz)

Fine mesh f (MHz)

137 292 31 1 385

I30 295 308 389

~

130.972 297.243 310.873 392.282

Nore: Fourier analysis of the estimate of E , given by the A potential ( E , = -aA,/ar) for the coarse (10 x 4 x 4) and fine mesh (20 x 8 x 8) at point Pz (0.05,0.075,0.1).

aa-i

t?i

W

-i

I";

4.a-1

O.aE4

,

FIGURE 31. Rectangular cavity loaded with a dielectric slab. spectra of E , = -aA,/al

at point

9 (x = 0.050 m.y = 0.075 m, z = 0.250 m) computed by means of Fourier analysis of the transient results obtained with coarse and fine meshes.

modes TEloland TE301as a function of x for different discretizations, as obtained from the A solution (E(A)= -aA/at) and from the F solution (Eg)= V x F I E ) . From these results, the role of the constitutive error in estimating the correctness of the numerical solution is clearly apparent. Furthermore, we report some results of the analysis of the frequency response. The cavity is excited with an arbitrary sinusoidal input of fixed amplitude. In Fig. 34, the reactive power is shown as a function of the frequency of the sinusoidal input. In Table 3, the first resonant frequencies for the coarse and the fine mesh, as given by the location of the energy peaks, are computed using the A and F variables, respectively. It can be seen that in this case no upper and lower bounds

76

R. ALBANESE AND G . RUBINACCI

-5.WE-2

1'

-1.OOE-I 6.OOE-O

3.00E-0

O.WE-8

nme

Rectangular cavily loaded WHh a dielectric slab Ey as a fundion of the lime at point P2 (Fine meah, d1=5e-1I ) l.WE-I

I

-1.00E-1

1

3.00E-8

6.OOE-8

0.00E-8

Time [wcl

FIGURE32. Rectangular cavity loaded with a dielectric slab. Transient analysis of E , at point = 0.250 m) obtained by the constitutive error approach with coarse and fine meshes. In both cases we have two estimates, namely E,(A) = -aA,/at and E,(F) = (aF,/aZ - a F z / a X ) / E . (x = 0.050 m, y = 0.075 m. z

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

77

TheTElO1 .ig.nfuncUonIf.num.-1.308dllasafunctkn TheTElOl aigsnfunnion lf.num.=1.378.81n~functbn u t EyiF)Ey.nux of x icoarw ~ h . d t = 5 a l l ~ I . E y ( A l / E y , m aand x E y l F 1 E y . m of x itin ~ h , d t - ~ l l ~ ) . E v l A ) / E v . m and

1.

2,

1-

w”

w”

0

0.00

0.20

0.40

0.20

0.00

x [ml

0.40

x [ml

0.00

0.20

0.40

x [ml FIGURE33. Rectangular cavity loaded with a dielectric slab. Electric field estimates E,(A) = -aA,/at and E,(F) = (aF,/az - a F z / a X ) / E and error field ,?,(A) - E,(F) as functions of x at y = 0.075 m, z = 0.250 m: (a) TElOl eigenfunction (coarse mesh); (b) TElOl eigenfunction (fine mesh); (c) TE301 eigenfunction (fine mesh).

are obtained. Our analytical and numerical values of the resonant frequency of the fundamentalmode are slightly different from those reported by Akhtazard and Johns (1975) and by Sadiku Matthew (1992). The same analysis has been performed on the inhomogeneously loaded cavity (0 < x < 0.200 m, 0 < y < 0.500 m, 0 < z < 0.300 m)described and studied, for instance, by Pichon and Bossavit (1993), Webb (1988), Bardi et al. (1992). The relative permittivity is er = 16 in the dielectric region (0 < x < 0.075 m, 0 < y < 0.125m,O < z < 0.175m)ander = lintherestofthedomain. Magnetic

R. ALBANESE AND G. RUBINACCI

78

TABLE 3 RECTANGULAR CAVITY LOADEDWITH A

DIELECTRIC SLAB.

Numerical Coarse mesh Mode

101 121 103 30 1

f (MHz) F variables

Analytical f (MHz)

A variables

130.972 297.243 310.873 392.282

I32 305 328 420

Fine mesh f (MHz)

A variables

F variables

131 299 315 40 I

131 299 315 400

131 303 326 41 8

Nore: The first resonant frequencies for the coarse and the fine mesh, as given by the location of the peaks of the reactive power, computed using the A and F variables.

O.OOE+O

2.00E+8

4.00E+8

6.OOE+B

Frequency [Hz] FIGURE34. Rectangular cavity loaded with a dielectric slab: reactive power (in arbitrary units) as a function of the frequency of the sinusoidal excitation.

walls are placed at x = 0 and y = 0, whereas the other walls are perfectly conducting. Finally, in Table 4, we present the value of the resonant frequency of the fundamental mode as obtained by the first peak of the reactive power given by the fine mesh.

E. Open Boundary Problems Any differentialapproach has to be coupled to a suitableprocedure for the treatment of the boundary condition at infinity. A detailed discussion of this open boundary

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

79

TABLE 4 INHOMOGENEOUSLY LOADED3D CAVITY. Formulation

A-F (Time domain) A (Reactive power peaks) F (Reactive power peaks) E (Golias et al., 1994) H (Golias et al., 1994) H (Webb, 1988) A-V (Bardi et al., 1992) F-I/I (Bardi et al., 1992)

f(MW 259 256 258 254.9 254.5 267 258 259

Nore: Values of the fundamental resonant frequency given by the Fourier analysis and by the first peak of the reactive power, compared with the results reported in Golias et al. (1994). Webb (1988), Bardi et al. (1992).

problem is outside the scope of this paper. Here, we briefly recall the philosophy of the main existing approaches. The most natural approach is truncation. The unbounded domain V is replaced by a bounded region VOclosed by an artificial frontier avo at a large but finite distance from the zone of interest, including all sources and material media, where homogeneous Dirichlet boundary conditions are usually imposed. This yields artificial sources at avo, namely, artificial magnetic charges or currents for stationary or quasistationary problems and unreal reflected waves for high-frequency problems. The main disadvantages of this simple approach are that a large number of degrees of freedom is required to describe the field behavior far away from the region of interest and that it is difficult to determine the minimum size of the bounded domain for a given acceptable error. Notice that to keep the complementary character of the two solutions of the magnetostatic problem, when using truncation, inequality (181) is preserved only if the artificial boundary conditions at a Voare V x A n = 0 for problem (177) and (T,- Vs2) x n = 0 for problem (178). In this way, assuming B = 0 and H = 0 outside avo, the canonical equations and interface conditions are verified everywhere (including a VO).Of course, the constitutive error does not approach zero at the inner side of a Vo. An alternativeapproach is based on coordinatetransformations. In low-frequency electromagnetics, it was first proposed with reference to the classic inversion with respect to a sphere VO of radius r, that encloses the conducting region by Albanese and Rubinacci (1988a) and by Xiuying and Guangzheng (1987). It was then generalized by Freeman and Lowther (1989), Imhoff et al. (1990), Lowther

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R. ALBANESE AND G. RUBINACCI

and Freeman (1992). Brunotte et al. (1992), and Stochniol (1992) to cope with an arbitrary shape of the boundary domain. Recently a similar approach was discussed again by Pinello et al. (1996), who proposed a modified from of Laplace’s equation based on coordinatestretching, mapping the open regions into rectangular domains. Related to the coordinate transformation is the use of infinite elements. Another family of approaches is based on the coupling between the finite element method-applied to the region occupied by material media-and boundary solution procedures like the boundary element method (BEM), well suited for the solution of the external homogeneous unbounded region, in which potential theory can be applied. The coupling between FEM and BSP was proposed by several authors in Cendes (1976), Zienkiewicz et al. (1977). and McDonald and Wexler (1980). The first significant application of this approach to computational electromagneticswas made by Bossavit and VCritC in their 3D eddy current code TRIFOU (Bossavit and V&it6,1982,1983). When using this approach, particular care should be taken to preserve the self-adjoint character of the operators. This is ensured in the method suggested in Albanese et al. (1986), and Albanese and Rubinacci ( 1988a),which intrinsicallyyields symmetric positive definite matrices. With this approach, the FEM analysis is carried out within a bounded domain V , of spherical shape enclosing sources, conductors, and magnetic materials. The procedure makes use of the Green function of the Laplace operator for the external region. It expresses the B . n term appearing in the boundary term of (178). or the corresponding term of (210), directly to the values attained by the scalar potential S2 on avo. In this way the boundary term involving both S2 and aS2/an can be transformed into a self-adjoint form consisting of the sum of two boundary integrals that depend only on the values of a. Suited for the solution of unbounded microwave problems is the use of absorbing boundary conditions (ABC) at the artificial boundary avo of the integration domain VO in which the FEM is applied. The most popular types of ABC are of local type, like the Engquist-Majda conditions (Engquist and Majda, 1977). which have recently been extended for their application to 3D vector edge element formulations (Yao Bi et al., 1995). The differential character of the local ABCs, which preserves the sparsity of the FEM system matrices, is obtained by asymptotic expansions of the solutions of wave equations. Therefore, if the artificial boundary is not sufficiently far from sources and material media, unreal reflected waves appear and the consequent error may become unacceptable. A powerful approach that overcomes the limitations of the local ABCs is the Perfectly Matched Layer (PML) technique recently proposed by Berenger (1994) and by Chew and Weedon (1994). This method is based on the use of an absorbing layer, made of an artificial matched medium designed to absorb without reflection any kind of electromagnetic wave, since the reflection factor of a plane wave striking the vacuum-layer interface is zero at any frequency and incidence angle. At the external side of the PML perfectly conducting conditions are applied. The

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conductivity and the thickness of the layer are designed so that the magnitude of the waves coming back to the computationaldomain reflected by the perfectly conducting boundary conditionsis sufficientlydamped by the double crossing of the PML.

VII. DISCUSSION AND CONCLUSIONS

The essential question is about the applicability of computationalelectromagnetics for the design and analysis of electromagneticdevices. It is by now demonstrated that the use of advanced computational tools is indispensablein scientificresearch in fields in which electromagnetics play a crucial role, such as nuclear fusion and particle accelerators and the design of complex devices like Magnetic Resonance spectrometers. As highlighted by Trowbridge (1996), the use of computational tools in industry is now becoming widespread for various reasons. First, the severe technical limitations of early computer hardware are now only a memory. Second, young professionals are now more familiarwith computertools. Third, a variety of reliable software tools, either general purpose package or specific programs for specific devices, are now largely available. Nevertheless, several problems remain that deserve serious attention from both users and developers of computational methods. To provide a wide and comprehensive list of these problems is outside the scope of this paper. Here we mention only those that in our view are on the top of the list. In several applications like Non-DestructiveTesting or Nuclear Magnetic Resonance, computational accuracy is a critical requirement. In principle, numerical methods converge to the correct solution as the discretizationincreases. However, the accuracy requirement of more than six significant digits (which is routinely required in Nuclear Magnetic Resonance) may lead to a number of unknowns that can hardly be treated with the present computer limitations in terms of time and memory. In these cases suitable techniques, like perturbative models (Albanese et al., 1995a) or the application of reciprocity (Albanese et al., 1990), may help. In any case, it is essential to q u a n w the numerical accuracy, possibly with error bars. Well-establishedtechniques for the error estimation are available (Fernandes et al., 1990; Babuska and Rodriguez, 1993), but, in our view, the most practical is the constitutive error approach, which provides global and local error estimates as well as upper and lower bounds for global (Rikabi et al., 1988a) and local (Greenberg, 1948; Albanese et al., 1996a) quantities. Nevertheless, the most critical limitation to the computational accuracy stems from the approximate computer models of the material media. In this respect, in the future, particular attention should be paid to the material modeling of hysteretic media, including vector hysteresis and modeling of minor loops.

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Coupled problems, i.e., problems in which the interaction of electromagnetic fields with phenomena of different nature cannot be neglected, are also a challenging task, involving interdisciplinary competence. Motion, thermal diffusion, continuum mechanics, fluid flows, and magnetohydrodynamics are only a few examples of classes of phenomena whose interaction with electromagnetic fields must be taken into account in the design of advanced devices. Visualizationor, more generally, presentationof the results is also a critical issue, since it is essential to transfer the results of the computational analysis, typically 3D field maps that permeate vacuum and material media, to the intuition of the designers of the devices. Finally, we should keep in mind that the computational analysis is sometimes aimed at understanding phenomena, but more often it is used for the design of devices in trial and error approaches. In this respect, optimization procedures and, more generally, approaches to the solution of inverse problems are gaining more and more interest. In the general framework of the future developments of computational electromagnetics, what role will be played by the FEM techniques described in this paper is hard to predict. However, some points are undeniable. Edge elements have provided a sound physical framework for discretized electromagnetic fields, which has proved to be the best environment for the constitutiveerror approach and the two-component vector potential formulations. Therefore, it is easy to imagine that the future achievements in the fields of error analysis will cross paths with this approach and that the 3D electromagnetic analysis with rigid body motion will probably make use of the integral formulation described in Section V, maybe taking advantage of the features of existing or next-generationparallel computers. It is clear that all methodologies should not only be an intellectual exercise but also be application driven. Applied research is significantif there is someone who intends to invest in it. It is important that managers and professionals involved in fields that have been marginally touched by the computer revolution realize this and push for specific developments.

ACKNOWLEDGMENTS This work was partially sponsored by Italian MURST and CNR.

REFERENCES Akhtazard, S., and Johns, P. B. (1975). IEEPmc. 22, 1349-1352. Albanese, R., and Rubinacci, G.(1988a). Proc. COMPUMAG, 6th, 25-28 August 1987, Graz, Austria; also in IEEE Trans. Magn. 24.98-101. Albanese, R., and Rubinacci, G. (1988b). IEE Proc. 135 (Pt.A), 457-462.

THE SOLUTION OF 3D EDDY CURRENT PROBLEMS

83

Albanese. R.. and Rubinacci, G. (1988~).Roc. Eddy Current Seminar (C. R. I. Emson, Ed.), Rutherford Appleton Laboratory, England, March 1988. pp. 81-85. Albanese, R.. and Rubinacci, G. (1990a). IEE Proc. 137 (Pt. A), 16-22. Albanese. R., and Rubinacci, G. (1990b). Int. J. Num. Meth. Eng. 29,515-532. Albanese. R., and Rubinacci, G. (1990~).IEEE Truns. Mugn. 26.650-653. Albanese, R., and Rubinacci, G. ( 1990d). In Industrial Application of Electromagnetic Computer Codes (Y.R. Crutzen, G. Molinari, and G . Rubinacci, Eds.), pp. 161-187, Kluwer Academic, Dordrecht, The Netherlands. Albanese, R., and Rubinacci, G . (1991). Solution of TEAM Workshop Problem 10. Proc. of 3rd Int. TEAM Workshop, Sorrento, 1991, pp. 27-36. Albanese, R., and Rubinacci, G. (1992). IEEE Trans. Mugn. 28, 1228-1231. Albanese, R., and Rubinacci, G. (l993a). Rev. Roum. Sci. Techn. Elecrrotechn. Energ. 38.45-52. Albanese, R., and Rubinacci, G. (1993b). J. Compuf.Phys. 108,236-245. Albanese. R., and Rubinacci, G. (1993~).Error estimation in a transient nonlinear three dimensional eddy current problem (TEAM Problem 10). Proceedings, 4th Int. TEAM Workshop, Miami, 1993, pp. 13-17. Albanese. R., and Rubinacci, G. (1994). Current Topics in MagneticResearch (J. Menon, Ed.), Vol. 1 pp. 281-293. Research Trends, Trivandrum, India. Albanese. R., Martone, R., Miano, G., and Rubinacci, G. ( I 985). IEEE Trans. Mugn. 21,2299-2302. Albanese. R., Blum, J., and De Barbieri, 0. (1986). Europh. Conf. Abstract, 8th Europh. Conf. on Computational Physics, Eibsee, Vol. 10D, pp. 4 1 A . Albanese, R., Martone, R., Miano, G., and Rubinacci, G. (1990). Proceedings, European TEAM Workshop and lnternational Seminar on Electromagnetic Field Analysis, Oxford, England, 23-25 April 1990 (Y.Crutzen, N. J. Diserens, C. R. I. Emson, and D. Rodger, Eds.), pp. 13-26. Albanese, R., Coccorese, E., Martone, R., Miano, G., and Rubinacci, G. (1991). E E E Truns. Mugn. 27,399G3995. Albanese. R., Hantila, F. I., and Rubinacci, G. (1992). IGTE Symposium, Graz, Austria, September 1992, pp. 101-106. Albanese, R., Fresa, R., Martone, R., and Rubinacci, G. (1994a). IEEE Trans. Magn. 30,2968-2971. Albanese, R., Fresa. R., Martone, R., and Rubinacci, G. (1994b). Proceedings of PIERS’94, Noordwijk, 1994. Albanese, R., Fresa, R., Martone, R., Miano, G., Rubinacci, G., and Verolino, L. (1994~).Int. Symposium on Electromagnetic Compatibility, 13-16 September 1994, Rome, Italy. pp. 872-876. Albanese, R., Fresa, R., Miano, G., Rubinacci, G., and Verolino, L. (1994d). COMPEL 13(Suppl. A), 257-263. Albanese, R.. Hantila, F. I., and Rubinacci, G. (1994e). EMF Workshop (Y.Crutzen, and E. Coccorese, Eds.), Capri, Italy, 29-31 August 1994, pp. 75-83. Albanese. R., Fresa, R., and Martone, R. (1995a). Int. J. Appl. Elecrmmagn. Mech. 6,73-88. Albanese, R., Fresa, R., Martone, R.. and Rubinacci, G. (1995b). Conference Proceedings, 11th Annual Review of Progress in Applied Computational Electromagnetics, 20-25 March 1995, Monterey, California, pp. 350-357. Albanese, R., Fresa, R., and Rubinacci, G. (1996a). Proceedings, 2d ENDE Workshop, October 1996, Tokyo, Japan. Albanese. R.. Hantila, F. I.. Preda, G., and Rubinacci, G. (1996b). Proceedings, IGTE Symposium, 23-26 September 1996, Graz, Austria. Albanese, R.. Hantila, F. I., and Rubinacci, G. (1996~).IEEE Trans. Magn. 32,784-787. Aymar, R. (1996). Proceedings, 19th SOFT (Symposium on Fusion Technology), Lisboa, Portugal, September 1996. Babuska, I., and Rodriguez, R. (1993). In!. J. Num. Merh. Eng. 36,539-567. Preis, K., Vrisk, G., and Richter, K. R. (1992). IEEE Trans. Mugn. 28,1142-1 145. Bardi, I., Biro, 0..

84

R. ALBANESE AND G. RUBINACCI

Barton. M. L.. and Cendes. Z. J. (1987). J. Appl. Phys. 61,3919-3921. Berenger, J. P. ( I 994). J. Coniput. Phys. 114, 185-200. Biro. 0..Ed. (1996). Proceedings of Compumag Conference on the Computation of Elcctromagnetic Fields. Berlin, Germany, 1 6 1 3 July 1995. IEEE Truns. Mugn. 32. Biro, 0..and Preis, K. (1990). IEEE Truns. Magn. 26, 418423. Biro, 0..and Richter, K. R. ( 1991). In Advances in Electronics I J J I ~Electron Physics (P. W. Hawkes, Ed.), Vol. 82, pp. 1-96, Academic, Boston. Biro. 0..Preis, K., Richter, K. R. (1996). IEEE Trans. Magn. 32, 651-654. Bloomberg. D. S., and Castelli, V. (1985). IEEE Trcms. Mtrgn. 21, 1174-1 180. Blum. J.. Dupas, L.. Leloup, C., and Thooris, B. (1983). IEEE Trans. Magn. 19, 2461-2464. Bossavit, A. ( 1 98 I ). Coniput. Methods Appl. Mech. Eng. 27,303-3 18. Bossavit. A. (1988). IEEE Trcms. Mugn. 24, 74-79. Bossavit, A. (1990). IEEE Trans. Magn. 26, 702-705. Bossavit, A. (1992). IEE Proc. 139 (Pt. A). 265-272. Bossavit, A. (1994). IEE Pmc. Sci. Meas. Technol. 139, 265-272. Bossavit, A. (1996). Int. J. Numerical Modelling: Elecrronic Networks, Devices and Fields 9, 19-34. Bossavit, A., and Mayergoyz, 1. ( 1989). IEEE Trufls.Magn. 25.28 16-282 1. Bossavit, A., and VBrite, J. C. (1982). IEEE Trans. Mogn. 18,431435. Bossavit. A., and VtritC, J. C. (1983). IEEE Truns. Magn. 19. 2465-2470. Brown, M. L. (1982). IEE Pmc. 129 (Pt. A), 46-53. Brunotte, X.. Meunier. G.. and Imhoff. J. F. (1992). IEEE Truns. Mugn. 28. 1663-1666. Bryant. C. F.. Emson, C. R. I., and Trowbridge, C. W. ( I 990). IEEE Trans. Mugn. 26.2373-2375. Carpenter, C. J. (1977). IEE Proc. 124, 1026-1034. Cendes, Z. J. (1976). IEEE Trctns. Microwave Theory Tech. 24,468-473. Cendes, Z. J. (1991). IEEE Trans. Magn. 27,3958-3966. Chew, W. C., and Weedon, W. H. (1994). Microwave Opr. Tech. Lett. 7,599404. Collie, C. J. (1976). Proc. COMPUMAG, Ist, Oxford, 31 March-2 April 1976, pp. 86-95. Davey. K. (1988). COMPEL 7, 11-27. Dular, P., Nicolet, A., Genon, A., and Legros, W. (1993). Proceedings of thc Fourth International TEAM Workshop, Miami, Florida, 5-6 November 1993 (0.Mohammed, Ed.), pp. 23-26. Dular, P.. Nicolet, A., Genon, A., and Legros, W. ( I 994). IEEE Trans. Magn. 30, 29862983. Enison, C. R. I. ( 1988). COMPEL 7.89-101. Engquist. B., and Majda, A. (1977). Math. Comp. 31, 629-651. Fernandes, P. (1995). IEE Pmc. Sci. Meus. Technol. 142,299-312. Fernandes, P., Girdinio, P.. Molfino, P., Molinari. G., and Repetto, M. (1990). IEEE Trans. Mugn. 26. 795-798. Forsman, K., Kettunen, L., and Nuutinen, J. P. (1996). IEEE Trans. Magn. 32, 1421-1424. Freeman, E. M.,and Lowther, D. A. ( 1989). IEEE Trans. Mugn. 25.4 1 3 5 4 137. Fujiwara, K. ( I 992). Proceedings, 5th International IGTE Symposium, Graz, Austria, 1992, pp. 185212. Gimignani, M., Musolino, A., and M. Raugi, M. (1992). IEEE Trans. Mag". 30, 3024-3027. Golias, N. A., and Tsiboukis, T. D. (1994). IEEE Trans. Mugn. 30, 2877-2880. Golias, N. A., Tsiboukis, T. D., and Bossavit, A. (1994). IEEE Trans. Magn. 30,3586-3589. Gomes, J., and Ida. N. (1993). Proceedings of the Fourth International TEAM Workshop, Miami, Florida, 5 4 November 1993 (0.Mohammed, Ed.), pp. 79-84. Greenberg, H. J. (1948). J. Math. Phys. 27. 161-182. Griffiths, H. B., and Hilton, P. J. (1970). A Comprehensive Text of Classical Mathematics: A Contemporary Interpretdon, Van Nostrand Reinhold, London, 1970. Gruber, R., and Rappaz, J. ( 1985). Finite Element Methods in Linear Ideal Mugnetohydmdynaniics, Springer Series in Computational Physics, Springer, Berlin.

THE SOLUTION O F 3D EDDY CURRENT PROBLEMS

85

Hanimond, P. ( 1989). IEE Proc. 136 (Pt. A), 207-216. Hammond, P., and Penman, J. (1978). IEE Pmc. 125,701-708. Hantila, F. 1. (1975). Rev. Roum. Sci. Techn. Electrotechn. Energ. 20, 397407. Holzinger, C. S. (1970). IEEE Trans. Magn. 6,6045 (1970). Kameari. A. ( 1981). J. Comput. Phys. 42, 124-140. Kameari, A. (1988). COMPEL 7.65-80. Kameari, A. (1990a). IEEE Truns. Magn. 26,466-469. Kameari, A. (1990b). Proceedings, 4th International IGTE Symposium and European TEAM Workshop. Graz, Austria, 10-12 October 1990,pp. 91-99. Karrnaker. H. C., and Robertson, S. D. T. (1973). IEEE Trans. PAS, 815-823. Kettunen, L., and Forsman, K. (19%). IEE Proc. Sci.. Meas. Technol. 143, 91-98. Kettunen, L., and Turner, L. (1992). IEEE Trans. Magn. 28, 1639-1642. Kettunen. L.. Forsman, K., Levine, D., and Gropp, W. (1995). Int. J. Num. Meth. Eng. 38, 26552675. Koizumi, M., and Higuchi, Y. (1995). IEEE Trans. Magn. 31, 1516-1519. Koizunii, M., and Onizawa, M. (1991). IEEE Trans. Magn. 27,40774080. Irnhoff, J. F., Meunier, G., and Sabonnadikre, J. C. (1990). IEEE Trans. Magn. 26,588-591. Li, C.. Ren, Z., and Razek, A. (1994). IEE Pmc. Sci. Meas. Technul. 141.25-30. Lowther, D. A., and Freeman, E. M. (1992). IEEE Trans. Magn. 28, 1667-1670. McDonald, B. H., and Wexler, A. ( I 980). In Finite Elements in Electrical and Magnetic Field Pmblrms (M. V. K. Chari, and P. P. Silvester, Eds.), pp. 161-190, Wiley, New York. Menges, J. B., and Cendes, Z. J. (1995). IEEE Trans. Magn. 31, 1342-1347. Moreau, J. J. (1966). Fonctionnelles Convexes, Seminaire Leray, Collbge de France, Paris. Mur, G., and de Hoop, A. T. (1985). IEEE Trans. Magn. 21.2188-2191. Nakata, T., Ed. (1990). COMPEL 9 (Suppl. A), 191-318. Nakata, T., Takahashi, N., Fujiwara, K., and Shiraki, Y. (1990). IEEE Trans. Magn. 26,434-437. Nakata, T.,Takahashi, N., and Fujiwara, K. (1991). Proc. 3d Int. TEAM Workshop, Sorrento (R. Albanese, E. Coccorese, Y. Crutzen. and P. Molfino, Eds.), pp. 21 1-221. Nedelec, J. C. (1980). Num. Math. 35,315-341. Penman, J. (1988). In Advances in Electronics and Electron Physics (P. W. Hawkes, Ed.), Vol. 70, pp. 315-364, Academic, Boston. Pichon. L.. and Bossavit, A. (1993). IEEE Trans. Magn. 29, 1595-1600. Pinello. W.. Gribbons, M., and Cangellaris, A. (1996). IEEE Trans. Magn. 32, 1397-1400. Polak, S. J., Wachters, A. J. H., and van Welij, J. S. (1983). lEEE Trans. Magn. 19,2447-2449. Reichert. K. (1970). IEEE Trans. Magn. 6, 283-288. Ren. Z. (1996). IEEE Trans. Magn. 32,655-658. Ren, Z., and Vkrite, J. C. (1989). Pmc. Beijing Int. Symp. Electromagnetic Fields El. Engng., 19-21 October 1988, Beijing, China, pp. 596-599. IAP, Pergamon, Elmsford, NY. Rikabi, J., Bryant, C. F., and Freeman, E. M. (1988a). Int. J. Num.Meth. Eng. 26, 1963-1987. Rikabi. J., Bryant, C. F., and Freeman, E. M. (1988b). IEE Proc. 135 (Pt. A), 208-216. Rodger, D. (1983). IEE Proc. 130 (Pt. A), 233-238. Rodger, D. (1988). COMPEL 7,27-63. Ruatto, k? ( I 992). Fusion Technology I992-Proceedings. 17th Symposium on Fusion Technology, Rome. Italy, 14-18 September 1992 (C. Ferro, M. Gasparotto, and H. Knoepfel, Eds.). pp. 14641468. Sabbagh, H. A., Miller, E. K., and Ludwig, A. (1990). Proceedings of Asian TEAM Workshop (Z. Cheng, K. Jiang, and N. Takahashi, Eds.), pp. 191-195. Sabonnadiere, J. C. ( I 995). Proceedings, The Sixth Biennal Conference on Electromagnetic Field Computation (CEFC94) (J. C. Sabonnadiere, General Chairman). IEEE Trans. Magn. 31, 13032191.

86

R. ALBANESE A N D G. RUBINACCI

Sadiku Matthew. N. 0. ( 1992). Nirrm,ricn/ Twhriiqirr.s iri E/c,c.tnirrrtr~rieric.s. CRC. Boca Rnton. FL. Sinikin. J.. and Trowhridge. C. W. ( l970), / / I / . J. Nitrrr. Mcj/h. ErrR. 14. 423440. Stochniul. A. (1992). lEEE Tnrrls. Mtigri. 28. 1679-1681. Takagi. T.. Nnrimatsu. S., Sugiura. T.. Hashimoto. M.. and Miya. K . (19x9). Proc. Team Wnrkshnp and Mecting on thc Applications of Eddy Currents Computations. Bicvrcs. 20-21 March I989 (J. C. Virile, A. Bossavit. J. Cahouet, and Y. Crutzen. Eds.). pp. 55-62. Trowhndgc. C. W. ( 1996). /EEL Trtrris. M t r q i . 32, 627-632. Uranknr. L. ( 1990). Ertgiricvririg Aritr/y.si,s Wirh Boitr~dtrrvE/ernerlr.s 7. 124- 128. lirata. K.. and Kamcari. A. (199s). lEEE Trtrrls. M n p r . 31. 144(&1443. van Wclij. J. S . (1985). lEEE Trirris. Mer,y7. 21, 239-2241, Veriti. J. C. (1984). COMPEL 3. 167-178. Webh. J. P. (1988). lEEE Trtrri~,M t r p t . 24. 162-165. Wehh. J. P.. and Forghani. B. ( 1993). lEEE T r m x M n p 29. 2461-2463. Whitncy. H. (I957). G'twrirrrit.lrirr,qrtrriori Tlirory. Princeton U.P., Princcton. NJ. Xiuying. Q.. and Guangzheng. N. (1987). Proc. COMPUMAG. 6th. 25-28 August 1987. Craz, Austria. Y d o Bi. J. L.. Nicolas. L.. and Nicolas. A. (1995). lEEE Trcois. M i r p i . 31, 1538-1541. Zienkiewic7. 0.C.. Kelly, D. W.. and Bettes. P. (1977). l r i r . ./. Nirm. Mrrk. Erlg. 11. 335-375.

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I . Introduction . . . . . . . . . . . I I . Nanofahrication Methods . . . . . . A. Electron-Beam Lithography . . . . B . Focused Ion Beams . . . . . . . C . Scanning Prohe Lithography . . . D. X-ray Lithography . . . . . . . . . . E . Othcr Fabrication Techniques F. Summary . . . . . . . . . . 111. Pattern Transler . . . . . . . . . A . Additive PatternTransfcr . . . . . B . Etching . . . . . . . . . . . C . Summary . . . . . . . . . . IV. Rcsolution Limit of Organic Resists . . A . Electron-Beam Exposure . . . . . B . Development of Exposed PMMA Resist . . . . . . . . . . C . Summary V. Applications of Nanostructures . . . . A . Quantum Interference . . . . . . B . Superlattices . . . . . . . . . C . Single Electronics . . . . . . . D . Quantum Dots . . . . . . . . E . Magnetic Nanostructures . . . . . F. Summary . . . . . . . . . . VI . Conclusion . . . . . . . . . . . Rct'crcnccs . . . . . . . . . . .

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I . INTRODUCTION The physics and engineering applications of nanostructures have created much interest in the subject of nanotechnology. Within the broad classification of nanotechnology one of the most important topics is nanoelectronics. It is primarily the relentless quest to reduce the size of integrated circuit elements that has been the driving force toward the development of nanotechnology. Nanotechnology in

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* Prescnt address: Department of Physics. State University of New York at Stony Brook Stony Brook NY 11794.3800. USA .

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microclcctronics dcscribes the process of malung minute artifacts in a material for electronic device applications and basic physics studies. It emphasizes the processes of pattern generation by which designed structures for devices are fabricated in a semiconductor substrate. Nanotechnology is particularly important bccausc it enables device miniaturization, which not only results in dramatic reductions in the unit cost per function but also leads to improved performance. As dcvicc dimcnsions decrease, the intrinsic switching time decreases and there is a reduction in power consumption per dcvice in the circuit, leading to more devices per chip and greater circuit functionality. For several decades there has been a drive to achieve smaller, faster, and less-expensive devices, which has demanded technology capable of reaching ever smaller dimensions. Today the production lines of the integrated circuit industry are working well into a submicron-length scale, and research work has entered a new phase-that of the nanometer-length scale. The regime of fabrication technology where minimum linewidths are less than or equal to 100 nm is described as nanofabrication. There are two distinct ways of working in this length scale, which is close to the basic structural size in many natural material systems. One way is the top-down method, which is based on lithography techniques and tries to make nanostructures from bulk materials. The other way may be characterized as a bottom-up approach to nanofabrication, basically a chemical synthesis method that deals with the fabrication of functional devices from atoms and molcculcs. Wc limit our discussion in this review to the top-down method, which is predominantly based on lithographic methods of nanofabrication. Nanometer-scalc fabrication with the potential for industrial applications was first demonstrated using electron-beam lithography (EBL) in the 1970s (Brocrs ct al., 1976). Minimum feature sizes around 10-15 nm were produced in an organic polymer resist using electron-bcam lithography more than 10 years ago (Beaumont et al., I98 1 ; Broers, 1981). The so-called conventional electron-beam lithography method was used, which is still the most practical method for nanomctcrscale structure fabrication. The resolution of conventional electron-beam lithography with organic rcsists was limited to about 10 nm, and devices with features significantly smaller than 10 nm were not fabricatcd by this method until very recently. Progress has becn madc recently in reducing the minimum feature size achievable by electron-beam lithography in organic resists by careful control of the development processes and exposure dose. Structures 5-7 nm wide have bccn fabricated in organic resists and transferred successfully to semiconductor substrates (Chen and Ahmed, 1993a. c; Vieu et al., 1997). Other resist systems as well as fabrication processes utilizing electron beams have also been studied and it has been demonstrated that inorganic resists have very high resolution, down to 1 nm (Isaacson and Muray, 1981). Electron beams have also been used for more unconventional fabrication techniques such as beam-induced deposition or etching. Although the electron-beam technique has been the dominant method in nanofabrication for over two decades and most of the useful nanostructure devices have

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been fabricated with this method, there are other fabrication techniques that are capable of nanometer fabrication and have certain advantages over the electronbeam technique. The focused ion-beam (FIB) based fabrication technique is one of the best alternative methods. It is applied in several ways to nanofabrication; not only via rcsist cxposure, but also through other techniques, such as implantation, milling, and deposition. Focused ion-beam techniques have some distinct advantages, including negligible proximity effect, where adjacent exposed areas interact and require a lower dose for correct exposure. But so far the resolution or minimum length scale achievable with FIB has been significantly greater than the results achieved with EBL. Scanning probe microscopes (SPM), such as the scanning tunneling microscope (STM) and atomic-force microscope (AFM), have been developed mainly to study the surface structures of solid materials. In general, the SPM has little effect on the sample under its normal working conditions; but when operational conditions are changed, STMs and AFMs can be used to fabricate nanostructures, potentially at atomic levels of rcsolution. It is this potentially h g h resolution that has attracted much interest in using the SPM for structure fabrication from not long after its invention. Resist exposure with the SPM is possible and structures down to 20 nm wide can be produced. Many chemical and physical reactions can be induced by SPMs that can be used to make nanostructures as small as a few nanometers wide. The SPM has also demonstrated the unique capability of manipulating single atoms and molecules, and nanostructures down to the atomic level have been produced by extracting and depositing single atoms. All methods mentioned so far work in a series manner and lack the necessary speed for mass production of nanostructures, although they possess the feature of high resolution. X-ray lithography, a natural extension of photolithography, has the advantage of having high resolution and at the same time a high throughput. However, many issues need to be resolved before it can find viable applications in nanofabrication. The most promising method of using X-rays is in-contact or proximity printing. Developments in mask making and X-ray sources are vital to the use of X-ray lithography in future large-scale nanofabrication for which there is no other viable technique at present. In nanofabrication, a planar process, where an irradiation-sensitive film (resist) is patterned by electrons, ions, or X-ray radiation, is generally used. The purpose of resist exposure is to use the pattern in the resist as a mask for producing structures in the substrate material. After the exposure, the pattern generated in the resist has to be transferred to underlying materials to make functional devices. There are many high-resolution pattern transfer techniques and they can be either additive, such as liftoff and plating, or subtractive, such as dry etching or wet chemical ctching. Structures less than 10 nm have been transferred to solid substrates with both additive and subtractive methods. With the extreme miniaturization of electronic devices the regime of classical diffusive transport, in which present day semiconductor devices operate, will

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eventually break down and quantum effects will becomc important. If the down scaling in the size of electronic devices in microelectronics is to continue, devices operating with different mechanisms, perhaps depending entirely on quantum mechanics might become necessary. As the structure size is reduced below some relevant physical length scale, many new physical phenomena occur, some of which are purely quantum-mechanical in origin. Examples of such phcnomena are quantum interference whcrc the wave nature of electrons is exploited (Webb et al., 1985), ballistic transport where the quantization of electron energy determines the basic feature of the devicc characteristics (van Wees et al., 1988; Wharam et al., 1988), and single electronics where the charge comes in quanta of the electronic charge e (Likharev, 1988). The branch of physics devoted to study these effects is often called “mesoscopic” physics, since on these short length scales the devices acquire unusual properties that are neither those of microscopic objects (atoms and molecules) nor those of macroscopic systems. The novel phenomena explored in mcsoscopic systems may provide options for innovation in future electronic devices. For example, sincc the experimental demonstration of quantum ballistic transport, many proposals for quantum dcviccs have appeared in the literature, varying from a novel principle of operation for a single transistor to entire computer architectures in which arrays of quantum dcviccs operate phase coherently. These devices usually have dimensions smaller than 100 nm and hence are called nanostructure devices. To be commercially attractive, nanostructure devices should not only offer definite advantages over conventional ones but also not impose any restrictions for the operation of thcsc devices. In this respect, one of the most stringent requirements is operation at room temperature. Most cxpcriments carried out in mesoscopic physics are prcscntly demonstrations in the domain of low-temperature physics and most nanostructure devices can operate only at cryogenic temperatures. Since the tempcrature at which quantum effects become significant is usually inversely proportional to some power of the device dimension, it is obvious that a further reduction of device dimensions is required to meet the operation conditions of these devices. In this review we discuss a fcw possible applications of nanostructure devices.

11. NANOFABRICATION METHODS

Thc process of miniaturization began as soon as the planar transistor was invented. This advance in the miniaturization of electronics made possible the invention of many solid-state devices, greater understanding of the physics of operation of these devices, cleverness in integrating them into complex circuits, and evolutionary improvcrnents in the technology of their fabrication. The motive for miniaturizing devices in integrated circuits is that smaller devices work faster,

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allow greater functionality, consume less power, and cost less. On the commercial market integrated circuits are now available with several million transistors on a single chip of silicon a few square centimeters in area. As the steady scaling down of microelectronic devices has proceeded, research into fabrication technology has progressed into the nanometer regime, where some of the device dimensions are less than 100 nm. In this sub-100-nm regime, the choices of fabrication methods become very limited. Although an optical microscope can detect a 100-nm object, it cannot resolve the object because the wavelength of the light used to image it is larger than the object size. For the same reason, optical lithography, the most commonly used tool in the integrated circuit industry, is capable of around 200-nm lithography. With shorter wavelengths and better lenses, the best optical lithography using phase-shift mask technology can produce structures with minimum linewidths down to about 125 nm. Optical lithography is not currently capable of nanometer fabrication and will not be a subject of this review. Fabrication methods capable of producing nanometer-size structures can be divided into two categories. One uses a mask to define the pattern and works in much the same way as optical lithography does; this category includes X-ray lithography, electron-beam proximity printing, and ion-beam proximity printing. Here the pattern resolution is mainly determined by the resolution of the mask. The other category is a more direct writing method, which does not need a mask to define a pattern and generally has a higher resolution than the mask-based indirect methods. The direct methods include electron-beam lithography, focused-ion-beam lithography, X-ray lithography, and STMs working in the lithography mode.

A. Electron-Beam Lithography Electron-beam lithography (EBL) machines have been studied for microstructure fabrication since the 1960s (Mackintosh, 1965). By the beginning of the 1970s, the principles of EBL had been established and different kinds of systems were developed for lithography applications. Most of the machines were made for fabrication at submicron or at best deep submicron resolution and their performance could not meet the requirement for nanolithography. For example, the electron-beam energy employed in submicron EBL machines was usually 20-30 keV, and although it is possible to obtain an electron-beam diameter less than 10 nm at a 20-kV electron acceleration voltage, proximity effects limit the type of nanostructures that can be made. To enable the practical realization of nanometer structures it is necessary to have a higher operating voltage of between 50 and 100 kV to give the process latitude needed for successful and reproducible development of patterns in a resist. Dcpending on the image forming electron optics, there are two basic kinds of EBL machines in use today. One is the shaped beam machine in which an uniformly

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illuminated aperture is demagnificd and imaged onto a sample by a series of clcctromagnetic lenses. The beam current has an almost uniform distribution with high current density and resolution is characterized by the edge resolution of the final image. The other is the Gaussian beam machine in which the beam crossover formed in the electron gun is demagnified and focused onto the substrate and the beam current density has a Gaussian distribution. Its resolution is determined by the spot size. Shaped beam clectron lithography was developed to provide a higher throughput due to its high current density. The resolution of a shaped beam machine is not as high as that of a Gaussian beam machine and is usually limited to about 100 nm. In nanolithography research the Gaussian beam machine is invariably used because of its smaller beam diameter. The electron optics in a Gaussian beam lithography system is very sinular to that in a scanning electron microscope (SEM) in which a very fine electron beam is generated and a crossover is focused onto thc surface of a sample by successive demagnification in an electron optical column. The significant differences between the two systems are that in an EBL the electron beam is scanned on the sample according to a certain designed pattern and the interaction of incident electrons and the sample is used to create the pattern, while in an SEM the beam is scanned in a raster fashion on the sample and signals generated by interaction of incident electron and sample are collected to produce an image. The electron beam in a EBL can be switched off (blanked) where no exposure is needed or the exposure is finished, while this is not ncccssary in a SEM. Finally, much attention is paid to the deflection system in EBL, with field size. scanning speed, and distortion of the beam being important issues. The operating voltage is also generally higher in EBL than the almost universal 30 kV uscd as the standard voltagc in SEM work. I. Electron-Beam Lithogruphy Systems

A typical schematic ofaGaussian beam EBL machine is shown in Fig. 1. It consists of an electron optical column, a pattern generator, a computer, and associated power supplics. In such a system, the main electron optical column usually consists of an electron-emitting gun, two to three dcmagnifing magnetic lenses, a defection system, and a beam-blanking unit as shown in Fig. 2 . In a practical EBL machine, there are additional optical elements, such as alignment defectors and stigmators. Depending on different writing strategies, there are two different kinds of EBL machines: the vector scan and the raster scan. In the vector scan system the electron beam scans only over the places where exposure is needed and is blanked as it moves between scanning localities, while thc raster scan method scans the entire ficld and blanks the beam where exposure is not needed. Thermionic electron guns with the combination of a cathode, a Wehnelt, and an anodc arc commonly uscd because they are reliable and simple to opcratc. Electrons thermally emitted from a cathode are accelerated by a voltage applied

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II

Interface

a

' 7 liGGGl ' I

I

I

Pattern Generator

Lens current power supply

43

I

I

Mag Box

I

Misc

Electron optical Column

n

--

p?-ADC

Display

buffer

04-J RT

alignment

'

Signal Amp

v t

stigmator

Backscatter detector and Pre-Amp I

C

Fl
poncnts

iirc l i o t

A hlock diagram of a typical clcctron hcain lithography machine. The vacuum cornshown.

bctween the cathode and the anode. A bias is applied to the Wehnelt electrode to control the emission beam current. A lanthanum hexaboride (LaB6)tip, either single crystal or polycrystalline, is usually used as the electron-emitting cathode for its high brightness and long lifetime compared to a tungsten hairpin tip. A field-emission gun, which is widely uscd in high-resolution scanning electron microscopes, has a higher brightness and a smaller beam diameter than a LaBh gun does, but its current is less stable than for LaB6.Another drawback is that a

100kV electron gun

Gun alignrncnl

First lens

Second lens Beam hlanking platcx Ohjectivc aperturc

Stigrnator First deflector of

SOIL Second deflector of SOIL Swinging Ohjectivc Irnmcrsion Lcns (SOL) FiGuRE 2. The electrnn optical coltinin of nanowriter. a high resolution electron heam coluiiin. Three electromagnclic lenses including the swinging oh,jcctivc imincrsion lens are uscd t o obtaiii a <S nni diaiiietcr heam siLe. (Reprinted with pcrmission from J. VNc. k i . Trc.h,io/. B6 ( 6 ) . 200') ( 1988). Chen. Z. W. et al. Copyright 1988 American Vacuum Society.)

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much better vacuum is needed for field emission compared to (lo-’ torr) for LaBh to operate so it is not so well suited for EBL. Most EBL machines used nowadays are equipped with thermionic LaBh cathodes. However, recent research has shown that the stability of field emission can be improved significantly especially when a thcrmal field emission gun is used. The thermal field-emission gun is now suitable for lithographic applications and there have been reports of the use of the gun for lithography applications (Gesley et al., 1988; McCord et al., 1992). Commercial systems using thermal field-emission guns are in advanced stages of development and production. It is very likely that as the technology progresses thermal fieldemission guns will replace LaB6 guns as the first choice in EBL systems. The crossover of the electron beam formed by the electron gun is focused onto the surface of a sample. The electron-beam spot size at the crossover for a LaBh cathode is usually 10-50 p m , hence it is necessary to demagnify the spot size considerably to achieve nanometer resolution. A set of magnetic lenses is normally used because of their low aberration coefficients compared to electrostatic lenses. The minimum beam size in an EBL machine is usually determined by the aberrations of electron optics and not by diffraction. The de Broglie wavelength of electrons is given by

where h is Planck’s constant, mo is the rest mass of the electron, e is the particle charge, and q5 is the electrical potential. For an electron beam of a few keV, the electron wavelength is smaller than 0.1 nm. For various reasons, which are discussed later, a high electron-beam acceleration voltage (350 keV) is normally used, thus the spot size of an EBL machine is not usually limited by diffraction. The total beam diameter d can be estimated using the expression

in which the term do is directly proportional to the square root of the current in the beam I and inversely proportional to (Y according to the following expression:

where B is the brightness of the cathode measured as current density per unit solid angle, and (Y is the semiangle of convergence of the beam at the sample plane. The term d, is the diameter of the disk of confusion caused by spherical aberration, d, = ;C,a3

(4)

where C, is the spherical aberration coefficient of the final lens. The term d, is the

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diameter of the circle determined by chromatic aberration, rl, =

C,(y).

(5)

where A V is the effective energy spread of thc electron beam, and C, is the chromatic aberration coefficient. Both spherical and chromatic aberration coefficients can be calculated for a given lens. Usually a short focal length lens is uscd as thc objective lens because of its lower aberration. With the optimum final aperture size and good design of the electron optics, it is possible to create a spot size of a few nanometers diameter on the optical axis of the electron-beam column. Further significant improvcment in the beam size will require the sample to be immerscd in the magnetic field of the objective lens, as in the case of scanning transmission electron microscopes. To write patterns on a sample it is necessary to move the electron beam over an area of the sample called the scan field. Deflection aberrations occur when the beam is moved off axis, which will also increase the beam diameter. In an SEM, apre-lens double deflection system is used because it allows a very small working distance to be used; but high resolution can be maintained only over a limited field. Outside this region off-axis deflection aberrations enlarge the electron beam and distort the shape of the scanned area. In nanometer-scale EBL machines, in-lens magnetic deflection systems are generally used. A magnetic deflection systcm has the advantage that the magnetic field of deflectors is combined with that of the final lens so the aberration coefficient can be minimized. Ohiwa et al. (197 1) proposed a deflection system where the axial position and rotation of the second deflector is optimizcd to reduce the aberrations. It was shown theoretically that the aberration of an in-lens deflection system can bc reduced to a level far lower than would be possible with a pre-lens system. However, the calculation involved was based on convenient analytic approximations and not on physically realistic lens and deflection fields and it was not possible to design a systcm that could satisfy the requirements. A Variable Axis Immersion Lens (VAIL) system based on the work of Ohiwa et al. using practical focusing/deflection fields has been developed for high-accuracy and highthroughput EBL (Sturans and Pfeiffer, 1983). A submicron resolution was achieved ovcr a 10 x 10-nim field. Chen ct al. (1988) proposed a Swinging Objective Immersion Lens (SOIL) system and employed it in a nanometer-scale EBL machine NANOWRITER, showing that a sub-10-nm spot sizc can be obtained over a 250 x 250-pm scanning field. The scanning of an electron beam in an EBL machine is controlled by a pattern generator, which in turn is controlled by a computer to write a given pattern. The electron beam is blanked by a beam-blanking unit when the pattern or stages of the pattern are finished. with the beam blanking controlled by the samc pattern generator. Thc position of the blanking unit is usually chosen such that the center of the deflection field eoincidcs with a crossover formed by a focusing lens. When

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the beam is blanked, the position of the beam on the sample will not be moved, which prevents a blanking ‘‘tail’’ being formed. The electron optical column is the most important component in any electronbeam lithography machine. Apart from this, there are many practical issues in the system design for nanometer-scale EBL machines. For more details, interested readers can refer to papers that appeared in earlier issues of this series. EBL systems designed specifically for nanometer-scale writing are now commercially available. In addition, add-on equipment is also available that can convert some SEM systems into less sophisticated and somewhat rudimentary writing systems. 2 . Electron-Beam Nanofabrication Methods

Electron-beam lithography is the most sophisticated and the most developed of the nanofabrication processes and the most practical and widely used direct pattern generation technique to date in the nanometer regime. Many different techniques have been developed for nanostructure fabrication using EBL machines, and fabrication of structures down to 1 nm has been possible. Some of the techniques are introduced in the following sections. a. Organic Resists. Electron-beam lithography using organic polymer resists, in other words the conventional electron-beam lithography method, is the oldest practical technique in nanofabrication. It uses the same pattern-defining technique as that used by optical lithography in integrated circuit production, the so-called planar process as illustrated in Fig. 3. In the planar process, a radiation-sensitive thin film, conventionally an organic polymer, is used as a pattern transfer medium, or resist. In this process, the substrate is first coated with a thin layer of resist by spin coating or spray coating, followed by the selective exposure of the resist to some form of radiation. After being irradiated by, for example, an electron beam, chemical reactions and physical changes occur in the resist. The exposed areas have a different solubility than the unexposed parts in a suitable solvent. The more soluble part can be removed selectively in the development stage so a pattern is obtained in the resist film, which can then be used as a mask for transferring the pattern to the substrate. Pattern-transfer techniques include deposition, etching, growth, and implantation, and are discussed later. There are two types of resists: positive resist where the electron-beam irradiated part becomes more soluble. and negative resist where the electron-beam irradiated part becomes less soluble. Among conventional organic polymer resists, poly(methy1 methacrylate) (PMMA) is known to have the highest resolution and was first identified as a high-resolution electron-beam resist in 1968 (Haller et al., 1968). At a low irradiation dose, PMMA works as a positive resist. Main-chain scission and sidegroup removal occur in resist molecules during electron-beam exposure. The exposed resist has a reduced molecular weight, hence a higher dissolution rate.

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Resist Substrate

Development

Positive resist

Negative resist

nnn Processing

deposit ion

etching

implantation

growth

after resist removal

FIGURE3 . Planar p r w e s . The substrate is coated with a thin layer of radiation sensitive liliii. This tilm is irradiated by a patterned radiation. Alter developrnent. a pattern is gcncratcd in the film. Thia pattern is transfered to the substrate by deposition. etching. implantation and growth.

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A positive tone pattern is generated. At a much higher dose, cross-linlung between polymer chains occurs in the resist and a three-dimensional network is formed and becomes less soluble; the insoluble residue can be used as a negative resist. PMMA resist in positive mode has a much higher resolution and is more generally used. In nanolithography, a thin resist layer and a high electron acceleration voltage are normally used to reduce the effect of forward scattering of electrons in the resist. Beaumont et al. (1981) and Broers (1981) demonstrated that sub-20-nm structures can be fabricated in PMMA resist on thin-membrane substrates. In the work of Beaumont et al. (198 1) an 8-nm-diameter 50-keV electron beam was used in the experiment. The thin resist layer was used to reduce the forward scattering of electrons, and a membrane substrate was used to reduce the backscattering of electrons. Metal lines less than 20 nm wide were fabricated on a thin membrane using a thin PMMA resist layer and a liftoff technique. The exposure distribution from an electron beam in PMMA resist was studied by Broers (1981) with a very finely focused beam of about I nm in diameter at 50 keV. In this study a thin layer of PMMA resist on a thin-membrane substrate was used. The exposure distribution was determined by measuring the exposure dose needed to open up lines with widths smaller than the width of the distribution. The width of the exposure distribution was measured to be between 11 to 14 nm. The spreading was assumed to be set by straggling of low-energy secondary electrons into the resist. Effects that may be attributed to the finite molecular size of the PMMA were also observed. It was further suggested that the secondary electrons can be excited remotely from the high-energy incident electron beam and may straggle farther into the resist before their energy is dissipated (Broers, 1981). Broers et al. (1989) later showed that the resolution of PMMA resist is about the same at a much higher electron accelerating voltage of 350 kV. Although thin membranes are very useful for resolution studies, most technological applications require the use of solid substrates rather than electron-transparent membranes. Usually the resolution on solid substrates is worse than that on thin substrates where the limitations imposed by the forward and backward scattering are substantially overcome. In solid substrates the proximity effect becomes severe when the size of the structure is comparable with the scattering range of electrons. However, for isolated structures in the nanometer regime, the range of electron backscattering becomes much larger than the size of the structures so it will not so much affect the resolution as reduce the exposure latitude. Using a trilevel resist system that separates the thin layer of high-resolution PMMA resist from the substrate, Jackel et al. (1981) produced 25-nm metal structures by the liftoff method. Craighead et al. (1983) studied the exposure properties of single-layer PMMA resists on solid substrates at different electron-beam energies and demonstrated that the exposure latitude increases significantly for higher electron energics. At 120 keV, metal lines as small as 10 nm can be fabricated on solid substrate

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FIGURE4. About 5 nm wide lines developed in a PMMA resist. Electron beam lithography and ultrasonic development were used to achieve the sub-10 nm resolution. (Reprinted with permission from Appl. Phys. Left. 62, 1499 (1993A). Chen, W. et al. Copyright 1993 American Institute of Physics.)

with an electron beam of about 2 nm in diameter. Emoto et al. (1985) made 8-nm lines in PMMA resist on a solid silicon substrate with an electron beam working at 50 kV. Later Chen and Ahmed (1993a) demonstrated that by carefully controlling the development process even smaller structures can be fabricated in PMMA resist. It was suggested that the intermolecular forces between the PMMA molecules and resist swelling during development limit the minimum structure achievable in experiments to about 10 nm unless special development methods are used (Chen and Ahmed, 1993a). By using ultrasonic agitation during development to aid the dissolution, sub-10-nm lithography is possible in PMMA resist. Figure 4 shows an SEM micrograph of lines made in PMMA resist on a solid substrate. The sample was coated with a 10-nm-thick AuPd layer prior to SEM examination. The thickness of the PMMA resist is about 60 nm and the lines are about 5-7 nm wide. It is not known if the lines were developed down to the substrate in the experiment. More recently, Vieu et al. (1997) fabricated -5-nm lines in PMMA with ultrasonic agitation. They subsequently used the patterned resist as a liftoff mask and made -5-nm monogranular metal wires, which shows that sub-10-nm lines can be fully developed in PMMA resists and used satisfactorily in the lift-off process provided that the deposited metal layer is not too thick. Vieu et al. suggest that the gaps

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formed in thicker metal lines formed by liftoff are not the result of incomplete development as suggested by Ryan et al. (1993, but are a consequence of strain in the metal film. Many negative resists have also been studied for lithography applications. PMMA working in a negative mode was demonstrated to have nanometer resolution (Broers et al., 1978). Metal lines 15 nm wide were fabricated by ion etching with a PMMA resist mask exposed in its negative mode. Later, Tamamura et al. (1982) studied the application of chloromethylated poly-a-methylstyrene for high-resolution lithography. It was shown that 20-nm-wide structures can be fabricated with this resist. More recently, the suitabilityfor nanolithography of the acid catalyzed resist poly(p-t-butyloxycarbonyloxystyrene)in both positive and negative modes was investigatedby Umbach et al. (1988). The exposure of this resist is based on a chemical amplification scheme, which enables it to have a higher sensitivity than that obtainable with resists like PMMA. Lines 40 nm wide in the positive mode and 18 nm wide in the negative mode were obtained in the resist. Contamination is usually an undesired phenomenon in a vacuum system. However, in certain conditionsit can be used as an electron-beamresist. Contamination resist is formed from the thin layer of hydrocarbon that condenses onto a sample that is placed in a vacuum system where a high partial pressure of hydrocarbons exists. The hydrocarbons at the sample surface are broken by the impact of the electron beam and a cone of material builds up under a stationary beam that can be used as an ion-milling resist. Lines as small as 8 nm wide were fabricated by using contamination resist and ion milling (Broers et al., 1976). Figure 5 shows a micrograph. A dose of 10’-104 C/m2 is required to produce an adequate resist thickness to protect a 100-nm-thick metal layer from ion etching. Such a high dose makes it difficult to fabricate large-area devices and limits the applications of contamination resist. Conventional EBL is the most versatile tool in nanostructure fabrication and most of the useful nanostructure devices have been made by combining conventional EBL with planar processing. The ultimate resolution of the technique is believed to be limited by the molecular size of the resist and the delocalized range of scattered electrons and secondary electrons in the resist. Because of this resolution limit, researchers began to look for other resists with higher resolution as well as other lithography processes. b. Inorganic Resists. Many inorganic materials, such as NaCl, MgF2, AlF3, LiF, and A1203. were found to sublimateunder electron-beamirradiation and have been studied for lithography applications. Broers and Sturgess (1978) discovered that some low molecular weight materials, e.g., NaCl and MgF2, vaporize under electron-beam irradiation. Holes of 5 nm diameter were formed in a 0.25pm-thick NaCl crystal by positioning a 1-nm-diameter 50-kV electron beam on the substrate. They also studied the pattern transfer of these inorganic resists.

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FIGURE 5. A 20 nm thick AuPd ring produced by ion milling through a contamination rcsist mask. The diamcter of the ring is 160 nm and the linewidth is about 15 nm. The line remains continuous. (Reprinted from Microelectronic Eng. 9, 187 (1989). Broers, A. N . et al. with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

A patterned MgF2 film was used to mask a gold film from ion milling and to produce slots less than 10 nm in width in a 20-nm-thick gold film. Later it was demonstrated by Isaacson and Muray (1981) that structures down to 1.5 nm in size can be fabricated in thin NaCl films by direct sublimation with an electron beam. The dose needed to vaporize NaCl is quite high, about lo2C/cm2 (Isaacson and Muray, 1981). Mochel et al. (1983) reported that holes and fine lines on a 2-nm scale can be made in metal p-aluminas. The equivalent dose is extremely high, about 5 x lot6C/cm2, compared to the normal dose for PMMA resist of C/cm2. Apparently resists requiring such a high dose are not likely to find any application in device fabrication. Muray et al. (1984) found that AlF3 can be vaporized at a much lower dose than NaCl and structures down to 2 nm were patterned in AIF3. The dose required is about 20 C/cm2. The patterned A1F3 films were used to transfer a pattern to a silicon nitride film by dry etching (Muray et al., 1985). On solid silicon substrates, Macaulay and Berger (1987) showed that sub-10-nm structures could be made in AlF3 films. It was also shown by Muray et al. that at certain conditions metallic A1 structures can be obtained in a AIF3 film after electron-beam irradiation (Muray et al., 1985). The dose required for A1F3 is still too high for the fabrication of useful devices. Muray et al. (1983) found that lithium fluoride has a resolution limit below

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FIGURE 6. Scanning transmission electron microscope (STEM) bright-field image of test pattern detail exposed into 50 nm LiF(AIF3)film on 20 nm Si3N4 membrane. The line exposure dose was 450 nC/cm. Pitch of the grating (upper right-hand comer): 37 nm; radius of the rings (lower right-hand comer): 25 nm; distance of dots (lower left-hand comer): 20 nm (Reprinted with permission from J. Vrrc. Sci. Techno/. B10 (6),2868 (1992). Langheinrich, W.et al. Copyright 1992 American Vacuum Society.)

10 nm with electron-beam sublimation and needs a much lower exposure dose (10-2C/cm2) than that for A1F3. It is known that LiF deposited at room temperature is polycrystal with an average grain size of tens of nanometers, which translates into a fluctuation in thickness of the film; therefore, LiF films deposited at room temperature are not suitable as high-resolution resists for device fabrication. Although structures as small as 1.5 nm were fabricated in a LiF film, the linewidth was not uniform due to the variation of film thickness. Langheinrich et al. (1992) showed that LiF film doped with A@ deposited at low temperature is quasiamorphous and sub-10-nm lithography is possible in such a film. Figure 6 shows a micrograph of sub-10-nm structures fabricated in LiF(AlF3) films. More recently, Fujita et al. (1995) studied the lithography properties of LiF and showed that there was no sublimation effect in a pure LiF film and only carbon contamination was observed. It was shown that sublimation can occur only when the LiF film is doped with AIF3.This does not agree with the results of Muray et al. (1983), and it is not possible to find the source of the discrepancy, because the experimental conditions are not identical. Fujita’s experiments were performed on a solid silicon substrate using a 50-keV electron beam instead of on thin membranes with a 100-keV electron beam as in the case the experiment of Muray et al. The contamination problem in the experiments of Fujita et al. was more severe because the vacuum during the pattern delineation was torr compared with < torr in the work of Muray et al. work. The LiF films were deposited by different methods and their properties could be quite different. Stress relaxation can also play a role in election-beam exposure of LiF films (Langheinrich et al., 1992).

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Although inorganic resist has been shown to have very high resolution, structures down to 2 nm being possible, the dose needed for this process is very high, except in the case of LiF doped with AlF3, which needs only a moderately higher dose compared with PMMA resist but has a higher resolution. The main problem associated with inorganic resists is that there is no suitable pattern transfer technique. In general, the patterned films do not stand up well to a variety of dry or wet etching techniques and it has not been possible as yet to produce any useful devices using inorganic resists. c. Electron-Beam-Assisted Mod@cation of Materials. Very high-energy electron-beam irradiation will damage materials and change their properties. Radiation damage has been used to fabricate nanostructures. Since, in this process, only electrons with sufficiently high energy can modify the material, the resolution will not be determined by the range of low-energy secondary electrons and can potentially be very high. As early as 1968 it was found that the bombardment of thermally grown Si02 with electrons would enhance its etch rate in p etch (15: 10:300 HF:HN03:H20); and trenches of 0.6 p m width were fabricated (0’Keeffe and Handy, 1968). The contamination buildup during electron-beam exposure blocked the etch and prevented smaller features from being made. Jones et al. (1986) studied the damage in silicon substrates caused by the irradiation of a 500-keV electron beam. It was shown that the damaged areas can have either an enhanced or a retarded etching rate, depending on dose and energy. Therefore, structures can be made by using a selective etching method. However, only submicron structures were fabricated in the preliminary experiment. High-resolution lithography using electron-beam-induced damage was realized in SiO2 (Allee and Broers, 1990). By using a sacrificial layer to overcome the contamination problem and a high electron energy (300 kV) to minimize the forward scattering of the electrons in the sacrificial layer and the oxide, arrays of lines were etched in Si02 with periods down to 21 nm (Allee and Broers, 1990). The difference in etch rate is not very high, so the grooves etched in Si02 are V-shaped. However, it is still possible to transfer the pattern in Si02 to a Si substrate by reactive ion etching, and lines of 10 nm width were produced in the Si substrate (Pan et al., 1991). Knotek and Feibelman (1979) discovered that if Si02 is irradiated by a lowenergy (1- to 3-keV) electron beam then oxygen is lost from the surface layers. A similar effect was found to exist for high-energy electrons. Using a very high current density electron beam at 100 keV Chen et al. (1993) demonstrated that Si02 in a thin membrane can be converted to Si after exposure to a dose of 3 x lo9Clm2. The change in the material is seen in the energy filtered STEM image and it was shown that the area irradiated by electron beam was changed from SiOz to silicon. Silicon dots, 2 nm in diameter and silicon wires 4-5 nm wide were produced.

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d. Electron-Beam-InducedDeposition and Etching. Apart from the exposure of resist, damage-assisted fabrication, and direct sublimation, electron beams can also be used to form patterns directly by inducing the deposition or etching of material. The electron-beam exposure is sustained with the introduction of another material onto the substrate. The smallest fabricated features reported so far are around 15 nm. A rod with a diameter of 15 nm was deposited from w F 6 using a 100-kVelectron beamof 3 nmdiameter (Matsuiet al., 1989).Morerecently 14-nmdiameter carbon dot patterns were deposited on a Si substrate (Ochiai et al., 1991). For dense patterns, lines of 30 nm width with a center-to-center separation of 50 nm were deposited from a gold organometallic compound (Lee and Hatzakis, 1989). Recently, Hoyle et al. (1996) studied the deposition characteristics of W(CO)6 at different electron-beam energy and a range of exposure doses. It was found that the use of low-energy electron beams results in faster deposition of material and in material with a lower resistivity and is therefore more suitable for the fabrication of electrical interconnects. However, the size of electron beam is increased as its energy is reduced, so the resolution of the method decreases with beam energy. Electron beams can be used to pattern a material directly in the presence of a reactive gas. In thls process, adsorbed gas molecules are dissociatedto form species that react with the underlying material to form molecules that are subsequently desorbed from the surface. Dissociation is usually induced by low-energy electrons such as the secondary electrons emitted from the surface as a result of inelastic collisions of the primary electrons with the substrate. Direct writing onto PMMA resist has been demonstrated by an electron-beam-induced surface reaction using a ClFl gas source (Matsui et al., 1989). Wang et al. (1995) demonstrated that a plasma-enhanced, chemical vapor-deposited, amorphous hydrogenated carbon film can be etched by a focused electron beam in the presence of oxygen gas. The structures in the film can then be transferred to underlying substrates.

B. Focused Ion Beams Focused ion beams (FIB) are potentially the most versatile of the means available for nanostructure fabrication. They can either be used for lithography on resist with subsequent pattern transfer on to the substrate, as in electron-beam-based processes, or they can be used to modify the substrate directly so that the need for using a resist as a pattern transfer medium is entirely eliminated. Theoretically, focused-ion-beam lithography has a higher resolution than electron-beam lithography due to the much shorter de Broglie wavelength of ions. In practice the focused-ion-beam machine is not as well developed as the electron beam and is not expected to have as high an ultimate resolution as that of the electron beam. The smallest beam size achieved in focused-ion-beam systems was reported to be

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Tetrode gun: Liquid-metalsource First accelerator Second accelerator Octopole stigmator plate set

Beam blaking plate set Alignment deflection plate set

i mJ I 1 Scan ampliiers

Beam curret monitor

Alignment deflection plate set Octopole stigmator plate set

Lens apeture

1-

CRT display

Probe forming einzel-lens

Scanning deflection plate set Secondafy-electron detector Target

FIGURE7. Schematic of a focused ion beam column including source, column and sample stage. The series of concentric electrodes between the source and the mass separator and between the mass separator and the beam deflector are the electrostatic lenses. (Reprinted from Microcircid Ella 9.3 132 ( 1 9 8 3 ~Cleaver, J. R. A. et al. with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25. 1055 KV Amsterdam, The Netherlands.)

about 8 nm (Kubena et al., 1989) compared with
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a two-lens system to provide flexibility in operation; a mass separator to remove unwanted ion species and preferably also to eliminate any neutrals from the ion beam; deflection of the beam across the final target; and, for pattern generation in lithography, a blanking system to turn the beam on and off. The key element in the development of FIB machines has been the liquid alloy (metal) ion sources, which offer a high brightness, typically exceeding 10' A/(cm2sr),and a small virtual source size of only tens of nanometers (Gamo et al., 1982). Liquid metal ion sources have a simple structure in which a reservoir of liquid metal is maintained at one end of a sharp needle; the metal wets the needle and flows down the tip and metal ions are extracted by a high voltage applied between the needle and an aperture placed below the tip. Following the initial use of gallium, the material that most easily provides a liquid-metal film, work has continued on developing a wider range of sources. In particular, alloys are being utilized to provide liquid metals that can yield a large variety of ions (Gamo et al., 1982). Many alloys can be used in the ion sources as long as they wet the needle. Typical ion species obtained by liquid metal ion sources are Be, As, Sb, Si, Ga, and Au. For some low-melting-point metals, such as Ga, it is possible to use a pure metal source, so there is no need to have a mass separator in the ion-beam column. However, when it is necessary to use the alloyed form for high-melting-point metals, a mass separator is necessary to select the ions wanted for the process. The mass separator is a critical component of a column and its presence must not degrade the optical properties significantly while providing adequate separation of ion species. Two types of mass separators have been proposed and implemented. One of them, E x B , uses a magnetic field with a matched electric field superimposed on it to provide sufficient resolution. Problems arise when a high-energy beam passes through the filter, and generally this type is best used when the ion beam is to be accelerated after filtering. Further, it separates ions according to their energy as well as to their mass-to-charge ratio, and this introduces significant chromatic aberration (Wang et al., 1981). This problem is overcome by the symmetry of the alternative scheme with four magnetic sectors (Cleaver et al., 1983). In this setup the beam is first deflected off axis by the first pair of magnets and separated into individual ion species, of which the desired ones are selected by an aperture that intercepts the remainder of the beam. The desired ion species are then restored to the original axis by the second pair. This configuration is effective in operation, having sufficient resolution and the capacity for use in high-voltage systems. In focused-ion-beam columns electrostatic instead of magnetic lenses and deflectors are used to focus and deflect the ions, because ions are much heavier and travel more slowly than electrons, and magnetic systems would not be suitable because of the difficulties in generating the strong fields that are required. In focused-ion-beam columns, chromatic aberration is the most serious practical limitation to their performance. The relatively large energy spread of 5-10 eV for

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%'&

0 ' 0

medium energy

9

t

(1)

emitted electron

"

bonds broken

dopant atom

low energy

,

0

t

(2)

high energy

e

7

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FIGURE8. Some physical processes occurring on ion impact on a substrate.

most liquid-metal ion sources, ion-ion interaction, together with the relatively large aberration coefficients of ion-optics components, make the minimum spot size around 10 nm, compared to c0.5 nm achievable in an electron-beam column. Pattern-generating and computer control systems are very similar to those required in electron-beam lithography. For a more detailed account of focused ion-beam lithography machines, see Melngailis (1987). When a beam of ions impacts on a substrate, different physical processes occur, as shown in Fig. 8. These processes can be used in many different ways to fabricate nanostructures, such as high-resolution milling, selective area implantation, direct deposition of nanoparticles, and other applications not possible with a beam of electrons. Many fabrication techniques utilizing these different processes have been investigated in the literature, and some are discussed below. 1. Focused-Ion-Beam Lithography Focused ion beams can be used to expose resist just like an electron beam does. Resist exposure with an FIB system was demonstrated and structures down to 30 nm were fabricated (Shiokawa et al., 1984). The smallest structures fabricated by a focused ion beam are sub-20-nm lines made in PMMA resist, which is very close to the smallest structure made by electron-beam exposure (Kubena et al., 1989). High-resolution lithography was also demonstratedin negative resist: Lines as small as 30 nm wide were fabricated using a bilevel resist system (Kubena et al., 1988). In this experiment, a 50-keV Ga beam of 8 nm diameter was used to

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expose a resist, which consisted of 40-nm-thick silicon containing negative-acting resist, polyphenylsilsesquioxane,as the top imaging layer and a 500-nm-thick AZ 1350B resist as the lower layer. With a bilayer system it is possible to achieve high resolution without introducing too much damage to the substrate by the ions. Inorganic resist can also be exposed by a focused ion beam. Koshida et al. (1992) fabricated metal lines as small as 40 nm on a Si substrate by focused ion-beam exposure of oxide resists. Because of the short penetration depth of ions in resist and substrate materials, there is negligible scattering in the resist and very low backscattering from the substrate. Therefore, there is almost no proximity effect for ion-beam lithography, hence it does not require the proximity effect corrections necessary in electronbeam lithography. Atkinson et al. (1992) demonstrated that 30-nmresolution or better is possible with no apparent proximity effects. Test patterns were exposed in PMMA resist on high atomic mass materials and no proximity effects were observed in the worst-case test pattern as shown in Fig. 9. There is almost no significant variation of linewidths. The energy dissipation rate of ions is much higher than that of electrons so it is more efficient for resist exposure. Depending on the mass of the exposing ions, resist exposure sensitivity for ion-beam lithography is two or more orders of magnitude higher than that for electron-beam lithography. However, one of the consequences of the high sensitivity of resist to ions is that statistical fluctuations in the number of ions in a given feature element of minimum size, often called a pixel, could limit the ultimate resolution achievable in a given resist. If the

FIGURE 9. Absence of proximity effect (a) in direct etching experiment and (b) in lift-off experiment using PMMA resist exposed by focused ion beam. (Reprinted with permission from J. Vac. Sci. Technol. BIO (6),3104 (1992).Atkinson, G.M.et al. Copyright 1992 American Vacuum Society.)

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average number of ions incident on a pixel necessary for exposure is N ,the statistical fluctuations in the number of incident ions will be N'/'.To be certain that there is a significant difference between an exposed pixel and a pixel that is not exposed, the number of incident ions must be larger than a certain value to avoid statistical errors. For a 50-keV Ga+ beam, the dose needed for PMMA resist on tungsten substrates is about ions/cm2. For such a sensitivity,each 100 x 100-nm pixel would need 1000ions incident and a fluctuation of 33 ions or a 3.3% exposure uncertainty is present. However, if the pixel is reduced to 10 nm, the number of ions landing on a 10-nm square pixel will be only 10. The exposure uncertainty will be increased to about 33%. Resists with sensitivityhigher than that of PMMA are not suitable for ion-beam lithography in the 10-nm regime due to these statistical considerations. This statisticalconstraint in effect removes the sensitivity advantageof ion beams over electron in the fabrication of nanostructures by conventional resist exposure. 2. Focused-lon-BeamEtching

The use of focused ion beams for the selective removal of material without the use of a patterned resist mask has been explored in recent years. Ion-beam sputtering has been used in the integrated circuit industry to repair photolithography masks (Wagner and Levin, 1989). The minimum feature size of a photo mask is such that the size of the focused ion beam in this application is usually not in the nanometer regime, typically 0.25 p m in diameter. As the downscaling of integrated circuits progresses into the nanometer regime, photolithography will no longer be able to provide the required resolution; alternative fabrication methods have to be used. X-ray lithography is a very promising candidate for nanometer-size mass production. The requirement for X-ray mask repair is much more stringent than that for photo masks and nanometer resolution is necessary. It has been demonstrated that focused ion-beam etching can provide sub-100-nm resolution in mask repair. In research laboratories,focused ion beams have been used to etch gold films directly and an etched linewidth of 38 nm was achieved (Seliger et al., 1979). By employing a post-objective lens retarding and accelerating field to achieve a small beam diameter, Kieslich et al. (1994) demonstrated that 20-nm wide lines can be etched in a gold-coated GaAs substrate by focused-ion-beam sputtering. The problem associated with focused ion-beam direct etching methods is that the sputtered material redeposits over the sample. The comers of etched structures are not well defined and the edges are rough. The etch rate is determined by sputtering so the etch rate is usually very low. The etch rate can be much increased by using gas-assisted focused ion-beam etching. Young et al. (1990a) found that up to 14 times enhancement in etching rate can be obtained for gas-assisted etching of a silicon substrate over pure sputtering and clearly defined comers and edges can

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FIGURE10. Groves etched into GaAs using focused ion beam etching and an un-patterned protective layer. The groves have near vertical side-walls and sharp edges at the GaAs surface. (Mat. Res. SOC.Symp. Proc. 199,205 (1990), Young,R. J. et al.)

be obtained using protective layers on top of the substrate material (Young et al., 1990b). PMMA resist was used as a sacrificial layer, so the comers are protected whle etching proceeds. This method gives very clearly defined lines and gratings as shown in Fig. 10. 3. Focused-Ion-Beam Implantation

Focused-ion-beam implantation can be used to change the properties of substrate materials directly for the fabrication of a variety of semiconductornanostructures, such as quantum wires, point contacts, and single-electron devices. One example of using focused-ion-beam implantation to fabricate nanostructures is to utilize the etch rate change of the implanted region. Shiokawa et al. (1990) showed that nanometer-size fabrication is possible with this technique. In their experiment a GaAs substrate was implantedwith periodic structuresby a 100-keVGa+ ion beam. The implanted region becomes amorphous, and therefore has a higher etch rate in HC1 solution. Periodic structures of 63-nm period with less than 20-nmlinewidths

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were fabricated. Atkinson et al. (1991) demonstrated that nanostructures can be made in InP substrates by a similar technique. Patterns were exposed in an InP substratewith 50-keV focused Ga+ beam and etched with an ammoniumhydroxide and water mixture (1:15). Lines as small as 15 nm and gratings of 20-nm-wide lines on a 50-nm pitch were formed. Another example of using focused-ion-beam implantation to fabricate nanostructures is to use the damage caused by the ion beams to create nonconducting regions on either side of a channel of high-mobility material. Although no material is removed, the crystal structure is damaged so the implanted region becomes nonconductive. The conducting channel between the damaged areas is further reduced by the depletion region and can be made as small as 60 nm (Fujisawaet al., 1994). There are problems with this kind of fabrication technique; the damage caused by the high-energy implantation energy and the unintentionally implanted ions in the vicinity of the conducting channel. Both of these degrade the electron transport properties of the conducting region. Lowering the implant energy keeps the implanted ions away from the conduction channel and reduces crystallographic damage. Itoh et al. (1992) demonstrated that electron mobility in quantum wires fabricated with focused-ion-beam shallow implantation is much higher than that for high-energy implantation. One of the advantages of the fabrication technique using implantation is that the device surface remains planar after fabrication and so enables the use of surface Schottky gates to confine the electrons or to change the density of the electrons (Blaikie et al., 1991, 1992). Blaikie et al. (1991, 1992) demonstrated that 1D wires can be fabricated and their width controlled by implanted gates. Subsequently, a metal gate was put on the surface of the device and used to control the electron concentration. Figure 11 shows a device fabricated by ion implantation. Electron-beam-indicated current (EBIC) was used to obtain the image. 4. Focused-Ion-Beam Deposition

Focused ion beams have been used to induce the depositionof submicron metal patterns, whch provide a maskless process of fabricating submicron metal structures (Blauner et al., 1988). This process was used in photo and X-ray mask repairing and minor circuit editing. Various materials, including Al, Ta, W, Fe, and Au, were deposited using different metal organics and focused ion beams. The deposited films usually contain oxygen and carbon, which come from the background atmosphere or insufficient decomposition (Gamo and Namba, 1985). The resistivity of the metal films obtained by ion-beam-assisted deposition is usually a factor ranging from 30 to 5000 times higher than that of the bulk metals. To overcome this problem, focused-ion-beam direct deposition has been used, which promises highpurity and low-resistivity metal films in contrast to induced deposition (Nagamachi et al., 1993).In this process a high-purity and high-current-densityion beam can be

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FIGUREI I . (a) Material layer structures; (b) schematic cross-section of a completed device; (c) EBIC micrograph of a multi-probe device. (Insf.Phys. ConJ Ser: No. 127, Chapter 4 (1992), Blaikie, R. J . et al.)

deposited directly in high-vacuum conditions. Gold lines 0.5 p m wide with resistivity close to the value of bulk gold were directly depositedby using this technique. The resolution of focused-ion-beam-induced deposition achieved so far is still in the submicron regime. Woodham and Ahmed (1995) proposed a single-atom lithography process in which a beam of ions is used for the direct fabrication of atomic-scale structures, By adjusting the ion landing energy it is possible to make atomic-scale structures. The landing energy of the ions is chosen such that it is high enough to provide an acceptable sticking probability on the surface, but not so high as to cause implantation or undue sputtering of material from the surface layers of the substrate. At a few tens to hundreds of electron volts of landing energy, ions are partially implanted and gold nanodots with size down to 1 nm were produced (Woodham and Ahmed, 1995). Figure 12 shows a SEM micrograph of gold nanodots deposited by a focused ion beam. The size of the dots is between 1 and 3 nm. Ultimately the deposition of single ions may be realized with this method if the dose is carefully controlled and a fast beam-blanking system is used. The key issue in single-ion deposition is the detection and control of single ions, and a preliminary investigation has been reported (Ahmed, 1996). The nanodots can find applications in fabricating the single-electron devices described in Section V. As mentioned earlier, the beam size of ion-beam machines is chromatic aberration limited due to the large energy spread of liquid metal ion sources, which makes it impossible to obtain beam sizes much less than 10 nm even at high beam energies. When the beam energy is reduced, chromatic aberration is increased so

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FIGURE12. The morphology for Au on GaAs deposited with single ion deposition machine at 50 eV. Island siLe is between 1-4 nm. (Reprinted with permission from J. Vuc. Sci. Tuchnul. B1Z ( 6 ) . 3280 (1995). Woodham, R. G. et al. Copyright 1995 American Vacuum Society. Photograph acknowledgment N. Cameron, University of Glasgow.)

the minimum beam size achievable is also increased. By employing a retarding field technique, a beam size less than 100 nm can be obtained, which is still too large for nanostructure fabrication. Recently, de Jager and h i t (1995) proposed a scheme where an energy filter and an optical component with a negative chromatic aberration coefficient are used in the ion optical column to reduce the effect of chromatic aberration. From their calculations, spot sizes of about 1 nm can be achieved. The maximum current in a 1-nm beam at a beam energy of 30 keV A. If this scheme is employed in the single-atom lithography will be 2 x machine, atomic-scale lithography could become possible. C. Scanning Probe Lithography

Since the invention of the scanning tunneling microscope (STM) (Binnig et al., 1982) and the atomic-force microscope ( A M ) (Binnig et al., 1986), it has been possible to study solid surface structures and processes down to the atomic level. The scanning tunneling microscope (STM) operates by positioning a very sharp tip within a few angstroms of the surface of a sample. A three-dimensionalpiezoelectric scanner is used to control the tip’s position in the x , y , and z directions. A voltage is applied between the tip and the sample causing a tunneling current to flow that depends exponentiallyon the tip-to-sampleseparation. The tip is scanned over the surface in a raster fashion to build up an image of the surface topography, often with atomic resolution. There are two principal modes of operation. One of them maintains the tip at a constant height above the sample surface and detects the

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tunneling current between the tip and the substrate, which is then used to construct an image of the surface. Another mode keeps the tunneling current constant by adjusting the distance between the tip and the sample surface and uses the height variation to build up an image of the surface. Other scanning probe microscopes (SPMs) are based on similar principles employing different interactions between the sample surface and the tip; for example, the atomic force microscope uses the interatomic force between atoms in the scanning tip and the sample to form an image. In normal operation an SPM has little or no effect on the sample. However, the working conditions can be deliberately arranged so that an SPM may be used to change a specific property of the sample surface. A variety of interactions between the tip and the substrate,ranging from resist exposure to manipulatingsingle atoms or molecules, have been utilized to modify the sample surface locally under the scanning tip. Because of the very high resolution of SPMs it should be possible to create nanometer-scale structures down to atomic-sized structures and functional units for scientific and perhaps practical applications. Scanning probe lithography (SPL) has attracted much attention because of its potentially high resolution. We shall consider some of the important developments in the rapidly advancing field of nanofabrication by SPM methods.

.

1. Exposure of Resist Materials

In a normal working STM,there is only a small tunneling current between the tip and the substrate, but if the voltage applied between the tip and the substrate is increased sufficiently, electrons are emitted from the tip through field emission. The voltage applied between the tip and the substrate in the field-emission mode is usually a few tens of volts instead of the few volts used in the imaging mode. The beam of electrons emitted from the scanning tip can be used to expose an electronsensitive resist in the same way as in conventional electron-beam lithography. McCord and Pease (1987) showed that patterns can be formed in contamination resist or metal halide films with a STM.Conventional organic electron-beam resists, such as PMMA, have also been investigated for STM exposure, and patterns are formed in the resists after development. AuPd wires as small as 22 nm wide were produced by the liftoff method using PMh4A resist and STM exposure (McCord and Pease, 1988). Other resist systems have also been studied in STM lithography. By exposing a very thin Langmuir-Blodgett film with an STM and subsequently etching the film using argon ion milling, Stockman et al. (1993) have made gold lines as small as 15 nm wide. Kragler et al. (1995) employed a two-layer resist system in which a layer of amorphous hydrogenated carbon was used as the bottom layer and a resist with chemical amplification properties as the top layer. Patterns less than 100 nm were exposed in the top resist layer by an STM. After development, patterns in the resist were transferred to the amorphous

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hydrogenated carbon bottom layer using reactive ion etching. The patterns in the bottom layer were then used as an etching mask to fabricate structures in a semiconductor substrate. The STM working in the electron-beam mode has a number of advantages over conventional electron beams. The current density of an STM can be orders of magnitude higher than in conventional electron-beam systems, hence the exposure time per pixel can be reduced. The low energy of the electrons reduces the proximity effect and also the radiation damage to any sensitive device in the substrate. STM exposure of conventional resist is not the same as a normal electron-beam exposure. Marrian et al. (1992)compared the exposureproperties of SAL-601 resist from Shipley by a conventional electron-beam machine and by an STM. Patterns in the resists were again transferred to a semiconductor substrate by reactive ion etching. It was shown that the minimum-size structures generated by the STM were smaller than those formed by the conventional electron beam machine, and there was no proximity effect in STM-fabricated patterns. However, it was not known if the larger minimum structure fabricated by conventional electron-beam lithography in this experiment was limited by the larger size of its electron beam. There are also disadvantages for STM exposure; for example, the imaging resist layer in STM lithography has to be very thin and it is necessary to work on conducting substrates, which limits the application of the method. The atomic-force microscope (AFM) can also work in the field-emission mode and expose electron-beam resists. PMMA resist was exposed both in positive and in negative modes by AFM,and 35-nm-wide lines were obtained in the resist (Majumdar et al., 1992).AFM exposure in other resists, such as poly(glycidy1methacrylate) (PGMA) negative electron-beam resist, was also demonstrated (Yamamoto et al., 1995). Less than 30-nm-wide structures were patterned in the PGMA resist and subsequently transferred to SiO;! substratesby wet chemical etching. Park et al. (1995) showed that a material called siloxene, commonly known as spin-on-glass, which is usually used as a planarization material in integrated circuit fabrication,can be exposed with the AFM.The etch rate of AFM-exposed areas in a buffered oxide etch is higher than that of unexposed areas, and the selectivity between the two is greater than 20. Hence SOG can be used as a positive resist and lines as narrow as 40 nm can be produced in the resist. A writing speed greater than 1 m m / s was achieved by Park et al., which is much higher than the normal writing speed in resist, which is in the range of a few microns per second. AFM exposure has some advantages over STM; e.g., mechanical damage can be eliminated by maintaining a constant force during exposure and the AFM can work with nonconducting substrates. 2. Localized Electrochemical Modijication Apart from the ability to expose a resist, an SPM is also capable of modifying a sample surface directly without using a resist through inducing chemical reactions

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(a)

(b)

FIGURE13. STM images of high resolution patterns written at -4.5 V tip bias, 2.0 nA, and a line dose of 2 x lo--’ C/cm. (a) 1 nm wide lines spaced on a 3 nm pitch and (b) the letters ‘UI ONR URI’ written with the same linewidth. (Reprinted with permission from J. Vuc. Sci. Technol. B12 (6), 3735 ( I 994). Lyding, J. W. et al. Copyright 1994 American Vacuum Society.)

locally under the SPM tip. When an SPM tip is in close proximity to the sample surface and a voltage is applied to the tip, an extremely high electric field is generated. The existence of this very localized and extremely high electric field under the tip presents new opportunities for nanostructure fabrication through inducing local surface structural and chemical changes. Dagata et al. (1990) discovered that the H-passivated Si (1 1 1) surface can be selectively oxidized in air with an STM as a consequence of the strong electric field generatedbetween the tip and the substrate. Because the chemical modification involves only the top one or two monolayers of material, extremely high lateral resolution is feasible, as demonstrated later by Becker et al. (1990) in ultrahigh vacuum, where areas as small as 4 nm were depassivated in this way. The etch rate of the oxidized area in a liquid etch is less than that of the unoxidized material, which means that the pattern can be generated in the substrate by selective etching. Lateral features as small as 25 nm were demonstrated (Snow et al., 1993). By working in UHV with the same technique, Lyding et al. (1994) have achieved linewidths of 1 nm on a 3-nm pitch. Figure 13 shows an STM image taken after the lines had been written. Similar experiments were performed with an AFM and nanometer-size structures were obtained (Day and Allee, 1993; Snow and Campbell, 1994). This process has been used to produce nanometer-scale side-gated silicon field effect transistors (Campbell et d., 1995). Recently Kramer and coworkers showed that hydrogenated amorphous silicon can also be oxidized by an STM tip and the oxide forms a mask so that only the nonoxidized area is removed with a wet etch (Kramer et al., 1995). Although silicon is the most important material because of its extensive use in microelectronics, the study of STM-induced oxidation processes is not limited to

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silicon; passivation processes in other materials have also been studied. Thundat et al. (1990) studied STM-tip-induced anodization of tantalum and found that anodization proceeded only when the metal was biased anodically in air or in an aqueous alkaline solution. Oxidation of a titanium surface induced by an STM tip was also reported (Sugimura et al., 1993) and dot or line patterns of oxide were fabricated on Ti with a spatial resolution of about 70 nm. Song et al. (1994) showed that a thin Cr film can be locally oxidized with the STM and chromium oxide lines as small as 25-nm width were obtained in the film. It was found that oxidized films have different physical and chemical properties, depending on the exposure dose. Films exposed with high dose can be etched with water or with a standard Cr etchant but at much higher etch rates than unexposed Cr to form a positive pattern. Film exposed at lower dose cannot be etched with either water or a standard Cr etchant, so a negative pattern is obtained if the film is etched. The extremely high electric field can induce other chemical reactions under the scanning tip, as shown by Utsugi (1990) for the case of silver selenide (AgSe). The AgSe compound is a typical mixed ionic electronic conductor. The high field moves Ag ions in AgSe, which then enables a chemical reaction between hydrogen and selenium ions to form hydrogen selenide gas. The high ionic conductivity enables the surface to be modified by the strong electric field just under the tip. By inducing this chemical reaction locally under the STM tip, structures less than 10 nm wide were fabricated (Utsugi, 1990, 1992). Because AgSe can be deposited in a film form using an electroplating technique, it can be used as a resist for subsequent processing of substrate materials. Another fabricationprocess using the SPM can be obtained when a voltage pulse is applied across the tip and the substrate. Direct modification of gold substrates by STM has been demonstrated by Abraham et al. (1 986). Later Schneir et al. (1993) carried out a more detailed study and discovered that lines as small as 5 nm can be written in a gold substrate if the substrate is first covered by a fluorocarbon grease. Further study by Li and coworkers (1989) showed that it is possible to make 2-nm-diameter features at the gold surface even without using a special coating (Li et al., 1989). However, the features written in the gold substrate are not stable due to the high surface diffusion coefficient of gold; patterns disappeared within a few minutes of generation.More permanent marks were generated with an STM on other materials, such as a graphite substrate, as demonstrated by Albrecht et al. (1989). Voltage pulses of variable width and amplitude were applied across the tip and the substrate and holes as small as 4 nm in diameter were produced in the graphite substrate (Albrecht et al., 1989). More recently, Moriarty et al. (1995) showed that atomic-scale modification of GaAs using an STM can be realized. AFMs have also been studied for direct modification of sample surfaces. Kado and Tohda (1995) demonstrated the possibility of recording processes on an amorphous GeSb2Te4 film with an AFM. In this experiment there is no topographic change in the recorded region, but the conductivityof the exposed areas is increased dramatically. This process opens possibilities of further patterning on the sample

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surface. The smallest region where the conductivity is changed by this process is about 10 nm. If this is used for information storage it corresponds to a storage density of 1 Tbit/cm2.

3. Direct Removal and Manipulation of Particles Another way of using SPMs to make nanostructures is to manipulate individual atoms and molecules directly. Not long after the invention of the STM, the controlled movement of atoms with the STM was demonstrated by Becker and colleagues (Becker et al., 1987). They transferred a single atom of germanium from the scanning tip to the germanium substrate by suddenly increasing the tipsubstrate voltage. A lot of work has been done on manipulating single atoms or molecules since then. Eigler and Schweizer (1990) demonstrated the ability of single-atom manipulation by moving xenon atoms on a Ni substrate to form a nanostructure, the letters IBM as shown in Fig. 14. The STM was operated in an ultrahigh vacuum environment at a very low temperature (4 K) (Eigler and Schweizer, 1990). This is a beautiful illustration of what can be done by controlled manipulation of atoms, although there is a long way to go to any practical application. Single-molecule manipulation at higher temperatures has also been demonstrated. Controlled movement of single Csomolecules on the Si (11 1)-7 x 7 surface by an STM tip has been shown by (Maruno et al., 1995). It was suggested that the movement of the Cm molecules by the STM tip results from the atomic forces between the tip and the molecule because there is a threshold gap distance between the tip and the substrate for moving the molecules and the threshold gap is different for different tip materials (Maruno et al., 1995). Apart from the

FIGURE14. A sequence of STM image taken during the construction of a patterned array of xenon atoms on a nickel (110 surface). Grey scale is assigned according to the slope of the surface. The atomic structure of the nickel surface is not resolved. The (171) direction runs vertically: The surface after xenon dosing; and after the construction. Each letter is 50 A from top to bottom. (Nature 344, 524 (1990). Eigler, D. M.et al.)

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STM experiments, controlled manipulation of nanometer-size particles by AFM was also demonstrated. Schaefer et al. (1995) showed that an AFM can be used to deliberately move Au clusters on a surface. Fully controlled manipulation of individual particles, similar to the atomic-scale positioning shown by Eigler and Schweizer using the STM, has also been demonstrated in which 30-nm GaAs particles were moved on the substrate to form nanostructures (Junno et al., 1995). Unlike the STM experiment,the experiment of Junno et al. was done under ambient conditions and with a nonconducting substrate. The STM can be used not only to manipulate individual atoms but also to extract atoms from a substrate. Hosoki and coworkers (1992) have successfully extracted S atoms from the surface of a MoS2 substrate (Hosoki et al., 1992). Words were formed by the holes left by the missing S atoms and each letter is only 1.5 nm high (Fig. 15). This still holds the record of writing the world’s smallest letters. Lyo and Avouris (1991) found that single silicon atoms can be removed from silicon’s (1 11) surface to the tip with a voltage pulse and redeposited on the substrate if the polarity of the voltage is reversed. In their experiment the separation between the tip and the sample is as small as 0.3 nm and the mechanism is assumed to be chemically assisted field evaporation. Huang et al. (1992) showed that silicon atoms can also be extracted from a substrate using different experimental settings, i.e., larger distance between the tip and the substrate. The ease of extraction of Si

FIGURE15. Letters formed in MoS2 after removing S atoms with STM. each of the letters is less than 1.5 nm in height. (Reprinted from Appl. Surf Sci. 60-1, 643 (1992). Hosoki, S. et al. with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands).

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atoms depends on the crystallographic sites of the Si atoms. Controlled extraction of single atoms was demonstrated in ultrahigh vacuum at room temperature. 4. Deposition

Field evaporation has been known for a long time where atoms are ionized and ejected from a surface due to the existence of a high electric field. Usually the potential required to generate a sufficiently high field is in the range of a few kilovolts in field evaporation experiments. Because of the close proximity of an STh4 tip to the sample, an extremely high electric field can be generated when only a few volts are applied between the STM tip and the surface. This high electric field has been exploited to evaporate locally a gold STM tip to deposit gold clusters of about 10 nm in size (Mamin et al., 1990). STMs can also be used to induce the deposition of nanometer-scalemetallic features onto a substrate from an organometallic gas by inelastic tunneling (McCordet al., 1988; Silver et al., 1987). SPM-based technology has extremely high resolution; nanometer-size and even atomic-level fabrication is possible. This might be the only viable fabrication method on the atomic level. However, it has not been employed in fabrication because of its lack of speed. Parallel operation might be essential to achieve the speed required for fabrication.The development of micromachining where thousands to millions of tips working in parallel can be fabricated in a single chip opens up ways for the application of SPM-based technology in nanofabrication (Xu et al., 1995b). D. X-ray Lithography X-ray lithography is an extension of photolithography. The wavelength of photons in X-ray lithography is much shorter than in photolithography and usually is in the 0.4-to 5-nm range. Spears and Smith (1972) reported the first experimentation with X-ray lithography. X-ray lithography was developed to achieve both high resolution and high throughput. It also has the advantages of being able to expose patterns in resist with high aspect ratio and of working on insulating as well as conducting substrates. In the early days of research in X-ray lithography, contact printing was often employed for nanostructure fabrication to minimize image blurring and to obtain a higher resolution. Lines and spaces of 20 nm were produced in PMMA using a special mask made by shadow evaporation (Flanders and White, 1981); and 50-nm lines and spaces using electron-beam lithography for the mask-making (Anderson et al., 1987). However, for large-scale production mask distortion becomes a serious problem for contact printing. In the event where the small gaps required for proximity X-ray nanolithography prove unacceptable, X-ray projection using arrays of zone plates may be a possible alternative. A scanning X-ray microscope with sub- 100-nm resolution has been successfully demonstrated in which a coherent flux of X-ray is focused by a zone plate to form a diffraction-limited spot on a

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sample that is mechanically raster-scanned to form an image (Rarback et al., 1988). If a resist-coated substrate is used with a beam-blanking mechanism, lithography exposures can be performed, but technical problems with beam blanking and the fabrication of high-resolution (sub-50-nm) zone plates still need to be solved. Proximity printing with soft X-rays (wavelength between 0.4 and 5 nm) is currently the only practical method for using X-ray lithography for mass production, because the field size and resolution of X-ray lenses are not yet good enough for projection imaging. Although the feasibility of the X-ray proximity printing method for submicron lithography has been established, its application to nanolithography is relatively new. In the nanometer regime, a compromise has to be made between the need to use short wavelength X-rays to reduce the effect of diffraction and the need to use long wavelength X-rays to increase the exposure sensitivity. It was assumed in the past that the resolution of X-ray proximity printing was limited by diffraction. However, Cerrina (1992) recently showed that this is not the case. The resolution is limited not by diffraction but by the fact that the diffraction process is nonlinear. Thus, the phase shift is different for different spatial frequencies, leading to false images at very high frequencies. Therefore, the resolution of proximity imaging can be strongly enhanced by controlling the phase behavior, opening the possibility of extending X-ray lithography well into the nanometer range. For many years it was believed that the resolution of X-ray lithography was set by the maximum range of photo and Auger electrons generated by the absorbed X-rays (Spears and Smith, 1972). However, work by Early and coworkers (1990) and Deguchi et al. (1990) showed that 30-nm linewidths on the mask were faithfully replicated in PMMA resist at wavelengths of 4.5, 1.3, and 0.8 nm (Fig. 16). The corresponding maximum photoelectron ranges are 4 2 0 - 3 0 , and

FIGURE16. 30-nm wide PMMA structures, standing alone in an otherwise completely exposed field exposed with a 30 nm wide gold absorber line and (a) CK (A = 4.5); (b) Cu L (A = 4.5); (c) AI K (A = 4.5) X-rays. (Reprinted from Microelectronic Eng. 11, 317 (1990). Early, K. et al. with kind permission from Elsevier Science-NL, SARA Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

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40-70 nm, respectively.It is now believed that exposure by X-ray is dominated by the very short-rangeAuger electrons and not the photoelectrons, which contribute only a low-level background. It is possible to create sizes down to 20 nm using wavelengths of 1 nm and longer. The ultimate resolution of X-rays is set by the mask in the case of proximity printing or by the zone plate for projection printing; in most cases, masks and zone plates are produced by electron-beam lithography. It is obvious that the resolution of X-ray lithography cannot surpass that of electron-beam lithography. The unique aspects of X-ray lithography are the absence of proximity effects, an intrinsic resolution below 30 nm, and parallel exposure. The major problem areas are the mask-sample gap (less than 5 p m for linewidths below 70 nm), absorber stress, which must be near zero to avoid mask distortion, and registration to nanometer resolution. E. Other Fabrication Techniques Imprint technology using compression molding of thermoplastic polymers is a low-cost mass manufacturing technology that has been around for several decades. Features with sizes greater than 1 p m have been routinely imprinted in plastics. Recently Chou et al. (1995) demonstrated that imprint techniques can be used to fabricate sub-25-nm structures in polymers. The mold used in this work was fabricated in a thlck SiO;, layer on a silicon substrate using electron-beam lithography and reactive ion etching. Feature sizes of 25 nm consisting of dots and lines were printed in a thin layer of PMMA resist. After the imprint, the PMMA resist was etched with oxygen R E to remove the residual resist at the bottoms of the imprinted region. Metal structures of about 25 nm were fabricated using a liftoff technique. This fabrication method may provide a low-cost, high-throughput lithography technique with sub-25-nm resolution. Structures down to nanometer size c,an also be fabricated even without using a nanometer-resolutiontool. One example is to use a granular film consisting of fine grains of nanometer size as a mask for reactive ion etching to produce nanometersize pillars. Deckman and Dunsmuir (1982) used polystyrene spheres dispersed over the substrate surface to form a monolayer of close packed spheres, so that when the substrate was etched by reactive ion etching, the spheres acted as a mask. Green et al. (1993) made nanometer-size pillars in various semiconductor substrates using irregular arrays of CsCl hemispheres. A very thin layer of CsCl was deposited onto the sample surface by vacuum evaporation, and the CsCl layer breaks up into hemispherical islands on exposure to a humid atmosphere. Nanometer-size pillars were obtained using the CsCl layer as an etching mask in a RIE machine. Gotza et al. (1995) demonstrated that a fine-grain film deposited on a silicon substrate by sputtering an A 1 2 0 3 target can act as an etching mask. Pillars were obtained by reactive ion etching and further thinned by rapid thermal

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FIGURE17. Pillars of about 30 nin in diameter etched in GaAdAlCaAs heterostructure substrate using a granular mask. (Reprinted with permission from Appl. Phys. Left. 66. (10). 1234 (1995), Alphenaar, B. et al. Copyright 1995 American Institute of Physics.)

oxidation to less than 10 nm. Metallic granular films were also used to pattern a substrate. Alphenaar et al. (1995) used thermally evaporated gold as an etching mask to obtain nanometer-size pillars in a GaAs substrate. Gold dots, a few tens of nanometers in diameter, were formed on the substrate by sintering after gold deposition. Reactive ion etching was used to transfer the pattern to the substrate, and very high aspect ratio pillars were fabricated. Figure 17 is an SEM micrograph showing pillars of around 30 nm in diameter.

E Summary Lithography methods with nanometerresolution are summarizedin Table 1. Among them, SPL has the highest resolution, and manipulation of single atoms or molecules with SPL has been demonstrated. However, it has not been possible so far to make any useful structures. For device fabrication, electron-beam lithography has the highest resolution and is the most versatile method. Experimental electronbeam methods, such as direct sublimation, need a very high dose, and a suitable

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NANOFABRICATION FOR ELECTRONICS TABLE 1 COMPARISON OF THE RESOLUTION OF DIFFERENT LITHOGRAPHY METHODS. Method Electron-beam lithography

Ion-beam lithography X-ray lithography SPL

Resist Polymer resist Direct sublimation Direct modification Etch and deposition

Dose (C/m2) XO.1

5

x105

2 2

=lo9 =I@’ x 10-2

%I Conventional resist Atomic manipulation

Resolution (nm)

xo. I

*

15

20 17 20 Single atom

* Writing speed is very slow, one atom at a time. pattern transfer method has yet to be found. Overall, electron-beam lithography with organic polymer resists still offers the best solution for nanometer-scale device fabrication. 111. PATTERNTRANSFER

The primary purpose of producing high-resolution patterns in resists with energetic particles is to use the resulting images as masks for fabricating microstructures in solid substrates, although in rare instances the pattern in the resist may be the main purpose of lithography. The most practical and versatile method of nanofabrication to date has been the planar method employing electron-beam lithography to make patterns in organic resists. The many ways of utilizing the patterns as masking media to fabricate structures in solid substrates can be categorized into two main types of pattern transfer methods: subtractive and additive. Subtractive methods employ different types of wet or dry etching techniques, and additive methods include electroplating and metal deposition followed by liftoff. Electron-beam sublimation methods can also provide high-resolution patterning of inorganic resists, but the transfer methods are not well established.

A. Additive Pattern Transfer 1. Liftoff

The liftoff technique is often used for the delineation of very high-resolution metal patterns. The process is described schematically in Fig. 18. A substrate is coated with a layer of resist and a pattern is defined in the resist using one of the pattern-formation techniques described in the last chapter. The substrate with the patterned resist layer is placed at some distance away from a metal evaporation

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FIGURE18. Lift-off process. An under-cut or straight sidewall are necessary for successful liftoff (a) patterning a layer of resist; (b) depositing a layer of material, usually metal; ( c ) after liftoff in a suitable solvent.

source. The metal is evaporated by means of thermal evaporation or other evaporation methods and deposited over the entire substrate. If the resist has an undercut or vertical profile, there will be no connecting material between the desired metal pattern and unwanted metal that lies on the surface of the resist. Stripping off the resist in a suitable solvent leaves only the desired metal pattern on the substrate. The liftoff process can be successful only when there is an undercut in the resist pattern or when the resist pattern has a straight side wall and the thickness of the material to be deposited is much less than the thickness of the resist. A ratio of 3: 1 between the thickness of the resist and that of the metal is usually enough for successful and reliable liftoff in the case of thick resists (>lo0 nm). For thinner resists, the ratio has to be increased to obtain successful liftoff. The liftoff technique has been successfully used to produce metal structures of dimensions less than 20 nm. On membrane substrates Beaumont et al. (1981) showed that metal lines as narrow as 17 nm can be fabricated by the liftoff method with electron-beam lithography and PMMA resist. On solid substrates the resolution is usually poorer due to the effect of backscattered electrons from the substrate. However, structures with dimensions similar to those obtained on membrane substratescan be fabricatedif the layout of the pattern is chosen such that

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the backscattered electrons can only generate a background exposure and reduce the contrast but not affect the resolution. Craighead et al. (1983) demonstrated that metal lines down to 10 nm in width can be made on solid substrates. Electron energies ranging from 20 to 120 keV were used in the experiment and it was shown that higher electron energies improve process latitude. Most recently Vieu et al. (1997) have shown that monogranular lines as narrow as 5 nm can be formed. The resolution of transferred patterns is usually determined by the resolution of the resist. As the best resolution is obtained with thin resist and the thickness of liftoff metal has to be thinner than that of resist, the aspect ratio of the resulting metal pattern cannot be high for successful liftoff if only a single layer of resist is used; for example, in the experiment of Beaumont et al. (198 l), the thickness of the metal film was only a few nanometers. Even in this case, some part of the metal failed to liftoff. To improve the quality of liftoff and to obtain higher aspect ratio patterns, multiple-layer resists consisting of a thin imaging layer on top of thicker underlayers can be used to create an undercut profile. The resolution of the resist pattern is then determined by the thin imaging layer so a high-resolution as well as high-aspect ratio can be achieved. Tennant et al. (1981) produced arrays of fine lines 30 nm wide and 330 nm high on 95-nm centers with a trilevel resist. A typical problem encountered with multilayer resist approaches is that different resist layers may intermix. In addition to a reduction in the effectiveness of either of the layers, the thin intermixed layer can in some case prevent proper exposure or development of the bottom layer. The intermixing problem can be eliminatedby using materials that do not react with each other or with the solvent of the other layer. Special structurescan also be fabricated with multilayer resists. Ahmed and Ahmed (1997) used a bilayer system consisting of a layer of PMMA resist on top of a thick layer of optical resist to fabricate high-aspect ratio patterns, such as gold airbridges. Several stages are usually needed to develop multilayer resists. More reliable and easier processes can be obtained if both layers can be exposed with a single exposure and developed with one development step. Double-layer PMMA resist systems with a low molecular weight resist at the bottom and a high molecular weight resist as the top layer is one such system. The dissolution rate of PMMA resist is a functionof its molecular weight in achosen developer, and the dissolution rate of lower molecular weight resist is higher than that of higher molecular weight resist; and higher molecular weight resist has a higher resolution. The difference in dissolution rate results in an undercut pattern profile in the resist after development, and at the same time a high resolution can be maintained. Rooks et al. (1987) demonstrated 30-nm-scale structures with double-layer PMMA resist. Conventional liftoff methods are not successful when trying to achieve sub-10nm structures. As the minimum size of patterned structures reduces to the 10-nm regime it often becomes comparable to the grain size of the metal to be deposited. It appears that during liftoff grains are removed as a whole, gaps appear in structures, and the grain size becomes one of the factors limiting the minimum feature size

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achievable by liftoff (Chen, 1994). Previously, ionized clustered-beam deposition was used to reduce the grain size and to deposit continuous metal films; 25-nmwide continuous Au wires were obtained (Huq et al., 1990). Recently Chen and Ahmed (1993~)demonstrated that an ionized beam deposition method can improve the adhesion of the deposited metal film to the substrate and reduce the grain size of the metal film. The metal structures were able to survive ultrasonic agitation during liftoff. A double-layer resist system was used because of its better liftoff characteristics. In the experiments a layer of low molecular weight (600 kg/mol) PMMA was spun on either silicon or GaAs substrates to a thickness of 50 nm and then baked at 120°C for 2 h, followed by spinning another layer of 50-nm-thick high molecular weight (1 150 kg/mol) PMMA on top. The samples were baked at 185°C for 8 h, then kept at 120°C before electron-beam exposure. The ionized beam deposition system is schematically shown in Fig. 19. In the ionized beam

/

/

Flux of ionized beam Acceeration electrode for ionization electrons

1 Electron emitter

FIGURE19. Schematic diagram of an ionized beam deposition system. The material to be deposited is thermally evaporated by resistive heating, ionized by an electron beam and then accelerated by a voltage applied between the sample and the evaporation source. (Chen, W., PhD Thesis. Cambridge University (1994).)

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FIGURE20. An SEM micrograph of 15 nm continuous PdPt lines fabricated with ionized beam deposition and liftoff process. (Chen, W.. PhD Thesis, Cambridge University (1994).)

deposition setup, the material to be deposited is thermally evaporated with resistive heating and ionized by an electron beam and accelerated by a bias voltage between the substrate and the evaporation source. An acceleration voltage of 100 V and 20-mA ionization current were used in the experiments. The background beam current when no material was evaporated was about 15 PA. After increasing the temperatureof the source, the beam current was increased to 20 p A for a deposition rate of 0.1 rids. An acceleration voltage of 500 V was used in some experiments for comparison with 50 V results, but liftoff failed because all of the metal stuck to the substrate. Sub-10-nm metal dots were fabricated using a AuPd film deposited with ionized beam method and liftoff technique (Chen and Ahmed, 1993~). Liftoff of continuous metal wires is more difficult than that of metal dots and it becomes very difficult when the linewidth is comparable with the size of the metal grain. Continuous PdPt wires with linewidth down to 15 nm have been fabricated using the ionized beam-deposited film and liftoff as shown in Fig. 20 (Chen, 1994). Attempts at making sub-10-nm wires were not successful and continuous wires were not obtained. If it is assumed that the metal particles land on the substrate in a random fashion, the particles should be distributed randomly along the resist gap. However, sub-10-nm wires always consist of large grains about the size of the wire width with large length of breaks between grains. It is speculated that it is energetically unfavorable for metal particles to align in a straight line. Metal

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particles are believed to migrate along the resist opening to form islands during the metal deposition in a recrystallization process. There is strain accumulation along the length, which could also break the wires. Another possible explanation is that the resist development was not complete, so that some resist molecules were left on the substrate and some of these molecules bridged across the resist gap. It is envisaged that when there are resist molecules left on the substrate surface, the adhesion of the metal film is reduced. For large patterns, metal can still stick to the substrate because the size of resist molecules is much smaller than the size of the pattern, but for smaller linewidths, the size of resist molecules can be comparable to the width of the lines. In the liftoff process, the lines break at points where the resist molecules are left behind. Recent results by Vieu et al. (1997) confirmed that discontinuity in the metal wires made by liftoff does not come from resist residue and development of the lines by ultrasonic agitation was complete to the substrate level. The discontinuities in the metal wires occur during the growth in thickness of the metallic film, especially at the onset of continuity in the film. If a thin enough layer of metal is deposited where its thickness is not enough to form a totally continuous film, lines are formed without any large gaps. Monogranular lines are formed by liftoff over long lengths without breaks. Figure 21 shows an SEM micrograph of 5-nm-wide lines obtained by liftoff a nominally 3-nm-thick, pure gold granular film. The liftoff technique can also be used to make structures that do not have a straight side wall, such as T-shaped or r-shaped structures. There are many different approaches for making such structures. Some use one exposure step while others use several exposure steps. Some employ single-layer resists and others use multiple-layer resists. One example was shown by Woodham et al. (1992), who used a combined electron- and focused ion-beam lithography and liftoff process to produce nanometer-size T-gates and r-gates. In this process a single layer of PMMA resist was exposed with a focused ion beam and an electron beam successively and developed in a single step. Because of the low and abrupt penetration depth of ions with less straggle than electrons, it is possible to expose the top layer of the resist by using a correctly chosen landing energy of ions. High-energy electron-beam lithography with a finer beam size can subsequently expose the entire layer of resist because the penetration depth is much greater for electrons. After development and liftoff of a metal layer, T- or r-shaped structures can be obtained. Figure 22 is an example of the fabricated structures.

2. Electroplating In comparison with the liftoff method where vacuum deposition equipment is needed, electroplating is much simpler and less expensive. Electroplating is

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(b)

FIGURE 2 I . SEM micrographs of Sub- 10 nm wide lines obtained by liftoff of a 3 nm nominally thick, pure gold granular film. In (a) the monogranular lines are 7 nm wide and in (b) their width is 5 nm. These monogranular lines are made of I-D chain of metallic grains. (Reprinted from Microeleclroriic Eng. 35,253, (1997). Vieu, C. et al. with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25, I055 KV Amsterdam, The Netherlands.)

essentiallya process of electrolyticdeposition of metal from solution and there are two basic variants: electrolytic and electroless. With electrolytic deposition, the surface on which the metal is to be deposited is used as the cathode, and deposition occurs when the metal ions travel under the influence of the electrical field in the aqueous solution of metal salts. A DC voltage is applied between the substrate and the metal to be deposited. The deposition rate can be controlled by the current density in the bath. The electroless process is a technique for deposition through the catalytic action of the deposit itself, without the use of a source of external current. Electroless deposition of gold, platinum, and nickel has been used in nanostructure fabrication. It is necessary to have a conducting substrate to enable the plating process and in many cases the plating base must be removed after

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FIGURE 22. T-gate structure fabricated with comhincd electron beam and focuscd ion beam lithograph and liftoff. (Reprinted with perniission from J. kic. Sci. Terhnol. B. 10.3280 ( I 992). Woodham. R. G. et al. Copyright 1992 American Vacuum Society.)

plating. These shortcomings limit the range of its application. However, this process does have certain advantages over the liftoff technique, such as in the fabrication of very high-aspect-ratio structures. Using electron-beam lithography and electroplating,arrays of Ni pillars of 35 nm diameter and 120 nm height on 100-nm pitch were fabricated (Chou et al., 1994). Figure 23 shows an SEM micrograph of an array of Ni pillars. One of the problems associated with electroplating is endpoint detection. The deposition rate cannot be easily controlled, so it is necessary to develop an in situ endpoint detection system that can be used to determine the optimum electroplating time. For example, overplating resulted in the deposition of mushroom-like structure due to the incorrect timing of the electrodeposition process (Xu et al., 1995a). Xu et al. (1995a) have developed an in situ electrical and optical monitoring technique to observe the electroplatingprocess that ensures reliable predictions of the metal deposition rate.

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FIGURE 23. SEM image of Ni pillar array of 35 nm diameter, 120 nm height and a 100 nm spacing fabricated by electron beam lithography and electroplating process. (Reprinted with permission from J. Appl. Phgs. 76 (10pt2). 6673 (1994). Chou, S. Y. et al. Copyright 1994 American Institute of Physics.)

B. Etching 1. Wet Etching

The process of wet chemical etching is simple and straightforward. It is widely used in the microelectronics industry. Usually wet etching is isotropic, but it is often possible to develop processes with a very high degree of selectivity with respect to composition or crystal structure. It also has the advantage of causing very little etching damage to underlying substrates compared with dry etching. However, these advantages are often outweighed in high-resolution applications by the generally isotropic nature of wet etching, which causes the undercutting of masking layers, making it difficult to control the dimensions of features that are of the same size as the etch depth. This makes it very difficult to transfer nanometersize patterns with wet chemical etching. For nanofabrication, therefore,dry etching is commonly used, but in some special circumstances wet chemical etching can be used to transfer nanometer-size patterns to a solid substrate. Katoh et al. (1990) employed a two-step wet chemical etching technique to achieve high-resolution pattern transfer. Patterns were defined with electron-beam

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lithography on 50-nm-thick PMMA resist and developed with 1:3 M1BK:IPA. The system Z O as patterns were transferred to GaAs substrates with a H ~ S O ~ - H ~ O ~ - H the first step of etching and with Br-CH30H as the second step etching. Gratings with equal lines and spaces of 25-nm width were fabricated in thick GaAs substrates. For crystallinematerials, it is possible to exploit the differentetch rate in different crystallographicorientations. Namatsu et al. (1995) demonstrated that sub- 10-nm structures can be fabricated in silicon substrates. It is known that the (1 11) planes of a silicon crystal have a lower etch rate than other planes in KOH solution. For example, the etch rate of the (1 10) plane is about 400 times faster than that of the (1 11) plane (Kendall, 1975). In the experiment of Namatsu et al. (1995) line patterns about 30 nm wide parallel to the (1 12) direction were first formed in a thin resist layer by electron-beam lithography. The Si surface in the exposed area was oxidized by an oxygen plasma after development to a thickness of 2 nm. This thin SO;?layer serves as an etching mask in the KOH-etching process. The resist was then removed with acetone and ultrasonic agitation. The lines were transferred to the silicon substrate by a KOH etch. Silicon structures of linewidth 17 nm and aspect ratio of about 20 were obtained. The structures can be further thinned with an alcohol-added KOH solution. After this step of etching, sub-10-nm ridges with very smooth side walls were obtained as shown in Fig. 24 (Namatsu et al., 1995). The etch rate difference in different crystal orientations has also been used to fabricate nanometer-sized free standing structures. Sub-micron size free standing GaAs wires have been fabricated with electron-beamlithography and wet chemical etching (Hasko et al., 1988).

2. DryEtching Although it is possible in some special cases to achieve nanometer-level pattern transfer using wet chemical etching, the majority of applications in nanofabrication use dry etching for pattern transfer processes because anisotropic etching is necessary to achieve adequate dimensional control, and dry etching can be a strongly anisotropic method, while wet chemical etching is basically isotropic. Any process that removes material from the surface of a solid in a gaseous environment can be called dry etching. In microelectronics,dry etching usually means a plasma-assisted etching process, which includes sputter etching, reactive ionbeam etching, reactive ion etching, and plasma etching. Plasma-assisted etching utilizes an rf-excited plasma of a chemically reactive species to etch the exposed material revealed through the developed resist images. Three types of reactors are commonly used: barrel, planar, and reactive ion etchers. For high-resolution pattern transfer a reactive ion etching (RIE) machine is usually used, because it is highly anisotropic. In a RIE machine, a low chamber pressure and a bias between the anode and cathode are used to make the incident radicals normal to the substrate surface, reducing resist undercutting. Because of the bias between the

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FIGURE24. SEM micrographs of sub-10 nm wide Si patterns after etching in propanol-added KOH solution: (a) original Si patterns; (b) additionally etched Si patterns. (Reprinted with permission from J. Vac. Sci. Technol. B12, 1473 (1995). Namatsu, H. et al. Copyright 1995 American Vacuum Society.)

electrodes, ions are accelerated toward the substrate and some sputtering of the substrate material can occur. The directionality of this process is very desirable in reproducing in the etched material the very high-resolution images obtained in an exposed resist. Scherer and Craighead (1986) demonstrated that structuresas smallas 40 nm can be fabricated in GsAs/AlGaAs multilayer materials. In this experiment, PMMA resist was used to obtain a liftoff pattern, which was then used as an etching mask. A 9: 1 Ar:BC13 gas mixture at a pressure of 15 mTorr at 50 W was used in a standard planar RIE system to accomplishthe etching. In etching experiments,the important

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processing considerations are the rate at which the material is removed, the etch rate of the resist, and the degree of isotropic etch. Paul et al. (1993b) investigated a number of different etch systems including CF4/02, SiC14, and SiC14KF4 to etch Si-based materials. It was found that CF4/O2 mixtures have a significant chemical effect, resulting in undercutting of resist. Anisotropic etch profiles were found from Sic4 and SiC41CF4 plasmas, indicating the physical etching nature of the chlorine-based plasmas. A mixture of 20 standard cubic centimetres per minute (sccm) Sic& and 20 sccm CF4 is suitable for etching a Si,-,Ge, material. Addition of CF4 to the Sick etch reduces the “grass” effect, where any dust or organic residue from resist or cleaning procedures can form a mask to the Sic14 etch creating pillars (Zhang et al., 1992). More recently, Chen and Ahmed (1W3a) demonstrated that sub-10-nm lines can be etched in a silicon substrate directly with R E using PMMA as a mask. It was found that a gas mixture of 20 sccm S i c 4 and 10 sccm CF4 was the optimum for etching the silicon substrate. Before etching the samples, the R E chamber was cleaned with an 0 2 plasma and conditioned with the working gas. A power of 400 W was used to etch the samples for 5 s. The chamber pressure was kept at 20 mTorr during the etching. The substrate bias was 715 V for an etching power of 400W. The samples were kept at 15°C during the etching to reduce resist deformation due to heating. After etching the samples were cleaned immediately with acetone and IPA in an ultrasonicbath to remove any residual resist and reactive gas molecules. The samples were directly examined in an S-900 SEM. Sub- 10-nm lines were transferred to a silicon substrate. Figure 25 shows that the linewidth

FIGURE25. SEM micrograph showing sub- 10 nm lines formed in Si by R E etching using PMMA as a mask. (Reprinted with permission from Appf. Phys. Letts. 62,1499 (l993a), Chen, W. and Ahmed. H. Copyright 1993 American Institute of Physics.)

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FIGURE 26. A AuPd wire of 5nm in width made by ion milling. The line remains continuous. (Reprinted from Microelecrronic Eiig. 9, 187, (1989) Broers, A. N. et al. with kind permission from Elsevier Science-NL. Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

of the etched lines is between 5 and 7 nm. Using the same etching process and a liftoff mask, high-aspect-ratio silicon pillars can be obtained (Chen and Ahmed, 1993b). Other dry etching methods have also been used in nanostructure fabrication. Broers et al. (1989) showed that AuPd lines about 15 nm wide can be fabricated in a 20-nm-thick AuPd film by ion milling using a contamination resist mask. A linewidth of 5 nm was obtained when the structure was overetched Fig. 26. Structures down to 7 nm have been fabricated by chemically assisted ion-beam etching where a liftoff mask was used (Van der Gaag and Scherer, 1990). In this experiment a double-layer PMMA resist with a high molecular weight PMMA resist on top of a low molecular weight PMMA resist layer was used. After electron-beam lithography, a high-resolution AISrF2 mask was obtained with the liftoff method. The mask pattern was transferred to a GaAs/AlGaAs substrate, and structures as small as 7 nm were obtained. Another example of using chemically assisted ionbeam etching in pattern transfer was demonstrated in etching InF', where a good anisotropy and smooth surfaces were achieved (Youtsey et al., 1994). It was also shown that surfaceroughness can be significantlyreduced at elevated temperatures.

C. Summary Structuresof dimensions as small as 10 nm can be fabricated using either additive or subtractive methods. Wet chemical etching is simple and cheap to use, but its isotropic nature makes its applications very limited in the nanometer regime. Reactive ion etching is very effective and has a very high resolution if suitable reactive gas and process are used. Liftoff and electroplating can provide high resolution and are usually used to generate metallic structures.

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Iv.

RESOLUTION

LIMITOF ORGANIC

RESISTS

The minimum size of structures obtainable on substrates is set by the resolution of the patterning techniques and pattern-transferprocess. Organic polymer resists (represented by PMMA) together with electron-beam lithography exhbit a high level of flexibility and high resolution, and this is the most widely used method in nanostructure fabrication. Although SPM-based processes can provide extremely high resolution, it is not possible at present to achieve the flexibility required for device fabrication using these processes. Most of the nanostructures used in experimental studies are fabricated by electron-beam lithography using PMMA resists, and it has been demonstrated that -5-nm structures can be fabricated in PMMA resists and subsequently transferred to substrate materials by etching and by liftoff processes. Devices with sizes in the sub-10-nm regime have shown interesting new physical phenomena. Further reduction in device size may be possible and this offers exciting opportunities for research into the physics of devices. To meet this challenge, a suitable tool for fabrication has to be found and it is necessary to study the resolution limit of organic resists to find out the limits of electron-beam lithography and organic resists. The minimum size of structures achievable in organic resists is decided by several factors, including the size of the electron beam, resist characteristics, interactions between electrons, resist and substrate molecules, and the resist development process. Among organic resists, PMMA is known to have the highest resolution. In this chapter factors limiting its resolution during exposure to electrons and during development processes are discussed. The discussion also applies to some extent to other high-resolution organic resists, but the details may not be exactly the same.

A. Electron-Beam Exposure

When a finely focused beam of electrons impingeson the surface of a thin resist film it passes through the resist and enters the substrate. In the passage of the incident electrons, collisions occur between electrons and resist molecules or substrate atoms, and incident electrons lose their energy through these interactions. Some of the incident electrons are scattered back and others penetrate into the substrate or pass through it in the case of a membrane substrate, and all generate lowenergy secondary electrons. The energy transfer from the electrons to the resist molecules causes physical or chemical changes in these molecules that differentiate them from unaffected molecules. This effect can be used to form a pattern in the resist following a development process that has contrasting effects on exposed and unexposed molecules.

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1. Scattering of Electrons in Resist and Substrate In a well-designed electron-beam column, the electron optical aberrations can be minimized and an electron beam of less than 1 nm in diameter on the optical axis can be generated. By using such a beam, the smallest structures fabricated in PMMA resist, however, was limited to about 10 nm (Broers, 1981); similar results were obtained on a different substrate with a slightly larger electron beam (Beaumont et al., 1981). It was concluded that the smallest size of the structure achievable in PMMA resist in resolution studies was not limited by the size of incident electron beam. The finely focused beam is broadened substantially after entering the resist by scattering events and the incident electrons can be scattered many times before entirely losing their energy. It was suggested that the energy loss during the passage of electrons through the matter arises mainly from inelastic scattering of the incident electrons with electrons in the target material as well as with the atomic nuclei of the target material (Lee, 1982). Exposure of PMMA resist with UV light has shown that electrons with energy of about 5 eV and greater can expose resist (Broers, 1988). Thus, the exposure of the resist is predominantly due to inelastic scattering and the generation of secondary electrons. Secondary electrons can be generated by both incident and scattered electrons and they can travel a short but significant distance before losing their energy. Therefore, the range of straggling distance of these secondary electronsin the resist is an important factor in determining the limit or the resolution of electron-beam lithography (Broers, 1988). The range of secondary electrons produced in PMMA during electron-beam exposure is not easily obtained from direct experiments, but can be deduced from some experimental data. Rishton et al. (1983) measured the exposure range of low-energy primary electrons, from 5 to 2500 eV, in PMMA resist and derived the numbers and energies of secondary electronsproduced during exposure using the data from electron energy loss spectra obtained by Ritsko et al. (1978). These data were combined in a Monte Car10 program to simulate the linewidth4ose relationship. Simulated results were compared with experiment. At a dose that results in an experimental linewidth of 10 nm, the simulation shows that the exposure linewidth due to the secondary electrons is much smaller (Rishton et al., 1983). It was found that although the spatial extent of the exposure due to high-energy secondary electrons is of the order of 10 nm, the level of exposure is only 10-15% of the required threshold energy density. The exposure range of secondary electrons is less than 10 nm. Apart from the secondary electrons, other energy diffusion processes occur in PMMA resists during electron-beam exposure (Partridge, 1973). In addition, it is possible that the inelastic scattering is not confined to the electron cloud of individual atoms, but is in the form of coherent excitation of valence electrons from many atoms (Lee, 1982). It is not easy to obtain the relative exposure intensity of

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various energy-transferprocesses experimentally, and a quantitativedetermination of these various energy-transfer processes on lateral exposure, which determines linewidth, is very difficult. Nevertheless, these various energy-diffusionprocesses that occur in PMMA resist could play a significant role in limiting the minimum linewidth observed in experiments, in addition to the effect of secondary electrons. The spatial extent of the chemical reaction that results from the exposure process can also potentially affect the resolution of lithography. Some resists are exposed by a chemical amplification process. However, there is no evidence available that this is the case for high-resolution resists such as PMMA, where no chemical amplification has been found. The chemical changes in PMMA resist after electron-beam exposure is restricted to the point at which the electron breaks or makes a chemical bond and does not trigger an extended reaction such as the crystal growth observed in silver halide emulsions (Broers, 1988).

2. Statistical Error The electrons in the incident beam arrive at the resist surface randomly in time and are distributed within any specified time according to Poisson's law. To ensure that statistical fluctuations do not leave any resist element under- or overexposed, it is necessary that the average exposure in an element exceeds some minimum number of electrons. Under the shot noise assumption that individual particles arrive independently of one another, the probability of any particular number n of particles arriving in any specified element is given by a Poisson distribution (Lepselter and Lynch, 1981):

m" f h m ) = -exp( - m ) n! x

1 (2nm)

exp

(- (n -2mm)' )

(6)

where m is the mean number of particles for the specified area element. The statistical problem is to determine the probability that an exposed area receives a dose n < nci,. where n,", is the minimum dose necessary to resolve the element and the probability that an unexposed area receives a dose n > n c i l . The statistical error only becomes significant when the linewidth is very small, as in the case of lithography in PMMA resist. If a minimum resolution area or pixel A = d 2 is to be exposed, the average number of exposing electrons m incident on that area must be much greater than the statistical fluctuation in this number, -m ' I 2 .For an 80-keV electron beam, the dose needed to open an area of (10 nm)' in PMMA resist is about 10 C/m2. The number of electrons needed to expose this area is m = 6000. This is much larger than the fluctuation m ' / * , which is about 77. It may be concluded that electron-beam exposure of PMMA resist does not

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suffer from statistical errors at the 10 nm level as far as the incident electrons are concerned. Because the incident energy is deposited in the resist mainly through secondary electron generation and it is these secondary electronsthat expose the resist, another statisticallimit relevant to the exposure process is the averagenumber of secondary electrons ii' created in a minimum resolution volume of d3. Assuming that all the energy dissipated in the resist is converted into secondary electrons of energy 100 eV, the number of secondary electrons can be calculated by using energy loss and absorption data (Ritsko et al., 1978). For exposure with an 80-keV electron beam at a dose 10 C/m2, the number of secondary electrons n created in the volume (10 nm)' is about 800, which is still much larger than the noise level given by n'/*(= 28). If the area of exposure is reduced to (3 nm)2,the incoming flux of electrons is still much greater than the fluctuation. However, there are only n = 30 secondary electrons generated in the exposure. This is not significantly larger than the fluctuation in the number of electrons n1l2,which is about 5 . This level of statistical fluctuation might cause some elemental areas to be under- or overexposed and hence affect the minimum linewidth achievable in the resist. 3. Size of the PMMA Molecules PMMA is a common plastic used in a variety of applications. It was first identified as a high-resolution positive electron-beam resist in 1968 (Haller et al., 1968). PMMA is a single-component polymer (homopolymer). Its repeat unit is

There are two functional groups, one methyl group and one ester group, in each monomer. Depending on the different orientations of each monomer, PMMA resists have different tacticities. The simple regular arrangement along a chain is an isotactic structure where the same side groups are located on the same side of the main chain plane. In a syndiotactic structure the side groups alternate from side to side. In an atactic or heterotactic structure, side groups are located randomly. The density of PMMA is 1.18 g/cm3 at 25°C. At room temperature PMMA is a glassy material. The glass transition temperature of PMMA is dependent on the tacticity of the sample, between41.5"C fora 19:l isotactic:heterotacticmixture and 120°C for a 1:2:7 isotactic:heterotactic:syndiotacticmixture. For normal radical polymerized PMMA, which is predominantly syndiotactic, the glass transition temperature is about 104°C.

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In a long-chain polymer such as PMMA, there is usually a distribution in its molecular weights. Usually the average molecular weights and molecular weight distribution are used to define the material. Four molecular weight averages are in common use: the number-average molecular weight, M,;the weight-average molecular weight, M,; the z-average molecular weight, M,;and the viscosityaverage molecular weight, M,.These are defined below in terms of the numbers of molecules Ni having molecular weights M ; ; or wi, the weight of species with molecular weights Mi :

The quantity a in Eq. (10) varies from 0.5 to 0.8 and is dependent on the solvent and the temperature; e.g., for PMMA dissolved in acetone, a is 0.73 at 20°C (Brandrup and Immergut, 1975). The two most important molecular weight averages are Mn and M,. For single-peaked distributions, M, is usually near the peak. The weight average molecular weight Mw is always larger than M,. Polydispersity is usually used to describe the randomnessof the distribution.For a random molecular weight distribution, the polydispersity index M,:M,:M, = 1:2:3. The relationship between different weight averages and the molecular distribution is illustrated in Fig. 27. The number average molecular weight of PMMA resist used in electron-beam lithography varies from 50,000to 1,100,OOOg/mol. In such a material, the molecules have very long chain lengths and allow for entanglement to occur as shown in Fig. 28b. In solid form, PMMA molecules are entangled and loosely coiled. The size of the polymer molecules, and thus the initial molecular weight, might be expected to be important in determining the minimum linewidth or minimum edge roughness achievable. If the molecules are assumed to have a spherical structure, the diameter of the molecule ball can be calculated as

where r is the density of PMMA that is 1.18 g/cm3, w is the molecular weight of the resist and A is the mass of an atom of the isotope carbon-12 and is equal to

A

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143

A

Molecular weight of fraction i FIGURE 27. Schematic of a simple molecular weight distribution, showing different molecular weight averages, M,,M,,Mw and M,.

(b)

(a)

FIGURE28. Entanglement of polymer chains: (a) low molecular weight, no entanglement; (b) bigh molecular weight, chains are. entangled. The transition between the two for PMMA is about 16000 atomic mass unit.

1.66 x lop2' kg. For a molecule of 500,00O atomic mass units (mu), D = 11 nm.

When in solution, a polymer molecule also assumes the shape of a random coil in the absence of monomer-monomer interactions in the solvent. The gyration radius R, of polymer molecules in a solvent can be related to its monomer size and molecular weight:

where I is the length of the monomer and n is the monomer number: n = -M"

MO

M , is the number averagemolecular weight of the polymer, and A40 is the monomer weight and is 100 amu for PMMA. The bond radius of the C-C bond in PMMA molecules is about 0.77 8, (Choi et al., 1988). A simple calculation gives a monomer length of 2.74 A. Thus, for a PMMA molecule of 500,000 amu, its

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W. CHEN AND H. AHMED

gyration radius is

R, x 7.8 nm that is, the diameter of the molecule is 15.6 nm. From the above calculation, it can be seen that both molecular sizes of PMMA resist as a solid and in solvent are already larger than the smallest structures that have been fabricated in PMMA resist in experiments. In addition, experiments on resists of different molecular weights show that the smallest linewidths in different resists are about the same, while the molecular size for a resist with a molecular weight of 1150 kg/mol is about 1.5 times that of 500 kg/mol molecular weight resist. Thus, the initial size of the PMMA molecule does not limit the resolution. Recent results by Khoury and Ferry (1996) also confirmed this conclusion. Comparable linewidths were written in three different molecular weight PMMA resists, with 15,000, 120,000, and 950,000 amu molecular weight, respectively. The smallest lines, about 7 nm wide, were written in the highest molecular weight resist. In electron-beam lithography, only the exposed molecules are dissolved into a solvent during the development process. The sizes of these fragmented molecules will determine the smallest structures that can be made in the resist. The molecules of an unexposed PMMA resist are usually entangled and have a very low dissolution rate. The entanglement threshold molecular weight for PMMA resist is about 16,000 amu (Sharma et al., 1984). For molecules below the entanglement threshold, the dissolution rate is much higher. The exposed resist is less entangled and easier to dissolve into a solvent. A molecular weight reduction of PMMA from 186,000 to 3300 g/mol for a typical electron-beam exposure was measured (Rishton, 1984). Molecules with 3300 amu will have a diameter of about 2 nm. It is expected that the resolution of PMMA resists imposed by the resist molecules will be at least twice that of the molecule size, that is, about 4 nm. The edge definition and surface smoothness after development will also be determined by the size of these fragmented molecules. 4. Degradation of PMMA Under Electron-BeamIrradiation When a film of PMMA is exposed to a sufficiently intense dose of radiation, it is fragmented through main chain scissions, chemical changes, and volatile fragment formation by cleavage of pendent methyl ester groups. These changes profoundly affect the dissolution rate of PMMA that enables an image to be developed in the resist film. The degraded part has a higher dissolution rate than that of unexposed areas in a chosen solvent. A positive image is generated after the more soluble part of the resist is dissolved into the solvent. At a much higher dose (about an order of magnitude higher), cross-linking between polymer chains occurs, and the insoluble residue can be used as a negative resist (Broers et al., 1978), but

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the resolution of PMMA working in a negative mode is not as high as that of the positive modes. When an electron beam interacts with the molecules in a PMMA resist, some chains of resist molecules are broken and at the same time some chains are crosslinked. The events that happen can be described by the radiation chemical yield G ( S ) , which is the number of chain scissions per unit of absorbed energy, and G ( X ) , the number of cross-linking events per unit of absorbed energy. It is not certain if the property of degradation of the PMMA resists is due to high G ( S ) values or to low G ( X ) values or both. The resulting molecular weight can be related to the main chain scissions and cross-linking by using the equation

where M n is the initial number average molecular weight, M fis the number average molecular weight after the irradiation, NA is Avogadro's number, G ( S ) and G ( X )are the number of chain scissions and cross-linking events, respectively, and D is the absorbed dose in electron volts per gram. Because the high-energy electrons interact weakly with the resist molecules, the absorbed energy is only a fraction of the incident energy. The absorbed energy density is, in general, a complicated function of position within the resist film and is a function of the exposure parameters such as beam energy, resist thickness, type of substrate, and geometry of the irradiated pattern. For high-energy beams and thin resist films, the absorbed energy density is approximately uniform (Greeneich, 1975). By using gel permeation chromatography with polystyrene reference standards, the molecular weight of irradiated PMMA can be obtained and it was shown that M, is inversely proportional to the incident dose d (Choi et al., 1988; Parsonage and Peppas, 1987). If it is assumed that cross-linking is absent, G ( X ) = 0, the number of main chain scissions, is proportional to the incident dose. The number of main chain scissions N, was determined by

and dose D is no proof that crossHowever, a linear relation between (Mf)-' linking is completely absent. One cannot be certain that the cross-linkingis totally absent in the irradiation of PMMA, because it is known that PMMA becomes a negative resist if the dose of radiation is sufficiently high (Broers et al., 1978). By making weight average molecular weight (Mw) measurements using light scattering methods after irradiating PMMA with an electron beam, Shultz et al. (1956) found that (Mw)-'was linear with dose. They also measured the viscosity average by means of intrinsic viscosity measurements. The linear molecular weight (M,) relationship between reciprocal M, and dose was observed. The fact that both of their M, and M, values fell on the same curve demonstrated that there could

146

W. CHEN AND H. AHMED

not have been a significant amount of cross-linking for the level of dose that they used in their experiment. It seems that at lower dose, cross-linking is negligible; but as the dose increase, G ( X ) increases faster than G ( S ) does and cross-linking becomes dominant at very high dose. In addition to a decrease in molecular weight, electron-beam irradiation also changes the molecular weight distribution. For a truly random scission, the resist polydispersity ( M w / M n )approaches a value of two at high exposure dose. The effect of molecular weight, molecular weight distribution, and tacticity on chemical yield (G value) of PMMA resists has been studied quantitatively (Gipstein et al., 1977). It was found that the G value is independent of the molecular weight (for Mn > 55,000), molecular weight distribution, and tacticity of the resist. Equation (14) can be rewritten as

and

K=-

G D 100N A

The energy dissipated in the resist can be calculated by using a Monte Carlo simulation. The threshold energy for development is 6.8 x lo2' eV/cm' to 2.4 x eV/cm3(Hawryluk, 1981). The chemical yield for PMMA is about 1.5; hence, K is between 1.56 x lop4and 5.48 x lop4. It can be seen from Eq. (17) for an initial molecular weight M,,> 50,000. that M fis almost independentof M,, In Fig. 29 the final molecular weight is drawn as a function of exposure dose at different initial molecular weights. The molecular weight of PMMA resists used in electron-beam lithography is usually greater than 100,000. Thus, the size of fragmented molecules is independent of initial molecular weight. This is why high molecular weight resists are used in high-resolution nanofabrication to achieve high contrast, hence higher process latitude. B. Development of Exposed PMMA Resist

For a PMMA resist, irradiation by electron beam at a moderate dose degrades the resist, and the exposed molecules are fragmented, resulting in a higher dissolution rate in a developer. The difference in dissolution rate between the exposed and unexposed areas permits images to be developed in the resist. The term development refers to the dissolution and removal of the more soluble regions in the resist film. In the development process, the intermolecular forces between resist and solvent molecules determine whether a resist molecule can be dissolved into the solvent. These forces can limit the smallest structures achievable in a PMMA resist. Since for polymeric materials the dissolution is always accompanied by swelling, resist

147

NANOFABRICATION FOR ELECTRONICS 7000

L

' '"""1

' '""''1

' ' """1

' ' "'""

'

'

"TI

6000

-z

+K = l . 5 ~ 1 0 - ~

0 3

g

4000

--O-

K=2x104 K=3x104

-A-

K=4x104

---t K=5x104

2000 1000

103

104

105

108

107

108

Initial molecular weight (amu)

FIGURE 29. Fragmented molecular weight as a function of initial molecular weight for PMMA resists irradiated by electron beam at different doses K in eV/g. The fragmented molecular weights are almost independent of the initial molecular weight if the initial molecular weight is higher than 5 x 105amu.

swelling during development can be important in determiningthe resolution of the resist. 1. Dissolution of PMMA Resist

Due to the complexity of the physics of polymer dissolution processes, the fundamental causes of the difference in dissolution rates between irradiated and original resist molecules remain unknown. At least three different phenomena occur during irradiation of a PMMA resist that can be used to explain the difference in solubility: main chain scission, chemical change, and gas evolution. There is no definitive evidence indicating which of these is responsible for the increased solubility of the irradiated resist. Usually, the main chain scission is used to explain the dissolution process because it is the most obvious phenomenon after irradiation and it is also the most convenient one to be correlated to the dissolution rate. The molecular weight of exposed resist is inversely proportional to the dose it receives. If the irradiation dose is high enough, the molecular weight in the irradiated part of the resist will be reduced to below the entanglement threshold of the resist molecules. Hence, the exposed molecules are less entangled and have a higher dissolution rate than the unexposed ones. In some solvents the exposed molecules can be selectively dissolved while keeping unexposed molecules intact.

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W. CHEN AND H. AHMED

FIGURE 30. The kinetics of polymer dissolution: (a) initial glassy polymer; (b) solvent diffuses into polymer and eorms swollen gel layer. The gel-polymer boundary B2 moves towards polymer. The solvent-gel boundary BI can move towards the solvent (swelling) in the early stages of dissolution; ( c )the solvent continues to penetrate further into the glassy region. In the gel region, polymer molecules disentangle and diffuse into the solvent and the solvent-gel boundary retreats from the solvent. (Adapted from Process Engineering Analysis in Srmiconductor Device Fabrication, McGraw Hill, Inc. ( 1993), Middleman, S. et al.)

Dissolution begins as solvent molecules diffuse into the glass-like polymer phase. The solventmolecules interact thermodynamicallywith the polymer chains. Individual chains change from their unperturbed state in the glass-like phase to a new solvated state. The solvated polymer swells, and the swollen region adjacent to the solution is called a gel and can be seen as a highly concentrated solution. Within this region polymer molecules become less entangled than they are in the glassy region and more mobile, permitting them to diffuse towards the solvent-gel boundary. As shown in Fig. 30 (Middleman and Hochberg, 1993). two distinct boundaries or interfaces, characterized by a sharp change in the concentration of the solvent, may exist. The first boundary, between the solvent and the gel, is particularly significant because it divides the dissolved and undissolved polymer. The second boundary lies between the gel-like and glass-like phases. In the early stages of dissolution the gel-polymer boundary moves toward the glass phase of the polymer, while the solvent-gel boundary moves toward the solvent. The dissolution of the polymer occurs once the swelling has reached a point where the polymer chains can disentangle and dissolve. If the polymer-polymer intermolecular forces can be overcome by the polymer-solvent interactions, the second step of the process, the true dissolution, can take place. During the true dissolution step, the gel-polymer boundary continues to move toward the polymer and the solvent-gel boundary also starts moving toward the polymer. As a result, polymer molecules begin to be dissolved into the solvent. The level of disentanglement effectively controls the duration of the swelling process and this level is a function of the molecular weight of the polymer. Higher

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molecular weight requires higher levels of disentanglementand thus a larger degree of swelling before dissolution can occur. For some solvents, such as methyl ethyl ketone (MEK), the dissolution rate of PMMA is believed to be controlled by the kinetics of disentanglement rather than by the convection and diffusion of the polymer molecules away from the surface (Middleman and Hochberg, 1993). Evidence for this is provided by an examination of the temperature dependence of the dissolution rate. The activation energy of dissolution can be calculated and it is more than twice the activation energy associated with diffusion. Because the dissolution is not limited by the convection and diffusion of polymer molecules away from the developer-resist boundary, development cannot be promoted by agitation of the development bath (Middleman and Hochberg, 1993). For other solvents, however, the dissolution of PMMA is not limited by the disentanglement only, and the dissolution process is much more complicated. 2. Solubility of PMMA

PMMA resists are completely soluble in good solvents. In a good solvent there are repulsive forces between the segments of PMMA molecules, so the molecule coil swells and becomes more expanded. In a poor solvent the molecule segments attract each other, and coils shrink. Often a poor solvent can be made into a good one by adding certain solutes or by raising the temperature above some critical value known as the theta temperature. The dissolution rate of a polymer in a given solvent is determined by many factors, including the molecular weight of the polymer, the stereochemistry of the material, the solvent properties, the temperature of the solvent, and the geometric characteristics of the sample (Gipstein et al., 1977; Parsonage et al., 1987). Doolittle (1965) demonstrated that the solubility of certain polymers in solvents of a homologous series depends on the molecular size, i.e., molecular weight, of the solvent. The solubilityof the polymer decreased as the molecular size of the solvent increased. This phenomenon was confirmed for the dissolution of PMMA in n-alkyl acetate solvents (Gipstein et al., 1977). The dissolution rate decreased with increasing size of the alkyl group in a series of n-alkyl acetate solvents. Developers used in electron-beamlithography usually consist of more than one component, typically a mixture of two solvents. With mixed solvents, dissolution becomes more complicated and the dissolution characteristics are not simply additive. An addition of some thermodynamically poor solvents into a good solvent will increase the solubility (Cooper et al., 1986). An addition of some alcohols, which are nonsolvents for PMMA, will increase the solubility rate of Ph4MA in MEK. The increase is not related to the thermodynamic properties of the alcohols. From the thermodynamic point of view, a small amount of methanol should be as effective as a larger amount of propanol since the difference in solubility parameters between alcohol and MEK is greater for methanol. The observed effect

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on dissolution rate was quite the opposite (Cooper et al., 1986). The enhancement of the dissolution rate can be explained as a plasticization effect. Because some thermodynamically poor solvents may penetrate a polymer matrix at a much faster rate than thermodynamically good solvents, the effective diffusion coefficient of the good solvent is increased, leading to an increased dissolution rate. The size of the nonsolvent molecules is important and the kinetic factors, such as diffusivity and viscosity, are rate-controlling in this regime. As the size of nonsolvent molecules or the concentration of nonsolvents increases, the thermodynamics becomes important and the disentanglement of polymer molecules determines the dissolution rate. The solubility rates R for various combinations of methyl isobutyl ketone (MIEiK) and isopropyl alcohol (PA) at several temperatures have been measured (Greeneich, 1975). The experimental data can be fitted to an empirical formula (Greeneich, 1975): R=Ro+-

B MY

where Ro, B. and y are fitting parameters. R is the dissolution rate in angstroms per minute, and M fis the fragmented molecular weight. Table 2 shows fitted parameters for various combinations of MIBK and IPA (Greeneich, 1975; Hatzakis, 1974; Hawryluk et al., 1974). TABLE 2 SOLUBILITY PARAMETERS FOR PMMA RESIST IN VARIOUS COMBINATIONS OF MIBK AND IPA AT DIFFERENT EMPERATURES. ~~~

Solvent

1:3 MIBKIPA 1:3 MIBK:IPA I:I MIBK:IPA 1:l M1BK:IPA MIBK MIBK MIBK MIBK 2:3 MIBK:IPA MIBK MIBK

Temperature

223°C 32PC 223°C 223°C 22.8"C 34.1"C 356°C 20.5"C RT RT RT

Ro 0.0 0.0 0.0 0.0 84 241.9 464.0 0

0 0

b

9.332 x 1.046 x 6.700 6.645 x 3.140 x 5.669 x 1.435 x 1.25 9.37 x 2.31 x 1.82 x

Nore: RT, room temperature; NA, not available. a

Greeneich (1975). Hatzakis et al. (1974). Hawryluk et al. (1974).

g

1014

3.86 10l6 3.86 2.00 109 lo6 1.188 10' 1.5 lo8 I .5 lo9 1.5 109 1.40 10'2 2.75 10' 1.45 10" 2.5

Mol wt range (2 x (2 x <5 x 52 x <2x <2 x (2 x

10' lo" 103 105

Id 105 105

NA <2 x Id 6 x lo3 - 3.2 x I@ 1 6 x lo3

Ref. a a a a a a a b C C

C

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From Table 2 and Eq. (15) dissolution rates can be calculatedfor resists of different molecular weights. In a standard developer used in electron-beam lithography, namely 1:3 MIBK:IPA, the dissolution rate for a PMMA resist with a numberA/min at 22.8"C; average molecular weight of 200 kg/mol is about 3.22 x because the development time used in electron-beam lithography is usually less than 1 minute, the resist thickness change is negligible. For a fragmented PMMA resist of a number average molecular weight of 2 x lo3, however, the dissolution rate is increased to 169 A /min. If R , is the solubility rate of the unexposed resist in the developer, the ratio of solubilities of the exposed to unexposed regions can be obtained from Eqs. (16) and (18). For an MIBK:IPA mixture, R,-, = 0, so R

- = (1

Ru

+ KM")?

It is desirable to keep the ratio R / R u as large as possible to obtain high-contrast development. It can be seen from Eq. (19) that high-contrast development can be obtained by using high molecular weight resists or large y values. That is why high molecular weight resists are usually used in electron-beam lithography. As shown in Table 2 the PMMA resist has a large y , hence a high contrast, in the standard 1:3 MIBK:IPA developer. Other developers, such as 3:7 celloso1ve:methanolmixture were also used for the developmentof PMMA resist (Howard et al., 1983). The solubility of exposed PMMA resist in 3:7 celloso1ve:methanol mixture was measured and the contrast was shown to be higher than that of the standard 1:3 MIBK:IPA developer (Chen, 1994). 3. Intermolecular Forces

While the entanglement of molecules, which is dependent on the molecular weight of a polymer, determines the disentanglement process in the dissolution of the polymer, the true dissolution is determined by the intermolecular forces between resist and solvent molecules. If the forces between the solvent and polymer exceed that between polymer and polymer, the polymer will be dissolved into the solvent. Many molecules cany no net charge, but many possess an electric dipole. If in a molecule the electron is drawn toward one of the atoms, the molecule will have a permanent dipole. Such a molecule is called a polar molecule. PMMA resist molecules are polar molecules. Three distinct types of forces contribute to the total long-range interaction between polar molecules, collectively known as the van der Waals force: the orientation force, the induction force, and the dispersion force. Each has an interaction-free energy that varies with the inverse of the sixth power of the distance. And the total van der Waals force for two dissimilar polar

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molecules in vacuum is (Israelachvili, 1992)

V ( r )= -

Cind

+ Corienc + Cdisp - --C rh

r6

where Cind, Corient, and Cdisp are the coefficients for induction force, orientation force, and dispersion force, respectively; C is the total van der Waals coefficient; and r is the distance between the two molecules. In the development process, resist molecules interact with each other through a solvent medium. This interaction can be very different from that of isolated molecules in vacuum or in a gas. First, the intrinsic dipole moment and polarizability of an isolated molecule may be different in the liquid state or when dissolved in a medium. Second, a dissolved molecule can move only by displacing an equal volume of solvent from its path; hence, the polarizability a in a medium must represent the excess polarizability of a molecule over that of the solvent and must vanish when a dissolved particle has the same properties as the solvent. For two dissimilar polar molecules, McLachlan (1963a, b) presented a generalized theory of van der Waals forces, which included in one equation the induction, orientation, and dispersion forces, and which can also be applied to interactions in a solvent medium. The van der Waals potential of two molecules 1 and 2 in a medium 3 is given by the series (McLachlan, 1963a, b)

V ( r )= -

6kT (47r e0)2r6

2

n =o. 1.2. ...

a1(iun)w(iun)

e:(iu,)

(21)

where k is the Boltzmann constant, T is the absolute temperature, oll(iu,) and a2(iun) are the polarizabilities of molecules 1 and 2, and &3(iu,,)is the dielectric permittivity of medium 3 at imaginary frequencies iu, ,where at 300 K

(22)

and where the prime over summation (C) indicates that the zero-frequency n = 0 term is multiplied by and h is Planck’s constant. For a molecule with one absorption frequency (the ionization frequency) nl, its electronic polarizability at real frequencies n , is given by the simple harmonic oscillator model (von Hipple, 1958):

i,

where a0 is the static polarizability. If pairwise additivity is assumed for the van der Waals forces, then the force exerted on an exposed molecule by unexposed resist walls and the substrate can be calculated. However, the van der Waals forces are generally not pairwise additive because the force between any two molecules is affected by the presence of other molecules nearby. Usually the contributionof many-body effects to the interaction of small gas atoms in vacuum is negligible. For large bodes interacting in a

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medium, however, the contribution of many-body effects is much larger and can be as high as 30% (Margenau and Kestner, 1969). The problem of additivity is avoided in the Lifshitz theory of intermolecular forces, where the atomic structure is ignored and the forces between bodies, now treated as continuous media, are derived in terms of such bulk properties as their dielectric constants and refractive indices. By using the Lifshitz theory, all the equations derived above for interaction energies remain valid and the only thing that changes is the way the interaction constant is calculated. For interactions between two macroscopic bodies, the Hamaker constant H = x2Cplp2

(24)

is normally used in the calculations. From the Lifshitz theory the Hamaker constant for two macroscopic phases 1 and 2 interacting across a medium 3 is given by (Israelachvili, 1992)

where k is the Boltzmann constant, T is the absolute temperature, and E I , & 2 , and are the dielectric constants of phase 1, phase 2, and medium 3, respectively. For two identical phases 1 interacting across medium 3 the above equation reduces to 83

Qpical values of Hamaker constants for the condensed phases, whether solid or liquid, are about lo-'' J for interactions in vacuum. The Hamaker constant for PMMA interacting in 3:7 celloso1ve:methanol at 300 K can be calculated and is about 6.8 x J. The interaction-free energy of a PMMA molecule of 1 nm diameter and a semi-infinite PMMA resist wall in 3:7 celloso1ve:methanolcan be J, and the force exerted on the molecule is 3 x 10-l2 N. calculated as 2 x The total intermolecular force between two molecules consists of two parts: the long-range attractive part and the short-range repulsive part. The total pair potential is drawn schematicallyin Fig. 31. Other important parameters describing the intermolecular forces are the separations at which the potential energy is zero; 6 ,the separation at which energy attains its minimum value r,; and the minimum energy itself, -8, where E is known as the well depth. When the separation between two molecules becomes very small and their electron clouds begin to overlap, there is a sharp repulsive force between the two molecules. For neutral molecules there is an attractive force, usually the van der Wads force at large distance. There are many empirical equations describingthe pair potentials. The best-known LennardJones, or 6-12, potential is often used to calculate the intermolecular forces. The

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W. CHEN AND H. AHMED

Potential, U(r)

Intermolecular

I I

FIGURE3 I . Intermolecular pair potential and force. There is an attractive force, van der Waals force at long distances and a repulsive force at short distances. When the intermolecular distance is r,,, the intermolecular force is zero and the potential energy attains its minimum value, --E.

potential is given by

or, equivalently: .(r)=4p[(;)12-

(31

where a(= 2-I'%,,,) is the intermolecular separation at which the energy is zero. The Lennard-Jones potential has no adjustable parameters other than CT and E , whose values can be determined by forcing agreement between experimental data for a physical property and a calculated value for the potential model. It is possible to ignore the repulsive part of the intermolecular force when we examine the dissolution process because of the extremely short range of this force. During developmentmany forces will be exerted on an exposed molecule, as shown in Fig. 32. It has to be energetically favorable for the exposed molecule to move from the resist to the developer. The attractive forces from substrate, unexposed resist, and other exposed resist will prevent the exposed molecule from dissolving into the developer. Experimentally, the attractive forces between polymer layers become significant within a distance of about twice the radius of gyration of

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FIGURE 32. Intermolecular forces exerted on an exposed molecule during development. Force FI is that between the exposed molecule and the substrate; F2 is that between the exposed molecule and unexposed molecules; F3 is that between the exposed molecule and other exposed molecules; F4 is that between the exposed molecule and solvent molecules.

the chains, which can extend to several tens nanometers for polymers (Pate1 and Tirrell, 1989). The gyration radius of high molecular weight PMMA is of the order of 10 nm,hence the interactions between molecular chains can extend to more than 10 nm. It was argued that during development exposed molecules in contact with or near the unexposed molecules are affected by the presence of the unexposed resist wall because of the intermolecular forces (Chen, 1994). If the linewidth is much larger than the size of the resist molecule (see Fig. 33a), the forces exerted by the unexposed resist molecules on an exposed molecule situated at the middle of the gap are very weak and can be ignored. The solubility is mainly determined by the interaction between the exposed resist molecules and the solvent molecules. Even if the exposed molecules that are in contact with the unexposed molecule walls are prevented from dissolution by the unexposed wall, the overall linewidth is not significantly affected. When the linewidth becomes comparable with the molecular size of the resist there are only a few exposed resist molecules between the unexposed resist walls (Fig. 33b). The forces from the unexposed molecules become significant and must be taken into account when examining the dissolution process. When the exposed molecules in contact with unexposed molecules are prevented from dissolution by intermolecular forces, the linewidth is reduced significantly. When the linewidth becomes comparable with the size of exposed molecules, intermolecular forces can completely prevent the opening of exposed lines during development. In practice, when the linewidth is only two or three fragmented molecules wide, its opening becomes impossible. As shown in

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W. CHEN AND H.AHMED

FIGURE33. Schematic of resist showing exposed and unexposed molecules: (a) linewidth is much wider than the fragmented molecular size; (b) linewidth is comparable with the size of the fragmented molecules.

previous section, the size of exposed PMMA molecules is about 2 nm; hence for lines less than 10 nm wide the intermolecular forces can prevent the opening of the line in the development process. To develop the line, the dose used to open a line has to be increased to such an extent that the molecules in the vicinity of the line also get enough dose to be developed. As a result, the lines are always wider than 10 nm. Lines narrower than 10 nm down to 5 nm width were fabricated in PMMA resists with the help of ultrasonic agitation (Chen and Ahmed, 1993a). It was claimed that ultrasonic agitation during development provided extra energy to overcome intermolecularforces. But results by Ryan et al. did not show significant improvement by using ultrasonic agitation (Ryan et al., 1995). Although 8-nm lines were obtained in PMMA resist it was suspected that they were not fully developed. However, Vieu et al. (1997) demonstratedthat -5-nm-wide lines were fully developed and that structures can be transferred to substrates by liftoff using the PMMA as a mask. They used ultrasonic agitation in their experiment to obtain lines in the resist that were fully developed. However, the exact role of ultrasonic agitation is not clear as yet and more experiments are necessary to clarify the conditions under which fully developed 5-nm lines and structures may be formed.

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4. Resist Swelling Resists swell due to the formation of a gel layer in the dissolution process. In marginally poor solvents, e.g., developers used in electron-beam lithography, solvation forces favoring mixing are insufficient to promote dissolution of high molecular weight polymers. However, appreciable swelling can occur, which could affect the resolution of the developed structures. Amorphous polymers of high molecular weight may typically experience between 0.1 and 30% swelling, without dissolution, depending on solvent quality and temperature (Park, 1968; Ueberreiter, 1968). Although PMMA suffers comparatively little swelling, which accounts partially for its ultrahigh resolution properties, some absorption of developer is necessary to aid the development process and swelling still exists to some extent (Weill, 1986). In adevelopmentprocess, the exposed resist molecules are dissolvedinto the solvent, and a gap is left behind to form a line in the resist. As the exposed molecules are dissolved, the developer molecules diffuse into the unexposed resist, causing the resist to swell and to make the gap narrower. In PMMA resist, the effect of resist swelling can be ignored for large exposed patterns. However, the linewidth of small patterns can be significantlyaffected by resist swelling. During development the gap between the unexposed molecule walls in small structures can be comparable to the size of unexposed molecules and the chains of these molecules are attracted toward the opposite wall and are interpenetrated. The interpenetration of the resist chains caused by the intermolecular forces can, in fact, impede complete separation of the walls after solvent evaporation. In practice, nominally unexposed areas adjacent to the exposed molecules can also receive some irradiation dose and have an increased dissolution rate because of the Gaussian profile of the electron beam. During development these molecules have an increased mobility and the swelling speed in this area is increased. It was shown that at the beginning of dissolution the thickness of the gel region is proportional to the time that the sample is in the solvent (Middleman and Hochberg, 1993). The swelling problem can, therefore, be reduced by using a shorter development time. Lines smaller in width than those obtainable with normal development can be fabricated with shorter development time, as shown by Chen and Ahmed (1993a), who used ultrasonic agitation to aid development while keeping the total development time to as short a period as possible, a few seconds. C. Summary

The ultimate resolution of electron-beam lithography in organic resist is not set by the size of the incident beam but by the size of resist molecules and the range of bond-breaking low-energy secondary electrons. Intermolecularforces strongly influence the development of very narrow lines. However, by careful control of

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development processes, it is possible to fabricate structures that are significantly less than 10 nm in size. It seems that 5-nm structures can be made, but it is not clear how much smaller we can go with PMMA. If our present understanding, as explained in this section is correct, the size of exposed resist molecules and the range of secondary electrons will limit the minimum achievable size in PMMA to a little less than 5 nm and new resists will have to be explored if we are to achieve sizes close to 1 nm.

v. APPLICATIONS OF NANOSTRUCTURES The continued reduction in size of electronic devices has led to a regime where quantum mechanics begins to play a major role in electron transport, and at nanometer sizes, quantum effects become dominant. It is now possible to observe and even to control quantum effects in a variety of materials and structures. Advances in nanostructure fabrication and material growth have permitted not only the investigation of the influence of quantum effects on device behavior, but also the construction of devices that depend on quantum effects to operate satisfactorily. High-quality material growth techniques, with which ultrathin (2-nm) layers of different materials with atomically sharp interfaces can be made, have led to the development of useful devices in which quantum effects play a significant part. In addition, many experiments have been carried out to study the novel physical properties associated with carrier confinement in a very thin layer, the so-called two-dimensional electron gas (2DEG). The carriers in this case have two degrees of freedom, but progress in nanostructure fabrication has made it possible to place additional lateral confinement on the 2DEG to study transport along a quantum wire in which there is 1 degree of freedom or even in quantum dots or OD structures (Fig. 34). Experiments have revealed novel quantum effects, opened nanowire quantum dot

-wwW

E

(a)

(b)

(C)

(d)

FIGURE34. Schematic of structures with the sorresponding density of states versus energy for (a) a hulk semiconductor; (b) ID confinement; (c) 2D confinement; and (d) 3D confinement, where lengths, t, w, and 1 are nanoscale sizes.

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up new ways of studying the fundamental questions of physics, and raised the possibility of radically new electronic devices that exploit the wave nature of the electrons. Structures with extremely small dimensions start to show some unique effects. The Coulomb blockade effect in small tunneling junctions is one example where nanostructuring plays a very significant part. Advances in fabrication techniques have also given us the opportunity to study magnetism in reduced dimensions. Although single-domain magnetic particles have been studied for a number of years due to their importance in recording applications, most experiments were carried out in an ensemble of many particles, and properties of a single particle cannot be obtained directly. It is now possible to control precisely the geometrical factors of magnetic materials using thin film technology and nanofabrication techniques. Studies can be carried out on structures with dimensions comparable to some fundamentallength scales in magnetics, such as domain size. In the following sections a few examples of the applications of nanostructures are discussed.

A. Quantum Interference At low temperatures, an electron propagating through a high-purity normal metal or a high-mobility semiconductormay travel up to afew microns before undergoing a collision that changes its energy randomly. Thus, it may behave as a coherent quantum-mechanical wave in that region, allowing the possibility of interference between parts of the wave that have been split up and subsequently recombined. In large devices, such interference effects are generally washed out by phasedestroying processes. Advances in nanofabrication have made it possible to build devices with lateral dimensions much smaller than the inelastic electron mean free path, which is the length scale for phase-destroying scattering,at low temperatures. The inelastic electron mean free path is inversely proportional to temperature and can be as long as a few microns in high-electron-mobility transistors at liquid He temperature. Because structures less than 100 nm can be fabricated routinely with nanofabrication techniques, studies on quantum interference effects can be carried out at low temperatures. The reason for the interest in studying quantum interference is that it might be possible to make high-speed, high-gain electronic devices. The high speed comes about primarily because of the small dimensions, so the transit time of electrons across the active region can be very small. The device can also have high gain if a small change in the electron wavelength will switch the device from a maximum in output to zero transmission. If the change in the wavelength can be induced by the gate voltage applied to a three-terminal device, the transconductance can, in principle, be large. An obvious structure for observing quantum interference effects is a ring, because it provides two alternative paths from the input to the output. The relative phase between the two paths can be controlled with a magnetic field perpendicular

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W. CHEN AND H. AHMED

to the plane of the ring. By changing the interference conditions, the conductance can be made to oscillate. This is the Aharonov and Bohm effect (1959). The effect was observed in metal rings where the conductance modulation is -0.1% (Washburn and Webb, 1986). To make metal structures, usually liftoff or etching methods are used. To define a ring structure physically in a semiconductor, it is only possible to use etching techniques. However, etching small structures will bring problems of surface roughness and surface states that reduce the quality of the devices, although shallow surface etching might work. Ford and Ahmed (1987) used a split-gate structure with a layer of PMMA as a spacer layer to define a ring structure in a semiconductor substrate. This method avoids using etching and has less electron-beam-induced damage to the substrate. Figure 35 shows (a) an SEM micrograph of an exposed resist ring prior to metallization; (b) a cross-sectional diagram of the ring. A much larger interference effect than that obtained in metal structures was observed in semiconductor materials where the conductance modulation was about 20% (Ford, 1988).

0.9pm

4

FIGURE 35. (a) SEM micrograph of an exposed resist loop prior to metallization (central dot is 0.1 fim in diameter); (b) cross-sectional diagram of the loop. (Reprinted from Microelectrorric Eng. 6. 169, (1987) Ford, C. J. B. and Ahmed. H. with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

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There is also the possibility of an electrostatic version of the Aharonov-Bohm effect. This effect is expected to exist in a ring structure in which the phase difference is introduced by changing the electrostatic potential of one arm with respect to the other. The conductance is expected to change periodically as the potential is changed, but such a periodic conductance modulation has not yet been observed. Because the phase shift is proportional to the transit time (Datta and Mclennan, 1990), the electrostatic effect is more difficult to observe than the magnetic version of the Aharonov-Bohm effect, where the phase shift depends only on the flux enclosed by the two alternative paths. De Vegvar et al. (1989) have fabricated a structure consisting of a gated ring with the intention of demonstrating the electrostatic Aharonov-Bohm effect. Although there was some evidence of a phase change due to the gate, it was not conclusive. More recently, Park et al. (1996) investigated a gated ring structure and discovered new h/e oscillations, in addition to the magnetic Aharonov-Bohm oscillations, which are attributed to the electrostatic Aharonov-Bohm effect. They also observed that the magnetostatic Aharonov-Bohm oscillations are shifted by the electrostatic potential.

B. Superlattices Advances in thin-film preparation techniques have made it possible to modify the electronic properties of semiconductors by imposing an artificial superlattice on the crystal that offers intriguing possibilities for physical investigations (Esaki and Tsu, 1970). One-dimensional superlattices in multilayered heterostructure consisting of a series of barriers and wells along the direction perpendicular to the layer plane are well known. The band structure in the conduction band of three-dimensional electrons in macroscopic semiconductor samples is modified in the vertical dimension, i.e., normal to the surface of the multilayer structure. Minibands are formed separated by minigaps analogous to the formation of energy bands in solids as a consequence of their periodic crystal lattices (Esaki and Tsu, 1970; Buttiker and Thomas, 1977). The conduction electrons are accelerated by an electric field until they reach the maximum energy in the minibands at the end of the minizones. If the scattering rate is sufficiently low, the electrons cycle through the minizone in momentum space and thereby oscillate in real space, an effect known as Bloch oscillations. If this occurs, then the current decreases with increasing electric field, leading to negative differential resistance (Kroemer, 1958). The negative differential resistance was experimentally verified in a grown GaAs/AlAs superlattice (Esaki and Chang, 1974). However, despite the precise control of layer thickness made possible by molecular beam epitaxy, multilayer structures have some major shortcomings. First, electrons can only see a superlattice in one direction; they behave as free electrons in the other two directions. This prevents the formation of true minigaps. The minibands in layered superlattices are not separated by real minigaps, but

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are connected by the two-dimensional transverse continuum of states (Kroemer, 1977). Second, the strength of the periodic modulation, or equivalently the size of the minigaps, which depends on the material composition, cannot be changed after the growth. Third, the mobility in such structures is relatively low, as compared to that in modulation-doped layers. These shortcomings can be partially or completely overcome in a lateral surface superlattice (Bate, 1977; Sakaki et al., 1976). A one-dimensional lateral surface superlattice was implemented in a Si/SiO;! field-effect transistor in which a modulation of the transconductance was observed (Warren et al., 1985). A negative differential resistance was reported for structures having two-dimensional grids patterned into the gate (Bernstein and Ferry, 1987). The application of magnetic fields to similar devices has revealed the existence of 1/B oscillations in the magnetoresistance (Gerhardts et al., 1989; Winkler and Kotthaus, 1989), while quenching of the Hall effect has also been reported (Smith et al., 1990). Evidence for the more complex predictions of Landau level splitting whenever the number of flux quanta per unit cell is a rational fraction (Hofstadter, 1976) has started to appear (Ismail, 1989) for devices in which the amplitude of the period modulation is large. More recently, periodic oscillations in gate voltage have been observed in the resistance of lateral superlattice devices close to pinch-off (Paris et al., 1991). In these studies there were several electrons per unit cell close to pinch-off, while Smith et al. (1992) investigated a superlattice where there is one electron per unit cell and the amplitude of the modulation is weak. Superlattices with dots of 20 nm in diameter in 60-nm pitch was fabricated on high-mobility GaAs/AlAs heterostructures. Figure 36 shows an SEM micrograph of a device. A sharp metal insulator transition is observed when the carrier concentrationis such that there is one electron per dot, which may indicate the electrons are condensing in a lattice commensurate with the applied periodic potential. Evidence of Landau level splitting was also observed at low periodic modulation potential. Lateral superlattices represent something of a limit to the density of devices in integrated circuits where device-device interactions and quantum effects will dominate operation. However, the quantum collective effects that arise in the lateral surface superlattice are interesting in their own right and offer new device capabilities, such as a tunable detector of infrared radiation and high-frequency oscillators and amplifiers. Nanofabrication technology has made it possible to investigate lateral superlattices and related physics.

C. Single Electronics One of the most exciting and potentially useful phenomena in nanostructurephysics is Coulomb blockade, which occurs in structures with extremely small capacitances, where the tunneling of single electrons leads to macroscopically observable effects. Electron transport through a metallic island, coupled to an external

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FIGURE 36. An SEM micrograph of an exposed superlattice showing the 50 nm period. The sample had a 20 nm thick layer of AuPd evaporated on top. (Reprinted with permission from J. Vac. Sci. Technol. B, 10,2904 (1992), Smith, C. G . et al. Copyright 1992 American Vacuum Society.)

circuit by two tunnel junctions, can directly reflect the quantization of the electron charge on the island (Fig. 37). To add charge Q to the island requires an energy Q 2 / 2 C ,where C is the total capacitance between the island and the rest of the environment. Adding an electron therefore requires a Coulomb energy e2/2C. The Coulomb blockade arises when the passage of a single electron into the island causes the F e d level in the island to rise by such a large amount, due to capacitive charging, that subsequent electrons are energetically unable to enter the island until an electron has left it. Hence, there is an energy gap in the tunneling density of states. For an electron to tunnel onto the island it must have an energy above the Fermi energy in the lead by e2/2C,and for a hole to tunnel it must have an energy below the Fermi energy by the same amount, so the gap has width e2/C. There are two conditions for Coulomb blockade to occur: The total capacitance C of the island to its environment should be so small that the charging energy of the island associated with an electron tunneling on or off must be much greater than the thermal energy, i.e., e2/2C >> ksT here k g is the Boltzmann constant. Otherwise, a considerable number of unoccupied states will be available below the Fermi level on both sides of a barrier and electrons will be able to tunnel when Coulomb blockade is desired. The resistances of the tunnel junctions should be much greater than the quantum resistance RK = h/e2 % 25.8 kQ. Otherwise, the electrons are not well localized on the island and their wave functions overlap with the other parts of the circuit.

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A --/

island

.

.

barriers Insulating substrate

'c

n n e'/c

Barriers

FIGURE 37. Coulomb blockade in an island: (a) an island with very small capacitance is connected to a voltage source through two tunnel junctions: (b) energy-level spectrum for this system, showing filled (hatched) and empty (shaded) levels. The Island has a gap of width e Z / C in its tunneling density of states. C is the total capacitance of the Island, e is the charge of an electron and E f is the fermi level. No current can flow until a bias (U)of at least e/2C is applied.

Coulomb blockade phenomena have been known since the early 1950s. when they were observed in granular metallic films (Gorter, 1951). Giaever and Zeller (1968) found that the conductance of evaporated tin particles embedded in an aluminium oxide film, at low temperature, decreases sharply at zero bias. Lambe and Jaklevic (1969) made similar observations. Their results consisted mainly of the observation of a negative dc offset in the current-voltage ( I - V )characteristic. They predicted conductance oscillations but did not observe them. Averin and Likharev published a number of papers in the mid 1980s, in which they pointed out the importanceof the interplay between continuous charging effects (where charge behaves as if it were indivisible) and discretechargingeffects (Averin and Likharev, 1985, 1986; Likharev, 1987). Thecharge stored by acapacitorin anordinary circuit can take any value: It is a continuous quantity. It is the result of the displacements of electrons throughout the circuit, and these displacements are continuous. The charge on an isolated island, however, is discrete (or quantized) and must be an integer multiple of e, the electronic charge. Conductance oscillationswere detected in granular films (Barner and Ruggiero, 1987; Kuz'min and Likharev, 1987). The enormous interest that has recently been generated in the field is because of the advances in nanotechnology. This makes it possible to use the Coulomb blockade to manipulate single electrons as desired. Fulton and Dolan (1987)

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FIGURE38. Schematic circuit diagram of an experimental single-electron memory device. The coulomb blockade electrometer detects the electron states on the memory node. (Reprinted with permission from J. Appl. Phys. 75 (IOptl), 5123 (l994), Nakazato, K. et al. Copyright 1994 American Institute of Physics.)

demonstrated the first single-electron transistor. Since then, a considerable number of groups have published results, demonstrating single-electron effects in a wide variety of devices and materials. Many device applications were proposed (Likharev, 1988). Recently a single-electron memory operating at 4.2 K has been demonstrated (Nakazato et al., 1993, 1994). The equivalent circuit diagram is shown in Fig. 38. The memory node capacitor is charged through a multiple tunnel junction. The charge in the node is detected by another multiple tunnel junction which acts as an electrometer; the current through it is dependent on the voltage at the memory node. A scanning electron micrograph is shown in Fig. 39. Clear hysteresis is observed in the current flowing through the electrometer as the voltage, Vg, applied to the memory capacitor is cycled repetitively. A typical result is reproduced in Fig. 40. The memory action of the cell can be exploited by following half the loop, for example, from D to A to B to write a bit or from B to C to D to remove it. At present the memory effect has been demonstrated for temperatures up to 4.2 K in such a device. Since the elementary charging energy e2/2C is inversely proportional to the capacitance C, which in turn is proportional to the size of the island, singleelectron tunneling has remained largely a low-temperature physical phenomenon (T 5 4 K). It is important if single-electron devices are to be practical that they should operate at liquid nitrogen or better still at room temperature. It has been shown by using an STM, with an island only a few nanometer across, that single-electron tunneling can persist up to room temperature (Schonenbergeret al.,

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FIGURE39. Mask layout and SEM micrograph of a single electron memory element, with electrometer. (Reprinted with permission from J. Appl. Phys. 75 (IOptl), 5123 (1994). Nakazato, K. et al. Copyright 1994 American Institute of Physics.)

...

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0.03

FIGURE 40. Memory operation characteristics, showing hysteresis at T = 0.03 K. The sweep rate of gate voltage is 70 pV/s. (Reprinted with permission from Appl. Phys. Letts. 75 (10ptl). 5 123 (1994). Nakazato, K. et al. Copyright 1994 American Institute of Physics.)

1992). Coulomb blockade effects were observed up to 50 K by using a more conventional side-gate technology and an island diameter of 20 nm in a silicon-based material (Paul et al., 1993a). Further reduction in the size of the island is needed to enable these devices to work at liquid nitrogen or higher temperatures. Yano et al. (1994) demonstrated the first single-electronmemorythat operates at room

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temperature by using ultrathin polysilicon film for the active region to achieve very small structures beyond the present lithography resolution. In metallic structures many experiments studying the Coulomb blockade effect and its device applications have been carried out after the first experiment demonstrating a single-electron transistor by Fulton and Dolan (1987). MAl203/Al tunnel junction structures fabricated by electron-beam lithography and shadow evaporation technique were used to construct the single-electron transistor. Manipulations of single electrons in devices based on A l / A l 2 0 3 / A l systems, such as single-electron box (Lafarge et al., 1991), single-electron turnstile (Geerligs et al., 1990), and single-electron pump (Pothier et al., 1991, 1992), have been demonstrated. However, these experiments were canied out at low temperaturesbecause the size of these structures is limited by the resolution of electron-beam lithography and it is still too big to operate at higher temperatures. Nakamura et al. (1996) recently showed that the size of tunnel junctions based on MAl203IAl structures can be reduced by an anodization process. They demonstrated singleelectron transistors operating at temperatures up to 30 K. By using a different fabrication technique to achieve smaller island capacitance, Chen et al. (1995) observed the Coulomb blockade effect in a lateral metal structure at temperatures up to liquid nitrogen temperature. The structure was made by a combination of high-resolution electron-beam lithography and ionized beam depositiontechnique. Figure 41 shows an SEM micrograph of a two-terminal device. Islands as small

r? Source

Drain

FIGURE41. (a) Schematic of a lateral multiple tunnel junction device; (b) SEM micrograph of a multiple tunnel junction device. The size of the islands is about 2 to 3 nm in diameter. (Reprinted with permission from Appl. Phys. Lerrs. 66,3383 (1995). Chen, W. et al. Copyright 1993 American Institute of Physics.)

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Voltage (V) I -V characteristicsshowing coulomb gap and staircases measured at 77 K. differential conductance is also shown in which the staircases can be seen mnre clearly. (Reprinted with permission frnm Appl. Phys. Letts. 66,3383 (1995). Chen, W. et al. Copyright 1993 American Institute nfPhysics.) FIGURE 42.

as 2 nm in diameter and a gap between source-drain contacts of about 30 nm were obtained. The small dimensions make possible the observation of Coulomb blockade effects at liquid nitrogen temperature (77 K). A source4rain I-V curve measured at 77 K is shown in Fig. 42. A clear Coulomb gap of about 0.1 V and Coulomb staircases are seen in the I-V characteristics. A single-electrontransistor based on this structure was later demonstrated (Chen and Ahmed, 1995). A gate can be put either on the side or underneath the islands (Fig. 43). Figure 44 shows the source-drain current as a function of the gate voltage measured at 77 K. There are clear oscillations in the I-V characteristics when the gate voltage is swept, indicating single-electroncharging effects.

D. Quantum Dots Microelectronicsnow pervades everyday life, thanks to the integrated circuit that controls everything from washing machines to personal computers. Silicon has become the dominant material of microelectronics. It has the key physical and chemical properties needed for microcircuit fabrication and is abundant and therefore cheap. As component sizes in electronic devices continuously shrink, the problems of crosstalk and interference loom large. This is why optical signal transmission and optical computing look so promising, and many scientists believe that in the next century electronics will be replaced by optoelectronics.

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(b)

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FIGURE 43. SEM micrographs of metal based single electron devices: (a) a side gate structure; (b) a side gate plus backgate structure.

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FIGURE44. Source-drain current (1 and 11) as a function of gate voltage with the source-drain voltage kept constant and the gate current (111) measured at 77 K in a metal based single electron transistor. curve (I) is offset by 5 pA for clarity. (Reprinted with permission from J. Vac. Sci. Technol. B. 13,2883 (1995). Chen, W. et al. Copyright 1995 American Vacuum Society.)

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FIGURE45. The electronic band structure (schematic)of silicon andGaAs. The uppermost valence

state is positioned at E = 0 eV. The bottom of the conduction band of silicon lies near the X point (Xl)whereas that of GaAs lies at r(rl).In order for an interband radiative recombination event to proceed in Si, there. must be another particle, such as a phonon ( h q )to carry away the large change of electron momentum or k-value. For GaAs the interband recombination can happen without involving a phonon. The probability of a three-body transition is much smaller than a two-body transition.

However, optoelectronics has until now had to rely on a more complex and much more expensive semiconductor, GaAs. In GaAs, which has a direct band gap, the transition of an electron from the minimum of the conduction band to the top of the valence band occurs with no change in momentum (Fig. 45). Silicon has an indirect band gap where the minimum separation of conduction and valence bands is between points at different positions in momentum space. Electronic transitions of exactly the same kind in silicon require a change in both the energy and the momentum of the carrier. The radiative transition across the band gap must be assisted by phonons with proper momentum (Fig. 45). The simultaneousproduction of a photon and phonons is a much less likely event than the simple release of a photon in the equivalent recombination event in a direct

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band gap material like GaAs. The quantum efficiency of Si is therefore considerably lower. Semiconductor researchers have spent many years trying to change the properties of silicon and make its band gap more direct, like that of GaAs, that is, to make silicon emit and detect light and be available as a material for optoelectronics. The discovery of luminescence from porous silicon fabricated by electrochemical dissolution revived the hope of using silicon to produce optoelectronicsdevices (Canham, 1990). A spectrum of visible light is emitted from porous silicon. However, the mechanism for the luminescence of porous silicon is still unknown and is still a hotly debated issue of semiconductorphysics. Many theories have been suggested, including a quantum confinement model (Canham, 1990), recombination over surface states (Xie et al., 1992), disordered silicon (Vasquez et al., 1992), and surface chemicals (Brandt et al., 1992). Although some theories are preferred over others, there is no definitive answer yet. There are also practical difficulties in making optoelectronicdevices with porous silicon. It was suggested that porous silicon can be adequately approximated by bulk material containing an array of noninteracting cylindrical pores of fixed radius that run perpendicular to the surface (Canham, 1990). The size of pores in luminescent porous silicon are typically a few nanometers. The silicon skeletons between these pores are about the same size. If structures with similar dimensions to that of porous silicon are made lithographically, it is possible to test the different theories and perhaps overcome the practical difficulties that arise in making optoelectronics devices with silicon. Research work has been carried out to investigate the possibility of fabricating silicon devices with lithographic techniques in the hope of finding the luminescence mechanism of porous silicon and to make practical optoelectronic devices. Si pillars with diametersin the sub-50-nm regime have been fabricatedby electronbeam lithography and reactive ion etching. Wet chemical etching was used subsequently to reduce the pillar size further to 10 nm (Liu et al., 1992; Fischer and Chou, 1993). Photoluminescence was not observed and it is possible that further reductions in pillar dimensions may be required. Fabrication of high-aspect-ratio sub-10-nm Si pillars has been achieved by using a combination of high-resolution electron-beam lithography and reactive ion etching (Chen and Ahmed, 1993b). Silicon pillars of 5-7 nm diameter and 7: 1 aspect ratio were fabricated. An SEM micrograph of fabricated silicon pillars is shown in Fig. 46. No photoluminescence was observed in these samples. A possible explanationwas that the depth of etching damage was so large that the silicon pillars were completelyconsumed and no crystalline material was left in the etched pillars. Recently Nassiopoulos et al. (1995, 1996) reported the observation of visible photoluminescence and electroluminescence from sub-10-nm silicon pillars. The structures were fabricated by a combination of conventional lithography and reactive ion etching.

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FIGURE 46. Sub-lO nm pillars fabricated with electron beam lithography and reactive ion etching. (Reprinted with permission from Appl. fhys. Letts. 63, 1 I16 (1993b). Chen, W. and Ahmed, H. Copyright 1993 American Institute of Physics.)

E. Magnetic Nanostructures Magnetic materials play an important role in our information age where much of the information is stored in some kind of magnetic storage devices. As the progress in miniaturization of microelectronics proceeds, computers with higher processing power become more generally available and the demand for information storage has increased. Further improvements in magnetic devices for information storage become necessary. Advances in lithography and pattern-transfer techniques has made it possible to structure thin films of magnetic materials laterally, giving rise to novel magnetic properties associated with reduced dimensions in the plane of the film. These novel properties could be used to make new storage devices. In magnetic epitaxial films the magnetic anisotropy is determined by shape and by crystal anisotropies. One-dimensional magnetic wire gratings have been fabricated with a combination of electron- and ion-beam lithography (Shearwoodet al., 1993, 1994). Figure47 shows a comparison between hysteresis loops for a planar film and for a wire of width 0.4 p m with the magnetic field applied perpendicular and parallel to the grating wires. For a wire structure, a uniaxial magnetic shape anisotropy is induced in plane, with the easy axis along the length of the wire and the hard axis perpendicular to the wire. This demonstrated that it is possible to use artificial lateral structuring to change the magnetic propertiesof magnetic

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Magnetic Field (k0e) FIGURE47. Comparison between hysteresis loops for a planar film and for a wire of width 0.4 p m with the magnetic field applied perpendicular and parallel to the grating wires. (Reprinted from Microelectronic Eng. 21, 431, (1993) Shearwood, C. et al. with kind permission from Elsevier

Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

films, with potential for device applications as well as basic studies in magnetism. Gratings of materials such as cobalt, Fe, and NiFe have been studied (Adeyeye et al., 1996; Shearwood et al., 1993, 1994). To understand the origin of the magnetic behavior of a recording medium and so to improve the performance of magnetic storage devices, it is essential to know the properties of its constituent, magnetic grains. Single-domain magnetic particles have been studied for some time. However, the propertiesof single particlesare not easy to obtain, because studies usually are carried out on an ensemble of magnetic particles. The advances in nanofabrication has opened up new ways of studying magnetic materials. It is now possible to fabricate magnetic nanostructures with dimensions of the same order of or even smaller than the size of a single magnetic domain so that the properties of single-domain particles can be studied in a systematic way. Smyth et al. (1988) prepared controlled arrays of submicron permalloy particles using electron-beam lithography and liftoff process. Different samples consisting of particles with different size, aspect ratio, and spacing between the particles were fabricated. However, the size of the particles was too big, so they were not single-domain particles. Recently, single-domain magnetic structures were investigated in the form of isolated Ni bars with a width varying from 15 to 300 nm and interactive Ni bar arrays with a spacing ranging from 200 nm with the strongest coupling in the bars’ short axis. (Wei and Chou, 1994). It was found that bars with a width smaller than 150 nm were single domain, and the switching field depends strongly on the width.

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Recently, Chou et al. (1994) fabricated Ni pillar arrays of 35 nm in diameter, 120 nm in height, and 100 nm in period. It was discovered that because of their nanoscale size, shape anisotropy, and separation from each other, each Ni pillar is single domain with only two quantized perpendicular magnetization states: up and down. A high-density storage device based on the pillar structures where each bit of information is stored in one pillar was proposed (Chou et al., 1994). E Summary Advances in nanotechnology have made it possible to study some previously unseen, and in some cases unknown, phenomena due to the reduced dimensionality. Understanding these phenomena and their applications requires further investigation and may need a further reduction in the fabricated structure size. As further progress takes place in nanotechnology, new physical phenomena may be observed and new type of electronic devices may be realized.

VI. CONCLUSION Nanotechnology for electronics has reached a stage of maturity where the basic physics and chemistry underlying the process of making nanostructures are beginning to limit further advances. Coincidentally, this is being reached at the same sage at which we note that the physical principles underlying the operation of conventional semiconductor devices will not be applicable for very long. For example, it has been asserted that CMOS devices significantly below 100 nm in gate length are unlikely to work satisfactorily. It is also believed that mass production methods of device manufacture, based on optical lithography, cannot advance below a threshold of around 100nm. At this size level of 100 nm, nanotechnology begins and in many research laboratories a number of techniques are being developed for making nanostructures and devices based on a size well below 100 nm, as described in this review. Below a size of 100 nm, electron-beam lithography is still seen as the dominant practical technology. The emergenceof research equipment working at voltages up to 100kV and with beam diameter reduced to less than 5 nm gives high resolution with sufficient process latitude to enable nanostructures in the 5- to 100-nm-size range to be fabricated in a number of centers worldwide. The limit of electronbeam-based nanofabrication in terms of resolution is likely to be around a few nanometers, set partly by the delocalization of energy dissipation from the point of entry of the beam,partly by the developmentcharacteristicsof the organic resist material irradiated by the electrons and partly by the transfer of the pattern into the device material.

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Nanostructures perhaps down to 1 nm are possible via the evolution of new resists, which have very high contrast. For this purpose, high-current electron beams with diameters less than 1 nm will be required. Resists of organic materials with long polymer chains will have to be replaced with inorganic materials with sufficientcontrast and sensitivity to be useful and having the required compatibility with material processing after development. Beam diameters for electron lithography of 1 nm or less will require sufficient current to give reasonable exposure times, and a crucial requirement will be the development of electron columns, probably based on field emission guns, with sufficient stability in operation to be compatible with lithography processes. Resolution in lithography goes hand in hand with the transfer of the lithography onto the underlying substrate. Here dry etching techniques with their strongly anisotropic properties come into their own. However, new ideas must emerge if 1-nm patterning is to be achieved. For nanostructures below 1 nm we are in the realm of the removal or attachment of only a few atoms of material and it is unlikely that this level of nanofabrication will be attempted by conventional particle-beam methodology based on resists. At this scale, we can consider nanofabrication in terms of atomic removal or placement and we approach nanostructures via the exciting possibilities opened by the scanning probe methods that have been developed following the invention of the STM. The ability of the probes to pick and place atoms one by one in defined locationswith subatomicprecision clearly places this technique at the peak of what is achievable in terms of the resolution of nanotechnology. Focused ion-beam methods provide an alternative means of fabricating nanostructures,where the intermediateresist exposure stage may be entirely eliminated. This technique has not yet reached its fundamentallimit. The development of new ion sources may give this method an edge in the fabrication of nanostructures in the future. More esoteric techniques with restricted applications are also under investigation in many research laboratories. For example, the direct deposition of nanoscale-size particles by ultra-low-energy deposition of single ions onto a surface is capable of producing ultra-small nanostructures. In an important approach, SPM methods can be applied to special surfaces such that the interaction between the surface and probe generates nanostructures on the surface. As we move deeper into nanotechnology and approach sub- 10-nm-size scales, new applications of nanotechnology arise in research into quantum effects in ultrasmall structures and the field of mesoscopic physics is entered. -0-dimensional systems have been studied extensively,thanks to the development of new material growth technology. With the advances in nanotechnology it is now possible to put additional confinement in electron transport, and new phenomena are observed because the dimensionality of the electrons is confined. Nanostructures have enabled the exploration of effects such as the quantization of resistance and the quenching of the Hall effect. Ultra-small island structures and tunnel barriers have enabled

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the study of effects such as Coulomb blockade that could form the basis of new electronics in the future. Small structures also create the possibilities of exploring new optical device phenomena, and nanostructures in magnetic material enable the exploration of anisotropy in magnetism. The impact of nanostructures in a wider range of physics and engineering is now being felt, and as the technology for nanofabrication advances, the applications are expected to grow. In science, new phenomena could be revealed via nanostructures and new discoveries may follow in the next decade. In engineering, nanostructures could become the basis of new devices, not only in electronics, but in many other branches of engineering.

REFERENCES Abraham, D. W., Mamin, H. J., Ganz, E., andClarke, J. (1986). Surface modification with the scanning tunneling microscope. IEM J. Res. Dev. 30.492. Adeyeye, A. 0..Bland, J. A. C., D a b , C., Lee, J., Ebels, U., and Ahmed, H. (1996). Size dependence of the magnetoresistance in submicron FeNi wires. J. Appl. fhys. 79 (8 pt 2b), 6120. Ahmed, H. (1996). Single-atom scale lithography for single-electron devices. fhysicu B 227, 259. Ahmed, M. M. and Ahmed, H. (1997). Novel electron beam lithography technique for submicron T-gate fabrication. J. Vac. Sci. Technol. B 15, 306. Aharonov, Y.,and Bohm, D. (1959). Significance of electromagnetic potentials in the quantum theory. f h y s . Rev. 115,485. Albrecht, T.R., Dovek, M. M., Lang, C. A., Quate, C. F., and Smith, D. P. E. (1989). Nanometer-scale hole formation on graphite using a scanning tunneling microscope. Appl. fhys. Lett. 55, 1727. Allee, D. R., and Broers, A. N. (1990). Direct nanometer scale patterning of SiOz with electron beam irradiation through a sacrificial layer. Appl. Phys. Lett 57,2271. Alphenaar, B. W., Durrani, Z. A. K., Heberle, A. P., and Wagner, M. (1995). Resistance bistability in resonant-tunneling diode pillar arrays. Appl. fhys. Lett., 66, 1234. Anderson, E. H., Kern, D. P., and Smith, H. I. (1987). Fabrication by tri-level electron beam lithography of X-ray masks with 50 nm linewidths. and replication by X-ray nanolithography. Microelectron. Eng. 6.54 1. Atkinson, G. M., Kubena, R. L., Larson, L. E., Nguyen, L. D., Stratton, F. P., Jelloian, L. M., Le, M. V., and Mcnulty, H. (1991). Self-aligned high electron-mobility transistor gate fabrication using focused ion-beams. J. Vac. Sci. Technol. B 9, 3506. Atkinson, G. M., Stratton, F. P., Kubena, R. L., and Wolfe, J. C. (1992). “30 nm resolution zero proximity lithography on high-2 substrates.” J. Vuc. Sci. Terhnol. B 10, 3104. Averin, D. V.,and Likharev. K. K. (1985). Possible coherent oscillations at single-electron tunneling. In SQUID’85 (H. Lubbig and H. D. Hahlbohm, Eds.), p. 197 de Gruyter, Berlin. Averin, D. V., and Likharev, K.K. (1986). Coulomb blockade of single-electron tunneling, andcoherent oscillations in small tunnel-junctions. J. Low Temp. fhys. 62, 345. Barner, J. B., and Ruggiero, S. T. (1987). Ohservation of the incremental charging of Ag-particles by single electrons. fhys. Rev. Lett. 59, 807. Bate, R. T.(1977). Electrically controllable supedattices. Bull. Am. Phys. SOC. 22,407. Beaumont, S. P., Bower, P.G., Tamamura, T., and Wilkinson, C. D. W. (1981). Sub-20-nm-wide metal lines by electron-beam exposure of thin poly(methy1 methacrylate) films and liftoff. Appl. fhys. Lett. 38,436. Becker, R. S.,Golovchenko, J. A., and Swartzentmber, B. S. (1987). Atomic-scale surface modifications using a tunnelling microscope. Nature 325,419.

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Becker, R. S., Higashi. G. S., Chabal, Y. J., and Becker, A. J. (1990). Atomic scale conversion of clean Si(l I I)-H-I x I to Si( I I 1 )-2x 1 by electron-stimulated desorption. Phys. Rev. Lett. 65. 1917. Bernstein. G., and Ferry, D. K. ( 1987). Negative differential conductivity in lateral surface superlattices. J. Krc. Sci. Technol. B5.964. Binnig, G., Rohrer, H., Gerber, C. H., and Weibel, E. (1982). Surface studies by scanning tunneling microscope. Phyr. Rev. Lett. 49.57. Binnig. G.. Quate, C. F., and Gerber, C. (1986).Atomic force microscope. Phys. Rev. Lett. 56,930. Blaikie. R. J., Nakazato, K., Fraboni, B., Hasko, D. G., Cleaver, J. R. A. and Ahmed, H. (1991). Fabrication of quantum wires and point contacts in GaAdAIGaAs heteroshuctures using focused ion-beam implanted gates. Microelectron. Engin. 13,373. Blaikie. R. J., Cleaver, J. R. A.. Ahmed, H.. and Nakazato, K. (1992).Variable width and electrondensity quantum wires in GaAs/AlGaAs with ion-implanted gates and a surface Schottky gate. Appl. Phvs. LRtf. 60,I618. Blauner. P. G., Ro, J. S., Butt, Y., Thompson, C. V., and Melngailis, .I.(1988).Focused ion-beam fabrication of sub-micron gold structures. J. Electrochem. Soc. 135,11. Brandrup. J., and Immergut, E. H. (Eds.). (1975). Polymer Handbook, 2d ed. Wiley, New York. Brandt, M. S.. Fuchs, H. D., Stutmann, M., Weber, J., and Cardona, M. (1992). The origin of visible luminescence from 'porous silicon': A new interpretation. Solid State Commun. 81,307. Broers, A. N. (1981).Resolution limits of PMMA resist for exposure with 50-kV electrons. J. Electrochem. Soc. 122,166. Broers, A. N. (1988).Resolution limits for electron-beam lithography. I E M J . Res. Dev. 32,502. Broers, A. N., and Sturgess, J. M. (1978).High resolution electron beam fabrication using a STEM. In Electron Microscopy 1978, p. 343.Microscopical Society of Canada, Toronto, Canada. Broers, A. N., Molzen, W. W., Cuomo, J. J., and Wittels, N. D. (1976).Electron-beam fabrication of 80-A metal strucrures. Appl. Phys. Lett. 29,596. Broers, A. N., Harper, I. M. E., and Molzen, W. W. (1978). 250-A linewidths with PMMA electron resist. Appl. Phys. Lett. 33,392. Broers, A. N., Timbs, A. E., and Koch, R. (1989).Nanolithography at 350kV in aTEM. Microelectron. Engin. 9,187. Biittiker, M., and Thomas, H. (1977).Current instability and domain propagation due to Bragg scattering. Phys. Rev. Lett, 38,78. Campbell, P. M., Snow, E. S., and Mcmarr, P. J. (1995). Fabrication of nanometer-scale side-gated silicon field-effect transistors with an atomic-force microscope. Appl. Phys. Lett. 66, 1388. Canham, L. T. (1990).Silicon quantum wire array fabrication by electrochemical and chemical dissolution of wafers. Appl. Phys. Lett. 57, 1046. Cemna, F. (1992).Recent advances in X-ray-lithography. Jpn. J. Appl. Phys. 31,4178. Cemna, F., Guo, J. Z. Y., Turner, S., Ocola, L., Khan, M.. and Anderson, P. (1992).Image-formation in X-ray-lithography-process optimization. Microelectron. Engin. 17, 135. Chen, G. S., Boothroyd, C. B., and Humphreys, C. J. (1993).Novel fabrication method for nanometerscale silicon dots and wires. Appl. Phys. Lett. 62,1949. Chen. W. ( I 994). Nanotechnology: Resolution Limits and Ultimate Miniaturisation. Thesis, Cambridge University. Chen. W., and Ahmed. H. (l993a).Fabrication of 5-7 nm wide etched lnes in silicon using 100 keV electron-beam lithography and poly-methyl methacrylate resist. Appl. Phys. Lett. 62,1499. Chen, W.,and Ahmed, H. (1993b).Fabrication of high-aspect-ratio silicon pillars of less-than-10-nm diameter. Appl. Phys. Lett. 63,I I 16. Chen, W., and Ahmed, H. (1993~). Fabrication of sub-10 nm structures by lift-off and by etching after electron-beam exposure of poly(methy1methacrylate) resist on solid substrates. J. Vac. Sci. Technol. B 11,2519. Chen, W.. and Ahmed, H. (1995).Fabrication and physics of -2nm islands for single electron devices. J. Vuc. Sci. Technol. E 13.2883.

178

W. CHEN AND H. AHMED

Chen, W.. Ahmed, H., and Nakazato, K. (1995). Coulomb-blockade at 77 K in nanoscale metallic islands in a lateral nanostructure. Appl. Phys. Lett. 66,3383. Chen, Z . W., Jones, G. A. C., and Ahmed, H. (1988). Nanowriter: A new high-voltage electron beam lithography system for nanometer-scale fabrication. J. Vac. Sci. Technol. B 6, 2009. Moore, J. A., Corelli, J. C., Silverman, J. P., and Bakhru, H. (1988). Degradation of poly Choi, J. 0.. (methylmethacrylate) by deep ultraviolet, X-ray, electron beam, and proton beam irradiations. J. Vac. Sci. Technol. B 6, 2286. Chou, S. Y., Wei, M. S., Krauss, P. R., Fischer, P. B., and B. A. (1994). Single-domain magnetic pillar array of 35-nm diameter and 65-gbits/in2 density for ultrahigh density quantum magnetic storage. J. Appl. Phys. 76 (10, pt 2). 6673. Chou, S . Y.,Krauss, P. R., and Renstrom, P. J. (1995). Imprint of sub-25 nrn vias and trenches in polymers. Appl. Phys. Letr. 67,3114. Cleaver, J. R. A., Heard, P. H., and Ahmed, H. (1983). Scanning ion beam lithography with a magnetic ion species filter. Microcircuit Engineering 83. In (H. Ahmed, J. R. A. Cleaver, and G. A C. Jones, Eds.), p. 134. Academic, London. Cooper, W. J., Krasicky,P. D., andRodriguez,F. (1986). Dissolutionratesof poly(methy1methacrylate) films in mixed solvents. J. Appl. Polym. Sci. 31,65. Craighead, H. G., Howard, R. E., Jackel, L. D., and Mankiewich, P. M. (1983). 10-nm linewidth electron beam lithography on GaAs. Appl. Phys. Lett. 42.38. Dagata, J. A., Schneir, J., Harary. H. H., Evans, C. J., Postek, M. T., and Bennett, J. (1990). Modification of hydrogen-passivated silicon by a scanning tunneling microscope operating in air. Appl. Phys. Lett. 56,2001. Datta, S., and Mclennan, M. J. (1990). Quantum transport in ultrasmall electronics devices. Rep. Prng. Phys. 53, 1003. Day, H. C., and Allee, D. R. (1993). Selective area oxidation of silicon with a scanning force microscope. Appl. Phys. Lett. 62,2691. de Jager, P. W. H., and Kruit, P. (1995). Applicability of focused ion beams for nanotechnology. Micrnelectron. Engin. 27, 327. De Vegvar, P. G. N., Timp. G., Mankiewich, P. M., Behringer, R., and Cunningham, J. (1989). Tunable Aharonov-Bohm effect in an electron interferometer. Phys. Rev. B 40,3491. Deckman, H. W., and Dunsmuir, J. H. (1982). Natural lithography. Appl. Phys. Lett. 41. 377. Deguchi, K., Ishiyama, T., Horiuchi, T., and Yoshikawa, A. (1990). Effects of photo-electron and Auger-electron scattering on resolution and linewidth in SR lithography. Jpn. J. Appl. Phys. 29, 2207. Doolittle, A. (1965). In Plasticizer Technology (P. Bruins, Ed.), Chapter 1. Reinhold, New York. Early, K., Schattenburg, M. L., and Smith, H. I. (1990). Absence of resolution degradation in X-ray lithography for lambda from 4.5 nm to 0.83 nm. Micmelectron. Engin. 11, 317. Eigler, D. M., and Schweizer, E. K. (1990). Positioning single atoms with a scanning tunnelling microscope. Nature 344,524. Emoto, F., Gamo, K., Namba, S., Samoto, N., and Shimizu, R. (1985). 8 nm wide line fabrication in PMMA on Si wafers by electron beam exposure. Jpn. J. Appl. Phys. 24. L809. Esaki, L., and Chang, L. L. (1974). New transport phenomenon in a semiconductor 'Superlattice.' Phys. Rev. Lett. 33,495. Esaki, L., and Tsu,R. (1970). Superlattice and negative differential conductivity in semiconductors. IBM J. Res. Dev. 14,61. Fischer, P. B., and Chou, S. Y.(1993). Sub-SO nm high aspect-ratio silicon pillars, ridges and trenches fabricated using ultrahigh resolution electron beam lithography and reactive ion etching. Appl. Phys. Lett. 62, 1414. Flanders, D. C., and White, A. E. (1981). Application of * I 0 0 Alinewidth structures fabricated by shadowing techniques. J. Voc. Sci. Technol. 19,892.

NANOFABRICATION FOR ELECTRONICS

179

Ford, C. J. B. (1988). Thesis, Cambridge University. Ford, C. J. B., and Ahmed, H. (1987). Fabrication of GaAs hezerojunction ring structures. Microelectmn. Engin. 6, 169. Fujisawa, T.,Bever, T., Hirayama, Y.,and Tarucha, S. ( 1994). Nanostructure fabrication and the science using focused ion beams. J. Vac. Sci. Technol. B 12,3755. Fujita, J., Watanabe, H., Ochiai, Y.,Manako, S., Tsai, J. S., and Matsui, S. (1995). Sub-lO nm lithography and development properties of inorganic resist by scanning electron beam. Appl. Phys. Lett. 66, 3065. Fulton, T. A. and Dolan, G. J. (1987). Observation of single-electron charging effects in small tunnel junction. Phys. Rev. Lett. 59, 109. Gamo, K.,and Namba, S. (1 985). Maskless focused ion beam-assisted deposition of metal-films. J. Electmchem. Soc. 132, c356. Gamo, K., Inomoto, Y.,Ochiai, Y., and Namba, S. (1982). Fine focused ion beam system using liquid metal alloy ion sources and maskless fabrication. In Tenth Inremational Conference on Electron and Ion Beam Science and Technology (R. Bakish, Ed.), p. 422. Electrochemical Society, Princeton, NJ. Geerligs. L. J., Anderegg, V. F., Holweg, F! A. M., Mooij, J. E., Pothier, H., Esteve, D.. Urbina, C., and Devoret, M. H. (1990). Frequency-locked turnstile device for single electrons. Phys. Rev. Lett. 64, 269 I . Gerhardts, R. R., Weiss, D., and Klitzing, K. V. (1989). Novel magnetoresistance oscillations in a periodically modulated two-dimensional electron gas. Phys. Rev Lett. 62,1173. Gesley, M. A,. Hohn, F.J., Viswanathan, R. G., and Wilson, A. D. (1988). A vector-scan thermal-field emission nanolithography system. J. Vac. Sci. Technol. B 6,2014. Giaever, I. and Zeller, H. R. (1968). Superconductivity of small tin particles measured by tunneling. Phys. Rev. Lett. 20, 1504. Gipstein, E., Ouano. A. C., Johnson, D. E., and Need, 0. U.(1977). Parameters affecting the electron beam sensitivity of poly (methyl methacrylate). IBM J. Res. Dev. 143. Gorter, C . J. (1951). A possible explanation of the increase of the electrical resistance of thin metal films at low temperatures and small field strengths. Physica X W . 777. Gotza, M., Saint-Cricq, B., Dutoit, M., and Jouneau, P. H. (1995). Natural masking for producing sub-10 nm silicon nanowires. Micmelectron. Engin. 27, 129. Green, M., Garciaparajo, M., Khaleque, F., and Murray, R. (1993). Quantum pillar structures on n+ gallium-arsenide fabricated using natural lithography. Appl. Phys. Left. 62,264. Greeneich, J. S. (1975). Developer characteristics of poly-methyl methacrylate electron resist. J. Elecrrochem. Soc. Solid-stateSci. Techno. 970. Haller, I., Hatzakis, M., and Srinivasan, R. (1968). High resolution positive resists for electron-beam exposure. IBM J. Res Dev. 12,25 I . Hasko, D. G., Pottts, A., Cleaver, J. R. A., Smith, C. G.. and Ahmed, H. (1988). Fabrication of submicrometer freestanding single crystal Galium arsenide and silicon structures for quantum transport studies. J. Vac. Sci. Technol. B 6, 1849. Hatzakis, M. (1988). Materials and processes for microstructure fabrication. IBM J. Res. Dev. 32, 441. Hawryluk, R. J. (1981). Exposure and development models used in electron beam lithography. J. Vac. Sci. Technol. B 19, 1. Hawryluk, R. J., Hawryluk, A. M., and Smith, H. I. (1974). Energy dissipation in a thin polymer film by electron beam scattering. J. Appl. Phys. 45,255 I. Hazakis, M.. Ting, C. H., and Viswanathan, N. (1974). In the sixth Intemtionul Conference on Electron and Ion Beam Science and Technology, (R. Bakish, Ed.), p. 542. Electro Chemical Society, Princeton, NJ. Hofstadter, D. R. (1976). Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14,2239.

180

W. CHEN AND H. AHMED

Hosoki, S., Hosaka, S., and Hasegawa, T. (1992). Surface modification of MoS2 using an STM. Appl. Surf: Sci. 60-1, 643. Howard. R. E., Jackel, L. D., Craighead, H. G., Mankiewich, P. M., and Feldman, M. (1983). Electron beam lithography from 20 to 120 keV with high quality beam. J. Vac. Sci. Technol. B 1, 1101. Hoyle, P. C., Cleaver, J. R. A,, and Ahmed, H. (I996). Electron beam induced deposition from W(C0)6 at 2 to 20 keV and its applications. J. Vac. Sci. Technol. B 14, 662. Huang, D. H., Uchida, H., and Aono, M. (1992). Fabrication of atomic-scale structures on Si( I 1 1)-7x 7 using a scanning tunneling microscope (STM). Jpn. J. Appl. Phys. 31,4501. Huq, S. E., Chen, Z. W., McMahon, R. A., Jones, G. A. C., and Ahmed, H. (1990). Fabrication of 25-nm-wide gold lines using nanometer scale lithography and ionized cluster beam depostion. Microelectron. Engin. 11, 343. Isaacson, M., and Muray, A. (1981). In situ vaporization of very low molecular weight resists using 1/2 nm diameter electron beams. J. Vac. Sci. Technol. 19, 11 17. Ismail, K.. 111, Smith, 111, T. P., Masselink, W. T., and Smith, H. I. (1 989). Magnetic flux commcnsurability in coupled quantum dots. Appl. Phys. Lett. 55, 2766. Israelachvili, J. N. (1992). Intermoleculur and Surface Forces, 2nd ed. Academic, London. Itoh, M., Saku, T.. and Tarucha, S. (1992). High-mobility quantum wires fabricated by Ga focused ion beam shallow implantation. Jpn. J. Appl. Phys. 31 (pt I, 12B). 4487. Jackel, L. D., Howard, R. E., Hu, E. L., Tennant, D. M., andcrabbe. P. (1981). 50-nm silicon structures fabricated with trilevel electron beam resist and reative-ion etching. Appl. Phys. Lett. 39, 268. Jones, G. A. C., Blythe, S.. and Ahmed, H. (1986). Direct fabrication of nanometre-scale structures in semiconductors with 500 keV lithography. Microelectron. Engin. 5, 265. Junno, T., Deppert, K., Montelius, L., and Samuelson, L. (1995). Controlled manipulation of nanoparticles with an atomic-force microscope, Appl. Phys. Lett. 66, 3627. Kado, H., and Tohda, T. (1995). Nanometer-scale recording on chalcogenide films with an atomic-force microscope. Appl. Phys. Lett. 66,2961. Katoh, T., Nagamune, Y., Li, G. P., Fukatsu, S., Shiraki, Y., and Ito, R. (1990). Fabrication of ultrafine gratings on GaAs by ciectron-beam lithography and 2-step wet chemical etching. Appl. Phys. Lett. 57, 1212. Kendall, D. L. (1975). On etching very narrow grooves in silicon. Appl. Phys. Lett. 26, 195. Khoury, M., and Ferry, D. K. (1996). Effect of molecular weight on poly(methy1 methacrylate) resolution. J. Vac. Sci. Technol. B 14, 75. Kieslich, A., Reithmaier, J. P.,and Forchel, A. (1994). Minimum feature sizes and ion beam profile for a focused ion beam system with post-objective lens retarding and acceleration mode. J. Vac Sri. Technol. B 12,3518-3522. Knotek, M. L., and Feibelman, P. J. (1979). Stability of ionically bonded surfaces in ionizing environments. Surf: Sci. 90, 78. Koshida, N., Watanuke, S., Yoshida, K., Endo, K., Komuro, M., and Atoda, N. (1992). Electrical properties of nanometer-width refractory metal lines fabricated by focused ion beam and oxide resists. Jpn. J. Appl. Phys. 31,4483. Kragler, K., Gunther. E., Leuschner, R., Falk, G., Hammerschmidt, A., Vonseggem, H., and Saemannischenko, G. ( I 995). Scanning-tunneling-microscopy based lithography employing amorphous hydrogenated carbon as a high-resolution resist mask. Appl. Phys. Lett. 67, I 163. Kramer, N., Birk, H., Jomtsma, J., and Schonenberger, C. (1995). Fabrication of metallic nanowires with a scanning tunneling microscope. Appl. Phys. Lett. 66, 1325. Kroemer, H. (1958). Proposed negative-mass microwave amplifier. Phys. Rev. 109, 1856. Kroemer, H. (I 977). Band-structure instability in long-period one-dimensional superlattices under strong population inversion. Phys. Rev. B 15, 880. Kubena, R. L., Joyce, R. J., Ward, J. W., Garvin, H. L.. Stratton, F. P., and Brault, R.G. (1988). Dot lithography for zero-dimensional quantum wells using focused ion-beams. J. Vac. Sci. Technol. B 6, 353.

NANOFABRICATION FOR ELECTRONICS

181

Kubena. R. L., Stratton, F. P., Ward, J. W., Atkinson, G. M., and Joyce, R. J. (1989). Sub-20-nm-wide line fabrication in poly(methy1methacrylate) using a Ga+ microprobe. J. Vac.Sci. Technol. B 7,1798. Kuz'min, L. S., and Likharev, K. K. (1987). Single-electron tunneling. JETPLett. 45,495. Lafarge, P., Pothier, H., Williams, E. R., Esteve, D., Urbina, C., and Devoret, M. H. (1991). Direct observation of macroscopic charge quantization. Z Phys. B 85,327. Lambe, J., and Jaklevic, R. C. (1969). Charge-quantization studies using a tunnel capacitor. Phys. Rev. Lett. 22, 137 I . Langheinrich, W., Spangenberg, B., and Beneking, H. (1992). Nanostmcture fabrication using lithium fluoride films as an electron beam resist. J. Vac. Sci. Technol. B 10,2868. Lee, K. L., ( 1982). Electron Beam Nanolithography, Thesis, Cambridge University. Lee, K. L., and Hatzakis, M. (1989). Direct electron-beam patterning for nanolithography. J. Vac. Sci. Terhnol. B 7, 1941. Lepselter. M. P., and Lynch, W. T. (Eds.). (1981). Resolution Limitations for Submicron Lithography. Academic, London. Li, Y.Z.. Vazquez, L., Piner. R.. Andres, R. P., and Reifenberger, R. (1989). Writing nanometer-scale symbols in gold using the scanning tunneling microscope. Appl. Phys. Lett. 54,1424. Likharev, K. K. (1987). Single-electron transistors-Electrostatic analogs of the dc SQUIDS. IEEE Trans. Magn. 23, 1142. Likharev, K. K. (1988). Correlated discrete transfer of single electrons in ultrasmall tunnel junctions. IBM J. Res. Dev. 32, 144. Liu, H. I., Maluf, N. I., and Pease, R. F. W. (1992). Oxidation of sub-SO nm Si columns for light emission study. J. Vac. Sri. Technol. B 10, 2846. Lyding, J. W., Abeln, G. C., Shen, T. C., Wang, C., and Tucker, J. R. (1994). Nanometer-scale patterning and oxidation of silicon surfaces with an ultrahigh-vacuum scanning tunneling microscope. J. Vac. Sci. Technol. B 12, 3735. Lyo, I. W.. and Avouris, P. ( I 99 1 ). Field-induced nanometer-to atomic-scale manipulation of silicon surface with the STM. Science 253, 173. Macaulay, J. M., and Berger, S. D. (1987). Nanofabrication using inorganic resists. Microelectmn. Engin. 6, 527. Mackintosh, I. M. (1965). Applications of the scanning electron microscope to solid-state devices. Proc. IEEE 53,370. Majumdar, A., Oden, P. I., Carrejo, J. ,.'F Nagahara, L. A., Grahma, J. J., and Alexander, J. (1992). Nanometer-scale lithography using the atomic force microscope. Appl. Phys. Lett. 61,2293. Mamin, H. J., Guethner, P. H., and Rugar, D. (1990). Atomic emission from a gold scanningtunneling-microscope tip. Phys. Rev. Leu. 65,2418. Margenau, H., and Kestner, N. R. (1969). Theory of Intermolecular Forces. Pergamon, Oxford, UK. Mamian, C. R. K., Dobisz, E. A., and Dagata, J. A. (1992). Electron-beam lithography with the scanning tunneling microscope. J. Vac. Sci. Technol. B 10,2877. Maruno, S., Inanaga, K., and Isu, T. (1995). Nanoscale manipulation of C-60 with a scanning tunneling microscope. Microelectron. Engin. 27,39. Matsui, S., Ichihashi, T., and Mito, M. (1989). Electron beam induced selective etching and deposition technology. J. Vac. Sci. Technol. B 7, 1 182. McCord, M. A., and Pease, R. F. W. (1987). Exposure of calcium-fluoride resist with the scanning tunneling microscope. J. Vac. Sci. Technol. B 5,430. McCord, M. A., and Pease, R. F. W. (1988). Lift-off metallization using poly(methy1 methacrylate) exposed with a scanning tunneling microscope. J. Vuc. Sci. Technol. B, 6,293. McCord, M. A., Kern, D. P., and Chang, T. H. P. (1988). Direct deposition of 10-nm metallic features with the scanning tunneling microscope. J. Vuc. Sci. Technol. B 6, 1877. McCord, M. A., Viswanathan, R., Hohn, F. J., Wilson, A. D., Naomann, R.,and Newman, T. H. (1992). 100 kV thermal field emission electron beam lithography tool for high-resolution X-ray mask patterning. J. Vac. Sci. Technol. B, 10, 2764.

182

W. CHEN AND H. AHMED

McLachlan, A. D. (1963a). Retarded dispersion forces between molecules. Pmc. R. Soc. Lond. Sec A 271,387. McLachlan, A. D. (1963b). Three-body dispersion forces. Mol. Phys. 6,423. Melngailis, J. ( 1987). Focused ion beam technology and applications. J. Vac. Sci. Technol. B 5,469. Middleman. S.. and Hochberg, A. K. (1993). Process Engineering Ana1.W in Semiconductor Device Fubricution. McGraw-Hill, New York. Mochel, M. E., Humphreys, C. J.. Eades, J. A., Mochel, J. M., and Petford, A. M. (1983). Electron beam writing on a 20-Ascale in metal B-aluminas. Appl. Phys. Lett. 42,392. Moriarty, P., Beton, P. H., and Wolf, D. A. (1995). Atomic-scale modifications of GaAs using a scanning tunneling microscope. Appl. Phys. Lett. 66,3527. Muray, A., Isaacson, M., Adesida, I., and Whitehead, B. (1983). Fabrication of apertures, slots and grooves at the 8-80 nm scale in silicon and metal films. J. Vac. Sci. Technol. B 1. 1091. Muray, A., Isaacson, M., and Adesida, I. (1984). AIF3-A new very high resolution electron beam resist. Appl. Phys. Lett. 45, 589. Muray, A., Scheinfein, M., and Isaacson, M. (1985). Radiolysis and resolution limits of inorganic halide resists. J. Vac. Sci. Technol. B 3, 367. Nagamachi, S., Yamakage, Y.,Maruno, H., Ueda, M., Sugimoto, S., and Asari, M. (1993). Focused ion beam direct deposition of gold. Appl. Phys. Lett. 62,2134. Nakamura, Y., Klein, D. L., and Tsai, J. S. (1996). AVA1203/AI single-electron transistors operable up to 30 K utilizing anodization controlled miniaturization enhancement. Appl. Phys. Len.68,275. Nakazato, K., Blaikie, R. J., Cleaver, J. R. A., and Ahmed. H. (1993). Single-electron memory. Electron. Lett. 29, 384. Nakazato, K., Blaikie, R. J., and Ahmed, H. (1994). Single-electron memory. J. Appl. Phys. 75 (10, pt. I), 5123. Namatsu, H., Nagase, M., Kurihara, K., Iwadate, K., Furuta, T., and Murase, K. (1995). Fabrication of sub-10-nm silicon lines with minimum fluctuation. J. Vuc. Sci. Technol. B 13, 1473. Nassiopoulos, A. G., Grigoropoulos, S.. Gogolides, E., and Papadimitriou, D. (1995). Visible luminescence from one-dimensional and 2-dimensional silicon structures produced by conventional lithographic and reactive ion etching techniques. Appl. Phys. Lett. 66, I 1 14. Nassiopoulos, A. G., Grigoropoulos, S., and Papadimitriou, D. (1996). Electroluminescent device based on silicon nanopillars. Appl. Phys. Lett. 69, 2267. Ochiai, Y., Baba, M., Wtanabe, H., and Matsui, S. (1991). Ten nanometer resolution nanolithography using newly developed 50 kV electron beam direct writing system. Jpn. J. Appl. Phys. 30,3266. Ohiwa, H., Goto, E., and Ono, A. (1971). Electron. Commun. Jpn. Sect. B 54,44. O’Keeffe, T. W., and Handy, R. M. (1968). In Solid State Electronics, p. 261. Pergamon, London. Pan, X.. Allee, D. R., and Broers, A. N. (1991). Nanometer scale pattern replication using electron beam direct patterned SiOz as the etching mask. Appl. Phys. ha.59,3158. Paris, E., Ma, J., Kriman, A. M., Ferry, D. K., and Barbier, E. (1991). Quantum hall effect in lateral surface superlattices. J. Phys. Condens. Mutter 3,6605. Park, G . (1968). The glassy state and slow process anomalies. In Diflision in Polymer (J. Crank and G. Park, Eds.), p. 140. Academic, London. Park, K., Lee, S.,Shin, M., Lee, E. H., and Kwon, H. C. (1996). New We oscillations and electrostatic Aharonov-Bohm effect in a GaAdAIGaAs-based mesoscopic ring. SUI$ Sci. 362,75 1. Park. S. W., Soh, H. T., Quate, C. F., and Park, S. I. (1995). Nanometer-scale lithography at high scanning speeds with the atomic-force microscope using spin on glass. Appl. Phys. Lett. 67,2415. Parsonage, E. E., and Peppas, N. A. (1987). Properties of positive resists, I: Irradiation-induced degradation and sensitivity of certain methyl methacrylate copolymers. Bc Polym. J. 19,469. Parsonage, E. E., Peppas, N. A., and Lee, P. I. (1987). Properties of positive resists, 11: Dissolusion charateristics of irradiated poly(methy1 methacrylate) and poly(methy1 methacrylate)-co-maleic anhydride. J. Vuc. Sci. Technol. B 5, 538.

NANOFABRICATION FOR ELECTRONICS

183

Partridge, R. H. (1973). In The Radiation Chemistry of Macromolecules ( M . Dole, Ed.) Academic, New York. Patel, S . S . , and Tirrell, M. (1989). Measurement of forces between surfaces in polymer fluids. Annu. Rev. Phys. Chem. 40.597. Paul, D. J., Cleaver, J. R. A,, and Ahmed, H. (1993a). Coulomb blockade in silicon based structures at temperatures up to 50 K. Appl. Phys. Lett. 63(5), 631. Paul, D. J., Cleaver, J. R. A,, and Ahmed, H. (1993b). Fabrication of wires in silicon-germanium material. Microelectron. Engin. 21, 349. Petroff, P. M. (1990). Carrier confinement to one and zero degrees of freedom. In Physics ofQuanrum Electron Devices (F. Capasso, Ed.). Springer, Berlin. Pothier, H., Lafarge, P., Ofila, P. F., Urbina, C., Esteve, D., and Devoret, M. H. (1991). Single electron pump fabricated with ultrasmall normal tunnel-junctions. Physicu B 169,573. Pothier, H., Lafarge, P., Urhina, C., Esteve, D., and Devoret, M. H. (1992). Single-electron pump based on charging effects. Europhys. Lett. 17,249. Rarhack, H., Shu, D., Feng, S. C., Ade, H., Kirz, J., Mcnulty, I., Kern, D. P. Chang, T. H. P., Vladimirsky, Y., Iskander, N., Attwood, D., Mcquaid, K., and Rothman, S. (1988). Scanning-X-ray microscope with 75-nm resolution. Rev. Sci. Instrum. 59,52. Rishton, S . A. (1984). Resolution Limits in Electron Beam Lithography. Thesis, Glasgow University. Rishton, S . A., Beaumont, S . P., and Wilkinson, C. D. W. (1983). Exposure range of low energy electrons in PMMA. In Proceedings of the Tenrh International Conference on Electrons and Ion Beam Science and Technology (R. Bakish, Ed.), p. 21 I. Electrochemical Society, Princeton, NJ. Ritsko, J. J., Brillson, L. J., Bigelow, R. W., and Fabish, T. J. (1978). Electron energy loss spectroscopy and the optical properties of poly-methyl methacrylate from 1 to 300 eV. J. Chem. Phys. 69, 393 1. Rooks, M. J., Wind, S., McEuen, P., and Prober, D. E. (1987). Fabrication of 30-nm-scale structures for electron transport studies using a poly-methyl methacrylate bilayer resist. J. Vac. Sci. Technol. B 5,318. Ryan, J. M.. Hoole, A. C. F., and Broers, A. N. (1995). A study of the effect of ultrasonic agitation during development of poly(methy1methacrylate) for ultrahigh resolution electron-beam lithography. J. Vac. Sci. Technol. B 13,3035. Sakaki. H., Wagatsuma, K., Hamasaki, J., and Saito, S . (1976). Possible applications of surface corrugated quantum thin films to negative resistance devices. Thin Solid Film 36,497. Schaefer, D. M., Reifenberger, R., Patil, A., and Andres, R.P. (1995). Fabrication of 2-dimensional arrays of nanometer-size clusters with the atomic-force microscope. Appl. Phys. Lett. 66, 1012. Scherer, A., and Craighead, H. G. (1986). Fabrication of small laterally patterned multiple quantum wells. Appl. Phys. Lett. 49, 1284. Schneir, J., Dagata, J. A., and Harary, H. H. (1993). Scanning tunneling microscope-based nanostmcture fabrication system. J. Vac. Sci. Technol. A, 11(4, pt. I), 754. Schonenberger, G., Houten, H. V., and Donkerslwt, H. C. (1992). Single-electron tunnelling observed at room temperature by scanning-tunnelling microscope. Eumphys. Len. 20,249. Seliger, R. L., Kubena, R. D., Olney, R. D., Ward, J. W., and Wang, V. (1979). High resolution ion-beam processes for microstructure fabrication. J. Vac. Sci. Technol. B 16, 1610. Sharma, V. K., Affrossman, S . , and Pethrick, R. A. (1984). Electron beam lithography-influence of molecular characteristics on the perfomance of positive resists. B,: Polym. J. 16, 73. Shearwood, C., Ahmed, H.. Nicholson, L. M., Bland, J. A. C., Baird, M. J., Patel, M., and Hughes, H. P. (1993). Fabrication and magnetisation measurements of variable-pitch gratings of cobalt on GaAs. Microelectron. Engin. 21,431. Shearwood, C., Blundell, S. J., Baird, M. J., Bland, J. A. C., Gester, M., Abmed, H., and Hughes, H. P. ( 1994). Magnetoresistance and magnetization in submicron ferromagnetic gratings. J. Appl. Phys. 75 (10, pt. I), 5249.

184

W. CHEN AND H. AHMED

Shiokawa, T., Aoyagi, Y.,Kim, P. H., Toyoda, K., and Namba, S. (1984). 30-nm line fabrication on PMMA resist by fine focused ion-beam. Jpn. J. Appl. Phys. Lett. 23, 1232. Shiokawa, T., Ishibashi, K., Kim. P. H., Aoyagi, Y.,Toyoda, K., and Namba, S. (1990). Fabrication of periodic structures in GaAs by focused-ion-beam implantation. Jpn. J. Appl. Phys. 29, 2864. Shultz. A. R., Roth, P. I., and Rathmann, G. B. (1956). Light scattering and viscosity study of electron-irradiated polystyrene and polymethacrylates. J. Polym. Sci. 22,495. Silver, R. M., Ehrichs, E. E.. Delozanne, A. L., and Binnig, G. (1987). Direct writing of sub-micron metallic features with a scanning tunneling microscope. Appl. Phys. Lett. 51, 247. Smith, C. G., Pepper, M., Newbury. R., Ahmed, H., Hasko, D. G., Peacock, D. C., Frost, J. E. F., Ritchie, D. A., Jones, G. A. C., and Hill, G. (1990). Transport in a superlattice of ID ballistic channels. J. Phys. Condens. Mutter 2,3405. Smith, C. G., Chen, W., Pepper, M., Ahmed, H., Hasko, D., Ritchie, D. A., Frost, J. E. F., and Jones, G. A. C. (1992). Fabrication and physics of lateral superlattices with 40-nm pitch on high-mobility GaAs GaAlAs heterostructures. J. Vuc. Sci. Technol. B 10, 2904. Smyth, J. F., Schultz. S., Kern, D., Schmid, H., and Yee, D. (1988). Hysteresis of submicron permalloy particulate arrays. J. Appl. Phys. 63,4237. Snow, E. S., and Campbell, P. M. (1994). Fabrication of Si nanostructures with an atomic-force microscope. Appl. Phys. Lett. 64, 1932. Snow, E. S., and Campbell, P. M., and McMarr, P. J. (1993). Fabrication of silicon nanostructures with a scanning tunneling microscope. Appl. Phys. Lett. 63,749. Song, H. J., Rack, M. J., Abugharbieh, K., Lee, S. Y., Khan, V.. Ferry, D. K.. and Allee, D. R. (1994). 25 nm chromium-oxide lines by scanning tunneling lithography in air. J. Vuc.Sci. Technol. B 12,3720. Spears, D. L, and Smith, H. 1. (1972). High-resolution pattern replication using soft X-rays. Electron. Lea. 8, 102. Stockman, L., Neuttiens, G.. Haesendonck, C. V., and Bruynseraede, Y. (1993). Submicrometer lithography patterning of thin gold films with a scanning tunneling microscope.App1. Phys. Lett. 62,2935. Sturans, M. A., and F'feiffer, H. C. (1983). Variable axis immersion lens (VAIL) In Microcircuit Engineering 83 (H. Ahmed, J. R. A. CleaverandG. A. C. Jones, Eds.), p. 107. Academic, Cambridge, UK. Sugimura, H., Uchida, T., Kitamura, N., and Masuhara, H. (1993). Nanofabrication of titanium surface by tip-induced anodization in scanning tunneling microscope. Jpn. J. Appl. Phys. 32,553. Tamamura, T., Sukegawa, K., and Sugawara, S. (1982). Resolution limit of negative electron resist exposed on a thin-film substrate. J. Electrochem. SOC. 129, 183I . Tennant, D. M., Jackel, L. D., Howard, R. E., Hu, E. L., Grabbe, P., Capik, R. J., and Schneider, B. S. ( I 98 1 ). 'henty-five nm features patterned with trilevel e-beam resist. J. Vuc. Sci. Technol. 19, 1304. Thundat, T., Nagahara, L. A., Oden, P. I., Lindsay, S. M., George, M. A., and Glaunsinger, W. S. (1990). Modification of tantalum surfaces by scanning tunneling microscopy in an electrochemicalcell. J. Vac. Sci. Technol. A 8, 3537. Ueberreiter, K. (1968). The solution process. In Di$usion in Polymers, p. 2 19. Academic, London. Umbach, C. P., Broers, A. N., Willson, C. G., Koch, R., and Laibowitz, R. B. (1988). Nanolithography with an acid catalyzed resist. J. Vuc. Sci. Technol. B 6, 3 19. Utsugi, Y. (1990). Nanometer-scale chemical modification using a scanning tunnelling microscope. Nature 347.741. Utsugi, Y. ( 1992). Chemical modification for nanolithography using scanning tunneling microscopy. Nanotechnology, 3, 161. Van der Gaag, B. P., and Scherer, A. (1990). Microfabrication below 10 nm. Appl. Phys. Lett. 56,481. van Wees, B. J., van Houten, H., Beenakker, C. W. J., Williamson, J. G., Kouwenhoven, L. P., van der Marel, D., and Foxon, C. T. (1988). Quantized conductance of point contacts in a two-dimensional electron gas. Phys. Rev. Lett. 60,848. Vasquez, R. P.. Fathauer. R. W.,George, T., Ksendzov, A,, and Lin, T. L. (1992). Electronic-structure of light-emitting porous Si. Appl. Phys. Lett. 60, 1004.

NANOFABRICATION FOR ELECTRONICS

185

Vieu, C., Mejias, M., Carcenac, F., Gaini, G., and Launois, H. (1997). Sub-lO nm monogranular metallic lines formed by 200 kV electron-beam lithography and lift-off in poly-methyl metbacrylate resist. Microelectronic Engin. 35, 253. von Hipple, A. R. (1958). Dielectric Materials and Applicufions.Wiley, New York. Wagner, A., and Levin, J. P. (1989). Focused ion-beam repair of lithographic masks. Nucl. Instrum. Methods Phys. B 3 7 4 2 2 4 . Wang, D., Hoyle, P. C., Cleaver, J. R. A,, Porkolab, G. A., and Macdonald, N. C. (1995). Lithography using electron-beam-induced etching of a carbon-film. J. Vac. Sci. Technol. B 13, 1984. Wang, V., Ward, J. W., and Seliger, R. L. (1981). A mass-separating focused-ion-beam system for maskless ion implantation. J. Vac. Sci. Technol B 19, 1158. Warren, A. C., Antoniadis. D. A., Smith, H. I., and Melngailis, J. (1985). Surface superlattice formation in silicon inversion layers using 0.2 p m period grating-gate electrodes. IEEE Electron Devices Lett. EDL-6,294. Washburn, S., and Webb, R. A. (1986). Aharonov-Bohm effect in normal metal quantum coherence and transport. Adv. Phys. 35,375. Webb, R. A,, Washburn, S., Umbach, C. P., and Laibowitz, R. B. (1985). Observation of Me Aharonov-Bohm oscillations in normal-metal rings. Phys. Rex Lett. 54,2696. Wei. M. S., and Chou, S. Y. (1994). Size effects on switching field of isolated and interactive arrays of nanoscale single-domain Ni bars fabricated using electron-beam nanolithography. J. Appl. Phys. 76 (10, pt. 2). 6679. Weill, A. (1986). Resists Patterning. In The Physics and Fabrication of Microstrucrures and Microdevices (M. J. Kelly and C. Weisbuch, Eds.), p. 58. Springer, New York. Wharam, D. A., Thomton, T. J., Newbury, R., Pepper, M., Ahmed, H., Frost, J. E. F., Hasko, D. G., Peacock, D. C., Ritchie, D. A,, and Jones, G.A. C. (1988). One-dimensional transport and the quantisation of the ballistic resistance. J. Phys. C21, 1209. Winkler, R. W., and Kotthaus, J. P. (1989). Landau-band conductivity in a two-dimensional electron system modulated by an artificial one-dimensional superlattice potential. Phys. Rev. Lett. 62, 1 177. Woodham, R. G.,and Ahmed, H. (1995). Fabrication of atomic-scale metallic microstructures by retarding-field focused ion beams. J. Vac. Sci. Technol. B 12,3280. Woodham, R. G.,Cleaver, J. R. A,, Ahmed, H., and Ladbrooke, P. H. (1992). T-gate, gamma-gate, and air-bridge fabrication for monolithic microwave integrated-circuits by mixed ion-beam, high-voltage electron-beam, and optical lithography. J. Vac. Sci. Technol. B, 10, 2927. Xie, Y. H., Wilson, W. L., Ross, F. M., Mucha, J. A., Fitzgerals, E. A., and Harris, T. D. (1992). Luminescence and structural study of porous silicon films. J. Appl. Phys. 71,2403. Xu, W., Wong, J., Cheng, C. C., Johnson, R., and Scherer, A. (1995a). Fabrication of ultrasmall magnets by electroplating. J. Vac. Sci. Technol. B 13,2372. Xu, Y., MacDonald, N. C., and Miller, S. A. (1995b). Integrated micro-scanning tunneling microscope. Appl. Phys. Lett. 67,2305. Yamamoto, S . , Yamada, H., and Tokumoto, H. (1995). Nanometer modifications of nonconductive materials using resist-films by atomic-force microscopy. Jpn. J. Appl. Phys. 34, 3396. Yano, K., Ishii, T., Hashimoto, T., Kobayashi, T., Murai, F., and Seki, K. (1994). Room-temperature single-electron memory. IEEE Trans. Electron. Devices 41, 1628. Young, R. J.. Cleaver, J. R. A,, and Ahmed, H. (1990a). Gas-assisted focused ion beam etching for microfabrication and inspection. Microelectron. Engin. 11,409. Young, R. J., Kirk, E. C. G.,Williams, D. A., and Ahmed, H. (1990b). Fabrication of planar and cross-sectional TEM specimens using a focused ion beam. Mat. Res. SOC.Syrnp. Proc. 199,205. Youtsey, C., Grundbacher, R., Panepu, R., Adesida, I., and Caneau, C. (1994). Characterization of chemically assist ion beam etching of InP. J. Vac. Sci. Technol. B 12, 3317. Zhang, Y., Oehrlein, G. S., and Defresart, E. (1992). Reactive ion etching of SiGe alloys using CFzClz. J. Appl. Phys. 71, 1936.

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ADVANCES IN IMAGING AND ELECTRON PHYSICS. VOL. 102

Miniature Electron Optics A. D. FEINERMAN AND D. A. C R E W Microfabrication Applications Luboratory, Universiry of Illinois at Chicago, Chicago, Illinois 60607-7053, USA

I. Introduction . . . . . . . . . . . . 11. Scaling Laws for Electrostatic Lenses . . . 111. Fabrication of Miniature Electrostatic Lenses . . . . . . . . . . . A. Review B. Stacking' . . . . . . . . . . . . C. Slicing2 . . . . . . . . . . . . D. LIGA Lathe . . . . . . . . . . . IV. Fabrication of Miniature Magnetostatic Lenses V. Electron Source . . , . . . . . . . . A. Spindt Source . . . . . . . . . . B. SiIicon Source . . . . . . . . . . VI. Detector . . . . . . . . . . . . . VII. Electron Optical Calculations . . . . . . A. A Tilted MSEM . . . . . . . . . VIII. Performance of a Stacked Einzel Lens4 . . A. MSEM Construction . . . . . . . . B. MSEM Operation and Image Formation . IX. SummaryandFutureProspects . . . . . References . . . . . . . . . . . .

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187 189 189 189 191 198

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217 220 224 225 225 228 232 233

I. INTRODUCTION

The term miniature electron optics is used here to refer to electrostatic lenses that are smaller than 10 cm. The technology to reduce the size of the lens is being used to reduce the beam voltage and miniaturize the scanning electron microscope (SEM). There are several applications for a miniature SEM (MSEM). An MSEM can be brought to the sample instead of bringing the sample to a standard SEM. This would be convenient when access to the sample is limited, for example, when inspecting the hull of a spacecraft or inside a fusion reactor, or when it is desirable to inspect objects in situ instead of bringing them to the analytical laboratory. In semiconductor processing there is a need for a low-voltage, high-resolution SEM that could observe integrated circuits in situ during each deposition and etching process. In biology the same instrument could observe specimens immediately after they were sliced with a microtone to minimize sample degradation. 187

Copyright @ 1998 by Academic Press. Inc. All righu of repduction in any form reserved. 1076-5670/97$25.00

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MSEMs can complement other analytical instruments like the scanning tunnel microscope (STM)or the atomic-force microscope (AFM). When the STM and AFM are operated at atomic resolution, their field of view is limited to a few tens of nanometers and the researcher can spend hours trying to determine if the atoms under view are the atoms of interest. An MSEM observing those instruments would allow the researcher to quickly locate the interesting areas of the sample. Miniaturization will speed up the stereo observationof three-dimensionalsamples, which at present proceeds in three steps: observation, rotation, and observation. Two or more MSEMs mounted at lo" with respect to each other can directly acquire a stereo image. Three-dimensional samples of interest range from the evaluation of the pore size and permeability of minerals in the petroleum industry (Huggett, 1990) to the submicron linewidth on an integrated circuit. The technology to make one MSEM could make an array of MSEMs, which would be useful for electron-beam lithography and wafer inspection: The present state of the art Dynamic Random Access Memory (DRAM) technology is 256 Mbit with a minimum feature size of 0.4 p m (Adler, 1994). In general, the size of a memory chip doubles and the smallest feature is reduced 70%every 3 years, quadrupling the amount of information that can be stored on a chip (Sematech, 1994). DRAMS are often developed with e-beam lithography and then manufactured with optical steppers (Larrabee and Chatterjee, 1991). The reason for switching technologies is the order of magnitude increase in throughput in the number of wafers an optical stepper can process in one hour. Optical steppers are faster because all of the pixels are exposed in parallel, whereas an electron-beam machine exposes pixels in a serial fashion. An array of N beams would reduce the total writing time by a factor of N , and would make electron-beam lithography economically competitive. In semiconductor processing the minimum feature size will soon be less than 0.1 pm across an 8"-diameter wafer. Determining the most economic method of patterning wafers is an active area of research, with X-ray lithography (Fleming et al., 1992), deep UV steppers with phase shifting (Lin, 1991), and arrays of electron-beam columns (Feinerman et al., 1992a) or STMs (Marrian et al., 1992) under consideration. Regardless of the lithography method chosen, a method will be required to rapidly inspect large wafers with a resolution of 1/ 10 the minimum feature size or 10 nm. This indicates that an inexpensive array of STMs, AFMs, or SEMs will be essential for the continued growth of this industry. The inspection problem will not be insignificant, however, and it might be simpler to fabricate an array of high-resolution SEMs with the methods discussed in this review than to process the data they will generate. For example, if we examine an 8" wafer consisting of 250 identical 1-cm2die with 250 parallel beams 100 x 100 nm, there will be 2.5 x 10l2 pixeldwafer and 10'" pixelddie. A 10-nm or larger foreign particle will vary the backscattered or secondary electron signal just enough so that when comparing 250 channels simultaneously the equipment can determine which areas might have particles or defects and must be examined at a higher magnification to resolve a 10-nm particle. If we assume that the data can be

189

MINIATURE ELECTRON OPTICS TABLE I

Lengths Potentials Fields Spherical aberration dcs= 0 . 5 C , ( ~ ~ Chromatic aberration dc = CCa(AV/V ) Interactions d, L / V’,’

-

Stray magnetic field deflection

Constant potential

Constant field

k 1

k k

k-’

1

k

k

k

1

k

k-.s

pi2

k312

processed as fast as it comes in on the 250 channels and 0.1 ps to examine each pixel, it would take at least 1000 s to observe the entire wafer. Three techniques are described below that can miniaturize electrostatic lenses operating in different voltage regimes. The integration of an electron source, deflector, and detector into the electrostatic lens in order to make an MSEM and a method to miniaturize a pancake magnetic lens is also discussed.

11. SCALING LAWSFOR ELECTROSTATIC LENSES

There are two common types of scaling: constant potential, where all the lengths are reduced by a factor k, where k is less than 1, and constant electric field, where both the lengths and the voltages are reduced by the factor k. The effect of scaling is shown in Table 1 where a is the maximum angle of emission of an electron that travels down the electrostaticcolumn (Chang et al., 1990). Constant potential scaling provides the largest improvement in resolution. The electric field increases in this case as l / k until a maximum electric field for a given gap size is reached. In our research we have held off 2.5 kV with 138-pm gaps or 18 kV/mm at 8 x lo-’ ton:

111. FABRICATION OF MINIATURE ELECTROSTATIC LENSES

A. Review

This section briefly reviews the SEM miniaturization methods developed by other research groups and the three methods developed at UIC: stacking, slicing, and

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the LIGA lathe (Feinerman et al., 1992a, 1994, 1996). Miniaturizing the SEM involves miniaturizing each component: electron source, deflector, detector, and one or more lenses to focus the beam. The lenses and deflector can be electrostatic or magnetic but electrostatic devices are more easily micromachined and do not dissipate power in vacuum (Trimmer and Gabriel, 1987). There are two main approaches to miniaturizing an electrostatic lens: Either assemble layers and then make apertures (method 1) or make apertures in the individual components and then assemble the components (method 2). The drawback of the first approach is the limited flexibility to vary the aperture size along the column. As discussed in Section VII, Einzel lenses perform better if one can make the second or focusing electrode aperture larger than the first and third apertures. The drawback of the second approach is the aperture alignmenterror when assembling the components. An example of the first miniaturization method is the proposed lithography wand that would be fabricated by thin-film deposition of several layers followed by reactive ion etching (WE) of the apertures (Jones et al., 1989). The maximum column length with this method is 10 p m and is determined by the thickness that can be reliably anisotropicallyetched to form the apertures and the maximum thickness of the conductor and insulatorthin films. A standard vacuum electrostatic design guideline is to restrict the maximum field between electrodes in a column to -10 kV/mm (Chang et al., 1990) and a 10-pm-long column can accelerate and deflect a 100-V beam, which would be capable of exposing only a very thin resist layer. Another problem with a very short column is that since the working distance is approximately half the column length, it would be difficult to mount two columns at lo" with respect to each other for stereo microscopy. An example of the second type of miniaturization method has been developed at IBM. Layers with pre-etched apertures are optically aligned to assemble a 2- to 3-mm-long column with a scanning tunneling microscope tip as the electron source (Muray et al., 1991). The disadvantages of this approach are the elaborate column fabrication method where a sophisticatedoptical inspection system allows the operator to manually align and then epoxy individual layers and the use of a large but well-characterized electron source. The maximum length of a column fabricated with this technique is 10 mm and is determined by the accuracy of the optical inspection system as it examines layers at different heights. In another example of the second method, an electrostatic lens is made from a perforated carbon film mounted to a TEM grid placed over a second TEM grid containing several hundred 20-pm-diameter holes where an 8-pm-thick insulating sheet of polyimide separatesthe two electrodes (Shedd et al., 1993). This technique relies on the random alignment of one of the several thousand perforations in the carbon film with one of the 20-pm holes and there is a small but finite chance of creating a well-aligned column. Our research program has developed three simple methods for manufacturing extremely accurate and inexpensive electron-beamcolumns: stacking, slicing, and

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MINIATURE ELECTRON OPTICS

191

the LIGA lathe. All three methods can vary the aperture size and location along the optical column, the electrode thickness and spacing, and the position of the deflector within the column. Stacking and the miniaturization methods discussed above approximate the SEM as a series of infinite planes with circular apertures separated by thin insulating layers (Feinermanet al., 1992a). In the slicing method the electrodes are not apertures in planes but are cylinders that are bonded to an insulating substrate where the common cylinder axis defines the electron optical axis (Feinermanet al., 1994).The maximum length of the column is limited by the size of the substrate and can be 300 mm or longer. Several sliced columns can be fabricated in parallel. As will be shown later, the LIGA lathe method is capable of fabricating electrodes with the widest variety of shapes (Feinerman et al., 1996). B. Stacking' In stacking, a (100) silicon wafer is anisotropically etched to create an array of die as shown in Figs. 1 and 2. On each die there is an aperture etched through a membrane and four v-grooves on the top and bottom surfaces of the die. Precision Pyrex fibers align and bond the v-grooves on both surfaces of the die. The structure can be designed to have the fibers rest either on the etched groove surface or on the groove's edges (Fig. 3). The relationship between groove width (W), fiber diameter (D), and gap between silicon die is given by the following equations (Mentzer, 1990), where I9 = c o s - ' ( m ) = 35.26". This is the angle between the normal to the (100) surface and a (1 11) plane.

W

IfDI cos(l9)' IfD

W >cos(79)'

W D Gap = -- sin(0) tan(79) Gap =

Jm'

If the v-grooves are allowed to etch to completion their depth will be W / a . We have found that for structural integrity the wafer thickness should be at least W and that a large gap can be obtained by choosing D = W / cos(b), which makes the Gap = W/&. The length of the column shown in Fig. 1 with these choices is then 5 W 3W/&, or 7.1 W. Adhering to a maximum electric field design guideline of 10 kV/mm, a 15-kV column would require 1.5-mm gaps, 2.1-mm-thick wafers, and would be 15 mm long. A 1-kV column would require 0.1-mm gaps, 0.14-mmthick wafers, and would be 1 mm long. The stacked design can be scaled to a wide range of voltages since silicon wafers and Pyrex fibers of almost any dimension can be commercially manufactured, processed, and assembled.

+

'

Portions of this section are reprinted, with permission, from J o u m l of Vucuum Science and Technology A, 10(4),61 1616, July 1992. Copyright 1992 American Vacuum Society.

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-I

Detector Deflector

... v ~ l

I Source

dL-

Membrane

FIGUREI . (a) Silicon die (DI-D4) are stacked with Pyrex fibers that align and bond to the dies' v-grooves. The v-grooves are staggered and truncated to increase the die strength. The top and bottom surfaces of each die are optically aligned during fabrication. The first silicon die contains a micromachined field-emission electron source and a gate electrode to generate the emitting field. The next three silicon die form an Einzel lens. The last die, D4, has an electron detector on the surface facing the sample. The MSEM is on a Pyrex die to provide electrical insulation between the electron source and the vacuum chamber. (b) The stacked design approximates the SEM as a series of infinite planes with circular apertures separated by thin insulating layers. A design guideline is to make the membrane surrounding the aperture 10 times larger than the aperture diameter. (c) One of the die is diced into eight electrically insulated sections (V 1 through V8)to generate a transverse electric field in the center of the die to deflect the electron beam. This die can also correct for astigmatism. The Pyrex washer holds the die together. The die are rectangular instead of square to facilitate electrical contact to the stack. As indicated in Fig. la, the contact region is to the right on DI, out of the page on D2, to the left on D3. and into the page on D4.

1. Description of Silicon Die Processing (Fig. 4 ) A silicon wafer was cleaned and then oxidized in steam at 900°C to grow 40 nm of SiOz. A 200-nm Si3N4 layer was then deposited over the SiOz in a low-pressure chemical vapor deposition (LPCVD) reactor. Both sides of the wafer were coated with photoresist and rectangular and square windows were opened in

MINIATURE ELECTRON OPTICS

193

FIGURE2. Silicon wafers are anisotropically etched to create four v-grooves on the top and bottom surface of each die (only three grooves are shown in the figure), and an aperture to allow the electron beam to pass through the die. One 4”-diameter wafer contains a hundred 7 x 9-mm die. Rectangular die are used to facilitate electrical contact to the column. Precision Pyrex fibers are diced to the proper length and placed in the v-grooves. The Pyrex fibers provide electrical insulation between the die, align the die in three directions, and are bonded to both die.

the photoresist on the bottom of the wafer after aligning the pattern to the wafer flat. The flat indicates the silicon (110) direction. The rectangular and square windows were processed to produce v-grooves and apertures,respectively. The Si3N4 was etched in a plasma etcher, then the photoresist was removed. The top surface of the wafer was aligned to the etched features on the bottom of the wafer with an infrared aligner, then plasma etched. An aluminum film was deposited on the bottom of the wafer and circular holes were etched into this film. The patterned aluminum film serves as a mask for a vertical plasma etch -30-60 p m into the silicon. The metal mask is removed and a thick Si02 layer -2 p m is grown on any exposed silicon. The oxide protecting the silicon on top of the aperture (on the opposite side of the wafer) is removed and the wafer is placed in an anisotropic etchant (44% by weight KOH in H 2 0 at 82°C). This solution etches the silicon (100) direction 400 times faster than the (1 11) direction (Petersen, 1982). The solution has a slight etch rate for Si02 and a negligible etch rate for Si3N4 (Bean, 1978). The wafers were kept in this solution until the KOH solution etched about halfway through the wafer. The Si02 protecting the v-grooves is removed and the wafer is placed into the anisotropic etchant until the silicon above the aperture is removed. The wafer is cut into individual 7x9-mm2 die and the SinN4 and SiO2 layers are removed with a 10-min immersion in 50%HF acid followed by a 5-min deionized H2O rinse. 2. Pyrex Fiber Processing Precision Pyrex fibers were drawn on a laser micrometer controlled fiber optic tower. Duran and Pyrex were chosen because their thermal expansion coefficients

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A. D. FEINEFWAN AND D. A. CREWE

FIGURE 3. The gap between silicon die is determined by the v-groove width ( W ) , and the fiber diameter ( D = 2R). The half-angle of the v-groove is 19 = 35.26”. and the depth is h = W/./2. (a) The center of a 308-pm fiber is positioned 76 p m above a 270-pm v-groove. The fiber contacts the silicon within the v-groove, 13 p m below the silicon wafer surface. (b) The center of a 450-pm fiber is positioned I80 p m above a 270-pm v-groove. The fiber contacts the silicon at the silicon wafer surface and rests on the groove’s edges.

of 3.2 x 10-6/”C closely match that of silicon at 2.6 x 10-6/”C. Both glasses have nearly identical chemical composition and are trademarks of Schott and Coming, respectively. The Pyrex fibers were waxed to a silicon wafer and cut to the desired length on a MicroAutomation 1006A dicing saw. The fibers were then solvent cleaned before being used in the MSEM assembly. 3. Stacked MSEM Assembly The die were aligned and anodically bonded together with 308-pmPyrex fibers as shown in Figs. 5 and 6 (Feinerman et al., 1992a). Pyrex can be bonded to silicon

MINIATURE ELECTRON OPTICS

I1

195

I

FIGURE 4. The process sequence for the silicon die used in the MSEM.The starting point is a silicon wafer covered with a dielectric layer consisting of Si3N4 on SiOz. ( 1 ) Rectangular windows (381 x 5000gm2) are opened in the bottom dielectric layer. (2) Rectangular and square windows (381 x 3400gm2 and 1500 x 1 5 0 0 ~ m 2are ) opened in the top dielectric layer. (3) A 30- to 50-gm-deep circular aperture is etched into the silicon using aluminum as the etch mask. (4) The aluminum is etched away and 2 ym of SiO2 is grown on the exposed silicon. The silicon below the Si3N4 is not oxidized. (5) The oxide protecting the silicon on top of the aperture is removed and the wafer is placed in an anisotropicetchant. The anisotropicetch is interruptedwhen the etch is about half the thickness of the wafer. (6) The oxide protecting the v-grooves is removed and the wafer is placed into the anisotropic etchant until the silicon above the aperture is removed. (7)The Si3N4 and SiO2 layers are removed and the wafer is cut into individual die.

at 250°C with a bond strength of 350 psi (Wallis and Pomerantz, 1969). The bond is strong enough (1 .O f0.5 pounds) to allow the die to be wire bonded. The glass deforms up to 1.6 pm during anodic bonding to silicon (Carlson, 1974; Carlson et al., 1974). This deformation will increase the fiber/silicon contact area and the increase will be larger if the contact point is below the silicon wafer surface (Feinerman et al., 1991) (Fig. 3a). The bond strength between the fiber and the silicon will increase as the contact area increases. Die have been stacked with 308and 450-pm-diameter fibers in 270-pm-wide grooves yielding 152- and 360-pm gaps between the silicon die, respectively. Attempts to bond 510-pm fibers into the 270-wm grooves have not been successful, possibly due to the small fiber/silicon contact area. The accuracy of the stacking technique is limited by the precision of the glass fibers, silicon die,and v-groove etching. Optical fibers have a diameter tolerance of f O . l%/kmof fiber (Gowar, 1984)or f 0 . 3 pm/km for a 308-pm fiber. A kilometer of fiber would provide enough material for several thousand microscopes. The total indicated runout (TIR), which is defined as the maximum surface deviation,

196

A. D.FEINERMAN AND D.A. CREWE

-

FIGURE5. n o silicon die are aligned and anodically bonded to a 308-pin-diameter duran fiber. The die are aligned to within the accuracy of the optical micrograph f 2 pm. The separation between the die is 152 pm.

on a 7 x 9-mm2 double-polished silicon die is much less than 1 pm. The etched v-groove (1 11) surfaces also have less than 1 p m of TIR (Feinerman et al., 1991). At present, the overall accuracy of the column is limited by the f 5 - p m infrared alignment of etched features in the top and bottom surfaces of the silicon die. This accuracy can be improved by exposing the bottom surface of the wafer with X-rays through a metal mask on the top surface of the wafer. The stacking technique should achieve submicron accuracy. 4. Stacked MSEM Electrostatic Deflector and Stigmator

A compact MSEM requires a micromachined electrostatic or magnetostatic deflector and stigmator integrated in the column (Figs. 1 and 7). Electrostatic deflectorhtigmators can be implemented by generating a transverse electric field with a single die (Fig. 1)or with two die (Figs. 7 and 8). The first design generates the field within a single die, which minimizes the column length. In the second approach a transverse field is generated between two successive die. This design has an advantage when building an array of MSEMs, because integrated circuit technology can be used to fabricate the multilevel interconnectsthat can drive all the electrodes in parallel (Fig. 8d). If a single die generated the transverse electric field then wire bonding or a similar technique would be required to drive all the electrodes. The beam deflection angle is given by tan y = L E,/2Vb, where L is the axial length of the deflector (thickness of D3 in Fig. 2 or the gap between D3 and D4 in Fig. 7a), vb is the beam energy as it enters the deflector, and E,, is the uniform transverse electric field created between the deflector plates. If the transverse field is 30 V/mm and the beam travels 2 2 0 p m through the gap between D3 and D4

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(b) FIGURE 6. (a) Optical micrograph of three 381-wm-thick silicon die stacked with glass fibers. V-grooves on the right are to check infrared alignment, while the rest are for fibers. At present the overall structure is limited by a f 5 - p m infrared alignment of the die’s top surface to its bottom surface. (b)Three silicon die will form an Einzel lens. The 0.16” vacuum pickup tool is visible in the micrograph, showing that the stack is self-supporting. The overhang of the die is rotated 90” between layers to facilitate electrical connections.

in Fig. 7a, then to a target 500pm beyond D5,a 5-pm beam deflection would be obtained with a 3-milliradian (mrad) deflection angle Deflecting the beam more than the 4-mrad convergence angle would introduce higher-order aberrations. As shown in Fig. 7, the minimum working distance for an MSEM is obtained when the deflector is inside the Einzel lens. A practical problem with this choice is that the deflector’s electronics operates at the Einzel electrode potential rather than operating at ground potential.

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A. D. FEINERMAN AND D. A. CREWE

7 d 1.... .....:

$ k.

rA

.:.:.:.:: ..-.

%

.yw>q

r

r

L

w

I

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

W

::z:

__ Source

.... r1oc

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

D1

FIGURE 7. (a) Deflecting the electron beam inside a decelerating dinzel lens increases the field of view and working distance at the expense of increased circuit complexity. (b) The beam is focused by a three-electrode Einzel lens and then deflected. The deflector operates near ground potential.

C. Slicing2 As discussed earlier the electrodes fabricated in the slicing method are not apertures in planes but are conducting cylinders bonded to an insulating substratewhere the common cylinder axis defines the electron optical axis (Fig. 9) (Feinerman et al., 1994). The maximum length of the column is limited by the size of the substrate and can be 300 mm or longer. The electrode inner and outer diameter, length, and aperture size can all be varied in the design. The slicing method can also produce an integrated electrostatic deflector (Fig. 10). 1. Slicing Processing

A (100) silicon wafer with a patterned Si3N4 film is bonded to a Pyrex (Corning 7740) wafer and anisotropically etched (Fig. 9). The anisotropic etchant removes silicon faster in the (100)direction than in the ( 111) direction and has a negligible Portions of Section IIlC are reprinted, with permission, from the Journal of k c u u m Science arid Technology B, 12(6), 3 182-31 86, November 1994. Copyright 1994 American Vacuum Society.

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199

etch rate for Pyrex and the nitride film. This etch creates v-grooves in a silicon wafer whose normal is parallel to a (100)direction and the depth of the v-groove is W/& where W is the opening in the nitride film. If the opening in the nitride is larger than f i t , where t is the wafer thickness, the etch will terminate on the Pyrex. The nitride etch mask is designed to create v-grooves in silicon islands on the Pyrex wafer. A Corning 7720 glass capillary is anodically bonded to the silicon and a dicing saw is used to create the required gaps in the capillary. Anodic bonding is a technique where glass is bonded to silicon at elevated temperatures by passing a current from the silicon into the glass (Wallis and Pomerantz, 1969). As discussed later in this section, the anodic bond is sufficiently strong that solid fibers and capillaries can be diced without any organic “potting” compound. In Section VII the resolution of the proposed sliced column is calculated when each electrode has 300- and 500-pm inner and outer diameters. The electrode aperture size can be varied along the column by bonding capillaries with different inside diameters into the v-grooves holding electrodes E2-FA (Fig. 9d).

FIGURE 8. (a) A silicon die at a single potential will have a uniform coating of metal on its top and bottom surfaces. (b) If a pair of silicon surfaces are used to deflect the electron beam and correct for astigmatism,one surfaceof each die will have eight independentlycontrolled metal electrodes insulated from the silicon with a thick high-quality SiO2 layer. (c) Cross-sectional view of deflector indicating the transverse electric field between the pair of die. (d) The deflectors for an array of MSEMs can be operated in parallel with integrated circuit interconnection technology. The interrupted lines indicate where a second level of metallization is required to avoid shorts between potentials. The contacts at the edge of the array (VI-V8) have been repeated for visual clarity. Only 8 contacts are needed to drive an N x N array of deflectors in parallel.

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A. D. FEINERMAN AND D. A. CREWE

FIGURE8. Continued.

After the structureis fabricatedthe glass surfacesthat will be exposed to the electron beam must be made sufficiently conductive to form an electrostatic column. Electrical contact to the conductive glass can be made by attaching leads to the silicon sections. A crucial question for the slicing method is the minimum conductive coating required for each electrode in the column. As is well known, insulators exposed to an electron beam will charge and the resulting electrostatic fields will have a deleterious effect on the electron beam itself. A starting point to determine the minimum conductive layer is to assume that after the first beam-limiting aperture no more than 1% of the beam will strike any surface. The stray current striking the middle of the electrode’s walls should not raise the potential of the wall by more than one-tenth of a volt, which is the variation in beam voltage expected from a cold field emitter source. If the glass surface has a coating of R,, Wsquare, the resistance of the electrode Re[ is given by the following formula, where D I ,DO, and L are the inner and outer diameters, and L is the axial length of the electrode:

MINIATURE ELECTRON OPTICS

20 1

FIGURE 9. Sliced MSEM. (a) A (100) silicon wafer with a patterned silicon nitride layer is anodically bonded to a Pyrex wafer and anisotropically etched. The nitride is removed with buffered HF acid. A dicing saw separates the silicon into electrically isolated electrode sections. (b) Precision GE772 capillary tubes are anodically bonded into the v-grooves. The glass has a thermal expansion coefficient of 3.6 x 10-6/"C and contains 2% PbO. (c) The capillary tubes are separated into electrodes with a dicing saw and a micromachined field-emission source is added to the column. (d) A three-dimensional view of a sliced electrostatic column. Electrodes El, E2, E3, and E4 are 1.5, 1, 1.5, and 1 nun long, respectively. A I-mm gap separates electrodes E l , E2, and E3, and a 1.5-mm gap separates E3 and E4. electrodes E2 to E4 have a 300-wm inner diameter and a 500-pm outer diameter.

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A. D. FEINERMAN AND D. A. CREWE

FIGURE10. Sliced deflector. (a) A ( I 00)silicon wafer with a patterned dielectric layer is anodically bonded to a Pyrex wafer and anisotropically etched. The dielectric is removed with buffered HF acid. A second layer is "stacked" over the first layer. (b) A preform consisting of GE772 and Pyrex is drawn and anodically bonded into the v-grooves. The structure is then diced as in the previous figure, with the blade electrically isolating the tube sections. (c) A second double-layer substrate is bonded to the top of the composite fibers. Electrical contact can now be made to each section of GE772 glass.

The above formula assumes the length of the v-groove is half that of the electrode, and ignores current bunching where the electrode makes contact to the v-groove. If L, Do, and DI are 1.5, 0.5, and 0.3 mm, respectively, then R,I = 0.56R,,. If there is a 1-namp beam, then R,, must be less than -1.8 x 10'' Wsquare to avoid a 0.1-Vvariation in the electrode's potential. This is an approximation and the assumptions will have to be confirmed by experiment. We have not yet made the glass conductivebut we have three proposed solutions. Our first solution is to use a glass containing PbO wherever the glass electrode surface will be exposed to the electron beam. This PbO could be reduced in a hydrogen ambient with the process used to create microchannel plates. A typical microchannel plate produced at Galileo Electro-Opticsin Massachusettsis 400 p m thick with an active area that contains 3.4 x lo6 capillaries, 10p m in diameter. The resistance of each capillary is approximately 13R,. The minimum microchannel plate resistance reached is 10-100 kQ using Corning 8161 (which contains 51% PbO) or Galileo MCP-10 glass (Laprade, 1989; Feller, 1990). This translates into a conductive layer of 3 x lo9 to 3 x 10" Wsquare, which is sufficient for the low beam currents used in imaging.

MINIATURE ELECTRON OPTICS

203

Another solution would be to metallize the glass by chemical vapor deposition. For example, a thin coating of polycrystalline silicon deposited on glass could be exposed to tungsten hexafluonde to form a tungsten film with a sheet resistivity of 2-100 Wsquare (Busta et al., 1985). An alternativeprocedure is to electroplate a thin layer of gold onto the reduced glass surface on the short capillary sections. The deflector shown in Fig. 10 will require that complex glass cross sections be drawn (Jansen and Ulrich, 1991) and selectively made conductive. Any metallic coating will lower the sheet resistance of the glass to a point where it can be used in an electron optical column. The overall accuracy of a sliced column depends on the accuracy with which a v-groove can be etched, and a capillary can be drawn and diced. The total indicated runout, or maximum surface deviation, on an etched v-groove (1 11) surface is less than 1 pm (Feinerman et al., 1991). Fibers can be purchased that are drawn with laser micrometercontrol and have a 1-pmor better tolerance on their diameter. The largest error is in the control of the length of each capillary section which is -3 pm. Most commercialelectronoptical columns have a dimensionaltolerance of -0.1 %. The errors in a sliced column are on the order of 0.3%(100 x 3 pm/ 10oO pm) and are expected to slightly degrade the performance of an electron optical column. The impact each error will have on the column’s resolution will have to be directly measured. D. LIGA Lathe3

As shown in Fig. 11, this lathe is capable of patterning the widest variety of electrode shapes on a micron scale, including shapes impossible to achieve with a conventional lathe (Feinerman et al., 1996). The electrode spacing and aperture size within an electron optical column can also be varied. The maximum length of the column is limited by the size of the X-ray exposure at a synchrotron, which is 100 mm at Argonne’s Advanced Photon Source. However, successive exposures can be stitched together. 1. LJGA Lathe Processing

In standard LIGA process (LIGA is a German acronym for lithography and galvoforming or electroplating), a planar substrate is covered with an X-ray-sensitive resist and exposed with a collimated X-ray source (Guckel et al., 1990). A typical X-ray resist is poly(methy1methacrylate),or PMMA (also known as Lucite). The exposed resist is removed in a developer (positive resist) and this process is the analog of a binary mill operating on a micron scale capable of creating twodimensional structures that are as thick as the PMMA. Metal is electroplated into Portions of Section IIID are reprinted, with permission, from the IEEE Journal of Microelecrmmechanical Systems, 5(4), 250-255, December 1996.

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A. D. FEINERMAN AND D. A. CREWE

FIGURE1 1. Three-dimensional vicws of electrostatic columns that can be produced on a LIGA lathe. The technique can create the widest variety ofelectrode shapes and can vary the aperture diameter along the length of the column. The technique used to create these structures requires that a cylindrical layer of X-ray resist be exposed and developed. After resist development, metal can be electroplated into the regions where the resist was removed or a conformal metal coating can be deposited around the structure.

FIGURE 12. A blank substrate ready for use on the X-ray lathe. An X-ray-sensitive resist surrounds an opaque core. A solid rod of X-ray sensitive resist could also be used as substrate.

the exposed and developed voids formed in the resist. The modifications developed in our laboratory extend LIGA into a variety of three-dimensional structures. A cylindrical core coated with an X-ray sensitive resist is schematically illustrated in Fig. 12. Nylon filament 460 p m in diameter has been coated with PMMA as has 125-pm gold-plated copper wire. The PMMA is built up to the desired thickness with multiple layers. This core is mounted with slight tension between the headstock and tailstock of a custom-builtglass blower’s lathe shown in Fig. 13. The two chucks on the lathe rotate simultaneouslyto avoid twisting the core during exposure. The lathe rotates at 1 rpm during 30-min and longer exposures.

MINIATURE ELECTRON OPTICS

205

FIGURE13. LIGA lathe prototype. Both ends of the substrate shown in Fig. 12 rotate at the same rate. Anti-backlash gears are used to prevent the. substrate from twisting during the exposure.

\ FIGURE14. X-ray mask used to create a two-level cylindrically symmetric surface. The X-ray resist exposed below the transparent regions of the X-ray mask is subsequently removed in the developer.

A two-level (binary) surface possessing cylindrical symmetry was fabricated by exposing the substrate with a mask consisting of opaque bars (Fig. 14). Micrographs taken after PMMA development are shown in Fig. 15. The current cylindrical resist layers are not as uniform as planar resist coatings. If the coating technology cannot be significantly improved, a uniform layer could be achieved by exposing the resist through a mask that absorbs all X-rays below the desired radius and removing the excess resist in a developer. The starting material can also be solid PMMA rod. A cylindrically symmetricpattern with a variable radius was fabricated as shown in Figs. 16 and 17. The radial penetration of the X-rays is determined by the shape

206

A. D. FEINERMAN AND D. A. CREWE

(b)

FIGURE15. (a) A -55-pm-thick PMMA coaling on a 125-pm-diameter Au-plated Cu wire. The PMMA cross-section thickness is not uniform with thick coatings. (b) A -15-wm-thick PMMA coating on a 125-pm Au-plated Cu wire.

MINIATURE ELECTRON OPTICS

207

FIGURE16. X-ray mask used to create a variable-level cylindrically symmetric surface. The separation of the transparent region of the mask and the rotating substrate axis determines the final radius of the resist. The exposure time can be reduced by 50% with a mask that exposes both sides of the substrate simultaneously.

Micrograph of a variable PMMA surface. A 460-ymnylon fiber was coated with -X-rays.FIGURE wm of PMMA. The substrate was intentionally overexposed and the nylon was damaged by the 17.

125

208

A. D. FElNERMAN AND D. A. CREWE

of the X-ray absorber. If the mask extends beyond the outer radius of the resist, no resist is exposed. Conversely, if the mask does not block the exposure all the resist will be exposed. The X-ray mask becomes the analog of the cutting tool of a conventional lathe. There are other possible modifications of LIGA technology where the X-ray exposure is modulated in time. As indicated in Fig. 18, octupoles for an electrostatic deflectorhtigmator can be created if the substrate is exposed through an aperture and the exposure is chopped synchronously with the rotation. The shutter motion in this case would have to be much faster than the time needed to make one complete rotation, which is 1 min with our current fixture.

FIGURE18. (a) Quadrupoles and other non-azimuthally symmetric shapes can be achieved with the LIGA lathe. (b) Cross-sectional view of a quadrupole exposure. The shutter position determines if X-rays will be transmitted through an aperture in an X-ray mask. The hatched areas represents the resist not exposed. The shutter motion has to be significantly faster than the time to make one 1 min. revolution, which is ,.-

MINIATURE ELECTRON OPTICS

209

2. L E A Lathe Dose Calculation

The binary exposure doses (Fig. 14) are compared to that of a planar slab with the same resist thickness. The exposure time calculation for the binary radius cylinder structure assumes an opaque core with radius Ri covered with resist to a radius R,. The variables are defined in Fig. 19a. This structure rotates with an angular speed of o while illuminated with collimated X-rays. The X-ray path length h at a particular radius r and angle 6 is given by the following formulas:

h = R, x cos(j3) - r x cos6

(5)

The exposure at a particular radius takes place between +zYc, where n = - cos-' 2 The exposure in one revolution of the substrate is given by

+

Ebinary(r) = Iincx

2

):(

1

w o

8,

lcos 19I exp

(-%)d6

(7)

where Iincis the incident powerlarea, and is the X-ray absorption length at a particular wavelength. If the core is not opaque and has the same .$ as the resist, the integral extends between 0 and n. The lcos 6 I factor in the integral takes into account the angle between the resist and the radiation. This calculation neglects refraction because the index of refraction for PMMA differs from 1 by less than across the range of X-rays used in the LIGA lathe exposures (Cerrina et al., 1993). In one revolution of the substrate, the bottom of a planar resist layer t p m thick would receive the following exposure: 2n E-planar(t) = Zinc x - x exp w

The ratio of the cylinder to planar exposure times for a comparable resist thickness varies slightly with the exact value of the X-ray absorption length, R, and R,. The longer exposure time is a consequence of the cylindrical geometry (longer path length and opaque core blocking the X-rays); however, there is more PMMA per unit mask opening in the cylindrical case 0.5n(R, Ri)/& or 2.4 with a 125-pm-diameter core covered with 5 0 p m of resist. As shown in Table 2, the exposure ratios range from 3 to 4 for the values of Ri, resist thickness, and wavelength used at the CXrL (Center for X-ray Lithography, Stoughton, WI,USA). The calculations in Table 2 assume an X-ray absorption length of 100pm, which corresponds to 0.36-nm X-rays. At the CXrL facility with a 1-GeV beam and a 25-pm beryllium window, the X-rays range from 0.25 to 0.5 nm with absorption lengths ranging from 310 to 40pm (Cerrina et al., 1993). A planar PMMA sheet 50 Fm thick is exposed in 6 min with a storage ring current of 150 mA, and

+

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A. D. FEINERMAN AND D. A. CREWE

111111111111111 x-ray

FIGURE19. (a) The X-ray dose at radius r with the mask shown in Fig. 14 depends on the X-ray path length h ( 8 ) and the radius of the opaque core. The exposure takes place from -0, to + 0,. The hatched area represents the resist, which is not exposed, since the core blocks the X-rays. (b) The X-ray dose at radius r with the mask shown in Fig. 16 depends on the X-ray path length h(I9) and the amount of the incident radiation the mask intercepts. The exposure takes place between B,j and 9If, on both sides of the substrate. The hatched area represents the resist, which is not exposed since the mask blocks the X-rays.

21 min were required to expose the 50-pm-thick layer of PMMA surrounding a 125-pm-diametercore. In a planar resist geometry the ratio of the exposure on the top and bottom surfaces is exp(t/e). In a cylindrical geometry this ratio increases by a factor of -1.2 because the core shadows the inner surface more than the outer surface during each revolution. Consequently, planar resist can be slightly thicker than cylindrical resist for any given X-ray energy.

MINIATURE ELECTRON OPTICS

21 1

TABLE 2 Core diameter (wm)

Resist thickness (m)

-

Cylindedplanar exposure ratio

~

3.20 3.23 3.31 3.45 3.21 3.26 3.54 3.81

5 10 50 124 5

125 125 125 I25 460 460 460 460

10

50 124

The exposure time for the variable radius cylinder shown in Fig. 16 is calculated next. In this structure incident radiation is blocked by the X-ray mask at radii less than R , from the substrate’s axis of rotation, as shown in Fig. 19b. The exposure at a radius greater than R, takes place on either side of the core between Omi and O,f, where

n

O,i = - - cos-’ 2

($)

(9)

The exposure in one revolution is then given by

There is no exposure at radius R , since z9,i = 6,f. The radius at which the resist is sufficiently exposed (R,) is determined by the actual exposure time. If the ratio of the exposure at the outer and inner surfaces was 5, then with a 125-pm-diameter core covered with 50 p m of resist and R , = Ri, R, = 1.16Ri. The mask must be undersized to achieve the desired radius.

Iv. FABRICATION OF MINIATURE MAGNETOSTATIC LENSES Researchers have developed impressive techniques to fabricate electromagnetic components on silicon wafers (Ahn and Allen, 1993). They include winding conductors around electroplated magnetic material or winding magnetic material around conductors (Ahnand Allen, 1994). These techniques are compatible with

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A. D. FEINERMAN AND D. A. CREWE

the stacked MSEM fabrication approach described in Section IIIB and could be used to create a magnetic pancake lens (Mulvey, 1982). As was already discussed, magnetostatic lenses have a disadvantage with respect to electrostatic lenses of power dissipation in vacuum but they have the benefit of lower aberration coefficients. It is highly unlikely that magnetic pancake lenses can be micromachined with the same tolerances as electrostatic lenses.

V. ELECTRON SOURCE A micromachined field-emission electron source is an essential component for a miniature scanning electron microscope (MSEM). As discussed in Section VIII, initial images in our laboratory were obtained with a commercial thermal fieldemission electron source that was an order of magnitude larger than the micromachined Einzel lens. A macroscopic source is also very difficult to align to the lens. The majority of the research on micromachined electron sources concerns stable current vs. voltage characteristics. A good review of existing work on micromachined sources can be found in an earlier volume in this series (Brodie and Spindt, 1992). An ideal source for an MSEM will produce at least 1 namp of electrons, which will travel down the electron optical axis. The electrons will appear to originate from a small source on the order of 1 nm in diameter and will have an energy spread no larger than 0.1 eV. The current from field-emission sources depends exponentially on the tip workfunction C#J and the proportionality factor between applied voltage and the electric field at the tip’s surface. A common source of variation of these parameters for microfabricated field emitters is a carbonaceous contamination and oxidation of the emitter surface (Somorjai, 1981), and fieldenhancing protuberances on the emitter surface. The micromachined electron sources discussed in this section can be easily incorporated into a stacked MSEM and are rugged enough to withstand processes that can remove these deposits from the emitting surface. A. Spindt Source

The fabrication of an array of Spindt sources is schematically shown in Fig. 20 (Spindt, 1968; Spindt et al., 1991). A 0.5-pm-thick thermal silicon dioxide (Si02) layer is grown on a high-conductivity silicon wafer -0.01 S2-cm. The SiO2 is then coated with a 0.25-pm-thick film of molybdenum. Submicron holes -0.8 p m in diameter are etched in the Mo film and electropolishedto smooth off any deviations from a circular opening. The Mo film serves as an etch mask when holes are being isotropically etched in the SiOz layer with buffered hydrofluoric acid. A second

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213

(c) SCANNING ELECTRON MICROGRAPH OF SPINDT CATHODE

Thin-film field-emmissionarray fabricated using anisotropic etching FIGURE20. Schematic of a field-emitterarray. The base is single-crystal silicon and the fieldemitter cathodes and gate film are vapor-deposited molybdenum. The insulating layer is thermally grown SiO2.

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A. D. FEINERMAN AND D. A. CREWE

mask opens rectangular patterns in the Mo and SiO2 films and the wafer is etched in KOH to form v-grooves for aligning the source to the electron optical axis. A third mask patterns the Mo film into a gate electrode. The wafer is then cut into individual die. Aluminum oxide (A1203) is deposited on a die at 60” with respect to the normal of the surface. A shallow deposition is used so the A1203 doesn’t reach the bottom of the submicron hole. A second Mo layer is deposited at normal incidence forming a cone in the hole. An immersion in KOH removes the A 1 2 0 3 and the second Mo layer everywhere except where it formed a cone in the hole. Microgaphs of an array of Mo cones are shown in Figs. 20b and c. The maximum temperature of this source is approximately 550°C and is limited by reaction between the molybdenum and silicon. The effects of a pure Hz and a 9/1 mixture of H2 and Ne plasma glow discharges on the I-V characteristics and emission uniformity of a single emitter tip have been investigated (Schwoebel and Spindt, 1993). The discharges were operated with a current-regulated direct current supply. During glow discharge processing the emitter tip and gate were electrically connected and served as the cathode in the discharge; thus, the emitter was bombarded with only positive ions. The anode of the glow discharge was -1 cm from the cathode. At this distance and with a pressure of 1 tom, operating voltages of 275 to 450 V were required to sustain a glow discharge in the gases employed. Total ion current densities at the emitter array were on the order of 1OI6 ions/cm2 s-I. Gases used were of Matheson research-grade purity admitted to the system from 1-L glass flasks. Following the glow discharge treatment the system was evacuated to UHV conditions prior to array operation. An in situ hydrogen plasma treatment to doses of 10l8 to 10’9/cm2has been shown to reduce the workfunction for a single tip from 0.5 to 1.5 eV and also to increase its emission uniformity. The emission patterns before and after the HZplasma are shown in Figs. 21a and b, respectively. The hydrogen cleaning allowed the tips to be immediately operated at 5 WAwithout the usual 50- to 100-h seasoning time. The hydrogen cleaning was followed by a H2/Ne plasma, which further improved the emission uniformity (Fig. 21c). The two plasma treatments lower the voltage required to achieve a given emission current compared to samples that have not been plasma cleaned, as shown in Fig. 22.

-

B. Silicon Source Conventional field emission sources for electron microscopes can either “flash” a cold tip (heat briefly to 1600 K) to clean and reform its surface or operate a zirconiudtungsten tip at 1800 K for thermally assisted field emission. Our laboratory is investigating a micromachined field-emission source on a single-crystal silicon microbridge that can be flashed or operated continuously at 1000-1200 K.

MINIATURE ELECTRON 0F"ICS

215

FIGURE2 1 . (a) Field electron micrograph of a single tip prior to hydrogen plasma treatment ( V = 175 V, I = 1 pamp). (b) Field electron micrograph of the single tip following a H2 plasma treatment ( V = 133 V, I = 1 pamp, total dose ions/cm2). (c) Field electron micrograph of the single tip after 9/1 H2/Ne plasma treatment (V = 130 V, I = 1 pamp, total dose -7 x 10'' ions/cm*).

The thermally assisted sources are not as bright as cold field-emission sources but have more stable emission characteristics. The energy spread of these tips is approximately twice that of a cold field-emission tip. Even operating a cold fieldemission tip at 500 K improves its stability. In the past, micromachined sources were developed for high-density displays or for vacuum integrated circuits, and designerscould not afford the space or heat dissipationrequired with this approach.

216

A. D. FEINERMAN AND D. A. CREWE

10-10

cu"

2

3n E

Y

6

8

10

12

I N x 103 [voitsyl FIGURE 22. Fowler-Nordheim data showing effect of pure hydrogen and hydrogen + 10% neon plasma treatment on a microfabricated single-tip field emitter. Line A: FN data prior to hydrogen plasma treatment. Line B: FN data after hydrogen plasma treatment (total dose ions/cm2). Line C: FN data after hydrogen +lo% neon plasma treatment (total dose -1 x ions/cm2). Line D: FN data following an additional hydrogen +lo% neon plasma treatment (total dose -7 x 10l8 ions/cm*).

However, as shown in Fig. 23, a stacked MSEM has the space to accommodate a miniature heater. In addition, a heat source would be valuable because silicon micromachined tips (Ravi and Marcus, 1991) are often contaminated with a thin Si02 layer and there are data showing that heating them to lo00 K in 1.5 x torr would desorb this layer (Yamazaki et al., 1992). A 15-km-high silicon tip without a gate has been fabricated in the center of a micromachined micro-bridge with a cross-section area of 1.1 x lop2mm2 and 5 mm long. The center of the bridge has been heated continuouslyup to 935°C. The field-emission characteristics of this tip at room temperature, 395"C, and 935°C have been investigated. The current fluctuations over 20 min of operation were

MINIATURE ELECTRON OPTICS

217

FIGURE23. Fabrication of a field emitter on bridge center. (a) Anisotropic etching of silicon to form an -100-hm-thick membrane. (b) Isotropic etch of silicon using a 30-pm-diameter SiOz circle island to form a 15-pm-high silicon tip on the membrane center. (c) Anisotropic etching of the membrane to form the bridge and v-grooves. (d) Oxidation sharpening of tip. (e) Device is bonded to Pyrex die. (f) Dicing saw electrically isolates bridge from the rest of die.

reduced from 150% at room temperature to 24% at 935°C. SEM micrographs in Fig. 24 indicate that after the tip operated at 935"C, it became rounded and some protrusions and concentric rings developed around the tip. Further investigation is needed before this tip can be used in an MSEM. VI. DETECTOR

,

A drawback of the MSEM design is that the short working distance reduces the number of choices for detecting secondary and backscatteredelectrons and X-rays. In the basic MSEM design shown in Fig. 1, the working distance is only 0.5 mm and the beam energy is 2.7 keV. The simplest electron detector is a Faraday cup, which collects electrons with a unity gain, which will limit the pixel acquisition time. An approximation to a Faraday cup can be achieved by grounding the last electrode (D4) shown in Fig. la. A metallic layer can be patterned and electrically isolated from the last silicon chip in the column in order to operate the detector at an accelerating potential with respect to the last die in the column to improve the electron collection efficiency. A standard silicon detector for secondary electrons is a reverse-biased PN or Schottky junction. A problem with implementing this in a miniature SEM is the

218

A. D. FEINERMAN AND D. A. CREWE

shallow penetration of low-energy electrons into silicon, with 1000-, loo-, and 10-eV electrons penetrating up to 20, 0.5, and 0.02 nm into silicon, respectively. Our laboratory is investigating a surface PN junction (Fig. 25), which should be able to detect low-energy secondary electrons. The energy of the electrons can be increased with the electrode arrangement shown in Fig. 26 (Crewe, 1994). A microchannel plate is another secondary electron detector under consideration for an MSEM.The typical microchannel plate consists of an array of glass capillaries that are 400 pm long, with a 10-pm diameter, which would not fit in

FIGURE 24. SEM micrographs of the 15-j~m-highsilicon tips after operation at (a) 25°C. (b) 395"C, and (c) 935°C.

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FIGURE24. Continued.

FIGURE25. Surface PN junction detector. Top and bottom square contacts are for the vertical N+ fingers. Left and right contacts are to the P-type bulk. Low energy incident electrons on the detector surface will be collected by the N+ fingers. Guard rings have been omitted for visual clarity.

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I

FIGURE 26. Interdigitated fingers on the detector surface at +I000 and -1000 V can boost the incident electron energy from 0-10 eV to several hundred eV. A circularly symmetric interdigitated pattern like two spirals rotated 180" would be preferred since it would introduce less astigmatism. The N+ fingers shown in Fig. 25 have been omitted to simplify the figure.

the 0.5-mm working distance. Researchers at Cornell's Nanofabrication facility have fabricated a microchannel plate into a silicon chip and achieved a gain of 10 (Tasker, 1990).

VII. ELECTRON OPTICAL CALCULATIONS

The resolution of an ideal electrostatic column is determined by the diffraction, spherical, and chromatic aberration limited probe size: dd. d,,, and d,. These quantities are given by the following formulas where a is the final convergence angle, v b and A V are the beam voltage and spread, and C,and C, are the spherical and chromatic aberration coefficients:

d,, = O.25C,a3 (cm) AV d, = Cca(cm) vb

22 1

MINIATURE ELECTRON OPTICS

The method used to calculate the aberration coefficients of a stacked lens was reported previously and is also used to calculate the properties of a sliced column (Crewe et al., 1992). The ultimate resolution is frequently reported as the root mean square of dd, d,,, d,, and source size, but this has been shown to be incorrect (Born and Wolf, 1980;Crewe, 1987).A more accuratedeterminationis obtainedby graphing dd, dcs,and d, versus u,the final convergence angle. Two resolution plots for a stacked and a sliced lens are shown in Fig. 27. The resolution of the column with this method is the maximum value of dd when it crosses d,, d,, or the source size. In general, electrostatic Einzel lenses above 5 kV are spherical-aberration limited, while below 5 kV, chromatic aberration dominates their performance. The

1

;.- .... .

I 10 Final h g k of Convergence (mr)

20

(a)

Final Angle 01 Convargencs (mr)

(b) FIGURE 27. (a) A stacked lens 3.2 m m long at 2.5 kV is chromatic-aberration limited to a resolution of 3.8 nm at a working distance of 0.5 mm. (b) A sliced lens 8.5 mm long at 15 kV is spherical-aberration limited to a resolution of 2.2 nm at a working distance of 1 mm.

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most accurate determination of the resolution happens to be the intersection of the diffraction curve with the line 0.15C,a3 when the system is spherical-aberration limited (Crewe, 1987). A stacked column 3.2 mm long with a 0.5-mm working distance has spherical and chromatic aberration coefficients of 18.2 and 2.2 cm, respectively. The calculations for a stacked lens assume a column constructed with 381-pm wafers separated by 220-pm gaps with the two-layer deflector, and is shown in Fig. 7a. The structure would have -1-, -2.25, -2.25, and 0-kV potentials applied to electrodes D2 to D5, respectively, making the maximum electric field between electrodes 10 kV/mm. This column would be chromatic-aberration limited and should have a resolution of 3.8 nm when operated at 2.5 kV with a 4-mad convergence angle. The stacked electron optical calculations assume a point electron source at -2.5 kV and 220 p m below D2. As mentioned in Section IIIA,there is an advantage to etching the apertures separately and then assembling the electrodes rather than assembling the electrodes, then making the apertures.The three-electrodelens shown in Fig. 7b with all apertures 100 p m in diameter operating at a beam energy of 2.5 keV and a working distance of 0.5 mm will have an optimal resolution of 5.25 nm when the final angle of convergence is constricted to be 3.25 mrad. If the focusing electrode aperture (D3 in Fig. 7b) is 200 pm in diameter the same lens will have an optimal resolution of 3.8 nm at the same working distance and beam energy at a final angle of 4 mrad. The need to use apertures of different diameters in the column is also apparent when one considers the above restriction on the final angle of convergence being limited to something on the order of a few milliradians. To restrict the beam to such a small angle a beam-limiting aperture on the order of 10 p m or less must be placed somewhere in the optical column. If the apertures in the column are fabricated simultaneously, then all the apertures would have to be -10 pm, resulting in high axial field gradients near the apertures and large aberration coefficients. With the ability to vary the aperture diameters the final electrode could be used to limit the beam angle and the above resolution calculations would not be affected, since that electrode is held at ground potential. A sliced column 8.5 mm long with a 1-mm working distance has spherical and chromaticaberration coefficients of 30.98 and 2.57 cm. The inner and outer diameter of all electrodes are 300 and 500 p m (see Fig. 9 for other column dimensions). The electron optical calculations for the sliced column assume a point electron source at -15 kV with a 0.1-eV energy spread 1 mm below E2. The structure would require -5-kV, -13.7-kV, and 0-kV potentials applied to electrodes E2 to E4,respectively, to focus the beam at the 1-mm working distance. This column should have a resolution of 2.2 nm when operated at 15 kV with a 3-mad convergence angle. Since the column operates with approximately unity magnification, the source size becomes important only when it exceeds the minimum attainable resolution of 2.2 nm shown in Fig. 27. A 125-pm-diameter tungsten wire at 25°C oriented along the (1 11) or (310) directions whose tip has been electrochemically

-

MINIATURE ELECTRON OPTICS

223

FIGURE28. An MSEM constructed from 7 x 9 mmz silicon die that are 38 I p m thick with 269-pm gaps will have a working distance of over 6 mm when tilted at 60" with respect to the sample. In this figure the wafer is 30' from beam and 60" from horizontal.

etched to a 0.1-pm radius will have a 1- to 2-nm source size and is therefore acceptable. The sliced electrode fabrication method proposed in Section IIIC makes the electrode's surface highly resistive, on the order of 10" ahquare. If stray current strikes the electrode's surface the electrode's potential will no longer be constant. To simulate a fraction of the beam symmetrically striking the electrode's inside surface, a voltage perturbation was chosen that linearly increased from zero at the edge of the electrode to a maximum at its center. If a 100-V perturbation occurred on electrodes E2 and E4 (Fig. 9), the effect on C, and C, for the column was less than 1%. A 10- and 100-Vperturbation of electrode E3 had a negligible effect on C, and increased C, by 5 and 11%. respectively.

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A. A Tilted MSEM

The stacked design can be easily modified to create an array of high-resolution MSEMs that can be tilted 60" or more with respect to the sample. Very large tilt angles are required for defect review and inspection. As shown in Fig. 28 the working distance with a 2-kV MSEM is over 6 mm when the sample is tilted 60" and the smallest probe size will be several microns. This hypothetical stack would be constructed from 7 x 9 - m 2 silicon die that are 381-pm thick with 269-pm gaps, and the aperture radii in D2, D3, and D4 (Fig. 1) are 100, 100, and 5 pm, respectively. A linear array of MSEMs could perform high throughput inspection of wafers. As shown in Fig. 29 the working distance can be reduced substantially to 1.082 mm by using anisotropic etching to reduce the width of D4. A dicing saw would trim D3 so that it was 1 mm from the electron optical axis or 10 times the 100-km aperture radius. The smallest probe size with a 2-kV beam is -5 nm in this case, ignoring any astigmatism introduced by the stack. The working distance can be reduced further to 649 p m by reducing the aperture radius to 75 p m and using a dicing saw to trim D3 so that it came within 0.75 mm from the electron optical axis. The smallest probe size with a 2-kV beam in this case is -4 nm, again ignoring any astigmatism introduced by the stack. A 20-kV MSEM could be also created along these lines that could achieve a 1.5-nm probe at a working distance of 2 mm and a 60" sample tilt. The higher voltage would allow the MSEM to also perform chemical analysis on the particles detected.

\ FIGURE 29. Trimming the MSEM shown in Fig. 28 will allow the working distance to be reduced to 1.08 nim when the sample is titled 60" with respect to the horizontal.

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225

VIII. PERFORMANCE OF A STACKED EINZELLENS4 A. MSEM Construction

The focusing properties of a stacked electrostaticelectron lens have been evaluated within a macroscopic assembly shown in Fig. 30 (Crewe et al., 1996). The entire miniature SEM (MSEM)test structure is a cylinder 7.5 cm in diameter and 10 cm tall. This assembly consists of a 2.5-kV Einzel lens, electron source, parallel plate deflectors, and a Faraday cup as an electron detector. The test assembly positions the electron source over the silicon lens. The beam will be electrostaticallyscanned over the sample and an image can be formed from a current signal taken either from the sample itself or from the detectorbelow the sample. The apparatus can be easily modified to incorporate the other micromachinedcomponents (deflectorkigmator, detector, and electron source) in the column as they are developed. The electron source is a macroscopic zirconiated tungsten thermally assisted Schottky field emitter operating at 1800K. The thermally assisted Z r O z field emission source available from FEiI Inc. was chosen because it can provide highly stable field emission in a desirable current range (1-25 p A ) at readily achieved vacuum levels ( torr). The chief drawbacks to this source are its large physical size (a cylinder 2 cm in diameter and 2 cm tall) relative to the micromachined lens, the need for a mechanism to align the emitted electrons to the optical axis of the micromachined lens, and the relatively high extraction voltage required to achieve field emission (>3 keV). The design of the test structure was dictated by the need to secure, align, and electrically insulate the source from an extractorelectrode. As recommended by the manufacturer, the source is placed 500 p m from an extractor electrode containing a commercially available 500-pm-diameter Pt-Ir aperture. We chose to machine a bulky stainless steel extractor electrode fit with a commercially available aperture for the purpose of absorbing most of the emitted electrons from the source. The source and extractor are placed 1 cm before the silicon lens. The test assembly consists of alternatingstainless steel and Macor (a machinable glassxeramic) rings. From the bottom up the structure consists of a Faraday cup to collect electrons, a sample holder designed to house a commercial 3-mm gold grid, a parallel plate deflector assembly, which must electrically isolate the deflectors from each other as well as the elements above and below, the micromachined electrostatic lens, which is mounted to a 16-pin Airpax header, an extractor electrode, and the FEI source. The assembly is stacked one ring above another and is held together under compression in a mu-metal exterior can, which provides both the structural integrity of the assembly as well as magnetic shielding of the optical column (Fig. 30). Portions of Section VIII are reprinted, with permission, from the Journal of Vacuum Science and Techno1og.v A , 14(6), 3808-3812, November 1996. Copyright 1996 American Vacuum Society.

(b) FiGum 30. (a) A commercial thermal field-emission TFE source can be aligned to a micromachined Einzel lens with this experimental arrangement. There are two Macor push rods that move the TFE source. Each rod is driven by a linear motion feedthrough and works against a UHV spring. Below the Einzel lens are electron-beam deflectors, a 3-mm TEM grid, which serves as the sample, and a Faraday cup to detect the transmitted beam. The entire arrangement is surrounded by a mu-metal can to shield the electrons from the earth's magnetic field. (b) Thermally assisted field-emission source positioned over micromachined silicon lens demonstrating electron beam focusing to a point on the wire grid sample.

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227

The critical alignment necessary in the structure is the alignment of the lens electrode apertures to one another and the alignment of the electron source to the lens apertures. The electrode to electrode alignment is accomplished through our micromachining technique and the electron source alignment is accomplished via two insulated linear-motion feedthroughs, which push on the FBI source at 90" with a return spring. This allowed the vast majority of the pieces in the assembly to be machined to fairly low tolerances (tolerances were specified to -f50 pm), which kept the machining cost low. The entire assembly is inserted into a commercially available 6-inch UHV vacuum chamber containing a 30-Us nonevaporable getter pump that is mounted to a 120-U~ion pump. The motion feedthroughs are attached, electrical connections are made, and the system is evacuated. A base pressure of 1 x torr is achieved in 48 h. The silicon lens was fabricated from 380-pm-thick silicon chips separated with 250-pm gaps. The performance of a 3-element lens using these physical parameters has been calculated and the results are shown in Fig. 31. These calculations indicate that the lens can produce a high-quality focus from a position near the exit aperture of the lens to a working distance of up to a few centimeters with potentials on the focusing electrode(s) that are allowed by the die to die gaps. The extractor aperture is optically aligned to the silicon apertures in the micromachined lens by placing the assembly under a microscope, using bottom illumination to view the bright circular spot formed by the apertures in the silicon, centering the 500-pm extractor aperture over that spot, and securing the extractor in place. vpical operating potential differences between the source and the extractor electrode are in the range of 2.5-3.75 kV for an emission current of

Final Angle 01 Convergence (mr)

FIGURE 3 1. Solid and dash-dot lines represent 4 and 0.5 mm working distance. ( 1 ) Current MSEM operating point; expected resolution of 425 nm. (2) 4-mm working distance; expected resolution of 6.2 nm. (3) 0.5-mm working distance; expected resolution of 2.3 nm.

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1-25 p A . The FEI source also contains a suppresser electrode, which is biased negative with respect to the tip to prevent thermally generated electron emission from escaping the source. Initially our micromachined silicon apertures were only 3.5 p m thick, which was probably not thick enough to take the bombardment of -30 p A of 3-keV emission. However, we subsequently improved the silicon process to give 100-pm-thick apertures and will later remove the extractor from the assembly. With the stainless-steel electrode in the system, the first two silicon electrodes can be operated in parallel as one optically long focusing electrode. This has been calculated to produce a higher-quality probe as well as to provide more flexibility in operation (Feinerman et al., 1994b). Calculations indicate that a stacked lens with 150-pm-diameter apertures will produce a 425-nm focus with a 2.5-kV beam at a working distance of 4 mm and a field-emission source 1 cm above the lens (Fig. 31). If the final angle of convergence is reduced from 10 to 2.6 mrad, the focus improves to 6.2 nm. If the working distance is reduced to 0.5 mm, a 2.3-nm resolution can be achieved at a final angle of convergence equal to 6.5 mrad. The efficiency of the electron detector will have to be increased, however, since the probe current is inversely proportional to the square of the convergence angle. Images of a 200- and 1000-mesh gold TEM wire grid at a working distance of 4 mm have been obtained in transmission. The beam is scanned over the sample using parallel plate deflectors. The silicon lens is 1.64 mm long and consists of three silicon die separated by Pyrex optical fibers as shown in Fig. 2. Images of the grid at magnifications above 7000 x are now being obtained.

B. MSEM Operation and Image Formation The potentials applied to the source and lens electrodes and the filament heating current are supplied by a computer-controlled set of electronics (Fig. 32). Three high-voltage power supplies and a constant current supply are floated with their virtual ground at the beam potential. Isolation from earth ground is achieved via optical couplers. The suppressor and focus potentials, and the filament heating current are controlled through an RS232 serial connection to a personal computer. The beam potential is manually set on an externally regulated high-voltage power supply. After initial conditioningof the extractorelectrode to allow for electron-stimulated desorption of gas ions, the total emission is increased to -3 p A and the sourceto-silicon aperture alignment is performed. Once a beam is brought through the lens, the focus electrode potential is optimized by comparing successiveline scans over the gold grid. The optimal focusing potential agrees well with calculated values, differing by less than 10%.

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m

FIGURE 32. Flow chart of MSEM control and image acquisition system. The use of two PCs is redundant and will be reduced to one computer controlling both the high-voltage gun control unit as well as the scan-generatodimage-acquisition electronics.

The deflection potential signals and the image data are generated and received by data acquisition boards in a personal computer. The low-voltage deflection ramps are the input to a high-speed, high-voltage amplifier capable of generating -500- to +500-V signals at a rate of 10 kHz. The faces of the deflectors that are perpendicular to the electron beam measure 1.5 x 1.5 mm and are spaced 1.25 mm apart. A simple time-of-flight deflection calculation predicted a beam deflection of 0.5 pmN of applied deflection signal. Experimentally, we have observed that one volt of deflection potential yields approximately 0.4 p m of beam deflection. Qpical deflection signals are staircase ramps in the range -150 to +150 V (for a field of view 120by 120 pm) generated at a line rate of 10 Hz.Image acquisition time for a 512 x 512-pixel image is then 51.2 s. The image data consist of the Faraday cup current (for a dark-field image) or the sample current (for a brightfield image) that has been put through a current-to-voltage amplifier with a gain of approximately 10" and a maximum pixel rate of 100 kHz. This 0- to 1-V signal is the input to a 12-bit analog-to-digital converter that acquires the image data synchronously with the deflection ramp generation. The raw image data are then normalized and imported into a commercially availableimage processing software package for viewing. The initial and final voltages of the X and Y deflection ramps can be software selected, and the magnitude of the amplified deflection signal can

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FIGURE33. This imageobtained with test apparatusdemonstrates theability of the micromachined silicon electron lens to focus on a 1000-mesh gold TEM grid. The grid wires are 6 p m wide and are spaced 19 pm apart, and the signal is from the Faraday cup current.

FIGURE34. Magnification is -2000 of 6-pm grid, and the signal is from the Faraday cup current. Defect in center of image is from screen saver turning on.

be varied, allowing the user to perform a dc offset high-magnification scan of a region of interest that is not in the center of a low-magnification image. Low- and high-magnificationimages of a 1000-mesh gold wire grid are shown in Figs. 33-36. The 10-90% rise time of the line scan shown in Fig. 37 covers

MINIATURE ELECTRON OPTICS

23 1

FIGURE 35. Magnification -3500 of 6-pm grid, and the signal is from the Faraday cup current.

FIGURE 36. High-magnihcation image ot aerecton wire p a . image has beenelectronically rotated to bring wire to a nearly vertical position. The cross wire is not at a right angle to the vertical wire, possibly due to a deformation of the sample when fit into the test assembly. The defect is approximately 0.5 p m wide.

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Scan Distance (pm)

FIGURE37. Line scan data from a high-magnificationimage of one period of the 1000-mesh grid. The scan signals were electrically rotated so that the beam was deflected perpendicular to the wires. The 1&90% rise time of 2.1 p m corresponds to a Gaussian probe sigma of 0.75 g m . In its present configuration the MSEM is spherical-aberrationlimited, so the Gaussian probe is a good approximation to the actual beam.

a lateral distance of 2.1 pm. This indicates that if the probe is Gaussian, it has a sigma of 0.75 pm. This is a worst-case estimation of the beam probe size, since the grid wires in reality have a finite slope, but does give a value that agrees well with calculations.

Ix. SUMMARY AND FUTURE PROSPECTS Microfabrication techniques have advanced to the point where conductors, semiconductors, and insulators can be positioned in complex three-dimensional arrangements with very high precision. This is equivalent to a conventional machinist operating miniature milling machines and lathes with micron-sized bits. This flexible machining capability allows electric and magnetic fields to be created that can accelerate, focus, steer, andor align charged particles, because the fields occupy a volume of space rather than simply existing next to a surface. Specific fabrication techniques developed at UIC include stacking silicon chips with Pyrex fibers, selective anodic bonding (slicing), and a LIGA lathe. These techniques are being used to integrate charged-particlesources, electrodes, and detectors into various miniature instruments including a subcentimeter scanning electron microscope, a 10-cm time-of-flight mass spectrometer, a 10-cm nuclear magnetic resonance instrument, and a 5-m linear accelerator/undulatorcapable of producing hard X-rays. Analytical instrumentsof this size will allow the analytical laboratory to be brought to the sample, which will be essential when the sample must be observed in situ, e.g., at a toxic waste site or in outer space.

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REFERENCES Adler, E., DeBrosse, J. K., Geissler, S. F., Holmes, S. J., Jaffe, M. D., Johnson, J. B., Koburger, C. W., 111, Lasky, J. B., Lloyd, B., Miles, G. L.. Nakos, J. S., Noble, W. P., Jr., Voldman, S. H., Armacost, M., and Ferguson, R. (1994). The evolution of IBM CMOS DRAM technology. IBM J. Res. Dev. 39( U2). Ahn, C. H., and Allen, M. G. (1993). A planar micromachined spiral inductor for integrated magnetic microactuator applications. J. Micmmech. Micmeng. 3(2), 3 7 4 . Ahn, C. H., and Allen, M. G . (1994). A new toroidal-meander type integrated inductor with a multilevel meander magnetic core. IEEE Trans. Magn. 30,73-79. Bean, K. E. (1978). IEEE Trans. Electron Devices ED-25, I 185. Born, M., and Wolf, E. (1980). In Principles ofOptics, 6th ed. Pergamon, Oxford, UK, p. 206. Brodie, I., and Spindt, C. A. (1992). Vacuum microelectronics. In Advances in Electronics and Electron Physics (P. Hawkes and B. Kazan, Eds.), Vol. 83. Busta, H. H., Feinerman, A. D., Ketterson, J. B., and Wong, G. K. (1985). J. AppL Phys. 58, 987989. Carlson, D. E. (1974). J. Amer: Cerum. SOC.57,291. Carlson. D. E., Hang, K.W., and Stockdale, G. F. (1974). J. Am. Ceram. SOC.57,295. Cenina, F., Turner, B. S., and Khan, M. (1993). Microelectron. Engin. 21, 103-106. Chang, T. H. P., Kern, D. P., and Muray, L. P. (1990). Microminiaturization of electron optical systems. J. Vuc. Sci. Technol. B 8, 1698-1705. Crewe, A. V. (1987). Ulframicroscopy23, 159-168. Crewe, A. V. (1994). Private communication. Crewe, D. A., Perng, D. C., Shoaf, S . E., and Feinerman, A. D. (1992). A micromachined electrostatic electron source. J. Vac. Sci. Technol. B 10,27542758, Crewe. D. A., Ruffin, M. M., and Feinerman, A. D. (1996). Initial tests of a micromachined SEM. J. Vac. Sci. & Technol. B, 14(6),3808-3812. Feinennan. A. D., Shoaf, S . E., and Crewe, D. A. (1991). Precision aligning and bonding of silicon die. In Patterning Science and Technology II/lnrerconnection and Contact Metallization for ULSI, PV92-6 (W. Greene, G. J. Hefferon, L. K. White, T. L. Herndon, and A. L. Wu, Eds.). Electrochemical Society Proceedings. Feinerman, A. D., Crewe, D. A,, Perng, D. C., Shoaf, S. E., andCrewe, A. V. (1992a). Sub-centimeter micromachined electron microscope. J. Vac. Sci. Technol.A 10, 61 1-616. Feinerman, A. D., Crewe, D. A., Perng, D. C., Shoaf, S. E., and Crewe, A. V. (1992b). In SPIEImuging Technologies and Applications 1778.78. Feinerman, A. D., Crewe, D. A,, and Crewe, A. V. (1994a). Microfabrication of arrays of scanning electron microscopes. J. Vac. Sci. Technol. B 12.3 182-3 186. Feinerman, A. D.. Crewe, D. A,, Perng, D. C., Spindt, C. A,, Schwoebel, P. R., and Crewe, A. V. ( I994b). Miniature electron microscopes for lithography. In SPIE-Microlithography '94 2194, 262-273. Feinerman. A. D.. Lajos, R., White, V., and Denton, D. (19%). X-ray lathe: an X-ray lithographic exposure tool for nonplanar objects. J. Microelectromechanical System, 5(4), 250-255. Feller, B. (1990). SPIE 1243, 149-161. Fleming, D., Maldonado, J. R., and Neisser, M. (1992). J. Vac. Sci. Technol. B 10, 25 I 1. Gowar, J. (1984). In Optical Communication Systems, Prentice Hall International, London, p. 99. Guckel, H., Christenson, T. R.. Skrobis, K. J., Denton, D. D., Choi, B., Lovell, E. G., Lee, J. W., Bajikar, S . S., and Chapman, T. W. (1990). Deep X-ray and uv lithographies for micromechanics. In Proceeding3 of IEEE Solid State Sensor and Actuator Workshop, Hilton Head, South Carolina, 4-7 June, pp. I 18-1 22.

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Huggett, J. M. (1990). In Advances in Electronics and Electron Physics, Vol. 77 (P. W. Hawkes, Ed.). Academic, New York, p. 139. Jansen, K., and Ulrich, R. (1991). J. Lighmuve Technol. 9 . 2 4 . Jones, G . W., Jones, S. K., Walters, M. D., and Dudley, B. W. (1989). IEEE Trans. Electron Devices 36, 2686. Laprade, B. (1989). SPIE 1072, 102-1 10. Larrabee, G., and Chattejee, P. (1991). DRAM Manufacturing in the 90s. Part 1 : The History Lesson, Semiconductor International, p. 84. Lin, B. J. ( I 99 I ). Quarter- and sub-quarter-micronoptical lithography.In Patterning Science and Technology II/Interronnecrion and Contact Metallization f o r U U I , PV92-6 (W. Greene, G. J. Hefferon, L. K. White. T. L. Herndon, and A. L. Wu, Eds.). Electrochemical Society Proceedings, pp. 3-15. Marrian, C. R. K., Dobisz, E. A., and Dagata, J. A. (1992). J. Vuc. Sci. Technol. B 10,2877. Mentzer, M. A. (1 990). In Principles of Optical Cimuit Engineering, Appendix IV. Dekker, New York. pp. 301-307. Mulvey, T. (1982). Unconventional lens design. In Magnetic Electron Lenses (P. W. Hawkes, Ed.), pp. 359-4 12. Muray. L. P., Staufer, U., Bassous, E., Kern, D. P., and Chang, T. H. P. (1991). J. Vuc. Sci. Technol. B 9,2955. Petersen, K. E., (1982). Proc. IEEE 70,422. Ravi, T. S., and Marcus, R. B. (199 I ). Oxidation sharpening of silicon tips. J. Vac. Sci. Technol. B 9. 2733-2737. Schwoebel, P. R., and Spindt, C. A. (1993). Glow discharge processing to enhance field emitter array performance. Appl. Phys. Lett. 63.33. Sematech, 1994. SIA National Technology R o a d m p f o r Semiconductors. Semiconductor lndustry Association, 181 Metro Drive, Suite 450, San Jose, California 95 1 10, http://www.sematech.org/ public/roadmap/doc/toc.html. Shedd, G. M., Schmid, H., Unger, P., and Fink, H.-W. (1993). Rev. Sci. Instrum. 64, 2579. Somojai, G. A. (1981). Chemistry in Two Dimensions: Surfaces. Cornell University Press, Ithaca, NY. Spindt, C. A. (1968). A thin-film field emission cathode. J. Appl. Phys. 39,3504-3505. Spindt, C. A., Holland, C. E., Rosengreen, A., and Brodie, I. (1991). Field emitter arrays for vacuum microelectronics. IEEE Trans. on Elect. Devices 38, 2355-2363. Tasker, G. W. ( I 990). SPIE 2640.58. Trimmer, S . N., and Gabriel, K. J. (1987). Sensors andActuators 11, 189. Wallis, G., and Pomerantz, D. I. (1969). J. Appl. Phys. 40,3946. Yamazaki, T., Miyata, N.. Aoyama, T., and Ito, T.(1992). Investigation of thermal removal of native oxide from Si(100) surfaces in hydrogen for low-temperature Si CVD epitaxy. J. Electrochem Soc. 139, 1175-1 180.

ADVANCES IN IMAGING AND ELECTRON PHYSICS. VOL.102

Optical Interconnection Networks KHAN M . IFTEKHARUDDIN Emken Research. 1835 Dueber Ave. S.U!. Canton. Ohio 44706

MOHAMMAD A . KARIM Department of Electrical and Computer Engineering. University of Dayton. Dayton. Ohio 45469.0226. USA

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Optical Interconnect Types . . . . . . . . . . . . . . . . . . . . . A . Free-Space Optical Interconnects . . . . . . . . . . . . . . . . . . B . Reconfigurable Optical Interconnects . . . . . . . . . . . . . . . . C . 3D Optical Interconnects . . . . . . . . . . . . . . . . . . . . . D . Guided-Wave Optical Interconnects . . . . . . . . . . . . . . . . . I11. Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . A . Free-Space Optical Interconnection Architectures . . . . . . . . . . . . B . Photorefractive Volume Holographic Interconnection Architectures . . . . . . C . 3D Interconnection Architectures . . . . . . . . . . . . . . . . . . D. Guided-Wave Interconnection Architectures . . . . . . . . . . . . . . IV. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . A . Free-Space Optical Interconnection Applications . . . . . . . . . . . . B . Reconfigurable Optical Interconnection Applications in Neural Networks . . . . C . 3D Free-Space Interconnection Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Guided-Wave Interconnection Applications V. Packaging of Optical Interconnects . . . . . . . . . . . . . . . . . . . VI . Problems and Possibilities . . . . . . . . . . . . . . . . . . . . . . A. Free-Space Interconnection Networks . . . . . . . . . . . . . . . . B . Photorefractive Holographic Reconfigurable Neural Interconnection Networks . . C. 3D Interconnection Networks . . . . . . . . . . . . . . . . . . . D. Guided-Wave Interconnection Networks . . . . . . . . . . . . . . . VII . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I . INTRODUCTION Ongoing advances in high-speed integrated circuit device technology along with increasing demand in data communication. digital signal and image processing. neural network. and machine vision systems necessitate the design of efficient interconnections topology. In electrical interconnections. in general. the scope of 235

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the systems is limited by interference and the planar nature of very large-scale integrated (VLSI) circuits (Kostuk et al., 1987). The chip speed, in particular, is limited by interconnect delays. Intrachip connections pose further difficulty. Optical interconnects, because of their inherent parallelism, high speed, and negligible mutual interference, offer a reasonable solution to this intricate problem (Goodman et al., 1984). Optical techniques may provide an alternative means for fast, secure interconnectionsof the devices directly to the interior of a chip. Recent optical computing research involving optical bistable devices (Gibbs, 1985), nonlinear Fabry-Perots (Jewel1et al., 1985), hybrid electrooptic devices (Lentine et al., 1990), and optical interconnects (Guha et al., 1990) has contributed to two distinct, parallel interconnect architectures. One approach uses integrated optics to interconnect optical logic devices, while the other involves 2D arrays of devices interconnected in free-space (Murdocca et al., 1988). Free-space interconnection, in turn, is classified as either focused (obtained by lens, beamsplitter, etc.) or unfocused (obtained by imaging light through holographic elements). The indexguided waveguide interconnects may be either space variant or space invariant. A number of studies have already used free-space space-invariant optical interconnection processors for realizing logic devices and functions (Caulfield et al., 1983; Cloonan and McCormick, 1991; Iftekharuddin and Karim, 1994; Sawchuck et al., 1987; Stirk et al., 1988). The free-space interconnection architectural approach is preferred because of its massive parallelism (Giglmayr, 1989).

11. OPTICAL INTERCONNECT R P E S Optical signal could be transmitted by photons in free-space rather than by electrons in guided media. Therefore, it is much easier to design a parallel architecture using optics than using electronics. Electronic parallel computers are already available, but they have certain drawbacks, The digital parallel processors can communicate directly only with those algorithmsthat are locally available(Goodmanet al., 1984; Rastani and Hubbard, 1992). Further, the VLSI-based electronics systems are inherently two dimensional in nature. Even in three-dimensionalVLSI technology, cross-talk and interconnect delay limitations often impose restrictions to nearestneighbor interconnections. Optical interconnections are able to input the entire two-dimensional data array in parallel using the third dimension, particularly in data propagation. Also, in case of an electronic parallel processor, the data can be input and output only along the edges of the two-dimensional array, one row or a single column at a time (Kostuk et al., 1985). In addition to space constraints, speed limitation plays a significant role in conventional digital computers. The velocity of signal propagation depends on the capacitanceper unit length. Thus, as more and more devices having capacitive

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components of certain admittance are introduced to an interconnection,the velocity of propagation decreases, which limits the overall speed of the device. On the other hand, optical signals, whether confined to waveguidesor free-space, can propagate at the speed of light in the medium independently of the number of componentsthat receive those signals. Furthermore, the stray capacitances that exist between the proximate electrical paths introduce cross-coupling of information to a degree that increases with the bandwidth of the signals of interest. Optical interconnectionsdo not suffer from such effects. Optical interconnectsprovide an extremelyhigh speed, causing fast switching of computing devices, which facilitates high bandwidth digital signal transmission. Possible ways to achieve intra- and interchipoptical interconnections(Goodman et al., 1984) include the following: 1. Index Guided, whereby interconnection mainly occurs by means of waveguides. The waveguides could either be provided by means of optical fiber or be integrated on a suitable substrate. 2. Free Space, whereby the propagation property of light in free space is used for necessary signal transfer. There are two main types of free-space interconnections: focused and unfocused. The focused interconnection is established by broadcasting the optical signals to the desired detector location by lens and/or beamsplitters. This may be either space variant or space invariant type. Some of the examples of architectures of space-invariant optical interconnections are perfect shuffle (Lohmann et al., 1986; Brenner and Huang, 1988), clos (Lin et al., 1988), and butterfly (Iftekharuddin and Karim, 1994). The unfocused interconnection, as the name suggests, is obtained by unfocused broadcasting of optical signal. A. Free-Space Optical Interconnects Among the various free-space interconnection networks currently being considered, the space-invariant regular interconnect method that uses beamsplitters has particular advantages over the space-variant hologram-based interconnects in terms of simplicity, extensibility, and high throughput (Kostuk et al., 1985). Already, considerableeffort has been carried out to realize logic devices and functions using different space-invariant interconnection networks such as crossover (Jahns and Murdocca, 1988), perfect shuffle (Lohmann et al.. 1986; Eichmann and Li, 1987), clos (Lin et al., 1988),banyan, and butterfly (Iftekharuddin and Karim, 1994).Although, the free-space interconnectionsnetworks are equally global (i.e.. these networks exploit the global nature of optics fully), crossbar is specifically suitable for the multiprocessor systems while perfect shuffle is preferable for the programmable logic arrays. The perfect-shuffle interconnection, in particular, may be used to realize complex algorithms, including those involving the fast

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Fourier transforms (Kostuk et al., 1985). The optical implementation of crossbar interconnection is prohibitively expensive, while that of perfect shuffle requires magnification of the input arrays by a factor of 2, which poses a rather nontrivial challenge for the diffraction-limited devices (Jahns and Murdocca, 1988). Further, the crossover is not used with optical devices primarily because of the limited fanout restriction of these devices. Interestingly, these problems can be overcome by using a butterfly interconnection network (Iftekharuddin and Karim, 1994). This latter interconnection may also be used to realize various logical functions, operations, and algorithms that were previously obtained using perfect-shuffle interconnection. The banyan, crossover, perfect shuffle, and butterfly are isomorphicinterconnections and are illustrated in Fig. 1. The banyan, butterfly, and crossover are spaceinvariant interconnections. The fan-out angle for each channel is constant over the array for these space-invariant interconnections. This type of interconnection is I n p u t variables

I n p u t variables

(b) Perfect Shuffle

(a) Banyan

O u t p u t variables

O u t p u t variables

I n p u t variables

( c ) Butterfly/Crossover

O u t p u t variables

FIGURE1. Optical interconnection topologies.

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readily implemented, with computer-generated holograms providing the angular control in a Fourier plane. An example of a space-variant interconnect is the perfect shuffle. In Fig. 1, the channels comprising the input array must all be routed at angles that are determined by their position in the array. The number and type of stages of a particular interconnect that are required to implement a multistage interconnection network (MIN) are also shown in Fig. 1. The perfect shuffle uses the same pattern for each stage (hence, it is stage invariant), which allows the same hardware to be used to recirculate the data for each stage of the routing. This property of the perfect shuffle minimizes both hardware and cost and simplifies the design and construction of systems. The space-invariantinterconnections (banyan, butterfly, and crossover)require a different interconnection pattern for each stage (i.e., they are stage variant). Some systems have been designed using these interconnections,with each stage being performed by a separate piece of hardware. This has the advantage of providing a pipeline architecture. However, this approach may become prohibitively expensive for large arrays. An alternative approach is to use arecirculating arrangementusing holographic interconnects. Reconfigurable interconnects provide a solution that is economical in both cost and complexity. Dynamically variable holograms can provide the interconnection for each stage of the MIN with only a single piece of hardware. At each stage of the MIN,the hologram reconfigures to provide the required routing pattern. Experimental demosntrationshave already been carried out to show a partial implementation of such a system using a liquid-crystaldisplay panel as the reconfigurable hologram (Prince et al., 1996). However, a drawback of using a liquid-crystal display panel as the reconfigurable interconnect is that the processing power of the system is normally determinedby the slowest component. In this case, the liquid-crystaldisplay panel runs at video rates, whereas the smart pixels can run at hundreds of megahertz. Clearly, for a system that would be competitive with existing electronics, the reconiigurableinterconnect must run at a speed similar to the smart pixels (Prince et al., 1996). Among the isomorphic interconnection networks, butterfly is a class of the log, N family of interconnection netwrok, where N is the number of inputs. This network connects N input ports of a switching system with the same number of output ports using a minimum number of interconnections. A butterfly on a string of length N(= 2') is described as an exchange of the least significant bit (LSB) and the most significant bit (MSB) (i.e., bit 0 and bit (r - 1)) in the binary address y of each element by of the string. This is shown in Fig. l c for the one-dimensional case. Thus, butterfly interconnection comprises three angles of connections such as a copy operation (for all elements of the string having a binary address with LSB = MSB), a shift to the right by N / 2 - 1 bits (for all elements of the string having a binary address with LSB = 1 and MSB = 0), and a shift to the left by N / 2 - 1 bits for rest of the elements (Murdocca et al., 1988). Note that in the case when all signals are allowed to shift right and/or left, respectively, the network

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results in a perfect shuffle. The optical implementation of the butterflies may be very similar to that of the banyan network (Murdocca, 1990) of Fig. la, since the latter is obtained by arranging butterflies in increasing granularity of log, N levels. Obtaining Boolean functions using any of the above free-space interconnection networks involves generation and a subsequent combination of possible unminimized minterms (Guilfoyle, 1984). For example, all 2' unminimized minterms of a function of r variables can be generated and then combined within a maximum depth of 2(r 1) levels. However, following the approach of Murdocca (1990), the fan-out (i.e., the last) level of the term generation stage may be actually combined with the fan-in (i.e., the first) level of the next stage. This reduces the overall gate count of the system. There exist two different optical hardware implementation schemes for free-space interconnection: prismatic mirror array implementation and self-electroopticeffect device implementation.

+

1. Prismatic Mirror Array Implementation

The optical hardware implementation of a free-space interconnection network (Cloonan, 1989)can easily be adapted for any particular interconnection topology. The implementation hardware details of a free-space interconnection network is shown in Fig. 2. The top section contains the hardware necessary for implementing free-space connections, while the bottom part combines the clock signal with the data signal coming out of the interconnection network. For the case of a butterfly implementation, the input (which is a circularly polarized image) is split into two images by the right-polarizing beamsplitter. One of these images is reflected back by a plane mirror, thus providing a no-shift connection of a 2D butterfly interconnection. The other copy is reflected by the prismatic mirror array, thus providing shifted (shift-left and shift-right) connections. The no-shift and shifted outputs of the interconnection network are combined and routed to the output patterned mirror reflector via a left-polarizingbeamsplitter. The reflected light from the patterned mirror is directed to the logic devices by the bottom beamsplitter. The clock signal is introduced onto the device arrays to read out the result of the necessary logic operation, which is then reflected onto the next interconnection stage by the combination of a beamsplitter and output patterned mirror reflector. The quarter-wave plate guarantees minimum optical loss for the system. Finally, the no-shift and shifted outputs of the operating cells are effectively passed through different masks (each corresponding to a specific shift operation) to mask out the unwanted signals. 2. Self-Electmoptic E$ect Device Implementation

The self-electrooptic effect device (SEED) technology takes advantage of the physics of multiple-quantum-well(MQW) modulators (Cloonan, 1989). The thin

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FIGURE2. The optical hardware details of prismatic implementation of interconnection network. (After Cloonan and McCormick (1991).)

and alternating layers of narrow- and wide-bandgap materials, such as GaAs and AlGaAs, produce the MQWs. The quantum wells confine carriers, absorb spectrum, and generate distinct excitationpeaks. The position of these peaks shifts as a result of electric field applied at a direction perpendicular to the plane of quantum wells. The electroabsorption phenomenon is known as quantum-confined Stark effect (Kostuk et al., 1985; Lohmann et al., 1986). A 5-Vchange in a 1-pm-thick MQW, for example, may inhibit changes in absorption coefficient of a factor of about two. This change in voltage can modulate light when the MQW material is placed in the intrinsic region of a reverse-biased p-i-n diode. The SEED may also be used for light-detection purposes. Some of the different variations of SEED technology include resistor-SEED, the symmetric-SEED(S-SEED), the logic-SEED (L-SEED), and field-effect-transistor-SEED (FET-SEED). The S-SEED consists of two electrically connected MQW p-i-n diodes as shown in Fig. 3a. The inputs to the S-SEED include set (s), reset (r), and clock (c) signals. These complementray s and r inputs are separated in time from the clock signal c. The appropriate operation of the device

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__ A

R2

C

diode

S State transition path f o r diode 1

V

R1

C

r

I

(a)

I

,Cr

(b)

FIGURE3. (a) S-SEED and (b) its power transfer curve. (After Hinton et al. (1994).)

depends on the state of the s and r inputs. For s > r , the lower MQW p-i-n diode is transmissive, while the upper diode is forced to be absorptive, causing the SSEED to attain a desirable state. Similarly, an opposite state prevails for r > s. In the absence of clock signal, low switching intensities for s and r inputs are enough to change the state of the device. Once the high-intensity clock signal is applied, the state of the device is changed to another stage. To ensure the highest differential gain, the ratio between the two clock beam powers is normally set to Unity.

The usual operation of an S-SEED follows the power-transfer curve as shown in Fig. 3b. Figure 3b shows optical reflected power (P,)in presence of the clock signal plotted against the total power impinging on the set-reset windows in the absence of clock signal. The output power is proportional to the reflectivity (Ri) when both the clock signals are incident. In Fig. 3b, Cr is the ratio of incident powers Ps and Pr, and Pp is the reflectivity of the diode. The absorption T is defined as T = (1 - R 1)/( 1 - R2). The respective power-transition points of the diodes are shown in Fig. 3b as well. The S-SEED-based modules, logic gates, and interconnection architecture have been widely used in the last few years. We discuss some specific SEED applications in the subsequent sections. The FET-SEEDS are the hybrid of optical communication and integrated electronics processing technology. The outcome of this effective integration of digital integrated circuit and optical detectors and modulators is the smart-pixel arrays (Guilfoyle, 1984) that allow for more connection-intensiveprocessing. The main components of a FET-SEED are doped-channel FEiTs, MQW modulators, and p-i-n MQW detectors integrated on a single common-layer substrate. The common layer provides the high-performance buffered-FET logic electronic circuits with optical inputs and outputs.

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B. Reconjgurable Optical Interconnects An attractive means for exploiting opitcal beam noninteraction property is to use free-space propagation but with a focused, i.e., holographic, technique (Goodman et al., 1984). Recent developments in the field of digital signal processing, data communications, neural networks, and image processing demand fast changing of interconnection patterns (dynamic configuration) as the processing proceeds. In electronics,the reconfiguration could be in MINs and in modifying the microcode of writable control store (WCS) for processors that use microcode. On the other hand, the parallel access capability for some optical technologies allows an entire interconnection pattern to be changed at a prespecified bit rate. This could be performed by sending signals to several channels simultaneouslyand then masking out the unwanted channels. This kind of reconfiguration may work best when the circuits are small. A powerful way to achieve this is to redirect channels by steering beams of light. But beam steering introduces losses or reconfigures at a speed much slower than bit rate (Hill and Smith, 1989). In comparison, the holographic interconnection holds much promise. Multiplexed volume holographic grating in different materials have been suggested as means of interconnecting optical information (Sawchuk and Jenkins, 1986). Most opitcal holographic interconnect architectures have been based on photorefractive materials. For these interconnection architectures, dynamic photorefractive holograms are used to perform routing of optical beams. With spatial light modulators (SLMs), routing configurations can be changed in fixed photorefractive holograms, which, guarantees a dynamic reconfiguration of the interconnects. The concept of reconfiguration in optical interconnects primarily emerged to overcome high intrinsic energy loss for fan out (Sawchuk and Jenkins, 1986; Yeh et al., 1986). For arrays of N lasers and N detectors, N 2 parallel interconnects may be obtained, leading to a fan out efficiency of 1/N. For M = N*N matrix (consisting of N lasers and N detectors), the interconnection is given by (Yeh et al., 1986) V’=MV

(1)

where V is the input vector representing the signals carried by the laser array, and V’ is the output vector representing the signals received by the detector array. Some space-invariant interconnections such as banyan and crossover require a different interconnection for each stage. The conventional means to design these stage-variant interconnections, and hence the processor, is to use a different piece of hardware for each stage. This approach, however, becomes expensive for a large array of interconnections. Alternatively, dynamically variable

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holograms may provide interconnection for each stage of the MIN processor with a single piece of hardware. We describe two different approaches for realizing the reconfigurable interconnections: real-time holographic implementation and reconfigurable liquid-crystal implementation. 1, Real-Time Holographic Implementation

The free-space unfocused imaging interconnection is realized by imaging light with the help of an optical element onto the detector in parallel. The required optical elements is often implemented using holograms, which acts as a complex grating and lens to generate focused grating components at the desired locations. The efficiency of such a scheme depends on the holographic optical element used as a recording medium. This particular method is flexible enough to realize any arbitrary connection configuration of dynamic nature. A real-time reconfiguration in holographic interconnection is usually achieved using photorefractive two-wave mixing combined with SLMs. Photorefractive materials act as dynamic holographic media, which provide a new hologram for each new interconnection pattern. As a result, such holograms contain the interconnection as prescribed by the SLM. After the hologram is prepared, the pump beam may be diffracted off the hologram and redirected into the array of detectors. A real-time holographic system controlled by SLM is shown in Fig. 4.This setup provides the exact interconnection pattern based on matrix vector multiplication of Eq. (1) (Yeh et al., 1986). In Fig. 4,a dynamically updated optical interconnectionis shown, which utilizes nonreciprocal energy transfer in photorefractive two-wave mixing. The combination of the beam facilitatesthe recording of the hologram inside the photorefractive crystal. A fraction of the incident laser beam Pi is coupled out of the beam by using a beamsplitter. This fraction of beam, called signal beam P, is then expanded

FIGURE4. A dynamically updated optical interconnection. (After Yeh et al. (1986)J

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by using a cylindrical lens and passed through SLM.The signal beam P, is recombined with the pump beam Ppinside a photorefractive crystal. As can be seen, the nonreciprocal energy coupling causes a large amount of energy to be transferred from pi to P,.which contains the interconnection pattern (Yeh et al., 1986). However, maximum energy efficiency is guaranteed by complete overlap of P, and Pp beams (Chiou and Yeh, 1990). Because of different interconnection patterns, P, and Ppmay have different beamlets, yet individual pixel shape may be indentical. The complete overlap of P, and P,,is achieved in the Fourier domain where all the pixels are indentical (Goodman et al., 1978). The Fourier transform of a beamlet of Ppand corresponding column of P, obtained by a lens, as shown in Fig. 5 . The resulting amplitude distributions at the focal plane of the lens are (Chiou and Yeh, 1986)

where s ( x , y) is the aperture of an individual pixel and o(u, u ) is its Fourier transform, ( u , u ) are the coordinates in the Fourier plane, and O(u, u ) is the corresponding phase. The shift invariance property of Fourier transform causes each P, beamlet to overlap completely with a Pp beam at the Fourier plane, thus yielding the maximum energy efficiency of the system. 2. Reconjigurable Liquid-CrystalImplementation

The pressing needs for a reconfigurable processor may be satisfied in part by dynamically interconnectingthe smart-pixel devices.A successfulimplementationof

FIGURE5. A Fourier transform lens.

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FIGURE6. System details of the dynamic interconnection between the two S-SEEDS. (After Prince et al. (1996).)

such a dynamic interconnection between two S-SEEDShas been reported (Prince et al., 1996). The reconfigurable interconnection is achieved using a hybrid microlens and a phase-only nematic liquid-crystal array. The typical fan-in and fan-out ratio is 2 x 1 to 8 x 8. The two polarizers on both sides of a commercially available amplitude-modulated liquid-crystal display (LCD) panel are removed to obtain phase-only operation (Prince et al., 1996). Phase modulation, in general, is more effective in displaying the gratings than its amplitude counterpart. Arbitrary displaying of the gratings on the LCD is achieved by interfacing a PC to the LCD. The hybrid lens, which images the S-SEED array, is designed to provide high resolution over a large field of view (Prince et al., 1996). The system details are provided in Fig. 6. C. 3 0 Optical Interconnects The demands in data communications for both increased clock rates and high parallelism and in telecommunications for improved bit rate and complexity in asynchronous transfer mode (ATM) switches have led to high-density optical interconnects. In addition, the concept of distributed computing and shared memory is pushing the limits of conventional electronic communication. Electronic communication often involves low bandwidth, high crosstalk, signal mismatch and reflection, high loss, and increased packing density of transmission channels.

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Future communication systems will require growingnumber of nodes on any given network and increasing bandwidth due to demand in video signal transmission. A network comprises two types of devices-switching elements and nodesinterconnected by communication links. Nodes are nonswitching devices such as processors, VO nodes, and gateways. Switching elements (also called routers in similar pipelined networks) contain multiple input and output ports, and provide means of passing data arriving at an input port to an appropriate output port. MINs, as the name suggests, comprise multiple stages of such networks. MINs are widely used for parallel processing and distributed computing. There are some networks that are bidirectional MINs. In a bidirectional MIN each communication link comprises two channels, which cany data in opposite directions. MINs are capable of scaling bisection bandwidth linearly with the number of nodes while maintaining of fixed number of communication ports per switching element. MINs with hundreds or thousands of nodes are densely connected and require some long communication links to support large numbers of nodes. Thus, the channels employ a protocol that maintains bandwidth over long links. Because the 3D MIN contains multiple stages, the hardware for both shuffling and switching in one stage may be duplicated log, N times to make the network full access, where N is the number of channels in the square array. The channels may be spatially arranged such that a single set of optics are used. The resulting system requires a feedback system and is an example of a space multiplexed optical MIN. It is evident that 3D optical interconnects may be used to input the entire 2D data in parallel, whereas its third dimensionmay be used for data propagation. Recent efforts have identified the implementation (Cloonan and McCormick, 1991) and classification (Giglmayr, 1989) of 3D interconnects. The advantages of using 3D interconnection architectures instead of its 2D and 1D counterparts have been explored. In particular, Iftekharuddin et al. (1994) compare two topologies, namely, folded perfect shuffle (Stirk et al., 1988) and butterfly (Iftekharuddin and Karim, 1994), as examples of 1D and 3D interconnection architectures,respectively. The 3D optical architecture is usually derived from a 2D (A4 x N) network that is replicated L times. The replication creates L parallel (i.e., L bit deep) M x N networks. Figure 7 shows an example of a 3D banyan interconnection network (Hinton et al., 1994). The optical details of the node connections are similar to those shown in Fig. 2, wherein the output of each node is directed to the pupil plane. The output signal at pupil plane is then split into three equal parts by the binary phase grating. These original copies of the signal are directed to the next stage for subsequent connections. One necessary condition for the independence of the split signals is that the splitting angle needs to be different at each stage. A simple technique for signal splitting is described in a subsequent section.

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FIGURE 8. Possible implementation of a guided-wave interconnect. (After Goodman et al. (1984).)

D. Guided-Wave Optical Interconnects Fibers and waveguides function are the “wires” for optical interconnects, with fibers being similar to copper wires and waveguides similar to microstrips and striplines. Commercially available fiber-optic networks are now being used for computer-to-computerinterconnects and computer-to-peripheral connections. As the need grows for higher performance systems, guided optical interconnects will start to be used within the computer, possibly as far down into the packaging hierarchy as MCM to MCM (multichip module, or wafer to wafer). Electrical interconnects achieve some bandwidth enhancement through timedivision multiplexing (TDM) and spatial multiplexing (e.g.. ribbon cables). But the immense bandwidth available in opitcal waveguides (in THz)has opened up additional avenues to bandwidth enhancement for optical interconnects, namely, wavelength division multiplexing (WDM) and subcarrier multiplexing (SCM). Novel schemes for combining spatialmultiplexing with TDM, WDM, and SCM for optical interconnection may be needed. Possible implementation of a guided-wave interconnection network is shown in Fig. 8. In Fig. 8, the holographic components may be replaced with any other suitable optical components. There may be other possible architectures for guided-wave optical interconnects implementation.

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111. ARCHITECTURES

A. Free-Space Optical InterconnectionArchitectures

SEED-basedfree-spacephotonic switching systemshave undergone major changes over the past few years (Hinton et al., 1994). The fabricated S-SEED mays have evolved from 16 x 8 size to 256 x 128 in four stages, while FET-SEED are going through considerable changes as well. The supported channels, system bit rate, and lifetime have respectively increased from 2 to 1024, 5 kb/s to 100 kbs/s, and less than 8 h to more than 5 weeks for S-SEEDS. At the same time, FET-SEED-based system has already achieved 32-channel capacity, 50-Mb/s system bit rate, and more than 8 weeks lifetime (Hinton et al., 1994). The system hardware has evolved from commercial off-the-shelf to custom-built products. The basic optical and optomechanical hardware components include polarizing beamsplitters (PBS), binary-phase gratings (BPG), beamsplitters, prism/mirror gratings, quarter/half-wave plates, mirrors, beam combines, lenses, lasers, analyzers, fiber optic cable, and SEED devices. The system module shown in Fig. 9 uses FET-SEED smart pixels as switching nodes (McCormick et al., 1994). The architecture is a packet-switching fabric that switches ATM-like cells of data. In Fig. 9, the s-polarized input signals are focused onto the FET-SEED array by PBS (Hinton et al., 1994). The p-polarized clock signal also passes through the PBS and onto FET-SEED array wherein it is reflection-modulated and finally directed as an output through another PBS. The first stage of the system is used to

FIGURE9. The system hardware that uses FET-SEED smart pixels as switching nodes. (After Hinton et al. (1994))

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FIGURE 10. Architecture 1 schematic using orthogonal destination addresses. (After Maniloff and Johnson ( 1990).)

latch and switch the incoming data that is passing through the single-mode fiber bundle. The subsequent stages are identical and may use different interconnection topology based on the problem at hand. The final output of the smart-pixel array is imaged onto the multimode fiber bundle for electronic processing. B. Photorefractive Volume Holographic Interconnection Architectures

In this section two optical interconnect architectures are discussed. The ferroelectric liquid-crystal (FLC)SLM and Li"bO3 photorefractiverecording materials are essential components in both these architectures. Architecture 1. The fundamentalbuilding block of this architectureinvolves the FLC SLM, which is placed in between the crossed polarizer and acts as an intensity modulator. The optic axis is aligned with the vertical incident polarizer to yield the off state. Ideally, no light is transmitted by the output analyzer. Switching the polarity of the applied electric field causes the optic axis to rotate by 45". The vertically incident polarized light is rotated by 90" to the horizontal and it could be transmitted through the output analyzer. Thus, the light transmitted by the FLC SLM is horizontally polarized. Figure 10 shows an optical system where a modulated laser beam illuminates a horizontal 1D SLM or a row of 2D SLMs. The SLM spatially encodes the address of the destination on the incident optical wavefront. This process, in effect, illuminates a static multiple-exposure hologram previously recorded in the volume SLM (Maniloff and Johnson, 1990). For interconnecting N processors, an N x N SLM is needed (Marom and Konfronti, 1987). This approach to holographic routing requires first programming of the LiNbO3 crystal with the routing holograms. The optical system shown in Fig. 11 records the multiple-exposure hologram. An argon ion laser with horizontally polarized light (Ai514.5 mm) is split into separate reference and object beams. The object beam illuminates an FLC SLM. Beyond the FLC SLM, a lens focuses the object beam in front of the

OPTICAL INTERCONNECTION NETWORKS

FIGURE1 I . Optical system to record multiple-exposure hologram. Johnson (1990).)

25 1

(After Maniloff and

photorefractivecrystal. Each exposure associates a particular referencebeam angle with a particular processor angle. Thus, an N-bit address appears on all rows of the FLC SLM. After each exposure, the reference beam is rotated in the plane of incidence such that it illuminatesthe crystal at a new angle. Another orthogonalpattern is programmed on the FLC SLM. The next holographic exposure is made between the new reference beam and the new destination address. Routing is accomplished by illuminating the hologram with a weak object beam. The response time of the photorefractivecrystal does not limit the reconfiguration speed of the holographic routing. But it is limited by the frame rate of this FLC SLM. Once the connection is established, high-speed data transmission occurs through the channel. Architecture 2. In the previous architecture, because it is a parallel interconnect system, cross-talk from multiple-inputchannels may add up in the output plane. In reality, the signal-to-noise ratio (SNR) of the network decreases, which increases the bit error rate. In this second architecture, cross-talk from different input channels arrives at different output detectors. To do this it is necessary to increase the dimension of the output plane and use electronic thresholding of the detected signals. The same kind of recording technique is used to record the holograms as in the architecture 1, i.e., the N destination addresses are recorded with N unique horizontal angles. A horizontal 1D SLM or a row of 2D SLM is illuminated by each processor. Thus, an SLM of N rows with an address length M is required. The value of M is determined by the allowable overlap between the destination address. In architecture 1, M = N, since therein orthogonal destination addresses are used. For architecture 2, however, M < N, since nonorthogonal destination addresses are used. At the output, each processor has a column of N detectors as shown in Fig. 12 (Maniloff and Johnson, 1990; Marom and Konfronti, 1987). In Fig. 12, different addresses may have transmitting pixels (1’s) in common since nonorthogonal addresses are used. The input light is diffracted not only into the horizontal angle, but also into any angles corresponding to addresses that have 1’s in common with the displayed address. The SLM determines the reconstructed

252

KHAN M. IFTEKHARUDDIN A N D MOHAMMAD A. KARIM

FIGUREI?. Oprical system for Archilecrurc 2. (After Maniloff and Johnson (1990).)

horizontal angle which is displayed by thc addrcss. A cylindrical lens with power in the vertical dircction is used to provide vertical scparation of the diffracted light. This architecture allows less cross-talk power than the signal power as the light from different input channels is mapped into different output rows. Thcrcfore. the cross-talk could be removed by thresholding the detector's electrical output.

Scveral 3D MIN architecturcs arc discusscd in thc literature (Lin ct al.. 1987). Figure 13 shows one such 3D topolopy. We present hricf dcscripLions on two frec-spacc MlNs below.

OPTICAL INTERCONNECTION NETWORKS

253

Omega (ShufleExchange) Network. This is a special type of 3D multistage network composed of shuffling stages on a square array of channels as fixed interconnections combined with dynamic switching elements at each stage. There exists an optically efficient implementation of the shuffling operation on 2D arrays. This method is called the one-copy algorithm and achieves 100%light power utilization theoretically, as opposed to conventional methods, which achieve only about 25% efficiency. Optical 2D shuffles using this method can be implemented by a hologram containing four facets. The 2D version of this shuffle/exchange network, where shuffling is performed on an 1D vector of channels, has been demonstrated to be equivalent to other multistage networks and has been studied in detail. The optoelectronic dynamic switching element has been designed systematically by considering its functional structure, signal modulation techniques, and practical implementation issues. Through-Wafer Interconnection. This special hybrid architecture is proposed based on the logical 3D mapping for processing elements in most parallel computers (Wills et al., 1995). The system consists of standard silicon chip components and thin-film optoelectronic device. The optoelectronicdevices operate at optical wavelength to which the silicon is transparent. Thus, the interconnection through the wafer and, hence, the 3D logical mapping of the processors are obtained. D. Guided- Wave Interconnection Architectures The split, shift, and combine operation shown in Fig. 2 is of importance in implementing free-space interconnection technology. Guided-wave interconnections have been successfully used to obtain such simple operations (Jahns and Brumback, 1990). The advantage of using guided optics for interconnection is that the precise alignment and mounting requirements are eliminated. This also incorporates the advances made in digital VLSI technology. The underlying system is composed of input objects, beamsplitters, lens, gratings, laser, and a glass substrate to guide the light beam. The system architecture may be similar to that shown in Fig. 8. The lens is diffractive in nature with four discrete phase levels and all the components are fabricated on one side of the glass substrate by using photolithography and thin-film deposition (Jahns and Brumback, 1990). This monolithic module is small and easily extendible. This architecture may also be useful in providing chip-to-chip interconnection.

IV. APPLICATIONS A. Free-Space Optical Interconnection Applications Herein, a butterfly interconnection approach is demonstratedto realize a 2 x 2-bit multiplier and then the technique is extended for obtaining sequential logic

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KHAN M. IFTEKHARUDDIN AND MOHAMMAD A. KARIM

operations. As an example, we show the design of a multibit parallel-in parallel-out (PIPO) shift register using such sequential logic elements. 1. Logic Circuit Implementation Using Butte rjly Interconnection Design

i. Multiplication Circuit. Because of its importance in digital systems, design of fast multipliershas been an area of active research. Related efforts have resulted in both hybrid and optical multiplicationschemes that use digital multiplicationby analog convolution (DMAC) (Karim, 1991)and truth-table look-up using symbolic substitution (Karim and Awwal, 1992; Cherri and Karim, 1989). However, in addition to having an inherently sequential architecture, DMAC has difficulty in reconverting the mixed-binary output to its binary equivalent (Cherri and Karim, 1989). On the other hand, real-time implementation of symbolic substitution is still somewhat handicapped because of the nonexistence of adequate electrooptic devices. In comparison,butterfly multiplicationarchitecture is relatively free from such implementation bottlenecks. Consider a 2 x 2-bit multiplicationcircuit where A1 A0 and B1Bo represent the and COrepresent the 4multiplicand and multiplier, respectively, while Ca, C2, CZ, bit product terms. The multiplication output may be expressed in a sum-of-product (SOP) form (Cherri and Karim, 1989) in terms of the binary inputs: Co = AiAoBlBo C1 = AlAoBlBo

+ AlAoBlBo + AlAoBlBo + AlAoBlBo + AlAoBlBo + AlAoB1Bo + AlAoBlBo

+ A ~ A ~ B+ A~ B~ ~A ~ B ~ B ~ C2 = AlAoBlBo

+ AlAoBlBo + AlAoBlBo

C3 = AiAoBiBo

(4) (5)

(6) (7)

Accordingly, a total of 9 different minterms need to be generated by the AND array of the butterfly network. Subsequently,these minterms will have to be appropriately ORed using an OR array. The schematic diagram for a 2 x 2-bit multiplier is shown in Fig. 14. First, the two bits of the multiplicand and multiplier are split into three identical copies and passed through the interconnection networks (IN), herein a butterfly network. In passing through the IN, the signals are shifted as shift-left, shift-right, and no-shift, correspondingto the definition of butterfly. Accordingly, three different masks M I , M2,and M3 (corresponding to shift-left, shift-right, and no-shift operations) are employed. The output of these butterflies generates the necessary minterms. They are then split again as before and passed through a second set of butterfly networks and another set of masks M4,Ms,and M6. These operations, combined with the final OR array generate the products COthrough C3. The details of the butterfly interconnectionnetworks and the masks corresponding to the shift-left, shift-right, and no-shift operations for AND and OR arrays

255

OPTICAL INTERCONNECTION NETWORKS

c1

I

IN

‘2

c3 T

121 IN 1 2 1 IN I2

IN - Interconnection network FIGURE14. Schematic of a 2 x 2 multiplier stage. (After Iftekharuddin and Karim (1994).)

are obtained next (Cherri and Karim, 1989). According to the butterfly interconnection topology (Murdocca et al., 1988), the width of the network is Z5, while the depth is 2 x 5. The extension of this arhitecture (shown in Fig. 14) for N x N bit multiplication is straightforward. ii. Flip-Flop and Shifi Register A flip-flop (FF)or a latch is the most basic memory device used in sequential systems (Karim and Awwal, 1992). Basically, a FF remembers its state Q even after the inputs have been withdrawn. In general, its inputs are used to control the output, which is then internally fed back as an input. Consequently, an FF may be considered a three-input device whose next state depends on its current state. For the case of a D FF, there are two inputs. Following the truth table for a D FF (Karim and Awwal, 1992), as an example, the entries are expressed in terms of time r such that At represents the finite duration. The SOP form of the next-state output is given by

Its optical betterfly interconnection implementation thus involves a width of 23 and step depth of 3. The schematic implementation of AND and OR arrays of a D FF would be similar to that of the multiplier in Fig. 14. Note that the next state output Q(r + At) is split into two copies, one of which is fed back as the present state input Q ( t ) .

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KHAN M. IFTEKHARUDDIN AND MOHAMMAD A. KAFUM

A shift register consists of a group of FFs connected so that each FF transfers its bit of information to the adjancent FF coincident with each clock pulse (Karim and Awwal, 1992). For a simple N-bit PIPO shift register that uses D FF as its basic buildingmodule,Dj(t)DZ(t). . . DN(t)and Q l ( t + A t ) Q ~ ( t + A t ) ... QN(t+At) respectively represent the parallel inputs and outputs. Note here, for the N-bit PIPO,the butterfly circuit becomes two dimensional, where one dimension takes care of the logic implementation of first input bits while the other dimension is used for parallel operation of all bits (Lin et al., 1988). The butterfly interconnections and the masks for the D FF and the shift register may be obtained as well. 2. Algorithm Implementation Optical interconnects may play an important role in realizing the computational and signalhmage processing algorithms. The 3D optical data flow to achieve the required algorithmic interconnection of a 2D data array for a specific operation may offer desired speed and efficiency. A number of possible optical interconnection algorithmic implementations are described in Goodman et al. (1984). One such operation is matrix multiplicationwherein optical interconnects enable fewer computations (i.e., addition and multiplication of two numbers) (Goodman et al., 1984). Global interconnection technology may play a significant role in equal and homogeneous distribution of the clock signal as well. Another realistic application of optical interconnects is implementation of different mathematical transforms, including Fourier and other isomorphic transforms. The transforms and sorting usually require global and dynamic interconnections (Goodman et al., 1984). The availability of different types of interconnects enables different applicationdependent architecture implementation of these transforms. One such implementation of Hartley transform is discussed in detail in a later section. Space and time-variant operations are essential in image restoration and pattern recognition applications. These operations involve studying structures of various sizes from a single point of reference and its nearest neighbors. The interconnectivity between each point and all object points in an image may be invariant to time and space. Hence, global natures of an optical interconnection may be a suitable candidate for such an application (Goodman et al., 1984). B. Reconjgurable Optical Interconnection Applications in Neural Networks

The use of photorefractive crystals as a volume-holographic material to implement a dynamically updated (neural) interconnect is associated with some specific problems. In Section IIB,this dynamic updating was achieved using SLM. However, by using the photorefractive crystals as a volume holographic material, more sophisticated adaptive optical networks (neural networks) could be implemented (Karim and Awwal, 1992). A schematic of the volume-holographic interconnect

OPTICAL INTERCONNECTION NETWORKS

257

Input

Training FIGURE15. Schematic of a volume holographic interconnects. (After Slinger (1991))

arrangement is shown in Fig. 15. Here, the optical interconnect is configured for the appropriate weighted interconnection network by interfering white light from the ith input mode with the kth mode of the training set. After all the desired connections have been written, replay of the hologram by the new input x; forms the weighted sum yk at the kth output mode as (Karim and Awwal, 1992)

Consequently, studies have been carried out to investigate performance of a realtime adaptiveholographic system in accordance withEq. (9) (Slinger, 1991). Such performance measurement is accomplished in terms of fidelity. For the case of prescribed learning, where the weights Wjk are precalculated before the interconnect is physically implemented and then written onto the hologram. Note that the interconnection is not trained, rather the real-time holographic media enable such a system to have its weights modified under the control of some overall learning algorithm (Psaltis et al., 1988). The weighted interconnection from mode i to mode k (as shown in Fig. 15) is established by interfering waves Ei and Et from the input and training array, respectively. The change in dielectric constant of holographic medium is then proportional to the intensity of the interference pattern and is given by (Slinger, 1991) A€ = c

where c is a constant and

N

N

;=I

j=1

C C IEi + Et

(10)

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KHAN M. IFTEKHARUDDIN AND MOHAMMAD A. KARlM

Here aik and bk are complex amplitudes, B = 2n/l is the propagation constant of the hologram, n is the hologram's bulk refractive index, 1 is the free-space wavelength of light, p; and 0.4 are wave vectors of the input and training waves, respectively, and r is the position vector. Introducing Eq. (11) into Eq. (lo), we have

+ aikbk exp[jB (pi - ok) . r]}

(12)

A faster way of recording the holograms is also available (Slinger, 1991). In this approach, each training mode is exposed in turn with the appropriate whole set of input waves. This procedure requires only N exposures to produce N 2 main gratings, while the previous scheme requires N 2 exposures. The resulting modulation is obtained as (Slinger, 1991)

where the subscript k refers to the training-wave terms. Now using the scaler wave equation for diffraction (Lin et al., 1988), the wave propagating inside the hologram is given by (Slinger, 1991) E=

c

+ C Bk exp(-jBok

A; exp(-jBpi . r>

i=l

. r)

(14)

k=l

where A; and Bk are the amplitudes of the diffracting waves originating from the input plane and those propagating to the output plane, respectively. SubstitutingEq.(14) into Eq.(13) and comparing exponential terms of the form exp(-jB p; . r) and exp(-jB o k r) yields the following coupled-wave differential equations (Slinger, 1991): dA;

dx -k COS 6; (a;kb;Bk

jg -k

cc

Uika;,

Am)

=0,

fori = 1 , 2,..., N

(15)

where g = Bc/2x, and 0; and 6, are the propagation angles of the input and output waves respectively. Also, 0; = 6-k has been chosen for simplicity. Equations (15) and (16) mean that energy from input waves is coupled to the output waves as

259

OPTICAL INTERCONNECTION NETWORKS

the light propagates through the hologram. There exists an analytic solution for this coupled wave equation provided the third term of Eq. (15) is assumed to be vanishingly small (Slinger, 1991). This is an ideal case, which suggests that the detrimental effect due to interference between input and training beams is absent. Under such an assumption, there are two cases to be considered. Case I: Coherent Replay (Restricted Weight). For the case when the input array wave is spatially coherent, applying proper boundary conditions and the power conservation law yields the solution of the differential Eqs. (15) and (16) as (Slinger, 1991)

This equation shows a close similarity to Eq. (9). The modulus of the output of the interconnect may be plotted to observe coupling effects at greater thicknesses. Case 11: Incoherent Replay (Restricted Weight). When the wave propagating from the input array mode is spatially incoherent, the system will be linear in intensity as opposed to amplitude (i.e., the coherent case). Imposing the proper boundary conditions to the differential Eqs. (15) and (16) and solving for intensity will yield (Slinger, 1991)

Equation (1 8) also shows close similarity to Eq.(9),but, as we can see, the inherent system is incapable of implementing negative weights. C. 3D Free-Space InterconnectionApplications To emphasize the advantage of using a 3D architecture,Iftekharuddin et al. (1994) implement the Hartley transform operation using a 3D butterfly. This may not be so implemented using an 1D folded perfect shuffle. 1. Hartley Transform Implementation

The discrete Hartley transform (DHT) was introduced by Bracewell (1984) as a better alternative to both the discrete Fourier transform (DFT) and discrete cosine transform (DCT). The DHT has a real kernel {cos(2pkfl/M) sin(2pkfl/M)}, whereas the DFT uses a complex kernel given by (exp(i2pfl/M)}. This makes DHT both simpler and faster than DFT since a multiplicationof a complex variable (in DFT) involves four real products. The regular structure of the DHT signal flow graph also offers simplicity for VLSI implementation. Furthermore, because of isomorphism, the DFT and hence also DCT may be transformed into DHT easily.

+

260

K H A N M . IFTEKHAKUDIIIN AND M O H A M M A D A. K A R l M

In the 1D case, for a data sequence {x,,; y1 = 0, I , 2, . . . , M - I ), the corresponding DHT data sequence {y,,; n = 0, 1,2, . , . , M - I } is defined by

?'a(M, x) = I

c (G) + (F)]

M-l

XI,

[cos

sin

(19)

,i=o

for k = 0, I , 2, . . . , M - 1. Note that M is the order of the DHT. To implement the DHT as a butterfly, however, the 2D DHT can be derived as

.

f o r k = 0, 1 , 2 , . . . M - 1 and I = 0, 1 , 2 , . . , M - 1. To illustrate the 2D DHT butterfly, let's consider that M is a power of 2 and accordingly explore Eq. (20) for M = 2 as follows: ,

As far as the I D case is concerned, this transformation can bc performed by using a butterfly interconnection network as shown in Fig. 1. This allows for a better modularity in computing the higher-order DHTs. Accordingly, a 2D DHT may bc implemented using a ID DHT matrix of order N'. In the matrix form of thc DHT formulation, the decimation-in-time (such that the overall DHT computation is decomposed into smaller and smaller subsequences of the input sequencc) ID DHT algorithm is given by (Hou, 1987)

I: [

T(M/2) KT(MI2) T(M/2) -KT(M/2)]

(22)

where y,, is the preceding half while y, is the rear half of the output data, respcctively, and x, and x,,arc the even and odd inputs in the bit-reversed order (Hino et al., 1994). T ( M / 2 ) is the rearranged DHT matrix, which arises from the inputs of the bit reversed order; and matrix K is given as

26 1

OPTICAL INTERCONNECTION NETWORKS

where j , = 2pk/M, “Diag” stands for diagonal matrix, and 1

0

.

....

. 0’ 1

1 0

I0

p=l:

(24)

Equations (22)-(24) provide a mechanism for generating higher-order 1D DHT transforms from the lower-order 1 D DHTs. On the other hand, Eqs. (19) and (20) yield the 2D DHT from the ID DHTs. For 2D DHT, the input is simply written = z,], with p = it rn, where L can as I D DHTs, by writing, for example, be either x or y. Then the precomputations of the necessary connections may be performed using the ID DHT algorithm of Eq. (21). For a given M , this allows for the determination of all the required Connections between the three planes of the 3D butterfly processor, such that the third dimension may be used for data propagation. To illustrate a 3D butterfly of a higher order, consider, for example, M = 4, which necessitates the ID DHT of order 16 (= 2J). This 3D buttcrfly (with the third dimension being used for data propagation) is shown in Fig. 16. Notc that this particular 3D buttcrfly uses a total offour 16th-order ID DHTs. However,

+

262

KHAN M. IREKHARUDDIN AND MOHAMMAD A. KARIM

due to the nonseparability of the kernels in the DHT, the 2D module (i.e., the 3D butterfly of Fig. 16) may not be used to generate the higher-order transform. 2. Computer Communications Applications In this section, we describe the applicationsof 3D MINs in AsynchronousTransfer Mode (ATM) switching computer communications circuits. Broadband switching fabrics are expected to be key parts of future network service systems. The ATM is fundamental to the emerging broadband ISDN. Since “broadband” brings with it new services requiring new bit rates, and because these services may have widely differing traffic characteristics as well as switching requirements, ATM was devised and standardizedto accommodate all these different needs simultaneously. ATM can multiplex voice, data, and video traffic, both packet-switched and circuit-switched. Furthermore, ATM switches can as readily switch any of these services as any other. This is unlike today’s switches, which are optimized for particular bit rates and services. Considering the various overheads of ATM control, such as loss for dividing bandwidth into virtual paths, the switching fabric will need to have a multi-Tb/s throughput. Such a switching fabric needs to consist of more than 100 ports, since the port speed is currently limited by the device performance to the lO-Gb/s range using the fastest available chips. For example, a practical l-Tb/s system using available chips with reasonable yields exceeds 400 ports. On the other hand, a switching fabric with a large number of ports requires a correspondingly large number of switching elements and interconnections. Even though modern LSI technology has reduced the space required by the switching elements (Hino et al., 1994), the space occupied by the interconnections is still a dominant cost factor.

D. Guided-Wave Interconnection Applications Guided-wave or integrated optical components (IOCs) in various materials are now being deployed in many components that are key to advanced transmitters in many fiber-optic-based CATV and long-haul telecommunicationssystems. The basic devices are often based on planar optical waveguides at the substrate surface. These channels through which light is routed on the chip are typically less than 10 pm across and are patterned using microlithographytechniques. Using appropriate optical circuits based on these channel guides, both passive functions (i.e., power splitting from one to several channels) and active functions (i.e., electricalto-optical signal conversion) are performed on the light signal. A closely related area that is still in its early stage is photonic integrated circuits, in which a variety of semiconductor optoelectronic devices such as lasers and modulators are monolithically integrated and interconnected with waveguides.

OPTICAL INTERCONNECTION NETWORKS

263

Until recently, IOCs have been predominantly used in analog, digital, and fiberoptic sensor applications. At present, however, major new applications are emerging. Perhaps the most significant new area is telecommunications, where IOCs will be used for multigigabit data transmission, signal splitting and loop distribution, and bidirectional communication modules. A second new application area is CATV where IOCs will soon be used for external modulation in fiber-optic-based signal distribution systems,In both telecommunications and CATV,IOCs will enable transmission of signal at higher data rates and over longer distances. The third major area is instrumentation development, wherein a significant application is fiber-optic gyroscopes. In telecommunications, in particular, IOCs are used for high-speed modulation, signal splitting, switching, and bidirectional communication. There is considerable interest also in higher degree of modulation. This could reduce both fiber dispersion and laser c h q problems that would otherwise limit maximum possible transmission distance and signal bit rate. L N O 3 modulators are being used in 2.5-Gbls systems to enable signal transmission over distances of greater than 100 km without any repeater. Typical performance characteristicsof these devices include a 4 - V drive voltage and 4-dB insertion loss. Similar devices are the technology of choice for underwater systems (e.g., transoceanic links operating at 5 Gbith), because they have either negligible or tunable chirp and enable the greatest distance between optical repeaters. Modulators for 10-Gb/s transmission are just now being considered for next-generation systems.

v. PACKAGING OF OPTICAL INTERCONNECTS Packaging technologies are fundamentally important to producing cost-effective photonic components. This is true for all high-volume, low-cost devices and also high-performance, limited-volume devices. For example, the cost of today's CD laser is determined more by the labor costs of the component optoelectronic packages. It is expected that future digital systems such as ATM switching systems and massively parallel processing computer systems will have large printed circuit board (PCB)-to-PCB connectivity requirements. Such requirements are vital to supporting large aggregate throughput demands (Hinton et al., 1994). Currently, electronic technology may not be able to simultaneously support the connection densities and the bandwidth required due to connector limitations as the PCB-tobackplane interface (Goodman et al., 1984). 2D free-space optical interconnects when implemented at the PCB-to-PCB level in the form of an optical backplane may provide greater connectivity at higher data rates than can be supported by current or future electronic backplanes (Gibbs, 1985). An optical backplane can be constructed using 2D arrays of passive, freespace, parallel optical channels which optically interconnectPCBs via smart pixels

264

KHAN M. IFTEKHARUDDIN AND MOHAMMAD A. KARIM

arrays. The smart-pixel optoelectronicsare two-dimensional device arrays capable of electrical-to-optical(WO)and optoelectrical (OE) conversion of digital data. In addition to E/O and O R conversions, these devices can perform processing operations, such as address recognition and packet routing. By interconnecting PCBs with 10,000 channelshoard (10 smart-pixel arrays per PCB at lo00 communication channels per smart-pixel array), each channel running at 100 Mb/s can support greater than a Tb/s of aggregate data traffic. The system demonstrator is generally based on FET-SEED smart pixels, PCB level optoelectronic packaging, diffractive microoptics, and base-plate optomechanics, which demonstrates a simple unidirectional PCB-to-PCB optical interconnection. Componentssuch as laser diodes and modulators,designedfor high-performance applications, are single-mode devices. In integrated systems, they must be connected using optical fibers or other types of waveguides with submicron alignment accuracy.Currently, optoelectronicpackaging is usually performed by highly skilled technicians looking through microscopes and manually adjustable submicron stages. Once the alignment is determined to be correct, the components are held in place using epoxy, solder, or other attachment techniques. This laborintensive process results in only a few packages being produced per day by each technician. The packaging costs are by far the largest fraction of the total cost of an assembled optoelectronic package. The consequences of this low-volume laborintensive process of packaging optoelectronic devices are readily apparent. The high cost is still a factor that limits the use of fiber optics in on-chip interconnects, interboard connections in computers, local area networks, and fibers to the home. Optoelectronic packaging needs to be automated to significantly reduce the costs of optoelectronic devices. In comparison, the electronics industry has successfully reduced the costs of its products through the massive use of automation, including robotics, parts handling, and feeding, and total quality management. Unfortunately, the submicron precision required for optoelectronic packaging greatly exceeds the requirements of the electronics industry. The automated systems developed to assemble integrated circuits cannot be readily applied to the problem of packaging optoelectronic circuits. VI. PROBLEMS AND POSSIBILITIES

A. Free-Space Interconnection Networks For the connection-intensive free-space interconnection networks to be widely acceptable, a realistic, repeatable, reliable, and cost-effective packaging technology, needs to be identified. The value of free-space interconnection in the processor environment that requires huge FLOPS can be appreciated by considering the following architectural scenarios (Jahns, 1994). To have ten thousand

OPTICAL INTERCONNECTION NETWORKS

265

100-GigaFLOPSCPUs, for example, on the order of 100 billion memory accesses per second will be required. Assuming a 64-bit word, this requires a 6400-Gb/s transfer rate between processor and memory, and, thus, 64 lOO-Gb/s optical interconnects. In addition, if we were to consider one of 10 CPU instructions to be UO with a peripheral related (implying a 64O-Gb/s transfer rate), the system will require seven lOO-Gb/s fibers. Further, if one of 100 CPU instructions involves communicating with another processor (implying a 64-Gb/s transfer rate), an additional 100-Gb/s link will be necessary. If the computer is packaged as 10 MCMs per board, 10 boards per card cage, and 10 card cages per rack, for example, then assuming a factor of 10 increase in connectivity from rack-to-card cage-to-board-to-MCM, 64 100-Gbh connections between processor and memory (rack-to-rack) implies 640 lO-Gb/s links between card cages, and 6400 1-Gb/s links between boards, and 64,000 100Mb/s connections between MCMs. Since there are 10,OOO CPUs (1 CPU/MCM), this figure translate into 100-Gb/s optical links and lO-Gb/s optical links, respectively, a packaging and manufacturing nightmare. For the architectural scenario of 100,000 10-GigaFLOPSCPUs, the above reasoning leads to a requirement for 800,120 100-Gb/s links and 7,000,000 lO-Gb/s links. Free-space implementation does not need a physical guiding medium for each individualbeam, thereby greatly simplifying typical interconnect problems.

B. Photorefractive Holographic Reconjigurable Neural Interconnection Networks The analytic solution for the coupled wave of Eqs. (15) and (16) shown in Section IVB assumed that the detrimentaleffect due to interference between input and training beam is absent. This assumption is nonideal, which may lead to an unrealistic situation. A generalized numerical solution may be pursued for the differential Eqs. (15) and (16). By applying appropriate boundary conditions for the coherent case, results (Slinger, 1991) may be obtained that largely deviate from the ideal case of Section IVB. This is the case here where interference term between the input and training beams has been neglected. Further, the interconnectionweights vary independently as a function of the thickness of the holographicmaterial. With the interference term present, the performance is even worse (Slinger, 1991), i.e., the interconnection weights become more undeterministic. Holographicinterconnects have the additional advantageof being reconfigurable en mass by modifying a beam steering element (e.g., a diffraction grating or hologram) in the media through which the interconnectbeams pass. This could lead to special-purpose machines becoming more general purpose, and to computing systems enhancing performance by selecting the interconnection structure most appropriate for the task at hand at any instant of time. Another obvious application

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KHAN M. IFEKHARUDDIN AND MOHAMMAD A. KARIM

could be a real-time implementation of reconfiguration network, which acts as cache memory in digital computer. Because of the poor optical efficiency, the SLMs could be replaced by appropriate software algorithm to accomplish the required updating instruction to be sent to the refractive material. C. 3 - 0 InterconnectionNetworks One rather nontrivial problem of MIN is that the bisection width grows almost linearly with the number of nodes. The bisection width is defined as the minimum number of connections that need to be removed to partition the network into two identical halves including processors. Some of the free-space MIN topologies may not be cost effective due to large bisection width as the network grows larger. Banyan-based switching circuits are one example of circuits that are limited in size due to large bisection growth (Lin et al., 1987). The routing of a set of messages through MINs can be another bottleneck for 3D optical interconnection networks. The most time-consuming problem is the generation of the control bits that define the routes for MIN. Interestingly, the routing problem for both the 2D electronic MINs and 3D optical MINs possesses similar intricacies. A neural network model will alleviate this problem (Giles and Goudreau, 1995) on a limited scale. The performance of the MINs is satisfactory for uniform network traflic. Several studies have indicated that the performance of the electronic MINs is degraded significantly when there is hot-spot traffic. The hot-spot traffic situation occurs when a large fraction of the messages is routed to one particular destination. To alleviate this shortcoming, MINs with two or more paths between all source and destination pairs are investigated (Wang et al., 1995). These multipath MINs can reduce the detrimental effects of hot spots using some specific routing strategies. The similar problems in optical MINs may be addressed in the same way. Switching networks of the multistage fabric have been studied extensively in the context of designing interconnection networks in at least four critical areas: telephony, parallel computing, high-performancelocal area networking, and widearea networking. Traffic characteristics vary considerably in each application; nevertheless, in each case the routing task gives rise to a complex integral multicommodity flow problem where data packets or connection requests between any two end terminals correspond to a single commodity. In this context, numerous results exist for the special cases where all link capacities are unit-valued, and the traffic pattern is either a permutation connection or a broadcast connection. D. Guided-Wave Interconnection Networks Guided optical interconnects suffer from one of the disadvantages of electrical interconnects, that is, the need for a physical guiding medium. This leads to several

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serious packaging limitations in applications requiring a large number of interconnections. First only one physical guide can occupy a given position in space, and second, attaching a large number of guides to an optoelectronic component is difficult and costly. Free-space optical interconnects, however, do not require a guide; therefore, a large degree of spatial multiplexing is possible. For example, an array of vertical-cavity surface-emitting lasers (VCSELs) could be used to transmit 10,000 signals simultaneously to an array of photodiodes in any specified pattern. This can be done far more compactly than through 10,OOO fibers.

VII. CONCLUSIONS

Evolutionary and revolutionary advances in electronic interconnectiontechnology have played a key role in allowing continued improvements in integrated circuit density, performance, and cost. Over the last two decades, circuit density has increased by a factor of approximately 10, while cost has constantly decreased. On the other hand, optical interconnects experienced reasonable growth, which paved the way for some real-life applications of this technology. The optical interconnection is not yet as cost-effective as its electronic counterpart. The promise and possibility depend on the appropriate growth of the optical device technology. The following is a summary of the optical interconnection applications discussed in this exploratory paper. As examples of free-space interconnection, multiplier, FF,and shift registers are described. The optical implementation of the multipliers does not require any feedback. However, an n x n-bit multiplier circuit for large n may turn out to be less than optimal in terms of circuit depth and gate count because of its size and single-layer characteristics(Murdocca et al., 1988). But this single-layer behavior may actually ovemde the aberration and diffraction limitations of the electrooptic devices (Sakano et al., 1990). On the other hand, design of parallelin parallel-out shift registers using D FFs involves a simple feedback circuitry. This multilayer design concept fully exploits the global nature of the free-space interconnection networks. Further, the gates (devices) involved in both cases have a maximum fan-in of 2 and fan-out of 2, which is rather easy to implementoptically. Finally, the specific optical implementation of sequentialdevice and register design opens up the possibility of designing many other optical memory devices using butterfly interconnects. The optical implementations described in this paper are closely related to the S-SEED technology (McCormick and Prise, 1990; Cloonan et al., 1990). However, because of the inherent isomorphism between butterfly and banyan networks (Murdocca et al., 1988), a Michelson and/or Mach-Zehnder type setup, which has already been used for banyan networks (Hou, 1987), can also be used. Further, the flip-flop structures may turn out to be important for the

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other isomorphically equivalent networks, e.g., perfect shuffle (Murdocca et al., 1988; Eichmann and Li, 1988) and, accordingly, for the space-invariant optical interconnectiontopology as a whole. This review paper briefly discusses the issue of reconfigurable characteristics of volume holographic photorefractive crystals like LiNbO3 and BaTi3 in adaptive neural networks. It may be noted that the overall efficiency of this kind of optical system depends on the material characteristics of the crystal material. Use of refractive materials like dichromated gelatin for the purpose of recording material yields efficiencies in excess of 99%, for a simple sine wave grating (Shudong et al., 1990).The study of neural network implementation using dynamic updating capability of refractive crystal reveals that the weighted sum is distorted as a function of position in the output array. A correction factor could be used to avoid this position dependency (Goodman, 1968). It is further noted that the inherent system is incapable of implementing the negative weights for neural networks. More importantly, the exact numerical solution shows that interconnectionweights vary independently as the function of thickness of holographic material. Supervised learning and self-organizing systems could be helpful in avoiding these problems (Paek et al., 1989). The optical interconnection is best exploited in 3D architecture. The 3D butterfly interconnection processor provides 100%connectivity between the input and output planes. For N = 64 and connectivity coefficient q = 1, the 3D architecture yields about three times better linear extent than does the ID folded PS network. Accordingly, the 3D interconnection processor is more preferable. The 2D implementation of the DHT (which is computationally more efficient than the DFT and DCT) is just one example of the many applications of 3D butterfly processor. The architectureof 1D folded perfect shuffle is not particularly suitable for the implementation of DHT. Thus, depending on the application area, a 3D architecture may turn out to be the natural choice in photonics. The electronic interconnection technology is undergoing a balancing of chip design requirements, with manufacturing process options available for metallization, insulators, planarization, and patterning. Chip interconnections serve as local and global wiring, connecting circuit elements and distributing power. Interconnects also function as the interface between chip and package, thereby also requiring stringent control of mechanical properties. High electrical and mechanical reliability and optimized resistivity are some of the bottlenecks for a successful electronic interconnection technology. On the other hand, optical interconnection technology is experiencing considerable growth based on its matured electronic counterpart. It is, therefore, logical to conclude that the current device development status only suggests the intertwining of electronic and optical counterparts to implement hybrid optoelectronic interconnection for the successful future of computing technology.

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REFERENCES Bracewell, R. N. (1984). The fast Hartley transform. Pmc. IEEE72, 1010. Brenner, K. H., and Huang, A. (1988). Optical implementations of the perfect shuffle interconnections. Appl. Opr. 27, 135. Caulfield, H. J., Neff, J. A., and Rhodes, W. T. (1983). Optical computing: the coming revolution in optical processing. Laser Focus 19, 100. Chem, A. K., and Karim, M. A. (1989). Symbolic-substitution-based operations using holograms: multiplication and histogram equalization.Opr. Eng. 28,638. Chiou, A.. and Yeh, P. (1990). Energy efficiency of optical interconnections using photorefractive holograms. Appl. Opr. 29, 1 1 1 1. C1oonan.T. J. (1989). Topological equivalence of optical crossover networks and modified data manipulator networks. Appl. Opr. 28, 2494. Cloonan, T. J., and McCormick. F. B. (1991). Photonic switching applications of 2-D and 3-D crossover networks based on 2-input. 2-output switching nodes. Appl. Opt. 30,2309. Cloonan, T. J., Herron, M. J., Tooley, F. A. P., Richards, G. W., McCormick, F. B., Kerbis, E., Brubaker, J. L., and Lentine, A. L. (1990). An all-optical implementation of a 3-D crossover switching network. IEEE Photon. Technol.Let!. 2,438. Eichmann, G., and Li, Y.(1987). Compact optical generalized perfect shuffle. Appl. Opr. 26, 116. Gibbs, H. M. (1985). Optical Bisrability: Conimlling Ligh! with Lighr. Academic, New York, Chapter 2, p. 21. Giglmayr, J. (1989). Classification scheme for 3-D shuffle interconnection patterns. Appl. Opt. 28, 3120. Giles, C. L., and Goudreau, M. (1995). Routing in optical multistage interconnection networks: a neural network solution. IEEE J. Lightwave Technol. 13, 1 11. Goodman, J. W. (1968). Inrmducrion to Fourier Oprics. McGraw-Hill, New York. Goodman, J. W., Dias, A. R., and Woody, L. M. (1978). Fully parallel, high-speed incoherent optical method for performing discrete Fourier transform. Opr. kfr. 2, 1. Goodman, J. W., Leonberger, F.. Kung, S.Y.,and Athale, R. (1984). Optical interconnectionsfor VLSI systems, Pmc. IEEE 72,850. Guha, A., Bristow. J., Sullivan, C., and Husain, A. (1990). Optical interconnections for massively parallel architectures.Appl. Opr. 29, 1077. Guilfoyle, P.S. (1984). Systolic acousto-optic binary convolver. Opr. Eng. 7 2 , 20. Hill, H. D., and Smith, A. J. (1989). Evaluating associativityin CPUcaches. IEEE Trans. Compur. 38, 1612. Hino, S., Togashi, M., and Yamasaki, K. (1994). Asynchronous transfer mode switchingLSIs with 10 Gbit/s serial inputs and outputs. IEEE Tech. Digesr VLII Circuits Symp. 7 3 . Hinton, H. S., Cloonan, T. J., McCormick, F. B., Lentine, A. L., and Tooley, F. A. P. (1994). Free-space digital optical systems. IEEE Pmc. 82, 1632. Hou, H. S. (1987). The fast Hartley algorithm. IEEE Trans. Compur. C-36,147. Iftekharuddin, K. M., and Karim, M. A. (1994). Butterfly interconnection network: design of multiplier, flip-flop and shift register. Appl. Opr. 33, 1457. Iftekharuddin, K. M., Jemili, K.,and Karim, M. A. (1994). Comparison between optical interconnection processors: folded perfect shuffle and 3D butterfly. Opr. Laser Technol. 26,265. Jahns, J. (1990). Optical implementation of the Banyan network. Opt. Commun. 76,321. Jahns, J. (1994). Planar packaging of free-space optical interconnections. Pmc. IEEE82, 1623. Jahns, J., and Brumback, B. A. (1990). Integrated-optical split-and-shiftmodule based on planar o p tics. Opr. Commun. 7 6 , 318.

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Jahns, J., and Murdocca, M. (1988). Crossover networks and their optical implementations. Appl. Opt. 27,3 155. Jewel], J. L.,Lee, Y. H., Warren, M., Gibbs, H. M., Peyghambarian, N., Gossard, A. C., and Wiegmann W. (1985). 3 pJ 82 MHz optical logic gates in a room temperature.GaAs-AIGaAs multiple quantum well etalon. Appl. Phy. Lett. 46.9 18. Karim, M. A. (1991). Smart quasiserial post processor for optical systolic systems.App1. Opt. 30,910. Karim, M. A., and Awwal, A. A. S. (1992). Optical Computing: An Introduction. Wiley, New York, Chapter 11, p. 292. Kostuk, R. K., Goodman, J. W., and Hesselink, L. (1985). Optical imaging applied to microelectronic chip-to-chip interconnections. Appl. Opt. 24,285 1. Kostuk, R. K., Goodman, J. W., and Hesselink, L. (1987). Design considerations for holographic optical interconnects. Appl. Opr. 26,3947. Lentine, A. L., Hinterlong, S. J., Cloonan, T. J., McCormick, F. B., Miller, D. A. B., Chirovsky, L. M. F., D’Asaro, L. A., Kopf, R. F., and Kuo, J. M. (1990). Quantum well optical tri-state devices. Appl. Opt. 29, 1157. Lin, S. H., Krile, T.F., and Walkup, J. F. (1987). 2-D optical multistage interconnection networks. Proc. SPIE 752,209. Lin, S. H., Krile, T. F., and Walkup, J. F. (1988). Two-dimensional optical clos interconnection network and its uses. Appl. Opt. 27, 1734. Lohmann, A. W., Stork, W., and Stucke, G. (1986). Optical perfect shuffle. Appl. Opt. 25, 1530. Maniloff, E. S., and Johnson, K. M. (1990). Dynamic holographic interconnects using static holograms. Opt. Eng. 29,225. Marom, E., and Konfronti, N. (1987). Dynamic optical interconnections. Opt. Lett. 12,539. McCormick, F. B., and Prise, M. E. (1990). Optical circuitry for free-space interconnections. Appl. Opt. 29,2013. McCormick, F. B., Cloonan, T. J., Lentine, A. L., Sasian, J. M., Morrison, R. L., Beckrnan, M. G., Walker, S. L., Wojcik, M. J., Hinterlong. S. J., Crisci, R.J., Novotny, R. A,, and Hinton, H. S. A. (1994). 5-stage free-space optical switching network with field-effect transistor self-electro-optic device smart-pixel arrays. Appl. Opt. 33,2603. Murdocca, M. (1990). Connection routing for microoptic system. Appl. Opt. 29, 1106. Murdocca, M. J., Huang, A., Jahns, J., and Streibl, N. (1988). Optical design of programmable logic arrays. Appl. Opt. 27,165 1. Oppenheim, A. V., and Schafer, R. W. (1989). Discrete-7ime Signul Processing. F’rentice-Hall, Englewood Cliffs, New Jersey. Paek, E. G., Wullert, J. R., and Patel, S. S. (1989). Holographic implementation of a learning machine based on multicategory perceptom learning algorithm. Opt. Lett. 14, 1303. Prince, S. M., Tooley, F. A. P., and Taghizadeh, M. R. (1996). Dynamic interconnection of S-SEED arrays. Appl. Opt., in press. Psaltis, D., Brady D., and Wagner, K. (1988). Adaptive optical networks using photorefractive crystals. Appl. Opt. 27, 1752. Rastani, K., and Hubbard, W. M. (1992). Large interconnects in photorefractives: grating erasure problem and a proposed solution. Appl. Opt. 31,598. Sakano, T.,Noguchi, K., and Matsumoto, T. (1990). Optical limits for spatial interconnection networks using 2-D optical array devices. Appl. Opt. 29, 1094. Sawchuck, A. A., Jenkins, B. K., Ragharendra, C. S., and Verma, A. (1987). Optical crossbar networks. IEEE Trans. Comput. C-20.50. Sawchuk, A. A., and Jenkins, B. K. (1986). Dynamic optical interconnections for parallel processors. Proc. Soc. Photo. Opt. Instrum. Eng. 625, 143. Shudong, W., Qiwang, S., Mayers, A., Gregory, D. A., and Yu, F. T. S. (1990). Reconfigurable interconnections using photorefractive holograms. Appl. Opt 29, 11 18.

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Slinger, C. (1991). Analysis of the N-to-N volume-holo-graphicneural interconnect.J. Opt. SOC.Am. A 8, 1074. Stirk, C. W., Athale, R. A., and Haney, M. W. (1988). Folded perfect shuffle processor. Appl. O p t 27,202. Wang, M. C., Siegel, H.J., Nichols, M. A., and Abraham, S. (1995). Using a multipath network for reducing the effects of hot spots. IEEE Trans. Purullel Distributed Proc. 6,252. Wills, D. S., Lacy, W. S., Ginestet, C. C., Buchanan, B., Cat, H. H., Wilkinson, S., Lee, M., Joerst, N. M., and Brooke, M. A. (1995). A three-dimentionalhigh-throughput architecture using through-wafer optical interconnect.IEEE J. Lightwave Technol. 13, 1805. Yeh, P.,Chiou, A. E. J., and Hong, J. (1986). Optical interconnection using photorefiactive dynamic holograms.Appl. Opt. 27,2093.

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ADVANCES IN IMAGING AND ELECTRON PHYSICS.VOL. 102

Aspects of Mirror Electron Microscopy S . A. NEPUKO* AND N. N. SEDOVt Institute of Physics, Technical University Clausthal, 38678 Clausthal-Zdlerfeld, FRG

I. Resolution of the Mirror Electron Microscope . . . . . . . . . . . . . A. Factors Determining the Microscope Resolution . . . . . . . . . . . B. The Influence of the Angular Spread in the Electron Beam . . . . . . . C. The Optimal Bias Voltage Applied to the Object . . . . . . . . . . D. Resolution Limitation Due to the Longitudinal Velocities of Electrons . . . E. Conclusions . . . . . . . . . . . . . . . . . . . . . . . 11. Distortion of Details of Object Image Under Observation in a Mirror Electron . . . . Microscope . . . . . . . . . . . . . . . . . . . 111. Limiting Sensitivity of a Mirror Electron Microscope for Observation of Steps . . . . . . . . . . . . . . . . . . . . . . . . on an Object IV. Image of Islands on an Object Surface in Mirror Electron Microscopy . . . . A. Calculation of the Trajectories of Electron Motion . . . . . . . . . . B. Calculation of the Brightness Distribution in the Image on the Screen . . . C. Analytical Calculation of Electron Trajectories . . . . . . . . . . . D. Measurement of the Contact Potential Difference Between a Spherule and a . . . . . . . . . . . . . . . . . . . . . . . . Substrate V. Calculation of Image Contrast in a Mirror Electron Microscope in the Focused Operation Mode . . . . . . . . . . . . . . . . . . . . . . . A. Geometrical Optics of the Immersion Objective in the Focused Operation Mode B. Calculation of Electron Deviation by Microfields . . . . . . . . . . C. The Current Density Distribution on the Microscope Screen . . . . . . D. Conclusions . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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Mirror electron microscopy (MEM) is very sensitiveto microfieldsand the geometrical roughness of objects under study. In a mirror electron microscope the primary electrons are retarded on their way to the object. They reflect not far from this object and then move in the opposite direction. Therefore, these electrons remain in the region of action of any microfields on the object surface for a long time. As a result, their trajectories are strongly distorted, which leads to contrast distortion and image deformation. The field distribution on the surface of the object being investigated (Dyukov et al., 1991). the real form and linear sizes of details on the object, * Permanent address: Institute of Physics, Ukrainian Academy of Sciences, pr.Nauki 46, 252022 Kiev, C.1.S ./Ukraine. t Permanent address: The Moscow Higher Military Command School, Golovachev str., 109380 Moscow, C.I.S./Russia. 273

Copyright@ 1998 by Academic Press, Inc. All rightsof repduction in any form reserved. 1076567W97 $25.00

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and other parameters can be restored by considering the above-mentioned effects. It was important to allow for the effects of the interaction with microfields and other object microroughness in studies of the problems examined in the present paper. The electron energy distribution and nonparallelism (divergence) of the electron beam illuminating the object have been taken into account.

I.

RESOLUTION OF THE

MIRRORELECTRON MICROSCOPE

The defocused operationmode has much in common with the principle of operation of the shadow microscope and is used in most MEM designs. In this regime each point of an object corresponds to a narrow electron beam, which is not focused to a point, although the image can be magnified by the converging lenses. Usually in MEM the object is a simple constructionconsisting of the plane object K, which is the cathode of the immersion objective, and of the accelerating anode A in the form of the diaphragm with beam hole (Fig. 1). Let us denote the distance between the object and the anode by 1, and the accelerating voltage of microscope by V,.As a rule, the diameter of the hole in the anode is smaller than the distance 1. The uniform electric field acting above the object surface is given by

VO Eo = 1

and the anode hole acts on electrons as a diverging lens with the focal length

f =-41

(2)

However, the objective may have another construction; in the following calculations, therefore, the focal length f of the objective lens is supposed to be arbitrary. The behavior of electron trajectories in such an objective was investigated in detail in Dyukov et al. (1991) and Sedov (1970). The electron trajectories in the X I

Y =0

v = vo

FIGURE1. A schematic of the immersion objective of MEM.

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275

objective are shown schematically in Fig. 2. The initial electron beam anives at the objective,then it is refracted by the lens and directed to the microscope screen, during which time the image can be additionally magnified by the intermediate lenses. The point on the object situated at the distance r from the optic axis is imaged on the screen by electrons arriving at the screen at the distance R from its center; in this case the microscope magnification M = R / r . If electrons suffer weak perturbation above the object surface, their trajectories are slightly deflected from the initial ones, and the electrons arrive at some point on the screen a distance A R from the initial point. This distance in the object plane corresponds to the shift S = R / M.The image contrast appears at the expenseof nonuniform redistribution of the electron beam on the screen, and it has been calculated quantitatively by Dyukov et al. (1991) and Sedov (1970). A negative bias voltage A V is applied to the object relative to the cathode of the electron gun. As a result, in the central part of the object in line with the system axis electrons reflect at the following height from the object surface:

However, electrons imaging noncentral parts of the object decrease somewhat the longitudinal components of their velocity owing to the change of direction of motion in the lens. Therefore, they reflect at a greater height above the object. This

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height is given by

+ (f +r212 0 2

AVI h(r) = ho + Ah = -

vo

(4)

This increase of height h for every distance r can be compensated by a corresponding change of the bias AV. In the following calculations, we can consider the value h to be determined only by the bias A V for a given part of the object. Another peculiarity of MEM should be also pointed out. With the simple optics shown in Fig. 2, the negatively charged or convex regions on the object surface look like dark spots on the screen and vice versa. Such image contrast is considered positive. However, as shown in Dyukov et al. (1991) in Fig. 27 the negative contrast of the image may be also obtained when converging lenses are present in the imaging optics. In the following, for simplicity, the contrast is assumed to be positive. We define contrast to mean the function of current density distribution on the screen j ( x , y), where all sizes are divided by the microscope magnification, that is, the modulus of apparent shift of every point of the object is given in the object coordinates.The plane ( x , y) of a system of coordinates coincides with the object plane; the z axis coincides with the optic axis of the objective.

A. Factors Determining the Microscope Resolution

The main factors influencing the geometrical resolution of MEM are as follows: 1. Diffraction effects connected with the finite value of de Broglie wave of electron A. 2. The finite height of the electron-beam reflection above the object h. 3. The energy spread of electrons in the primary beam. 4. The angular spread of electrons in the illuminating beam, which is connected with the finite source size of electrons. 5 . Aberrations of the imaging lenses. 6. Other effects connected with the distortion of electron trajectories close to the object surface.

Let us analyzethe factorslisted above. Diffraction effects in MEM have been frequently calculated, for example, in Wiskott (1956) and Gvosdover and Zel’dovich (1973). Therefore, in the present work diffraction is not calculated, and the radius of the diffraction spot should be taken into account in the general equation for resolution. We can only point out the order of magnitude of the diffraction spot. The spread of energies in the primary beam is assumed to be about 0.1 eV, and there are also transverse velocities which never become equal to zero. The de Broglie

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277

wavelength of these electrons is

In the point of return of electrons, their wavelengths are the same order of magnitude, therefore the minimum diffraction spot will be about the same:

& % 40A

(6)

The finite height of electron reflection h limits resolution to the same order of magnitude, because any roughness on the object surfaceresults in a perturbation of electrons reflected not only above this point, but at the distance of the same order in the lateral direction as well. Therefore, it is impossible to obtain a diffraction spot less than h . The spread of the longitudinal electron velocities results in the same effect. As a result of this velocity spread, electron reflection takes place in reality not in the plane placed at the height h above the object, but in some layer of thickness AW Awl Ah=-=(7) eEo eVo where AW is the spread of energies in the electron beam in the longitudinal direction, and e is the electron charge. Deterioration of resolution due to this factor will be the same order of magnitude. The diameter of the diffraction spot will be made more precise by the calculation of the image contrast from a point roughness on the object surface. In reality the two last factors are closely related. To make the majority of the electrons reflect as close to the object in a given place as possible, an operator of MEM may change the bias hV on the object to obtain the highest resolution in such a way that some part of electrons touches the object surface, in this case h = Ah. However, this question should be considered in more detail because of the presence of higher-speed electrons in the beam. The influence of the angular spread of electrons in the beam 8 on the resolution was calculated in Dyukov et al. (1991), Fig. 29 and Eqs. (208x215). The order of magnitude of resolution deterioration was obtained there:

where 8 is the angular aperture of the incoming beam, p is the size of the primary electron source,for example, the crossoverof the electrongun,and L is the distance between the electron source and the objective. This equation will be made more precise later. Aberrations of the imaging lenses are expressed in terms of the well-known equations,therefore they are not calculatedin this paper. In calculatingthe total size

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of diffraction spot it will suffice to take into account the spot owing to aberrations p. Lastly, the influence of other effects related to microfields on the object surface is an additional factor which may be considered individually. The diffraction spot is now calculated, taking into account the distribution function of the initial velocities of electrons. This function is close to the Maxwellian distribution. The influenceof the longitudinalinitial velocities on the height of the electron-beamreflection above the object is considered as well as the influence of angular spread of electrons due to the transverse initial velocities. These factors are important, because they determine the microscope resolution. B. The Influence of the Angular Spread in the Electron Beam With a Maxwellian distributionof the initial velocitiesof electrons, the distribution function of the current density in the cross section of the crossover of the electron gun is Gaussian:

where pl =,-/,lf fi is the focal length of the immersion objective of the electron gun, and WOis the most likely initial energy. For thermoionic cathodes W O= k T , where k is Boltzmann’s constant, and T is the absolute temperature of cathode. Let us assume that the illuminatingsystem of the microscope includes one more condenser lens with the focal length f2 placed at the distance L 1 from the electron gun and at the distance L2 from an observed object. Such a system is shown in Fig. 3. The condenser lens decreases the crossover by a factor of L 1 /f2; therefore,

FIGURE 3. The optical scheme of the microscope illuminating system.

ASPECTS OF MIRROR ELECTRON MICROSCOPY

279

the size this secondary source is

and the angular aperture of the illuminating beam before the entrance into the objective is given by

Using Eq. (8) we obtain for the diffraction spot at the expense of beam convergence

In this case the current distribution in this spot corresponds to a Gaussian curve

As an example let us take 1 = 0.5 cm, f = 2 cm, fl = 0.5 cm, f2 = 1 cm, L.1 = 30 cm, L2 = 50 cm, Wo = 0.3 eV, and Vo = 30 kV. Then we obtain 60 = 105 A. On decreasing the focal lengths down to f, = 0.3 cm and f2 = 0.5 cm we obtain = 32 A. This spot can be decreased still further by introduction of the second condenser lens. However, the smaller is po, the smaller is the current density of the illuminating beam; that is, the image brightness is greatly lowered.

C. The Optimal Bias VoltageApplied to the Object

Let us calculate now the major contribution to the size of diffraction spot on the image owing to the longitudinal components of electrons of the primary beam. For precise calculation of this effect it is necessary to know the distribution function of initial energies. This function will be approximated by a function corresponding closely to Maxwellian distribution:

where WOis the most likely energy determined by the cathode temperature. This function satisfies the condition of normalization rbo

Jo

N(W)dW=I

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S. A. NEPIJKO AND N. N. SEDOV

0

2

1

3

WIW,

FIGURE 4. Distribution function of the initial electrons’ energy N ( W ) (1) and its integral func-

tion J ( W ) (2).

The relative number of electrons having energies from zero to some value WI is

Functions N ( W) and J (W) are shown in Fig. 4. The real energy distribution of electrons in the beam is determined not only by the initial energies but also by the Boersch effect, after passing the beam through all crossovers as well. Therefore, it may be wider than the thermal distribution, and it may be slightly modified. For the most precise calculations it is necessary to take the distribution function obtained experimentally for each kind of microscope. Assume that the electron beam approaches the object surface as shown in Fig. 5. With negligible bias voltage between the object and the gun cathode A V and with considerationof the contact potential difference, the lowest velocity electrons of the beam turn back at the distance from the object determined by Eq. (3), but electrons having an additional energy W approach closer to the object, and for them h=

A V - W/e EQ

Electrons with energy W 2 e A V reach the object surface and cause secondary emission from it, thus not taking part in the mirror image creation. If the microscope works in the focused operation mode, then a mixed image appears created by mirror-imaged and secondary electrons. In the defocused operation mode secondary electrons are responsible for the appearance of a blurred background weakening the image contrast.

ASPECTS OF MIRROR ELECTRON MICROSCOPY

,I I I

28 1

I%

h,

4

FIGURE5. Groups of electrons with different energies within the energy distribution of electrons N ( W ) reflect at different heights above the object.

The question now arises about the optimal value of bias voltage A V such, that the highest resolution for the mirror image is obtained. On the one hand, by decreasing A V the distribution function maximum approaches the object surface, resulting in resolution improvement. However, when it approaches for closely, the fraction of the electrons reaching the object increases. The question about the optimal bias voltage A V can be the subject of an individual investigation. In the present paper it is suggested that the optimal bias is such that 75%of the electrons of the beam are mirror image from the object. In this case the image is still mirror, and resolution is close to the optimal one. From the plot of function J ( W) shown in Fig. 4,it follows that for this purpose the bias must be

WO

A V = 2.7-

e

D. Resolution Limitation Due to the Longitudinal Velocities of Electrons

To calculate resolution limited by the longitudinal velocities of electrons, the following procedure should be adopted. Let us assume that on the object surface there is a point feature in the form of small positive or negative charge that is correspondingly equivalent to a microscopic recess or protrusion on the surface. These features cause deformation of electron trajectories as shown in Fig. 6. Redistribution of the electron beam on the microscope screen results in an image of the feature as a light or dark spot. The calculationof the current density distribution

282

S. A. NEPIJKO A N D N.N. SEDOV

bject

~FIGURE 6. Distortion of electron trajectories by positively and negatively charged peculiarities on the object.

on the screen j ( x , y) should be performed for each group of electrons with energies between W and W A W separately, and then the obtained result should be summed up with consideration for the contribution of each group according to the curve of the energy distribution. From the resulting curve we can obtain the resolution in conformity with Rayleigh's criterion. Dyukov et al. (199 1) and Sedov (1970) have shown that quantitativecalculation of the image contrast should be done as follows. The functions of the potential distribution on the object surface correspond to the object microfields. These functions should be taken as q ( x , y) for two-dimensional microfields and q ( x ) for undimensional ones. The potential distributionin the space above the object is given by the solution of Dirichlet's problem for a half-space as

+

and for a half-plane as

By solving the problem about the electron trajectories in such a field with consideration for the accelerating field E above the object, we obtain the following functions of shift of electrons from their initial position on the screen in the

ASPECTS OF MIRROR ELECTRON MICROSCOPY

283

x direction, for example. For two-dimensional microfields we get

and for unidimensional microfields

Here coordinates (x , y ) are divided by the microscope magnification. The value of the Euler’s function r ( f )= 3.6256. In the following, to simplify the calculation we will deal with unidimensional microfields, that is, we will calculate resolution along lines. Taking into consideration the fact that reflection of electrons does not take place at zero height, but at the distance h from the object, Eq. (22) is slightly changed:

S(X)

=

,/i(f+ 21) 2Jzvo

bp’(x - 4 ) J

1, w

m

d

dQTp

~

(23)

In the case of a point perturbation on the object we take the function q ( x ) in the form of Dirac’s S-function: q ( x ) = S(x). After substitution we obtain

For plotting the curves of the density distribution on the image, we need to know the distribution with respect to the function S(x): dS _ -dx

d <+f 21)

2l/zVo 4(h2 + x2)5I2 (h + 4

+J

x [x2(2h

-2 d w (

m ) 3 / 2

+7 d m ) h+d m ) ( h + 2 2x2 + d m ) ] m ) ( 6 h

(25)

Plots of these functions according to Eqs. (24) and (25) are shown in Fig. 7. The coordinate x is plotted in relative units x / a , where a = W/eEo. To plot the curves of the current density distributionon the screen, it is essential to use the following equation from Dyukov et al. (1991) and Sedov (1970):

which corresponds to the rule of conservation of field tubes for electron flow.

284

S . A. NEPIJKO AND N. N. SEDOV

-3

S

Iis

0.5

I

dx

-2

\

\

-0.5

FIGURE 7.

Curves of shift S ( x ) (1) and derivative d S / d x (2) for a point peculiarity on the object.

Here j o is the current density value in the absence of perturbations. It is clear from Eq. (26) that the shift of coordinate x at the distance S ( x ) takes place simultaneously with the current density change. This must be taken into account when plotting curves. Results of calculations according to Eqs. (24)-(26) in graphical form are shown in Fig. 8. Increasing values h correspond to numbers 1 4 . These curves are obtained for the positively charged peculiarity on the object. It is seen from curves 3 and 4 that in the center there is maximum increase as h decreases, and there are weak maxima at the edges. At some critical value h the central maximum increases ad infinitum and then splits into two parts that corresponds to the appearance of a caustic on the screen owing to intersection of electron trajectories. The total curve is plotted taking into account the contributions from different electron groups according to Eq. (14). In this case sharp maxima disappear because they become weaker, as the energy interval is reduced and there are a large number of them. In fact, each sharp maximum corresponds to a smoother curve according toEq. (13). The resultant curve for the positively charged feature is shown in Fig. 9. Using the Rayleigh criterion we can conclude from this curve that two features spaced

285

ASPECTS OF MIRROR ELECTRON MICROSCOPY

1 -1

-0.5

I

0

I

0.5

1 x/a

FIGURE8. Individual cases of brightness distribution on the screen for a positively charged peculiarity. Increasing values of height h correspond to numbers 14.

-1

-0.5

0

0.5

I x/a

FIGURE9. Resultant curve of the brightness distribution on the screen for a positively charged

peculiarity.

at the distance

are observed as separate. This distance is the resolution. For example, for WO= 0.3 eV, 1 = 0.5 cm, and VO= 30 kV we obtain SW = 250 A.

286

S. A. NEPUKO AND N. N. SEDOV

FIGURE 10. Individual cases of the brightness distribution on the screen for a negatively charged peculiarity. Increasing values of height h correspond to numbers 1-4.

-1

-0.5

0

0.5

1

./a

FIGURE1 1 . Resultant curve of the brightness distribution on the screen for a positively charged peculiarity.

Corresponding curves for a negatively charged feature are shown in Figs. 10 and 11. From these figures it is obvious that the resolution for a negatively charged feature is worse by a factor of about two than for a positively charged one. However, as the perturbation becomes weaker, this difference decreases. In this case the contrast depth also weakens, so that these features become difficult to discern on the light background. The calculationspresented are carried out for comparatively

ASPECTS OF MIRROR ELECTRON MICROSCOPY

287

weak features, for which caustics in the form of very bright and sharp points and lines, or in the form of large black spots on the screen are not visible. In these cases the resolution can get worse. E. Conclusions Resolution in MEM is determined by

where Sx in the diffraction spot due to other factors influencing resolution. Here SQ is determined by Eq. (12), and the optimal value SW is determined by Eq. (26). For the above-mentioned numerical examples, for suitable values of SA, So, and SW we obtain S = 275 A. It is also clear from these examples that in real devices the maximum contribution to 6 is made by SW while the spread of the longitudinal velocities of electrons, SQ and 61, come next. For positively charged features, looking like light spots on the screen, the resolution is about twofold better than for negatively charged ones. For weak contrast this difference decreases. Resolution in MEM depends strongly on the relative force of these features and can get considerably worse when their intensity is increased. Thus, it is possible to give a definite value of resolution only for weak perturbations. Resolution depends highly on the bias voltage between the object and the electron gun cathode. There is an optimal bias such that the resolved distance is minimum. Resolution of an emission electron microscope is written in a form similar to Eq. (26). Equation (26) corresponds to the known BrUche and Recknagel expression

S=k-

& Eo

where E = Wo/e in our designation, and dimensionlesscoefficient k was obtained by Briiche as k = 1.2 for the minimal cross section of the electron beam and by Recknagel as k = 4 for the Gaussian plane of the image. Thus, resolution of MEM is in essence determinedby the same equation as for the emission microscope resolution, but the spread of energies of the primary beam is taken instead of the initial energies of electrons from the object. One major conclusion can be drawn from Eqs. (12) and (26). To improve resolution of MEM, it is necessary to use the source of electrons with the minimum angular aperture and maximum brightness, consequently a field emission electron gun should be used.

288

S. A. NEPIJKO AND N. N. SEDOV

11. DISTORTION OF DETAILS OF OBJECT IMAGE UNDER OBSERVATION IN A MIRRORELECTRON MICROSCOPE

The observation of an object in the MEM is accompanied by a shift of the object details in their image on the screen. As a matter of fact, the image contrast appears on the screen due to such a shift. This shift may be rather large, even bigger than the size of details themselves. Therefore, this effect should be taken into account when interpreting images in M E M . The shift and distortion of details in the image can be calculated. Such a calculation enables the real sizes of details and their arrangement relating to other object features to be restored. The general equation (23) is used to calculate the image shift S(x) on the screen, but this shift value is divided by the microscope magnification; that is, the modulus of the apparent shift of every object point is given in the object coordinates. This equation is applicable for the potential distribution on the object surface depending on only coordinate x. A similar but more complicated formula can be also written for fields depending on two coordinates, x and y. In the case when the object surface is equipotential, but is not plane and is described by a function H = H ( x ) , Eq. (23) is transformed by substitution d x ) = EoH(x>

(30)

where Eo = Vo/ 1 is the field intensity of the accelerating uniform field above the object, which is created by the anode of the immersion objective of microscope. Then Eq. (23) takes the following form:

; 1,

S(x) = -

w

~ ’ ( x

Jm

t)

JW

de

(31)

For a two-dimensional distribution of microfields ~ ( xy) , on the plane object surface, the equation, analogous to Eq. (23), takes the following form for shift along the x axis:

Here the value of the Euler’s function r(:) = 3.6256. The same equation can be written for the shift along the y axis. In the case of geometrical roughness the following relationship can be used:

d x ,Y ) = E o H ( x , Y )

(33)

Equation (32) is written for h = 0, because at h # 0 this equation proves to be very complicated.

ASPECTS OF MIRROR ELECTRON MICROSCOPY

289

All the above-mentioned equations are valid in the general case; for a specific case, the corresponding integrals must be evaluated analytically or numerically. These equations may be used to estimate the image distortion for the object under study. However, in practice the image is known in most cases, but the real object geometry or the distribution of microfields on its surface is unknown. This raises the question of whether the real picture can be restored from the image, which is certain to contain distortions as compared with the real object. The solution to this question is given by Dyukov et al. (1991). The solution of this so-called inverse problem involves two steps: (1) Find the shift value S ( x , y ) for each small element of an object by means of an image of the current density distribution on the screen; and (2) calculate the real position and shape of object details. For this purpose the numerical integral transformation reported Dyukov et al. (1991) is appropriate. Let us consider the various cases of distortions on the image of object details. First we calculate the value of the image shift S ( x ) for a step with height H and the coordinate on the object surface x = 0. This step separates two semiplanes on the object. We use Eq. (31) for calculation. For such a step the derivative is defined as Dirac’s 8-function multiplied by H. In this case the integral in Eq. (31) can be evaluated and the equation is given by

The derivative of this function with respect to the coordinate x is also necessary for image plotting on the MEM screen: Sl(x) = -

H(f +20 4@h3/2

( 2 + J m ) x / h [1

+ (x/h)2]3/2d-

(35)

The function of the current distribution on the MEM screen j ( x ) is calculated according to Eq. (26). The results of this calculation are shown in Fig. 12. A relative coordinate x / h is plotted on the abscissa. Curves 1-4 correspond to the various heights H of steps. We introduce the dimensionless coefficient

This determines the relationship between the step height H and microscope parameters governing the contrast depth of the image. The coefficient values 0.5, 1, 1.6, and 5 correspond to curves 1 4 . Let us take the following MEM parameters: the accelerating voltage V = 18 kV, l = 4 mm, f = 41, h = 0.33 mm, which correspond to the negative voltage bias on the object relating to cathode of

290

S. A. NEPIJKO AND N. N. SEDOV

j/h1

-6

-4

-2

0

2

4

6

8

x/h

FIGURE12. Brightness distribution on the screen for a geometrical steps of different heights: K = 0.5 ( I ) , I (2). 1.6 (3), and 5 (4). Coefficient K gives relative height of a step.

the electron gun, which is equal 1.5 V. Then curves 1-4 correspond to the step heights H = 7.2, 14.3,23, and 72 A. The step on the screen looks like a double strip-wide weakly defined dark strip and a narrower light one, which shifts toward the lower part of the object. At iC = 1.665 the current density at a maximum tends to infinity, the shift S = 2.35h. For the previous example this shift amounts to 0.78 mm. With a further increase of the step height H,the maximum splits into two parts, and a caustic appears because of the crossing of electron trajectories. The shift of the step image visible on the screen increases further. Consider now the image of the narrow strip of width 2a on the object surface, which protrudes above this surface for short height H or, on the contrary, which is a recess of height H. The protruding strip of width 2a can be considered two steps of height H and of opposite sign with the coordinates x = -a and x = a. Thprpfnrp

S(x) = Sl(X + a ) - S,(x - a )

(37)

ASPECTS OF MIRROR ELECTRON MICROSCOPY

-8

a

-4

-2

0

2

4

29 1

6

8

xlh

FIGURE13. Curves of the contrast distribution for protruding geometrical strips of different heights: K = 0.5 (l), 1 (2), and 2 (3).

and correspondingly S’(x) = S { ( X

+ a) - s;<x - a )

(38)

Curves j ( x ) plotted for the protruding strip accordingto these equations are shown in Fig. 13. On the abscissa the relative coordinate x / h is plotted, and the ordinate is the current density on the screen in relation to the initial current density in the absence of object roughness. Curves 1, 2, and 3 are plotted for various relative values of the strip height H.Here the coefficient K takes the values 0.5,l. and 2 for the strip heights H = 7.2,14.3, and 28.6 A with the above-mentioned numerical parameters of microscope. In this case the strip width on the object amounts to 2a = 2pm. It can be seen from this example that the apparent width of the protruding strip is more than the real one, and that the apparent width increases as the step height H increases. This apparent width can be estimated correspondingly as 2.7, 3.3, and 4 pm. On a further increase of H the apparent width of the strip continues to rise indefinitely, and caustics appear at both ends of the image.

292

-8

S. A. NEPIJKO AND N. N. SEDOV

-6

4

-2

0

2

4

6

FIGURE 14. Curves of the contrast distribution for strip recess of different heights: 1 (2), 2 (3), and 3 (4).

8

x/h

= 0.5 (1).

The opposite picture is observed for the opposite sign of H, that is, when we deal with a strip recess. The corresponding images are shown in Fig. 14. The apparent width of the strip decreases as the strip becomes deeper. Curves 1 4 are plotted for the coefficient K = 0.5, 1, 2, and 3. At K = 3 the strip image contracts up to one bright line for the given parameter values. On a further increase of depth, the caustic appears again; the image looks like two bright parallel lines but the distance between them is not equal to the real width of the strip and is determined by the microscope parameters and the width and depth of the strip. A similar picture can be also obtained for other object geometries. For example, for a protrusion or a recess taking a round form on the object surface the functions of current distribution on the screen will be practically the same as for strips, but the ordinate will be the distance from its center. The type of image detail described above can be realized if there are only the divergent lenses in the optics of the imaging part of MEM. The image contrast may become negative with converging lenses (Dyukov et al., 1991). However, in all these cases, the real picture on the object surface can be reconstructed by the numerical processing of the image.

ASPECTS OF MIRROR ELECTRON MICROSCOPY

293

The following conclusions can be drawn: 1. In the MEM the shift of the image details as compared with their real arrangement on the object takes place. This shift may be quite large. 2. The sizes of details in the images of the protrusion type on the object surface can be several times greater than their real sizes. 3. The sizes of details in the images of the recess type on the object surface can be several times less than their real sizes, to the extent that such a detail can look like a light point. 4. It is possible to transform the image contrast to a negative image with the opposite distortion of sizes of the object details. An additional investigation of microscope optics is necessary to observe this phenomenon. 5. The real picture of the details on the object surface can be reconstructed by the mathematical processing of the image.

111. LIMITING SENSITIVITY OF A MIRRORELECTRON MICROSCOPE FOR OBSERVATION OF STEPS ON AN OBJECT

The mirror electron microscope is highly sensitive to object micro-geometry and to microfields on its surface. It is desirable to explain quantitatively this fact, and hence to calculate the limiting sensitivity of the MEM and to restore the real picture of features on the object surface on the basis of the observed image. This question is important because strong distortion of the image compared to the real picture of an object surface is inherent in M E M . In the present work the picture of the current density distribution on the screen of MEM in the presence of a step of small height on the smooth object surface is quantitatively calculated.The contrast depth of such an image is calculated, taking into account the step height and the microscope parameters. It allows us to calculate the minimum step height that can be observed and to establish which microscope parameters should be changed to improve the sensitivity. The equations obtained are also applicable to the potential step on an object formed, for example, by the contact potential difference between different object regions. The calculation is carried out for the defocused operation mode of MEM which is in most common use. The sensitivity of the MEM to microfields is higher in this regime than in the focused operation mode. Consider now the simplest type of MEM objective used in most devices. It consists of the plane cathode as an object K and the diaphragm with beam hole A as an anode (Fig. 1). Calculationof the distortionof electron trajectories above the object is conveniently carried out for the potential relief p = p(x, y) on the object surface, and the transition to object microgeometry H = H ( x , y) is conveniently

294

S. A. NEPIJKO AND N. N. SEDOV

effected by means of Eq. (30). Here Cartesian coordinates ( x , y ) lie in the object plane, and the z axis lies in the optic axis of the objective. For the quantitative calculation of the image contrast in MEM due to a onedimensional microfield p(x) on the object, the function (23) should be calculated. Since the shift S ( x ) is different for various points of the object, this shift results in contrast on the image, and the entire current of the electron beam is kept constant. The calculation of the current density distribution on the screen j ( x ) is carried out according to Eq. (26). For the potential step of value on the object surface, the derivative qo’(x) is expressed in terms of the Dirac 8-function:

Substitution of this function in Eq. (23) yields S(x) =

Jt(f + 2Z)po Jh + JFT? 2dVO

JE-2

(40)

The derivative S’(x) = d S / d x is also required for the calculation of image contrast:

Plots of functions S and S’ are shown in Fig. 15. On the abscissa dimensionless relative values x / h are plotted. The factor K: depending on the microscope parameters and on 90

is taken to be unity. Curves of the current density distributionon the screen, plotted for steps of various heights according to Eq. (26), are shown in Fig. 16. On the abscissa, relative units x / h are plotted, and the ordinate is the relative density of the electron current at the screen j / j o . Curves are plotted for K: = 0.5, 1, 2 and which represent the relative step-heights. The following conclusions can be drawn from these plots. A step or a potential jump on the object is observed as double strip that is wider and darker on one side ( j / j o < 1) and narrower and lighter on the other ( j / j o > 1). The boundary observed between the region of darkening and of increasing brightness on the screen does not correspond to the real position of the step at x = 0 and is shifted toward the light edging. The shift value can be calculated from the condition determining the transition point from the dark strip to the light one j / j o = 1. It

ASPECTS OF MIRROR ELECTRON MICROSCOPY

-6

-4

-2

295

0

1-03 FIGURE15. Curves of shift S ( x ) ( 1 ) and derivative S'(x) (2) for a potential step on the surface of the object.

takes place according to Eq. (26) at S' = 0. Consequently, for the point

Substituting f = 41 and cpo = HVo/l in Eq. (43) gives the simpler equation 3Hd

S(0) = -

4

The minimum height of electron reflection above the object h ~ ,is, determined from the condition s - El hnun . --- = (45) Eo Vo where E is spread of the initial energies in the electron beam expressed in volts. Let us consider a numerical example. If 1 = 4 mm, VO = 18 kV,and E = 0.3 V, we obtain h = 6.7 x lo-' m = 670 A. Then, for the step of height H = 50 A the shift value of its image is S(0) = 3.7 pm. Hence it follows that the image shift,

296

S. A. NEPUKO AND N. N. SEDOV

J

I

-4

-2

0

2

4

x/h

FIGURE16. Curves of the current density distribution on the screen for a step are plotted for K = 0.5 ( l ) , 1 (2). and 2 (3) respective to relative heights of steps.

that is, in essence its distortion, may be rather large. The real position of the step relative to other details on the object can be restored by means of the equations above. This shift decreases as the step height decreases, and for a monatomic step with H = 1 A the shift value is equal to a mere 0.074 p m under some conditions. It is also clear from Fig. 16 that, as the step height increases, the dark strip becomes narrower and, at last, becomes very narrow and bright, corresponding to the appearance of a caustic on the screen. In this case, the maximum image contrast is obtained. The caustic then splits into two parts, and the maxima move apart. The caustic appears when the coefficient K: in Eq. (42) takes the value 2.402. Let us now determine the connection between the depth of image contrast and the height of observed step H. It is convenient to use the moment at which the caustic appears at S ' ( x ) = -1 as the criterion of the contrast depth. It is clear from Eq. (41) that the function S ' ( x ) reaches a minimum value equal to -0.416 at x / h = 0.7265 without taking into account the coefficient before the function. After substitution of this value we obtain the equation for the minimum:

ASPECTS OF MIRROR ELECTRON MICROSCOPY

297

Equating this expression to - 1 one can obtain the minimum value of potential step on the object (PO, at which this step can be still observedin MEM with high contrast:

To calculate the limiting sensitivity of MEM, hminfrom Eq. (45) should again be substituted:

= 1.4 x V,This calculation If € = 0.3 V and Vo = 18 kV, we obtain pDmin confirmsthe high sensitivityof MEM to microfields and shows the possibility of observation of various regions on the object owing to the contact potential difference. Let us calculate now the maximum sensitivity of MEM to the object. To do this, Eq.(30) should be substituted for (PO in Eq. (48):

For the microscope parameters indicated above, we obtain Hmin = 3.1 x lo-'' m = 3.1 A.

If a weaker contrast is acceptable, islands of even one atomic layer thick on the object, that is, at Hmin= 1 8,will be observable. From the examples cited above it is also seen that the contrast increases considerably when surface roughness is added to the contact potential difference. Taking into account the energy distribution of primary electrons does not radically change the estimation performed earlier, but it permits us to refine the calculation of image contrast. The presence of spread of energies causes the electrons of different energies to reflect at the different heights h above the object, as shown in Fig. 5. Therefore, for the calculation of the final image, individual images produced by each group of electrons with different energies must be calculated. The resulting curves of current distribution for each group should be multiplied by the corresponding weight coefficient depending on the energy distribution. The curves thus obtained should then be added to give the final picture. The quantitative calculation of the image of the object was performed under following assumptions. The energy distribution of electrons is represented by

where the most likely energy WO = 0.5 eV. The negative bias voltage on the object A V relative to the cathode of the object gun is taken equal to 2.5Wo/e, that

298

S. A. NEPIJKO AND N. N. SEDOV

-0.4 -0.2 0 0.2 0.4 0.6 x [PI FIGURE17. Individual curves of the current density distribution on the screen for the monochromatic electron beam at negative bias voltage on the object. A V = 0.1 (I), 0.4 (2), and 0.7 V (3). Curves 1-3 correspond to different heights of reflection from the object.

-0.6

is, 1.25 V. In this case about 70% of the electrons of the beam are reflected before reaching the object surface. Let us take following parameters to make an estimate: the accelerating voltage Vo = 18 kV;the distance between object and anode 1 = 4 mm; f = 41; the step height of the object H = 2 8. Some individual curves of the current distribution on the screen are shown in Fig. 17. Curve 1 corresponds to a monochromatic electron beam which reflects at the height h = 220 8, above the object. It would take place at the negative bias voltage on the object A V = 0.1 V. Since the height of reflection is small, the contrast depth is so great that caustic is created on the image. Curve 2 corresponds to the height of reflection h = 890 8, and A V with the bias A V = 0.4 V. Curve 3 correspondsto h = 15508 and A V = 0.7 V. As the height of reflection increases, the contrast depth decreases. This suggeststhat the major contributionin the image formation is made by electrons that approach the nearest object, but their specific contribution is not the largest. Nevertheless, the resultant curve in Fig. 18 shows rather strong image contrast. Thls curve is obtained taking into account that 30% of the electrons reaching the object result in secondary emission and illuminate the screen uniformly, decreasing the contrast. The coefficient of secondary emission for these electrons is taken equal to unity. The results obtained confirm the possibility of observing steps

ASPECTS OF MIRROR ELECTRON MICROSCOPY

299

0.5-

I

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 x [PI FIGURE18. Brightness distribution on the screen for a step. The energy distribution of primary electrons is taken into sccount. Vo = 18 kV, 1 = 4 mm, H = 2 A. At the applied voltage 70% of electrons reflect and 30%of electrons reach the object.

several angstroms in height in the presence of the real spread of energies in the primary electron beam in MEM. From the calculations the following conclusions can be drawn: 1. To increase the sensitivity of MEM, it is of course, desirable to decrease the spread of energy of electrons & by monochromatization of the electron beam either before reflection from the object or after it. Decreasing & by half increases the sensitivity by a factor of 2a. 2. Under observation of the potential steps the sensitivity does not depend on the distance 1 between the object and anode. It improves slightly on increasing the anode voltage VO,because the electrons approach closer to the object. 3. Under observation of the geometrical steps the sensitivity improves with decreasing distance I , and it improves significantly with increasing anode voltage. In this case, the distortion of electric field near the object enhances the effect. Finally, one would think that the order of magnitude of the minimum height of the observed step can be easily determined by its comparison with the resolvable distance to MEM, which is equal in order of magnitude to

300

S. A. NEPIJKO AND N. N. SEDOV

For the above-mentioned numerical example S takes on the value 670 A. Since many of the electrons of the beam approach the object for this distance, the minimum observed step height would be expected to be the same order of magnitude. However, the quantitative calculation of the image contrast shows that the sensitivity of MEM is much better in reality, and it is defined by other equations. The reason is that microfields of features on the object surface, such as steps, spread for quite a considerable height above the object and act on electrons in the tangential direction along their whole path to the object and from it. Also, the potential steps have a potential difference less by several orders of magnitude than the voltage corresponding to the initial energies of electrons in the beam.

Iv.

IMAGE OF ISLANDS ON AN OBJECT SURFACE IN

MIRROR

ELECTRON MICROSCOPY Properties of small particles are actively investigated in physical electronics (Nepijko, 1985). Observation of small particles in the MEM is one way to study them. By investigating the image created by small spherical particles positioned on the smooth object surface in MEM, it is possible to determine some parameters of these particles, their sizes, and the contact potential difference between a spherical particle and a substrate. There is a spherule of diameter d = 2R on the smooth object surface. Suppose now that its potential is equal to the surface potential, i.e., there is no contact potential difference (CPD). In a uniform electric field with intensity ED.the conducting spherule of radius R creates an additional electric field around it. This field coincides with the field of an electric dipole having the following moment (Govorkov and Kupalian, 1970): p = 4n&oR3Eo

(52)

However, this situation becomes more involved because of the phenomenon of electrostatic reflection from the smooth substrate surface. To the uniform field Eo should be added the total field of a negatively charged spherule and a positively charged spherule that is its mirror-image (Fig. 19). The field configuration of this system is more complicated, but this field is also a dipole field to a high degree of accuracy, and it has another value of the electrical moment. The potential of dipole field, considering the influence of a uniform field Eo, can be represented in the form

Here the z axis coincides with the symmetry axis of the immersion objective, and r is the distance between the observation point and the axis. Components of the

ASPECTS OF MIRROR ELECTRON MICROSCOPY

30 1

I

I FIGURE 19.

Calculation of the electric moment of spherule with consideration for the electrostatic

reflection.

field intensity along the r and z axes are

and

Here EO is the dielectric constant. The value of electrical moment p of this system can be determined from the condition that the top of the real spherule has the same potential as the substrate, i.e., equal to zero (contact potential difference is not considered). In this case, electrons of the primary beam, approaching the spherule exactly along the system axis, are reflected from the spherule top as well as from the plane substrate. How to account for and measure the contact potential difference is discussed later. ) 0 and Eq.(53)it follows that From the condition q ~ ( 0 , 2 R=

p = 4I7&od3Eo

(56b)

An additional check shows that the dipole with terminating distance between charges could possess the same electrical moment, for example, if these charges are located in the centers of real and mirror-image spherules.

302

S. A. NEPIJKO AND N. N. SEDOV

A. Calculation of the Trajectories of Electron Motion Solution of the equation of electron motion in the coordinates (I, z ) d2r e -- _ - Er dt2 m d2z dt2

e Ez m

-- _ _

in the field describedby Eqs. (54) and (55)cannot be expressed in terms of elementary functions. Here e and m are the charge and mass of the electron, respectively. The problem can be solved in two ways: (1)by numerically calculating the electron trajectories, and (2) by using successive approximations. Numerical calculation can be done with high accuracy. The picture can be calculated with consideration for various factors, such as the spread of initial energies of electrons in the primary beam and the negative bias on the microscope object relative to the electron gun cathode. An example of this calculation is shown in Fig. 20. The calculation was done for the followingparameters: the accelerating voltage of MEM V, = 18 kY the anode-object distance in the objective 1 = 4 mm; the spherule radius R = 1000 A; and the negative bias on an object A V = -0.15 V. The initial electron energies in the beam were not taken into account

-6

-4

-2

0

2

4

FIGURE 20. Numerically calculated electron trajectories in mirror electron microscopy if there is a spherule on the object surface.

ASPECTS OF MIRROR ELECTRON MICROSCOPY

303

in this case. The form of trajectories of the initially parallel beam of electrons moving from the anode is shown in the figure. Due to the action of the law of similarity in electron optics, the picture of electron trajectories remains also the same for other values of the initial parameters, for example, under changing the accelerating voltage VOor the spherule radius R. In this case it is necessary to change the negative bias on the object VOproportionally to the accelerating voltage AV. Thus, in essence this picture in Fig. 20 depends on the only parameter A V / VO.

B. Calculation of the Brightness Distribution in the Image on the Screen

On the basis of these numerical calculations,the picture of brightness distribution in image, observed on the screen, can be constructed. By the brightness is meant the plot of the current density distribution on the microscope screen. For an object with circular symmetry the current density redistribution on the microscope screen can be calculated. Let j d r ) denote the initial current density at the screen in the absence of perturbations. In the presence of perturbations, electrons at a distance r from the center are shifted in the radial direction through a distance S ( r ) (Fig. 21). Then for infinitely narrow rings with radii from r to r Ar one can write the equation of conservation of the electron current:

+

+

+

2njor = 21r[j(r S)](r S)d(r

+ S)

(58)

FIGURE 21. Calculation of the redistribution of current density on the screen of mirror electron microscope in the case of circular symmetry.

304

S. A. NEPIJKO AND N. N. SEDOV

+ +

Here j ( r S ) is the new current density for points separated from the center by a distance r S. Hence,

We will convert all coordinates on the screen to a scale related to the object, i.e., these coordinates will be divided by the magnification of the microscope. Dyukov et al. (1 991) proved that the shift value Son the screen of MEM owing to the perturbation of the electron trajectories near the object surface can be calculated as follows: Av S ( r ) = -(f 2UO

+ 21)

where A v = v, is the additional velocity acquired by an electron in the radial direction due to perturbations, and

is the velocity acquired by an electron in the electron gun of MEM for the accelerating potential difference VO. f is the focal length of the objective lens of MEM, which is adjacent to the immersion objective. In most constructions of MEM the beam hole of the anode serves as such a lens. It acts as a diverging lens with the absolute value of the focal length equal to f = 41 (2). In this case Eq. (60) is simplified and takes the form

Figure 22 presents the curve of the current density distribution of the screen for the case of calculation of the electron trajectories shown in Fig. 20. It is seen from the figure that on the screen of MEM the spherule image appears as a black circle. Its dimensions far exceed the dimensions of the spherule itself. This black circle is fringed with a bright border. It is impossible to distinguish any details within the black circle. However, the size of the observed circle is uniquely determined by the spherule diameter; the spherule diameter can therefore be obtained from its image. The correspondence between these magnitudes is fully calculated later. As the numerical calculations show, the image observed on the screen does not depend in practice on the spread of initial energies of electrons or on the value of negative bias on the object unless it is too large for electrons to be reflected at the height many times higher than the spherule diameter. Therefore, these parameters were not taken into account in the following analytical calculation of image.

ASPIX'I'S 0 1 : MIKKOK El . I X " ) N

-6

-4

-2

MICROSCOPY

2

0

305

4

'Ihc cumwt ikiisity tlistrihition 011 the wrccii ol'iiiirn~rclcu'tn~aiiiicnwccyw il'thcrc is ;I sphcnilc on the ohicct wrhcc. The curves dciicitcil hy piiiits ;1rc ohtilined iit the cx~xiiw ol'clcu'tnins rcllcctcd dinu'tly ohovc il sphcnilc. LI is ;I convcnlion;1l length which is tlcpcntlciit ti11 the particular coiitlitions ol' prohkii~. M(it'itli 22.

C. Ariulytic*cilCulculrtiori of Elfctmri Trrjectorics

Thc molion of clcctmns in thc immcrsion objcctivc of MEM undcr thc action of thc uniform accclcratingficld Eo ( I ) is takcn a! thc zcro approximation of thc law. In this caw

d'; CE~) _-dt' m

(63)

"=/q dt

dt =

d;

(Ma)

(Mb)

J2Eo: ( e l m )

Thc motion of clcctrons. allowing for thc ndial ficld intcnsity of thc pcnurbation ficld Er. givcn by Eq. (54). is takcn as thc first approximation. Thc clcmcntary incrcmcnt of thc clcctron vclwity dv, acquircd within the scction d; in thc r dircction is f

dvr=-Erdt= m

Er d ;

306

S. A. NEPIJKO AND N. N. SEDOV

The full value of velocity vr acquired by an electron is O0

E,(z)dz

Here the upper limit of the integral is taken equal to infinity because for values z 1 the additional perturbation field Er is practically equal to zero. The coefficient 2 in Eq. (66) takes into account the motion of electrons to the object and in the opposite direction. It is possible to proceed to further approximations in the calculation of electron motion, but comparison with the numerical calculations shows that even the first approximation gives results that reflect quantitatively the picture of the current density distribution on the microscope screen very well. In accordance with Eqs. (60) and (62), for calculation of the shift of electrons S on the screen under the action of perturbations, it is sufficient to calculate only the radial component of electron velocity. Therefore, only the first motion integral (66) must be taken for solution of the problem. Substituting the expression for the field of an electric dipole (54) into Eq. (66), one obtains

ur =

O0

& dz (r2 z 2 ) 5 / 2

+

Calculation of this integral and the following transformations give

vr =

3P45Gm 1 4 E 0 a

[r($)3 2 , 3 1 2

(68)

:.

Here r(:) = 3.6256 is the value of Euler’s r-function at the argument equal to Let us calculate now, according to Eq. (60) and with consideration for Eq. (61), the shift S, which is visible on the microscope screen and is caused by the perturbation. One obtains K S(r) = (69) r5/2 where

The current density distribution on the microscope screen can then be obtained from Eq. (59). Curves of the current density distribution, plotted in this case, coincide very closely with the curves obtained by the numerical calculation. These curves are shown in Fig. 22. High accuracy even of the first approximation can be

ASPECTS OF MIRROR ELECTRON MICROSCOPY

307

explained by the relative weakness of fields, perturbing the electron trajectories, as compared with the field Eo. For practical use of the obtained results, it is most convenient to measure the diameter D of the black circle, observed on the screen, on the photograph, or by any other means, for example, by scanning the image. This diameter D corresponds uniquely to the diameter d of spherule itself. Below is shown how the contact potential difference (CPD) between the spherule and the substrate can be calculated. From Eq. (59) it follows that the radius R of the black circle on the screen can be obtained from the condition that the current density j tends to infinity and forms the light border around the black circle. This takes place when

dS l+;T;=o Differentiating Eq.(70)and substitutingthe resulting expression into Eq. (71).one can determine the radius of the black circle r, on the microscope screen:

After evaluating the numerical coefficient, we obtain the formula for the diameter of the black circle:

Equation (73)has a general character, and can hence be used for calculation in the case when there is CPD as well as in the absence of it. When the spherule potential is equal to the substrate one, the expression for the spherule electrical moment p from Eq. (56a) should be substituted into the formufa. Then we obtain D = 3.638

3n3f2d3( f

+ 21)

(74)

Condition (2), i.e., f = 41, is satisfied in the most MEM designs. Substituting this condition into the formula and putting together all numerical coefficients,one can obtain a very simple expression:

D = 5.3331’f7d6f7

(75)

Once again, the black circle diameter D is related to the object in scale. The numerical coefficients in all equations are taken in the International System of Units, i.e., the spherule diameter d and value of 1 should be taken in meters. Simplicity of the formula and absence of dependence on the accelerating voltage V ,from it, as mentioned above, can be explained by the action of the similarity

S. A. NEPIJKO AND N. N. SEDOV

308

law. The fact that there is no dependence on the bias voltage on the object and on the initial energies of electrons can be attributed to the fact that the diameter D is virtually unaffected by any changes. Of some interest is the dependence of the visible diameter D on the diameter d of spherule itself. The ratio of these diameters is given by

_Dd -- 5.3331‘1’ d1/7

(76)

This “magnification coefficient” of the visible diameter of spherule as a function of the diameter d of the spherule itself is shown in Fig. 23 for various values of the parameter of microscope 1. It is seen from these curves that when the spherule diameter is equal to 100 A the visible diameter of the circle is greater by a factor of 50-60, and at the diameter equal to 1000 A it is greater by a factor of 3 0 4 0 , depending on the value of the parameter 1. For a particular value of this parameter, for example, 1 = 4 mm = 0.004 m, an even simpler expression is obtained:

D = 2.42d617

(77)

The dimensions in this expression appear wrong because concrete dimensional values have already substituted. All lengths are expressed in meters here.

’\

*

I

0

2

4

6

8

igd [A1

FIGURE23. Ratio of the visible diameter D of spherule to its real diameter d for various values of parameter 1 of the microscope: I = 20 (l), 10 (2). 6 (3). and 4 mm (4).

ASPECTS OF MIRROR ELECTRON MICROSCOPY

309

These equations can be used to calculated the real spherule diameter. For example, from (75) we obtain d=

0.142D7l6 1’16

or for 1 = 4 mm from Eiq. (77) d = 0.356D7/6

(79)

D. Measurement of the Contact Potential Difference Between a Spherule and a Substrate Let us assume that the spherule has potential A 9 relative to the substrate in the presence of CPD.Under calculation of electron motion this results in a change of the electric moment p corresponding to the spherule. Let us calculate this new moment, taking into account the fact that the potential in the point on spherule top at z = d, r = 0 must now be equal to Ap. In this case from Eiq. (53) one can obtain

Hence the new spherule moment is p = 4n&0d2(dEo- A(P)

(81)

W o methods for calculating CPD by means of the spherule image in mirror electron microscope can be proposed: 1. It is possible after measuring the diameter of the black circle on the image

to deposit a thin conducting layer on the object. This layer eliminates CPD, but does not result in any essential change of the spherule diameter. The geometrical sizes of spherule can then be calculated on the base of the image. CPD is calculated from the difference between these images. This method is rather simple but the object is destroyed. 2. It is possible to obtain two images of the spherule at different accelerating voltages of microscope. In principle, other parameters can be varied, such as the distance 1 between the object and the anode of the immersion objective. If there is CPD,the black circles of the images have different diameters. So,CPD can be calculated by the use of these data. This method is better because the object remains intact. Let us consider now both methods in more detail. 1 . Let D1 denote the diameter of the circle observed on the screen if there is a

CPD;the diameter of circle observed after deposition is designated as DO.Using

310

S. A. NEPUKO AND N. N. SEDOV

Eq. (73), which was taken for the sake of simplicity for the case f = 41, one obtains

D = 3.638

{

9pl3I2,/5

I')

4EOVo [r( f

}

217

Putting all numerical coefficients together in brackets gives

(

D = 3.638 0.303-;:;)217 Then the electric moment p can be expressed in the form

Equating it to the moment from Eq. (81)yields

dVo Arp=--

2.85 x 10-3D0:12V~ d213/2

(85) 1 The spherule diameter d can be determined after deposition by means of Eq. (78). Substituting it into Eq.(85) and performing simple rearrangements gives the expression for calculating (CPD): Arp =

0 . 1 4 2 V 0 ( 0 ~-/ ~DYI2) 17/6D7I3

(86)

0

2. Let us now derive the formula for determining CPD by varying the accelerating voltage of the microscope VO. To do this, two different accelerating voltages, Vol and V02, should be taken. Different electric moments of the same spherule correspond to these voltages:

These expressions should be substituted into Eq. (83) after determining the diameters of black circles, D1 and D2. Derived Eqs. (87a) and (87b) should be considered as two equations in two unknowns d and Ap:

ASPECTS OF MIRROR ELECTRON MICROSCOPY

311

Omitting details of the solution of this system we present the resulting expressions:

Under the higher accelerating voltage, the measured dimension of the black circle on the image agrees more closely with the geometricalspheruledimension because of the relative decreaseof CPD influence on the image. If the spherule is positively charged relative to the substrate, then distortion of the uniform field above the spheruledecreases, and its visible diameterdecreases, too. Conversely, the visible diameter of the positively charged spherule increases. Thus, if the diameters of the black circles on the image are first measured at the low accelerating voltage and then at the higher one, it is possible to draw conclusions about the sign of CPD. If at the higher voltage VOthe visible diameter of the spheruledecreases,then this spheruleis negatively chargedrelative to the substrate. Conversely,if the visible diameter decreases as the acceleratingvoltage increases, then this spherule is positively charged relative to the substrate. The value of the CPD is calculated by Eq.(90), and the spherule size is calculated by Eq.(89).

v. CALCULATION OF IMAGE CONTRAST IN A MIRROR ELECTRONMICROSCOPE IN THE FOCUSED OPERATION MODE

In a MEM the defocused operation mode, having much in common with the operation mode of a shadow microscope, is usually used. However, the focused operation mode, when the object plane is optically conjugate to the microscope screen plane, can be realized, too (Dyukov et al., 1991). For this purpose the converginglenses should be placed in the path of electrons between the object and the screen, whereas it is not obligatory in the defocused regime. There are two types of MEM constructionin which the focused operation mode is realized: 1. The immersion objective of MEM is focusing. For example, this is the electrostatic objective of Briiche-Johanson’s type or the objective with the magnetic lens, which is analogous to objective of an MEM. Focusing the electron beam reflected from the objects is carried out by this objective, and the projection lenses only transfer this image on the screen (Fig. 24).

312

S. A. NEPUKO AND N. N. SEDOV

RWRE24. An optical scheme of MEM with the focused operation mode. EK is the electron gun; KLI and KL2 are the condensing lenses: MF is the deflecting magnetic field; K is the object (cathode); A is the anode; L is the immersion objective lens; PLI and PL2 are the projective lenses; E is the screen.

2. The immersion objective of the microscope is not focusing. An example of this is the two-electrode electrostatic objective containing an object and anode in the form of diaphragm with the beam hole. In this case, focusing the image is fulfilled by the first projection lens PLl (Fig. 24), which must be a long-focus lens for this purpose. Magnification of the image is carried out by the second projection lens PL2. Each of these systems has its own advantages and disadvantages. However, in all cases the focused regime is possible only with magnetic separation of the primary and reflected beams. Otherwise, as consideration of the optical paths of the rays shows, the image on the screen coincides with the hole in this screen for transmitting the initial beam. This occurs because the initial and reflected electrons go through the same lenses and retrace the path of each other with the mirror image about the optical axis. It is possible to verify that focusing the mirror image does indeed occur by the changeover from the mirror operation mode to the secondary emission one. To do this, the negative bias of the object voltage about the cathode of the electron

ASPECTS OF MIRROR ELECTRON MICROSCOPY

313

gun should be changed to a positive bias. Then, in the defocused operation mode the mirror image disappears in the screen center at first, and the diffuse light spot appears due to nonfocused secondary electrons. It is edged by the dark ring. In the focused regime, the image of object details must remain in the same place on the screen, but it changes its appearance. Here, we examine the quantitative calculation of the current density distribution on the MEM screen under the focused regime depending on arbitrary nonuniformities on the object surface, which may be geometrical roughness or electrical microfields. The first type of optical design of MEM is adopted. Equations for calculating the image contrast in MEM of the second type are different, and are the subject of separate investigation. Let us introduce a Cartesian coordinate system in which the plane ( x , y) coincides with the object surface, and the axis z coincides with the optic axis of the immersion objective. The uniform electric field close to the object is EO = Vi/l(l). Let us assume that electrical microfields on the plane surface of the object are described by the function of the potential distributionon this surface p = p(x, y). If the images is created as the result of geometric relief on the object surface h = h ( x , y), for image calculation we use the equation of conformity, which is true with reasonably high precision:

The distortion of the electrical field above the object will be practically the same (Dyukov et al., 1991). After calculating the distortion of the electron trajectories owing to the object microfields, one can determine the redistribution of the electron-beam density on the screen. By the image contrast in meant the distribution function of the current density on the screen jo(x, y). In this case, the coordinates ( x , y) correspond not to the screen, but to the object. That is, the coordinates on the screen are divided by the microscope magnification. The initial current density on the screen was j o in the absence of microfields, and without loss of generality we can consider that jo =jdx, Y). A. Geometrical Optics of the Immersion Objective in the Focused Operation Mode Let us regard the immersion objective as a thick lens with different focal lengths in the object and image spaces. However, the objective optical arrangement in the mirror regime has the unusual feature that the focal length in the object space is zero: The energy of electrons under reflection and, consequently, the corresponding

3 14

S. A. NEPIJKO AND N. N. SEDOV

FIGURE25. A scheme of electron beams in the focused regime of the immersion objective. a, b, c are incoming beams; a‘, b’, c’ are outgoing beams; K’ is virtual cathode; HI and H2 are the main planes of lens with focal lengths fl and fz.

coefficient of reflection tend to zero. To avoid this difficulty, let us single out a plane situated before the object at a distance b in such a way that the electron trajectories in this plane can still be considered to be paraxial. The uniform field region adjoining the object images the object situated at a distance 2b from the marked plane. Such an image of the object is designated as K’ in Fig. 25. The final result of plotting does not depend on the choice of the value of b. There are no regions in the remainder of the immersion objective in which the potential tends to zero, so the usual construction based on the laws of geometrical optics laws can be used in this case. Let us consider the part of the objective field far from the object as a thick lens with the focal lengths of the object space and image one equal to fI and f2 and with the corresponding principal planes H1 and H2. These planes are crossed as is usually the case in electron optics, and

”=/” f2

v2

where Vl is the potential of the plane marked above. In the case of the focused image electron beams coming from the distant source will be focused again on the distant screen. Such a path of beams is shown in Fig. 25. Changes of the beam path caused by the fact that the source and the screen are not really infinitely far.

ASPECTS OF MIRROR ELECTRON MICROSCOPY

315

Some conclusions can be drawn from plotting the beam paths in Fig. 25: 1. The spacing between an immersed point of the objective and the objective axis is related to the aperture angle 8 of beams arriving at the objective:

Therefore, the radius of the object region imaged on the screen is determined by the aperture angle of the illuminating beam. 2. The tangential velocity of electrons reflected above the object is determined by the angle a! (Fig. 25) and depends on the incident radius ro of beams at the objective. This velocity expressed in terms of the voltage corresponding to the velocity is

v, = v,tan2a! = vo

M2

(94)

Therefore, the radius of the beam arriving at the objective determines the illumination conditions, the image brightness and the microscope resolving power. The diaphragm placed in the focus plane F 2 restricts only the beam aperture, not the visual field, as in the case of the emission microscope. The diaphragm shift to the side restricts incoming and outgoing beams simultaneously and, consequently, one-sided restriction of the beam cannot be obtained in such a way. 3. The immersion objective magnification is defined only by the focal length of lens in the image space f 2 . 4. Outgoing beams are parallel to incoming ones. That is a reason why the defocused regime can be realized only with magnetic separation of the primary and reflected beams. From the above it follows that in the plane of focus F 2 the illuminating system of MEM must produce a beam with small radius, but large aperture. This can be achieved by means of two condenser lenses KL1 and KL2. The image character depends strongly on the assymmetry of the beam coincident on the objective, giving one-sided illumination. At the radius ro of the illuminating beam, electrons are reflected at various heights above the object as a function of their tangential velocities at the beam refraction. Owing to this, the microscope resolution is additionally limited by

Therefore, to obtain high resolution, the aperture diaphragm can be used. Resolution is also deteriorated because of the initial energy spread of the electron beam

316

S. A. NEPIJKO AND N. N. SEDOV

WO= eE; in this case, the Briiche-Recknagel formula (29) with k sz 1 is also valid:

B. Calculation of Electron Deviation by Microjields Consider now the trajectory of electrons reflected from the object in the region of uniform field Eo (Fig. 26). In the absence of microfields on the object, it will be a parabola, shown as a solid line. Microfields above the object result in the trajectory deviation shown by dotted line. Let us calculate the electron deviation in the directions x and y, which is visible on the screen. When an electron is at a distance z from the object, we take the plane that is parallel with the object surface and is placed at the distance z, but behind the object. It is clear from Fig. 26 that for this plane the shift observed in the direction of the axis x equals x Sx(z)= A x - ~ z ~ ~ ~ u = A x - ~ z : 2

(97)

where a! is the angle between the tangent to the trajectory and the axis z , Ax = x - xo. and xo is the coordinate of the observed point.

c" I

il

K FIGURE26. Calculation of the visible deviation S under perturbation of electron trajectories by microfields.

ASPECTS OF MIRROR ELECTRON MICROSCOPY

317

As a first approximation, assume that electron motion along the z axis depends on the field E only:

where e and m are as usual the charge and mass of electron, and t is the time. Substitution of these expressions into Eq.(97) gives Sx = AX - t i

(99)

Let us differentiate this expression with respect to time:

w e now calculate d S x / d z assuming z = z ( t ) : dS,- -- dS, d t -

dz

dt dz

m S, d S x / d t -,/2(e/m)Eoz eEot

Substituting Eq. (100) into Eq. (101) gives

The total shift observed on the screen is

where indexes 1 and 2 correspond to the electron motion toward the object and away from it. To a first approximation, the two integrals in Eq. (103) are equal in value but opposite in sign; that is, the image contrast owing to the effect of a first-order quantity disappears because of the electron motion to the object and away from it. Therefore, the image of the object details appears only due to effect of a secondorder quantity. It arises from the fact that during motion above the object the electron moves in the tangential direction, hence different forces influence it in the process of motion to the object and back. The second approximation can be found by expansion in series:

In this case

ax -- -e a2v _ ax

m ax2

318

S. A. NEPUKO AND N.N. SEDOV

where V = V ( x ,y, z) is the potential distribution function in the space above the object. For the second approximation let us assume that x - xo = uxOt

(106)

where ux0 is the electron velocity in the direction of axis x. After all substitutions in Eq. (103) we obtain

In this equation, integration over z to 00 can be used instead integration to 1 because object microfields do not usually spread out far from it. The potential distribution in the space V = V(x, y, z ) can be expressed in terms of the potential ~ ( xy), on the object surface as the solution of a Dirichlet problem for the half-space (19) taking into account the accelerating field Eo:

After substitution of this expression in Eq. (106) and integration over z as well as after series of transformations described in [ 11 we obtain &(x, y) = -

(109) Have the value of Euler's r function r($)= 3.6256. By integrating Eq.(109)by parts one can get rid of the second derivative a2V/ax2 under the integral: &(x, Y ) = -

>

[r ($ 12 ~ x o 42/2n3/2Ei'2m

@/ax) bp (x - 6, Y - tl) de dtl

e2+ V 2 Y 4

(110)

This expression is the starting point for the image contrast calculation. For microfields or microgeometry of the concrete kind, the required function ~ ( xy), should be substituted in Eq. (1 10). Then one can evaluate this integral analytically or numerically. ) on the Transition to the case of unidimentional microfields ~ ( x depending coordinate x only is carried out by integration of Eq. (1 10) over q:

ASPECTS OF MIRROR ELECTRON MICROSCOPY

319

The following conclusions can be drawn from this equation. Various kinds of image contrast are possible depending on the method of object illumination. Indeed, if the illuminating beam is unsymmetrical, then velocities U,O of some direction predominate (Fig. 25), and this results in image contrast of the first order. Under movement of the illuminating beam or under object shift, the dominant values u,. can change their sign, whereupon the image contrast becomes negative. This case can be observed in MEM construction made in accordance with the second design, that is, with divergent objective lens action. Finally, under absolutely symmetrical illumination of the object, positive and negative values ux0are equally probable. Therefore, contrastof this kind will not be seen. However, this does not mean that there is no microfield image on the screen. There are low tangential velocities caused by the reflected microfield itself, that is, effects of the next-order approximation arise. These velocity values are calculated by a method similar to the one for obtainingEqs. (1 10) and (1 11). Expressions for them are given as Eqs. (53) and (54) in Dyukov et al. (1991). For two-dimensional microfields p(x, y)

For unidimensional microfields the following expression is obtained:

Then we obtain the final expression (1 12) for the contrast of this kind:

A similar expression can be written for electron shift in the direction of the y axis. In the case of unidimentional microfields

An interesting feature of this kind of contrast is its nonlinearity with respect to the microfield amplitudes. The square-law relationship between the shift and the microfield potential value (P follows from Eqs. (114) and (1 15). As we shall see later, it results in the observation that under such a contrast any potential jump on the object independently of polarity looks like the dark line edged with light strips on its sides. Since the contrast of this kind is determined by the effects of a second-order quantity, it is weaker than the image contrast of the same object

320

S. A. NEPIJKO AND N. N. SEDOV

in the defocused operation mode of MEM. Therefore, the defocused regime is more advantageous to obtain the maximum sensitivity to small roughnesses on the object. As an illustration of the application of the equation obtained let us give the calculation of the shift S(x) on the screen for the geometrical height step ho with the coordinate x = 0. In accordance with Eq. (91) this step is changed to the potential step boo = Eoho (in modulus). One should also take into account that in MEM a negative voltage bias V is always applied to the object relative to the electron gun. In this case, electrons reflect at a height a = A V/E above the object. At this height the potential jump becomes smoother, and the potential distribution function is given by X

d. x. ,) = Eoho arctan -

l7 a When Eq. ( 1 16) is substituted into Eq. (1 15) integrals can be calculated analytically:

S(x) =

Jm[Jm-Jm

2 J k

1

+ (x/a)*

(1 17)

To calculate curves of the current density distribution on the screen we also need the derivative of this functions S’(x) = d S / d x . Plots of function S ( x ) and its derivative are shown in Fig. 27. For plot S(x) we set the coefficient

equal to 1, and for plot S’(x) coefficient

is taken equal to 1. C. The Current Density Distribution on the Microscope Screen

If there are no diaphragms in the path of electrons reflected from the object, then the full current of the electron beam is retained. Because of this the image on the screen appears only at the expense of redistribution of the electron beam. The new current density distribution on the screen j ( x ) may be calculated with the help of Eq. (26) at known functions of displacement S ( x ) . (The two-dimensional case is described by Dyukov et al. (1991).) As it follows from Eq. (26), it is necessary to

ASPECTS OF MIRROR ELECTRON MICROSCOPY

32 1

FIGURE27. The displacement function S ( I ) and its derivative S’ (2) in the case of a step on the object surface.

know both the displacement S(x) and the derivatives of this function with respect to coordinate S’(x). Figure 28 gives the curves of the current density distribution calculatedby means of Eqs. (117) and (26) for a step for values of the coefficient K = 0.5, 1, and 2. It corresponds to various step heights in accordance with Eq. (117). From this figure we notice that the step does look like a dark strip (minimum density is at the center) with brighter edging on its sides. An interesting problem involves calculating the maximum sensitivity of MEM to a geometrical step in the focused regime using reported data. The criterion of high contrast is S ;,, = 1. In this case,

S. A. NEPIJKO AND N. N.SEDOV

322

,

I

4

-2

0

I

2

4

xfa

FIGURE 28. Functions of the distribution current density j in the case of a step on the object. Coefficient K gives relative height of a step.

Equating it to 1 yields fiAV h nun. -Eo In practice it is possible to take A V z &, and so according to Eq. (96) we have

h nun. -

a& - As, EO

that is, the height of the steps observed in the focused regime and the MEM resolution are of the same order. For example, for VO = 18 kV, 1 = 4 mm, and A V = 0.3 V, we obtain h ~ ,=, 0.1 pm. The results are considerably better than in the defocused regime. Steps of lesser height can be observed but with lower contrast. Similar calculations can be made for details having any other form on the object surface.

D. Conclusions 1. The focused operation mode of MEM can be realized only in devices with magnetic separation of the primary beam and the reflected beam.

ASPECTS OF MIRROR ELECTRON MICROSCOPY

323

2. Two types of h4EM construction with the focused regime are possible: first, with the immersion objective acting as a converging lens as in an emission microscope; and second, with diverging objective lens action. Only the first type of microscope is discussed in this paper. 3. The contrast depth from uniform details on the object surface in the focused regime is considerably less than in the defocused regime. 4. The image contrast depends on the symmetry of the illuminating beam, and contrast can become negative with symmetry perturbation. 5. With symmetrical illumination of the object, the contrast depth is proportional to the square of the value of the microfield voltage on the object or of roughness depth on the surface. 6. To calculate the contrast as a function of the microgeometry on the object surface, one can use the same equations as for the electrical microfields on the object. 7. With symmetrical illumination, geometrical or potential steps on the object surface look like a dark line edged by light strips independently of their direction.

ACKNOWLEDGMENTS

The authors thank E. Bauer for his interest and stimulatingdiscussions. One of us (S.N.) gratefully acknowledges support by the Deutsche Forschungsgemeinschaft.

REFERENCES Dyukov, V. G., Nepijko, S. A., and Sedov, N. N. (1991). Electron Microscopy of Local Potentials. Naukova Dumka, Kiev (in Russian). Govorkov, S . A., and Kupalian. S. D. ( I 970). Theory of Electromagnetic Field in Exercises and Pmblems. Vysshaja shkola, Moscow (in Russian). Gvosdover, R. S . , and Zel’dovich, B. Ya. (1973). The quantum theory of image contrast formation of electrical and magnetic microfields in mirror electron microscopy. J. Microsc. 17, 107-1 30. Nepijko, S . A. (1985). Physical Properties of Small Metal Particles. Naukova Dunka, Kiev (in Russian). Sedov, N. N. (1970). ThBorie quantitative des sysPmes en micmscopie tlectronique B balayage, B miroir et B Bmission. J. Microsc. 9, 1-26. Wiskott, D. ( 1956). Zur Theorie des Auflicht-Elektronenmikroskops,II: Wellenmechanische Elektronenoptik in der Umgebung des Objekts. Oprik 13,481-493.

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Index

A

logic circuit implementation using, 254-256

Aberrations of imaging lenses, 276, 277-278 Absorbed boundary conditions (ABC), 80 Additive pattern transfer, 125-132 AIF3 films, 102-103,104 Algorithm implementation, 256 Ampere’s law, 5 constitutive error approach and, 9 neglecting displacement current in, 8 Angular spread, 276,277,278-279 Applied Computational Electromagnetic Society (ACES), 2 Atomic-force microscope (AFM),89, 114 exposure of resist materials, 116 localized electrochemical modification, 118-119

C CARIDDI, 33,37-39 Cartesian coordinate system, 14 Cavities, analysis of resonant closed, 69-78 Chromatic aberration, 96,22&222 Computer communications applications, interconnections and, 262 Constitutive equation, 6 , 7 Constitutive error approach, 3,9,50 analysis of resonant cavities, 69-78 eddy current problem, 54-63 electromagnetic problem, 63-69 magnetostatic problem, 5 1-54 open boundary problems, 78-8 1 Contact potential difference (CPD), 300, 307,309-3 1 1 Coordinate transformations, open boundary problems and, 79-80 Coulomb blockade, nanofabrication and, 162-168 Coulomb gauge, 12 Crossover interconnections, 237,238,239 Current density distribution, 320-322 Current density vector potential, 11

Banyan interconnections, 237,238, 239 Biot-Savart law, 41 Boundary conditions, edge elements, 25-30 Brightness distribution, calculating, 303-304 Butterfly interconnections, 237,238, 239-240

325

326

INDEX

D Degrees of freedom, edge elements, 23, 24-25 Deposition, 97 electron-beam lithography, 105 focused ion beam, 112-1 14 scanning probe microscopes, 121 Discrete Hartley transform (DHT), 259-262 Dynamic Random Access Memory (DRAM) technology, 188

Eddy current problems computational electromagnetics and 3D background of, 3-4 basic eddy current problem, 6-9 conclusions, 81-82 constitutive error approach, 3,9,50-81 edge elements, 3 4 , 15-30 field equations and material properties, 4-9 fields and, 10-1 1 gauges, 12-15 linear, 3 1-39 nonlinear, 38-50 vector and scalar potentials, 11-12 Edge elements boundary conditions, 25-30 degrees of freedom, 23,2425 development of, 3-4, 16 shape functions, 17-22 tree-cotree decomposition, 16,20-25 for vector potentials and 3D field problems, 15-30 Whitney, 19-20 Einzel lens, performance of stacked, 225-232

Electromagnetic model, classic, 4-5 Electromagnetic problem, constitutive error approach and, 63-69 Electron-beam exposure, 138-146 Electron-beam lithography (EBL), 88-89 damage and modification of materials, 104 deflection aberrations, 96 deposition and etching, 105 development of, 91-92 inorganic resists, 101-104 lanthanum hexaboride tips, 93,95 organic resists, 97-101 systems, 92-97 thermal field-emission guns, 95 types of machines used, 91-92 vector versus raster scan, 92 Electron deviation by microfields, calculating, 3 16-320 Electroplating, 130-132 Electrostatic lenses fabrication of, 189-21 1 scaling laws for, 189 Etching, 133 dry, 134-137 electron-beamlithography, 105 focused ion beam, 110-1 11 wet, 133-134

F Faraday’s law, 5,8-9 constitutive error approach and, 9 Ferroelectric liquid-crystal (FLC) interconnections,250-25 1 Field-effect-transistor-SEED (FET-SEED), 241,242,249 Field equations and material properties, 4-9 Fields, eddy current and, 9-10

327

INDEX

Finite element approach to linear eddy current problems, 31-38 to nonlinear eddy current problems, 44-50 Finite Element Method-Boundary Element Method (FEM-BEM), 16 open boundary problems and, 80 Focused ion beams (FIBS),89 deposition, 112-1 14 etching, 110-1 11 implantation, 111-1 12 lithography, 108-1 10 mass separators, 107 uses for, 105-107 Free-space optical interconnections, 237-242 applications,253-256 architectures,249-250 problems and possibilities, 264-265

Gauges, 12-15 Guided-wave optical interconnections,248 applications,262-263 architectures,253 problems and possibilities, 266-267

Incomplete Cholesky Conjugate Gradient (ICCG), 53-54 Index guided interconnections,237 International ThermonuclearExperimental Reactor (ITER), 39 Inorganic resists, 101-104 Iterative procedures, nonlinear eddy current problems and, 4 1 4

L Laplace transforms, 55 Lennard-Jones potential, 154 LiF films, 103-104 Lifshitz theory, 153 Liftoff, 125-130 LIGA lathe, 203-21 1 Limiting sensitivityof MEM, calculating, 293-300 Linear eddy current problems finite element approach to, 32-39 formulation, 3 1-32 Lipschitz condition, 43 Liquid-crystal implementation, 245-246 Lorentz gauge, 12

M H Hamaker constant, 153

I Image contrast in focused operation mode, calculating, 3 11-322 Image of islands, calculating, 300-3 11 Implantation, 97,111-1 12 Imprint technology, 123

Magnetic nanostructures, 172-174 Magnetostaticlenses, fabrication of, 211-212 Magnetostatics constitutive error approach and, 5 1-54 nonlinear eddy current problems and, 3841 Maxwell equations, quasi-stationary analysis of resonant cavities and, 74 eddy current problem and, 9 electromagneticproblem and constitutiveerror approach, 63-65

328

INDEX

Mesoscopic physics, 90 Miniature scanning electron microscope (MSEM), 187 applications, 187-189 chromatic aberration calculation, 220-222 construction, 225-228 detector problems, 217-220 electron optical calculations, 220-224 electron source, 2 12-2 17 fabrication of electrostatic lenses, 189-2 11 fabrication of megnetostatic lenses, 211-212 future of, 232 LIGA lathe, 203-21 1 miniaturizationmethods, types of, 190-191 operation and image formation, 228-232 performance of stacked Einzel lens, 225-232 scaling laws for electrostatic lenses, 189 slicing, 198-203 stacking, 191-197 tilted, 224 Mirror electron microscopy (MEM), 273 aberrations of imaging lenses, 276, 277-278 angular spread, 276,277 angular spread, impact of, 278-279 brightness distribution, calculating, 303-304 conclusions, 322-323 contact potential difference, 300, 307, 309-3 11 current density distribution, 320-322 diffraction, 276-277 distortions, 288-293 electron deviation by microfields, calculating, 316-320 energy spread, 276,277 finite height, 276,277

geometric optics, calculating, 3 13-3 16 image contrast in focused operation mode, calculating, 31 1-322 image of islands, calculating, 300-3 11 limiting sensitivity of, calculating, 293-300 longitudinal velocities, impact of, 281-287 optimal bias voltage applied, 279-281 resolution of, 274-287 trajectories of electron motion, calculating, 302-303,305-309

N Nanofabrication electron-beam lithography, 88-89, 91-105 focused ion beams, 89, 105-1 14 other fabrication techniques, 123-124 pattern transfer, 125-137 resolution differences, comparison of, 125 resolution limits of organic resists, 138-1 58 scanning probe microscopes, 89, 114-121 top-down method, 88 X-ray lithography, 89, 121-123 Nanofabrication applications, 158 Coulomb blockade, 162-168 magnetic nanostructures, 172-174 quantum dots, 168-171 quantum interference, 159-161 superlattices, 161-162 Nanotechnology conclusions, 174-176 development of, 87-90 Nonlinear eddy current problems finite element approach to, 44-50

INDEX

iterative procedures for, 4 1 4 4 magnetostatics and, 39-41

Omega (shuffle/exchange) network, 253 Open boundary problems, constitutive error approach and, 78-8 1 Optical interconnections applications, 253-263 architectures, 249-253 banyan, 237,238,239 butterfly, 237,238,239-240 clos, 237 conclusions, 267-268 crossover, 237,238,239 free space, 237-242,249-250,253-256, 264-265 guided-wave, 248,253,262-263, 266-267 index guided, 237 liquid-crystal implementation, 245-246 packaging of, 263-264 perfect shuffle, 237-238,239,240 prismatic mirror array implementation, 240 problems and possibilities, 264-267 real-time holographic, 244-245, 250-252 reconfigurable, 243-246,256-259, 265-266 role of, 236-236 self-electrooptic effect device implementation, 240-242 3D, 246-247,252-253,259-262,266 types of, 236-248 Optoelectronics, 168-171 Organic resists description of, 97-101 resolution limits of, 138-158

329

P Pattern transfer techniques, 97 additive, 125-132 electroplating, 130-132 etching, 133-137 liftoff, 125-130 purpose of, 125 Perfectly Matched Layer (PML) method, 80-8 1 Perfect shuffle interconnections,237-238, 239,240 Photorefractive volume holographic i n t e r c ~ ~ e c 250,251-252, ti~~~, 265-266 Picard-Banach procedure, 42 44,49 Planar process, 97 Polar molecules, 151-156 Poly(glycidy1 methacrylate) (PGMA) resist, 116 Poly(methy1 methacrylate) (PMMA) resist, 97 degradation of, under irradiation, 144-146 development of exposed, 146-157 dissolution of, 147-149 electron-beam exposure, 138-146 electron-beam lithography, 99-101 electron scattering in resist and Substrate, 139-140 focused ion beam, 108-1 10 intermolecular forces, 151-156 liftoff, 126-130 LIGA lathe, 203-21 1 molecule size, 141-144 solubility Of, 149-15 1 statistical error, 140-141 swelling, 157 Poynting theorem, 42 Prismatic mirror array implementation, 240

330

INDEX

Q Quantum dots, nanofabrication and, 168-1 7 1 Quantum interference, nanofabrication and, 159-161

Reactive ion etching (RE), 134-135, 190 Real-time holographic interconnections, 244-245 architectures, 250-252 Reconfigurable optical interconnections, 243-246 applications, 256-259 problems and possibilities, 265-266 Reverse-biased PN, 217-218

S Scalar potentials, 11-12 Scaling laws for electrostatic lenses, 189 Scanning probe lithography (SPL),115 Scanning probe microscopes (SPM), 89 atomic-forcemicroscope, 89, 114 deposition, 121 direct removal and manipulation of particles, 119-121 exposure of resist materials, 115-1 16 localized electrochemicalmodification, 116-1 19 operation of, 114-1 15 scanning tunneling microscope, 89, 114 Scanning tunneling microscope (STM), 89, 114 exposure of resist materials, 115-1 16 localized electrochemicalmodification, 116-1 18 Schottky junction, 217-218 Self-electroopticeffect device (SEED) implementation, 240-242

Shape functions, edge element, 17-22 Silicon source, 214-217 Slicing, 198-203 Spindt source, 2 12-214 Spurious modes, 69-70 Stacking, 191-197 performance of stacked Einzel lens, 225-232 Stationary fields, 5-6 Superlattices, nanofabrication and, 161- 162 Swinging Objective Immersion Lens (SOIL) system, 96

T Testing of Electromagnetic Analysis Methods (TEAM), 2 3D optical interconnections. 246-247 applications, 259-262 architectures, 252-253 problems and possibilities, 266 Through-wafer interconnections, 253 Tokamaks, 3,39 Trajectories of electron motion, calculating, 302-303,305-309 Tree-cotree decomposition, 16.23-28 Truncation, open boundary problems and, 79

v van der Waals force, 151-154 Variable Axis Immersion Lens (VAL) system, 96 Vector potentials, 11-12 edge elements for, 15-30 popularity of, 15

X X-ray lithography, 89, 121-123

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