Advances in Electronics and Electron Phisics. Vol. 49

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 49

CONTRIBUTORS TO

THISVOLUME

Georges J. BtnC F. T. Chan G. Foster M. Lieber Edward F. Ritz, Jr. A. V. Rzhanov K. K. Svitashev A. van der Ziel W. Williamson, Jr.

Advances in

Electronics and Electron Physics EDITEDBY L. MARTON AND C. MARTON Washington, D .C.

EDITORIAL BOARD E. R. Piore T. E. Allibone M. Ponte H. B. G . Casirnir W. G . Dow A. Rose A. 0. C. Nier L. P. Smith F. K. Willenbrock

VOLUME 49

1979

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London Toronto Sydney San Francisco

COPYRIGHT @ 1979, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDINO PHOTOCOPY, RECORDINO, OR ANY INFORMATION STORAOE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NW1 IDX

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 49-7504 ISBN 0-12-014649-5 PRINTED IN THE UNITED STATES OF AMERICA

19 80 81 82

9 8 16 5 4 3 21

CONTENTS CONTRIBUTORS TO VOLUME 49 FOREWORD . . . . . .

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

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

Ellipsometric Techniques to Study Surfaces and Thin Films A . V. RZHANOV AND K . K . SVITASHEV I . Introduction . . . . . . . . . . . . . . . . . I1. Calculation of the Relative Reflectivity Coefficient for Multilayer Reflecting Systems . . . . . . . . . . . I11. Experimental Techniques for Determining the Relative Reflectivity Coefficient . . . . . . . . . . . . . . . IV. Examples of Experimental Ellipsometric Studies of Some Systems . . V. Conclusion . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

vii ix

1 6 20 43 75 78

Foundations and Preliminary Results on Medical Diagnosis by Nuclear Magnetism GPORGESJ . B ~ ~ N I ? Part 1 : A New Method in Medical Diagnosis . . . . . . . . I . Introduction . . . . . . . . . . . . . . . . . I1. Information That Can Be Obtained by Nuclear Magnetism . . . . I11. The Relaxation of a Magnetic Dipole in the Water of Biological Substances . . . . . . . . . . . . . . Part 2: Application of the Techniques of NMR to the Study of Biological Tissues . . . . . . . . . . . . . . . IV. The Techniques . . . . . . . . . . . . . . . . . V . Application to the Study of Biological Tissues in Vitro (Biopsis) . . . VI . Application to Measurements in Situ . . . . . . . . . . . VII . Final Remarks . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . Note Added in Proof . . . . . . . . . . . . . . .

109 109 112 119 129 130 132

Applications of the Glauber and Eikonal Approximations to Atomic Collisions F. T . CHAN. M . L m W . G . FOSTER. AND W . WILLIAMSON. JR. 1. Introduction . . . . . . . . . . . . . . . . . I1. Eikonal Approximations for the Scattering Amplitudes . . . . . I11. Electron Scattering from Neutral Atoms . . . . . . . . . IV. Ionization ofNeutral Atoms by Electron Collisions . . . . . . V. Electron Scattering from Ions . . . . . . . . . . . . . VI . Other Eikonal-Type Approximations . . . . . . . . . .

134 137 148 183 199 205

V

86 86 87 95

vi

CONTENTS

VII . Applications of the Glauber Approximation to Electron-Molecule Collisions . . . . . . . . . . . . . VIII . Summary and Conclusions . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

216 217 219

Flicker Noise in Electronic Devices A . VAN DW ZIEL I . Introduction . . . . . I1. Unusual Examples of l/fNoise 111. Flicker Noise in Vacuum Tubes IV . Flicker Noise in Resistors . . V . l/fNoise in Solid-state Devices VI . Miscellaneous Problems . . References . . . . . .

. . . . . . . . . . . . 225 . . . . . . . . . . . . 231 . . . . . . . . . . . .

234

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

292

. . . . . . . . . . . . 244 . . . . . . . . . . . . 269 . . . . . . . . . . . . 290

Recent Advances in Electron Beam Deflection EDWARD F. RITZ.JR . I . Introduction . . . . I1. Magnetostatic Deflection . 111. Electrostatic Deflection . IV. Traveling-Wave Deflection V . Scan Magnification . . . VI . Conclusion . . . . . References . . . . .

AUTHOR INDEX . SUBJECT INDEX .

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

299 303 . . . . . . . . . . . . . 323 . . . . . . . . . . . . . 345 . . . . . . . . . . . . . 349 . . . . . . . . . . . . . 354 . . . . . . . . . . . . . 355

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

. . . . . . . . . . . . . . . . . . 359 . . . . . . . . . . . . . . . . . . 37 1

CONTRIBUTORS TO VOLUME 49 Numbers in parentheses indicate the pages on which the authors’ contributions begin.

GEORGESJ. BBNB, DCpartement de Physique de la Matike CondensCe, Section de Physique, UniversitC de Genke, 24, Quai Ernest Ansermet, CH-121 I-Genkve-4, Switzerland (85) F. T. CHAN,University of Arkansas, Fayetteville, Arkansas 72701 (133) G. FOSTER,*Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309 (133) M. LIEBER, University of Arkansas, Fayetteville, Arkansas 72701 (133) F. RITZ,JR., Tektronix, Inc., Beaverton, Oregon 97005 (299) EDWARD Institute of Semiconductor Physics, Academy of Sciences A. V. RZHANOV, of the USSR, Siberian Branch, Novosibirsk, USSR (1) K. K. SVITASHEV, Institute of Semiconductor Physics, Academy of Sciences of the USSR, Siberian Branch, Novosibirsk, USSR (1) A. VAN DW ZIEL,Electrical Engineering Department, University of Minnesota, Minneapolis, Minnesota 55455 (225) W. WILLIAMSON, JR., University of Toledo, Toledo, Ohio 43606 (133)

* Present address : University of Connecticut at Torrington, Torrington, Connecticut 06790. vii

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FOREWORD

Five excellent papers make up Volume 49. Surface studies and ellipsometry are part of one of the most active fields of research. The paper by Rzhanov and Svitashev reviews the applications of the techniques of study and provides a comprehensive bibliography that encompasses research done in the USSR as well as in the West. The promise of rapid and precise medical diagnosis by nuclear magnetism is presented in the second article by BCnC. The principles of the subject are developed for the engineer and the scientist involved in biomedical research. Chan, Lieber, Foster, and Williamson deal with Glauber and eikonal approximations in atomic scattering, concentrating on the determination of atomic excitation cross sections in collisions with electrons. Dealing with a problem in electronics itself, van der Ziel provides an analysis of flicker noise in modern electronic elements. The applications of electron beams have increased in recent years. A review of the current analysis techniques for controlling the beams is given by Ritz. Following our custom, we now list the titles of future reviews, with the names of their authors. The listings are given in three categories: first, regular critical reviews; second, as usual, supplementary volumes; and third, a special listing of Volume 50 of this serial publication. This fiftieth volume, marking a kind of anniversary, will be devoted entirely to historical presentations of different subjects in electronics and electron physics. Critical Reviews: The Gum-Hilson Effect A Review of Application of Superconductivity Sonar Electron Attachment and Detachment Electron-Beam-ControlledLasers Amorphous Semiconductors Electron Beams in Microfabrication. I1 Design Automation of Digital Systems. I and I1

Spin Effects in Electron-Atom Collision Processes Electronic Clocks and Watches Review of Hydromagnetic Shocks and Waves Beam Waveguides and Guided Propagation Seeing with Sound Large Molecules in Space Recent Advances and Basic Studies of Photoemitters Josephson Effect Electronics ix

M. P. Shaw and H. Grubin W. B. Fowler F. N. Spiess R. S. Berry Charles Cason H. Scher and G. Pfister P. R.Thornton W. G. Magnuson and Robert J. Smith H. Kleinpoppen A. Gnadinger A. Jaumotte & Hirsch L. Ronchi A. F. Brown M. and G. Winnewisser H. Timan M. Nisenoff

X

FOREWORD

Signal Processing with CCDs and SAWS Present Stage of High Voltage Electron Microscopy Noise Fluctuations in Semiconductor Laser and LED Light Sources X-Ray Laser Research Energy Losses in Electron Microscopy The Impact of Integrated Electronics in Medicine Design Theory in Quadrupole Mass Spectrometry Ionic Photodetachment and Photodissociation Electron Interference Phenomena Electron Storage Rings Radiation Damage in Semiconductors Solid-state Imaging Devices Particle Beam Fusion Resonant Multiphoton Process Magnetic Reconnection Experiments Cyclotron Resonance Devices The Biological Effects of Microwaves Advances in Infrared Light Sources Heavy Doping Effects in Silicon Spectroscopy of Electrons from High Energy Atomic Collisions Solid Surfaces Analysis Surface Analysis Using Charged Particle Beams Low Energy Atomic Beam Spectroscopy Sputtering Photovoltaic Effect Electron Irradiation Effect in MOS Systems

Light Valve Technology High Power Lasers Visualization of Single Heavy Atoms with the Electron Microscope Spin Polarized Low Energy Electron Scattering Defect Centers in 111-V Semiconductors Atomic Frequency Standards Interfaces Reliability High Power Millimeter Radiation from Intense Relativistic Electron Beams Solar Physics Auger Electron Spectroscopy Fiber Optic Communication Systems Microwave Imaging of Subsurface Features Novel MW Techniques for Industrial Measurement! Diagnosis and Therapy Using Microwaves

W. W. Brodersen and R. M. White B. Jouffrey H. Melchior Ch. Cason and M. Scully B. Jouffrey J. D. Meindl P. Dawson T. M. Miller M. C. Li D. Trines N. D. Wilsey E. H. Snow A. J. Toepfer P. P. Lambropoulos P. J. Baum R. S.Symous and H. R. Jory H. Frohlich Ch. Timmermann R. Van Overstraeten D. Berknyi M. H. Higatsberger F. P. Viehbiick and F. Riidenauer E. M. Horl and E. Semerad G. H. Wehner R. H. Bube J. N. Churchill, F. E. Holmstrom, and T. W. Collins J. Grinberg V. N. Smiley J. S. Wall D. T. Pierce and R. J. Celotta J. Schneider and V. Kaufmann C. Audoin M. L. Cohen H. Wilde T. C. Marshall and S.P. Schlesinger L. E. Cram P. Holloway P. W. Baier and M. Pandit A. P. Anderson W. Schilz and B. Schiek M. Gautherie and A. Priou

xi

FOREWORD

Electron Scattering and Nuclear Structure Electrical Structure of the Middle Atmosphere Microwave Superconducting Electronics

G. A. Peterson L. C. Hale R. Adde

Supplementary Volumes: Image Transmission Systems High-Voltage and High-Power Applications of Thyristors Applied Corpuscular Optics Microwave Field Effect Transistors

G. Karady A. Septier J. Frey

Volume 50: Early History of Accelerators Sixty Years of Electronics Thermoelectricity to 1850: A New Phenomenon Ferdinand Braun: Forgotten Forefather From the Flat Earth to the Topology of Space-Time History of Noise Research Power Electronics at General Electric: 1900- 1941 Evolution of the Concept of the Elementary Charge

M. S. Livingston P. Grivet B. S. Finn Ch. Siisskind H. F. Harmuth A. van der Ziel J. E. Brittain L. Marton and C. Marton

W. K. Pratt

As in the past, I have enjoyed the friendly cooperation and advice of many friends and colleagues. Our heartfelt thanks go to them, since without their help it would have been almost impossible to issue a volume such as the present one.

L. MARTON C. MARTON

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL. 49

Ellipsometric Techniques to Study Surfaces and Thin Films A. V. RZHANOV

AND

K. K. SVITASHEV

Institute of Semiconductor Physics Academy of Sciences of the USSR Siberian Branch, Novosibirsk. USSR

I. Introduction.. ........................................................... 11. Calculation of the Relative Reflectivity Coefficient for Multilayer Reflecting Systems .............................................. A. Procedure for Calculating the Relative Reflectivity for a Multilayer Reflecting System Containing Optically Inhomogeneous-in-Thickness Layers ........... B. Relative Reflectivity Coefficient for Several Types of Thin-Film Systems ...... C. Analysis of the Situation; Horizons for Further Investigations ........ 111. Experimental Techniques for Determining the Relative Reflectivity Coefficie A. General Expression for the Operating Light Beam Intensity at the Output of an Ideal Ellipsometer. ............................................... B. Formulas for Estimating Angles of Polarization )I and A by Experimentally Measured Values. ............................. .................... C. Automation of Ellipsometric Measurements .............................. D. Present State of the Question; Problems and Horizons. ..................... 1V. Examples of Experimental Ellipsometric Studies of Some Systems ............... A. Measurement of Thin-Film Parameters. .............................. B. Studies of Adsorption--&sorption Processes by Means of Ellipsometry . . . C. Determination of the Optical Characteristics of Materials ............... V. Conclusion .............................................................

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

1 6

6 15

22

28 36 39 43

75 78

I. INTRODUCTION The continuing development and refinement of measurement techniques is the major factor determining progress in any field of natural science. Every new piece of information is connected, in some way, with an increase in the sensitivity, precision, and localization of the measurements and with the expansion of the range of conditions under which the measurements can be carried out. At the present time additional demands are being made on measurement techniques. Of great importance, for example, are the theoretical possibility and technical realizability of measuring process automation. Of no less importance is the opportunity to measure many different object characteristics under the same conditions, thereby improving essentially the general informativeness of the study. 1

Copyright Q 1979 by Academic Rcss. Inc. Ail rights of reproduction in any form reserved. ISBN 0-12-014649-5

2

A. V. RZHANOV AND K. K. SVITASHEV

On the other hand, progress in computing techniques has reduced drastically the requirement for simplicity of the functional connections between the directly measured quantities and those that are of interest to an experimenter. The processing of the measurement data allows such techniques to be used successfully where these connections are highly complicated and require cumbersome calculations and transformations. In many cases these functions can be performed by minicomputers and microprocessors fitted directly into the measuring equipment. The rapid increase in use of the method of reflective polarimetry, or, as it is now commonly known-ellipsometry, is a striking example of the tendencies mentioned above. The method involves the measurement of the changes that occur in light beam polarization upon its reflection from the object under study. The work of Malus and Brewster initiated the studies in this field more than a century and a half ago. Drude (1887, 1889, 1890) and Strutt (1871, 1892, 1907, 1912) made fundamental contributions to the development of the general physical notions and to the creation of the theory. Later on separate and frequently rather successive attempts were made to apply the method to the solution of specific problems of surface phenomena in physics and physical chemistry. However, the use of reflective polarimetry has not been wide spread over a long period of time. The complexity and cumbersomeness of the mathematical treatment of the measurement results were the main obstacles. The general theoretical relations of ellipsometry may be simplified only for the simplest reflecting systems, so that it becomes possible to determine directly the quantities of interest through the measured polarization characteristics of the reflected light beam. In more complicated and interesting cases such direct determinations of the quantities to be measured appear to be impossible. As a rule, it is necessary to make preliminary calculations of the polarization characteristics for the model of the rcflecting system corresponding to the object under study. Only comparison of the experimental results with the calculation data permits us to determine the object characteristics that are of interest. It is clear that such complexity of the measuring procedure severely inhibited the expansion of the method's areas of application and did not stimulate the perfection of its instrumentation. On the other hand, the ellipsometric research method has one very important feature that places it beyond compare with regard to other optical techniques. Since the surface layer of the object under study plays the determining role in light reflection, this method can be used to obtain unique information on the structure of this layer. In this sense, ellipsometry has been a pioneer in the whole trend, which nowadays also covers LEED,

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

3

RHEED, ESCA, SIMS, Auger spectroscopy, and a number of other techniques. The evolution of adsorption studies and the chemistry of surface reactions has prompted investigators to address ellipsometry from time to time in spite of the difficulties mentioned above. However, the determining role in the intensive development of the method was played, as usual, by the practical requirements for technological progress. The state of surface and semiconductor presurface layers turned out to be of great importance in the early stages of the developmen; of semiconductor electronics. Further progress toward the miniaturization of semiconductor devices only redoubled the requirements for the control and passivation of their surfaces. Finally, the development of microelectronics transferred the center of attention of all or almost all major electronics research to semiconductor presurface layers and the interfaces between semiconductors and dielectric films. In this connection, the problems concerning the study of the interfaces between the layers of different materials and the measurement of the characteristics of the layers came to be of paramount importance. Ellipsometry with its unique capabilities for studying and measuring thin and superthin films during the process of their growth acquired special significance. It should be noted that this significance became even greater with the emergence and development of new directions, such as optoelectronics, acoustooptics, and integrated optics. The values of the optical constants and the changes of these values under different ambient influences are directly controlled by ellipsometry. The propagation of surface acoustic waves in crystals and multilayer systems and of electromagnetic waves through waveguides depends substantially on the surface treatment of solids and the interface structure within them and can be studied by this method as well. It is essential that ellipsometric measurements be noncontact and nondestructive. They can be carried out over a wide temperature range, under conditions of vacuum or reactive ambients. The sensitivity and precision of these measurements are extremely high, and if high-class instruments are applied and some special modifications of the method are used, they are practically beyond compare as regards other methods. Together with the practical applications of ellipsometry, one should also note its role in the development of theoretical optics. The phenomenon of light reflection itself cannot be regarded as having been studied comprehensively. The intrinsic mechanism of this phenomenon and a great deal of its details are still under study and review. Not all is apparent, for example, with regard to the problems concerning the reflection of light by anisotropic media and surfaces with submonolayer adsorbed coverings.

4

A. V. RZHANOV AND K. K. SVITASHEV

The study of nonlinear effects on the reflection of powerful laser beams has only just begun. Of great importance is the problem dealing with reflected wave formation. One can formulate a number of other problems that are still far from comprehensive solution and to the elucidation of which ellipsometry could contribute. To characterize the general nature of the problem of ellipsometry, let us consider the simplest type of interaction of a plane homogeneous reflecting system with a quasi-monochromatic light beam, homogeneous over its cross section. The change of the beam’s polarization upon reflection can be described by introducing two complex reflectivity coefficients : R , = - Esref -

Es inc

-

r, eias,

R , = __ E p ref = r p . eiap E p inc

(1)

where ESref,ESinc,Epref,Epincare the complex amplitudes of the electric vectors in the reflected and incident light beams for s and p components, respectively; r, and r, characterize the change of the complex amplitudes modules; 6, and 6, are the phase shifts upon reflection. The component of beam polarization with the electric vector oriented normally to the plane of incidence is implied to be the s component, and with the electric vector oriented parallel to the plane of incidence to be the p component. It is obvious that all the magnitudes mentioned above are time independent under stationary conditions. It is convenient to introduce the relative reflectivity coefficient as the reflecting system characteristic amenable to experimental determination : P = R,/R, = (rp/rs)* exp[@, - 6,)l

(24

Of conventional use is the following notation : rJr,

= tan$,

6, - 6,

=A

(2b)

where $ and A, usually called the polarizing or ellipsometric angles, can be determined experimentally. Then, the fundamental equation of ellipsometry takes the following simple form: p = tan $ eiA

(3)

The light’s relative reflectivity from some reflecting system p is determined by the parameters of the system, as well as by the experimental conditions, such as the optical characteristics of the environment and the angle of light incidence on the specimen under study.

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

5

Equation (3) connects the coefficient p with two experimentally measured ellipsometric angles II/ and A. In principle, the opportunity appears for determining two parameters of the reflecting system provided the other parameters and the conditions of the experiment are known. However, it should be emphasized that ellipsometric measurements are not direct, since the connection between the measured and to be determined data depends on the specific type of reflecting system. That is why in the usual case it is necessary to solve three interconnected problems when working with ellipsometry :

1. To choose those types of reflecting systems that are presumably most similar structurally to the real object and to estimate the theoretical values of p at the prescribed parameters of the chosen models. 2. To determine experimentally the values of angles II/ and A, and hence p, when changing one or several parameters of the real object. 3. To compare the calculated and experimentally measured values when changing the same parameters of the model and the object under study. The agreement between the measured and calculated data with the prescribed accuracy will prove the success of the choice of reflecting system model. Of course, the equivalence of the model and object under study may be considered as establishing only over that range of parameter changes within which it has been verified experimentally. But within this range ellipsometric measurements can be applied with confidence to control or investigate the changes of the object’s parameters under the influence of environmental conditions or in terms of the quantitative changes of its characteristics. The qualitative character of the object under study should certainly not be changed. In accordance with the foregoing description of the fundamental problems of ellipsometry, the present review consists of three main sections. The first deals with the problems concerning the calculation of the relative reflectivity coefficient p for a number of the reflecting system models that are of most practical interest. The second section is devoted to an analysis of the experimental methods of determining the angles II/ and A. Finally, the third section cites a number of typical examples illustrating the facility of ellipsometry in measuring thin-film parameters or studying adsorptiondesorption processes, as well as in determining the optical characteristics of the material. The survey concludes with a brief section containing recommendations for those readers who wish to study the problems and applications of ellipsometry at greater length.

6

A. V. RZHANOV AND K. K . SVITASHEV

11. CALCULATION OF THE RELATIVE REFLECTIVITY COEFFICIENT FOR MULTILAYER REFLECTING SYSTEMS

As already mentioned in the introduction, a preliminary calculation of the relative reflectivity for a set of models of the reflecting system under study is the starting point of any ellipsometric investigation. Analysis of the presently known results of ellipsometric investigations shows that the real objects are, as a rule, complex in their structure. Usually, the reflecting system model corresponding to the real objects should involve both homogeneous and inhomogeneous layers, the optical properties of which are continuously changing in thickness. In this connection, the development of the procedures for calculating the relative reflectivity for a multilayer reflecting system is of great importance in practical ellipsometry. A. Procedure for Calculating the Relative Reflectivity for a Multilayer Reflecting System Containing Optically Znhomogeneous-in-Thickness Layers

The most universal procedure for calculating the relative reflectivity for a multilayer reflecting system is that of the matrix, proposed in Abeles’s classic work (1950). This procedure is based on a set of Maxwell macroscopic equations for a monochromatic electromagnetic wave. rot H

=

(iEw/c)E,

rot E = ( - i w / c ) H

div D = 0,

(4)

div H = 0

It is assumed that there are no free charges and conduction currents in the components of the reflecting system and that permeability for all the materials forming this system is equal to unity. The optical properties of these materials are described by means of the complex permittivity : E(O)

= q(w)

+i~~(w)

(5)

or by the complex index of refraction :

N(o)= n ( o ) - ik(w)

(6)

where n(w) is the real index of refraction, k the nondimensional extinction coefficient, and = n2 - k2,

c2 = 2nk

(7)

ELLIPSOMETRIC STUDIES OF SURFACES A N D THIN FILMS

7

The value k is connected with absorption coefficient c1 (cm- ') by the relation : c1

= 4nk/A,,

(8)

where A, is the wavelength in vacuo. The essence of the matrix procedure for calculating relative reflectivity is described in detail in the work by Rzhanov et al. (1978) and consists in the following. Let a plane monochromatic electromagnetic wave be incident at the angle 'po on the surface of a multilayer reflecting system. In this case some electromagnetic field is induced in all the elements of the reflecting system. Within every homogeneous layer cj (or homogeneous sublayer E j + I , P of inhomogeneous layer E ~ + this ~ ) field is a sum of two plane monochromatic waves. One of them is the superposition of the wave entering the layer ej through a plane boundary zj-l = const and all the secondary waves reflected from this boundary onto the layer cj. The other wave is the superposition of all the waves reflected from the boundary zjP2 = const onto the layer ej. The incident wave E , and the wave E,, reflected by our system are summed in medium E,. Only the refracted wave El exists in medium (Fig. 1). Because of a complete homogeneity in the x y plane, the electric and magnetic fields of all these waves must not depend on the coordinates x and y . Since k(k,, k,, kz), the wave vectors of all the waves under consideration, lie in the plane of incidence, i.e., in the xz plane, then for all these waves k, = 0, and the coordinate y is automatically eliminated from all the equations. As for the x coordinate, the dependence on it is excluded only when the exponent e-ikxx,entering the expression for every wave under consideration, is common for all these waves. These circumstances dictate the following condition : k, = const = k,, = k , sin 'p, = (w&/c)

sin 'p,

(9)

where ko = w&/c is the module of the wave vector of the incident wave. Thus the dependence of the total field on time and coordinate x at any point in space should be described by the factor exp[i(wt - ko,x)], and hence the expressions for the electric and magnetic fields in our case should be of the form: E = E(z) exp i(wt - k,,x)

(10)

H(z) exp i(wt - k,,x)

(1 1)

H

=

where E(z) and H(z) are the vector functions of the z coordinate only.

8

A. V. RZHANOV A N D K. K. SVITASHEV

FIG.1. Schematic view of the multilayer reflecting system containing inhomogeneous layer Ej+

1(z).

For the s-polarized incident wave, the electric field at any point in space has only the s component, and the magnetic field has only the p component ; i.e., E , = E,,

Ex = E,

=

0,

H, = 0

and the vector equations (4) are reduced to four scalar equations:

(12)

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

dH,(z) dz

9

0 + iko,Hz(z) = k ( z )E,(z) C

Substituting H, and H , from (13) and (14) into (15) and taking into account (9), we obtain d2E(z) dz2

0’ +7 [ ~ ( z-) c0 sin2 cpo]Es(z)= 0 c

(17)

The general solution of Eq. (17) is of the form E,(z) = C&)(Z)

+ c,Ey)(z)

(18)

where EL5”(z)and ELz)(z)are linear-independent solutions of Eq. (17). The constants C, and C, are determined from the boundary conditions

The magnitude E,(zo) is the total electric field of the s-polarized wave in zo = const plane. For the quantities C, and C, to be determined from the set of Eqs. (19), it is necessary that the determinant of the set be different from zero at any z o . This demand is satisfied if the solutions g l ) ( z ) and E$’)(z) are linear independent. Thus, the tangential component of the electric field is determined, at least formally, at any point in space. The tangential component H,(z) of the magnetic field can be directly derived from Eq. (13):

where j?, = -icJw = -iLoJ2n

(21)

Let us introduce into consideration the column Q(z), which comprises the tangential components of the electric and magnetic fields in some

10

A. V. RZHANOV AND K. K. SVITASHEV

arbitrary z = const plane. The column can be transformed by using (20) and (21) and the general rules of matrix multiplication:

The relation (22) provides a highly convenient procedure for determining E,(z) and H,(z) in any z plane using the linear-independent solutions of Eq. (17) and the values of E,(zo) and H,(zo) in some fixed plane zo. Indeed, since the linear-independent solutions Eil)(z)and E;’)(z) are determined to an accuracy of an arbitrary multiplier, we always have the opportunity to choose this arbitrary multiplier so that the following conditions should be satisfied :

c2=

Psyl

= H,(zo)

20

For example, it is sufficient to set

E!’)(zo) = 1, dE:” dz

--(Zo)

= 0,

E ! z ) ( ~ o=) 0 dEi2) dz

-- ( 2 0 )

1 =P S

Thus, the relation (22) can be rewritten in the following form:

where

It is easy to make sure that when z = zo and (24) is satisfied, the matrix M(z, z o ) transforms into a unit one. To construct matrix (26), it is necessary to dispose of two partial solutions of Eq. (17). However, it is very difficult to solve this equation for the

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

11

case of a laminated medium containing, among others, inhomogeneous layers. In this connection one can act in the following way: Having solved Eq. (17) for a homogeneous plane-parallel layer, one can construct the matrix that will connect the column values at the lower and upper boundaries of this layer. And then, using the matrix of a single layer, it is possible to construct the matrix of the system containing any number of layers, including the matrix of the inhomogeneous layer. Let us consider the homogeneous plane-parallel layer zj-2 I z Izj-l of thickness d j = dz, = zj-l - z j P 2 .Within the layer the permittivity is constant and has the value cj (see Fig. 1). For this layer Eq. (17) is of the form d2E,(z) dz2

o2 +[cj - e0 sin2 cpo]E,(z) = 0 c2

Let us construct the partial solutions of Eq. (27) on the basis of the exp( - iwgjz/c) and exp(iwgjz/c) functions satisfying this equation if gj = J E j

- Eo sin2 cpo

(28)

The functions gJ(zj-l

- z)]}

(29)

are the partial linear-independent solutions of Eq. (27). It is a simple matter to see that the functions (29) and (30) are the solutions of Eq. (27) within the layer zj-l I z I zjb2 only. The matrix M(dzj), constructed on the basis of these functions, is valid within the same limits and permits connecting the values of the Q column at the lower and upper boundaries of the E~ layer. , Q(zj-2) = M s ( ~ j - 2 zj-l)Q(zj-l),

where

(31)

12

A. V. RZHANOV AND K. K. SVITASHEV

and

6,

w

2n

c '

A0

= - g. dz, = - dzj,/Ej -

co sin'

'po

(33)

When we take into account the continuity of the Q(z) column at the interface between any two media, the generalization of the result for the case of a multilayer reflecting system consisting of the homogeneous layers presents no difficulties. Using the notation of Fig. 1 let us write down a set of matrix equations relating the Q(0) column at the lower boundary of the reflecting system to the Q(zo)column at the upper boundary. Evidently,

and

where

Thus, Abeles's complete matrix for a multilayer reflecting system is the product of Abeles's matrices for the homogeneous layers forming this system. If the thickness of the homogeneous layers is small compared to the light wavelength in vacuo, i.e., if

and the value of the permittivity of the cj layers is not too large, the condition 6, 4 1 is obviously satisfied and Abeles's matrix for a multilayer reflecting system can be cast in the explicit form. Indeed, we shall expand the exponents in matrix (32) in a power series in dz, and limit ourselves to the members of the first and second order of smallness. As a result of such an operation, Abeles's matrix for the system of the homogeneous layers takes the form

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

13

-

M s t h i n layers system@, ~ 0 )

[l

m

c bjdzj 1 + C cjdz; + c b

+ c c j d z i + c a j d z j [ i l bkdzk) m

m

j=2

k= 1

j= 1

j= 1

m

m

j= 1

j=2

L

(38) where aj = -wigf/c, bj = -wi/c, cj .0 2 g j / 2 c 2 , and c = light velocity. From the complete Abeles' matrlx it is rather easy to pass to the matrix for the inhomogeneous layer within which the permittivity is a continuous function of the z coordinate. For that it is sufficient to assume in (38) dzj -+ 0 and m + co. It is apparent in this case that m

lim

cjdzi = 0

(39)

m-tm j = 1

dz-0

while the other sums in (38) are transforming in appropriate integrals. The final results can be represented in the following form: M s inhorn layer

-[:

21

where [E(z)- c0

sin2 poldz

0

dz = - i - d j f l C

[E(z)- e0 sin2 p o ] z dz 2,-

dz

[

*f r

[E(u)- E~ sin2 p o ldu

(41)

Thus, if a reflecting system contains one or more inhomogeneous layers, then when calculating Abeles's complete matrix for this system, it is sufficient to utilize the matrix type of (40) instead of the matrix type of (32) for the appropriate inhomogeneous layers.

14

A. V. RZHANOV AND K. K. SVITASHEV

We shall denote Abeles’s matrix for the cj homogeneous layer in the case of p-polarized incident wave by M p ( ~ j - 2z j, - l ) . We shall denote the column made up of the tangential components of E and H vectors by R(z). Then

where

& = ic/o. ~/E(z)

(43)

and

where

and M p ( z j - 2 , ~ j - l= )

M(zj-1 - ~

=[

1

-(e-iaj

j - 2 )=

Mp(dzj)

+ eiaj)

2

-‘i ( e - i a j -

eiaj)

29j

- 9j( e - i a j - e i a j ) 2Ej

1

- (e-iaj

2

+

eiaj)

]

(46)

The quantities gj and dj in (46) are determined by the relations (28) and (33). For the case of the inhomogeneous E ~ layer, + ~ the matrix (46) takes the form M p inhorn layer

where

-[-;

3

(47)

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

15

Disposing of Abeles’ complete matrices for various reflecting systems, an investigator is able to calculate all the reflectivities for this system. The problems dealing with light reflection from anisotropic media and thin-film systems containing anisotropic layers are considered in the literature (Bockshtein, 1973; Kuzjmin, 1976; see also Table I). The generalization of the matrix procedure for calculating the relative reflectivity for the case of laminated anisotropic media is given by Rzhanov et al. (1978). However, much yet remains to be done in this field. TABLE I BIBLIOGRAPHY ON ELLIPSOMETRY OF ANISOTROPIC MEDIA Theoretical studies

Experimental studies

Abeles et al. (1973) Azzam and Bashara (1972a,b) Azzam and Bashara (1974) Azzam and Bashara (1975) DeSmet (1973) DeSmet (1 974) DeSmet (1975a,b) Elshazly-Zaghloul et al. (I 976) Engelsen (197 1) Holmes and Feucht (1966) Semenenko (1977) Semenenko and Mironov (1976a,b) Tomar (1975)

Clayton and DeSmet (1976) Henrion and Osswald (1969) Kawabata and Ichiji (1976) Meyer et al. (1973) Wyatt (1975) Yamaguchi el al. (1 972)

B. Relative Reflectivity Coeficient for Several Types of Thin-Film Systems The algorithms and programs for calculating the relative reflectivity for quite a number of types of thin-film systems have been developed by one of the authors of this survey and his associates (Burykin et al., 1979). Presented below are the calculation results for those systems that are of most practical interest. 1. Reflecting System of the Type “Substrate-Znhomogeneous Layer- T w o Homogeneous Layers”

The expression for calculating the relative reflectivity for the reflecting

16

A. V. RZHANOV AND K. K. SVITASHEV

system under study (see Fig. 2) can be cast in the form:

X

+ r04sr43se-zia4+ r +r ro4s + r43se

1

2ibs

R 3 1 s e-2ia3 + 0 4 s R 3 1 s e - 2 i ( h + b 4 ) (49) 0 4 s r4 3 s R 3 1 s e-2ia3+ R 3 1 s e-2i(63+64) 43s

where rjkp,rjks are Fresnel coefficients for waves of suitable polarization reflected from an abrupt boundary between ej and &k.

The magnitudes g j and S j are determined by the relations (28) and (33), respectively. The magnitudes A,, B,, Cs,Iss, A,, Be, Cp, and Dp should be calculated following formulas (41) and (48).

FIG. 2. Schematic view of the reflecting system containing substrate layer E ~ ( z )and , two homogeneous layers e, and e4.

inhomogeneous

17

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

The quantities R,, and Rjksplay the role of Fresnel coefficients when reflecting from an interface between the media with ej and E~ permittivities, if this interface is not abrupt and represents an inhomogeneous layer within which E = E ( Z ) and varies smoothly from the value of E~ at the interface with the E~ medium to the value of &k at the interface with the &k medium. 2. Reflecting System of the Type “Substrate-Two Homogeneous Layers” In order to turn to such a system, it is sufficient to set z1 = 0. The inhomogeneous layer vanishes in this case : A S = A P = B S = B P = CS = CP = D S = D P = O and the quantities R3ip and R31stransform into Fresnel coefficients r31, and r 3 l S , describing the light reflection from an abrupt interface between the E~ and media. 3. Reflecting System of the Type “Substrate-Inhomogeneous Layer”

In this case P = ROlp/ROls

In a model free from an inhomogeneous layer, the values R o l p and R O l s transform as in the previous case into the ordinary Fresnel coefficients r n l P and rols, and expression (52) describes the process of light reflection from the interface of two semiinfinite c0 and c1 media. It is a simple matter to show (see Sokolov, 1961) that if the c0 medium is transparent, then

n2 - k’ = ni sin’ cpo 2nk = ng sin’ cpo tan’ q0

cos’ 2$ - sin’ 2$ sin’ A] (1 - sin 2$ cos A)’ 2 cos 2$ sin 214 sin A (1 - sin 21) cos A)2

(52a)

(52b)

4. Reflecting System of the Type “Substrate-Inhomogeneous

Layer- Homogeneous Layer” In this case d3 = 0, and expression (49) takes the form

In the model without an inhomogeneous layer, R41, = r41p,R4is = r4iS, and expression (53) transforms into the expression for the relative reflec-

18

A. V. RZHANOV A N D K. K. SVITASHEV

tivity coefficient for the system containing the substrate and a single homogeneous layer :

where the values ro4p, ro4s, r4lP, and r4lS are Fresnel coefficients for p- and s-polarized waves with 0 indexing the environment, 1 the substrate, and 4 the film. In the future we shall replace the index 4 by f (abbreviation of the word “film”). Then for 6 we can write

6 = (w/c)JEf - c0 sin’

‘po*

(55)

df

where df is the film thickness. For transparent films Ef and 6 are the real magnitudes and p is the periodic function of the film thickness with the period do determined by the relation 6 = R or do = 1,/2Jn: - n; sin’

where no =

and nf =

‘po

&.

5. Rejecting System Containing Substrate and Very Thin Transparent Film

The term “very thin” means that the inequality d f / l << 1 is satisfied. The satisfaction of this inequality denotes that all the exponents in expression (54) can be replaced by the expansions in a power series limited by the members containing the first powers of df/l only. Taking this into account, after simple transformations we can reduce Eq. (54) to the form : p = p(1

+ i C A * d+ C , * d )

(57)

where n2,)cos ‘po sin’

‘po

M“l/nr2) - .I - a1N M’ N’

+

(58)

4R C , = - no(# - n2,)cos ‘posin’

‘po

N [ ( l / n : ) - a] - a,M M2 N2

(59)

4R cA -- no($ 1

1

-

+

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

19

and ji is the relative reflectivity coefficient for a “clean” film-free surface. Separating the real and imaginary parts in (57), we obtain

3)cos(A - E ) = 1 + C, . d (tan $/tan $) sin(A - A) = C, d Taking cos(A - A) 1, cos $ cos $, and sin($ - 3) (tan $/tan

(63) (64)

*

N

N

N

$ - $, we ob-

tain from (61)

$ - $ =3C$.dsinZ$

(65)

Taking into consideration that for very thin films IC, . dl 4 1 and tan(A - L\) 1: (A - A), we find from (63) and (64)

A - 5 = CA* d

(66)

The approximate equations (65) and (66), were first obtained by Drude (1889). However, the coefficients CA and C, were determined inaccurately. More precise values of CAand C,, which were in agreement with expressions (58)-(61), were obtained by Archer and Gobeli (1965). It was Saxena (1965b) who made a detailed comparison between the approximate equations (65) and (66) and the exact equation (54). In particular, Saxena showed that it would be expedient to replace Eq. (65) by a more accurate one:

$ - 3 =$Ci.d2sin2$

(67)

Equation (67) follows from Eqs. (63) and (64). The assumption of the smallness of the value of (A - A) is of no use when deriving Eq. (67). The expressions for the relative reflectivity coefficient given in the present section can be used directly to carry out the numerical computations of the value of p for quite a number of concrete problems. C. Analysis of the Situation :Horizons for Further Investigations At first sight the foregoing calculation procedure seems to be quite complete. But in reality, the case is somewhat different. First of all, Abeles’s matrices permit calculation of the reflectivities for a plane monochromatic wave. But generally speaking, these waves can be met in the pages of textbooks and monographs only. At best experimenters

20

A. V. RZHANOV AND K. K. SVITASHEV

deal with quasi-monochromatic and quasi-parallel light beams. In this connection, when carrying out ellipsometric investigations, one should clearly realize the approximate character of reflectivity calculations performed by the above-mentioned procedure and should consciously follow one of two possible courses: 1. To carry out experiments using a carefully collimated, highly monochromatic light beam and to use the values of the relative reflectivity coefficient, calculated in approximation of a plane monochromatic wave. 2. To determine quantitatively the spectral composition and angular aperture of the operating light beam and to interpret the results of measurements taking into account the dependences p = p(cpo) and p = p&).

The first approach can be realized when working with laser light beams. The necessity and usefulness of the second approach are not only connected with the temporary absence or unreliability of laser light sources for some spectral regions. The fact is that utilization of an operating light beam with a sufficiently large aperture and wide spectrum expands the possibilities of ellipsometry. Such an approach, for example, permits removal of the periodicity of the dependence p = p(d) for transparent films and thus expansion of the region of unambiguous determination of the film thickness by means of ellipsometry (Semenenko et al., 1978). The second problem consists in substantiating the applicability of the macroscopic Maxwell equations to describe the processes of light reflection from samples with thin (100-100OA)and superthin (from units up to some tenths of angstroms) films present on their surfaces. This problem also includes the question concerning the interpretation of the results of the ellipsometric study of adsorption-desorption processes.

TECHNIQUES FOR DETERMINING THE RELATIVE 111. EXPERIMENTAL REFLECTIVITY COEFFICIENT

The main goal of experimental ellipsometry is to measure the angles of polarization with as high an accuracy as possible. An instrument that is used to carry out these measurements is called an ellipsometer. Experimental techniques for determining ellipsometric angles can be divided into two groups: 1. Null techniques based on the measurement of the azimuth angles of the polarizer, analyzer, and compensator, corresponding to the extreme (minimum as a rule) values of the operating light beam intensity at the output of the ellipsometer.

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

21

2. Techniques based on the measurement of the operating light beam intensity at the output of the ellipsometer (photometric techniques).

The first group of techniques utilizes an ellipsometer containing two linear polarizers (a polarizer and an analyzer), a compensator (usually a quarter-wave plate), and a photodetector (see Fig. 3a, b). The polarizer, analyzer, and compensator can rotate freely about the axis of the operating light beam, and suitable azimuth angles are fixed with the aid of special angle-measuring units. The essence of all null techniques consists in the establishment of the interconnections between the $ and A angles of polarization and the azimuth angles of the polarizer, analyzer, and compensator. The latter correspond to the extreme values of the operating light beam intensity at the output of the ellipsometer.As a rule, this intensity is minimum. When photometric techniques are used, a compensator is eliminated from the optical scheme of the ellipsometer (see Fig. 3c). And angles t,h and A are calculated following the formulas containing the ratio of the light intensities at the output of the ellipsometer, measured at some fixed positions of the polarizer and analyzer.

FIG. 3. Three modifications (a, b, c) of the basic schematic diagram of an ellipsometer: L, light source;W,, operating light beam; P, polarizer; C, compensator;A, analyzer; D, photodetector; S,sample under study.

22

A. V. RZHANOV AND K. K. SVITASHEV

Thus, all the experimental techniques for determining the angles of polarization are in the end based on the measurement of the operating light beam intensity at the output of the ellipsometer. In this section we shall consider the procedures for measuring ellipsometric angles, as well as the problems concerning utmost accuracy and the ways of automating measurements on the basis of the analysis of the general expression for the operating light beam intensity at the output of the ellipsometer. A . General Expression for the Operating Light Beam Intensity at the Output of an Ideal Ellipsometer

Figure 3% b, and c shows schematically the three modifications of ellipsometer arrangement that we are going to consider. Usually, the interaction of a light beam with any stationary linear optical system can be described by the relations (van de Hulst, 1957; Prishivalko, 1963) E,(t) Ep(t)

+ a2E,o(t) = c13Es0(t) + a 4 E p O ( t ) = a, E,o(t)

(68) (69)

where Es0(t), EpO(t),E,(t), and E,(t) are the instantaneous values of the strengths of the s and p components of the electric field in the light beam at the input and output, respectively, and a l , az, a,, and a4 are some complex numbers characterizing a linear system. For the case of a quasi-monochromatic light beam at any fixed point of the beam (Born and Wolf, 1964) E,(t) = a,(t)exp{i[4,(t) - 2

7 4

E,(t) = ap(t)exp{i[4,(t) - 2 7 4 )

(70) (71)

where a&), a#), 4,(t),and Ql,(t) are the independent, slowly changing (as compared with the period of light oscillations) amplitudes and phases of the s and p components of the light beam electric vector, Just these four functions determine entirely the state of polarization of the operating light beam at any fixed point on the beam. Today the functions a#), a,(t), &(t) and +,(t) cannot be directly measured experimentally. The four functions that are amenable to direct experimental measurement are the Stokes parameters So, S , , S,, S , (Born and Wolf, 1964; Shurklie 1962):

ELLIPSOMETRIC STUDIES OF SURFACES A N D THIN FILMS

23

where the brackets ( ) denote, as usual, time averaging. The relation between the values of the Stokes parameters at the input of a linear optical system and their values at the output of this system can be represented in the following form (van de Hulst, 1957; Prishivalko, 1963);

where [Mik] is the 4 x 4 matrix. The general view of the 4 x 4 matrix, describing the transformation of the Stokes parameters by an arbitrary linear optical system, is presented in Fig. 4. Let us consider the case when the operating light beam passes through a succession of optical devices each of which performs a linear transformation of the relations (68) and (69). Then, in order to find the Stokes parameters at the output of the entire system, it is necessary to apply the matrices ikf!k, M:!, . . ., MZ to the column made up of the Stokes parameters at the input of the first device; i.e.,

The matrices Miik,M:!,.. .,MZ characterize every linear device, affecting consecutively the operating beam. The 4 x 4 matrix for the main elements of the optical system of the ellipsometer-polarizer (analyzer), compensator, and system under studycan be obtained as particular cases of the 4 x 4 matrix for an arbitrary linear optical system (Shurkliff, 1962). Using the rule for reading the azimuth

24 A. V. RZHANOV AND K. K. SVITASHEV

_.. 4

1

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

25

FIG.5. The rule for reading the azimuth angles yA, yp, yc. x1 is the direction of the polanzet and analyzer transmission or the L‘fast”axis of a compensator.

angles of the polarizer, analyzer, and compensator (see Fig. 5), these matrices have the following form : (1) An ideal linear polarizer (analyzer) :

‘ 1

cos 2yp

cos 2yp

cos2 2yp

WYP) = 4 sin 27,

sin 2yp

0-

sin 2yp cos 2yp 0

sin 2yp cos 2yp

sin2 2yp

0

0

0

0.

. o

where yp is the angle between the s direction and “transmission” direction of the polarizer. When turning from a polarizer to an analyzer, the angle yp transforms into yA, the angle between the s direction and “transmission” direction of the analyzer. (2) Reflecting system :

+

9,” r i

r,” - r i

0

+ r;

0

r i - r i rz 0

0

2rsrp cos A

- 0

0

-2r,rp sin A

2r,rp sin A

A. V. RZHANOV A N D K. K. SVITASHEV

26

(3) A compensator (birefringent crystal plate) :

-1

M(YC,J)

=

0

0

0

o

6 1 - 2 sin2 yc sin22

6 -sin 4yc sin22

sin2yc sin 6

o

6 -sin 4yc sina 2

-0

-sin 2yc sin 6

1

6 1 - cos2 2yc sin2 - cos 2yc sin 6 2

- cos 2yc sin 6

cos 6

(80)

where yc is the angle between the s direction and a “fast” axis of the comare the phase shifts pensator. 6 = A&ow - A4fast and A4slow and that take place when the light beams pass through the compensator and are polarized along its “slow” and “fast” axes. If we neglect the multiple light reflections within the compensator, then

where d, is the thickness of a birefringent crystal plate, and blmand n,,, are the refractive indices of the light beams linearly polarized along the “slow” and “fast” axes of the compensator. The effects caused by the multiple light reflections within the compensator are discussed in detail by Rzhanov et al. (1978). The final calculation results for the 4 x 4 matrices for the three basic ellipsometer schemes (see Fig. 3) can be presented in the following form:

where

and the expressions for Mtsp, MtFp, MZCpcoefficients are MtSP = 4r:

COS’

yA COS’ yp

+ 4rf sin2yA sin2 y p

+ 8rSrpsin yA sin yp cos yA cos yp cos A

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

M;4fSP= (r:

+ r;) 1 + sin 27,

+ cos 27,

cos 2y, sin 4yc sin’

27

6 2

-

cos 2yp 1 - 2 sin2 27, sin’ cos 2y,

+ sin 2y,

6 sin 4yc sin22

+ 2rsrpcos A sin 2yp + cos 27,

sin 4yc sin’

7

-

2

+ 2rsrpsin A sin 27, sin(2yc - 27,) M;qScp= (r:

sin 6

+ r;) 1 + cos 2yA sin 27, sin 47,

+ cos 27,

sin’

(84)

6

-

2

cos 2yp 1 - 2 sin2 2yc sin’ cos 27,

+ sin 27, sin 47,

- 2 sin’ 2yc sin2

+ 2rsrpcos A sin 2yA

6 sin22

i2)} 1 - 2 cos2 2yc sin2

$1

!) 2

+ cos 2ypsin 4yc sin2 + 2rsrpsin A sin 2y, sin(2yc - 2y,) sin 6 Now one can easily calculate the Stokes parameters for the operating light beam at the output of the ellipsometer provided that the So,, Sl0, S2,, and S,, Stokes parameters, characterizing this beam at its input for all the ellipsometer schemes under consideration, are known.

28

A. V. RZHANOV A N D K. K . SVITASHEV

In practice, the only parameter of the operating light beam that is measured at its output is the total intensity of this beam IOU, or the parameter Soout,which is the same. Let us denote the operating light beam intensities at the output of the ellipsometer for three considered schemes of the instrument by It;", I:,' and Taking into account (76), (81), and (82) and applying the general rules of matrix multiplication, we obtain

It:'.

+ Slocos 27, + S20 sin 2yp)

(86a)

+ SlO cos 27, + s 2 0 sin 27,)

(86b)

IO",~' = ~ M A1S1 P( S o 0 + SlO cos 2 Y P + s 2 0 sin 2YP)

(86c)

I$' ACSP

I,",

= ~Mffcp(Soo = LMACSP 2

11

(So0

In the case when the operating light beam at its input is completely depolarized or has a circular polarization, Sl0 = SzO= 0 and the expressions (86) take the form ASCP(ACSP, ASP)

I,",

= ( I ~ / ~ ) M ~ S C P ( A C S PASP) .

(87)

where Soo = I , is introduced to denote the total intensity of the operating light beam at its output. Expression (86) or (87) completely describes the dependence of the operating light beam intensity at the output of an ideal ellipsometer on the azimuth angles of the polarizer yp, analyzer yA and compensator yc, as well as on the parameters of the reflecting system under study. It should be noted that the previously published expressions for the operating light beam intensity at its output (Cahan, 1969; Jasperson et al., 1973; Aspnes, 1976) are particular cases of expressions (86). B. Formulas for Estimating Angles of Polarization $ and A by Experimentally Measured Values In this section we shall derive the formulas for calculating the angles of polarization when using both the null and photometric measuring techniques, proceeding from the general expressions for the operating light beam intensity at the output of the ellipsometer.

1 . Null Techniques for Determining the Angles of Polarization In order to utilize null techniques for determining the angles of polarization, it is necessary to have an ellipsometer assembled in ACSP sequence (Fig. 3a) or in ASCP sequence (Fig. 3b). Since they follow directly from expressions (84b (85), and (86), both ACSP and ASCP ellipsometer arrange-

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

29

ments are quite equivalent. Indeed, (84) transforms into (85) by a simple substitution of yA angles for yp and yp angles for yA. Further consideration should be made for the ACSP arrangement only. After the foregoing angle substitutions, all the results obtained can be used directly when working with the ellipsometer assembled in ASCP sequence. We shall analyze the null techniques to determine the angles of polarization for the case of an ideal ellipsometer with quasi-parallel and quasimonochromatic operating light beams. Furthermore, let us assume that the light beam at the input of the ellipsometer is completely depolarized or completely circularly polarized. Then, the intensity of the operating light beam at the output of the ellipsometer is described by relation (87) and is a periodic function of three independent variables, yp, yA, and yc, the azimuth angles of the polarizer, analyzer, and compensator. From relations (84) and (87) one can see that the values of angles yp, yA, and yc at which takes the extreme values, depend to a considerable extent on the values of A, i,b, and 6 . The quantity 6 is determined by adjustment of the ellipsometer (Azzam and Bashara, 1977; Rzhanov et al., 1978). As to the values of the angles of polarization that characterize the reflectingsystemunder study, two equations are necessary for their determination. They should relate the values of these angles to the values of ypc, yAe, and yce, which correspond to the extreme intensity of the operating light beam at its output. In principle, there exist three pairs of such equations : (1) yp = const (polarizer is fixed) :

(2) yA = const (analyzer is fixed):

(3) yc = const (compensator is fixed):

Each pair of equations gives rise to its own specific technique for measuring the angles of polarization. Relation (87) permits analysis of all these procedures from common positions. However, such an analysis would occupy a great deal of space. As a typical example, let us dwell on the analysis of Eqs. (90).

A.

30

V. RZHANOV AND K. K. SVITASHEV

When using the fixed-compensator procedure, it is expedient to set the compensator so that yc = f45". Then expression (87) takes the form

ltzSp= I&[1 + cos 27, cos 27, cos 6 + cos 249(cOs 2yp + cos 27, cos 6) + sin 2$(sin 2YA cos A f cos 27, sin A sin 6) sin 2yp (91) and the system of Eqs. (90) can be cast in the following form:

+ cos 249 sin 2ype- sin 249 cos 2ype(sin2yAe cos A + cos 2yAe sin A sin 6) = 0 sin 2yAe cos 2ypecos 6 + cos 249 sin 2yAe cos 6 - sin 249 sin 2ype

cos 2yAe sin 2ypecos 6

(92)

x (cos 2yAe cos A f sin 2yAe sin A sin 6) = 0 (93)

In expressions (91) and (92) when a double sign precedes a term, the upper sign corresponds to yc = +45" and the lower one to yc = -45". In practice, when employing the null technique, the angles of polarization are estimated by the values of yAminand yPmin, corresponding to the minimum intensity of the operating light beam at its output. In this case the intensity of the completely polarized component of the operating light beam at its output, defined by Eq. (91), is equal to zero. Therefore, it would be worth adding one more equation to relations (92) and (93): 1

+ cos 2ypmin cos cos 6 + cos 2J/(cOs 2ypmin + cos 2yAmincos 6) + sin 249 sin 2ypmin(sin2yAmincos A f cos 2yAminsin A sin 6) = 0 2yAmi"

(94)

Relations (92-94) are the basis of the procedure for determining angles 49 and A by means of an ideal ellipsometer where a quasi-parallel, quasimonochromatic light beam is used, and a compensator is fixed so that yc =

f45".

The working formulas for the estimation of the sought angles of polarization by the measured values of yAminand yPmincan be easily obtained when using relations (92) and (94). Without dwelling at length on the details of the calculations (for greater detail, see Rzhanov et al., 1978), we shall give the working formulas for estimating the values of the ellipsometric angles by the measured values Of YAmin and YPmin:

$ = (-1 P{ YPmin

+ P - [2NP - ( -

- l](n/4)}

A = (-1)Nc{2yAmin+ v - (-1P(n/2) & 2nn}

(95) (96)

where n = 0,1,2,3,. . . , Np is the number of that quarter of the trigonometric circle where yPminis located, and Nc is the number of that quarter of

31

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

the trigonometric circle where the “fast” axis of a compensator is located (see Fig. 6).

I

v = arctan [sin 6 - 13 x

[ (; 7 tan

-

)

2yAmin sin 6

I-’>

+ tan(n/2 T 2yAmin)

(98)

Figure 7 represents the dependencies p = p(z/2 - yPmin) for 6 = 89” and 6 = 70’. The values of the (z/2 - SAmin)angles from which the proper curve was obtained are given as the parameters near each curve. Figure 8

Compensator Limb

Polarizer l i m b

FIG.6. Zones of ellipsometric measurements:C is the location of the “fast” axis of a compensator; P is the acceptable region of yp angle change.

32

A. V. RZHANOV A N D K. K. SVITASHEV

0

20

40

AnqLe

60

80

[x/2-TpmLn] (deqrees)

FIG.7. Angle p as a function of (n/2 - yPmin). (a) 6 = 89"; (b) 6 = 70".

Angle

[v-&,4

(degrees)

FIG.8. Angle v as a function of ( n / 2 - yA

shows the dependencies v = v(n/2 - yAmin)for a number of 6 values. It is evident from the figures that at the same values of 6, ,u 9 v for a majority of yA angle values. When using an ellipsometer assembled according to the ASCP arrangement, it is necessary to replace relations (95) and (96) by the following: $

= (-ONA{yArnin

+~

1 -[ ~ N A -

- 1IW4))

(99)

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

A = (- 1)Nc[2ypmi,+ v - (- 1)NA(n/2)} i-2nn

33 (100)

where N A is the number of that quarter of the trigonometric circle where angle YAmin is located. Usually when deriving the working formula for estimating the values of angles and A by the azimuth angles of the polarizer, analyzer, and compensator, yAmin, yPmin, and yc, four measuring zones should be introduced into consideration. For an ellipsometer assembled according to the ACSP arrangement, the measuring zones are determined in the following way (see Fig. 6): Zone I : yc =

+45", 0" Iyp I 90" or 180" Iyp I 270", yA, arbitrary (101)

Zone I1 : yc = +45",

90" I yp I180" or 270" Iyp I360", yA, arbitrary (102)

Zone I11 : yc = -45",

0" Iyp I 90" or 180" Iyp I 270", yA, arbitrary (103)

Zone IV: yc =

-45", 90" 5 yp I180" or 270" Iyp I360", yA, arbitrary (104)

For an ellipsometer assembled in ASCP sequence, the measuring zones are determined by the positions of the compensator or analyzer, while the position of the polarizer is arbitrary. In order to obtain the table of the measuring zones of such an ellipsometer, it is necessary to replace angle yp in (101)-(104) by angle yA and yA by yp. The quantities N,, Np,and Nc, introduced into formulas ( 3 9 , (96), (99), and (loo), determine unambiguously the number of the traditional measuring zones prescribed by the relations (101)-( 104). For an ellipsometer assembled in ACSP sequence, the number of the measuring zones N can be calculated by N =

5

+ (-1)"p + (2

1)"C

and for an ellipsometer assembled in ASCP sequence by N=

5

+ (2

1)NA

+ (-

l)N"

The quantities p and v can be excluded from the formulas for the calculation of the polarizing angles by the derivation of such working formulas

34

A. V. RZHANOV AND K. K. SMTASHEV

that contain the values of angles yPminand yAmin only, obtained as a result of measurements in different zones. This follows directly from the relations (99, (961 (99), and (100) and from the above rules of sign option when calculating p and v following formulas (97) and (98). Quite a number of authors (Archer and Gobeli, 1965; McCrakin et al., 1963; Smith, 1972b) have pointed out that for the correct determination of the ellipsometric angles, it is necessary to carry out ellipsometric measurements in all four zones. The above analysis shows that the correct values of A and $ ellipsometric angles at any 6 value can be obtained by ellipsometric measurements carried out in one zone only. Moreover, the choice of a zone is quite arbitrary. The effects due to the difference of a compensator from a quarter-wave plate can be easily taken into account by introducing p and v corrections. To our mind the suggested “one-zone” procedure facilitates substantially the approach to the automation of ellipsometric measurements and simplifies and speeds up the process of spectroellipsometric measurements as well. One more modification of the null techniques for measuring the angles of polarization is in principle possible and is being used today. In this modification the operating light beam is extinguished by the rotation of one optical element only and by the variation of q0 (angle of light incident on the specimen under study) or no (refractive index of the environment). In this case the dependences A = A(cpo, no) and $ = $((po, no) are utilized. Figure 9 presents one of the variants of ellipsometer configuration intended for carrying out this type of null technique for measuring ellipsometric angles (for greater detail, see Kinosita and Yamamota, 1976). 2. Photometric Techniques for Determining the Angles of Polarization The ASP ellipsometer arrangement that does not employ a compensator is presented in Fig. 3c. This ellipsometer is intended for carrying out the photometric techniques for measuring angles $ and A. The general expressions for the operating light beam intensity at the output of such an ellipsometer are of the ASP form of Eq. (87). We shall illustrate the possibility of the experimental determination of ellipsometric angles by the photometric technique by a simple example. Let a polarizer be set up so that yp = +45” and is fixed and let an analyzer be fairly free to rotate :

z:ip

Z

= 2 R(COS’$ COS’ yA

2

+ sinZI+!/ sinZyA + 2 sin $ sin yA cos $ cos yA cos A)

(107)

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

35

S FIG.9. Schematic diagram of the principal angle-of-incidenceellipsometer; L, light source ; W,, operating light beam; R, splitter; P(A), polarizing prism; S, specimen under study; M, mirror; D, photodetector.

Now let us determine the values of It: at

= 0",45", and 90". Apparently,

z:3450, 0") = &Z,R cos2 $

(108)

Z:,'(45", 45") = $Z,R( 1 + 2 sin I) cos I) cos A)

(109)

~2:'(45", 90") = +Z,R sin' $

(110)

It follows directly from (108)-(110) that $ = arctan

A = arccos

2Z:iP(45", 45") - [Z~~'(45", 0") + Z,qsP(45",go")] ,/Z2iP(45", 0°)l~~p(450, 90")

(112)

The relations (111) and (1 12) are the working formulas, permitting calculation of A and $ by three directly measured values of the operating light beam intensity Z,qSp(45", 00), Z::'(45", 45"),and Z2,'(45", 90"). 3. Threshold Sensitivity of the Ellipsometer

By the threshold sensitivity of the ellipsometer, we mean those changes of the polarizing angles S k i n and GI)min that still can be detected by the instrument. The value of the signal-to-noise ratio at the output of the ellipsometer naturally has the determining influence on akin and S$min.Besides, S k i n and St,bmindepend on the properties of the reflecting system itself. In particular, it is evident, for example, that the value A cannot be measured at all, provided that the s or p component is absent in the reflected beam. (Such situations are realized at $ -+ 0" or $ +goo).

36

A. V. RZHANOV AND K. K. SVITASHEV

All kinds of instabilities in the optomechanical and electronic systems of the instrument can seriously affect its threshold sensitivity. The influence of these instabilities is especially important when employing photometric techniques for measuring the angles of polarization (Mardezhov et al., 1977). Thorough analyses of the problems concerning the threshold sensitivity of the ellipsometer are given by Alexandrov and Zapassky (1976), Aspnes (1975b, 1976),Azzam (1976%b), Jerrard (1960a),Smith and Hacsaylo (1963), and Svitashev et al. (1977b). C . Automation of Ellipsometric Measurements

The problem of automation of ellipsometric measurements is extremely urgent today. Utilization of automated ellipsometric systems reduces the duration of a single measurement to some tens of milliseconds, as compared with 2-3 minutes when measuring by an ellipsometer with a hand control. Such a shortening of the duration of a single measurement turns an automatic ellipsometer into an effective instrument for studying the kinetics of the different physicochemical processes on solid or liquid surfaces. An automatic ellipsometer can also be employed for the continuous control of quite a number of technological processes. 1. Automation of Null Techniques for Measuring Polarizing Angles Nowadays, two ways of automating ellipsometric measurement null techniques are under consideration. The first (Ord and Wilis, 1967; Ord, 1969) represents an automatic imitation of the hand-operated null measurement technique. In this case the role of the hands is played by stepped reversible motors, linked, in turn, to the rotators of a polarizer and an analyzer. The main disadvantages of this method are the cumbersomeness of the device and the presence of the rotating mechanical units, which limit the system’s speed and measurement accuracy. The second proposal for null technique automation was made by Winterbottom (1963). The functional diagram, proposed by Winterbottom, is a static system of automatic control, where the error signal appears when the operating light beam intensity at the output of the ellipsometer deviates from its minimum value owing to the change of rs, rp, and A parameters of the reflecting system. In order to transform the values and signs of these deviations into electric signals, corresponding to them in amplitude and phase, Winterbottom suggested modulating the turn of a linear polarization plane of the operating light beam at the output of a polarizer and at the input of an analyzer by means of Faraday cells (see Fig. 10). Now we shall consider the operation of such an automation system

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

1

1

BSS

37

W BT l

I

FIG. 10. Schematic representation of the Winterbottom principle of ellipsometric measurement automation: Fcp,F,, are Faraday cells used as compensators, effecting the rotations of the polarization plane of the operating light beam, necessary to compensate the changes of A and II/ parameters and to obtain the minimum of light intensity at the output of the ellipsometer; F,,, F,, are Faraday cells used as modulators, carrying out the test swinging of a linear polarization plane of the operating light beam; G,, G, are modulator generators; BTS are the signal transformation blocks (I, 11, 111); and BSS is the signal selection block. The other notation is identical with that in Fig. 3.

using an ideal ellipsometer assembled in ACSP sequence as an example. Let us assume a compensator to be an exact quarter-wave plate which is fixed so that yc = +45". At some initial moment let the reflecting system to be characterized by the parameters t,b0 and do, and a polarizer and an analyzer be fixed so that the relations A. = (3n/2) - 2yA,in and $, = (n/2) - yPmin are realized; i.e., the intensity of the output light equals 0. Further let us suppose that at t > to, $ = $o = const, yp = yPmin = const, and yA = yAmin= const, but A = A. + Al(t), where Al(t) remains a small magnitude at all t > to. Then by virtue of Eq. (87) at all t > to, we have for the output light intensity (see Fig. 11): I:;"

= $ZoR sin'

2ypminx [Al(t)]'

(113)

On the other hand, if at t > to, $ = $o = const, A = A, = const, but yA = yAmin bYA(t), where 8yA(t) is a small magnitude, then by using Eq. (87) it is easy to show that (Fig. 11):

+

122'~ = ~

Z sin2 ~ 2yP,, R

x [byA(t)]2

(114)

From Fig. 11 one can see that at i,b = $o and A = A. (case a in Fig. ll), the test swinging of a linear polarization plane at the input of an analyzer with Xestfrequency leads to the appearance of the signal with 2Xmt fre-

38

A. V. RZHANOV AND K. K. SVITASHEV

c

a

b

FIG.11. Schematic diagram showing the operation of the automatic ellipsometer presented in Fig. 10.

quency at the output of the ellipsometer. At II/ = II/o but A = A. f A1 (cases b and c in Fig. ll), the test swinging of a linear polarization plane at the input of an analyzer with Jest frequency leads to the appearance of the error signal with the same frequency at the output of the ellipsometer. Besides, the signal amplitude is proportional to lA1l, and the phase (0" or 180") depends on the sign of A l . An analogous situation takes place at the simultaneous change of both the II/ and A parameters of the reflecting system. The electric signal of the A,,, frequency is the operating signal of error of the automatic monitoring system and appears at the output of a photodetector when the values II/ and A deviate from their initial values rc/o and Ao. After necessary transformations this signal is introduced into the coils of Faraday cells Fcpand F C A (see Fig. 10) and causes the turning of the linear polarization planes of the operating light beam at the output of a polarizer and at the input of an analyzer, ensuring the minimum light intensity at the output of the ellipsometer. Modulated null ellipsometry using Faraday magnetooptical effects has recently begun to be practiced on a wide scale. Practically complete analyses of the performance characteristics of an automatic null ellipsometer are given by Algazin et al. (1977) and Archipenko et al. (1977).

2. Automation of Photometric Techniques for Measuring Polarizing Angles The simplest way of automating photometric techniques is to fix a polarizer so that yp = +45" and to measure the output light at yA = 0",

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

39

yA = &45", and yA = 90" uniformally rotating an analyzer. Then, one can

calculate the values of $ and A following formulas (111) and (112) with the aid of a minicomputer (Aspnes, 1973). This method is extremely simple and convenient in those cases when the requirements for the sensitivity of automated devices are not too high. The particular merit of this method of automation is its spectral nonselectivity. In this connection it was exactly this measuring technique that was realized when an automatic ellipsometer-spectrometer was created (Aspnes, 1975c; Hunderi and Ryberg, 1976). Quite a number of interesting modifications of the considered method of automating ellipsometric measurements are suggested by Aspnes and Hauge (19761 Hauge and Dill (1973, 1975), Hauge (1976), Jasperson and Schnatterly (1969), Jasperson et al. (1973), and Treu et al. (1973). These modifications are connected with compensator rotation and the simultaneous rotation of a compensator and an analyzer, as well as with the phase modulation of the operating light beam when using electrooptical and photoelastic effects in a compensator or in a special additional plate of a birefringent material. Detailed comparisons between the characteristics of null and photometric automated systems are given by Azzam (1976b), Aspnes (1975b), and Algazin et al. (1977). D. Present State of the Question; Problems and Horizons By our estimate the various laboratories and firms have developed to the present time more than 50 different types of manual and automatic ellipsometers intended for work in the visible, near-infrared, and far-infrared spectral regions (see Table 11) Lasers, high-pressure mercury lamps, or powerful thermal sources are utilized as the light sources in ellipsometers. Thermal sources are most often used when working in the infrared. Polarizing prisms (usually GlanFoucault prisms) are used as polarizer and analyzer in high-perfection ellipsometers. The compensator is a quarter-wave quartz plate. Photoelectric multipliers serve, as a rule, as photodetectors. In high-perfection ellipsometers the measurement precision of the azimuth angles of the polarizer, analyzer, and compensator and the angle of incidence of the operating light beam on the specimen under study is about 0.01". In precise research-type ellipsometric arrangements intended basically for the study of physicochemical processes on the surface of solids, the accuracy is equal to 0.001". For work in the infrared polarizing prisms should not be used (iceland spar is transparent up to I r 2 . 5 ~only); instead the transmission gratings should be utilized as polarizer and analyzer. As a compensator one can use I/4 plates of mica or quartz (when working at wavelengths greater

40

A. V. RZHANOV AND K. K. SVITASHEV

TABLE I1 BIBLIOGRAPHY ON EL LIP SO MET^ CONFIGURATIONS ~~~

Ellipsometer type Manual ellipsometer

Ellipsometric microscope Automatic ellipsometer

~

References Archer (1968) Aspnes (1975a) Azzam (1 9764 Demichelis ef al. (1 969) Edwards et al. (1975) Hazebroek and Holscher (1 973) Henty and Jerrard (1 976) Jerrard (1969b) Kudo ef al. (1968) Lihl et al. (1967) Monin et nl. (1968) Rothen (1957) Wilmans (1969) Yamamoto (1974) Antziferov et al. (1 972) Dokuchaev et al. (1972) Algazin et al. (1977) Aspnes and Hauge (1976) Blumkina ef nl. (1976) Cahan (1 969) Greef (1970) Hauge and Dill (1 973)

Ellipsometer type Automatic ellipsometer (continued)

Differential ellipsometer Spectrometerellipsometer

IR-ellipsometer

References Hauge and Dill (1 975) Jasperson ef al. (1 973) Kasai (1976) Lluesma et al. (1 976) Mathieu et al. (1974) Meulen and Hien (1 974) Ord (1 969) Ord and Wills (1967) Roberts and Meadows ( 1974) Takasaki (1966a,b) Treu (1974) Aspnes and Studna (1975) Hunderi and Ryberg (1 976) Matheson et al. (1976) T. Smith (1976) DeNicola (1971) Hilton and Jones (1966) Mardezhov et al. (1977) Mishnev ef al. (1 972) Stobie et al. (1975) Svitashev et al. (1977~)

than 35pm). Usually vacuum thermal elements are used as photodetectors. As a result, the measurement precision of the ellipsometric angles in IRellipsometers is somewhat lower than that in the ellipsometers for the visible and is about 0.1"-0.5". The perfection of the polarizing optics and the application of more powerful and stable lasers, as well as the optimization of the electronic circuits of the signal processing, will in the near future probably increase measurement precision when necessary, by at least an order of magnitude. Nowadays, much prominence is given to the problem dealing with the increase of ellipsometric measurement locality. To our mind a highly felicitous way to solve this problem is proposed by Antziferov et al. (1975). The essence of the matter is clear from Fig. 12. Figure 13 shows the photograph of an LPE-3M ellipsometer assembled in ASCP sequence. The main idea of the cited work is realized in such an ellipsometer design.

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

41

6

FIG.12. Schematic diagram of an LPE-3M ellipsometer: 1 . laser; 2, circular polarizer; 3, modulator; 4, polarizer; 5, compensator; 6, specimen; 7, microobjective; 8, analyzer; 9, specular diaphragm; 10, photodetector; 1 1 , screen to observe visually the image of the specimen under study in polarized light.

FIG. 13. LPE-3M ellipsometer.

42

A. V. RZHANOV A N D K. K. SVITASHEV

All procedures for estimating angles $ and A by directly measured quantities can be obtained on the basis of the analysis of the expression for the output light intensity derived in Section 111-A of the present survey. When deriving this expression, we neglected the effects of multiple light reflection in the optical components of the ellipsometer. However, such an idealization is justifiable, since the experimental values of A and $, calculated following formulas (95) and (96), are in close agreement with the theoretically predicted ones in those cases when the equivalence of the reflecting system model to the real object is beyond any doubt. Of great interest is the generalization of expression (87) given by Semenenko (1975), Semenenko and Mironov (1977), Semenenko et al. (1978), and Svitashev et al. (1971, 1973b) for the case of the operating light beam of arbitrary aperture Aip and any spectral width A l . To our mind, worthy of note is Semenenko’s introduction of the concept of the “generalized” angles of polarization and $ (Semenenko, 1978). The basic equations for determining angles and $ (at any values of Acp and M of the operating light beam, as well as at any values of the f and 6 parameters of a compensator) are

$ = (- l)Np{ypmin- [2Np - (- 1)“ - 11 n/4} The same equations should be used in determining B and $ experimentally. The “generalized” angles of polarization a and $ are reduced to the ordin*

ary ellipsometric angles A and $ when employing a quasi-monochromatic, quasi-parallel operating light beam and an ideal compensator (6 = n/2, f = 1). Under the same conditions, but for 6 # n/2 we have by virtue of (99-100): A=A+p,

$=$+v

(1 16)

In the general case the magnitudes and $ depend not only on A and $ and the parameters of a compensator, but on Aip and M of the operating light beam as well. The procedure for calculating the generalized relative reflectivity =

tan$eiZ

(1 17)

consists in solving the integral equations describing the conditions for minimizing the output light intensity when using an operating light beam of arbitrary aperture and any spectral width. In our opinion, ellipsometry based on an operating light beam with arbitrary values of Aip and M opens fairly interesting possibilities in measuring thin-film parameters (Semenenko et d., 1978).

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

43

In addition, ellipsometric spectroscopy seems able to become the application sphere of “generalized” ellipsometric angles. In traditional spectroscopy one tries to diminish the spectral interval as much as possible when working at a given wavelength. In ellipsometry the method of the “generalized” ellipsometric angle makes it possible, in principle, to calculate whole portions of the dispersion curves, referring to wide spectral intervals (8001200A and even more) by the values of A and $ measured at different angles of incidence (or at a continuous angle-of-incidence scanning). Wavelength scanning is replaced by that of angle of incidence! This sort of approach to ellipsometric spectroscopy is currently being carried out at our Institute. But a great deal of work needs to be done in this field. OF EXPERIMENTAL ELLIPSOMETRIC STUDIES Iv. EXAMPLES OF SOME SYSTEMS

In this section we shall not try to give a comprehensive survey of the experimental work on ellipsometry. We consider it our task to list a number of typical examples that will allow a clear concept to be formed about the real possibilities of ellipsometric research methods and about the nature of the results that can be obtained when using these methods. A. Measurement of Thin-Film Parameters

Nowadays, the measurement of thin-film parameters is one of the chief concerns of applied ellipsometry. In effect, two conditions are necessary for applying ellipsometric techniques to the measurement of thin-film parameters. They are as follows:

1. The existence of “optical contrast” between a film and substrate, i.e., the difference from zero of at least one of the magnitudes An = n,,, - n, or Ak = ksub - k f . 2. Not too heavy light absorption in the film, i.e., satisfaction of the condition udf N 1, where u is determined by relation (8). Since ellipsometric measurements can, in principle, be made over a wide wavelength range (approximately from 0.3 to 100pm), it is clear that in practice it is always possible to find the part of the spectral region where the “optical contrast” between a film and substrate exists, and the film itself is transparent or weakly absorbing. The most universal and reliable way to determine the thickness and complex refractive index by ellipsometric measurements is by comparing the results of measurements with those of the numerical calculation of the

44

A. V. RZHANOV AND K. K. SVITASHEV

ellipsometric angle values for a number of the reflecting system models and a set of the model parameters. Essentially, the results of calculation supply an experimenter with a set of hypotheses on the behavior of the ellipsometric angles in some specific situations. Comparing experimental results with those of calculation permits us to reject incorrect hypotheses and to create a model equivalent to the system under study. The matter of a model corresponding to the system under study being settled, ellipsometry becomes really a highly effective method for controlling the system parameters and can be successfully utilized when studying the changes of these parameters under the influence of various ambients. In this section we shall cite a number of concrete examples demonstrating the facility of ellipsometry in determining thin-film parameters.

1. Measurement of the Parameters of Dielectric Films on Semiconductor Substrates

For systems of this type, “optical contrast” exists over a fairly wide spectral region (particularly in the visible) so that the “operating” wavelength can be chosen without much trouble. The fundamental equation of ellipsometry for the system “substrate (absorbing or transparent)-dielectric film“ when there exists an abrupt interface between a film and substrate is of the form of Eq.(54). The totality of the solutions of this equation for the prescribed values of A, (po, no, n , , and k, and for a set of film thickness values df and the index of refraction n, forms a set of curves (nomogram) on the plane (A, $). This set of curves can be directly used to estimate d, and n, by the measured values of A and $. Every value of n, corresponds to one curve. Every point of the curve corresponds to the definite (to within do period) value of the film thickness d, . To illustrate a real situation, Figs. (14)and (15)represent the numerical solutions of Eq. (54) for the systems “germanium substrate-dielectric = 6328 & q0 = 70”, film” and “silicon substrate-dielectric film” at loperat no = 1, n, = 1.3-2.48.The same figures present the results of the experimental determination of the polarizing angles for the systems : “silicon-silicon dioxide (Si-SiO,),” “germanium-germanium monosulfide (Ge-GeS),” and “germanium-germanium dioxide (Ge-GeO, )” for the different values of dielectric-film thickness. From the figures one can easily see that for the Si-Si02 system the experimental values of the ellipsometric angles A and $ lay well over the calculated curve corresponding to n, = 1.46.As regards the systems Ge-GeS and Ge-GeO,, the situation appeared to be so complicated that in order to create calculated models equivalent to reality, it was necessary to study

45

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

40

0

20

30

40

50

60

Ellipsometric angle

70

80 90

(degrees)

FIG.14. Numerical solutions of the fundamental equation of ellipsometry for the system "silicon substrate-dielectric film" (I = 6328 A; 'p0 = 70";n , = 3.9, k , = 0.02, n, = 1.3 + 2.48). Dashed lines are the lines of equal thickness, N is the number of lines of equal thickness; GL is the proportionality factor, different for different n,; I , 2, 3 are the results of measurements for the system Si-Si02.

0

40

20

30

40

50

ELlipsometric angle

60 70 80 (degrees)

!I

FIG.15. Numerical solutions of the fundamental equation of ellipsometry for the system "germanium substrate-dielectric film" (I = 6328 A, 'po = 70", n , = 5.4, k , = 0.77, n, = 1.3 + 2.48). Dashed lines are the lines of equal thickness; N is the number of lines of equal thickness; ci is the proportionality factor, different for different n,. 1,2 are the results of measurements for the system Ge-GeS; 3 , 4 are the results of measurements for the system Ge-Ge02.

46

A. V. RZHANOV AND K. K. SVITASHEV

360 c a “38

3 no Q a4

e

--I

U

u .-1 L

180

b: E 0

a

.J

=! w

90

0

20

30

40

ElLipsometric angle

50

60

(degrees)

FIG.16. Results of measurement of the ellipsometric angles and numerical solutions of the fundamental equation of ellipsometry for the system “germanium-four-layer film with the different values of the layers’ refractive index.” The figures on the curves correspond to the total film thickness in angstroms.

these systems in detail with regard to both consecutive increases and gradual etching of the film thickness. Figure 16 shows good enough agreement between the experimental results obtained for germanium sulfide film and the theoretical curves calculated under the assumption that this film consists of four sublayers with different optical properties. The first sublayer is inhomogeneous, of thickness 80-110 k The refractive index of this layer is complex and changes with layer thickness from the (5.4- i0.8), typical for germanium at Aopcrat = 6328 A, up to (1.9-i0). The absorption in this layer is evidently connected with the enrichment of the layer by germanium. The values of the refractive indices and the thicknesses of other film sublayers are shown in Fig. 17. The data of Figs. 16 and 17 refer, naturally, to quite definite thicknesses of germanium sulfide film The thicknesses of all “sublayers” change with the change in total film thickness. However, the qualitative picture holds good (Vasiljeva et al., 1974). The results mentioned above show just how complicated are such systems as dielectric layers grown on the surface of semiconductors, and how much we have to do to understand thoroughly all physicochemical and electron processes occurring in these layers. It is of importance from the methodological point of view that for transparent films the mutually unambiguous correspondence between the

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

47

FLlm thickness d

(4

FIG.17. Model of the system "germanium-germanium sulfide film."

Generalized elLipsometric angle

p(degrees)

FIG.18. Dependence of the generalized ellipsometric angles $ and 3 on the thickness of Si,N4 film on silicon substrate for monochromatic (solid curve) and nonmonochromatic (dashed curve) operating tight beams (cpo = 70", N,,= 4.12 - i0.03, N.,,N, = 2 - i0). Dark circles and light triangles indicate calculations-numbers near points indicate the film thickness in angstroms. Crosses and tight circles are the results of measurements of the generalized angles at M = 50 A and A I = 1200 A, respectively. (a): 1 is the spectral dependence of the transmission coefficient of the filter 3C - 8 of 3-mm thickness and 2 is the spectral curve assumed when calculating over the wavelength range from I, to 12.&, is the operating wavelength.

48

A. V. RZHANOV A N D K. K. SVITASHEV

values of the polarizing angles A and $ and the df and nf film parameters exists within the range 0 5 df 5 do only. At df > do, the values of the film thickness are determined with an accuracy to period do, i.e., df = d o d,. Here m is the integer and d , < d o . In this connection we consider the results presented by Semenenko et al. (1978) to be of great interest. From this work one can see that at least in some cases the range of unambiguous determination of the film parameters can be substantially extended when changing from a quasi-monochromatic light beam to the operating light beam of a sizable spectral width. Figure 18 shows what occurs in this case. It should be noted that one of the important applications of ellipsometry is the measurement of the thickness of superthin dielectric films (d < 50 A) on semiconductor and metal substrates. The real advantages of ellipsometry in such measurements are illustrated in Fig. 19, which presents the results of measurements of angles A and t,b in the process of growing natural oxide film on a germanium surface. As one can see from the figure, the angle A changed approximately by 1.5” during 32 hr, while the angle $ remained unchanged to within 1-2’. If one considers the change of the effective thickness of the oxide film adeE,then the results presented in the figure yield

+

adeR= 5,9 f 0,3 A

0 2 4 6

8 i0

22 24 26 20 30 32

Time ( hours) FIG. 19. Change of the polarizing angles and A during the growth of oxide film on an etched germanium surface in air. (1, 2, 3 are the specimen numbers.)

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

49

where d,, is determined by the relation

4,

= CA(A - Ainitial)

(118)

This relation follows directly from expression (66), where for the system under consideration CA = 0.0656 AJmin. As is evident from the foregoing examples, the accuracy of ellipsometric measurements of the thickness and refractive index of thin transparent films is very high. However, it is necessary to emphasize the appreciable difference between the accuracy in measuring the increase of the film parameters and in measuring the absolute values of these parameters. In the works by Jerrard (1969a), Matheson et al. (1976), and Schueler (1969), the only subject under discussion was the problem dealing with the effect of the experimental errors that arise while determining the angles of polarization on the measurement results. The analyses given in these works show that when using up-to-date techniques it is quite practicable to observe, for example, a change in transparent film thickness on the order of 0.1 8, or even of several hundredths of an angstrom. Of course, in the present case the mean film thickness within the operating light beam cross section is meant. In the work by Svitashev et al. (1972b), it is shown that the basic sources of error in measuring the absolute values of the film thickness and refractive index are the errors in prescribing the presumably known parameters and experimental conditions, when creating a nomogram necessary to determine df and n,. These parameters cover the optical characteristics of the substrate, the refractive index of the environment (no), the wavelength of the operating light beam (;lopera,), the angle of light incidence on the specimen (qo),etc. It turns out, for example, that it is sufficient to make a mistake in the value of qo by 10, in order for the absolute accuracy in measuring SiOz film thickness on silicon to be no more than 10-15 A over the entire thickness range from 0 to do. In measuring film thicknesses to ensure absolute precision to the order of several angstroms, the index of refraction and the absorption coefficient of a substrate should be known up to the fourth and sometimes to the fifth decimal place. It should be noted that in the range of small film thicknesses (d, c 200 A), the simultaneous determination of the film thickness and the index of refraction becomes impossible if employing ordinary techniques, since the solutions of the fundamental equation of ellipsometry corresponding to the different values of n, and d, practically coincide. The most effective way of tackling this problem was found by Egorova et al. (1974). To determine the film thickness and index of refraction, the authors suggested using the dependences of the values of the function A = A(d,, n,) on the ambient index of refraction no. In this case the results of ellipsometric measurements are interpreted by means of the nomograms that represent the combination

50

A. V. RZHANOV AND K . K. SVITASHEV

E llipsometrcc angLe A

170 1ao {go 200

FIG. 20. Phase nomogram A , , A2 for the system “silicon substrate-dielectric film.” A, at no = 1 1 (air); A, at no = 1.457 (CCIJ. The figures in parentheses are the values of nr. The figures without parentheses are the film thicknesses in angstroms. Dashed lines are the lines of equal thickness. 1, 2, 3 are zero positions for k , = 0, kl = 0.02, and k , = 0.05, respectively. The experiment was carried out for different oxide films on silicon: 4, thermal; 5, pyrolytic; 6, natural oxide.

of two functions, A , = A,(d,,n,) and A2 = A 2 ( d , , q ) . The function A , is calculated for the case of no = nil) and the function A, for the case of no = n f ) . The example of this type of nomogram is given in Fig. 20. In particular, when using these techniques, Egorova et al. (1974) managed to show that the index of refraction of SiO, films on silicon prepared through thermal oxidation does not change down to film thickness of the order of 15-20 A. A similar technique for measuring the thickness and refractive index of superthin films has been proposed by Shklyarevsky et al. (1971). It is based on using the value of qo,the angle of incidence, as a free parameter.

2. Measurement of the Thickness of Epitaxial Semiconductor Films on Semiconductor Substrates

To solve this problem, it is necessary for ellipsometric measurements to be made in the middle or far infrared, since only in these spectral regions does the optical contrast between a film and substrate arise. This optical contrast is caused by different concentrations of the carriers in the film and substrate (Pankov, 1971). The first experimental data on ellipsometric measurements in the infra-

51

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

red were obtained by Hilton and Jones (1966). The precision of their results was restricted by an insufficiently powerful radiation source and the low luminosity of the experimental setup. IR-ellipsometry was further developed by Svitashev et al. (1972%1973a). It was found possible to make measurements by employing a convergent light beam, which permits an increase in the signal-to-noise ratio at the output of the instrument. While interpreting results, a model of the reflecting system with a smooth junction between the epitaxial film and substrate was used, thus providing the opportunity for obtaining reliable quantitative results on the thickness of epitaxial silicon layers on silicon. DeNicola et al. (1972) succeeded in applying an H,O-laser to IR-ellipsometry, but the lack of experimental data prevented complete corroboration of the advantages inherent in the method. In the work by Svitashev et al. (1977~)results were given allowing the precision of IR-ellipsometry in measuring epitaxial layer thicknesses when using an H,O-laser as a light source (Lop,,, = 118 pm) to be judged. The accuracy in measuring A and i,b by the present ellipsometer was determined from a number of independent measurements on a silicon specimen with resistivity of about 0.01 ohm cm. The obtained root-mean-square deviations were o[A] = 25’ and o[$] = 15’, respectively. The optical characteristics of the system “epitaxial film Si(n-type)-substrate Si(n-type)” were under study. The optical constants of the silicon substrate of 0.01 ohm - cm resistivity were determined from the measurements of the ellipsometric angles for a large number of specimens. Table I11 presents the results of the measurements of angles $ and A and the values of the index of refraction and extinction coefficient estimated by these data following formulas (5% and b). The measurements were made at the angle of incidence cpo = 65”. TABLE 111 IR-ELLIPSOMETRIC MEASUREMENTS OF A, I), n, AND k ( p = 0.01 n . C M )

ON

SILICON

N”

A (deg)

6 6 (deg)

I) (deg)

SI) (deg)

nI

k,

1 2 3 4 5 6 7

157.3 156.8 155.9 156.8 156.7 154.0 156.5 156.3

- 1.0 -0.5 -0.4 +0.5 -0.4 -2.3 +0.2 0.75

37.1 37.5 38.1 38.9 38.4 36.4 36.2 37.6

-0.1 -0.1 +0.5 +1.3 +0.8 -1.2 -0.4 0.63

4.6 4.4 4.1 4.1 4.2 4.1 4.1 4.3

6.8 6.6 6.7 7.3 7.0 5.7 5.8 6.6

Average

A. V. RZHANOV AND K. K. SVITASHEV

52

The epitaxial film thickness is determined by the theoretical curve. To calculate it, one needs to know not only the optical constants of a substrate but the optical constants of a film as well. Besides, one has to choose the reflecting system model. The optical constants of the film were determined by comparison of the experimental results, obtained by a gradual etching of the epitaxial film, with the theoretical curves calculated for high-resistance silicon (Moss, 1959). Good agreement between one of the theoretical curves and the experimental data was obtained when comparing the set of A and $ points, found during the gradual etching of the film, with the curves calculated from the constants of the given substrate and some values of the n and k of the film close to the known data for high-resistance silicon ( p = 1.5 ohm. cm, nf = 3.3, k, = 0.01). The curves were calculated on the basis of a reflecting system model with an abrupt interface between the substrate and epitaxial layer. These curves were calculated by using the values of the optical constants of a substrate obtained by averaging the values measured for a large number of specimens with p = 0.01 ohm-cm. As mentioned above, the procedure for determining the epitaxial film thickness is reduced to a comTABLE IV A

N 1 2 3 4 5

6 7 8 9 10 11 12 13 14 15 16 17

$ ELLIPSOMETRIC ANGLES AND CORRESPONDING d FILM THICKNESS VALUES OBTAINED BY IR-ELLIPSOMETRY (dl), INTERFEROMETRY (dJ, AND TECHNOLOGICAL DATA(d3)

AND

dl

d2

A

IL

Ocd

Ocm)

123"38' 124"19' 146"18' 145"12' 146"O 143"20' 252"O 287"O' 268"O' 300"O 154"O 134"13' 146"6' 191"O 129"lS' 2640' 130"O

34"O 1' 34"3O 36"3O 37"54' 37"3O 34"15' 35"12' 39"48' 35"55' 45"O 3Y4O 33"53' 36"O 35"3O 34"23' 36"O 36"12'

2.7 2.65 0.9 1.o 0.75 1.2 10.3 8.7 9.3 8.2 0.15 2.1 0.8 15.7 2.2 9.6 2.2

3.2 2.9 -

d3

2-2.5 2-2.5 1 1

-

0.7 0.7 5 5 10 10 3 3 1 16

-

-

-

-

10.7 8.9 9.8 8.5 -

9.8

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

53

parison of the numerical solutions of the fundamental equation of ellipsometry (54) with the measured values of the ellipsometric angles. The applicability of Eq. (54) is, in the present case, stipulated by the fact that when working in the far infrared, the system “poorly doped semiconductor epitaxial film-heavily doped semiconductor substrate” is a particular case of the system “absorbing substrate-transparent film.” The film thicknesses for a set of studied specimens are given in Table IV, where they are compared with the results of measurements made by IR-interferometry and the technological data on the thicknesses of the grown films. From the table one can see that the values of the film thicknesses determined by ellipsometry are always smaller than those obtained by means of IR-interferometry. The results of the work by DeNicola et al. (1972) also prove the decrease in the

EUpsometric an$e

(degrees )

FIG. 21. Theoretical curve A - $ for the system “silicon substrate (N,= 4.3 - i6.6, - iO.Ol).” The numbers near the arrows indicate the thickness of the epitaxial layer in micrometers at the corresponding points of the curve. I-experimental points for the specimen treated by gradual etching. The numbers at the experimental points show the serial number of the etching. The length of the prime along the coordinate axes is equal to a random error in measuring the corresponding angles of polarization. Roper., = 1 18 pm. p = 0.01 ohm, cm)-silicon epitaxial film (N2= 3.3

54

A. V. RZHANOV A N D K. K. SVITASHEV

values of the film thicknesses when measured by ellipsometry at I = 118.6 pm as compared with the data of IR-interferometry. This systematic divergence could not be attributed to measurement errors. From the reproducibility of the results obtained, one can easily evaluate the accuracy when measuring with a given ellipsometer. From Fig. 21 it is evident that there is at least a lo” change in angle A for 1-pm change in thickness. Hence, the accuracy of measurement on this ellipsometer is not worse than 0.05 pm when the reading accuracy of the angle measuring unit is 30. The discrepancy between the results of the two methods under consideration can be explained by a deeper penetration of radiation at the frequencies used in IR-interferometry (A = 10-35 pm) into an interface, while the radiation at 1 = 118.6 pm begins, apparently, to reflect from the edge of the inhomogeneous layer with a lower carrier concentration, i.e., from the boundary between the film and the inhomogeneous layer. The technological data on the thicknesses of the grown films (see Table IV) have an appreciable error and, naturally, do not always agree with the results of measurements made by other techniques. The above results permit the following conclusion to be drawn. Laser ellipsometry in the far infrared, being a nondestructive, sufficiently universal technique for measuring the thicknesses of epitaxial semiconductor films, approaches in accuracy ellipsometry in the visible. A very high degree of accuracy in measuring epitaxial film thicknesses on heavily doped substrates can be obtained when using a C0,-laser (Aopera, = 10.6 pm) as a source of the operating light beam (Batavin et al., 1974). Additional data on the facilities of ellipsometry in the infrared can be obtained from the review by Rezvyi and Phinarev (1977). 3. Determination of Absorbing Film Parameters ; “Search” Program

In the case of absorbing films, the functions $(df) and A(df) lose their periodicity with period do, and the appropriate curve in the A$ plane takes the shape of a “helix” (see Fig. 22). As a result the opportunity arises to determine unambiguously the absorbing film thickness at d, > do. This opportunity is most simply realized at the known values of the film and substrate optical constants (Semenenko et al., 1972). But the difficulties that arise are caused by the large role played by the value of the film extinction coefficient k,. When the values of kf are large, the “helix” rolls rapidly up into a point, and starting with a definite film thickness, it is already impossible to determine df with sufficient accuracy. When the values of kf are small, it is, in principle, possible to make measurements of df over a wide range of film thicknesses df %- do. But in this case higher demands are made on

ELLIPSOMETRIC STUDIES OF SURFACES A N D THIN FILMS

55

180 c

ul 0) 0)

kn

-2

90

0

30

60

Ellipsometric angle

90

(degrees)

FIG.22. Numerical solutions of the fundamental equation of ellipsometry for the system “quartz substrate-monocrystalline silicon film” (A = 5461 A, cpo = 60”, no = 1.75, k , = 0, n , = 1.46, k , = 0, n, = 4.05, k, = 0.028), The figure represents only separate portions of the calculated curves. The number at the beginning of each portion indicates the layer thickness of monocrystalline silicon at the starting point of this portion and translates to micrometers as follows: 0-0, 1-0.265, 2-0.575, 3-0.865, 4-1.115, 5-1.445, 6-1.735, 7-2.025, 8-2.315, 9-2.605, 10-2.995, 11-3.285, 12-3.575, 13-3.865. The change in silicon layer thickness when passing from one point to another equals 50 A. The arrows show the direction of thickness increase. to indicates the massive silicon specimen.

the instrument (on its adjustment and on its precision in determining A$), as well as on the prescribed accuracy of the film’s optical constants. While calculating the nomogram, even slight deviations of the prescribed values n, and kf from their true values will lead, at large enough thicknesses, to crude errors in determining df. In practice, we cannot often guarantee the accuracy of the film’s optical constants, and sometimes their values are merely unknown. In such a case there arises the problem of dealing with the simultaneous determination of three parameters of the reflecting system : d,, n,, and kf. Quite a number of works (Malin and Vedam, 1976; Oldham, 1969; So, 1976), devoted to the ellipsometry of absorbing films have been recently

56

A.

V. RZHANOV AND K. K. SVITASHEV

published, but all of them are of a purely computational character. Dagman et al. (1978) analyzed a procedure for determining absorbing film parameters based on the application of the maximum likelihood method. In this case it is necessary to determine some experimental pairs of A and $ polarizing angles. Then, the determination of d,, n,, and kf parameters is possible when utilizing the minimization procedure of some function dependent on these parameters. This is performed by means of the program “parameters search by fitting method.” Used as input data to the program are some original values of the parameters sought, the regions of their changing, and the initial searching step, as well as the experimental values of A and $ (with the experimental errors of their determination), measured at different angles of incidence cpo and, in the general case, for different film thicknesses. In the “Search” program the finding of the unknown parameters is reduced to minimization of the aiming function S and is performed in accordance with the following algorithm: first, find 4 (at rn = 2) or 2rn - 1 (at rn > 2) coordinate extrema by means of coordinate descent. Here rn is the number of parameters sought. These particular extrema serve as the reference points in the sought parameter space by which one can calculate analytically the minimum position of the aiming function. If the level curves of this function are elliptical, then by the present procedure the minimum position can be determined at once (Fig. 23a). If the level curves are not elliptical, then the calculated value will be a certain approximation to the minimum. Then, the searching procedure will be advanced ; i.e., one will find the new 2m - 1 reference points by which a new minimum position will be calculated (Fig. 23b); i.e., the considered searching procedure turns out to be self-guiding. The aiming function S is to within a factor of 1/N the likelihood function (Hudson, 1964) for the ellipsometric angles, dependent on the thickness d,, indices of refraction n,, and extinction coefficients k,.

where the summation is over a set of measuring situations. and Ai, are the experimental values of the ellipsometric angles $ and A in the i-measuring situation; t,bic and Aic are the calculated values of $ and A in the i-measuring situation at some values of the 4, kf, and df parameters under determination; s$,, and 6Ai, are the experimental errors in determining angles $ and A ; N is the number of measuring situations. The set of measuring situations is most often determined by the totality at which the ellipsometric of the angles of incidence cpo (cpol, qo2,.. ., qON)

ELLIPSOMETRIC STUDIES OF SURFACES A N D THIN FILMS

Sought parameter

57

p1

(b) FIG.23. Level lines of the aiming function S in the space of the sought parameters (twodimensional case). Numbers denote the coordinate extrema of the S function. (a) Elliptical level lines: Po-the starting point of search; P,,,-the final point of search. (b) Nonelliptical level lines: P‘-the intermediate point of search.

angles are measured. Evidently, in order to determine three parameters of an absorbing film, it is necessary to make measurements at two angles of incidence at least (the number of measuring situations N = 2). But the set of measuring situations can also be determined by the choice of some specimens differing in film thickness df only: d;’), dr2), . . ., d:”. In this case when adding each new specimen, we obtain one new unknown parameter (film thickness dp)) together with two experimental values of the polarizing angles lClie and Aie. The calculation values of the polarizing angles entering the S function in the search process are estimated by the values of the unknown parameters, gradually approximating the sought ones. The satisfaction of the condition S > 1 serves as a criterion for the end of the search

58

A. V. RZHANOV A N D K. K. SVITASHEV

process. As it follows directly from the work by Hudson (1964), the function S is determined by the expression S = x2/4N,where x2 = (xi - Xi)’/02 ; xi is an independent normally distributed random magnitude; Xi is the mean value of the random magnitude x i ; o is the root-mean-square deviation; and N is the number of the distributional degrees of freedom of x2. Thus, rejecting the solutions satisfying the condition S > 1, we reject the solution with n,, k,, and d,, satisfying the experimental data with confidence level p, which corresponds to the values x2 > = 4N. The quantity p is estimated by the tables for the prescribed values of the distributional degrees of the freedom x’ and the chosen 2; (Hudson, 1964). In the examples given below the number of measuring situations becomes equal to 3 or 4, and the values of p in these cases equal roughly 6 and 4%, respectively. This means that we reject the correct solutions in approximately 5% of the cases, which is quite acceptable. Since the function S is, in fact, the likelihood function for the angles of polarization, then there exists the opportunity for approximate evaluation of the root-mean-square error in d,, n,, and kf. This is illustrated in Fig. 24, where the function S is plotted against one of the film parameters under determination. Two other parameters are fixed and correspond to the values at which the S function is minimum. The evaluations of all three parameters are supposed to be independent in this case. Hence, one can see that the precision of the film parameter determination depends on the minimum shape of the aiming function. This shape is in turn determined by quite a number of factors and, in particular, by a felicitous choice of (poi, at which the measurements are made, and by the values of the experimental errors. By the experimental errors we mean the statistical errors in determining the polarizing angles. The statistical errors include both the measuring eigenerrors in determining angles and A and the errors related to inaccuracies in determining the angle of incidence, the optical constants

+

2

t

0

Swqht parumeter P, FIG.24. Aiming function versus one of the parameters under determination in the minimum vicinity. opiis the mean-square error in determining the i parameter at 68% significance level.

59

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

of the substrate, and other parameters included in the ellipsometricequation. As an example, we cite the parameteric values for titanium films on quartz substrates and vanadium dioxide films on sital substrates obtained by means of the "Search" program. All the measurements were made on a laser photoelectric ellipsometer LPE-2 at )lopera,= 6328 A (Antziferov et al., 1972). The values of the calculated ellipsometric angles for the determined parameters of titanium films coincide, within the measuring errors, with the experimental values of I(/ and A for all the angles of incidence at which the measurements were made (Table V). The values of the films' optical constants agree with the values of n, = 1.95 and k, = 1.34 determined from the measurements on thick titanium film. The experimental points for the specimens presented in Table V lay well beyond the theoretical dependencies of the ellipsometric angles on the film thickness at four angles of incidence for the reflecting system "quartz substrate-film with the optical constants nf = 1.95 and k, = 1,34" (Fig. 25). In the lower part of Table V the parameters of the same films are presented, but they were determined in another way. Since in the present case films with different thicknesses had approximately the same optical constants, the experimental values of the polarizing angles, measured at the same angle of incidence for three specimens, were introduced in the "Search" program in order to determine the thicknesses of all three films under study and their optical constants. TABLE V EXPERIMENTAL ELLIPSOMETRIC ANGLEVALUESAND RESULTS OF CALCULATION OF A, $, d,, nf, k , BY "SEARCH" PROGRAM FOR TI FILMON QUARTZ SUBSTRATE

N 1

2

3

4

90

4

60" 63" 65" 70"

12Y35' 119"05' 111"lO 86'21'

23'20' 21"36 20"50 1938'

12Y27'33" 119"05'08" 110"53'01" 86"04'11"

23'22'35 21"40'29 20"45'57" 19"56'08"

55"

60" 63" 65"

150"25' 138"48' 128'54' 120"19'

25"24' 21"23' 18'58' 17'36'

150"27'50 138"52'50 128O51'47" 120"2003

25"21'45 21'2039 1Y00'32 17"37'40

60" 65" 70"

150"21' 91'29' 28"55'

9"02' 4"38' 9"27'

149"50'05 91' 12140 29"07'57"

Y03'59 4"43'05" 9"29'55"

*e

*o

d, (A)

n,

kf

730

1.950

1.338

420

1.948

1.348

118.8

1.974

1.314

Five parameters calculated simultaneously by the "Search" program at fixed cpo = 65" are equal to d,, = 735 A, df2= 427 A, d,, = 116 A, nf = 1.968, and kf = 1.342

A. V. RZHANOV AND K. K. SVITASHEV

60

(a)

420 Rlm thickwss d

400

(c 1

(b)

i.93 c95 1.97 L33 i35 i.37 Refractlye index nr Extinctan coefficient ki

A) FIG.25. Aiming function versus the sought parameters of the absorbing films in the minimum vicinity for Ti specimen N2. (a) S = S(d);(b) S = s(",); (c) S = S(k,). 1, $ = $,, A = Ae; 2, the introduction of a systematic error in $ ($ = $e + 10, A = Ae ). 3, the introduction of a systematic error in A (A = Ae + 3 0 , $ = $e ).

Vanadium dioxide films have phase transition of the first kind at 68°C accompanied by a drastic change in optical properties (Verleur et al., 1968). Dagman et al. (1978) determined the values of the thicknesses and optical constants of VO,, as well as the changes in the optical constants with temperature in the region of the phase transition (Beresneva et al., 1977). Table VI presents the parametric values for two vanadium dioxide films at room temperature and at a higher temperature than that of the phase TABLE VI EXPERIMENTAL ELLIPSOMETRIC ANGLEVALUES AND RESULTS OF CALCULATING A, $, df, AND k, BY "SEARCH" PROGRAM FOR VO, WITH FILMS AT DIFFERENT TEMPERATURES d,

(4

"r

7"05'22 l"13'48" 8"27'36

45

2.014

0.3

179"48'17" l"41'38" 360"08'14

6"5028 O"45'31" 8'33'39

41

1.782

0.4

21"18' 1Y42' Y14'

168"2056 16T32'35 148"06'49

21"12'43" 15'3659 9"22'31"

731

2.260

0.276

16"29' 1w36' 6"16'

160"02'40 145"13'04" 99"55'38"

16"21'52 10"31'08" 6"16'38"

736

1.925

0.332

N

Po

A,

1, r = 20°C

55" 60"

188"29 298"21' 352"38'

7"10 1'13' 8"28'

1 18"25'55" 298"11'23" 352"41'57"

60" 65"

179"33' l"00 359"50

6"50 O"46' 8"34'

55" 60" 65"

168"OS' 162"43' 148"lI'

55" 60" 65"

159"55'

65" 1,

r = t,,,",

2,r = 20°C

2, r = r,,,

55"

145"03' 100"15'

*e

A,

*c

kf

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

61

transition. These parametric values were determined by the “Search” program. The calculated values of the polarizing angles agree, within the experimental error, with the measured values. The values of the film thicknesses determined at two different temperatures are also in attractive agreement for both specimens. Since it is not the accuracy in determining the original parameters but the real experimental errors in determining the polarizing angles that are introduced into the program as input data, the accuracy in determining n,, k,, and df is evaluated a posteriori. Figure 26 shows the values of the S function versus three parameters of titanium film (sample N2). In this case &,be and dAe include the errors in determining $ and A proper and the errors related to the inaccuracy in measuring the angle of incidence. Mean-square errors of the parameters obtained are o(d,) = + 3 & a(+) = k0.007,o(k,) = k0.005.The same figure also represents the data on the effect of a systematic error on the values of the parameters sought. It is a simple matter to see that this effect can be substantial. The proper choice of the region where these parameter changes take place is an important aspect of the matter when determining parameters by the

Eltipsometric angle

(degrees)

FIG.26. Solution of the fundamental equation of ellipsometry for the reflecting system ”quartz substrate-absorbing film” at four values of cpo (n, = 1.95, n, = 1.34: 1 - cpo = 50”, 2 - cpo = 60’, 3 - rpo= 65”, 4 - cpo = 70”). Numbers indicate the values of df in angstroms, dashed lines are the lines of equal thickness; 5 , 6 , 7 indicate experimental points for three specimens ; A,,,, = 6328 A.

A. V. RZWANOV AND K. K. SVITASHEV

62

TABLE VII EXAMPLE OF INCORRECTPARAMETER DETERMINATIONS FOR TI FILMBY THE "SEARCH" PROGRAM DUE TO BAD CHOICE OF SEARCH STARTING POINT

N

cpo

Ae

IL.

4

2

55" 60" 63" 65"

150"25' 138"48' 128'54' 120"19'

25"24' 21"23' 18"58' 17"36

150"25'44" 138"45'03" 128"39'33" 120"04'50

df

ILC

25"22'46" 21"21'51" 1Y02'20 1T4010

(4

nf

kf

305

1.830

1.579

"Search" program. In particular, the several sets of kf, nf,and 4 can enter the region restricted by experimental errors. Thus, there are presented in Table VII the results of the wrong determination of the parameters of titanium film (sample N2) by the "Search" program. Experimental values of the ellipsometric angles and their errors are quite the same as in the example given in Table V. But in the present case the starting point of "Search" was unfortunately chosen when the sought parameters were changing over a wide range. The values of the ellipsometric angles calculated to within experimental errors agree with the experimental values, but the values of n,, k,, and 4 obtained in this case differ sharply from the results of Table V. The existence of a great number of minima of the S aiming function in the space (nf,k,, df) and the presence of the real experimental errors are the principal restrictions of the procedure. Thus, if the parameters of the system under study are unknown and one has to prescribe the wide ranges of their change within which the solutions are searched, then having obtained the results formally satisfying the condition S 5 1, it is necessary to analyze it. When investigating titanium films, several films from the same technological process were used. The false minimum from sample N2 was rejected on the ground that the optical constants in this case should have close values. In the case of vanadium dioxide films the condition of equality of the film thickness before and after the phase transition was used. The application of such extra conditions reduces practically to the limitation of the space volume (nf, kf, df), within which the solutions are searched, and thus it decreases essentially the number of false minima, though it does not eliminate completely the ambiguity of the solution. In conclusion we note that the real deviation in the values of absorbing film parameters for the measured systems exceeded essentially the requisite accuracy for determining parameters by the above techniques. Thus, the estimations of the mean-square errors for the optical constants of titanium films for all the measured specimens give a much lower value than the differ-

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

63

ences in n, and kf obtained on the different specimens. This seems to be connected with the existence of a thin oxide film on the titanium surface. The effect of this film turns out to be stronger on thinner specimens (Table V). Thus, the real errors in determining parameters are caused in this case by a lack of correspondence between the model entered into the program and the real reflecting system. The “Search” program can be employed to solve a wide spectrum of problems of applied ellipsometry. B. Studies of Adsorption-Desorption Processes by Means of Ellipsometry The high sensitivity of ellipsometric measuring techniques provides the opportunity to fix reliably rather small adsorption coverages accounting for hundredth and even thousandth parts of a monolayer. Modern highclass automatic ellipsometers allow high-precision measurements of the angles-up to some angular seconds-with a very short duration-for about 10msec. The last value is extremely important because it ensures the possibility of studying the kinetics of adsorption-desorption processes, at least when their rates are not too high. At the same time it should be emphasized that there exist some serious theoretical difficulties in interpreting ellipsometric measurements at small adsorption coverages. The common theory employing macroscopic parameters is not evidently realistic in this case. The attempts to develop the microscopic theory by proceeding from consideration of the elementary interaction of the separate adsorbed atoms and molecules with the light beam are far from being completed yet. The difficulties arising in this case are caused both by the complexity and cumbersomeness of the mathematical calculations and by the lack and uncertainty of the data for constructing the physical analogs corresponding to reality. However, it is exactly this circumstance that must in our opinion stimulate the setting up and carrying out of experimental studies of the adsorption-desorption processes by means of ellipsometry. To our mind, in this way only can one obtain the necessary facts. The first quantitative experimental studies in this field were performed by Archer and Gobeli (1965)who investigated oxygen adsorption on a freshly split silicon face (111). In interpreting the results of their measurements, they proceeded from ideas about the equivalence of a submonolayer adsorption coverage and a two-dimensional lattice of the scattering centers with different values of the coefficients of parallel and normal light scattering to the surface. On the basis of these ideas, one can give good grounds for the assumption about the proportionality of the polarizing angle changes to the degree of surface adsorption coverage. Archer and Gobeli (1965)obtained the thickness value of a monolayer

64

A.

V.

RZHANOV AND K. K. SVITASHEV

coverage and the value of the oxygen-sticking coefficient at different stages of the adsorption process. Later on in the works by Bootsma and Meyer (1969%b, c) the assumption concerning a linear dependence of the polarizing angle A on the degree of coverage of the germanium surface by the adsorbed krypton or methane was experimentally verified up to a degree of coverage of the order of 0.1 monolayer. This result permits us to introduce the relation connecting the change of the polarizing angle A with the effective thickness of the adsorbed layer : 6A =

c, d,, *

(120)

Here deR = dmonolayer * 0, where 6 is the degree of surface coverage in a submonolayer region. The most surprising fact is that the value of the C, coefficient can be calculated on the basis of extrapolation of Drude classical theory. In this theory the effective refractive index of the adsorbed layer is estimated through the values of the atomic or molecular polarizabilities and the dimensions of the adsorbed atoms or molecules by means of the Lorenz-Lorentz relation (Slater, 1967) :

where a, is the electron polarizability and V, is the molecular volume of the adsorbed particles. A good agreement between the experimental evidence and the calculated values of coefficient C, was observed for the cases of krypton and methane adsorption on germanium. At the same time in the cited works an anomalous behavior of the $ angle was discovered when changing the degree of silicon surface coverage in the course of chemisorption. The authors came to the conclusion that this phenomenon relates to the changes of the state of the adsorbent presurface layers upon chemisorption. Algazin et al. (1977) and Rzhanov (1977) managed to get new data on the character of the anomalous behavior of $ in the region of submonolayer coverages. The experiment in this case consisted in the thermodesorption of a germanium sulfide thin layer. This layer was first grown on a germanium surface in the flow of H2S (Vasiljeva et al., 1971, 1974). The experimental results are given in Fig. 27. As one can see, the angle of polarization A increases monotonously in the course of the thermodesorption and reaches its highest value at the total desorption of the germanium sulfide layer. The atomic cleanliness of the germanium surface was in this case verified by the experiments on the LEED technique (Rzhanov et al., 1973). The angle of polarization $ behaves in a complicated manner: at first it de-

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

65

ELLipsometrlc

angles AlCl

I

G O 9 . 22’00’

14@-21°00‘

1359. 20°30’

i309~2O0O0’ O b

iL

io 3’6

io i o

60

Time (minutes)

FIG.27. Change of the ellipsometric angles II/ and A and photomultiplier “dark” current I in germanium sulfide thermal desorption from germanium surface.

creases, then goes through minimum, increases again, and at last reaches its highest value, typical for atomically clean germanium surfaces. This phenomenon can be connected with a reconstruction of the top layers of germanium itself when removing the last molecules of germanium sulfide. This reconstruction is equal to the surface phase transition, and during the reconstruction a heavy disorder of the germanium surface should take place. This disorder is manifested by a sharp peak in the “dark” current of a photomultiplier recording light beam at the output of the ellipsometer (see Fig. 27). The “dark” current is caused by the partial depolarization of light as a result of its scattering by the disordered surface. As it appears from the experiment, the “dark” current has a drastic maximum in the region of the anomalous behavior of the polarizing angle Ic/ and then drops sharply when approaching the atomically clean surface. Ellipsometric studies under conditions of partial reversibility provide highly valuable and in many instances even unique inlormation on the nature of adsorption-desorption processes. As a typical example of this kind of study, let us analyze the results of Baklanov et al. (1976, 1978), who investigated the adsorption of bromine vapors on the face (111) of an atomically clean germanium surface. The studies were carried out by an automatic ellipsometer at room temperature.

66

A. V. RZHANOV A N D K. K. SVITASHEV

Germanium specimens with surfaces cleaned by the evaporation of thin-film germanium sulfide at temperatures of the order of 400°C in vacuo at 1 * 10-'-5 lo-' torr (Rzhanov et al., 1973) were cooled down to room temperature. The criterion for setting the temperature at 20°C was the values of the polarizing angles (see Section IV-C of our survey). Then the reactor was filled with bromine vapors and the changes in the polarizing angles and the ionic current af a manometric sensor were continuously registered. The typical experimental results are presented in Fig. 28 where one can see that the changes in the ellipsometric angles during bromine adsorption on the germanium surface are entirely irreversible and have the shape of the saturated curves. The diagram presents logP as a function of t. The cross-hatched area in this diagram indicates that at the given bromine injection, the pumping out of the reactor to the pressure corresponding both to the upper area boundary and the lower one does not affect the subsequent change in the ellipsometric angles. When estimating the refractive index of a bromine-adsorbed layer on a germanium surface, Baklanov et al. (1978) used Eq. (121) assuming that the value of bromine molecular polarizability changes little upon chemisorption. The molecular volume upon monolayer adsorption was taken to be equal to the molecular volume in the liquid bromine.

Time (minutes) FIG. 28. Typical results of ellipsometric angle change with injections of bromine into the reactor.

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

67

That is why an analysis was made of possible errors in determining the adsorbed layer thickness at the fixed value of a, = 6.87 and V, ranging from 70 to 100 (A)3. It was shown that the error in determining the thickness does not exceed 20% at V,, varying in the above range. Employing the foregoing calculation procedure and experimental results, the dependence of JA and d on the exposure to bromine vapors was obtained (Fig. 29). Using this dependence, one can calculate the values of the bromine molecules’ sticking coefficient S on the germanium surface. The value of the sticking coefficient is determined as a portion of the adsorbing molecules from the total number of molecules colliding with the germanium surface. And so N,

de dt

-=

P

,/ws

where N , is the concentration of the surface adsorbing centers ; P/,/2nmkT is the number of gas molecule collisions with the unit of surface area in the unit of time; S = (1 - e)a*e-EIkT;CT is the condensation coefficient of the adsorbing gas molecules on the free centers of an adsorbent ; 8 is the degree of coverage; and E is the activation energy of adsorption. Taking into

FIG.29. Dependence of 6A and thickness of the adsorbed layer on exposure to bromine. Insert the values of the ellipsometric angle changes upon bromine adsorption on a germanium surface (1 11): t = 20°C; 1 , 2 , 3 denote the experiments.

68

A. V. RZHANOV AND K . K. SVITASHEV

Thickness of adsorbed Layer

(8)

0

& A (minutes) 30. Sticking coefficient of bromine molecules S on germanium surface ( I 1I ) as a function of adsorption layer thickness. I = 2 0 C . FIG.

account that 8 = SA/SA,,,, where SA,,, is the maximum change of A upon adsorption, it is easy to show, that

and one can find the form of the function S(6A) or S(d). The determined values of the sticking coefficient at different thicknesses of the adsorption layer are presented in Fig. 30. At 8 = 0, S = oe-E/kT.Therefore, proceeding from the data represented in Fig. 30, it is possible to evaluate the extreme values of the activation energy of adsorption and the values of the condensation coefficient of bromine molecules on a germanium surface: rre-E’kTN 1 lo-’; E I2.7 kcal/mol; 1 . lo-’ Irr I1. Hence, the considered adsorption process at room temperature runs practically irreversibly and without activation. To our mind the foregoing examples demonstrate well enough the efficiency of ellipsometry in studying adsorption-desorption processes.

C . Determination of the Optical Characteristics of Materials Ellipsometry permits the measurement of the refractive indices n and extinction coefficients k of very different materials with high accuracy over a wide range of temperatures, pressures, and wavelengths. In this case practically no restrictions are placed on the magnitudes n and k . In this we shall analyze the real capabilities of ellipsometry, citing as

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

69

an example the measurement of the optical characteristics of germanium and silicon within a wide temperature range, and then we shall give a bibliography of the literature devoted to the measurement of the optical characteristics of quite a number of other materials. Ellipsometric spectroscopy is a very interesting and nowadays an intensively developing branch of ellipsometry. The subject of ellipsometric spectroscopy is the experimental determination and theoretical interpretation of n(A) and k ( l ) functions over as wide a wavelength range as possible. However, we shall not deal with this range of problems, referring the interested readers to the recently published review work by Aspnes (1976). This work is specially devoted to the problems of ellipsometric spectroscopy.

1. Optical Characteristics of Germanium and Silicon When determining the optical constants of germanium and silicon, different authors have employed different approaches as regards both measuring techniques and methods of preparing the object under study. Thus, together with traditional ellipsometric techniques for estimating optical constants by the measured values of angles A and $ (Archer, 1958 ; Baklanov et al., 1975; Ibrahim and Bashara, 1972b; Hopper et al., 1975; Meulen and Hien, 1974; Meyer et al., 1971), the parameteric ellipsometric measurements have also been made. For these measurements the immersion medium refractive index (Egorova et al., 1974) or the angle of incidence of the operating light beam (Hopper et al., 1976) served as a parameter. In addition, the optical constants were determined by Brewster’s pseudoangle (Potter, 1966), as well as by the transmission and reflection coefficients of thin samples of the material under investigation (Dash and Newman, 1955 ; Philipp and Taft, 1959,1960). As to the state of the surfaces of these materials, there were used in the above measurements both surfaces with a natural oxide film (after one or another type of surface treatment) and surfaces covered with a specially grown protective film. Meyer et al. (1971) carried out studies on germanium and silicon surfaces heat cleaned in vacuo. The authors also made an attempt to determine n and k of such surfaces. However, because of essential experimental errors, the authors were forced to use in all the calculations and considerations at room temperature (no studies were made at higher temperatures) not their own values of n and k but the more reliable ones reported in the works by Philipp and Taft (1959, 1960) for silicon and germanium, respectively. The latter were obtained in the system “substrate-natural oxide film.” For that reason, the results of the work by Meyer et al. (1971) can be referred to with good reason as studies on the reflecting system of the type “substratefilm.”

70

A. V. RZHANOV A N D K. K . SVITASHEV

390 W

3 ~

e 185

z 07

0.9

ExtinctLon coefficient h

3.80

0

OD2

Qo4

Extinction coefficient k

(b) FIG. 31. Optical constants of (a) germanium and (b) silicon at room temperature for ,Iapcrat = 6328 A. 1, 2, 4, 5, 6,9, 10 represent the data of Archer (1958), Potter (1966), Ibrahim and Bashara (1972a), Dash and Newman (1955), Egorova et al. (1974), Philipp and Taft (1959), Meulen and Hien (1974), respectively, 7, 8 are the results of measurements on the system Si-Si02 at different angles of incidence (7) according to the data of Rzhanov er al. (1978) and at different thicknesses of SiO, film (8)-60, 80, and 120 A, after Algazin et al. (1978). for atomically clean surfaces. For germanium, n = 5.1, k = 1.1; and n = 4.3, k = 0.9, respectively according to the work of Meyer ef al. (1971); Philipp and Taft (1959). For silicon, n = 3.8 and k N 0 after Meyer er al. (1971). Solid lines are the region-of-confidence values obtained by the authors of the original works. Dotted lines are our evaluations of the region-of-confidence results.

Of note is the work of Ibrahim and Bashara (1972a), who have performed ellipsometric parametric measurements (at three angles of incidence) on silicon surfaces cleaned by ionic bombardment with subsequent annealing up to 800°C. However, as subsequent studies have shown (see, for instance, Hopper et al., 1976), the silicon surface was not atomically clean after such a treatment and the reflecting system presented a complex structure. This is pointed to in the works by Ibrahim and Bashara (1972a). In Figs. 31 and 32 one can see the results of n and k measurements (a) for germanium and (b) for silicon at room temperature (Fig. 31) and at higher temperatures (Fig. 32). The representation of these data in the nk plane permits us to compare them among the work of the different authors, excluding the errors caused by inaccurate temperature measurement. On the other hand, a comparison of Fig. 32b with Fig. 33b, where the temperature dependences of n and k are given for silicon, illustrates clearly the crude errors made by the individual authors when measuring temperature. The lack of reliable data on the optical characteristics of atomically clean germanium and silicon surfaces over a wide temperature range ex-

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

40 15 20 Extinction coefficient

k

Extinction cwfficient

71

k

FIG.32. Values of (a) germanium and (c) silicon optical constants over a wide temperature range. 1, 3, 4 represent the data of Baklanov er al. (1975); Hopper er al. (1975), Meulen and Hien (1974). 2, 5 represent the data of Algazin er al. (1978) for atomatically clean surfaces. (b) and (d) are the experimental values in the region-of-confidence values of n and k for germanium ( t = 370°C) and silicon (r = 640°C) according to the data of Algazin er al. (1978). A,,, = 6328 A.

0.3

1

c

aJ .>

. U -I

0.25 0

c

3 0*4j, C .A

% JsO

Temperature,("C) (a)

800

400

Temperature,

1200

0

W

("C)

(b)

FIG.33. The temperature dependences of (a) germanium and (b) silicon optical constants. 1, 2, 3, 4 represent the data of Meulen and Hien (1974); Hopper et al. (1975); Ibrahim and Bashara (1972b), and Algazin er al. (1978). A,, = 6328 A.

72

A.

V. RZHANOV AND

K. K. SVITASHEV

tremely hampers the carrying out of ellipsometric studies of the initial stages of the growth of dielectric and semiconductor films on these surfaces. These studies are of importance in both scientific and applied areas, as well as for the development of ellipsometric techniques for controlling a number of technological processes. In the present situation an accurate enough determination of atomically clean germanium and silicon surfaces in the direct ellipsometric experiment becomes the barest necessity. In this connection the authors of the present survey initiated work on the determination of n and k for germanium and silicon faces (111) over a wide temperature range (up to values close to the melting temperatures). This work was done on an automated ellipsometric setup. The wavelength of the operating light beam was equal to 6328 A (Algazin et al., 1977, 1978). In the latter work the atomically clean germanium surfaces were obtained by sublimation in vacuo (at 300-400°C) of germanium monosulfide films of 150-200 A thickness, grown by the technique described by Rzhanov et al. (1973). The atomically clean silicon surfaces were prepared by heatcleaning in vacuo at 1250-1300°C. The atomically clean surfaces prepared in this way were utilized to determine the optical constants over the entire temperature range under study. The stabilization of A and I(/ values under stationary temperature conditions chosen for this series of measurements served as a criterion for the final readiness of the surface for the present series of measurements. The usage of the automated ellipsometric setup with a high threshold sensitivity and measurement accuracy (Algazin et al., 1977) created the conditions necessary for carrying out the united and unbroken cycle of investigations : the process of film sublimation or surface cleaning; the control of the surface state and reliability in determining n and k for atomically clean germanium and silicon surfaces. In this case the threshold sensitivity in measuring A and $ values in a submonolayer region of coverage corresponded to 0.01 of a monolayer for the system “germanium-germanium monosulfide” and to several thousandths of a monolayer for the system “silicon-silicon dioxide.” Relative resolution in n and k was for germanium. With silicon it was l o p 5 in n and in k. The temperature measurement of the specimen under study was made by means of a thermocouple or pyrometer, as well as by the magnitudes of the specimen resistivity determined by a four-probe method. The use of a thermocouple as a temperature sensor permitted easy automation of the processes of heating and cooling the specimens, as well as that of exactly maintaining the temperature of the chosen specimen. The application of the four-probe method made it possible to obtain the required high accuracy in determining absolute values of the temperature, ranging from several hundreds degrees Celsius and higher (for greater detail, see Algazin et al., 1977).

ELLIPSOMETRIC STUDIES OF SURFACES A N D THIN FILMS

73

Together with the usual methods, the ellipsometric method of temperature control by the value of angle $ was used. (Ibrahim and Bashara, 1972b). It should be emphasized that this method requires the primary plotting of the calibration curve $ = $ (t”C). When employing the angle of light incident on the specimen (cpo = 72”05’), we found a linear dependence $ = kt”C for atomically clean silicon surfaces, where k = (23-24)’/10O0C. When determining the optical constants n and k, the values of the polarizing angles for every germanium and silicon specimen were measured under conditions of both a gradual temperature rise and its decrease over the whole working range (one set of measurements). Two to three sets of such measurements were made on the same sample. The results of these measurements proved that no irreversible changes of surface properties arise in temperature cycles. Similarly, the values of n and k were derived for 7 germanium and 12 silicon specimens. The values of the n and k pairs of the optical constants were estimated by the values of the corresponding pairs of $ and A, measured by the four-zone method. The calculation results are shown in Figs. 31 and 32 in n and k coordinates and in Fig. 33 as the functions n = n (t”C) and k = k (t”C). The temperature dependences (Figs. 32 and 33) present the mean values of all the measured optical constants. Now let us dwell at greater length on the analysis of the errors in determining germanium and silicon optical constants. Figure 32a and b present the data on n and k at room temperature for each germanium and silicon specimen under study, separately. These data characterize the dispersion in the values of the optical constants obtained for the different specimens. Of interest is the comparison of these data with the results of error calculations in determining the complex refractive index N . These errors are caused by the errors in measuring the polarizing angles A(6NA) and $(6N,), the angle of light incident on the specimen cpo(6N,,), and the specimen temperature t”(bN,o). The errors 6n, 6k, and 6 N , caused by the foregoing, were calculated in the usual way (Shchigolev, 1969) at the following values of the errors of the experimentally measured magnitudes : 6A = 6$ = kl’,

6cpo = +0.5’,

6t = +4“C

(124)

The results of calculation are shown in Fig. 31a, b as the vectors 6NA, 6N,, 6N,,,,,, and 6N,,.For the sake of convenience the above vectors are enlarged (see the scale in the figure). To illustrate the accuracy in determining n and k on one specimen, Fig. 33b,d represents the results of measurement of these values at temperatures of about 370°C for germanium (b) and 640°C for silicon (d). The dots in the figures were obtained as a result of carrying out the following thermal cycles : “temperature of the experiment-room temperature-temperature of the experiment” or “temperature of the ex-

74

A. V. RZHANOV AND K. K. SVITASHEV

periment-temperature close to the melting temperature of the materialtemperature of the experiment.” Furthermore, in these figures one can see the mean values of n and k at the temperatures of the experiment and the calculated values of 6N,, 6N,, 6Nv0, and 6Nto,for the considered case at the values 6 4 a$, 6qo, and 6t0 determined by the relations (124). The figures also present the values Bncompleteand 6k,o,,,pl,t, calculated following the formulas 6nfomplete= JSni + 84 Sn& Sn:o (125a)

6kcomplete = J6ki

+ 6k;

+ + + 6ki, + 6k;o

(125b)

A comparison of the scatter of the experimental points (Fig. 32b,d) with the values of intervals nmean 6ncompleteand k,,,, i-6kco,pl,C, shows that the errors in measuring temperature and the values A, and ‘pocorrespond to reality. Besides, as one can see from the figures, the error in measuring temperature is the main reason for the dispersion of the experimental values of n and k. On the other hand, an analysis of the data presented in Fig. 31 shows that the dispersion of the measured values of n and k when passing from one sample to another exceeds appreciably the intervals hean k Bncomplete, k,,,,,, f 6kco,p,,le calculated from (125) and (124). We have no grounds for assuming that the errors in measuring temperatures and polarizing angles, taken into account when calculating 6N,, 6N,, and SN,, are valid, as mentioned above, at raised temperatures, can increase essentially at room temperatures. Obviously, one should look for the reason for the dispersion of germanium and silicon optical constants when passing from one sample to another in the insufficient reproducibility of the specimen adjustment by the angle of incidence of the operating light beam on the surface, as well as in the lack of reproducibility of the object under study itself. The main reasons for the nonreproducibility of the object under study are evidently the errors in the orientation of the surface and the emergence of the uncontrollable adsorbed layers on it. Of independent interest is the analysis of the general view of Fig. 31a, b, where on the one hand, the values of the optical constants of the atomically clean germanium and silicon surfaces obtained by Algazin et al. (1978) are given and on the other, the results of the determination of these constants by measurements on the system “substrate-film” are presented. The striking difference between the values of the optical constants of the atomically clean germanium and silicon surfaces and those of the same but filmed materials proves the highly different state of germanium and silicon surfaces in both types of the considered systems. As already mentioned in the previous section, the first studies of the

+,

ELLIPSOMETRIC STUDIES OF SURFACES AND THIN FILMS

75

adsorption of different gases on atomically clean germanium or silicon surfaces (Meyer et al., 1971) and the adsorption of oxygen on silicon (Ibach et al., 1973) uncovered experimentally the anomalous (from the point of view of the ideas existing at that time) behavior of the angle of polarization-its decrease in the region of a submonolayer coverage. To explain the newly discovered “anomalous” effect, Meyer et al. (1971) supposed that there exists some surface layer with disturbed optical constant values, which leads to the variations of k values on atomically clean surfaces, as compared with the bulk values for the same material. This notion is apparently correct in principle. However, in order to determine unambiguously the parameters of the “disturbed” layer, it is necessary to make high-accuracy measurements by means of ellipsometry, as ‘well as the LEED method, Auger-spectroscopy, superhigh-voltage electron microscopy, etc. As far as possible, the measurements should be made on the same or identical specimens, in contrast to Meyer (1976). This work discusses and considers the separate studies carried out by means of the different techniques, some of which are characterized by appreciable experimental errors (for example, Meyer et al., 1971). In other words, up to now there has been no possibility of determining unambiguously the combinations of the effective parameters of the “disturbed” layer. For this reason the selected values of these effective parameters at the fixed thickness of the inhomogeneous layer d = 5 A (Meyer et al., 1971) can only be considered as one of the possibilities of such selection.

+

2. Bibliography on Optical Characteristics of Various Materials The determination of the complex refractive index of the material under study is the first step in any ellipsometric study dealing with the particular material. In this connection we present, in Table VIII, a bibliography of works devoted to the determination of the magnitudes n and k for various materials, which are objects of great attention in semiconductor and solid state physics. V. CONCLUSION

Our survey is over. Naturally, we did not manage to throw light on all the aspects of ellipsometry. Such methodological problems, for example, as the analysis of the threshold sensitivity of ellipsometry (Svitashev et al., 1977b), the ellipsometry of rough surfaces (Svitashev et al., 1977a), the problems dealing with ellipsometer adjustment are beyond the scope of our survey, which primarily concerns the applications of ellipsometry.

TABLE VIII ON OPTICAL CONSTANTS BIBLIOGRAPHY

Material Na A1

Si

TiO, Ti-AI V VO, Cr

Spectral range 0.5-0.4 eV 0.45-0.65 pm

References

2.0 pm 2-12 pm 6328 A 6328 A 6328 A 5490 A 5461 A 250-650 A 0.54-0.67 pm 0.34- 1.8 pm 2.5-12 pm 0.5-5.5 pm 1 .l-2.0 pm 1.35-1 1.O pm

N. V. Smith (1969) Shklyarevsky and Milaslavsk y (1957) Beattie and Conn (1955) Beattie (1955) Halford et al. ( 1 973) Nyce and Skolnick (1975) Allen (1976) Fane and Neal ( 1970) Mertens et al. (1963) Hass and Waylonis (1961) Lisitza and Tzwelych (1959) Meyer et al. (1971) Icenogle et al. (1 976) Thutupally and Tomlin (1 977) Primak (1971) Salzberg and Villa (1957)

5461 A, 5893 A 5000 A 0.64-6.60 eV 0.25-5.0 eV 6328 A 0.64-6.60 eV 5461 A 5500 8,

Kucirek ( 1973) Johnson and Tao (1969) Johnson and Christy (1974) Verleur et al. (1 968) Beresneva et al. (1 977) Johnson and Christy (1974) Sirohi (1969) Genshaw and Sirohi (1971)

Material Si

SiO,

SiO,N, K KCI Ti

cu

Spectral range 6328 A 5461 A 5461 A 5461 A 5461 A 3655 A I-lOeV 1400-800cm-' 7-11 pm 5461 A 1000-1650 A

References

1-8 eV 0.5-4.0 eV 10.6 pm 5000 A 0.64-6.60 eV 5461 A. 6328 A 1-12pm

Algazin et a/. (1978) Burge and Bennett (1964) So and Vedam ( 1972b) Vedam ez a/. ( 1969) Archer (1957) So and Vedam ( 1972a) Philipp and Taft ( 1960) Zolotarev ( 1970) Rakov et al. (1968) Claussen and Flower (1 963) Lamy ( 1977) Agajanian ( 1977) Philipp (1973)' N. V. Smith (1969) Pedinoff et a/. (1977) Johnson and Tao ( 1969) Johnson and Christy (1974) T. Smith (1972a) Shklyarevsky and Padalka (1959)

1-12 pm

Padalka and Shklyarevsky (1962)

0.45-0.65 pm 7.0 pm 0.36-2.50 prn 0.52-3.31 eV

Shklyarevsky et al. (1966) Beattie and Conn (1955) Roberts ( 1960) Stoll (1969)

Fe Ni

Inconel Ge

GeO, GeSe Ag

Ag-Cd, Mn, Sn Hg Bi

3 pm, 9.5 pm 0.36-2.65 pm 5461 A, 6328 A 0.405-0.612 pm 2.06-16.0 pm 300-500 A 5461 A 6328 a 6328 A 3600-7000 A 0.5-3.0 eV 1.8-2.7 eV 1400-800 cm-' 1080-600 cm0.5- 1.5 eV 1-12pm 1-6 pm 4 Pm 0.45-0.65 pm 1.15, 0.34, 10pm 5461 a 1.40-2.52 eV

Johnson and Christy (1974) Yolken and Kruger (1 965) Johnson and Christy (1974) Studna (1975) Shklyarevsky and Padalka (1959) Beattie and Conn (1955) Roberts (1959) T. Smith (1977) Goodell er 01. (1973) Salzberg and Villa (1957) Feuerbacher et al. ( 1968) Archer (1 957) Baklanov et al. ( 1975) Algazin er al. (1978) Archer (1958) Phillipp and Taft (1959) Schmidt ( 1969) Zolotarev (1970) Zolotarev and Morozov (1973) Lukes (1 968) Shklyarevsky et al. (1958) Motulevich and Shubin (1957) Beattie k i d Conn (1955) Shklyarevsky et al. (1966) A d a m et al. ( 1975) Yamaguchi et al. (1972) Flaten and Stern (1975)

5461 A 5461 8, 0.4-1 .O pm

T. Smith (1 967) L. E.Smith and Stromberg (1966) Usoskin er al. (1973)

0.646.60 eV 3670-6970 A 0.64-6.60 eV 1.8-3.5 eV 1-12 pm

Ga GaAs Ge

InSb

Sb Ta. TaO

W W-Re Au

Hg Bi,Te,S5

5461 I% 5461 A, 6328 a 0.45- 1.54 pm 5461 A 5461 A 0.63-0.72 pm 2.0-2.5 pm 0.38-0.98 pm 2.5-12 pm 0.62-2.0 pm 5461 5461 A

5461 A 1-12pm 6.0, 12.0 pm 4960, 5487, 6049 0.36-2.65 pm 3900-6500 A 0.3-16 pm 1-12 pm

Mertens and Plumb (1 964) T. Smith (1977) Leluk et al. (1 964) Dell'Oca et al. (1971) A d a m and Pruniaw (1973) Lisitza and Tzwelych (1958) Avery and Clegg (1953) Meyer er al. (1971) Icenogle et al. (1976) Denton and Tomlin (1972) Saxena (1965a) Harkness and Young (1966) Kurdiani (1 964) Shklyarevsky ef al. (1959) Beattie and Conn (1955) Aguado Bombin and Neal (1977)

0.45-0.65 pm 1, 15, 3.4, 10 pm 5461 a 6328 a 0.45-1.54 pm 6.0 pm

Roberts (1959) Carol1 and Melmed (1971) Kirillova ez al. (1975) Shklyarevsky and Padalka (1959) Shklyarevsky et al. (1966) A d a m et al. (1 975) Mertens and Plumb (1 964) Sirohi and Genshaw (1 969) Leluk et al. (1964) Beattie and Conn (1955)

4358, 5461 ! I 2000- 1OOOO A 13837 A

Jones et al. (1974) Malm et al. (1969) Abeies et al. (1973)

78

A. V. RZHANOV AND K. K. SVITASHEV

And no wonder, since the spectrum of ellipsometric applications has been extended noticeably in recent years, and the total number of volumes devoted to the different problems of ellipsometry exceeds lo00 today (Institute of Semiconductor Physics, 1977). When selecting the material for our survey, we also proceeded from the fact that fairly serious analyses of a great number of ellipsometric problems were presented in the monographs by Azzam and Bashara (1977), Gorshkov (1970), Uryvsky (1971), as well as in the surveys by Rezvyi and Phinarev (1977), and Aspnes (1976). We tried to have our survey supplement rather than repeat these works. Finally, before making the final point, we should like to place special emphasis on the following. Ellipsometry in an explicit form answers only the question dealing with the optical properties of surface coverage upon adsorption. But using these characteristics, it becomes possible to answer indirectly questions concerning both the composition and structure of massive specimens and thin films. For this it is necessary to draw a comparison between the data obtained not only by ellipsometry, but by the other techniques as well. Ellipsometry possesses a number of invaluable advantages over the other research methods. These advantages concern not only its sensitivity, but the range of materials covered and the simplicity of specimen preparation. But the chief thing is that ellipsometric research methods are nondestructive. This provides an investigator with the opportunity to make a set of nonstop measurements, to observe continuously the physical and chemical processes on the surface, and to repeat many times the experiments on the same specimen. But we feel that the unique capabilities of ellipsometry are not entirely used to advantage nowadays. The applications of ellipsometry are far from being exhausted. Many new experiments can be set up.

ACKNOWLEDGMENTS The authors wish to express their sincere gratitude to Master of Science Mrs. R. L. Shchekochihina, Mrs. L. P. Kibireva, Mrs. E. D. Voloshkevich, and Mrs. S. N. Svitasheva for translating the present survey and greatly assisting in its planning. The authors only are responsible for the content.

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Semenenko, A. I., Semenenko, L. V., Mardezhov, A. S., and Svitashev, K. K. (1978). Ukr. Fiz. Zh. 23, 504. Semenenko, L. V., Svitashev, K. K., Semenenko, A. I., and Sokolov, V. K. (1972). Opt. Spektrosk. 32, 1204. Shchigolev, B. M. (1969). “Mathematical Treatment of Observations.” Nauka, Moscow. Shklyarevsky, I. N., and Milaslavsky, V. K. (1957). Opt. Spektrosk. 3, 361. Shklyarevsky, I. N., and Padalka, V. G. (1959). Opt. Spektrosk. 6,776. Shklyarevsky, I. N., Starunov, N. G., and Padalka, V. G. (1968). Opt. Spektrosk. 4,792. Shklyarevsky, I. N., Avdeenko, A. A,, and Padalka, V. G . (1959). Opt. Spektrosk. 6,528. Shklyarevsky, I. N., Yarovaya, R. G., Kostuk, V. P., and Leluk, L. G. (1966). Opt. Spektrosk. 20, 1074. Shlyarevsky, I. N., El-Shazly, A. F. A., and Idezak, E. (1971). Solid State Commun. 9, 1737. Shurkliff, W.A. (1962). “Polarized Light.” Harvard Univ. Press, Cambridge, Massachusetts. Sirohi, R. S. (1969). Er. J. Appl. Phys. 2,468. Sirohi, R. S., and Genshaw, M. A. (1969). J. Electrochem. SOC.116,910. Slater, I. C. (1967). “Insulators, Semiconductors and Metals.” McGraw-Hill, New York. Smith, L. E., and Stromberg, R. R. (1966). J. Opt. SOC.Am. 56, 1539. Smith, N. V. (1969). Phys. Reu. 183,634. Smith, R. C., and Hacsaylo, M. (1963). Natl. Bur. Stand. ( U . S . ) ,Spec. Publ. 256,83 and 93. Smith, T. (1967). J. Opt. SOC.Am. 57, 1207. Smith, T. (1972a). J. Opt. SOC.Am. 62,774. Smith, T. (1972b). J. Electrochem. SOC.119, 1398. Smith, T. (1976). Surf. Sci. 56,212. Smith, T. (1977). J. Opt. SOC.Am. 67,48. So, S . S. (1 976). Surf Sci. 56,97. So, S. S., and Vedam, K. (1972a). J. Opt. SOC.Am. 62, 16. So, S. S., and Vedam, K. (1972b). J . Opt. SOC.Am. 62, 596. Sokolov, A. V. (1961). “Optical Properties of Metals.” Fizmatgiz, Moscow. Stobie, R. W., Rao, B.. and Digman, M. J. (1975). Appl. Opt. 14,999. Stoll, M. P. (1969). J. Appl. Phys. 40,4533. Strutt, J. W. (1871). Philos. Mag. [4] 42, 81. Strutt, J. W. (1892). Philos. Mag. [ 5 ] 33, 1. Strutt, J. W. (1907). Proc. R. SOC.London, Ser. A 79, 399. Strutt, J. W. (1912). Proc. R. SOC.London, Ser. A 86,207. Studna, A. A. (1975). Solid State Commun. 16, 1063. Svitashev, K. K., Semenenko, A. I., Semenenko, L. V., and Sokolov, V. K. (1971). Opt. Spektrosk. 30,532. Svitashev, K. K., Semenenko, A. I., Semenenko, L. V., Sokolov, V. K., and Suchorukov, 0. G. (1972a). Mikroelekironika (Akad. Nauk SSSR) 1, 153. Svitashev, K. K., Semenenko, A. I., Semenenko, L. V., and Sokolov, V. K. (1972b). Opt. Spektrosk. 33,742. Svitashev, K. K., Semenenko, A. I., Semenenko, L. V., and Sokolov, V. K. (1973a). Mikroelektronika (Akad. Nauk SSSR) 2,454. Svitashev, K . K . , Semenenko, A. I., Semenenko, L. V., and Sokolov, V. K. (1973b). Opt. Spektrosk. 34,941. Svitashev, K. K., Semenenko, A. I., Semenenko, L. V., and Shwartz, N. L. (1977a). O p f . Spektrosk. 42,969. Svitashev, K. K., Semenenko, A. I., Semenenko, L. V., Sokolov, V. K., and Filatova, E. S. (1977b). Opt. Spektrosk. 42,1142.

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Svitashev, K. K., Semenenko, L. V., Sharshunov, A. G., and Feoktistov, A. M. (1977~).Mikroelektronika (Akad. Nauk SSSR) 6,258. Takasaki, H. (1966a). J . Opt. SOC.Am. 56,557. Takasaki, H. (1966b). Appl. Opt. 5,769. Thutupally, G. K. M., and Tomlin, S.G. (1977). J . Phys. C. [I] 10,467. Tomar, M. S. (1975). Optik (Stuttgart)42, 297. Treu, J. I. (1974). Rev. Sci. lnsfrum. 45, 1462. True, J. I., Callender, A. B., and Schnatterly, S. E. (1973). Rev. Sci. Instrum. 44, 793. Uryvsky, Y. 0. (1971). “Ellipsometry.” University Press, Voronezh. Usoskin, A. I., Shklyarevsky, I. N., Gerchikov, A. S., and Verminsky, Y. S. (1973). Opr. Spektrosk. 34,954. Van de H u h , M. C. (1957). “Light Scattering by Small Particles.” Wiley, New York. Vasiljeva, L. L., and Repinsky. S. M. (1974). Izo. Akad. Nuuk SSSR, Neorg. Mater. 10, 1389. Vasiljeva, L. L., Kushkova, A. S., and Repinsky, S. M. (1971). Zh. Fiz.Khim. 45,1132. Vasiljeva, L. L., Svitashev, K. K., Semenenko, A. I., Semenenko, L. V., and Sokolov, V. K. (1974). Opt. Spektrosk. 37,574. Vedam, K., Knausenberger, W. H., and Lukes, F. (1969). J . Opt. SOC.Am. 59,64. Verleur, H. W., Barker, A. S., and Berglund, C. N. (1968). Phys. Rev. [2] 172,788. Wilmans, I. (1969). Surf. Sci. 16, 147. Winterbottom, A. B. (1963). Natl. Bur. Stand. (US.), Spec. Publ. 256,97. Wyatt, P. W. (1975). J . Electrochem. Soc. 122, 1660. Yamaguchi, T., Yoshida, S., and Kinbara, A. (1972). J . Opt. SOC.Am. 62,634. Yamamoto, M. (1975). Jpn. J . Appl. Phys., Suppl. 14-1,413. Yolken, H. T., and Kruger, J. (1965). J . Opt. SOC.Am. 55,842. Zolotarev, V. M. (1970). Opt. Spektrosk. 29,66. Zolotarev, V. M.. and Morozov, V. N. (1973). Opt. Spektrosk. 34, 319.

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL . 49

Foundations and Preliminary Results on Medical Diagnosis by Nuclear Magnetism GEORGES J . BENE D&partmenrde Physique de la Mati6r.e Condensbe Section de Physique Universitd de GenPve Geneva. Switzerland

Part I . A New Method in Medical Diagnosis ..... I . Introduction ......................... .............................. A . Medical Diagnosis: Physical Methods .............................. B. Appraisal of New Methods .......................... ...... C . A New Challenger: Nuclear Magnetism ............... ...... I1. Information That Can Be Obtained by Nuclear Magnetism .................... A . Identification of Isotopes and the Molecules Containing Them ............. B. Abundance or Density of the Corresponding Atoms ....................... C . Structure of the Environment .......................................... D . Dynamics of the Environment .......................... I11. The Relaxation of a Magnetic Dipole in the Water of Biological A . Pure Water .......................... B. Aqueous Solutions ................... ......................... C . Inhomogeneous Substances ............................................ Part 2: Application of the Techniques of NMR to the Study of Biological Tissues ................................................. IV . The Techniques ......................................................... A . Introduction ......................................................... B Techniques Based on Nuclear Magnetic Resonance . . . . .......... C . The Free Precession in the Earth's Field ................................. V . Application to the Study of Biological Tissues in Vitro (Biopsis) . . . . . . . . . . . . . . . . A . Physiological Fluids .................................................. B. Soft Tissues .............................. .............. VI . Application to Measurements in Situ ....................................... A . Projection Reconstruction Method ...................................... B. Single-Point Spin Mapping Techniques ................. C . Mansfield's Technique ................................................ D. Other Techniques .................................................... E . Applications of the Zeugmatography to Medical Diagnosis ................. F. In Situ Diagnosis in a Weak Field ...................................... VII . Final Remarks .......................................................... References ....... ........................ ................... Note Added in Proof .....................................................

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86 86 86 86 87 87 87 88 89 91 95 95 100 106 109 109 109 109 110 112 112 114 119 119 121 123 125 126 127 129 130 132

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Copyright 0 1979 by Academic Press Inc. All rights of reproduction in any form reserved. ISBN 012-014649-5

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PART 1: A NEW METHOD IN MEDICAL DIAGNOSIS I. INTRODUCTION A . Medical Diagnosis: Physical Methods

The identification of diseases, which is the object of medical diagnosis, is often a difficult task that is in consistent search of new methods. Diagnosis is traditionally based on the results of external observation and an interview with the patient. These means are quite often insufficient : External observation is necessarily relatively superficial and the examination has a subjective character which might modify reality. This has forced the development and diversification of more objective and better methods of diagnosis : amplification and quantitative measures for immediate observation (stethoscope, thermometer, etc.) ; biochemical analysis of the products eliminated by the body or that are easy to extract, in order to obtain information about the pathological tissue; recording of internal data (electric tensions) that might be related to the functioning of the organs (electrocardiograms, etc.). B. Appraisal of New Methods The complexity of biological organisms extends from the molecular to the macroscopic level. It appears to be a spatial as well as a chemical and physical heterogeneity. At present, there exist no methods or techniques that are able to detect and follow simultaneously all the processes that take place in a living organism. Several methods, however, enable a successful exploration of more limited parts. The most elaborate methods usually demand considerable investment in space, apparatus, or staff (Pullan, 1975). A new technique must thus be really justified to be introduced into an already crowded market. Part of such a justification of a new method is the requirement that it satisfy some of the following conditions : (1) It enables a more complete or more precise exploration of the human body. (2) It can identify and not only localize a pathological disease. (3) It is cheaper, faster, easier to operate. (4) It eliminates difficult interventions, uncertain results, and time wasted in laboratory analyses. (5) It eliminates long and painful examinations for the patient.

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C . A New Challenger: Nuclear Magnetism

The method of nuclear magnetism (Abragam, 1961) is based on the study of the magnetization of the nuclei of certain atoms when they are subjected to the action of an external magnetic field. This magnetization is quite weak, only about one thousandth of that produced by electronic paramagnetism; however, radioelectric techniques and particularly magnetic resonance, make it possible to observe it in isolated atoms as well as in condensed matter (solids, liquids, and gases). Note that not all the nuclei possess a permanent magnetization but only those whose mass number M and charge Z are not both paired. Their magnetic moments are always associated with an angular momentum or spin I. Those of spin >$ have, moreover, an electric quadrupole moment that can be related to the nonspherical symmetry of the electric charge density. The nuclei carrying a magnetic moment behave as “informants” situated inside a material; their behavior is quite sensitive to the nature, the structure, and the dynamics of their environment. They thus enable one to observe small changes in a sample under study. The fact that nuclear magnetism is the basis of the study presents some advantages: There is neither an absorption nor a diffusion that is not entirely due to the studied phenomenon. A consequence of this fact is that the energy involved in nuclear magnetism is quite weak. This has certain drawbacks. First, the sensitivity is quite small, which considerably reduces the range of possible applications. Second, it makes it more difficult to make a spatial distinction between the atomic nuclei of the same kind in the sample. These two aspects will be studied in the following.

11. INFORMATION THATCAN BE OBTAINED BY NUCLEAR MAGNETISM A . Ident8cation of Isotopes and the Molecules Containing Them

Any atom, the nucleus of which carries a magnetic moment,-and there are many such-can be identified by the techniques of magnetic resonance, which easily distinguish between the information provided by two magnetic nuclei of different natures. The nuclei most easily observed are obviously those of high magnetic moment. Hi belongs to this class and to a smaller degree also Fi9, N:4, and P:: . For others like D:, Ci3, N:’, and O;’, it is necessary to operate with a higher sensitivity or to make isotopic enrichment. The pair-pair nuclei like He:, Ci2, and 0t6are not detectable by this method; however, we are mainly interested in biological tissues where Hi plays an essential role, especially the Hi of water molecules. Nuclear magnetization enables the

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GEORGES J.

BBNB

identification of nuclei by the fact that the Larmor pulsation w, = 27117, of a magnetic nucleus in a constant magnetic field Bo is a characteristic property via the relation 0 0

(1)

= YBO

where y, the gyromagnetic ratio, is an intrinsic quantity of the nucleus that is directly related to its structure. For the set of existing nuclei, the values of y extend over quite a wide range; for example, the gyromagnetic ratio of H: is about 130 times higher than Ce:;’. On the other hand, the values of the gyromagnetic ratios closest to that of the protons are given by H: ( 7%) and Fi9 (-6%).

+

B. Abundance or Density of the Corresponding Atoms The nuclear equilibrium magnetization M , of a sample in an external field B, (I1 telsa) and at temperature T (- 300°K) is proportional to the number of nuclei contained in the sample and to B,, in accordance with Curie’s law. This is because the interaction energy of the nuclear magnetism is small under such conditions. Moreover, the fact that one can select the information provided by a given kind of nucleus makes it possible to evaluate the contribution to M , for each kind. Accordingly, one can, for example, separate the Mo contribution from Hi from that of the other nuclei, and even from that of electronic origin, which is always bigger (diamagnetism and paramagnetism). In the case of natural water, however, the magnetization is almost entirely due to the protons of the water molecules HZO,since the nucleus Oi6 is without magnetization, and the concentration of other nuclei (D: , 0;’) is too small to have a detectable influence on the total M,. Consequently for natural water, the amplitude Mo (Hi) is proportional to the number of water molecules. The method used to isolate and amplify selectively the given frequency and phase signals and thus to observe and measure the nuclear magnetization is based on radioelectric techniques. The sensitivity of the apparatus is such that it is possible to observe the nuclear magnetization of the protons in 1 cm3 of water at ordinary temperatesla). It is this sensitivity that tures in the earth’s magnetic field ( 5 . makes it possible to envisage the construction of a nuclear magnetograph, which should work in the “opposite way” to that of the x-ray apparatus: The latter works principally on the absorption of x-rays in the heavy atoms (Caand P of the bones, for example), while a nuclear magnetograph would selectively excite the nuclear magnetism at the Larmor frequency on which the receptor has been fixed (mainly H atoms). In this case, the soft tissues show a much higher nuclear magnetization than the bones, as is easily under-

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89

FIG.1 . Water content of various human tissues. From Mansfield and Pyrett (1978).

standable from Fig. 1, which gives the proportion of water present in some human tissues (Mansfield and Pykett, 1978). C . Structure of the Environment

I. Ordered Systems It is well known that nuclear magnetism, like x-rays, can be used to study the structure of ordered systems. The information obtained from the study of nuclear magnetism is complementary, however, to that obtained from the absorption spectra of x-rays, since nuclear magnetism is particularly important in the light atoms like hydrogen. The study of nuclear magnetism of the light atoms in a given lattice makes it thus possible to fix the positions and relative orientations of these atoms in the lattice. Mansfield and Grannel (1973) have shown that diffraction techniques can be applied to study periodic structures (1 -2 mm of recurrence) by means of the magnetization of the protons present in the structure. We shall not discuss this technique because it has not yet been applied to the study of biological tissues. We mention also the case of high resolution, which has permitted elucidation of the structure of the complex organic molecules from a study of the protons or Ck3 nuclei in mutual interaction or interacting with other magnetic nuclei of the molecule in a solution that is sufficiently diluted to eliminate intermolecular interactions. 2. The Local Field

The size and spectral distribution of the nuclear magnetization of the proton is evidently sensitive to the intensity and the orientation of the

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GEORGES J .

BBNB

applied field B, (constant or variable). Moreover, they are sensitive to the magnetic field produced by other elementary systems of the sample, as for example, other magnetic nuclei, paramagnetic ions, or free radicals. This local field is at the origin of (i) the decomposition of the Larmor frequency in a discrete or continuous ensemble of frequencies distributed in a finite spectrum and which is observable in solids as well as in fluids; (ii) the size of the time constants that characterize the return to equilibrium of the magnetization when one has changed the external conditions, as, for example, the direction or the amplitude of the applied field. The local field acts by means of the coupling it produces between the magnetized elementary systems. Usually this field is magnetic, but if there are nuclei carrying an electric quadrupole moment, they will be sensitive to the gradient of a local electric field. In our case, the spectral decomposition is of little interest; however, the time constants are highly important because they are related to the correlation times z, which characterize the dynamics of the environment.

3 . Relaxation Times The most important time constants characterizing the nuclear magnetism are (a) Spin-lattice relaxation time TI , which characterizes the exchange of energy between the spin system and the degrees of liberty of the lattice (thermal energy, defined by the absolute temperature of the sample). In fact, if one abruptly changes the strength but not the direction of the external field applied to a homogeneous sample of uniform temperature, then the return to equilibrium follows an exponential law, the time constant being the relaxation time q . (b) Spin-spin relaxation time T,, which characterizes the exchange of magnetic energy inside the spin system, i.e., without a global change of the equilibrium magnetization. It characterizes the mean life of the spins in a given energy state, as well as the exchange of magnetic energy between the spins. It can be determined from the curve describing the loss of coherence when one abruptly changes the direction of the applied field B,, without changing its size. In fact, this coherence is equivalent to the existence of a finite magnetization in a direction different from that of B, . The transient phenomenon that takes place when one changes the direction of B, is a precession at the Larmor frequency, the envelope being an exponential with

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

91

time constant T2. In the case of nuclei possessing an electric quadrupole moment, the gradient of the local electric field (which can be quite strong in crystals or polar liquids) influences and by diminishing their duration. Similarly, the local magnetic field of paramagnetic ions or free radicals, carrying magnetic moments about 1000 times as big as those of the nuclei, acts in a similar way on the nuclear relaxation times. Generally, one can say that the nuclear relaxation times are quite sensitive to the concentration and the dynamics of the magnetic nuclei considered; to the nature of their environment, in particular, to its microscopic geometry ; in the case of the protons, to the pH of the environment, which acts on ionic exchanges, and evidently also to the temperature and the influence of the walls limiting the homogeneous regions of the sample. D . Dynamics of the Environment

Nuclear magnetism makes it possible to study the dynamics of materials on the molecular level, without appreciable perturbations. This is contrary to most of the techniques by means of which similar information can be obtained (x-rays, visible light, or ultrasound). The origin of this dynamic is thermal. It is isotropic in liquids and more or less anisotropic in structured tissues. Notice that in the case of biological tissues, the movements in solids and in relatively rigid systems (bones, macromolecules) are almost unobservable. In fact only protons of the more or less free water molecules in soft tissues or physiological liquids may be easily studied. Biological tissues are highly complex on the molecular level, as well as on the macroscopic level, and it is worthwhile to consider the description of these two levels separately.

I . Description of the Microscopic Structure There exist more or less elaborate models that allow identification of the molecular motions causing observable quantities and thus enable us to describe them in a precise way. The molecules in liquid water are subject to the Brownian motion in several ways : (a) By rotational motion, the velocity of which depends on the temperature and the internal friction of the molecules. Such a motion is aleatory, and the rotation includes all speeds between 0 and a maximum value v,, being determined by the absolute temperature of the lattice (James, 1975).

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GEORGES J. BkNh

FIG.2. Spectral density of the correlation function as a function of

0,.

From James (1975).

Such a maximum speed defines a rotation correlation time z, = z, the time needed by a molecule to make an eigenrotation of -33". It is often useful to calculate the Fourier transform of the velocity spectrum and determine the amplitude J(w) of the component corresponding to a given frequency w. In Brownian motion such a spectrum has the shape of a white spectrum between the values 0 and o,(0, = l/zc) (Fig. 2). This correlation time is related to the rotation diffusion coefficient D, by the relation z, = (6Dr)-'. In pure water at ordinary temperatures, this correlation time is about lo-'' sec. In a simple model where the molecule is considered as a rigid sphere in an environment of viscosity q, the DebyeStokes theory gives z, = 4na3q/3kT with K = Boltzmann's constant

(2) (b) By translational motion, which is characterized by a correlation time z,, the time needed for a molecule to make a displacement corresponding to its diameter 2a. This correlation time is related to the translation diffusion constant D,, by the relation z, = a2(3Dtr)-'

For water at ordinary temperatures D,, = 1.85 1O T 5 sec/cm2

According to the theory of Brownian motion, z, = 2nqa3/kT

(3)

The correlation times vary with the temperature T according to Arrhenius's law zc

=

C' ex~(-EactIRT)

where Ea,, denotes the activation energy.

(4)

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

93

We shall not insist on the fact that the flow velocity of the fluids can be determined by its effect on the nuclear relaxation, since this effect has not yet been sufficiently studied for biological tissues. On the other hand, we mention that the water molecules in aqueous solutions (ions, proteins, etc.) can be bound to the molecules of the dissolved substances for certain periods of time. When the molecular weight of the dissolved molecules is much higher than that of water, the correlation times for the bound water molecules are much longer than for the free molecules. Relaxation measurements enable one to evaluate

(1) the relaxation times of bound water molecules ; (2) the time duration of water molecules in this new site; (3) the proportion of bound water molecules. When the circulation of liquids is constrained by walls of small dimension, as in capillary blood vessels and for intracellular water, the rotational and translational motions of the molecules are considerably modified. This may imply a modification of the correlation times that can be observed via nuclear magnetism. The water in the cells might be in an environment that is more or less structured. Not only may the constants z, be Werent, but the translational diffusion constant may depend on the form of the cell. The corresponding anisotropy in a muscular cell has been observed. In complex tissues there might be a nonhomogeneous environment such that the dynamic constants of water are different in each compartment that is sufficiently well isolated from the neighboring compartments. Moreover, the variation of these constants might be different for different tissues. In any case, observation of the global tissue gives the superposition of the effects from each compartment. It is worthwhile noting that nuclear magnetism may allow information about each compartment to be extracted. This is the case when the intensity of nuclear magnetism is sufficiently strong. It quite often happens that it is impossible to give such a microscopic description of a given system. In such cases one has to take a phenomenological approach, which is accomplished by starting with the same fundamental interactions. 2 . Phenomenological Description

It is possible to evaluate the characteristics of dynamic constants (correlation times, etc.) of a given tissue even if one does not possess a microscopic model that describes the system with precision and by means of which one can interpret the observed quantities. One might, for example, observe several correlation times in a given solution, the origin of which is impossible

94

GEORGE J.

BBNB

to identify by means of microscopic models. The same thing might happen for a number of soft biological tissues, of which it is impossible to make even a primitive model. Thus one is only left with the possibility of making phenomenological models, an approach that has turned out to be useful when one has to deal with systems whose internal dynamics can be described by a continuous spectrum of correlation times. One is led to describe the distribution of correlation times by (a) a mean correlation time; (b) a form factor that describes the distribution in the neighborhood of the mean correlation time. In spite of its artificial character, this approach has turned out to be quite fruitful.

3. Pathological Interest The importance of dynamic quantities (z,, etc.) is considerable even when one does not possess a precise interpretation. Their importance, moreover, stems from the fact that they change when the environment is altered. In a biological tissue, such a change may be the effect of a pathological modification, which might imply a change of temperature, chemical composition, geometrical structure, or microscopic dynamics. In the case of physiological fluids, which are usually aqueous solutions of proteins, the modification of the physiochemical conditions may lead to a variation of the number of water molecules that are bound to a protein, as well as in the strength of the bound. This again might imply an appreciable modification of the rotational and translational diffusion constants. Two important questions arise : (a) Is it possible or easy to measure the dynamic constants in a biological tissue? (b) Do they characterize a given biological or pathological state. I.e., do they differ between two different healthy tissues, or do they differ for a healthy and a pathological state of the same tissue?

4. Evaluation of the Dynamic Constants The samples that have been studied Gontain a large amount of water

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

95

molecules, either in a liquid state (diluted solutions of mineral ions and proteins) or more or less constrained, as in the soft tissues. The relation between the dynamic and structural constants of a liquid and the relaxation times determined by nuclear magnetism depend on the kind of interaction involved between the nuclei themselves, and a nucleus and its environment. In most of the cases that are encountered, the magnetic dipole interaction is dominant. We shall thus describe the principal properties of this interaction in the aqueous solution; about the other possible interactions, which are of less importance, we shall only mention their essential effects. The interactions that are less frequently encountered or are weaker, are the contact interaction, which is important only if there are paramagnetic electronic centers; the electric quadrupole interaction for nuclei of spin Z > 3. 111. THERELAXATION OF A MAGNETIC DIPOLE IN OF BIOLOGICAL SUBSTANCES

THE WATER

A . Pure Water

1. The Magnetic Dipole Interaction: Expressions for 7; and T2 In order to understand the perturbations caused by the environment on the water molecules in biological substances, it is first necessary to consider pure water. The interaction between the two nuclear protons of a water molecule is described by the Hamiltonian =

-

- (h2y,ys/b3)[3(I n)(S n) - I S ]

(5)

(in which h = h(2n)-' with h = Planck's constant), where b is the distance between the two protons of the molecule, Z and S denote their spins, n is the unit vector along the b direction, and yr ,ys are the gyromagnetic ratios of these two protons (Fig. 3). One may assume that the distance b = b,, = const, and that the action of the local field on the S proton is essentially due to the magnetic induction associated with the magnetic moment of I and vice versa. This last spin should induce transitions only if it varies in time and if the Fourier transform spectrum of this variation contains a component with a frequency equal to one transition frequency of the given four-level quantum

96

GEORGES J . BbNk

b

FIG.3. Interaction of two nuclear spins.

system. In the case of liquids, the Brownian motion can produce such transitions. This is described in a precise way by the correlation time z, and its spectral distribution. The transitions induced by the spectral components of the local field may not only exchange energy inside the spin system but can even change its total energy. In the simple case when one has only one correlation time, the transition probabilities that define the relaxation constants are thus given as functions of the Larmor frequency ooof the nuclei in the applied external field and the quantity z,. In fact, 2, defines the spectral amplitude of the local field. Thus one obtains (I = S = y I = ys = y )

4,

The coefficients of these two expressions are proportional to the “rigid lattice second moment” 0; = (9/20)(h2y4/bX). For water, it is about 2.5 * lolo sec-’. 2. Variation of TI and T2 as Functions of 2, and wo

(a) The variation of Tl and 5 as functions of z, is given in Fig. 4,in which are indicated the singularities that limit the following three zones (Outhred and George, 1973) : (1) mobile fluid: & = T2 (left); (2) viscous fluid: and Tzvarying in opposite directions (center); (3) solid state: Tz = constant (right). Tl has a minimum when oo= 7,- and becomes constant when zC-’ x Am, the line width. This is particularly important since the correlation times for a given fluid are inversely proportional to the absolute temperature.

97

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

FIG. 4. Relationship between relaxation times and nuclear correlation time Outhred and George (1973).

T ~ .From

Finally we notice that the nature of the environment and its mobility have a great influence on the dynamics of nuclear magnetization. For water we have Liquid, 20°C; z, x lo-'' sec; Tl z 3.5 sec. sec; Tl z 2 sec; & z Solid, -5°C; z, x

sec.

In most of the biological systems the observed values lie between these extremes. (b) The variation of the relaxations rates T;' and T;' as functions of the amplitude of the applied field (and accordingly of the Larmor frequency w0/2z)is even more interesting because it makes possible an estimation of the correlation times. This variation is displayed in the semilogarithmic diagram of Fig. 5, which gives the graphs of T; and T; in the unity o! = hZy4/bg, as functions of the Larmor pulsation in the unity l/zc. This diagram also gives the variation of Tlp(see Section 111-Bl). On the diagram one can observe that

'

' '

'

'

T; and T; approach the same value 1.5 when oiz,' 4 1. T; goes to 0 and TT goes to 0.45 when oizz %- 1. T; x 0.44T;' for wiz,' x 1.

'

Thus q-' is a finite function of z, for all frequencies; however, TI goes to 0 in the high-frequency domains. T;' can thus be used to evaluate z, in

98

GEORGES J.

0.l

0.25

0.5

i

BBNB

2

3

4 5

10

FIG.5. Dispersion of relaxation rates.

the domain where the relation wtz," % 1 is not satisfied. On the other hand, because of the large variation of T; with wo, the measurement of &for only one value of w,, gives no information about 7,. The determination of &-' (or TC1)contains, however, in a larger frequency domain, a lot of information about molecular dynamics.

'

'

3 . Relaxation Times of Pure Water

In the case of pure water at ordinary temperatures, the correlation times - lo-'' sec. This gives NN T2x 3.6 sec for the experimental determination at 25°C. It is, moreover, worthwhile noting a small dispersion of T2 with temperature in the high-frequency domain (Krynicki, 1966). This variation of Tl with temperature deviates slightly from the Arrhenius behavior. The activation energy diminishes between 0 and 100°C (Fig. 6). Because of the great mobility of the ions H + and H - , there is also an exchange mechanism (correlation time z,) the importance of which varies with the pH of the substance (Glick and Tewari, 1966). The correlation time for this mechanism has been evaluated to be 0.7 msec at 23°C. In the neighborhood of pH = 7, the decomposition of the proton resonance by 0:' ( I = 8) at its natural concentration gives a T2 of about 2.5 sec without changing & . Of course, we do not take into account the relaxation between the

z, and zD (see pp. 91 and 92) are of the order of 10- l 2

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

TEMPERATURE IN 1

1.51

OC

5

I

3.6

I

I

99

1

I

I

34 3.2 3.0 2.8 FIG.6. Proton spin-lattice relaxation time T, , for a h f r e e water, as a function of temperature. From Krynicki (1966).

spins and the magnetic field generated by the molecular rotation. This mechanism, though important in gases, is not appreciable in fluids. Thus for water at 270°C,Tl = 30 sec, and 2, = 1.5.1O-l4 sec (Hausser, 1963). In Fig. 7 we give the main processes in water (Packer, 1977). According to these correlation time values, the variation of the relaxation time of water is quite small in the range of Larmor frequencies. This is, however, not generally the case for physiological substances. For human blood at ordinary temperatures we have, for example, FrequencyIMH,

Tl Is

Tz is

50 10 0.1 0.01

1 0.28 0.14 0.13

0.4 0.12

Accordingly, for such substances, it becomes necessary to measure Tiand eventually T, in a larger range of frequencies. The technical difficulties caused by such measurements has lead to the study of other parameters more accessible to experimental determination, for example, the spinlattice relaxation time Go,measured in a rotating frame of reference.

100

GEORGES J. BhNk

b FIG.7 . Dynamic processes in water.

T~ is a correlation time for translational diffusion; time for reorientation; and T,; the lifetime of a hydrogen atom in a water molecule between hydrogen exchange events. From Packer ( 1 977).

T ~ a, correlation

B. Aqueous Solutions

1 . Relaxation in a Rotating Frame of Reference

The rather large area of variation of the spin-lattice relaxation as a function of the frequency, the importance of the long correlation times in a large number of biological substances (7, between and sec), and the difficulties of measuring T2 with precision in a strong field because of the effects of the inhomogeneities of the applied field have made it necessary to consider a new parameter, the spin-lattice relaxation time in a rotating frame of reference (Jones, 1975). (a) If the frame of reference rotates with the Larmor frequency, the relaxation occurs along the radio-frequency field H1 -4 H o , and the relaxation time q pis given by the expression (a1= y H , )

The variation of Tlpl is similar to that of T ; ;however, its maximum value corresponds to the value of T; at wo (see Fig. 5). When wo % ml, the maximum of Tlpl coincides with the minimum value of TT1. (b) It has also recently been observed that the T l pcan be determined in a nonresonant rotating frame. If w is the frequency of the applied RF

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

101

W1

FIG.8. Effective field, out of resonance.

field, we have a nutation frequency of nuclei, with an angular frequency given by (Fig. 8) we = J(wo - 0 ) 2

+ 0:

(9)

Cornell and Pope (1974) have described an rf pulse sequence that measures TIPin the presence of a large rotating rf field distant from resonance by 10’ - lo7 rad/sec. In the weak collision approximation z, < T,, this relaxation time provides a direct measure of z, in the range lO-’-lO-’ sec difficult to obtain by conventional techniques that measure ?; or T p . When the substance under study is characterized by several correlation times of different magnitudes, T; becomes sensitive to all the correlation times and in particular to the longest ones in the whole domain of the Larmor frequencies wo. On the other hand, T i becomes sensitive only to correlation times that are short compared to 0;

’.

2 . Relaxation in the Presence of Paramagnetic Systems (ions,free radicals) Paramagnetic ions in a diluted solution have a dominating effect on the relaxation, not only owing to the dipolar magnetic interaction but also to the scalar interaction depending on the electronic density at the position of are given the nucleus. For nuclei of spin their contributions to Ti and by the equations of Solomon (1955) and Bloembergen (1957). This effect of the paramagnetic ions and the free radicals is appreciable only in some biological substances. On the other hand, in solutions where the relaxation times are sufficiently long (52 sec), the effect of the dissolved oxygen becomes important. In the normal molecular state ’Z the oxygen molecule has a magnetic moment due to the two noncoupled spins. Even

4

102

GEORGES J.

BCNB

for the equilibrium concentration with the surrounding air at ordinary temperatures, this effect changes Tl from 3.6 sec to about 2.3 sec, as well as causing a small change of Tz. It is thus essential to take into account this effect in the measurements made in vitro or to avoid the dissolution of 0’ in the extracted samples.

3 . General Case a. Exchange of molecules between two kinds of sites. When water contains substances in solution or even in suspension, it is possible for the water molecules to be bound to the foreign objects present: ions, molecules, cells, etc. One must then distinguish between different “kinds” of water molecules, according to the site occupied. Since the correlation time is determined by the mobility of the water molecules in those sites, it is thus different for different kinds of sites; and consequently so are and Tz for the corresponding protons. When considering the case of two sites, there are two important limiting cases.

1. If the lifetime of the water molecules in each of the two sites is long compared to and & for these sites, the system behaves as a two-phase substance. One can then determine two spin-lattice relaxation times and two spin-spin relaxation times. The relative importance of the contributions of the two “phases,” A and B, is related to the relative population PA/PBof water molecules in these two sites. Moreover, the duration of the relaxation is diminished by the lifetime zA and zB of the water molecules in each site. One has, for example,

TL (measured) = T ; :+ z; +

*.

2. If on the contrary, the lifetime of each of the two phases is short compared to the relaxation times Tl and G,the substance is characterized by only one value of Tl and T2, which is a function of the relative lifetime of a water molecule in sites A and B. If, for example, ?;A and TB are the spin-lattice relaxation times in the two sites A and B, then, with PA and PB the relative lifetimes of a water molecule in these sites, PA + PB = 1, the measured 17; leads to the value Tl

=-+TIA

pB TIB

In the intermediate case, when the exchanges are not too rapid, T2 is given with a good approximation by the expression

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

103

in which oAand w, are the Larmor frequencies corresponding to sites A and B. There exists a corresponding expression for 1/T, (Zimmerman and Brittin, 1957; McConnel, 1958). One encounters this situation in ionic solutions, molecular solutions with proteins, in particular, and in certain suspensions or heterogeneous substances.

Case of nonparamagnetic ionic solutions : In biological tissues one is mainly concerned with the ions C1-, Na', and K'. The influence of these ions on the longitudinal relaxation of pure water has been studied in detail by Engel and Hertz (1968). The concentration encountered in biological tissues is quite small. For ClNa the concentration is maximally about 9 gm/liter. The difference between the T;' of pure water at 25°C and that of water containing dissolved ions is (sec-') Na' :

+ 0.06

c1- : - 0.01 Kf: - 0.01

(structure breaking) (structure forming)

For NaCl of the above concentration, Tl is thus 3.5 sec compared to Tl = 3.6 sec for pure water. For ClK the difference is much smaller. We can thus often neglect this contribution, the origin of which is due to a modification of the thermal motion of the water molecules. This motion is more rapid for negative hydration than for positive. A situation that is frequently encountered is that of solutions of proteins. This is an example of a two-phase situation where the water molecules are either free or fixed to the proteins. Since the lifetimes in the two sites are small compared to the relaxation times 7 ' and T2 of the two phases, we have the situation described in the case of formula (10) for TI and T2:

The relaxation times T,and & can also be given as the following function of the relaxation times of the two sites A and B (Daszkiewicz et al., 1973):

where m is the mass of the solvent containing 1 gm of big molecules and c the concentration of these big molecules. This formula also shows that the observed relaxation rates T; ' decreases linearly with the concentration c for c 4 1. If one takes for TA the observed values for the solvent alone and expresses TB as a function of the correlation time zp of the quasi-hydrated

104

GEORGES J. BhNf!

proteins by means of formulas (6) and (7), one can determine the slopes of the lines q-l = f (c),and from these m, T ~ as, well as 14;\ and TB.One should notim that T~ is highly dependent on the molecular weights and on the shape of the protein, which can thus be studied by this method. b. Discontinuous distribution of the correlation times in complex systems. Already, for solutions of proteins we must introduce two correlation times, one for the free water molecules and one for those fixed to the proteins. In general, we have a distribution of the form

a,b(z - z i ) with

P(T)=

C ai= 1 i

i

There are n fractions, the ith being in the proportion ai and having the correlation time zi. We might have a discrete distribution of the correlation times in order to obtain the optimal values of T', Tip, and T, observed in a frog gastrocnemius muscle (Finch and Homer, 1974). Another example is given by aqueous solutions of MnZ+ where the variation of with frequency on the domain 2 kHz-2 MHz implies the existence of at least three correlation times (Bloom, 1956) : for the solvent, -lo-" sec; for the protons fixed at the ions, - 5 . lo-' sec; for the lifetime of a water molecule in the first shell of hydration around the ion, 2 sec (Fig. 9).

1'2

c .--

t

o.2 0' lo3

--.

\

\

I

lo4

I

lo5

\

I I

106

10'

V, (HP)

FIG.9. Proton relaxation time ratio as a function of Larmor frequency for aqueous solutions of Mn2+ (TZe= T2in the earth field): 0 ,25°C. From Bloom (1956).

105

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

c. The continuum, It happens in some cases that it is necessary to take into account a continuous distribution of correlation times in order to explain the variation of Tl with the Larmor frequency. A particularly simple method, devised by Davidson and Cole (1951), for the dielectric relaxation may be used here for this magnetic case. We thus have

T;

= T;d

+ D + A Re(l/[l + ( i ~ ~ / v , ) f l / ~ ] }

(13)

where T;,,! is the relaxation rate of the solvent and vo the proton Larmor frequency (Re = real part). D, A, v,, and @ are parameters to be determined. For a solution of yeast alcohol dehydrogenase, the above expression takes into account a weaker dependence of Tl on vo than for the Lorentz distribution (@= 2). A comparison between these distributions is given in Fig. 10 (Hallenga and Koenig, 1976). The introduction of a distribution p(z) of the correlation times entails, in general, a modification of Eqs. (6) and (7). Then we have

-001

0.1 1 10 PROTON LARMOR FREQUENCY (MHz)

100

FIG. 10. Dispersion of the solvent water protein magnetic relaxation rate 1/T, for a 65mg/ml solution of yeast alcohol dehydrogenase, molecular weight 160,000, at 5.9”C.The solid circles are the experimental data points. From Hallenga and Koening (1976).

GEORGES J.

106

B~NB

Prob

FIG. 11. Plot of the distribution function versus log T~ assuming a bimodal log-normal distribution of correlation times. From Zipp ci nl. (1976).

with p ( z ) d z = 1. The integration only goes over states for which there is a rapid exchange between the two phases, and the integration limits for T;' and T;' are not necessarily the same. Zipp et al. (1976) have suggested using a bimodal log-normal distribution to describe the distribution of correlation times for bound water in erythrocytes at low temperature with two maxima and nine adjustable parameters. The distribution of correlation times is given in Fig. 1 1. Andrasko (1975) has identified at least three bands of correlation times in agarose gels: those of the protons of the solvent, zcl = lo-'' sec; those of the water molecules that are strongly bound to the agarose molecules, zcz = 5 . sec; and those of water molecules that are less free than the sec < tcj < molecules of the solvent but not bound to agarose, 6 . sec. C . Inhomogeneous Substances 1. Presence of Cells Suspended Inside the Fluid

A number of physiological substances are constituted by a dispersion of cells in a liquid, which itself might be a solution containing ions and proteins. When the number of cells is high and their dimension small (- 1 pm), one assumes that they produce the same effects as those of the proteins of the solution: The water molecules can be bound to the surface of the cells. If the lifetime of the bound water molecules is long compared to TI

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

107

and T,, they are practically rigid and their proton signal has a large line width. If the lifetime is short, it has the effect of diminishing the TI and T, of the solvent according to the mechanism described in Section III-B3a. Only a few quantitative investigations have been performed in order to study this phenomenon. Lindstrom and Koenig (1974) have studied the variation of TI in a suspension of erythrocytes and a solution of hemoglobin with the same concentration. The results for the two samples are quite similar and show that the membrane has only little effect on the interaction between the water of the solvent and the hemoglobin. An important case to investigate would be that of serosities rich in leucocytes. On the other hand, the study of small balls dispersed in water has shown that the probability of a water molecule of the solvent changing its site from the solvent to be bound to a cell varies with the distance between the two sites (Pfeifer and Michel, 1964; Glasel, 1970). Instantaneous magnetization is thus not only a function of time, as in the Zimmermann-Brittin equation (10), but also of the spatial coordinates (Garwin and Reich, 1959). In a biological substance, this effect seems to be more important in the capillary vessels than in the homogeneous dispersions.

2. The Soft Biological Tissues The soft biological tissues are quite complicated substances, and a detailed study of the relaxation mechanisms of such samples therefore falls outside the framework of this paper. We only mention that one encounters here the same kind of problems as for the physiological fluids; however, they are in addition inhomogeneous on the microscopic level with well-defined compartments of different contents between which there might be an exchange. It is possible to make a schematic representation of such an inhomogeneity (Fig. 12) on the macroscopic level (Packer, 1977).

FIG.12. Schematic of large-scale heterogeneity. From Packer (12).

108

GFDRGES J.

Region a

BBNB Region P

H

Af Y

FIG. 13. Schematic of small-scale heterogeneity and various dynamic processes that may be experienced by water molecules in such a system. From Packer (12).

Two neighboring biological tissues A and B can have different chemical compositions, different geometrical forms, and different dimensions. They can also differ in the degree of ordering or internal anisotropy. Water molecules may transfer between regions A and B at a rate that is a function of the self-diffusion coefficient of water in each region and of the potential barrier constituted by the interface separating the two regions. This macroscopic heterogeneity is completed by a heterogeneity on the microscopic level. Of interest for us is that it influences the time constants of the different dynamic processes that characterize the water molecules in such systems. Figure 13 gives a schematic representation of two interacting regions a and p (Packer, 1977). The macromolecular structures represented by the cross-hatched regions are characterized by dimensions d, x, y, etc., and orientations O,, O,, etc., with respect to an external fixed axis. The relative rotations between the two regions are characterized by correlation times zM . The water molecules that are not influenced by the macromolecules make translational and rotational motions, and exchange their nuclear protons. These processes are characterized by the correlation times z, z, , z, , respectively. The water molecules that interact with the macromolecules have anisotropic rotational motions characterized by a correlation time z; . The mean duration of the interaction is denoted by 7;. Finally, the water molecules may diffuse from region a to region j?, their lifetimes in a given region being re( -&/2D,), with D, the self-diffusion coefficient, while the exchange of protons with the macromolecules corresponds to a lifetime of 7.: This microscopic heterogeneity is described by a set of correlation times and by an anisotropy in the diffusion constants. We shall give some examples in Section V.

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

109

PART 2: APPLICATION OF THE TECHNIQUES OF NMR TO THE STUDY OF BIOLOGICAL TISSUES

IV. THE TECHNIQUES A . Introduction The study of biological tissues is mainly concentrated on the dynamic constants of the water situated in complex but relatively mobile media. Thus, neither the wide-line nor high-resolution techniques of NMR are easily applicable. The methods of NMR, based on the dynamic behavior of the systems, as, for example, the observation of transients and their Fourier transform, however, have been successfully applied to the study of biological tissues extracted from living beings. Similarly, the free precession of the nuclear protons in the earth's field after a prepolarization seems to have important implications for possible applications to medical diagnosis in situ. B. Techniques Based on Nuclear Magnetic Resonance

In the techniques based on NMR, the signal observed at the Larmor frequency is usually a quite complicated function of nuclear quantities like equilibrium magnetization M,, relaxation times Tl and T2 measured in an applied field B,, and the quantities related to the experimental conditions, such as the frequency and the intensity of the RF field, the spatial inhomogeneity of the applied field, and the characteristics of the reception circuit. The experimentalconditions can of course be adjusted such as to optimalize the received signal; however, the determination of the nuclear quantities poses in general a number of difficulties. The determination of M, by determination of the surface of the resonant signal is based on the hypothesis that the observation is made far from saturation. The different techniques for measuring ,"I by spin echo are generally good if the calibration of the intensity of the HF field and the timing of the pulses have been done correctly. This measurement is in general slow unless one can apply the impulsion program of Csaki (1963). Then the measurement of Tl has a time duration of 2 to 3 Tl . Since it is important to know Tl on a large domain of the values of B,, several methods must be used for its determination : (1) a method by which Tl is determined for a large range of B, values, but the polarization and the Tl measurements are made in a strong constant field (Koenig and Schillinger, 1969; Graf et al., 1977); (2) the measurement of Tlpfor which the results must be correlated with the Tl and & corresponding to B, (see Fig. 5).

110

GEORGES J .

BBNB

The development of these methods is related to the fact that T, must be measured in the domain of variation for which o,z, x 1 in order to enable a determination of 7,. It is also important to know T2 because it is always sensitive to the slowest correlation times. Unfortunately, T2 is difficult to obtain exactly in resonance experiments: The line width and the decrease in the free precession are highly sensitive to the amplitude of B, , the homogeneity of B,-and only a correct use of spin-echo techniques is able to overcome the errors' causes. For a discussion of these problems we refer the reader to the literature. Finally, note that a homogeneous magnetic field on which one has superposed a given gradient along an axis enables the determination of the diffusion constants of the systems carrying the magnetized nuclei. Since the realization of such gradients is rather complicated, we have only a small number of measurements of the diffusion. It is, however, such measurements that enable us to approach the specificproblems posed by a study of the water in the soft biological tissues. C. The Free Precession in the Earth's Field (Packard and Varian, 1954).

A sample containing nuclear spins is submitted to a constant and intense magnetic field Bp,The direction of this polarizing field is chosen orthogonal to the measuring field B, (which in our case is the earth's magnetic field) (Fig. 14). As Bp 9 B,, the equilibrium magnetization Mpobtained by leaving the sample for a sufficiently long time (9c)in the field B, + Bo is practically parallel to Bp. One then cuts the field Bp [the duration of the switching being small compared to the Larmor period v, = 27c(yB,)- 'I. The magnetization Mpis then submitted only to the perpendicular field B,; we have a free precession of Mpin the B, field: (1) The free precession oscillates with pulsation

0,.

t'

FIG.14. Block diagram of free-precession apparatus.

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

111

(2) The damping of this precession has a time constant T2which contains the transverse relaxation T,,as well as the broadening due to the inhomogeneity of the field and possibly other sources of damping. (3) The initial amplitude depends only on M p . It thus gives information on the number of nuclei present and permits an indirect determination of the value of & in the polarizing field. Even if the prepolarization is made in a strong field (10 m tesla), the sensitivity of the apparatus based on free precession is much smaller than that of the NMR apparatus for which the external field may reach the order of 10 tesla. The duration of the cutting of the polarizing field, before the observation of the free precession, must be of the order of a period of the free precession in the earth’s field, i.e., 5 . low4sec. This constraint imposes an upper limit to the strength of the polarizing field, as well as a dead time between the moment of the cutting and the beginning of the observation of the precession. The maximum of the amplitude of the free precession gives a relative measure for M , without any effect of saturation. If the apparatus is placed in a region where the earth’s field is homogeneous and if the reception circuit is not too selective, the decrease of the signal from the free precession permits a direct determination of T2 in the external field, i.e., the earth’s field. This maximal amplitude A , depends only on the duration t of the prepolarization. Moreover, the study of the variation of A , as a function o f t provides a simple way of determining TI in the polarizing field (Fig. 15). The impossi-

112

GEORGES J.

BI?NI?

bility of realizing strong gradients in the weak field thus excludes the measurement of too small diffusion constants. One can evidently measure the relaxation in even weaker steady fields to obtain the effects of particularly long z, (2, x 1 msec if Bo x 4p tesla) or determine Ti and T, from measurements in a rotating frame of reference.

V. APPLICATION TO THE STUDYOF BIOLOGICAL TISSUES IN VITRO(BIOPSIS) The motivation for the application of nuclear magnetism to medical diagnosis stems from the results of the investigation by such methods of a number of healthy and pathological biological tissues. The purpose of these studies has, in general, been fundamental in nature-mainly to interpret the variation of the measured parameters by means of structural or dynamic microscopic models. They constitute an important contribution to medical diagnosis, although the actual efforts are not focused on that area.

A . Physiological Fluids The simplest biological substances are the physiological fluids concentrated in certain regions and organs of living beings. In spite of their great variety with respect to composition and function, they all present certain similar features. They are, in general, aqueous solutions with a small concentration of mineral ions (isotonic solution ClNa 9 gm/liter) which contain proteins of highly different molecular weights in greatly varying concentrations. They might also contain red blood cells, leucocytes, cells, etc., in suspension. The nuclear magnetism of the water molecules in such fluids is sensitive to the nature and concentration of the proteins present, of the pH of the fluid, and to the presence of objects in suspension if they are in a nonnegligible concentration. The proteins already have an influence on the relaxation times at a concentration of the order of 1%. In the following we shall discuss some results obtained on healthy and pathological physiological fluids. 1. The Blood The blood of mammals (including human blood) has been the object of numerous investigationsby means of the techniques of nuclear magnetism. The variation of TI as a function of Larmor frequency measured in a large domain (Lindstrom and Koenig, 1974), completed by measurements of q pperformed in order to have TI measurements at even lower values of the

113

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

magnetic field, have ma& it possible to determine the spectrum of the 2, for whole blood, as well as for the red blood cells. To do this, we use either the distribution of Davidson and Cole (1951) or the bimodal log-normal distribution (Zipp et al., 1976). The presence of a sickle cell disease has the effect of increasing the relaxation rates. The presence of methemoglobin decreases TI and T2 in a way that is characteristic of the presence of Fe3 ions (Blicharska and Dmitriev, 1976). Also analyzed have been the coagulation of the blood characterized by the polymerization of the fibrinogen (Daszkiewicz et al., 1965) and the variation of T2 as a function of the concentration of red blood cells in the mixture (plasma red blood cells) in the weak-field range. This variation which has been studied for ox blood (Hiltbrand et al., 1978), follows that of the viscosity and is well described by the equations of Daszkiewicz et al. (1973) (Fig. 16). Thus, by the methods of nuclear magnetism, one is able to detect the presence of an abnormal viscosity of the blood, or of abnormal quantities of hemoglobin or its pathological derivatives, as well as of heavy proteins of the plasma. +

+

2. Amniotic Fluid (BBn&et al., 1977b) The human amniotic fluid has been studied in the weak-field range. In its normal state, such fluid is mainly an isotonic liquid solution diluted with

A -a

6

- 16.3 - 12.2 ^n a - 90 - 6.5 f=

- 4.6

-

3.2

- 2.1 - 13 0

10

30

50

FIG. 16. Measurement of cow centrifuged blood. theoretical curve; 0 viscosity.

70

90 (Vol Ht'/d

+ 7"' experimental values; - T;'

GEORGES J.

114

2

+ N

BBN~

1

1.5

TI (sed FIG. 17. Relaxation times of human amniotic fluids. El, isotonic solution; 0 , normal; 8 ,meconium; +, hydramnias.

proteins (a few grams per liter). The range of values of T2 measured in the earth’s field as a function of TI measured in a polarizing field of 5 m tesla is quite different from that of an amniotic fluid polluted with meconium or corresponding to an hydramnios disease. The high ratio I;/& corresponds to a longer z, for amnioticfluid with meconium at ordinary temperatures (Fig. 17). 3 . The Pleural Fluid

The pleural fluid, a simple solution of proteins in transudates, with a nonnegligible quantity of leucocytes and other cells in the case of exudates, has also been studied in the weak field (BCnC et al., 1978). For the transudates it has been found that the relaxation rate increases proportionally with the concentration of proteins. For the exudates the values obtained for T
The possibility of making a medical diagnosis by an examination of the

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

115

nuclear magnetism in biological tissues is based on the results of a number of fundamental studies of healthy homogeneous tissues. The majority of these investigations has been made on muscular tissues of vertebrates, mainly mammals. In the following we shall give a brief review of the results obtained in this area of research and their possible consequences for medical diagnosis. a. Existence of a spectrum of relaxation and correlation times. The study of muscular tissues of the rat squelette at 25 and 50 MHz (Hazlewood et al., 1974) has shown that there are at least three distinct values for the relaxation time T2corresponding to the tissue water (1) extracellular water : 180 msec, 10%; (2) intracellular free water: 45 msec, 82%; (3) water bound to proteins or membranes: c 5 msec, 8%. These results, which have been confirmed by other workers (Diegel and Pintar, 1975a), are situated at the border between the macroscopic and microscopic heterogeneities. The microscopic “heterogeneity” in a given compartment manifests itself by the existence of several correlation times. Three correlation times have been observed in several tissues from the mouse (Knispel et al., 1974): 7d

z, t,

rotational and translational diffusion of the free water molecules : < lO-’Osec (z, and zD on the Fig. 13); rotational diffusion of the water molecules bound to the macromolecules: 2.10-* sec (7:on Fig. 13); slow exchange of water molecules between the hydration shell sec (7: on Fig. of the macromolecules and the solvent -7. 13).

The nature of the intracellular water has been the object of serious controversy. More than 90% of this water has a correlation time that is quite close to that of pure liquid water ( < 10- sec), which would seem to exclude a possible crystal structure of this water. However, recent investigations performed by Hazlewood (1977) have shown that the diffusion coefficient D@) of the nuclear water protons of a muscle of the squelette is anisotropic with respect to the longest axis of the cell; the anisotropy is 0,(Oo)/D,(90”) 1 1.39. This shows clearly that the intracellular water is not simply a diluted solution. From a practical point of view, the existence of several correlation times and the importance of the longest, which causes Tl to change from 1 sec at high frequencies to about 50 msec at the null field, show that (1) TI is a quite unimportant characteristic in the strong field. Thus,

116

GEORGES J.

BBNB

for a mouse muscle (Knispel et al., 1974) there are two correlation times that characterize the substance : z, x 2.10-*

sec,

z, x 7. l o w 6 sec

The transition frequencies for the variations are w, x 8

MHz,

o,x 22 kHz

With respect to Fig. 5, the variation gets reduced by about 95% for the shortest correlation time at w,, = 30 MHz. A measurement made at o,,> 30 MHz is not sensitive to any of the more important mechanisms concerning the water of such tissues. (2) The measurements made between 2 and 30 MHz are sensitive for z, but not for z, ;moreover, a unique measurement of Tl has no clear molecular significance. (3) It is necessary to measure Tl in the range 5-80 kHz in order to identify the correlation time z, , which seems the most significant. (4) T2 is sensitive to all of the correlation mechanisms; in particular, to those characterized by the longer correlation times [Eq. (7)]. The dispersion is smaller than for those of Tl but must, nevertheless, be determined. 2. Pathological Tissues a . Larger values of relaxation times for cancerous tissues. The investigations whose results have been described in this paper were all motivated by the observation due to Damadian (1971) that the relaxation times of cancerous tissues were longer than those of the same kinds of healthy tissue. This result has been confirmed for numerous tissues; one has only a few examples where the ratio Tl (cancerous tissue) Tl (same healthy tissue)

is close to or smaller than 1. An increase in Tl has also been observed even for healthy tissues if there is a tumor in the organism providing the tissues (Frey et al., 1972). On the other hand, the results from cancerous and healthy samples extracted from human breasts show no difference with regard to the measurements of Tiat 60 MHz (Bovee et al., 1978). This indicates a limitation of the application of this technique to pathological analysis; however, the preceding remarks concerning the dispersion of the relaxation times of biological tissues show that this negative result is not really surprising. At 60 MHz we are at too high a frequency to have an appreciable absorption for low correlation times. b. Influence of the rate of evolution of the tumor. It has been observed

117

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

that the TI of cancerous tissues depends on the rate of growth of the tumor. If the tumor is growing rapidly, then the TI becomes longer than for the corresponding healthy tissue; if, however, the growth is slow, there is no marked difference (Inch et al., 1974). Moreover, it has been possible to relate the duration of TI to the quantity of liquid water present or to the ratio water bound to the macromolecules free water

r=

The water molecules are in fact being rapidly exchanged between these two sites. Figure 18 shows the relation between & and the percentage of tissue water. We see that fetus tissues and tumor tissues have the longest relaxations times and water content. There are only few results as concerns T2. This is unfortunate since T2 is significant for molecular dynamics because of its dependence on the long correlation times. This explains the remark of Diegel and Pintar (1975b) that measurements at low frequencies (2 instead of 30 MHz) would improve the possibility of distinguishing between healthy and cancerous tissues. It also explains why a diagram of the T, measured at two frequencies may permit the separation of the ranges of variation corresponding to pathological tissues (Coles, 1976). Moreover, it shows the 100-

-s

7 WATER

80-

v

5 60W !-

z

0

u

u

40 -

W

3 20

I

I

I

I

-

I ‘FAT f

0

0.2

I

04

I

I

I

0.6 08 1.0 RELAXATION TIME (sec)

I t

I

I

If

2.0

30

FIG.18. Water content (%) versus T, for tissues from mice measured at 30 MHz at room temperature. From Inch et al. (1974).

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GEORGES J.

B~NB

2016=, A c

& U ;I2V 8-

-.

\ \

’‘\, \

40

I

- - _ _ __ _ _ _ T$ (HI=’)

-- -- - -_--------I

kf

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I

FIG.19. Dispersion of T, and TIPin samples of mouse spleen tissue: -, _ _ _ _ ,tumerous tissue. From Knispel (1974).

healthy tissue;

importance of measuring Tl or TlPin their ranges of dispersion (Fig. 19) (Knispel et al., 1974). These measurements show the importance of determining by nuclear magnetism (1) the spectrum of the correlation times z, or its weighted mean Z,; (2) the spreading of the relaxation rate, in particular, when there are no appreciable variations of z, . It is also possible to use the model of Daszkiewicz et al. (1973) by taking as TiA values those of the isotonic liquid. Equation (12) is then an equation with three unknown variables (c, m, qB)and can be solved with the use of three measured relaxation times (Tl or T2) within the dispersion range. The comparison between normal blood, the packed red cells of normal blood, and a hemoglobin solution in the case of sickle cell diseases shows similar z, but different spread of relaxation rates (Lindstrom and Koenig, 1974).

3. Critical Comment As was mentioned by Swartz (1978), it is presently not clear what possibilities the use of NMR offers for the diagnosis of cancer. If one restricts the meaning of the word diagnosis to the distinction between healthy and cancerous tissues,

(a) The measurements of Tl made at only one frequency may be insen-

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

119

sitive to important parameters or, in the opposite case, be without quantitative significance. (b) For the few measurements of T2,sometimes supplied with measurements of q,,, there is an uncertainty due to the dispersion in the magneticfield range (Fig. 5). (c) The most important results are the measurements of TI and over the whole important dispersion range. These results enable us to estimate the complete range of variation of the only important molecular parameters, the rates of relaxation and the durations of the correlation times. Only progress in this direction can make the technique of NMR competitive with histopathological analysis. However, one can hope to be able to apply NMR to the screening of certain diseases or as a complementary diagnosis (adjuvant). It is in this perspective that the in situ methods of exploration must be considered. VI.

APPLICATION TO

MEASUREMENTS IN SITU

An initial observation of Damadian (1971) indicated the possibility of applying this technique to the search for tumors by measurements performed in situ on animals or even human beings. This possibility was confirmed by Weisman et al. (1972), whose measurements of a mouse tail were able to localize a tumor. They obtained TI = 0.31 sec for the normal tissue and 0.7 sec for the tumor. The problem posed by this method is to extract a signal from the nuclear magnetism of a given region that is small compared to the size of the organism of which it forms a part. The most important development in this direction has been to superpose a spatial magnetic gradient on the constant field B,. Thus, the applied magnetic field and also Larmor’s frequency become functions of the spatial coordinates, since the distribution of the obtained frequencies of an inhomogeneous magnetic field also has a spatial distribution. Gabillard has already shown that by applying the magnetic gradient on a sample of a certain volume, one will obtain an NMR spectrum that gives the profile of the density of the resonant nuclei in the direction of the gradient (Gabillard, 1951, 1952). A large number of variants have been proposed over the last few years.

A. Projection Reconstruction Method (Lauterbur, 1973)

The projection reconstruction method consists of making a large number of one-dimensional projections with different angles of directions of the

GEORGES J.

120

B~NB

Y

t Z=

I

i

i

i

FIG. 20. Relationship between a three-dimensional object, its two-dimensional projection along the Y axis, and four one-dimensional projections at 45" intervals in the XZ plane. The arrows indicate the gradient directions. From Lauterbur (1 973).

gradients and Bo constant field, the differences being small. By means of a computer with enough capacity, one can then rebuild an image from the projections according to the scheme of Fig. 20. The process of reconstructing images from projections is well known. It is encountered in electronic microscopy and astronomy. It has also been applied in the computerized tomography of x-rays. By obtaining a large number of two-dimensional images, one can obtain, moreover, a threedimensional lattice of the intensities of the NMR signals. If, however, the sample has high symmetry, a small number of one-dimensional profiles is sufficient. Only 1 for a cylinder of circular base, 2 for a prism of rectangular base, etc. In the general case, a two-dimensional decomposition in n2 elements of a given lattice is completely determined by n projections. A smaller number of projections diminishes the resolution or leaves out important information. By means of this technique, Lauterbur has produced the image of the thoracic cavity of a mouse by means of 12 projections, changing the angle by 15" each time. This method scans successively each point of a given object and is thus relatively slow.

121

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

B. Single-Point Spin Mapping Technique In the single-point spin mapping technique due to Hinshaw (1 976), one applies the gradient on the whole sample, except for a small region which is the only one observable. Thus, by varying this, one can observe all the points of a bigger region.

reversed

P

(1

t

I I

I I

I I I

0

20

z

FIG.21. Diagram of the spatial variation of the total magnetic field when a localized magnetic-field gradient is added to a uniform magnetic field B, . From Mansfield (1 976).

-Y

-B FIG.22. Diagram of the sample chamber geometry in the “multisensitive point method.” From Hinshaw et al. (1978a).

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GEORGES J. BkNh

This method cannot be realized with static gradients ; the superposition of two static gradients in different directions gives a static gradient in a third direction. In the most simple version, each gradient is modulated at a different frequency. A simple detector cuts all modulated tensions and receives only the signal from the region where the gradient is zero. Since the gradients are produced by inverse wound coils, the region of zero gradient is inside the sample explored (Fig. 21) (Mansfield, 1976). By changing the current in the three pairs of coils that generate the three modulated gradients, one can go through all the sample. This method is thus also quite slow. In an alternative version of the method, the “multisensitive point method,” one keeps only two alternating gradients with different frequencies (or with the same frequency, but with a phase difference

FIG. 23. Thin transverse NMR image of live human wrist and corresponding anatomy. From Gardner et al. (1975).

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MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

of 42). The third gradient is a static one. The two alternating gradients define a line of sensitivity (Hinshaw et al., 1978a) (Fig. 22), along which one applies the static gradient. The NMR signal following an RF-pulse of 90" contains information about the whole line, which a computer can transform into an intensity profile. The line is displaced by varying the current producing the alternative gradients. This method has made it possible to obtain an anatomically detailed image with 128 x 128 points of a human wrist in 9 min. Figure 23 (Hinshaw et al., 1978b) shows the image obtained and the corresponding anatomy (Gardner et al., 1975). C . Mansjield's Techniques 1. Line-Scanning

The sample is placed in a strong static field B which produces a polarization of all the spins. One then superposes on the static field different gradients along two or three axes (Fig. 24). Consider a thin slice (thicktess Az) of the sample; the method works as follows (Mansfield and Maudsley, 1977): (1) The spins in the slice Ay at the ordinate yo are submitted to the static field By0on which is superposed a gradient Gy. (2) One then applies a sequence of pulses, the spectral distribution of which is confined to a small interval (for example, those of the slice Ay

sy plane

chamber

'Scan

direction

FIG.24. Sketch illustrating the principle of line scanning. From. Mansfield and Maudsley (1977).

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GEORGES J .

B~NB

corresponding to the distance yo). The effect of these pulses is to turn the mean value of the direction of the spins by 90” and to partially saturate the magnetization. (3) One then replaces the gradient Gy by a gradient Gx. In the slice of section AyAz the local values of the field B changes along the axis Ox. Moreover, the magnetization of all the elements of the volume AxAyAz is perpendicular to Oy. The magnetization m, for each of the elements precesses by its own Larmor frequency, and these frequencies vary along the Ox axis because of the gradient Gx. (4) The superposition of all the elementary precessions is the freeinduction decay, the Fourier transform of which gives a signal proportional to m, along the Ox axis. Since the frequencies of the signals vary along the Ox axis, we have in one record the density of protons along Ox, which can be read off an oscilloscope. The choice of Az can be made by the gradient G z ; the authors used for that a coil 8 mm thick. Note that if TI is sufficiently long and if the pulse is strong, it saturates the magnetization. This saturation increases when the inverval z between the successive pulses diminishes. Often, the concentration of protons in various soft tissues does not change appreciably: By varying B, , we make a supplementary discrimination by the saturation due to relaxation time TI. 2. Planar and Multiplanar Imaging Method (Mansfield and Maudsley, 1976) The basic idea of the planar and multiplanar method is to superpose a “discrete lattice structure” on the density distribution of spins originally continuous. By means of this method, the signals from all the spins of a plane layer in the three-dimensional sample can be recorded simultaneously and distinguishedfrom each other. This reduces considerably the time needed to produce an image. Let us simplify by considering the example of a twodimensional sample (Fig. 25) : (1) In phase A of duration t A , on which is applied a gradient G y and, at the same time, a pulse that excites all the spins in a slice of thickness Ay and recurrence b, we obtain a nutation of the magnetization, which rotates by8 = 90”. (2) At the end of pulse A, one applies in addition a gradient Gz such that the spins are submitted to Gy Gz. One then receives the free-induction decay from all the elements of the volume AxAyzm, spaced by y = y o + mb in the xo plane, and the Fourier transform gives the density distribution of spins in the volume Ax * y .z .

+

This method has been applied to picture an “oil-filled annulus.” The

MEDICAL DIAGNOSIS BY NUCLEAR MAGNETISM

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Y

/

Ax FIG.25. Sketch showing multistrip or grid selection in Gy within one layer of magnetization of thickness Ax. The cross-hatching corresponds to the perturbed spin regions. From Mansfield and Maudsley (1976).

extension of this method to three-dimensions has been envisaged, but up to now no picture has been realized (Mansfield and Pykett, 1978). The authors assume that the method should make it possible to obtain a cross section of the human body at 4 MHz with a signal-to-noise ratio of 10 in a few seconds. D . Other Techniques In this section, we shall only give a brief description of the essential characteristics of some other methods about which no precise information has been published or which have not yet been developed with a view toward possible application to biology or medicine.

(1) Damadian's Fonar uses a parabolic gradient field, and only the region corresponding to the top of the parabola is sufficiently homogeneous to give a signal by resonance excitation. By this method it has been possible to picture the thorax of a human being by means of a superconducting magnet with a diameter of 53 in. in 3 hr (Damadian et al., 1977). ( 2 ) Fourier transform zeugmatography is a method that has been proposed by Kumar et al. (1975). It combines the advantage of total detection of the global irradiation with that of the Fourier transform. After the 90" pulse, one applies successively the gradients Gx, Gy, and Gz and varies their durations tx, ty, and tz. To obtain three-dimensional pictures, one must make N 2 free-induction

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GEORGES J. B h d

decays (FID) of this kind for all the values of tx and ty, with fixed tz = TI max ( N = number of points of a FID). The signal-to-noise ratio is the same for all the pictures; however, since we only look at two dimensions, there is too much information. The inconvenience is that one must reach TI max between two successive pulses. This method has not yet been applied to the study of biological systems. [See Note Added in Proof, p. 132.1 (3) To conclude, let us mention that the method of Hutchison et al. (1974) is also based on the use of the Fourier transform, but only after the detection of the NMR by spin echo. This makes it possible to suppress the dead time following the 90" pulse. By this method, they have been able to distinguish regions of different TI. This result is obviously of interest for medical diagnosis.

E. Applications of Zeugmatography to Medical Diagnosis Zeugmatography has not yet been extensively applied to the study of pathological systems. In addition to the observations of Weisman et al. (1972), an image of a tumor in a living mouse has been obtained by Damadian et al. (1976) and more recently, also by the Damadian group, on the human chest (private communication). It is clear that the application of this technique to medical diagnosis implies a strong (50 m tesla) uniform magnetic field of large dimensions; computers with large capacities for realizing pictures ; and use of techniques that permit the determination of relaxation times in the dispersion region. At first one tries to obtain pictures of the same kind and with similar contrasts as those obtained by x-rays, by using as a parameter the density of the nuclear water protons and the relaxation times. This is already a quite considerable step to take, although the quality of the pictures that have been obtained on animals of medium size give hope that the aim can be reached by appropriate technical means. It is precisely with regard to this point that the challenge is at present the strongest. The next step will imply the use of methods that permit the determination and separate showing of the most important parameters that can be extracted from nuclear magnetism (Mo 3 TI Tz). This will represent great effort and it is therefore only a remote possibility that cheaper apparatus than that of the x-ray scanner will be obtainable in the next few years. For low frequencies ( Iseveral megahertz) the penetration of the tissues is still sufficient (Bottomley and Andrew, 1978), and it seems that static magnetic fields < 1 kOe and oscillating magnetic fields I 1 gauss are much less dangerous than ionization radiation (Andrew, 1976). 9

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F. In Situ Diagnosis in a Weak Field

We shall close this section with a discussion of a different approach based on measurements of nuclear magnetism in a homogeneous weak magnetic field like the earth’s field, the nuclear magnetization being amplified by prepolarization in a field of 5 to 10 m tesla. The aim of this method is not to give a reconstruction of pictures providing detailed structural information of a given invisible region of the human body. Our interest is in sufficiently big and homogeneous regions, in order not to have too big a problem with the sensitivity in spite of the weak specific magnetization of the nuclei. The aim is to give a precise identification of the biological medium explored. Measured are the constants characterizing a given pathological state, by determination of the parameters M o , TI, T,, etc., accessible by nuclear magnetism. 1. Experimental Results The preliminary work has concentrated on the measurement of the T2 of common and well-localizedphysiological fluids-urine in the bladder and blood in the cardiac region. The experiment has been to place a prepolarization (and reception) coil as close as possible to the explored region, and has consisted in determining the T2from the signal obtained and comparing the result with the corresponding value obtained for the same physiological fluid at the same temperature immediately after the extraction. For urine, the identification of T2measured in situ has been obtained by means of a new measurement in situ after micturition. For the blood, this was obviously not possible, and we only compared measurements of blood in situ and after extraction. The discrimination of the exponentials corresponding to the accompanying tissues is easy because of the time constants, and their intensities are very different from those of the fluids. For measurements in situ, one has thus determined the time constants of the dominating exponential. a. Results obtainedfor human blood T2extracted from the signal obtained from the anterior median part of the chest (on E. H. and G. B.): E. H.: 141 msec f 3%;

G. B . : 158 msec f 3%

T, measured on the blood extracted immediately after the above-mentioned measurement :

E. H.: 140 msec & 3%;

G . B.: 157 msec & 3%

This excellent agreement is explained by the facts that the blood in this region is quite close to the surface of the body and that the time constants of the venous and arterial blood are about the same.

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b. Results obtained for human urine. We obtained

in situ before micturition : 1.90 sec; in situ after micturition: 0.17 sec; in vitro after micturition : 2.20 sec. The difference between in situ and in vitro results (15%) is now well understood (Borcard et al., 1979). The free precession signal is, in the in situ measurements, a superposition of two or three exponentials : (1) the damping of the measuring coil itself (time constant -25-45 msec) ; (2) the precession of the fluid protons around the bladder (T, 200 msec) ; (3) the urine signal (T, 2.2 sec).

-

-

The signal obtained after micturition is a superposition of the (1) and (2) exponentials; the in situ signal with urine present is a superposition of the three exponentials. The values obtained for urine T, are in good agreement with in vitro measurements. 2. Comments (a) The direct measurement of T, by free precession presupposes among field of measurement. This other things a highly homogeneous (again means that the laboratory must be well located. This requirement can be avoided if the time constants of the free precession in a rotating frame are measured at the Larmor frequency (- 2 kHz). Then one does not obtain T,,but [(l/T,) + (1/T2)]-' in the range where T, T, (Torrey, 1949). It is sufficient that B, , which is quite weak, has the required homogeneity. (b) The measurement of q9in a nonresonant rotating frame in a weak field permits a very large extension of the range of dispersion of the correlation times. This might be of importance for the diagnosis of certain infections. (c) The aims to be achieved are an improvement in the sensitivity and treatment of information ; precise measurement of other parameters like Mo , TI , and qpr in adequate ranges of variation; the study of techniques that permit the creation of purely homogeneous fields of measurements. (d) Even if the study of physiological fluids has turned out to be of considerable interest, an improvement of the sensitivity, for example, by increas-

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ing the intensity of Bo, will without doubt be necessary for the application of this method to the examination of the soft tissues. VII. FINALREMARKS

To make an objective judgment of this new diagnostic method, one must first realize that only the preliminary steps have been accomplished. The possibility of using nuclear magnetism in medicine was first pointed out by Damadian (1971). Shortly after it was proposed to use this NMR to make pictures of the interior of the human body (Damadian, 1972; Lauterbur, 1973). The possibility of exploration in the weak field was announced later (Btnt et al., 1977a). On the other hand, the domain of investigation represented by the identification of pathological states of the tissues and organs of the human body is quite vast. The technical processes necessary for the application of zeugmatography to the human body is developing rapidly. Though this new method is still in its exploratory phase, it is already clear that it opens up new possibilities. A . What Possibilities?

This method should provide a possible substitute for computerized x-ray tomography (CXT), so as to make it possible to obtain information of the same kind and with a similar precision without ionizing radiation. One may add that the price should be of the same order. This method also offers a substitute for echography, with which it shares the advantage of being without danger. Its higher price would be compensated for by a somewhat better resolution. B. Problems to Solve

(1) Even if zeugmatography were to become competitive with CXT, one has to take into account that the medical profession has had long experience with x-rays, while nuclear magnetism is a newcomer in its arsenal. This implies necessary investments in competence, space, and time. (2) It is necessary to improve the sensitivity and resolution of the information and the speed by which it is obtained. This requires stronger fields, the inherent danger of which is supposed negligible; however, this has not yet been sufficiently well tested. (3) The research teams working on zeugmatography have been able to explore systems up to 10-20 cm in diameter with good resolution. It is clear

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that the following step, the exploration of a human being with the same resolution, presents considerable difficulties, for the solution of which it will be necessary to consider the time of development and the cost of apparatus. (4)The optimal use of nuclear magnetism with a view toward medical diagnosis implies a three-dimensional mapping of the principal parameters M , , TI, Tz of nuclear magnetism. The author was recently informed that Pykett and Mansfield (1978) have published the first two-dimensional TI map of a rat leg. The knowledge of these parameters permits the determination of the time constants that characterize the healthy tissues and their pathological alterations. As a start, it is indispensable to make a considerable effort to analyze the tissues; in fact, it is clear that each of the parameters seen alone is not very significant. There is thus still much to be known about the significance of the parameters, as well as the technical problems associated with their determination. This situation justifies a variety of different approaches to the problem, a diversification that can lead to apparatus quite different from what we have in mind at the present time.

ACKNOWLEDGMENTS The author is grateful to Dr. T. Aaberge for translating the text into English and to Mr. P.-E. Bisson for drawing the figures. We obtained kind permission to reproduce figures from the following publishers and authors. Rockefeller Univ. Press: Outhred and George (1973), Fig. 4. North Holland Pub. Co.: Krynicki (1966), Fig. 6. Royal Society: Packer (1977), Figs. 7, 12, and 13. American Institute of Physics: Bloom (1956), Fig. 9. American Chemical Society: Hallenga and Koenig (1976). Fig. 10. US Dept. of Health: Inch, McCredie, Knispel, Thompson, and Pintar (1974), Fig. 18. Macmillan Journals Ltd. : Lauterbur (1973), Fig. 20. Andrew, Bottomley, Hinshaw, Holland, and Moore (1977), Fig. 23. Taylor and Francis Ltd.: Mansfield (1976), Fig. 21. British Institute of Radiology: Hinshaw, Andrew, Bottomley, Holland, Moore, and Worthington (1978). Fig. 22; Mansfield and Maudsley (1977). Fig. 24. Saunders: Gardner, Gray, and O’Rahilly (1975), Fig. 23. The Institute of Physics: Mansfield and Maudsley (1976), Fig. 25. Academic Press: Mansfield and Pykett (1978), Fig. 1; James (1975), Fig. 2; Zipp, Kuntz, and James (1976), Fig. 1 1 ; Knispel, Thompson, and Pintar (1974), Fig. 19.

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GEORGES J.

BBNB

Hinshaw, W. S., Bottomley, P. A,, and Holland, G. N. (1978b). Nature (London) 270,722. Hutchison, J. M. S. (1976). Proc. L. H. Gray Con5 Univ. Lee& Med. Images, 7th, 1976 pp. 135-141. Inch, W. R., McCredie, J. A., Knispel, R. R., Thompson, R. T., and Pintar, M. M. (1974). J . Natl. Cancer Insr. 52, 353. James, T. L. (1975). “Nuclear Magnetic Resonances in Biochemistry,” p. 37. Academic Press, New York. James, T. L., Matson, G. B., Kuntz, I. D., Fisher, R. W.,and Buttlaire, D. H. (1977). J. Mugn. Reson. 28,417. Jones, G. P. (1975). In “Magnetic Resonance in Chemistry and Biology” (J. N. Herak and K. J. Adamic, eds.), p. 171. Dekker, New York. Knispel, R. R., and Pintar, M. M. (1975). Chem. Phys. L e t f .32, 328. Knispel, R. R., Thompson, R. T., and Pintar, M. M. (1974). J. Mugn.Reson. 14,44. Koenig, S. H., and Schillinger, W. E. (1969). J. Biol. Chem. 244, 3283. Krynicki, K. (1966). Physica 32, 167. Kumar, A., Welti, D., and Ernst, R. R. (1975). J. Magn. Reson. 18,69. Lauterbur, P. C. (1973). Nature (London) 242, 190. Lauterbur, P. C. (1978). In “N. M. R. in Biology” (R. A. Dwek et al., eds.), p. 323. Academic Press, New York. Lindstrom, T. R., and Koenig, S. H. (1974). J. Magn. Reson. 15, 344. McConnel, H. M. (1958). J. Chem. Phys. 28,430. Mansfield, P. (1976). Contemp. Phys. 17,553. Mansfield, P., and Grannel, P. K. (1973). J. Phys. C [I] 6, L422. Mansfield, P., and Maudsley, A. A. (1976). J. Phys. C [I] 9, L409. Mansfield, P., and Maudsley, A. A. (1977). Br. J. Radiol. 50, 188. Mansfield, P., and Pykett, 1. L. (1978). J . Mugn.Reson. 29, 355. Outhred, R. K., and George, E. P. (1973). Biophys. J. 13,83. Packard, M., and Varian, R.(1954). Phys. Rev. [2] 93,941. Packer, K. J. (1977). Philos. Trans. R . SOC.London, Ser. B278, 59. Pfeifer, H., and Michel, D. (1964). NMR Relax. Solih, Proc. Colloq. Ampere, 1969 p. 414. Pullan, B. R. (1975). Phys. BUN.447. Pykett, I. L., and Mansfield, P. (1978). Phys. Med. Biol. 28,961-967. Solomon, I. (1955). Phys. Rev. 99, 559. Swartz, H. M. (1978). J. Magn. Reson. 29, 393. Torrey, H. C. (1949). Phys. Rev. [2] 76, 1059. Weisman, I. D., Bennett, L. H., Maxwell, L. R., Sr., Woods, M. M.. and Burk, D. (1972). Science 178, 1288. Zaner, K. S., and Damadian, R. (1977). Physiol. Chem. Phys. 9,473. Zimmerman, J. R., and Brittin, W. E. (1957). J . Chem. Phys. 61, 1328. Zipp, A,, Kuntz, I. D., and James, T. L. (1976). J. Mugn.Reson. 24,411. Zipp, A., Kuntz, I. D., and James, T. L. (1977). Arch. Biochem. Biophys. 178,435. NOTEADDEDIN PROOF(concerning Section VI,D, pp. 125-126) D. I. Hoult (J.Mugn.Res. 33, 183-197, 1979) has recently proposed a new method of obtaining NMR images, which retains the inherent sensitivity of the two-dimensional Fourier transform while obviating the need for any changes of the field gradient. The conditions applied in the laboratory frame (homogeneous field plus field gradient) are applied in the rotating frame. A general discussion of sensitivity and performance time in NMR imaging was recently published by P. Brunner and R. Ernst J. Mugn.Res. 33,83-106, 1979).

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL. 49

Applications of the Glauber and Eikonal Approximations to Atomic Collisions F. T. CHAN

AND

M. LIEBER

University of Arkansas, Fayetteville, Arkansas

G. FOSTER* Joint Institute for Laboratory Astrophysics University of Colorado and National Bureau of Standards Boulder, Colorado

AND

W. WILLIAMSON, JR. University of Toledo, Toledo, Ohio

I. Introduction ........................................................... A. Qualitative Description and the Importance of Atomic Collision Phenomena B. Previous Review Papers on Glauber and Eikonal Approximations Applied to Atomic Collision Problems .............................. C. Purposes of this Review 11. Eikonal Approximations fo e s . . .....................

.

134 134

137

B. Electron-Helium Scattering ........................ C. Electron-Lithium Scattering IV. Ionization of Neutral Atoms by Electron Collisions. A. General Discussion ................................

174

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

199

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

204

C. Comparison with Experimental Data and Evaluation

183

* Present address : University of Connecticut at Torrington, Torrington, Connecticut 06790. 133

Copyright 0 1979 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-014649-5

134

F. T. CHAN ET AL.

VI. Other Eikonal-Type Approximations ................................ A. The Blankenbecler-Goldberger Approximation .......................... B. The Glauber-Angle Approximation .................................... ............ C. The Two-Potential Eikonal Approximation D. The Modified-Glauber Approximation of Birman and Rosendorff VII. Applications of the Glauber Approximation to Electron-Molecule Col VIII. Summary and Conclusions .................... References ...............................................

205 205 206 209 213 216 217 219

I. INTRODUCTION

A . Qualitative Description and the Importance of Atomic Collision Phenomena

The determination of cross sections for charged-particle-impact excitation and ionization of atoms and for excitation of ions has been a problem of long-standing interest to both theoreticians and experimentalists. The experimental difficulties involved in accurately measuring cross sections for many atoms and excitation processes of interest (I-.?), in addition to the astrophysical significance of excitation cross sections, have spurred much effort to develop reliable quantum theoretical techniques for cross-section calculations. The problem of determining atomic excitation cross sections is not only of interest in its own right, but the resulting cross sections will be useful to the astrophysicist in understanding the atomic reactions that occur in gaseous pebulae and stellar atmospheres. In particular (4, quantitative information on the cross sections associated with excitation and ionization by electrons is necessary for an understanding of the observed spectral characteristics of aurorae; such information is also useful in connection with investigations of the solar corona. Several authors (5,6) discuss the importance of having accurate transition probabilities and collisional cross sections for the proper interpretation of the properties of nebulae. Cross sections for various excitation and ionization processes are also useful in the diagnostics of laboratory plasmas. As early as 1912, attempts were ma& to compute ionization cross sections classically (7). Some of the earliest work in quantum mechanics was also concerned with the problem of atomic excitation calculations. The papers of Born (8) and Oppenheimer (9) are particularly noteworthy. In fact, the Born approximation has been used extensively over the years to calculate atomic excitation cross sections at high energies. However, the Born approximation generally fails at low (near threshold) and intermediate energies; hence there has been much work in the recent past devoted to the development of low- and intermediate-energy methods.

THE GLAUBER AND EIKONAL APPROXIMATIONS

135

With the advent of high-speed computers, low-energy techniques, such as the close-coupling approximation, have been applied extensively to the calculation of cross sections at low energies, often with very good results. However, low-energy methods, such as the close-coupling approximation, generally become impractical to use when the incident energy is too high. Hence there has been much effort in recent years to develop techniques applicable in the intermediate-energy range. In particular, eikonal-type methods, such as the Glauber and eikonal approximations discussed in this review, have been widely used in the past eight or ten years to calculate atomic collision cross sections. In classical optics at short wavelengths (10, 11) and in quantum mechanical potential scattering at high energies (12) eikonal approximations give the solution to the wave equation in terms of path integrals along the geometric rays perpendicular to the wave fronts. On this basis, the Glauber and eikonal approximations used in atomic-scattering calculations can be traced back to the nineteenth century work of Hamilton (13) via the analogy between mechanics and geometrical optics (14). In the quantum mechanical domain, the connection between the eikonal approximation and the WKB approximation for potential scattering is well known (15, 16). Applications of the Glauber approximation were originally confined to high-energy nuclear physics (17-23). As Glauber pointed out, the theory is traceable to work by Moliere (24) on the elastic scattering of fast charged particles in various materials. Glauber’s earlier papers are not very transparent, but a reasonably understandable exposition of his approximation was given by Glauber (25) in a series of lectures titled “High Energy Collision Theory” delivered at the 1958 Boulder Summer School. Application of the Glauber (25) and eikonal approximations to atomic collisions is relatively recent (26-31). It began in 1968 with the pioneering work by Franco (31) on the elastic scattering of electrons by atomic hydrogen. Franco recognized that the five-dimensional Glauber amplitude for e- + H (Is) elastic scattering could be reduced without further approximation to a one-dimensional integral that can be evaluated easily on a computer [a method similar to Franco’s original work was independently proposed by Birman and RosendorlT(32)I.Since then, there has beena great deal of interest in applying the Glauber, eikonal, and related intermediate-energy theories to atomic collision problems. B. Previous Review Papers on Glauber and Eikonal Approximations Applied to Atomic Collision Problems

In the eight or ten years since the Glauber and eikonal-type approximations have been utilized extensively for atomic collision calculations, a number of excellent reviews have appeared in the literature. These reviews

136

F. T. CHAN ET AL.

either concentrate primarily on the Glauber and eikonal-type methods themselves, or discuss these methods as part of a broader review dealing with charged-particle atomic-scattering theories in general. One of the earliest reviews on the application of the Glauber method to atomic collisions is the brief review of Gerjuoy (26). A much more comprehensive review of the Glauber approximation is that of Gerjuoy and Thomas (29).The Gerjuoy-Thomas review covers in detail the application of the Glauber approximation to atomic collision calculations (up to July 1973). It reviews the various Glauber-type calculations performed and critically compares them to available experiment in order to assess the utility of the Glauber approximation in the atomic collisions domain. Their paper is noteworthy in that it presents several derivations of the Glauber scattering amplitude, for both potential scattering and composite collisions, and compares the various derivations. Various analytic properties of the Glauber scattering amplitude are also discussed. In a recent article by Joachain and Quigg (33),the problem of quantum collisions involving several-particle systems (such as electron-atom and electron-ion systems) is reviewed within the framework of multiple-scattering theory. The eikonal and Glauber methods are discussed here, as are other eikonal-type methods (such as the eikonal-Born series method) which have been formulated as an improvement on the Glauber approximation. In a recent very comprehensive survey Byron and Joachain (30) discuss recent developments in the application of eikonal methods to the field of electron-atom and positron-atom collisions. Their review first analyzes the foundation of the eikonal approximation within the framework of potential scattering; the theory is then generalized to atomic collision processes. The review next discusses various many-body applications of the eikonal method including the Glauber approximation, the eikonalBorn series method, and optical model theories, among others. Finally, numerous applications of the different methods are considered (including elastic scattering and various inelastic processes), and the results of many recent calculations are compared with experiment and critically discussed. Another very recent review is that of Bransden and McDowell (34). They cover in some detail most theoretical models used in the calculation of atomic collision processes, particularly at intermediate energies. The article discusses the close-coupling approximation and its variants, optical potential models, extensions of the Born approximation, various distorted wave methods, many-body theories, and in particular, the Glauber approximation and its variants. The scattering of electrons by atoms and molecules is the subject of another very recent survey by Burke and Williams (35).This article discusses most of the theoretical approaches to the electron-atom scattering problem,

THE GLAUBER AND EIKONAL APPROXIMATIONS

137

but generally in somewhat less detail than, for example, the BransdenMcDowell review. The excitation of positive ions by electron impact is covered in a recent article by Seaton (36). Although his review was completed before any Glauber electron-ion scattering calculations were reported in the literature, it is noteworthy because of its detailed coverage of other theoretical approaches to the problem of electron-ion scattering, as well as its review of experimental results. C . Purposes of this Review

In this review, we discuss the application of the Glauber and eikonal approximations to atomic collision problems. We concentrate on the conventional Glauber (also known as the restricted Glauber) and the straightline eikonal (unrestricted Glauber) approximations; however, a discussion of other eikonal methods is also included. In this paper we develop the theory leading to the various Glauberand eikonal-type scattering amplitudes, and then review various applications of these Glauber-eikonal methods to important atomic collision problems. 11. EIKONAL APPROXIMATIONS FOR THE SCATTERING AMPLITUDES A . General Formulation

In this section we give a brief derivation of the eikonal approximation beginning with the Lippmann-Schwinger equation (37). Although the derivation may be extended to multiple-electron atoms, we will consider a system that consists of a hydrogen atom as the target and an electron as the projectile. The initial state of the system, corresponding to t - t - co, is described by the equation Hil4a) = Eil4a)

(2.1)

where the subscript a describes the state of the system. The scattering state that develops out of the free state 14a) is + I): and satisfies the equation, HI+:)

=

&I+:)

(2.2)

where H = Hi + F and 6 is the potential operator that describes the interaction between the incident particle and the target system in the initial state. The formal solution to Eq. (2.2) is known as the Lippmann-Schwinger equation and is given by

I+:)

=

I4a)

+ (Ea - Hi + i&)-'F[+:)*

(2.3)

F. T. CHAN ET AL.

138

An equivalent formulation of the scattering problem is given in terms of the state ($I; which develops into the final free state ( 4 b l , where the subscript denotes the final state of the system. The state vectors ( 4 b l and ($;[ satisfy the equations =

(4bIHf

(4blEf

(2.4) ($,IH = ($bI(Hf + 6 ) = ( $ , I E b where 6 is potential operator that describes the interaction in the final state between the projectile and target system. The Lippmann-Schwinger equation for ($;I is

($bl

=

(4bl

+ ($;16(Eb

- Hf +

(2.5)

iE)-l

The Y-matrix element (on the energy shell) for the process is given by

( q ~ p )= (bp-la)

- (P(b)lqa)

where (bl and la) denote the properly normalized and symmetrized state vectors describing the system. P(b) is the permutation operator that acts on the target particle (subscripted by a zero) and the identical bound particle. The direct Y-matrix element is

(blYla) = ( 4 b l vd + &(E and the exchange $-matrix element is

+ iE)-’

&/+a>

(2.6a)

(P(b)191a) = ( + P ( b ) l & (Or h) + h(E- + j E ) - ’ vdI40) (2*6b) In the above, & represents the direct interaction potential operator. If a momentum-conserving 6 function is factored out of the above 9-matrix element, the reduced T-matrix element (on the energy-momenturn shell) can be defined as

(6lYla)

= 6(P, - Pb)(b)T)G)

The direct and exchange reduced matrix elements become ( b ( T ( a )= &: = ( 4 b l &

+ &(E - H + k)-’

&(+a)

(2.7a)

vd (or b) + h(E- H + iE)-’ 6/40) (2.7b) may be rewritten as In the direct channel 6 = 5 and cO (P(b)lTla) =

=

(4P(b)l

=

(4bl

K/$:)

(2.8)

The exchange-matrixelement may be written in two different ways, which can be identified as the “post” and “prior” forms of the element. First we

THE GLAUBER AND EIKONAL APPROXIMATIONS

139

take the overall factor & to operate on the ket 14,) and take the operator 1 + V,(E - H + i~)-’ to operate on the bra (+p(b)l. Using the appropriate symmetrized Lippmann-Schwinger equation corresponding to Eq. (2.5) in Eq. (2.7b), the prior form of the exchange T-matrix element may be expressed as, Thy (Prior) = Tb;; = ($P(b)l&l+cr) (2.9a) Taking the overall factor V, in Eq. (2.7b) and applying it to the bra (+p(b)l, and applying the operator 1 + ( E - H + i E ) - l & to the ket the post form of the exchange T-matrix element can be written Td,” (Post) =

6; = (+P(b)l&l$T)

(2.9b)

At this point it is important to note that it may be shown that &; is identical to T i , and a post-prior discrepancy cannot arise until an approximation is made for ($&)I or I+,‘). For electron-atom scattering, the direct and exchange amplitudes are related to the reduced T-matrix elements in the following way :

A,

= - (2n)2[rn/h2]Q

gb’, = -(2n)Z[rn/hZ]$:

(2.10a) (2.10b)

where the hats over the T’s indicate that the spin dependence has been removed from the reduced T-matrix elements. Before discussing approximations for the scattered state vectors, we will give the appropriate Lippmann-Schwinger equations in the coordinate representation. The completeness and closure relations for the target and projectile systems are r

(2.1 la)

s

j l r , ) dr, (rol = Iko)

a 0

(kol = 1

(2.11b)

Inserting the above into Eq. (2.3) leads to the following, (rorl$T) where

=

(ro, rl4,)

+

s

drb dr’ F(rb, r’)G[(ro, r ; rb, r’)$;(rbr‘)

(2.12a)

F. T. CHAN ET AL.

140

Similarly the coordinate representation for

($;I

is

, l

where G:(. *.) = C

s

(r’rblkN) dk(kNlrr,) Eb - EN - (h2k2/2m)+ ie

(2.12d)

Up to this point all of the above equations describing the scattering of electrons from atoms are exact. In the next subsections we discuss the eikonal and ($p(b)lror), as well as a and Glauber approximations for (r,rl$:) limited number of other approximations.

B. The Eikonal Approximation

The eikonal or unrestricted Glauber approximation may be derived in a number of different ways (38-42). We will derive it by linearizing the Green’s function (43-46). We begin by inserting the following into Eq. (2.12b), (rolko)

exp(ik, * r,) = (zn)3/2 ’

h2kf E, = -+ Ei (target) 2m

G+(r,, r ; rb, r’) = (2.13) If we assume that (2rn/h2)[(EN- Ei)/kf] 4 1 and neglect this term in the denominator, then the sum over the intermediate states of the target system may be done using the closure relation. Redefining the intermediate momentum as Q = k, - ki , equation (2.13) becomes,

-

exp[iki (r, - r;)]

G;(r,,r;r&,r‘) = x 6(r

s

- r’) dQ ex~[iQ.(ro - r6)I] Q2

+ 2 Q * k i- ie

Neglecting the Q2 term in the denominator, and defining the components of r, that are parallel and perpendicular to ki as z, and b,, respectively,

141

THE GLAUBER AND EIKONAL APPROXIMATIONS

we find G:(r,,r;r;,r‘)

1

exp[ik, (r, - rb)] S(b, - bo)O(zo- zb)6(r - r’) 2ki

=

The above linearized Green’s function may now be put into Eq. (2.12a), which becomes

2m h2

x exp( - ikizb)- v(rb, r)+:(rbr)

Now let

+: (ror)

=

(2.14)

4a(ror)F(ro)= [ex~(ikizo)/(2~)~’~] +i(rF(ro)

where F(ro)is determined by the following Volterra integral equation : 2ki

dzb &, 6(bo - b,)

2m h2 v(rb, r)F(rb)

which may easily be solved. Inserting the solution into Eq. (2.14) gives the eikonal approximation to i,b: (Tor):

[

iki

2m h2

:m j

x exp ik, * ro - -

dzb mD0 6(bo - b,) - V(rb, r)]

(2.15)

Combining Eqs.(2.8) and (2.lOa) with the above we find the following for the direct eikonal scattering amplitude :

27r h

dr, exp(iA * r,)

x e x p [ 2ki - l r -oo

s

dr @(r)#+(r)VJr,, r) 2m h2

dzb &, 6(bo - b,) - v(rb, r)

1

(2.16)

where A = ki - k, is the momentum transfer to the target system from the projectile. In the eikonal approximation one normally chooses as the z axis. However, this is not the conventional choice of the z axis in the wide-angle Glauber approximation.

142

F. T. CHAN ET AL.

The derivation for the eikonal approximation to (t&,)lrOr) is analogous to the above (47-50). It is found that,

2m h2

1

dz' db' 6(b - b ) - Vf(rb, r)

x exp[ - ik, * r -

(2.17)

Combining Eqs. (2.9a), (2.9b), and (2.10b) with the above and Eq. (2.15) gives the following for the prior and post forms of the eikonal exchange amplitude :

dz' db' 6(b - b)

(2.18a)

-

ik, ro - ikf r -

dzb db6 6(bo - b,)

(2.18b) Let us recall that the approximations in deriving the direct and exchange eikonal scattering amplitudes for electron-atom scattering problems are (1) EN

- Ei

h2k2/2m

'

and (2) the intermediate momentum transfer is both small in magnitude and nearly parallel to k,, which allows Q2 to be neglected in the integrand of . : G In principle, one can evaluate symmetrized differential cross sections within the context of the eikonal approximation by combining Eqs. (2.16) and (2.18a, b). In practice the amplitudes must be reduced to simpler analytic forms before any numerical work can be accomplished. C . The Glauber Approximation The small-angle Glauber approximation (GA) may be written down

143

THE GLAUBER AND EIKONAL APPROXIMATIONS

immediately using Eq. (2.16) if the longitudinal component of the momentum transfer (AII)is set to zero. The zo integral in this case becomes a perfect differential, and the direct small-angle GA is given by

s

fG(i+f) = (ki/2ni) dboexp(iq.bo)dr~~(r){exp[i~G(bo,r)] - l}4i(r) (2.19a) where

xG(bo, r) = - k;

rm

(2.19b)

dzb mD0 6(bo - bo)[m/h2]v(rb, r)

and q is the component of the momentum transfer that is perpendicular to the axis. However, as it is currently used, the formal wide-angle GA does not require that the longitudinal component of the momentum transfer be zero. The i axis is simply chosen to be perpendicular to the momentum transfer. Thus the derivation of the wide-angle GA does contain a swindle. A comprehensive discussion about the choice of the i axis for the direct Glauber amplitude is given by Gerjuoy and Thomas and will not be restated here (29).Unless otherwise specified, we will always assume that the GA is calculated from Eqs. (2.19a, b) with q lying in the scattering plane and perpendicular to the 2 axis; i.e., 2 is not chosen along k,. Equations (2.18a, b) may be used for a small-angle Glauber exchange amplitude if All is set to zero; however, easier analytic reductions are possible within the Glauber-Ochkur (GO) context and have been given for simple systems (47-49, 51, 52). If we consider electron-atom scattering where

+ (e2/lr - rol)

(2.20a)

WO, r) = - (e2/r) + (e2/lr - roll

(2.20b)

W O , r) =

-(e2/ro)

and

then, to the lowest order of k;', if the k, or kf dependence in glected, the GO amplitudes become g20(i+f)=

--

kz

&,

is ne-

[ h2 f -

dr 4:(r) exp(iA r) 4i(r)exp(ix2,) (2.21a)

In the above,

(2.21c)

144

F. T. CHAN ET AL.

and

(2.2 1e) The post-prior GO amplitudes may be written in a condensed notation as

e2m

x exp(iA r) 4i(r) - ( I

hZ

T .zPiq+

(2.22a)

q.z=o

with

q+ = (ezm/hz)k;'

(2.22b)

q- = (e2m/h2)k;'

(2.22c)

Two things should be noted about Eq. (2.22a) : (1) One cannot combine this with the direct Glauber amplitude to calculate symmetrized amplitudes unless the indeterminate phase factor is set equal to unity. (2) The consistency of the approximation in inverse powers of ki is questionable. (This is most easily seen by expanding the accumulated phase in a power series.) We will elaborate on these points later in the review. The exact range of validity of the Glauber approximation is difficult to assess, but it is essentially the same as the eikonal approximation with the possible extension to wider scattering angles because of the choice of the z axis. D . The Inclusion of Second-Order Corrections

The eikonal and Glauber approximations have been reasonably successful in predicting intermediate-energy theoretical differential cross sections for electron-atom scattering processes within limited angular ranges. Both approximations have difficulty in accurately predicting large-angle scattering (say 8 2 207, and for some processes they give divergences in the forward direction. Attempts to remedy these deficiencies have followed three closely associated but different trends of thought. The main object of these approximations is to derive an expression for the scattering amplitude that

145

THE GLAUBF!R AND EIKONAL APPROXIMATIONS

is correct to order k;’. All of them are related to the second Born approximation. Therefore, a short discussion of the Born and eikonal series will be given before a summary of each of the second-order approximations is presented. If the Lippmann-Schwinger equation (2.3) is iterated and substituted into Eq. (2.10a), the resulting Born series for the scattering amplitude is (37)

f = -(2n)2

3

+ (4blvGi+ v l 4 a ) +

[(4bl v l 4 a )

-I- (4bIvGi+v*’*G’KI4,)

4-

**.I

* ’

’

(2.23)

Let (2.24a) where J B ~=

-(2.)Z(m/hz)(4bl

vGi+

’

*

G:

514,)

(2.24b)

is called the nth Born term in the Born series. appears n times, while G‘ appears (n - 1) times in fBn. The nth Born approximation is given by terminating the Born series with the nth Born term fBn

n

fBn =

(2.25)

3Bj j= 1

The second Born approximation becomes fBZ = fB1

+ 3BZ

(2.26a)

where 3B1 =

-(2.)Z(m/h2)(4blv14a)

(2.26b)

and fBZ = -(2n)2(m/hz)(4blvG:

vl4a)

Using the completeness relations given by Eqs. (2.11),

X

(4b(rOr), &(rOr)

4kN(rOr))<4kN(rbr’),

kZ - k:

(2.26~) fBz

may be written

v(rbr’)+t~(~b~’)) (2.27)

-!-(2m/h2)(EN - Ei) - i&

The contribution of a finite number of intermediate target states to

fB2

146

F. T. CHAN ET AL.

may be included using the following approximation : - kz

-kz

+ (2m/h2)(EN - Ei) = -k i , + (2m/h2)(EN- Ei) = -(kz - 26,) = - P 2 ,

N s n N 2n

+1

(2.28a)

N 2n

+1

(2.28b)

where

6i = (m/h2)(EN

-

Ei)av,

Using the completeness relation

and the above approximate expression in jB2, this term becomes

(2.29a) where the simplified Born approximation (j&) is

An eikonal, or Glauber, series may also be defined by using Eq. (2.19b) in Eq. (2.19a) and expanding the Glauber phase in a Taylor series. The Glauber, or eikonal, series is defined as follows:

c a l

fG =

j= 1

fGj

(2.30a)

THE GLAUBW AND EIKONAL APPROXIMATIONS

147

and the nth Glauber, or eikonal, approximation becomes n

fGn

=

c

j= 1

(2.30~)

fGj

The second-order approximations we present below consist of judicious choices for fB2 and an approximation for the exchange amplitude. 1. The Eikonal-Born Series (EBS)

The EBS is essentially the following approximation (53-55) : fEBS

=fBl

gEBS

= 90

+ f B Z + RefGf,,

(2.31a) (2.31b)

where go is the ordinary Ochkur exchange amplitude :

[

-

ezm d r 4:(r) exp(iA r) 4i(r) kz hZ

go=---

(2.32)

2. The Modified-Glauber Approximation (MG) In the MG approximation the scattering amplitude is approximated as (5657) fMG = f G

gMG

- fGZ

= gEIA.=O

+ fBZ (Or

(Or

90)

&BZ)

(2.33a) (2.33b)

3. The Fixed-Scatterer Approximation (FS) In the fixed-scatterer approximation (58) fFS

= fB1

+ fFSZ

(2.34a)

where (2.34b) We refrain from further discussions of the above approximations until later sections where they are compared with experimental data for specific scattering processes.

148

F. T. CHAN ET AL.

111. ELECTRON SCATTERING FROM NEUTRAL ATOMS

Since the pioneering work of Franco (31), a number of atomic theorists have spent much time and labor in reducing the multiple-dimensional Glauber and eikonal amplitudes to tractable form.

A . Electron- Hydrogen Scattering 1. Glauber Theory

+

a. e H (Is) scattering. The Glauber direct scattering amplitude (1s + 1s);q) describing the elastic scattering of an electron with velocity ui (momentum hki) by a ground-state hydrogen atom is given by (25,29, (31, 32) [see also Eq. (2.19a)l fG

where &,s

=

e-./&

is the hydrogen ground-state wave function, T ( b ; r )= 1 - (Ib - sl/b)2iq

(3.2)

and 0 = l/ui = l/ki (in atomic units). In Eqs. (3.1) and (3.2), b and s are the respective projections of the position vectors of the incident electron and the bound electron onto the plane perpendicular to the Glauber path. The Glauber amplitude is evaluated by taking the Glauber path integral along the direction perpendicular to the momentum transfer q. We note that the amplitude fG(ls + 1s;q) of Eq. (3.1) can be written in terms of a generating function; in particular, 1s; q) = 21%A, 4)11=2

fG(lS

where we define

s

iki q) = 7exp( -Ar)r(b; r) exp(iq b) d2b dr (24

(3.3)

(3.4)

Franco (31) then showed, by manipulations in cylindrical coordinates, that the five-dimensional integral (3.4) reduces without further approximation to the following one-dimensional integral :

THE GLAUBER AND EIKONAL APPROXIMATIONS

+

x [I - (cos B')-2iqlcos28'12iq+'2~1(9 iiq, 1

149

+ i$q; 1 ; sin2 2@)]

(3.5) In Eq. (3.9, ZF1is the usual hypergeometric function. The differential and integrated (over scattering angle) cross sections can be obtained from the elastic amplitude (3.3) in the usual way. Using Franco's reduction method, results for inelastic excitation to the 2s, 2p, 3s, 3p states (59,60)and to the 3d state (61)by electron impact have been reported. In an important publication (62), Thomas and Gerjuoy approach the amplitude integral in a slightly novel way by introducing another generating function with an extra factor of r - l . They write

where T-G I0

iki (24

L1 d = 7

sr

exp(-Ar)

T(b; r) exp(iq b) d2b dr

(3.7)

(Henceforth the upper index T-G will be dropped.) With considerable manipulations of special functions, including derivation of a new integral representation involving Bessel functions ;

ii I

rzn

Jo

d q exp(irnq)(l

+ s2 - 2s cos

Thomas and Gerjuoy are able to reduce the generating function Eq. (3.7) to closed-form hypergeometric functions :

After some manipulation (63),one can reduce I& q) to a form suitable for the high-incident-energy limit (q 0) : --+

I O U , 4)

= 134 4) + i Y ~ l ( k4 ) + O(V2)

(3.10)

150

F. T. CHAN ET AL.

where (3.1 1)

and (3.12)

In the limit of q +0, substituting Eq. (3.11) into Eq. (3.6), the Glauber direct scattering amplitude is found to be identical with the Born direct scattering amplitude. In fact, the generating function Eq. (3.9) can account for the class of transitions ns + n's including elastic and inelastic collisions. A simple selection rule for arbitrary transition amplitudes is also derived by Thomas and Gerjuoy (62):

fG(nlm+ n'l'm'; q) -= 0 when 1 + 1'

(3.13)

+ Iml + lm'l = odd integer. Hence f&s

+

n'p, m' = 0; q) = 0

(3.14)

The nonzero components of the Glauber amplitudes for the class of transitions ns n'p, fG(ns + n'p, rn' = -t 1 ;q) can be written in terms of another generating function (62), --+

ik,

s

e-Ar

7T(b; r) exp(iq

4) = 7

(24

b)s d2b dr

2 - iq, 1 - iq; 1 ; -

- (1

+ q 2 ) 2 ~ 1 ( 2- iq, 1 - i q ; 2 ; - q2

A2)]

(3.15)

The Thomas-Gerjuoy reduction method has been extended to obtain closed-form expressions for scattering amplitudes for 1s 3d by Chan and Chang (64), for ls+nlm by Thomas and Franco (65), and for arbitrary nlm + n'l'm' by Toshima (66); we do not quote their results here. For transitions involving low-lying states, these expressions containing a simple sum of hypergeometric functions are very tractable in computations and are particularly suitable for studying the asymptotic behavior as k, + co --+

THE GLAUBFX AND EIKONAL APPROXIMATIONS

151

and q + 0. However, for transitions involving high-lying initial n and/or final n’ states, the Thomas-Gerjuoy method leads to expressions containing a very large number of hypergeometric functions (due to many repeated parametric differentiations) and hence is not really useful. Furthermore, by exploiting the known relations between the generalized Legendre functions P , (v is a complex number) and the hypergeometric functions, Thomas and Gerjuoy (67) have shown that the closed-form expressions for the Glauber amplitudes can be integrated analytically over scattering angle to give closed-form expressions for the integrated cross sections. In particular, the aG(ls+ 1s; k,) can be expressed as a comparatively simple s u m of products of P, (or z F , ) which are as easy to compute as the amplitude itself. [Details are in Eq. (25) of Thomas and Gerjuoy (691. For k,-+ coy Thomas and Gerjuoy obtain a simple expression for total cross section : aG(ls 1s; ki) z $(z/kT) --+

(3.16)

Thomas and Gerjuoy have further found that the Glauber predicted aG(ls 1s; k,) is well represented by the very simple Eq. (3.16), within a few percent of the exact results, for all incident energies. This is a very surprising result, which is not true in the Born and other approximations. Although similar closed-form expressions for integrated inelastic e- + H (1s) cross section are derivable, the detailed analysis is very involved [even for H (2s)I. No other calculations following Thomas-Gerjuoy reelastic eduction method have been reported. It is well known that at intermediate and lower incident energies (especially when the velocities of incident and bound electrons are about the same), exchange effects in electron-atom collisions become important. In Section 11-C we have pointed out that a general expression for a Glauber exchange amplitude has not yet been given in the literature, but approximations within the Bonham-Ochkur context (referred to as the GlauberBonham-Ochkur approximation) have been proposed by a number of people (48,49,51,52,68). The Bonham-Ochkur method (69,70) was originally proposed as a means of simplifying the Born-Oppenheimer exchange amplitudes for scattering problems (such as electron-hydrogen scattering). Tenney and Yates (68)appear to have been the first to apply the Bonham-Ochkur method to the Glauber exchange amplitudes (for electron-hydrogen scattering). However, as Madan (47) pointed out, the calculation of Tenney and Yates was incorrect. The correct post and prior (closed-form) expressions for the Glauber-Bonham-Ochkur exchange amplitudes for electron-hydrogen scattering were first given by Madan (48), and later (independently) by Dewangen (49) and Khayrallah (51). Khayrallah’s work is noteworthy in --+

+

152

F. T. CHAN ET AL.

that it includes a derivation of the closed-form expression for the total exchange cross section, in addition to extensive numerical calculations. The Bonham-Ochkur method essentially consists of expanding the exchange amplitude in inverse powers of k, or kf (k,and kf are the initial and final momenta, respectively, of the scattered electron) and retaining only the leading term in the expansion. A detailed discussion of the method can be found in the work of Bonham (69).For electron-hydrogen scattering, the Glauber-Bonham-Ochkur (GBO) exchange amplitude is given by Madan (47,429, Dewangen (49),and Khayrallah (51).An explicit expression for the GBO exchange amplitude is given in Section II-C of this review. As mentioned above, for electron-hydrogen scattering the GBO exchange amplitude can be reduced to a closed-form expression (48,49,51):

x {[p2

+ (q + y12]i**-1[p T i(q, + y,)]-i**}

(3.17)

In the above expression, +/- indicates post/prior, and D,, is a differential operator that generates the bound-state target wave function product as follows : (3.18)

Up to the present time, the GBO exchange amplitude has only been used to correct for exchange in electron-hydrogen elastic scattering (47,51,63). b. e- + H (excited states) scattering. Although study of electron scattering by atoms initially prepared in an excited state, in particular, the longlived metastable states, has important applications in astrophysics, plasma physics, and various gaseous phenomena, relatively little work has been done even on the scattering from excited states of simple atoms. Several theoretical calculations have been performed to study the elastic (63,71) and inelastic (72, 73) scattering of electrons by metastable hydrogen. The Glauber direct scattering amplitude fG(2s + 2 s ; q) describing the elastic scattering of an electron with velocity ui by a metastable hydrogen atom can be expressed in a closed and compact form in terms of the generating function Eq. (3.9)(63):

The Glauber-Bonham-Ochkur exchange amplitude for e-

+ H (2s)

THE GLAUBER AND EIKONAL APPROXIMATIONS

153

elastic scattering is given by (48,49,51,63)

where g(n,4) =

[PyP + 42)1 -iv]

-

(3.21)

If we let q +0 in Eqs. (3.20) and (3.21), the Glauber-Bonham-Ochkur exchange amplitude is immediately recognized to be the Born-BonhamOchkur exchange amplitude. The direct, exchange, and symmetrized differential and total cross sections are found from the direct and exchange scattering amplitudes in the usual way. It is found (63) that the Born and Glauber integrated elasticscattering cross section (ignoring exchange effects that are small) is well represented by oBVG(2s + 2s ;ki) = (4162/105)(~/k?)

(3.22)

for Ei 2 20 eV. It is remarkable that oG(2s+ 2s ;ki) - 4162/105 x 17 oG(ls+ 1s; ki) - 7/3

(3.23)

is close to 16, the ratio between 2s and 1s orbital geometric cross sections. In a recent paper, Chan, Chang, Lieber, and Kim (73) have calculated the differential cross section for the process e- + H(2s)+e- + H(3p) using the Glauber approximation, and compared several features of the result with the Born approximation result and, qualitatively, with experiment. According to the prescription outlined above, the Glauber amplitude for 2s + 3p, rn' = 1 is

(3.24) where Z, is given by Eq. (3.15), assuming q Iki, and the quantization axis is along ki. The amplitude for the m' = - 1 substate differs by an unimportant phase factor while the rn' = 0 amplitude vanishes according to (3.14). The differential cross section is thus

F. T. CHAN ET AL.

154

but the resulting expression, while easily evaluated on a computer, is too cumbersome to be written here. The Born approximation can again be recovered by taking the limit q 0 in Eq. (3.15); i.e., we insert --+

Z:(A,

q) = lim ZJA, q) = - 8/q(A2

+ q2)2

(3.26)

$-+O

into Eq. (3.24), in place of I,. The simple result is k, 21° (q4 - %A2q2 (42

+

+ *A4)’l

A2)l0

(3.27) A=5/6

In comparing results for inelastic processes it is conventional to use the generalized oscillator strength (GOS) (74) in place of the differential cross section. The GOS is defined by

k. q2 do q2 lf(2s +3p, m’)I2 G = (AE) 2 -- = (AE) k, 2 d R 2 m’

(3.28)

where AE is the energy difference between the initial and final atomic energy states. The GOS reduces to the optical oscillator strength in the limit q + O . In the Born approximation the GOS is independent of the incident energy and depends only on the momentum transfer q. This permits a ready determination of the validity of the Born approximation. In the case of the process in question here, e- + H(2s)+e- + H(3p), we find for the GOS G(2s + 3p), = 2

AE 4

I(3pm’l exdiq r)12s)I2 m’

where AE = +($ - 6) = for the 2s --+ 3p transition. Chan et al. then make the following observations: In the Born approximation G(2s + 3p)Bhas two minima occurring at the roots of the numerator of Eq. (3.27a), i.e., at q2 = 0.139 and q2 = 1.16; furthermore G(2s 3p)B vanishes at these two minima. These zero minima, which arise from the coincident vanishing of the amplitudes contributing to the summation in Eq. (3.28), are fixed, independently of q, in the Born approximation. While experimental data for comparison are not available for hydrogen, the presence of minima in the GOS has been inferred in experiments on resonance transitions in mercury and rare gases (75). These experiments indicate that, in contrast with the Born approximation predictions, the minima do not occur at fixed q2, but rather vary with the incident energy, shifting to smaller --+

THE GLAUBW AND EIKONAL APPROXIMATIONS

155

q2 values as q is increased. Furthermore the value of the cross section at the minimum is not zero but also depends on the incident energy, increasing as q increases. The Glauber result is in qualitative agreement with this behavior but exhibits some unexpected features. At q = 0, G(2s + 3p)G = G(2s --t 3pb; so there are two zero minima. As q increases, the first minimum occurs at progressively smaller q2 values, and the value of GO changes. At q approximately 0.2, a third minimum appears at q2 z 20. The second minimum disappears shortly thereafter, and the third minimum disappears soon afterward. From q z 0.3 to q z 1.2 there is only one minimum. At q z 1.2 a second minimum reappears and persists to threshold at q = 2.68. See Fig. 1. Unfortunately the experimental difficulty in verifying this complex behavior appears enormous. The authors of Chan et al. (73) suggest that the deficiencies of the Born approximation pointed out above arise from the vanishing of the nuclear charge contribution because of the orthogonality of the initial and final atomic wave functions. Indeed, a calculation by Bonham (76) of a portion

FIG.1 . Trajectory of the minima and maxima in the generalized oscillator strengths as a function of the momentum transfer (in a.u.) K and q = k-' where k is the incident electron momentum. The outermost curve marked and Kz,, are the lower and upper limits of the momentum transfer. Note that for a narrow range of q values, near 0.22, there are three pairs of extrema, while for 0.3 < 1 < 1.2 there is only one pair. From Chan et al. (73).

c,"

F. T. CHAN ET AL.

156

of the second Born approximation (SBA), including a nuclear contribution, indicates similar improved behavior, namely, nonzero minima moving in the correct manner with incident energy. Other possible mechanisms are also discussed. Finally, Chan et al. (73) study the behavior of the amplitudes for large q at fixed but large incident energy (fixed q , but with q sufficiently small so that the first two terms in a power series in q may be retained). They find that as q + co, the behavior is fG A/q7 + Bq/q3. The first term is the same as the Born approximation prediction and agrees with the general results of Rau and Fano (77). The second term will clearly dominate the first when q # 0. The fact that the Born approximation predicts too rapid a decrease with q had been noted earlier (78). However, second Born approximation calculations suggest the dominant behavior should be still slower : B q / q 2 . Gau and Macek (79), using an “unconstrained Glauber” approximation, in which the requirement q Ikiis dropped, also deduce a B’q/q2 behavior, which arises from the m’ = 0 substate (whose amplitude is nonzero in their method). It should be noted that for q # 0, q2+co is an unphysical limit, since q2 varies only between the finite limits (ki - kf)2 and (ki + k,)’ as the scattering angle varies from 0 to n. c. Polarization fractions and collision parameters ( I , x). The Glauber scattering amplitudes f@( 1s + nl, m,;q) described in the previous sections for electron impact excitation of hydrogen atoms, using the direction perpendicular to q as the quantization z axis, have concentrated on the predicted differential and total cross sections [the superscript (c) represents the q-dependent coordinate system cB(F)]. However, for calculating the polarization fraction and the parameters (1,~) it is necessary to have the Glauber amplitudes 1s nl, m,;q) calculated in the coordinate system (29, 80) quantized along the direction kiof the incident electron [which we denote by C(ki)].The connection between these two sets of Glauber amplitudes is found by the following transformation (64,80): --+

fa(

f&ls

+

--+

nl, m,;q) =

D!,!:mi(a, /I, y)f$f(ls + nl, mi;q)

(3.29)

mi

In Eq. (3.29), D!,!L‘,. is the usual representation of R,[=R,(a,/I,y)] on the space spanned by eigenvectors of L? with angular momentum number 1. The representation DiL,(a, /I, y) is related to the matrix d!,!!,,,by (81)

D!,!L,(a,p, y ) = eimy&!,,,(/I) eim’a

(3.30)

where a = 4q,/I = Oq - in, and y = - $ q are the Euler angles (80) [O, and 4, are the angular coordinates of q in C(ki)]. Using Eq. (4.1.15) of Edmonds (81)one can easily find the matrix d!,!L,(jl)and hence the f&ls + nl, m,;9).

THE GLAUBER AND EIKONAL APPROXIMATIONS

157

For radiation lines emitted by the hydrogen atom following electron excitation to the nl states, the polarization fraction PdEi) = (111 - Z L ) / ( ~ I I + 11)

(3.31)

according to the theory of Percival and Seaton (82) is related to QnImr(Ei), the total cross section for exciting the hydrogen atom from the ground and I, are the instate to nlrn, sublevels in C(ki) system. In Eq. (3.31), Ill tensities, observed at 90" to the incident-electron-beam direction, of the respective lines having electric vectors parallel and perpendicular to the incident-electron-beam direction. As an example, let us consider e- + H (1s) +e- + H (2p). The ls-2p Glauber amplitudes, evaluated in Cg(& are given by (62) f$j)(ls + 2p, rn, = 0;q) = 0

(3.32)

and

=

fexp(T id,)h,(q)

(3.33)

By using d$!,,,(p) ( d l ) , and Eqs. (3.29), (3.30), (3.32), and (3.33), we obtain fg)(ls-+2p,rn, = 0;q) = JZcos0;h,(q)

(3.34)

and f$(h-P 2p, rn, = & 1;q) = kexp( T id,) sin e4 * h,(q)

(3.35)

The Glauber total cross sections are of course independent of whether the quantization axis is chosen along ki or along an axis perpendicular to q. The parameter A0(2p)is defined via

By Eqs. (3.34) and (3.35) we obtain (3.37) The phase difference x between fg)(ls + 2p, rn, ;q) with rn, = 1 and - 1, vanishes in Glauber theory. Indeed, as pointed out in Eminyan et al. (84, the first Born approximation, or any theory that results in a Am, = 0 selection rule along the momentum transfer direction in the excitation, implies that A0(2p) = cos' O,, and x = 0.

158

F. T. CHAN ET AL.

The polarization fraction P,(E,) for Lyman-cr radiation in e- + H(1s) collisions (80), according to the theory of Percival and Seaton (82), is given by (3.38) In Eq. (3.38), Qpm,(ml = 0 and 1) are the total cross sections for exciting the hydrogen atom from ground state to 2pm, sublevels and are found via Eqs. (3.34) and (3.35) in the usual way. The parameters (A, x), the polarization fractions of 3p - 2s, 3d - 2p, and the Balmer-cr line in e- + H (1s) collisions are defined and calculated by Chan and Chang (64); we do not quote their specific expressions here. 2. Unrestricted Glauber Approximation (or Eikonal Approximation)

As we have seen from previous sections, the equation for Glauber scattering amplitudes is very tractable for simple atoms. However, the Glauber approximation suffers from a number of difficulties (28): (1) The cross sections for e- + atom scattering is the same as that for e+ + atom collisions. (2) In each order of perturbation theory, the Glauber amplitude is either purely real or purely imaginary, alternating from order to order. (3) For inelastic processes, the choice of trajectory at small angles is rather unphysical.

In 1971, Byron (38) suggested that the eikonal approximation (or unrestricted Glauber approximation, i.e., use of the Glauber wave function but without the 2 Iq assumption) might solve all these problems in one shot. If we consider the scattering of electrons from hydrogen atoms within the context of the eikonal approximation, the direct and exchange amplitudes are given, respectively, by Eqs. (2.16), (2.18a), and (2.18b). The appropriate potentials are given by Eqs. (2.20a) and (2.20b). If the phase factors are integrated, six more integrals must be evaluated in order to reduce the appropriate scattering amplitudes to analytic form. No one has been successful in accomplishing this reduction; however, using a series of very clever mathematical techniques, Gau and Macek (79) successfully reduced the direct amplitude down to a two-dimensional integral. Madan (48) and, independently, Foster and Williamson (50) extended the Gau-Macek technique to the exchange amplitudes and reduced them to two-dimensional integrals. The problem is thus reduced to thc evaluation of a two-dimensional integral that can be numerically evaluated.

THE GLAUBER AND EIKONAL APPROXIMATIONS

159

We will not present the entire technique, but merely outline the produre for the reduction ,and quote the final results. It consists of: (1) Writing the bound state wave functions in the form

4*(r) = D*(P*Y

Y*)C* exP(-P*r

+ ir*

r)lr* = o

(3.39)

where the plus sign indicates an initial state and the minus sign a final state. C, is a normalizing constant, and D*(u,,y,) is a differential operator that generates the appropriate angular dependence from the components of 7,, which is then set equal to zero. In the direct amplitude, the product of the wave functions may be written in the form of Eq. (3.39), but for the exchange amplitudes the initial and final states must be independently written in the form of Eq. (3.39). (2) A term that comes from the phase factor is rewritten as an integral representation of the gamma function. (3) The terms that contain (r - r,l are Fourier transformed. (4) The integral over the bound electron coordinate is evaluated. (5) The integral over the Fourier transform variable is evaluated using Feynman parametrization. (6) The integrations over the coordinates of the incident electron are evaluated using parabolic coordinates. The remaining two integrals, which are introduced by the Feynman parametrization, must be numerically integrated. The direct amplitude is given by the following expressions (atomic units) :

x j:dAA-iq-'

jo'

dxx-'[9(1,0,0,0) - F(l,l,O,l)]

(3.40)

where

+ q'2)"'-'"(A - iqL)-iq-r

(3.41)

+ 2ilx(1 - x)yz + p2x + y2x(1 - x)]'/~ q' = A - iA(1 - x).2 + x7

(3.42a)

9 ( m ,p , r, s) = AS(l - x)"A-P(A2 In Eq. (3.41), A and q' are defined by A = [A2(1 - x)2

(3.42b)

160

F. T. CHAN ET AL.

The exchange amplitudes are given by the somewhat more complicated expressions,

(3.43) where

3. Comparison with Experimental Data and Evaluation The elastic and inelastic scattering of electrons by atomic hydrogen is among the most fundamental of atomic collision problems. The electronhydrogen scattering problem is the simplest one; furthermore, the target wave functions are known exactly, so that any error introduced into the calculation is the result of the approximation used to obtain the scattering amplitude which is utilized for the calculation. Hence, the electron-hydrogen scattering problem is a natural first candidate for testing any new intermediate-energy scattering theory. Thus, there have been many recent Glauber/eikonal-type calculations for electron-hydrogen scattering. In this section we compare the results of the various approximations considered in this review with the recent experimental data. However, before comparing and discussing the results for electronhydrogen scattering, we briefly consider positron-hydrogen scattering. Although no experimental results for positron-atomic hydrogen scattering are available as yet, the calculation of positron-hydrogen scattering cross

THE GLAUBER AND EIKONAL APPROXIMATIONS

161

TABLE I e ' + H AND e- + H ELASTIC TOTALCROSS SECTIONS AS CALCULATED IN THE EIKONAL APPROXIMATION

Total cross section (mi) Incident energy (ev)

Positrons

Electrons

50 100 200

0.95

0.44 0.24

0.72 0.39 0.22

sections is of interest. The conventional Glauber approximation predicts identical cross sections for electron or positron scattering, at the same incident energy, from an atomic target; one expects these cross sections to be different on physical grounds (29). Hence, one important test of any theory that attempts to improve on the conventional Glauber method is that it predicts different cross sections for electron and positron scattering. Furthermore, positron-scattering cross sections can be scaled, in many cases, to provide proton-scattering cross sections. A limited number of eikonal calculations for positron-hydrogen scattering cross sections have been done. In Table I above we compare the e+ + H and e- + H elastic total cross sections as calculated in the eikonal approximation (84). As can be seen, the e+ + H and e- + H eikonal cross sections differ, as expected. Furthermore, the eikonal e+ + H and e- + H cross sections tend to each other as the incident energy becomes high; this is also expected, since at very high energies the eikonal results should tend to the Born. We remark here that Byron and Joachain (53)used the eikonal-Born series (EBS) method to calculate e+ + H elastic cross sections at 100 eV over a limited angular range; they also found the e+ + H and e- + H cross sections to differ. In Table I1 we compare the Is+ 2s e+ + H and e- + H total cross sections as calculated in the eikonal approximation (85) with the corresponding e+ + H eikonal Monte-Carlo cross sections of Byron (38). It can be seen here, for the 1s+2s case also, that the e+ + H and e- + H eikonal cross sections differ and tend to each other as the incident energy becomes high. We now consider electron-hydrogen elastic scattering and compare the results of various Glauber and eikonal-type calculations with recent experimental data. Calculations for e- + H scattering using Glauber or eikonal methods have usually been done over an intermediate-energy range of 50-500 eV, with a few calculations being carried out on energies down to about 12 eV or so. We consider here differential cross section-results at 100 eV; this

F. T. W A N ET AL.

162

TABLE I1 e ' + H AND e- + H IS +2s TOTALCROSS SECTIONS As CALCULATBD IN THE EIKONAL APPROXIMATION Total cross section (xu;) Incident energy (ev) 50

100 200

104

Positrons (eikonal MonteCarlo) 0.097 0.053

-

Positrons (eikonal)

Electrons (eikonal)

0.122 0.054 0.041

0.084 0.048 0.037

11 11 1111I0111 1111 20 4 0 0 0 l m 1 2 0 a

FIG.2. Differential cross section du/dR versus scattering angle 0 for the elastic scattering of 100-eV electrons from atomic hydrogen. The eikonal results corrected for exchange (-) are compared with the eikonal results without exchange (----).the Glauber results without exchange (----). the Born results without exchange (------ ), and the recent experimental results of Williams (0).du/& is in units of a;/sr, and 0 is in degrees. From Foster and Williamson (5-0).

THE GLAUBER AND EIKONAL APPROXIMATIONS

163

intermediate energy is common to most of the calculations and is roughly at the midpoint of the intermediate-energy range usually considered in such calculations. The qualitative conclusions drawn at 100 eV are generally true at other intermediate energies also. In Figs. 2 and 3 below, we compare the differential cross sections for e- + H elastic scattering at 100 eV, as calculated using various Glauber and eikonal-type methods and as determined experimentally. Several sets of experimental data are available (86-88); we compare the theories with the recent data of Williams (88), which seem to be the most accurate. In Fig. 2 we present the results of the eikonal approximation with exchange (50) and without exchange (89), the Glauber results without exchange (90), the first Born results without exchange (91), and the data of Williams. In Fig. 3 we present the results of the eikonal-Born series (53), the modified-Glauber results corrected for Glauber exchange (52), the

e

FIG.3. Differential cross section &/&I versus scattering angle 6 for the elastic scattering of 100-eV electrons from atomic hydrogen. The recent experimental results of Williams (0) are compared with the eikonal-Born series results (-), the modified Glauber results corrected for exchange with the full Glauber exchange amplitude (----). and the Glauber results corrected for exchange with the Glauber-Bonham-Ochkur exchange amplitude (------). du/& is in units of at/sr, and tJis in degrees.

164

F. T. CHAN ET AL.

Glauber results corrected for exchange with the Glauber-Bonham-Ochkur exchange amplitude (54, and the data of Williams. Examination of Figs. 2 and 3 leads to the following conclusions. The Glauber and eikonal results are reasonably good at lower angles but fail rather badly at large angles. In fact, for elastic e- + H scattering, even the first Born results are better at large angles than the Glauber or eikonal results. We note here that the Glauber and eikonal methods are expected to fail at large angles because they contain a poor representation of the second Born term, which is known to be important for large-angle scattering (53). Turning now to a comparison of the Glauber and eikonal results with and without exchange, it is seen that the inclusion of exchange effects generally improves the results at the lower intermediate energies (100 eV or less). Exchange effects are expected to be appreciable at lower energies, as discussed previously; hence most second-order Glauber or eikonal methods make allowance for exchange in some way. Examining now the results of the second-order methods, it is seen that the eikonal-Born series and modified-Glauber calculations are a noticeable improvement over the Glauber and eikonal results, especially at large angles. The modified-Glauber results, corrected for Glauber exchange, are seen to be in particularly good agreement with the experimental data. These results show the importance of accounting for the second Born term in any intermediate-energy scattering theory. We note here that the modifiedGlauber calculations were also performed without exchange (56); these results (which we do not quote here), when compared with the exchangecorrected modified-Glauber calculations, show again the importance of exchange at the lower energies. The modified-Glauber results with exchange are a noticeable improvement over the nonexchange modified-Glauber calculations. The calculations for elastic e- + H scattering at 100 eV in the fixedscatterer approximation (58) are not shown as they are nearly identical to the exchange-corrected modified-Glauber results and hence agree very well with experiment (88). We remark here that the fixed-scatterer calculations did not include exchange effects (58). We now turn our attention to the total e + H elastic cross sections. In Table 111, we compare the e- + H elastic total cross sections, as predicted by the various Glauber/eikonal methods considered in this review. We also include the first Born cross sections, in addition to the total cross section at 50 eV inferred from the experimental data of Teubner et al. (86). We note here that no total cross sections were given in the eikonal-Born-series or fixed-scatterer approximations; we would expect the eikonal-Born series and fixed-scatterer total cross sections to be an improvement over both the conventional Glauber and eikonal cross sections.

TABLE I11 e-

+ H ELASTICTOTALCROSSSECTIONS, IN UNITS OF xu;.

Incident energy (eV)

Glaubef

50 100 200

0.64 0.29 0.15

a

Glauber with Glauber exchangeb 0.79 0.38 0.18

AS

PREDICTED BY VARIOUSGLAUBER AND EIKONAL-TYPE &PROXIMATIONS

Born"

Eikonal"

Eikonal with post exchange"

0.51 0.29 0.15

0.72 0.39 0.22

0.84 0.43 0.24

Values quoted from Foster and Williamson (50). See Gien (52). See Khayrallah (51). See Gerjuoy and Thomas (29) and Teubner et ul. (86).

Eikonal with prior exchange" 0.94 0.44 -

Modified Glaube?

Modified Glauber with exchangeb

Glauber with Glauber-Ochkur exchange'

1.05 0.43 0.19

1.26 0.53 0.21

0.83 0.37 0.17

Experimentald 1.20

-

166

F. T. CHAN ET AL.

Examination of Table I11 shows that the total cross sections are generally improved by the inclusion of exchange, regardless of which Glauber or eikonalLtype theory is used. This is just what one expects. It is also evident from examination of Table I11 that the inclusion of second-order corrections (by taking account of the second Born term) to the Glauber or eikonal amplitudes also improves the cross sections. We note here that the exchangecorrected modified-Glauber method seems to agree particularly well with the inferred experimental total cross section (at 50 eV). We now consider excitation of atomic hydrogen to the n = 2 level by electron impact and compare the results of the various Glauber and eikonaltype calculations with recent experimental data. Again, we consider here the differential cross section results at 100 eV; similar conclusions can be drawn at other intermediate energies. In Fig. 4 below we compare the differential cross section for e- + H

0 lkll FIG. 4. Differential cross sections for ls+2s excitation of atomic hydrogen by electron

impact at 100 eV. The differential cross sections have been multiplied by sin0 for convenience. the exchange-correct eikonal result, . . . ., the eikonal Born series result, ----, the first Born approximation, ----, the Glauber approximation, .-.- , the eikonal approximation without exchange; and --, the Glauber approximation corrected with Glauber exchange.

-,

THE GLAUBER AND EIKONAL APPROXIMATIONS

167

1s -+ 2s scattering at 100 eV as calculated using various Glauber and eikonal-type methods. We show the results of the exchange-corrected eikonal approximation (92),the eikonal results without exchange (93,the Glauber results without exchange (94, 99,the eikonal-Born series results (96),and the first Born results (96). Examination of Fig. 4 shows that all of the various theories predict similar small-angle differential cross sections. However, at large angles, large differences can be seen. In particular, the large-angle Glauber and eikonal results are seen to be much lower than those of the second-order eikonal-Born series method. This was also observed in the elastic scattering case and again points out the importance of accounting for the second Born term. The first Born results are seen to fail badly at large angles. Looking at the Glauber and eikonal results with and without exchange, it can be seen that the inclusion of exchange increases the intermediate-angle cross sections; hence the inclusion of exchange tends to improve the results.

it

e

FIG.5 . Differential cross sections for Is -+(2s + 2p) excitation of atomic hydrogen by electron impact at 100 eV: the eikonal result without exchange; ----, the Born approximation; ....., the eikonal-Born-series result, ----, the Glauber result without exchange, .-.- ,the Glauber result with exchange; --, the modified Glauber result with exchange; 0 ,the experimental data points, h / d R is in units of ai/sr and 0 is in degrees.

F. T. CHAN ET AL.

168

We next direct our attention to the e- + H 1s +(2s + 2p) differential cross sections at 100 eV. In Fig. 5 we present the recent experimental data of Williams and Willis (97),along with the eikonal results without exchange (94,the Born results (98),the eikonal-Born series results (96),the Glauber results without exchange (98),the Glauber results with exchange (98),and the modified-Glauber results with exchange (98). Examination of Fig. 5 shows that the Glauber, eikonal, and Born methods all fail badly at large angles with the Born results being particularly poor. On the other hand, the modified-Glauber and eikonal-Born series results are in quite good agreement with experiment. This again stresses the importance of accounting for the second Born term. Comparing the Glauber results with and without exchange, it is again seen that the inclusion of exchange effects tends to improve the cross sections at the lower intermediate energies. No calculations for e- + H ls-+(2s + 2p) scattering at 100 eV have been performed in the fixed-scatterer approximations (58); however, the fixed-scatterer results at 54 eV (which we do not quote) are in good agreement with experiment (97). Since the experimental data for e- + H 1s+(2s + 2p) excitation extend down to only 20°, whereas the majority of the integrated (i.e., total) cross section comes from small angles less than about 20", we cannot compare total cross sections with experiment. However, it is of interest to compare the total cross sections with each other. In Table IV we compare the available 1s +2s results for the total cross section in the Glauber and eikonal approximations, and in Table V we show the available 1s +2p results. It is seen that the various calculations predict similar total cross sections at all energies considered; this is expected, since the calculations give similar results for small-angle scattering, which comprises the bulk of the total cross section. TABLE IV e-

+ H Is

2s TOTALCROSS SECTIONS (IN UNITSOF xa:)

+

BY

Incident energy (eV) 50 100 200

AS PREDICTED

VARIOUSGLAUBER AND EIKONAL-TYPE METHODS

Glauber"

Eikonalb

Exchangecorrected eikonal"

0.078

0.072 0.048 0.028

0.076 0.048 0.035

0.050

0.028

Taken from Foster and Williamson (92). See Gau and Macek (93). ' Interpolated from Fig. 7 in Byron (38).

Eikonal MonteCarlo'

Born"

0.080 0.052 -

0.060 0.030

0.110

169

THE GLAUBER AND EIKONAL APPROXIMATIONS

TABLE V e-

+ H IS +2P TOTALCROSSSECTIONS (IN UNITS OF nu:)

AS

Indicent energy (eV) 50 100

200

PREDICTED BY THE GLAUBER AND EIKONAL METHODS

Glauber"

Eikonal"

0.83 0.67 0.47

0.73 0.64 0.46

Eikonal MonteCarlob

Bomb

0.73 0.64

1.05 0.75

-

-

Taken from Gau and Macek (93). Interpolated from Fig. 6 in Byron (38).

We note that the Born results are generally somewhat higher than the Glauber or eikonal results. We continue the discussion of e- + H scattering by noting that several of the Glauber and eikonal calculations have addressed themselves to the prediction of Ly-a polarization fractions and orientation and alignment parameters. A detailed discussion of these parameters for e- + H 1s-2p scattering is given by Gau and Macek (93). The conventional Glauber approximation incorrectly predicts the Ly-a polarization fraction unless the z axis is redefined after the Glauber amplitudes have been computed (80), this method has been used (80) by Gerjuoy et al. to calculate Ly-a polarization fractions for hydrogen. The Ly-a polarization fractions for e- + H 1s +2p excitation have been calculated in the Born (80), Glauber (80), and eikonal (38, 93) approximations. The results of these various calculations (which we do not quote here) are compared in the work of Gau and Macek (93).They find that the various calculations are all in fairly close agreement and agree fairly well with experiment (97). Gau and Macek also find that the orientation and alignment parameters predicted by the eikonal approximation are markedly different from the corresponding predictions of the Born and conventional Glauber theories (93). The Glauber predicted differential and integrated cross sections for the excitation of n = 3 levels of ground-state atomic hydrogen by electron impact have been calculated by Tai et al. (59), Bhadra and Ghosh (64, and Chan and Chang (64). The Glauber and Born curves for the differential cross section of 1s-31 excitation fall monotonically with increasing scattering angle, and the higher the energy the steeper the fall (59, 61). The Glauber do/dR starts out above (i-e.,more sharply peaked) the FBA in the forward direction, then falls below the FBA at intermediate scattering angles, and finally becomes greater than

170

F. T. CHAN ET AL.

171

THE GLAUBER AND EIKONAL APPROXIMATIONS

the FBA at large-angle scattering ( 230-50"). Since there is no experimental data for (da/ds1),,- 31, we do not display corresponding figures here. The individual n = 3 cross sections and the Balmer-cc cross section of hydrogen atoms by electron impact with incident energies from 18 to 500 eV are represented in Figs. 6 and 7. Also shown in the same figures are the experimental data (99-101) and other theoretical results using the Born approximation of Vainshtein (102), the Born-Oppenheimer approximation of Morrison and Rudge (103), the distorted wave method of Vainshtein (106, the close-coupling calculation of Burke et al. (105), the polarized orbital distorted wave calculation of McDowell et al. (106), and the impact parameter calculation of Jamieson (107). We note from Figs. 6 and 7 that the present n = 3 experimental data would agree fairly well with the Glauber prediction, except for the 1s-3d excitation cross section. 7.0

I

I

I \ I

1

1

1

1

,

I

I

I

1

,

1

1

1

1

I

1

I

I l l l l l

#

k2.0

-

ID

-

0

1

Oil

1

x) I00 Electmn Impact Eneqy (ev)

1

1

500

1

1

1

loo0

FIG. 7. Total 2s - Ha excitation cross section. The theoretical results are denoted as in Fig. 6, with the addition of the impact parameter calculation (-----) of Jamieson (207).The experimental data are the results of Kleinpoppen and Kraiss (99) (a),the experimental measurements of Mahan et al. (101) (O), and of Walker and St. John (100) ( x ). From Chan and Chang (64).

FIG.6 . (a) 1s - 3s, (b) 1s - 3p, and (c) 1s - 3d excitation cross sections of hydrogen by electron impact. The theoretical curves are as follows: -, GA; ----,Born approximation, Vainshtein (202); ----, distorted wave, Vainshtein (104); .-.- , Morrison and Rudge (103); polarized orbital distorted wave, McDowell et al. (106);close coupling, Burke et al. (105). The experimental results of Mahan el al. (101) are given by x and 0 using the inphase and in-phase-plus-out-of-phase fits.

172

F. T. CHAN ET AL.

1

50. \

25

1 I

I

-

1

I

I

I

I 1 1 I l l

l

1

I 1

I

1

I

I I

I ,

1

1

I l l 1

1

,

( b)

\

\

',

Born '\

0

L

25

0

.'z

-

(J

.

-

'

-zz-1

I

I

I

I 1 1 1

1

I

I

I

1

l

1

1

1

FIG.8. Polarization fraction of (a) 3p, (b) 3d, and (c) Ha as functions of the electron energy. The theoretical curves are -, GA and ----,Born approximation, Mahan et al. (101). The experimental data of Kleinpoppen and Kraiss (99) are given as x . From Chan and Chang (64).

1

0

I

I

30

10

I

50

&dog)

FIG.9. The parameter A,, (3p) as a function of scattering angles for SO-, 80-, and 100-eV incident electron energies. From Chan and Chang (64).

THE GLAUBER AND EIKONAL APPROXIMATIONS

173

The polarization fractions of the resulting radiation from 3p to 2s, from 3d to 2p, and of the Balmer-a line emitted by hydrogen atoms following excitation to n = 3 states are shown (64) in Fig. 8. We note from the figure that the Glauber curve closely resembles those obtained from the Born calculations for the polarization fraction of radiation from 3p to 2s. However, we see from Fig. 8b that for the polarization fraction of the 3d-2p line, the Glauber values show a large deviation from the Born calculations at lowincident impact energies although they both predict almost the same value at 500 eV. The parameters A, (3p), A, (3d), and ,IL have been calculated by Chan and Chang (64) in GA as functions of scattering angles for 50-, 80-, and 100-eV incident electron energies. The results are shown in Figs. 9 and 10. However, both Glauber and Born approximations predict the zero-phase differences between the excitation amplitudes for different magnetic sublevels of the substates. Experimental measurements of (A, x) using the coincidence techniques would provide an important check for the Glauber approximation.

0.6

8 -x

04

a2

ae

-

0.6

0

4 04

x

< 2 a 0

30

10

50

8 (dog)

FIG.10. The parameters I , (3d) and I1 (3d) as functions of scattering angles for 50-,80-, and 100-eV incident electron energies. From Chan and Chang (64).

F. T. CHAN ET AL.

174

B. Electron- Helium Scattering Neglecting exchange and spin effects, the Glauber amplitude for e- + He scattering is very similar in form to the amplitude for corresponding collisions involving atomic hydrogen. In particular, the Glauber amplitude for elastic scattering by ground-state He atoms with wave function qi(rl, r,) is given by (25,108)

x exp(iq b) d2b dr, dr,

(3.46)

where (3.47)

To illustrate the analytic methods of reducing the associated multidimensional integral in Eq. (3.46), it is sufficient to take qi(r,, r,) to be the simplest Hylleraas wave function for the He ground state; namely (109), (3.48) with CI = 1.69. We note that the Glauber amplitude of Eq. (3.46) can be written in terms of a generating function ; in particular, (3.49) where we define

-

x exp(iq b) d2bdr, dr,

(3.50)

Following the same analytic methods for e- + H scattering, Franco (108) was able to reduce the associated eight-dimensional integral in Eq. (3.50) to a three-dimensional integral (again involving hypergeometric functions), which was evaluated numerically [in fact, Franco (108) used a more complex Hartree-Fock wave function (110) for the helium ground state]. Furthermore, Yates and Tenney (111) showed that this three-dimensional integral can be reduced to a two-dimensional integral and reported numerical results for inelastic excitation of the 2% state of helium by electron impact. (As will be discussed immediately, Franco (112) at this time had already shown how to reduce the integrals to one dimension, but had not yet published his calculations.)

THE GLAUBER AND EIKONAL APPROXIMATIONS

175

It is clear that for atoms with Z 2 2, further reduction of the Glauber amplitude expressed as a (32 2)-dimensional integral is essential if the approximation is to be useful in the theory of collisions involving multielectron target atoms, assuming their analytic (Slater-type) wave functions of initial and final states are available. A general procedure for such reductions has been proposed by Franco (112) in 1971. For simplicity, we again consider e- + He(1 ' S ) elastic scattering, although all the results are equally valid for more complex atoms. First, Franco drops the term equal to unity in Eq. (3.50), since its contribution is proportional to a 6 function in q and therefore vanishes for inelastic scattering or for elastic scattering with q # 0. Then, on the basis of the Thomas-Gerjuoy integral representation Eq. (3.Q Franco has developed a method for reducing the multidimensional integral (eight dimensional for He) to a one-dimensional integral representation. However, Franco's final integral representation involves the calculation and integration of the differences between strongly (exponentially) divergent functions, as well as the numerical calculation of 6 functions whenever elastic scattering is considered. For inelastic scattering, a very accurate integration routine has to be used, as Franco, in a subsequent publication (Z13), applied his method to obtain 1'S-2'S excitation cross section in e- + He scattering. In an attempt to generalize the Thomas-Gerjuoy reduction technique for e- + H collisions, a new analytic method for reducing the Glauber amplitude to a one-dimensional integral representation involving modified Lommel functions was proposed by Thomas and Chan (ZZ4) in 1973. In analogy to Eq. (3.7) for eH scattering, Thomas and Chan introduce a different generating function for e- + He elastic scattering with an extra factor (rl 1,)- ; namely

+

+

'

s

ik, exp( -Alrl) exp( -Azr2) Io(A1, 12; 4) = 2 r-1

[

1-

(Ib

sll)z'q(Ib

s,\)"~]

~

12

x exp(iq b) d2b dr, dr,

(3.51)

Hence, (3.52)

J&(l'S+l1S;q) = l*=.i2=Za

Thomas and Chan show that the first term (independent of q) under the integral in Eq. (3.52) which, dropped by Franco, leads to a 6 function, is exactly canceled by a similar factor stemming from the second term (dependent on q). Therefore the S(q) function is explicitly removed in computations. Using the Thomas-Gerjuoy integral representation Eq. (3.8), Thomas

F. T. CHAN ET AL.

176

and Chan obtain a one-dimensional integral for Zo(Il,I z ; 4 ) ; namely,

x (i&b)-ziq-z5Yziq-l,o(iI,b)

(3.53)

where J and 5!'p,vare the usual Bessel and modified Lommel functions (224), respectively. Equation (3.53) is understandable if we decompose the full r of Eq. (3.47) in a fashion analogous to the Glauber multiple-scattering expansion (25):

T(b; r,, r,) = T(b; rl)

+ T(b; r2) - T(b; r,)T(b; r,)

(3.54)

If expression (3.54) is used in Eq. (3.50),using the procedure of Thomas and Gerjuoy (62) we immediately obtain in Eq. (3.50) the terms involving the hypergeometric functions arising from the integrals over the sum T(b; r l ) + T(b; rz). At high energies and small scattering angle where single-particle scattering dominates the amplitude, we may expect the scattering amplitude to be well approximated by the terms involving only the hypergeometric functions, which diverge as ln(q2) as q --t 0, whereas the integral involving the products of modified Lommel functions is well defined and finite as q 0. The generating function Eq. (3.53) has since been used to calculate the elastic and excitation cross sections for electron collisions with helium atoms involving spherically symmetric S states (224-117). The Thomas-Chan reduction method has also been applied, including derivation of similar generating functions, to electron impact excitation of helium to nonspherical states (228-220) (n'P, n'D). Since the differential cross section for proton-helium scattering is more sharply peaked in the forward direction than for electron-helium collision, the Glauber approximation is expected to be more suitable in the former case. Only results on 1'S 2'P excitation for proton impact on helium atoms have been reported (222). The Glauber theoretical differential and integrated (elastic as well as inelastic) cross sections for e- + He collisions have been calculated by a --+

--+

177

THE GLAUBER AND EIKONAL APPROXIMATIONS

number of authors; they include Franco (108,113),Yates and Tenney (ZZZ), Thomas and Chan (114), and Chan and co-workers (115, 118-120). Byron and Joachain (53, 54, 122) have applied the eikonal-Born series method to analyze elastic scattering and the excitation of the 2's state of helium by electrons of energy E > 100 eV. Byron and Joachain (123) and Saha et al. (124) have used the eikonal optical-model approximation to investigate the elastic scattering of electrons by helium atoms. Recently, Kurepa and Vuskovic (125) have successfully measured the absolute elastic differential cross sections for electrons on helium for incident energies of 100, 150, and 200 eV in the angular range between 5 and 150". Their measurements are in best agreement with the data of Byron and Joachain obtained by using the eikonal-Born series method (53, 54). In particular, the theoretical curves (53, 54) predicted by Byron and Joachain are the only ones having the same shape at small angles as the experimental curves. A more quantitative picture is provided by Table VI (53, 54, 123, 125-127). We note from Table VI that the Glauber predicted results are too low at all angles (except very near the forward direction, where it diverges). For e- + He collisions, it is not clear how important the exchange effects TABLE VI AND EXPERIMENTAL DIFFERENTIAL CROSS SECTIONS, COMPARISON OF VARIOUS THEQRETICAL I N ATOMIC UNITS,FOR ELASTIC e- + He SCATTERING AT AN INCIDENT-ELECTRON ENERGY OF 200 eV

Experimental values Theoretical values

0 (deg)

Born

Glauber

EBS

Optical eikonal

(54)

(54)

(53,54)

(123)

6.27 ( - 1) 6.05 ( - 1) 5.44 ( - 1) 3.73 ( - 1) 2.25 ( - 1) 7.80 (-2) 3.15 (-2) 1.56(-2) 9.18 (-3) 6.31 (-3) 4.97 ( -3) 4.36 (-3)

00

1.15 6.97 ( - 1) 3.30 (-1) 1.70 ( - 1) 5.07 (-2) 1.88 (-2) 9.48 (-3) 5.66 (-3) 4.00 (-3) 2.88 (-3) 2.59 (-3)

3.16 2.12 1.34 5.83 (-1) 2.88 (-1) 8.76 ( -2) 3.61 (-2) 1.97(-2) 1.32 (-2) 1.01 (-2) 8.60 (-3) 7.88 (-3)

2.97 2.03 1.28 5.81 (-1) 2.90 ( - 1 ) 8.42 (- 1) 3.08 (-2) 1.41 (-2) 7.85 (-3) 5.17 (-3) 3.95 (-3) 3.41 (-3)

Chamberlain et al. (126)

Vriens et al. (127)

Kurepa and Vuskovic (125)

~

0 5 10 20 30 50 70 90 110 130 150 180

-

1.64 1.04 4.58 ( - 1) 2.24 ( - 1)

-

-

2.04 2.00 1.28 1.23 5.61 ( - 1) 5.57 ( - 1) 2.75 ( - 1) 2.92 ( - 1) 9.53 (-2) 3.61 (-2) 2.1 1 (-2) 1.39 (-2) 1.07 (-2) 9.28 ( -3)

F. T. CHAN ET AL.

178

3

Y

C

h 0

FIG.11. Differential cross sections (in a i / s r ) for the excitation of the 2lS state of helium by 200-eV electrons (a) at small angles and (b) at larger angles. -, EBS calculations (122); ----, first Born approximation; .-.-, Glauber approximation (111). Experimental data: 0,Vriens et al. (127); A , Chamberlain et al. (126); 0, Opal and Beaty (128); 0 ,Suzuki and Takayanagi (129); A,Dillon and Lassettre (130). From Byron and Joachain (122).

(a)

-s 0

55.5eV

(b)

100eV

Al.

0 4 -

a

0

10

5 ),(

I5 0 5 Swlterlng angle (dog)

10

I5

20

(b)

FIG.12. Differential cross sections for 2lP excitation of helium by electrons at (a) 55.5 eV, Glauber (118); 0 , Vriens et al. (129, renormalized at 5" to Chamberlain (b) 100 eV. -, el al. (126); I, Chamberlain et a/. (126); x , Truhlar et al. (131). From Chan and Chen (118).

THE GLAUBW AND EIKONAL APPROXIMATIONS

i~~

Scottaring angle (drg)

179

ib)

FIG.13. Differential cross sections for 2'P excitation of helium by electrons at (a) 200 eV, (b) 400 eV. -, Glauber (118);0, Wens et al. (129, renormalized at 5" to Chamberlain et al. (126); I, Chamberlain et al. (126). From Chan and Chen (118).

Elntron

500 energy ( B V )

1000

FIG.14. Total cross sections for the 11S+2'P excitation of helium by electron impact. Curve G, Glauber (118); Curve S, (3). BBC (132); Curve B1, first Born approximation, Holt et al. (133); Curve SB2, second born approximation, Woollings and McDowell (134); Curve HG, Hidalgo and Geltman (135); 0 , Van Eck and de Jongh'(138); x , Moustafa Moussa er al. (136); 1, Donaldson et al. (137). From Chan and Chen (118).

180

F. T. CHAN JiT AL.

Electron lmpoct Energy (eV )

FIG. 15. Total cross sections for 1 'S-3'D excitation of helium by electron impact. -, Glauber (130); .-.- , Born approximation; ----,Ochkur approximation (139); ----, Woollings-McDowell approximations (140); 0 , St. John et al. (141); x , Moustafa Moussa et al. (136).From Chan and Chang (120).

are since their inclusion does tend to put the Glauber prediction for e- + H elastic scattering in much better agreement with experiment (51). For e- + He inelastic collisions, the present 1's - 2% and 1's - n'P experimental data agree fairly well with the Glauber prediction (112, 113, 115,118,119).In contrast, the Glauber values lie below experimental points, as well as the Born values for 1's - n'D excitations. The patterns follow that of the e- + H case. We have selected a few curves shown in Figs. 11- 15 for the purpose of illustration (111,118,120,122,126-141).We notice from Fig. 11 that significant differences develop between the Glauber and EBS results at very small angles (0 c k - 2 ) and also at large angles where precise absolute measurements would be valuable. Even with proper renormalization, all the theoretical predictions for 1'S - 3 'D excitation still lie below experimental data at around 50 eV. Chan and Chang (120) have also calculated the polarization fraction P(Ei) at incident energies from 50 to 1000 eV for the 6678-a helium line. The results are shown in Fig. 16. Although the shape of the Glauber curve resembles the experimental values of Moustafa Moussa et al. (136), we see that the agreement with experiment is also poor and further investigation seems desirable. Anderson, Hughes, and Norton (142) have studied the excitation of the

THE GLAUBER AND EIKONAL APPROXIMATIONS

181

Electron Impact Energy (eV)

FIG.16. Polarization fraction of the 6678-A helium line excited by electron impact. -, Glauber (120);0 , Moustafa Moussa et al. (136). From Chan and Chang (120).

4ls3Flevels of helium by electron impact using time-resolved spectroscopy technique. Their data are much larger than predicted by the Born approximation. To date, there have been no applications of Glauber calculations to such high angular momentum states. Byron (38) utilized the Monte-Carlo integration method to perform the nine-dimensional integrations involved in evaluating the (unreduced) eikonal direct amplitudes for the 1'S -+ 2lP excitation of helium by electron impact. He obtained results (which we do not quote here) in reasonable agreement with experiment, although the accuracy of his numerical results may suffer somewhat due to the statistical uncertainty of the Monte-Carlo method. Byron and Joachain (143) looked at the 1'S 23S (exchange) transition in helium, using the unrestricted Glauber (UG) exchange amplitude. Again, they utilized the Monte Carlo method to evaluate the nine-dimensional integrals involved and found reasonable agreement with experiment. Their results (which again we do not quote) were a noticeable improvement over the Born and Ochkur approximations. The only other application (up to the present time) of the UG amplitude to electron-helium scattering is the work of Kelsey and Macek (144). They studied the 11S+21P excitation of helium by electron impact using the reduced form of the UG amplitude [reduced from a nine-dimensionalintegral to a four-dimensional one, using the Gau-Macek (79, 145) reduction method]. They confined their work to high-energy scattering at large angles -+

182

F. T. CHAN ET AL.

and hence were able to expand the UG amplitude, retaining only the leading term which they could then reduce to closed form. They found oust as in the case of ls+2p excitation of hydrogen) that, for large angles and high energies, the electron-nucleus interaction dominates and produces a Rutherford q-4 dependence in the differential cross section. They also calculated the so-called orientation parameter (79) Oy!,and found it to be large; a large value for Oy! was also found for electron-hydrogen 1s +2p excitation using the UG approximation (79). C . Electron-Lithium Scattering

Stimulated by the pioneering work of Franco (113), Mathur et al. (146, 147) have tried to extend the Glauber approximation to study the interaction of electrons with the alkali-metal atoms. Within the framework of the frozen-core approximation (i.e., only the interaction with the active electron and a nucleus of charge unity is considered), they have calculated the Glauber cross sections of the elastic scattering (146) and the resonance transition (147) (2s - 2p) in lithium by electron impact. Using the numerical results reported in Mathur et al. (140, Tripathi et al. (148) have evaluated the polarization fraction of the Li resonance line, following the procedure of Gerjuoy et al. (80). Using Thomas-Gerjuoy analytic expressions, Walters (149) has recalculated the Glauber scattering amplitudes of eLi collisions. He has employed the same wave functions used by Mathur et al. (146, 147) but has ended up with significantly different Glauber and Born cross sections for the resonance transition. In regard to the discrepancy, Walters offers convincingarguments for preferring his results. Therefore the polarization fraction for the lithium resonance line reported in Tripathi et al. (148) should also be repeated. Although the alkali atoms, by and large, behave like one electron system, the practicality and utility of the Glauber approximation without using the frozen-core approximation have to be demonstrated. In attempting to answer these questions, several publications (150-152) on reducing the full Glauber amplitudes (expressed as an eleven-dimensional integral in e- Li collisions) to tractable form have been reported. Kumar and Srivastava (150) have used Franco’s procedures (112) for reducing the Glauber amplitude to one-dimensional integrals to study e- + Li elastic scattering. They have been able to avoid the encounter with the divergent functions appearing in Franco’s final expression by stopping a step earlier. The concealed 6 function in the momentum transfer q also presents no numerical problem. The price to be paid for this simplification is that their final expression is a two-dimensional integral representation. In a subsequent publication (151), Kumar and Srivastava have applied their methods to calculate the Glauber

+

+

THE GLAUBER AND EIKONAL APPROXIMATIONS

183

cross sections for the resonance transition (2s +2p) in lithium by electron bombardment. Their results are always a bit smaller than the frozen Glauber results, but approach them as the energy increases. However, the differences are not significant, indicating that the inner 1s electrons are rather inert. On the other hand, Chan and Chang (152) have reported calculations on the elastic e- + Li collision in Glauber approximation using the integral reduction techniques of Thomas and Chan (114).They have obtained the scattering amplitude in terms of a sum of hypergeometric functions and the integrals over products of two and three modified-Lommel functions [recall the discussion after Eq. (3.54)]. The criterion for using the frozen-core approximation is also discussed in Chan and Chang (152). Using an electron-impact spectrometer technique (153, 1 5 4 , Williams, Trajmar, and Bozinis (155) recently successfully measured, for the first time, the differential cross sections for elastic scattering and for the excitation of the 2p2P, 3p2P, 4p2P, and 3s2S states of lithium at 5.4-, lo-, 20-, and 60-eV electron impact energies. Differential cross-section measurements are more basic than those for the total cross sections. Their measurement, therefore, is expected to provide an additional test of theoretical models. Although the Franco [one-dimensional (112) or two-dimensional (150, 151) integral representation] and Thomas and Chan (114) analytic methods appear to be practicable for predicting the differential cross sections in e- + Li collisions, no actual calculations following their formulations have yet been reported.

Iv.

IONIZATION OF

NEUTRAL ATOMSBY ELECTRON COLLISIONS A . General Discussion

At first glance it might seem that the previous results could be immediately carried over to the ionization problem with little alteration, e.g., simply insert a continuum wave function in place of the bound-state wave function for the appropriate final-state electron, depending on whether direct or exchange scattering is being considered. But on reflection it is clear that this is an oversimplification. Indeed, there is, and can be, no physical distinction between the two emerging electrons. Several questions then arise: (1) Is the separation into direct and exchange scattering meaningful? (2) Can the use of a plane wave to describe one emerging electron, which assumes essentially perfect screening of the Coulomb potential by the second electron, be justified? (3) How can the possibility of interference between the emerging electrons be accounted for? A thorough discussion of these questions would be out of place here, and the reader is referred to the excellent review article by Rudge (156) and to the monographs by Bransden (157), Geltman (158),

184

F. T. CHAN ET AL.

and Peterkop (159) for the details. However, without the answers to these questions it is not possible to even define the scattering amplitude as in Section 11-A. Therefore, some discussion is unavoidable. Assuming the two electrons to be, for the moment, distinguishable, with electron 1 being the incident and electron 2 the initially bound electron, we can define, for convenience, direct scattering to be the case in which electron 1 emerges with a greater velocity than the ejected electron 2, while in exchange scattering the reverse is true. For these two processes one can, by analogy with the cases of elastic and bound-state excitation scattering, define scattering amplitudes f(kl, k,) and g(kl, k,) so that the cross sections for the two processes are proportional to and 1gI2, respectively. Note that f is defined only for k; 2 k', and g for k; I k,, via integral expressions to be given below. Using these expressions it can be shown that if normalization and phase factors are properly chosen, and suitable extensions of the range of definitions off and g are made, one has

')fI

f(k1, k;) = g(k2, k l )

(4.1)

which is simply an expression of the lack of real physical distinction between the direct and exchange processes. In order to discuss the integral expressions for f and g, it is necessary to address the second of the above questions. If the interactions were of short range, we would proceed as in Section 11, i.e., as in Eqs. (2.8)-(2.10). Explicitly, one would find the exact expression

with a similar expression for g having the subscripts on CP exchanged. In Eq. (4.2) is the exact solution to the Schrddinger equation, (H- E)Yll, which represents the collision with the atom of electron 1, incident with momentum hk, , and satisfying outgoing-wave boundary conditions in all regions. It includes the elastic, excitation, and ionization possibilities with the amplitudes defined as appropriate coefficients in the asymptotic regions as rl 0' 0 or r,+ co, or both. On the other hand, a k i k i represents the final unperturbed state and is to a degree arbitrary, except that it must specify the final momenta of the emerging pair of electrons to be hki and hk2 by means of a boundary condition at infinity. We have written (H - E)CP here instead of VCP as in Section 11-A, although (H - E)CP = VCP where V is that part of the potential not included in CP. This facilitates the present discussion. For example, we can take 0 to be the product of two plane waves, i.e., exp(ikl * r l + ik, sr,) (apart from normalization). Then V is the sum of all the two-particle potentials. This form makes transparently clear the

185

THE GLAUBER AND EIKONAL APPROXIMATIONS

identity of the direct and exchange processes and Eq. (4.1). Alternatively we could choose 0 in the manner mentioned in the opening paragraph of this section, i.e., a plane wave in electron 1 times a continuum wave function for electron 2 for the direct process, or the opposite way for the exchange process. In this case V consists of the sum of the two potentials not included in the continuum wave function. The validity of Eq. (4.1)in this case is not so transparent. It should be noted that the relevant two-body continuum wave functions must have, at large distances, a plane-wave part, exp(ik; r,) or exp(ik; *r2), plus an incoming spherical wave. Finally, one can choose CD to be the product of two continuum wave functions-one for electron 1 and one for electron 2. V is then just the interelectron repulsion. When Coulomb potentials are present, the above discussion must be modified. In general, each emerging electron will be moving in the Coulomb potential of the remaining ion, partially screened by the other electron. The faster moving electron will see a better screened atom As is discussed in Rudge (156) and Bransden (157), electron 1 is considered to move in a Coulomb potential owing to an effective charge Z, located at the origin, and electron 2 is similarly assumed to move in a potential owing to an effective charge Z2.These must be chosen so that the electronic potential energy has the correct form at large distances. This, in turn, can be expressed in terms of the momenta, using the fact that for large times r is proportional to k, with the result (156,157):

-

Note that if k; %k2,the right-hand side is approximately 1/k2, so that Z, is approximately 1 and Z, is approximately 0; i.e., electron 1 sees a nearly perfectly shielded potential. Note also that Z, and Z2 are dependent on k’, and k;, and that one may be chosen arbitrarily, but the other is then fixed by Eq. (4.3). The formula for f(k,, k2), which is the revised version of Eq. (4.2),can then be established:

As before, is the exact wave function, while CD is arbitrary, except that it must have asymptotically the same behavior as the product of two incoming Coulomb wave functions, one for electron 1 with wave vector k; and effective charge Z,, and one for electron 2 with effective charge Z, and wave vector k2.In particular, we can choose CD to be precisely this product : 0iti,k5(rl> r2) =

zl)6 5 @ 2 , z2)

(4.5a)

186

F. T. CHAN ET AL.

where

[- -;l;

x exp(ik*r),F,

-i(kr

+ k*r)

and ,F,(a;#?;z)is the confluent hypergeometric, or Kummer function. In this case the convergence of Eq. (4.4)has been proved, and we note that (H- E)Q = VQ, where

v = - (1 - 2,) - (1 - 2 2 ) +-

1 rl 12 lr1 - 121 In Eq. (4.4)the phase factor A is, in atomic units,

where E is the total energy: E = &kf

+ G2)= ik; - I

(4.8)

with I the binding energy of electron 2 before ejection. With the choice of phase (4.7)and the normalizations and form of Eq. (4.5),we can establish Eq. (4.1),and then a separate integral expression for g need not be written. In approximate evaluations of f using Eqs. (4.4), (4.5),it is generally the case that Eq. (4.3)is not retained. Thus, for example, the most common form of the Born approximation consists not only of inserting into Eq.

(4.4)the approximate wave function, ytl(rl, r2)

= [exp(ik1 r 1 ) / ( 2 P I ~ d r 2 )

(4.9)

but setting Z1 = 0 , Z 2 = 1 (i.e., assuming perfect screening),and also setting A = 0. The wave function (4.5a)reduces to a plane wave in rl times a Coulomb wave function for r2. This is also the approximation used in the Glauber calculations to be discussed in the next subsection. To conclude this lengthy introduction, we address the third question of the opening paragraph. For distinguishable electrons the total ionization cross section would be

The restriction on the values of k’, is to avoid double counting. Alternatively,

THE GLAUBER AND EIKONAL APPROXIMATIONS

187

we may drop this restriction but must then divide the right-hand side by two. If we then use Eq. (4.1),we can eliminate g and by a change of dummy variables deduce that the expression can be reduced to CJ

=

SdL; sdn;

If(k;,

(4.11)

k;)12

a.

where k; runs over the full range from 0 to (In these expressions it should be recalled that k;2 + k;z = 2E in atomic units.) For indistinguishable electrons we follow the procedures used in Section I1 ; namely, we define singlet and triplet scattering amplitudes :

f,(k;,k;)

= f(k;,k;)

+ (-1)S~(k;, k;),

s = 0,1

(4.12)

Then for an unpolarized beam, and using Eq. (4.1) and symmetrization of the integrand as in (4.11), we get (256-158)

The first term is identical to (4.11) and is then the same as the distinguishable electron case. The second term is the result of indistinguishability and represents an interference between the emerging electrons. The range of the k2 integration is 0 to @ in (4.13),just as in Eq. (4.11).Note that the phase factor A of Eqs. (4.4) and (4.7) is of great importance in evaluating the interference. We have taken pains to stress the range of integration in the various expressions because it has been a matter of some controversy. In particular, in calculating 0 from the “Born approximation,” as discussed previously, the interference term has been discarded because it is much more difficult to calculate. Considerations of consistency of this approximation require that the upper limit on the k; integration must also be changed to as discussed in Rudge (156) and Geltman (158). This is not the same as “ignoring exchange,” which is often asserted. That would amount to setting g = 0, which cannot be done consistently in ionization in view of Eq. (4.1). Rather, it is a case of truncating f at k; = k;, where exchange effects are most important.

fi

B. Eikonal Approximations in Electron-Hydrogen Ionization

Because of the greater technical complexity of the ionization problem, both in the formulation as outlined above and in the calculation [witness the five-dimensional integration needed to compute the total cross section in Eqs. (4.11) and (4.12) and the unavoidable complication caused by the

F. T. CHAN ET AL.

188

use of Coulomb continuum wave functions (4.5a)], there have been published relatively few calculations, even for the simplest systems, that go beyond the above-mentioned Born approximation. The only eikonal-type approximation calculations relevant to this review have been a few since 1972 in the context of the Glauber approximation. The first of these, on ionization of atomic hydrogen from the 1s ground state, was reported in a brief note by Hidalgo et al. (160), with details (but without the results) in a subsequent paper by McGuire et al. (161).Their derivation follows the standard Glauber approach, as outlined in Sections 11-B and C above, but requires some additional justification. They use Eq. (4.4) but set A = 0. For Yt, they insert a standard Glauber-approximation wave function as in Eq. (2.15): ytl(rl, r2) = ( W 3 / ' exp(ik, rl) (po(r2)

-

+

x exp[-

V(b z'&, r2) dz' kl -m Here (po = e-)./,/k (in atomic units) is the ground-state wave function. The z axis is chosen along k, ; the choice of potential V will be discussed in a moment. In arriving at (4.14), McGuire et al. (161) show that recoil of the target atom is neglected. This is really justifiable in the case of elastic or excitation scattering, but in ionization it reduces the maximum energy that can be carried off by the ejected electron. Therefore they justify ignoring recoil by observing that, experimentally, the ejected electron spectrum is strongly peaked at relatively low energies. There are no data for hydrogenic systems; so in this case they rely on Born calculations which show peaking at about the binding energy of the ejected electron. They then argue that this same observation permits them to ignore the partial screening requirement discussed above in Eqs. (4.3) and (4.5).Instead they use Z,= 0, Z2 = 1, i.e., full screening. The wave function (9 that they are thus led to insert is the product of a plane wave for electron 1 and a Coulomb wave for electron 2:

4

rz) = ( 2 7 v 2exp(ik1

%i,k&l,

where p(;

r1) G ( r 2 )

(4.15)

is given by Eq. (4.5b) with Z = 1. The potential V is thus

V(r,, r2) = -r;'

+ Ir, - r21-'

(4.16)

just as in the cases discussed in Section 11. The same procedures as previously described lead to Eq. (2.19a) but with an incoming continuum Coulomb wave function, Eq. (4.5b), replacing the final bound-state wave function :

f&

k) =

ikl

s

(p;(r)*T(b, s) (po(r)exp(iq b) d2b dr

(4.16a)

189

THE GLAUBER AND EIKONAL APPROXIMATIONS

T(b,s) = 1 - (Jb- sJ/b)2i"

(4.16b)

where q = l/kl, q = k, - kl, k = k2, r = r2, and b, s are, as in previous sections, the projections of rl and r2 on the x y plane. Again the z axis has now been chosen perpendicular to q. In order to apply the reduction techniques of Thomas and Gerjuoy (62), they next expand qc in a series of partial waves:

where

x exp(-ikr),F,(l

+ 1 + i/k;21 + 2;2ikr)

(4.17b)

McGuire et al. ( I64 argue that, just as in Born approximation calculations, the I = 1 partial wave should dominate the cross section and that contributions from higher partial waves should rapidly decrease, so that only the first few should suffice. This is borne out by their calculations (160) for 1 = 0, 1, 2, and 3 for q small; the reader is cautioned, however, that there are numerical errors in this paper, although later work has not invalidated the conclusion. When the confluent hypergeometric function is written out as an explicit power series in r and (4.17) is inserted into (4.16), the result can be reduced to a form similar to those of Section 111: (4.18a) with

(4.18b) The generating functions in (4.18b) are the five-dimensional integrals

11, =

s

rl

-

-Km(3) exp(iq b)r(b,s) d2b dr :e

(4.19)

but, using the methods of Thomas and Gerjuoy (62),these may be evaluated analytically in terms of hypergeometric functions. The coefficients k) can be easily evaluated by means of a two-term recurrence relation. The results will not be given here because of their complexity and because

F. T. CHAN ET AL.

190

they have been superseded (see below). Thus, the evaluation of fa has been reduced to a doubly infinite sum (over 1, n) of hypergeometric functions. The total cross section is then reduced to a double integral, three angle integrals being elementary : r

m

1

(4.20) in units of zut. Unfortunately, a difficulty now appears : The sum of Eq. (4.18b) diverges for k 2 0.71k0,,,, where hk,,, is the orbital momentum of the electron before ejection. This limits the usefulness of (4.18) and (4.20), although the problem

I\

e+ H-p+e+e

O 0.1. 1 l

0

I'0

I

I

I

I

I

I

I

I

I

I

I

1

I

I

I

I

I

20 30 40 50 60 70 00 90 100

INCIDENT ENERGY (eV) FIG.17. Total cross section for the ionization of atomic hydrogen by electron impact versus projectile energy. The electrons are considered distinguishable. Data were extracted from Fite and Brackmann (163) and Boyd and Boksenberg (164); x , observed, Fite-Brackmann; 0 , observed; Boyd-Boksenberg. From Golden and McGuire (162).

191

THE GLAUBER AND EIKONAL. APPROXIMATIONS

arises generally beyond the peak of the double differential cross section (d%/dqdk), which occurs at about OSk,,, as a function of k. It should be recalled that k = k; has been assumed small in the derivation of (4.18). Nevertheless, this difficulty was overcome in subsequent publication (162), in which a Padk approximation to the series in Eq. (4.20) is used to extend the region of convergence in k to about 2 or 3 times the previous limit, and a Born approximation is used to continue from these to infinite. The results for the total cross section are illustrated in Fig. 17 where they are compared with the pure Born approximation and two sets of experimental data (162-164). As mentioned in the end of Section IV-A, the correct upper limit when the interference term is not calculated should be not f l E . In Fig. 17, the upper limit was but in Fig. 18, @ has been used in both the Born and Glauber approximations. This somewhat reduces the values of both and puts the Glauber in good agreement with the data for incident electron energies above about 35 eV. Recently a notable advance was made in the ionization calculations by

fi

fl,

1.1

-

1.0

-

o;N

k z 0

v

0

- p+e+e

-

090.8-

F

y

etH

-

0.7-

8 0.60

-

[r

0 0.5-

z

E

2

Q

0 OA: .3

0.2-

-

0.1 -

-

0'

I

I

I

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

10 20 30 40 50 60 70 80 90 100

INCIDENT ENERGY (eV) FIG. 18. Total cross section for the ionization of atomic hydrogen by electron impact versus projectile energy. Exchange effects are treated approximately (cf. text). Data are the same as in Fig. 17: x , observed, Fite-Brackmann; 0 , observed, Boyd-Boksenberg. From Golden and McGuire (162).

F. T. CHAN ET AL.

192

workers in the United States (165), Japan (166), and India (167).All three groups succeeded in replacing the troublesome infinite sum of (4.18b) by a one-dimensional integral that converges for all k and can be easily evaluated numerically. The methods of all three groups are quite similar although the details naturally differ somewhat. Golden and McGuire (165) express Am directly in terms of an integral:

- J;E

Im

x I;,(P)*T(b, s) exp(iq b) d2bdr

(4.21)

which is closely related to an infinite sum of the generating functions (4.19) if the confluent hypergeometric function were to be expanded in an infinite series in kr. Instead J , is reduced to a finite sum of one-dimensional integrals :

- i)mr(m- iq) - 274r( e x ~ ( - i ~ qC ) - iq)m ! q2

1+2

Jl,

=

Jfm

(4.22a)

p= 0

where JPm= ( I x

:f

+ 3 - p)!

G(q/b)

i AfmC-p-4(C/b)p2Fl 1 + 1 - -, 1 + 4 - p ; 21 + 2 ; -2ik/C ( k + (- 1)""' 'Armd-'-4(d/b)p

[

+ 1 - -,i 1 + 4 - p ; 21 + 2; -2ik/d

I)

db (4.22b) ( k Here (pq is the azimuthal angle of q, C = 1 - ik - iqb, d = 1 - ik iqb, the Arm are simple numerical coefficients. The function G is itself a hypergeometric function. Further details are omitted. Narumi et al. (166) and Jain and Srivastava (167)use similar techniques but employ an integral representation for the confluent hypergeometric function in Eq. (4.17) at an early stage. Narumi et a1 (166) present their results in an explicit form, while Jain and Srivastava use the form of derivatives of a generating function. Golden and McGuire (165) have repeated their earlier calculation (162) using the integral representation and find that the results agree to within 1%. Furthermore, the new technique is applicable to problems of collisional ionization by heavy-particle impact, where the large values of ejected momenta render their earlier Pad6 approximation method inapplicable. Narumi et al. (166) display the cross section D as a function of Eino(Fig. 19) and the single differential cross section da/dk as a function of k (Fig. 20). x 2F1 1

+

rm GLAUBER

AND EIKONAL APPROXIMATIONS

193

€, (ey)

FIG.19. The total cross section for the ionization of atomic hydrogen versus the incident electron energy. The results in the Glauber approximation are compared with those of absolute (+) and relative ( 0 )measurements (163) and with the Born estimate for reference. From Narumi, Tsuji, and Miyamoto (166).

FIG.20. The differential cross section doldk for the ionization of atomic hydrogen versus the ejected electron momentum or energy at the incident electron energy E, = 50 eV. From Narumi, Tsuji, and Miyamoto (166).

The total cross section is compared with the Born approximation and with experimental data, and the agreement between the Glauber result and the data appears excellent at all energies, even near threshold, except for about a 10% discrepancy near the 50-eV peak. The Born approximation is poor until about 200 eV. Their curves for dcr/dk are evaluated in the Glauber

F. T. CHAN ET AL.

194 20

15

5

0 0

20

80

60

40

100

4 (OV) FIG. 21. Integrated cross sections of Hzs (e, e’e)H+: -, Glauber; ----, Born; ----, close coupling; @ Dixon ef al. (169).From Kotegawa et al. (168).

and Born approximations at 50 eV, where the difference between the Glauber and Born total cross sections is greatest. It appears from their discussion that, in evaluating 0, they used an upper limit of ,/% on the k integral. The same group recently presented results of ionization from the 2s states (268)(Fig. 21), with the agreement again being quite good (268,269). Finally, Jain and Srivastava (267) present no calculations. C . Ionization of Multielectron Atoms by Electron Impact

Golden and McGuire (270) have recently applied the Glauber approximation (or, more accurately, an approximation to the Glauber approximation) in calculating the total cross section for ionization of K-shell electrons in multielectron atoms. They have especially studied the case of helium targets and have tested the sensitivity of the calculated cross section to variations in several parameters. The cross section, in units of nu;, is

4kl)

=W

Y

s

d4 d4q

41f(% k)I2

(4.23)

where atomic units have been used, as usual, for the parameters. The notation is the same as in the previous section; i.e. q = k, - k,, k = k2, dq = dq dqq, etc. In the Glauber approximation, we have, by the methods of Section I [cf. Eq. (2.19a)l:

x qji(rl,.

..,r,) d2bodr,

* *

dr,

(4.24)

THE GLAUBER AND EIKONAL APPROXIMATIONS

195

where rl ,. . .,r, represent the electrons of the target atom, 4i and 4f appropriate unperturbed initial- and final-state wave functions (see below), and bo is the projection of T o , the incident electron's position vector, onto the (4.25a)

(4.25b) where Xi(bo,ri) = q

jm($ - F) dz

ro

ril = 2q l n ( y )

(4.25~)

--oo

Golden and McGuire now decompose r in the same fashion as in Eq. (3.54), the multiple-scattering Glauber expansion : (4.26) where ri(b0, ri) = 1 - exP[iXi(bo, ri)] = 1-

(Ib - ~ i l / b ) ~ ~ "

(4.27)

They now make the single-scattering approximation ; i.e., they drop all but the single-scattering summation in (4.26), which they argue should make an error of less than 10% for low-2 atoms. For the wave functions in Eq. (4.24) Golden and McGuire make the following choices. Focusing on helium, they choose a product form 4i(ri, r2) = cpil(r,) qiAr2)

(4.28)

for the initial state, where the pil, cpi, will be chosen to be hydrogenic wave they assume a singlet functions with an effective nuclear charge. For wave function where only one orbital in (4.28) has been changed; i.e., 4dr1, r ~ =)

5

[cpil(rl) cpG(rJ + ViI(r2) (m;(r1)1

(4.29)

where cp; is an incoming-wave continuum Coulomb wave function Eq. (4.5b). With these assumptions the Glauber amplitude reduces to a singleparticle form :

F. T. CHAN ET AL.

196

For +i, Golden and McGuire use a 1s hydrogen wave function with effective charge 1;i.e., cpi(r)

= A3I2 ,-“I&

(4.31)

Two values of 1 are used: 1 = 1.687, as predicted by a variational fit to the helium ground state, and A = 1.618, as obtained from a Hartree-Fock fit. As they point out, the important regions of configuration space in making these fits may not be the important regions in calculating the scattering amplitude. For cp; they use two choices of effective charge: Z* = 1 (perfect screening) and Z* = A. The latter has the advantage of guaranteeing orthogonality of the bound and continuum wave functions. When Z* = 1 is used, they nevertheless put Z* = 1in the 1 = 0 term of the partial-

co0.5

-k 6

-

k0.4

‘0.3

-

lo, -

OJ

I

I

1.0

1.5

I

I

I

I

1

2.0

2.5 Loao Eo W.1

3.0

3.5

4.0

FIG.22. Total cross sections for the ionization of helium by electron impact versus log,, (projectile energy): 0 , Smith; 0,Harrison; x , Rapp, Golden; Asundi, Kurepa; A, Schram, de Heer, Wiel, Kistemaker. The single-particle Glauber results labeled by GM1 and GM2 were calculated using target wave-function parameters 1 = Z* = 1.618 and 1 = 1.618, Z* = 1.O, except 1 = 0, where Z* = I = 1.618, respectively. The Born result shown was computed using 1 = Z* = 1.618. From Golden and McGuire (170).

+,

THE GLAUBER AND EIKONAL APPROXIMATIONS

@4

197

.

i 0 3 .

10.2.

I

0.1.

I

I I I 2.5 3.0 35 Eo (MI FIG.23. Total cross sections for the ionization of helium by electron impact versus log,, (projectile energy): 0, Smith; 0,Harrison; x , Rapp, Golden; +, Asundi, Kurepa; A, Schram, de Heer, Wiel, Kistemaker. Exchange effects have been treated approximately. The single-particle Glauber results are GMEI, R = Z*= 1.618; GMEZ, 1 = 1.618, Z*= 1.0, except I = 0; and GME3,R = 1.687, Z* = 1.0, except I = 0. From Golden and McGuire (170).

1.0

I

I

1.5

2.0

wave expansion of cp; ; the orthogonality of the spherical harmonics takes care of the rest. Comparisons of the results with the Born approximation and with experimental data are shown in Figs. (22-24). Golden and McGuire (170) also take a brief look at atoms with more than two electrons, especially lithium, boron, neon, and argon (but no results are presented for the latter pair). They point out that there are difficulties in normalizing the Coulomb wave function with an effective charge Z*, since at large distances one should have 2 = 1, while the latter choice gives the incorrect electron flux. Making the same assumptions for the Glauber and Born cases, they plot the ratio of the Born to Glauber total cross sections for four target atoms. The difference between Born and Glauber results becomes negligible for higher Z targets (Fig. 25). They report that for argon and neon the two approximations differ by less than 3%. The major problems are the uncertainties in the choice of q&-, the lesser problems in the choice of cpi and of use of the single-scatteringapproximation. The two other papers are worth mentioning. Mohan and Vidhani (171),

F. T. CHAN ET AL.

198 0.6

e+M(ls*)- 2e+H&ls) Be1I-Kingston

A

0.5 CO

-: 50.4 t-

w

v)

80.3

8

H

50.2

8 0.1

E ,

1.0

1.5

I

1

I

2.0

2.5

3.0

Log&

(a1

FIG.24. Total cross sections for the ionization of helium by electron impact versus log,, Asundi, Kurepa; A, (projectile energy): 0 , Smith; 0,Harrison; x , Rapp, Golden; 0, Schram, de Heer, Wiel, Kistemaker. Exchange effects have been treated approximately. The calculation of Bell and Kingston uses the Born approximation. The single-particle Glauber results are GME3, L = 1.687, Z* = 1.O, except I = 0 where Z* = L = 1.687; GMEZ, L = 1.618, Z* = 1.0, except I = 0. From Golden and McGuire (170).

using methods parallel to the early work of McGuire et al. (161), reduce the helium ionization problem to single integrals over Meijer G functions. The series expansion of these can be evaluated term by term in closed form but unfortunately is beset with the same divergence difficulty as the work of McGuire et al. Jain and Srivastava (172) attack the same problem using the methods of Thomas and Chan. They deduce a formula for fG as an infinite sum of one-dimensional integrals involving modified-Lommel functions, which is better behaved than the result of Mohan and Vidhani. Neither paper presents numerical results, but computations would appear to be exceedingly tedious.

THE GLAUBER AND EIKONAL APPROXIMATIONS

199

vo/ z FIG. 25. Ratio of Born to single-particle Glauber cross sections versus a scaled velocity. In both calculations Z* = 1 and exchange contributions are ignored. For helium, lithium, and boron, uo/Zt 0.7 at ionization threshold, and the peak of the ionization cross section occurs at 1 c v,/Z < 2. From Golden and McGuire (170).

V. ELECTRON SCATTERING FROM IONS Consider an ion with Z protons and 2'bound electrons. The coordinate representation of the Hamiltonian describing the system is

H

= Hi

+

=-

h2 Zoe2 - hZ ' '' Ze2 -v; -c v ; - j12m ro 2m =l rj j=l

where Zo = 2 - Z is the ionic charge. If we choose

F. T. CHAN ET AL.

200 and

(5.2b) then the states of the projectile particle are described by an outgoing Coulomb wave function (37). (rolko) = qko(ro)= e-'*/'r(l

r,) + ia) exp(iko (2n)3/2

x lFl(- ia ; 1; ikoro

- ik, r,)

(5.3a)

where j(k'lr) dr (rlk) = 6(k - k)

(5.3b)

and CI

= -(Z,/k,)(e2m/h2)

(5.3c)

The eikonal amplitude which describes the scattering of electrons from ions may be derived using procedures analogous to those described in Section 11. However, Coulomb wave functions must be used to represent the intermediate states of the projectile rather than plane waves. By using the addition theorem for the Kummer functions and keeping the lowest order terms in the expansion, one eventually finds the following expression for the direct Coulomb-eikonal (CE) scattering amplitude, fCE =

"S

- ( 2 ~) ~ hZ dro drj q&(ro)d$(rj) &(royrj) qCi(riJ 4i(rj) (5.4)

In the above qci(ro)and qcf(r,) are the Coulomb wave functions given by expression (5.3a). The prior and post forms of the Coulomb-eikonal exchange amplitudes also follow from the procedures in Section 11. They are found to be,

THE GLAUBER AND EIKONAL APPROXIMATIONS

20 1

and

x exp[ -

4 ki

s"

-m

The Coulomb-Glauber (CG) approximation that is used in the literature has.not been derived directly from Eq. (5.4), but by redefining Hi and K as follows (65,173,174): (5.6a) (5.6b) The advantage of using a screened Coulomb potential is that the planewave Green's function may be used in the derivation of the CG amplitude. This leads to the following modification of Eq. (5.4):

and

A mnemonic derivation of the CG approximation, in the same spirit of Section 11, can be given if we choose A to be perpendicular to the zo axis, and then do an integration by parts over zo. The resulting expression is fcGe

is s

= -ki

2a

db exp(iq b) dr, +ft(rj)

By restricting the use of fCG to inelastic scattering processes, i.e., (+&#+) = 0, using Eq. (5.6b) in Eq. (5.7), and taking fcG = fCGe, then within an

F. T. CHAN ET AL.

202

indeterminate phase factor the CG amplitude becomes

fa

=

2

- 2(iZo/ki)(e2m/ti2)

- fi]

j d b exp(iq b)[ me2

j d r j 4t

The derivation of a Coulomb-Glauber exchange amplitude has not been given in the literature because difficulties analogous to those in deriving the Glauber exchange amplitude for neutral atoms are encountered. However, a Coulomb-Glauber-Ochkur (CGO) amplitude may be derived (175). The derivation is similar to the derivation for neutrals given in Section I1 of this paper, the difference being that Coulomb waves are used. The prior and post forms of the CGO amplitude (within an indeterminate phase factor) are found to be

x

jdrk41(rk)exp(iA rk)4i(rk)[rk T zk]-izq*[r k -+ zk]-izOq*

(5.9)

Up to the present time, the scattering of electrons from positive ions has not been considered within the context of the EBS, MG, or FS approximations, and only two processes have been considered in the CG or CGO approximation. In Sections A and B these calculations are briefly described, and in Section C they are compared with experimental data. A . Coulomb-Glauber Approximation

Inserting 6 from Eq. (5.2b) into the CG amplitude given by Eq. (5.8) and evaluating the Glauber phase gives the following (65,173,174):

The CG amplitude given above has been further reduced for particular processes.

1. e

+ He+(ls) -+ e + He'(2s)

Narumi and Tsuji (173) and, independently, Ishihara and Chen (174)

THE GLAUBER AND EIKONAL APPROXIMATIONS

203

considered this process in the CG approximation. They showed that Eq. (5.10) could be reduced to a one-dimensionalintegral which they numerically evaluated. Narumi and Tsuji give the differential cross section for Ei = 108.8 eV and the total cross section from threshold to approximately 300 eV. Ishihara and Chen give the differential cross section for Ei= 100, 200, and 300 eV, and the total cross section from about 50 to lo3 eV. 2. e

+ He+(ls)+e + He’(2p)

Narumi and Tsuji have calculated the total cross section for this process. They quote results from incident energies ranging from about 50 to lo3 eV. 3. e

+ A(’-”(ls)

+e

+ A(Z-’)(n’l‘m’)

By using some clever mathematical techniques Thomas and Franco have reduced the CG amplitude for this inelastic process down to an analytic form in terms of the Meijer G functions (65). Although they did not present any numerical results in their paper (176), they made the following important observations : (1) The theoretical total cross sections for the above processes go smoothly to zero when the energy of the incident electron is equal to the excitation threshold energy. This behavior is in contradiction to experimental results; however it may be understood because a cutoff was used for the Coulomb potential and the electrons do not “fall” into the charged ion. This is not a serious defect in the theory since the CGA is not valid near threshold. They also point out that Narumi and Tsuji’s results do not go smoothly to zero at threshold, and their calculation is therefore in question near threshold. (2) By choosing the quantization axis perpendicular to the momentum transfer, one imposes restrictive symmetries on the CG amplitude, which in turn lead to selection rules. For ls+nlm excitations, the CG amplitude vanishes when (1-m) is odd. In the process Hef(ls+2p) the selection rule dictates that the m = 0 amplitude vanish, and they show that the m = f 1 amplitudes behave as q-6 for large scattering angles rather than as the typical Coulombic 4-4 behavior. (3) The restrictive symmetries imposed on the CG amplitude precludes the possibility of the approximation reliably predicting line radiation that could be observed in coincidence with the excitation process. B. Coulomb-Glauber-Ochkur Approximation The CGO amplitude given by Eq. (5.9) has been reduced to an analytic form by Williamson, Foster, and Kwong (275). They considered the process

F. T. CHAN ET AL.

204

e + He+(ls) +e + He'(2s) and calculated direct, exchange, and post/prior symmetrized differential cross sections. They have also calculated symmetrized total cross sections for a range of incident energies. The symmetrized total cross sections again go smoothly to zero at threshold energy in contradiction to observations. It should also be noted that the ambiguity associated with the indeterminate CGO phase factor was utilized in a manner so that an overall indeterminate phase factor could be factored out of the symmetrized amplitudes. C . Comparison with Experimental Data and Evaluation

The differential and total excitation cross sections for the processes e + He+(ls+2s) and e + He+(ls +2p) have been calculated using a variety of approximation methods besides those mentioned above (106, 177-280);

l0Jl

'

100

I

I

I

400

MmI m [BV]

I

I

I

100

FIG.26. The e + He' (Is) -+ e + He' (2s) total cross section in units of nu;. -, CB approximation(177,178);----.the CBO (177,178) approximation; -.-., the CG approximation (173-176); and -.-., the post CGO approximation (175). The circles are the experimental data points of Ref. 181, and the lowest energy shown on the plot is the threshold energy. From Williamson et al. (175).

THE GLAUBER AND EIKONAL APPROXIMATIONS

205

however limited experimental data are available for comparison between theory and recent experiments (181).Seaton has discussed the former process in his review paper (36), and up to the present time there are no experimental data available for the latter process. The total cross section for e + He+(ls+2s) versus incident electron energy is displayed in Fig. 26. The CB, CBO, CG, and symmetrized CGO cross sections are compared with the experimental data of Dolder and Peart, who corrected for cascades and normalized their data with the CB approximation at high energies (11). It appears that the CG or the CGO approximation gives reasonable results for intermediate energies, but it seems premature to draw any definite conclusions. It would be more satisfying if comparison of experimental and theoretical differential cross sections could be made. VI. OTHEREIKONAL-TYPE APPROXIMATIONS There have been many attempts to improve the eikonal approximations, some of which we have already described-so many, in fact, that we cannot attempt to discuss them all thoroughly in this review. Therefore, what follows is a brief survey of a few of these. A . The Blankenbecler-Goldberger Approximation

The Blankenbecler-Goldberger approximation (to be referred to as B-G), initiated by Blankenbecler and Goldberger (182)in 1962, has recently been applied by Kamal, Richardson, and Teshima (183)to the elastic e- + H and e- He scattering problems. In the language of Green's functions, the Green's function that leads to the Glauber amplitude is, for potential scattering [cf. Eq. (2.13b)l

+

Gf(r, r') = (- i/2k)S2(b - b)O(z - 2 )ei"'-'')

(6.1)

In Eq. (6.1) the (Heaviside) step function O(z - z') embodies the same basic assumption employed by Glauber, that significant propagation occurs only in the forward direction. Relaxing this restriction, one obtains the B-G Green's function ; namely, G:-G(r, r') = ( - i / 4 k ) d2(b - b')eik('-'')

The B-G amplitude that results from Eq. (6.2) is

(6.2)

206

F. T. CHAN ET AL.

The function X(b) is the same eikonal that appears in the Glauber approximation via

s

r,(b) = l~ls(r)12[1 - (Ib - sl/b)2iqdr = 1 - exp[ih(b)]

(6.4)

for e- + H scattering. A closed form expression of T,(b) in terms of the modified-Lommel functions is given in Thomas and Chan (114). Similar expressions for the e- + He elastic scattering eikonal can be obtained and are not any more difficult to evaluate than those for e- H problem (183), To illustrate their methods, Kamal et al. (183) present numerical results for e- + H, e- + He elastic h / d Q at Ei = 100 eV in both Glauber and B-G approximations. They notice that for q2 4 1, there is hardly any difference in da/dn between two approximations. For q2 % 1, the difference becomes more marked with the Glauber prediction staying higher (recall that the Glauber values are already below the experimental points). However, as da/& drops by three orders of magnitude at q2 z 10, the B-G prediction for a, will not be much lower than that of the Glauber prediction. Although the B-G approximation seems applicable to inelastic eH, e- + He, and possibly to e- + Li (including elastic) collision processes, no conclusion can be drawn at the present time.

+

+

B. The Glauber-Angle Approximation The so-called Glauber-angle (GA) approximation is an outgrowth of several years of study by J. C. Y. Chen and his associates (184). Since it has already been reviewed (29),we shall be brief here. In the eikonal approximations discussed heretofore, the trajectory of the incident particle has been taken as a straight line. In the Glauber approximation, the additional assumption is made that this trajectory is perpendicular to the momentum transfer q. For elastic scattering this is a compromise between the incident and final direction, ki and kf, and may be responsible for the effectiveness of this approximation at relatively wide angles. Chen et al. (185) argue that this choice of direction, which appears very questionable for inelastic cases, actually compensates for some of the errors of the linear trajectory assumption, thus giving good results for lower energies, where the straight trajectory assumption is clearly invalid. Nevertheless, the trajectory employed should ideally be, in the spirit of semiclassical approximations, as close as possible to the actual classical trajectory (184,185). In this spirit they take as a trajectory two semi-infinite

THE GLAUBER AND EIKONAL APPROXIMATIONS

207

straight-line segments intersecting near the target (185,186). They take the incident segment parallel to ki and the emergent segment parallel to kf. This choice preserves much of the simple structure of the Glauber approximation. At the same time they assume, as in the Glauber case, that the longitudinal part of q may be omitted. The resulting approximation is the GA approximation. Unfortunately while the formalism appears simple, the reduction techniques that have significantly advanced Glauber calculations, such as those of Thomas and Gerjuoy, mentioned often previously, are not available in the GA case, and multidimensional integrals must be evaluated. Chen et al. (186) show that in the GA case the only alteration to the Glauber formulas is in the eikonal phase. In the scattering of an electron by an n-particle target, this phase retains its additivity property : XGA =

c xy

(6.5)

i

where xy is the GA phase shift due to the electron's interaction with the jth particle of the target. The two-particle phase xyA differs from the corresponding straight-line Glauber by a small correction :

xy = xy + Axj

(6.6a)

where the Glauber phase is (6.6b) with V, the two-particle potential, while the correction is (6.6~)

In the latter 28 is the classical scattering angle and d is the distance of closest approach of the angle trajectory to the origin. For e- H scattering, for example

+

xG = (2/ki)ln(lb - sl/b)

(6.7a)

as usual, while Ax = -

[Ib

- sIz +

b tan j? - sin fi (b tan j? - z ' ) ~ ] ~ ' ~

where z' is the z component of the position vector for the target electron. The four-dimensional integrals have been evaluated numerically (186)for both total and differential cross sections for both elalstic e H(1s) scatter-

+

F. T. CHAN ET AL.

208

I

0

20

l

\

60 8 (dog)

l

90

I

120

FIG. 27. Comparison of angular dependence of the differential (e, H)(n = 2)-excitation cross section at 50 eV in the straight-line and angle Glauber-eikonal approximations and in the first-order Born approximation with experimental data (187) that are normalized to the Glauber-eikonal result at an angle of approximately 42", where the two theoretical curves corresponding to the straight-line and angle approximations come together: -, Glauber (angle); ----,Glauber (straight); ----, Born, a, experimental. From Hambro, Chen, and Ishihara (186).

ing and inelastic scattering to the n = 2 levels. At the same time the Ax = 0 (pure Glauber) case was evaluated as a check on the numerical integration. Sample results are shown in Figs. 27 (186,187) and 28 (186,188). It is seen that the GA differential cross section is somewhat larger than the standard Glauber at wide angles and lies closer to the experimental data. The total cross sections are only slightly different at lower energies ( 5100 eVj and indistinguishable at higher energies where the straight line is expected to be a better approximation. The difference in wide-angle behavior does not significantly alter the total cross section because the numerical value is small at wide angles.

209

THE GLAUBW AND EIKONAL APPROXIMATIONS I

'

I

1

'

1.2

-

1.0

-

-

co 0.0 -

-

-

0

b

Y

-

0.6-

-

0.4

A

-

0.2

o.o10 ~

15

20

30 40 50

70

100

150 200 300 400500700

E (eV) FIG.28. Comparison of the energy dependence of the total (e, H) 2p-excitation cross section in the straight-line (-- - -) and angle (-) Glauber-eikonal approximations with experimental data (A)of Long et al. (188).From Hambro, Chen, and Ishihara (186).

C . The Two-Potential Eikonal Approximation

Ishihara and Chen (189) have identified another possible source of difficulty in applying the Glauber approximation to atomic collision problems and have suggested a cure. They note that the nuclear potential - Z / r is the dominant contributor to wide-angle scattering at high energies and dominates all scattering at lower energies. This is because at high energies and wide angles the electron penetrates close to the nucleus, while at lower energies only the lowest partial waves contribute to the scattering, and these correspond to close approach to the nucleus. However, as a semiclassical approximation, the Glauber approximation requires VI < E, and this may be violated as the electron approaches the nucleus. In fact, in elastic scattering, where the Born approximation is often superior to the Glauber at low energies at all angles, Ishihara and Chen suggest using a two-potential approach, i.e., separating the offending nuclear potential for a strict quantum mechanical treatment, while treating the remainder by eikonal techniques. That is, the total potential V is written in the form

I

v = v, + v1

(6.8)

where Vl is assumed to be spherically symmetric and to contain the problematic singularities, and so to require careful treatment in the full quantum

F. T. CHAN ET AL.

210

mechanical manner (Ishihara and Chen use a partial wave expansion). V,, on the other hand, is assumed to satisfy IVol 4 E for all r and can be treated by semiclassical means. Assuming V, also spherically symmetric, the partial-wave expansion for f can be written, in the case of potential scattering, as

Je)

=

k-1

c (21 + i)q;P,(cos e)

(6.9)

1

where, corresponding to the decomposition (6.8) :

'I; = exp(id',')) sin 61')

+ exp(2id1°))exp(i6l')) sin 61')

(6.10)

Here Sl') is the phase shift for the potential V, while 61°) = 6, - Si') with 6, the phase shift for V. This is the two-potential theorem in partial-wave form (190). It should be noted that 6io) depends on V, as well as on V,. In the semiclassical spirit, for d',') they replace the sum over 1 by an integral over b, the two being related by b, = (1 + 4)/k. The result is Ad) = ik

:J

b db (1 - exp[ix(b)])Jo(2kb sin 90)

+ k-' C (21 + 1) exp[ix(bl)] exp(i8l')) sin 6l')

(6.11)

1

where the eikonal phase ~ ( b=) 26i0),and can be calculated by eikonal or Glauber methods. The potential scattering results can be generalized to electron-atom scattering using the frozen-core approach in a natural way. Ishihara and Chen give the explicit results for elastic electron-hydrogen scattering, including exchange ; i.e., (6.12a) where the frozen-core "static" potential is

+ r-')e-2r

Kt = -(1

(6.12b)

and the exchange part is an integral operator : r

(upper sign for singlet, lower for triplet states). Kxchis short range and may be ignored in calculating x. That is, we take

- -r-'

0 -

+ Ir - r'l-1 -

Kt

(6.13)

21 1

THE GLAUBER AND EIKONAL APPROXIMATIONS

which yields, by standard methods [as in (6.B.3a)], the major part of the eikonal phase : [K0(2b)

+ bKl(2b)]

(6.14)

The first term is the familiar Glauber phase for the full V (less exchange). We note that the cancellation of the I = 0 singularity in Vo gives a smooth xo-the singularity at b = 0 is gone. In fact, x is smoothly varying, which is an advantage in numerical work. In general, x also depends on V,, but the above xo did not contain this. The full x is

x

= xo

:j

+ -2k V0(z = 0)

d2{1

zr

[z12

- (r2Vl/E)]1’2

(In the present case V,, is used for V,) The scattering amplitude is then 2xi

d2b exp(iq. b)[Tfi(b) - 11

+ -k1 , (21 + 1)exp(i8f”)sin G~’)P,(cosO)Tfi(br)

(6.16a)

where

r,i(b) = (flexdi~)li)

(6.16b)

In a recent paper, Kumar et al. (191) proposed a simplified version of the above procedure. They do not use a partial-wave expansion, but instead use a distorted-wave (DW) approach. The full two-potential amplitude is (192, 193)

f = - (2.)2[(@fl

Vol$i+ )

+ ($F I VI p: >I

(6.17)

where Y’ is an eigenfunction of the hamiltonian with both potentials, Qf with neither, and $,; $; with just V,. As in the DW Born approximation, the exact wave function in (6.17) is replaced by $,: the wave function with potential Vo only. Thus f

f‘” + f‘2’

(6.18a)

where

P)= -(242(@fI &I$:>

(6.18b)

f2)= -(2x)2($:Iv11$:)

(6.18~)

212

F. T. CHAN ET AL.

8 (dql

FIG. 29. Differential cross sections for e - H elastic scattering at incident electron energy of 20 eV: Present calculation (---) compared with Born calculation (----), Glauber calculation including exchange effects in Ochkur approximation (--.--. -), two-potential eikonal calculation of Ishihara and Chen (189) (--..--..), and the experimental data of Williams (88) ( 0 ) .Relative angular-distributiondata of Teubner et al. (86) ( 0 )are also indicated. From Kumar et al. (191).

Since Vo is assumed smooth, and IVol 4 E everywhere, &, $2 can be calculated by semiclassical means, and f") is just the amplitude for the pure Vo potential, evaluated by the Glauber approximation in the present work (i.e., they also ignore the longitudinal part of q) Kumar et al. (191) illustrate using the same example as Ishihara and Chen (189): elastic e- + H(1s) scattering. For Vo they use precisely Eq. (6.13) (they do not use V,,,,,) and deduce (6.14).The techniques of Franco (31) are used to reduce the resulting integrals in Eq. (6.18) to two-dimensional form. Further reductions could not be achieved because of the K O and K, Bessel functions in Eq. (6.14). Exchange is included in the Ochkur approximation. In Figs. 29-32 we show some results of the calculations of Ishihara and Chen (189) and Kumar et al. (194, for the differential cross section at low energies (20,30 ev) and higher (50, 100 ev) energies. It is seen that the Kumar method is not as satisfactory as the Ishihara-Chen, but in view of its simplicity, it makes a worthwhile improvement on the Glauber-

213

THE GLAUBER AND EIKONAL APPROXIMATIONS

8 (dcql

FIG.30. Same as in Fig. 29 but for 30 eV. Open circles correspond to the experimental data of Lloyd et al. (86).From Kumar et al. (191).

Ochkur at lower energies. At the higher energies, the Ishihara-Chen fits rather well, but the Kumar result is scarcely better than the Glauber-Ochkur. D. The Modified-GlauberApproximation of Birman and Rosendorfl As we have seen, the Glauber approximation has several deficiencies. Birman and Rosendorff (194) have focused, in particular, on the logarithmic divergence of the scattering amplitude in the forward direction in elastio and some inelastic-scattering processes. This divergence implies that the optical theorem is not satisfied in these cases. To treat this defect, they have proposed a modified approximation which, however, has not yet been applied to any explicit calculations. Consider, for simplicity, the case of electron-hydrogen scattering. The wave function "2, satisfies the integral equation (2.12). The Green's function, Eq. (2.12b) can be written in explicit form

h2 G?(r0, r ;rb, r') = - -2n2 C pN(r')*pN(r)exp(ikNIro- rbl 2m N Iro - rbl

(6.19)

F. T. CHAN ET AL.

214

FIG.31. Same as in Fig. 30 but at 50 eV.----, corresponds to the eikonal-Born series calculations of Byron and Joachain (55,193). From Kumar et al. (191).

where qNis a hydrogen wave function and kN =

(6.20)

J(2m/hz)(E - EN)

The s u m in Eq. (6.19) includes the continuum states; where EN > E, kN is taken positive imaginary. Now, to simplify this expression, Birman and Rosendorff approximate kN by a constant, independent of N, which they call ki - Ak, hki being the momentum of the incident electron. Ak is then a free parameter, related to a mean excitation energy. The outgoing spherical wave in (6.19) may be removed and the summation performed via the closure relation : GI+)(ro,r;r&r')z

-

[3 -

2n2

exp[i(ki - Ak)lro - rbl] 6(r Iro - fol

- r')

(6.21)

THE GLAUBER AND EIKONAL APPROXIMATIONS

215

8 (dql

FIG.32. Same as in Fig. 31 but at 100 eV. From Kumar et al. (191).

They next approximate the wave function 'Pii by a modulated plane wave 1 Yi,(ro,I) = 03/2exdiki ro) qi(r)P(ro, r)

(6.22)

and, using Eqs. (6.21,22) in Eq. (2.12) and expanding to first order in l/ki, it is found that p satisfies the one-dimensional integral equation :

x exp[

- iAk(zo - io)]p(bo, zb ;r) dzb

(6.23)

where we have written ro = bo + zoh, etc. The solution to this Volterra

216

F, T. CHAN ET AL.

equation is easily seen to be

p(ro,r) = 1 - iexp[i(x - zoAk)] x exp{i[zbAk - x(bo, zb ;r)]} dz;

(6.24)

where

x(ro,r) =

m 1 - --

V(bo, zb ;r) dzb

(6.25)

We note that if Ak = 0, Eq. (6.24) reduces to

dro, r) = expCix(b0, zo ;r>3

(6.26)

which, when inserted in Eq. (6.22), is just the usual Glauber wave function [Eq. (2.15)]. Therefore, the standard Glauber approximation corresponds to zero mean excitation energy. We shall not trace the derivation any further. Suffice it to say that with Ak # 0, it is shown (194) that the scattering amplitude in this approximation does not diverge in the forward direction. The parameter Ak can be adjusted so that the optical theorem is satisfied, that is, so that Im f(0)agrees with the experimental total cross section.

VII. APPLICATIONS OF THE GLAUBER APPROXIMATION TO ELECTRON-MOLECULE COLLISIONS In spite of numerous applications of the Glauber approximation to electron-atom scattering problems, there have been only a few attempts to apply it to the atom-atom and electron-molecule scattering. This is because of the multicentered nature of the targets involved and the complexity of possible excitation channels. The difficulty of the problem increases enormously and rapidly with the increasing number of particles in the system. Although Chen (195) has proposed a method for treating problems of multicenter targets and of composite incident particles, no actual calculations following his formulations have been reported yet. However, the complication of these collision processes can be reduced with further simplifying assumptions, and a few theoretical results (196-202) have been published. Among these, the works of Byron et al. (196) on H (2s) quenching in H (2s) + He (Is2) collisions, Chang et al. (199) on e- + H, vibrational excitation, and Yates and Tenney (297) on high-energy scattering of electrons by N, , I,, and V, have been reviewed by Gerjuoy and Thomas (29). Bhattacharyya and Ghosh (200) have investigated the elastic scattering

217

THE GLAUBER AND EIKONAL APPROXIMATIONS

of electrons from molecular hydrogen by using the Glauber approximation within the framework of potential scattering. These authors replace the actual electron-molecule interaction by an optical potential, which includes the static potential and the effects of polarization and exchange. This approximation reduces the original multidimensional integral (e- H,) to the two-dimensional Glauber potential scattering integral, which is further reduced to a one-dimensional integral over b (the final integration over b was done numerically). Bhattacharyya and Ghosh (200) have calculated the total and the differential cross sections in the energy range 9.4-100 eV and compared the results with the experimental values (203, 86). They find that their theoretical results are reliable for incident electron energies 220 eV. Huang and Chan (202) have also studied the same collision process by using the Glauber approximation (including the exchange effect) within the framework of independent scattering centers (204, 205). The predicted values in Huang and Chan (202) for differential cross sections also give reasonable agreement with experimental data at incident energies of 30, 50, 100, and 200 eV. However, the “reasonable agreement” should be regarded as “accidental” since the independent atomic model is a high-energy approximation (the incident de Broglie wavelength must be smaller than the internuclear separation). In this regard, the numerical results for EiI100 eV reported in Ref. 200 are more reliable. It should be pointed out that both calculations (200,202),do not reproduce the dip structure of the experimental observations in electron angular distribution around 100”. Also, the extension of the elastic scattering using potential scattering and independent atomic model to excitations is not in sight at the present time. Bhattacharyya and Ghosh (201) have attempted to calculate the Glauber amplitude to rotational excitations of molecular hydrogen by electron impact in the framework of the adiabatic approximation. In this approximation, the target molecule is a rigid rotator, and only the amplitude of the elastic scattering of an electron from the target molecule held fixed in space is considered. By 40-eV electron impact, these authors obtain a minimum in rotational cross section at 35”, whereas the observed minimum is at 40”. However, the calculated cross sections are, on the average, 2.5 times smaller in magnitude than the observed values. The discrepancies are probably due to the exchange effect, which is not taken into account.

+

VIII. SUMMARY AND CONCLUSIONS Applications of the Glauber and eikonal approximations in atomicscattering theory have made substantial progress during the past decade. Many authors have contributed to this progress and in this concluding sec-

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tion of the paper we will highlight the general progress that has been made rather than discuss specific work of specific authors. Closed-form expressions for Glauber direct and Glauber-Ochkur exchange amplitudes have been derived for elastic and inelastic electron scattering from hydrogenic systems. In principle, limiting forms of the scattering amplitude can be studied as a function of momentum transfer, but in practice one is limited to studying this behavior for low-lying states because of the large number of terms that may be present in it. The unrestricted Glauber or eikonal direct and exchange amplitudes have been reduced to the numerical evaluation of a double integral for hydrogenic systems. When both of the above approximations are compared with the limited experimental data available for electron hydrogen scattering, it is found that they give reasonable to good results (from low to high energies) for differential cross sections up to moderate scattering angles (excluding the H (ls+3d) case). The inclusion of exchange effects are found to make small, but significant, improvements in the differential cross sections. However, large-angle scattering seems to be best described by the inclusion of second Born terms in the scattering amplitude. In this regard eikonal-Born series, and the modifiedGlauber approximation are especially noteworthy. The Glauber and Glauber-Ochkur approximations have also been reasonably successful in describing electron scattering from moderately complex target systems such as helium and lithium. Large-angle scattering again seems to dictate the inclusion of second-order Born terms in the scattering amplitude in order to have better agreement between theoretical and experimental differential cross sections. This is particularly true at lower energies. The inelastic scattering of electrons from He' has recently been described within the Coulomb-Glauber, and Coulomb-Glauber-Ochkur context. The comparison of experimental and theoretical total cross sections is encouraging, but comparisons of differential cross sections would be a more rigorous test of the theory. The success of the Glauber and eikonal approximations in atomic scattering theory has been substantial during the past decade, but there is still room for improvement. This is particularly true at lower energies and larger scattering angles. The success of the eikonal-Born series and modifiedGlauber approximations present strong arguments for the inclusion of second Born terms in the scattering amplitude. Additional calculations and the extension of these theories to ionization and electron scattering from ions seems desirable. It seems premature to state that the Glauber, eikonalBorn, or modified-Glauber approximations are the final answer for an intermediate-energy scattering theory. Perhaps the next decade will provide us with sufficient experimental and theoretical data to make a definite statement.

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219

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+

224

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P. K. Bhattacharyya and A. S. Ghosh, Phys. Rev. A [3] 12,480 (1975). P. K. Bhattacharyya and A. S. Ghosh. Phys. Rev. A [3] 14, 1587 (1976). J. T. J . Huang and F. T. Chan, Phys. Rev. A [3] 15, 1782 (1977). S. Trajmar, D. G . Truhlar, and J . K. Rice, J . Chrm. Phys. 52,4502 (1970). L. D. Landau and E. M. Lifshitz, "Quantum Mechanics," 2nd ed., Sect. 148. AddisonWesley, Reading, Massachusetts, 1965. 205. P. G . Burke, "Atomic and Molecular Collisions," Lect. Notes, p. 190. U. K. At. Energy Auth., Aldermaston, England. 1967.

200. 201. 202. 203. 204.

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL. 49

Flicker Noise in Electronic Devices A. VAN DER ZIEL Electrical Engineering Department University of Minnesota Minneapolis, Minnesota

..

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Statement of the Problem. History of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . B. Divergent Integrals . . . . . . . . . 11. Unusual Examples of l/f Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Open-Circuit Noise Voltage of a Lossy Capacitor . . . . . . . . . . . . . . . . . . . . . . B. Flux Flow Noise in Type I1 Superconductors. . . . . . . . . . . . . . . . . . . . . . . . 111. Flicker Noise in Vacuum Tubes . . . . . . . . . . A. Flicker Noise in Field Emission Diodes B. Flicker Noise in Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... C. Flicker Noise in Secondary Emission D. Flicker Noise in Vacuum Diodes and E. Flicker Partition Noise in Positive Grid Triodes and Pentodes . . . . . . . . . . . . . . . IV. Flicker Noise in Resistors A. Fundamental Experim B. Noise in Metal-Meta C. Flicker Noise in Semi D. Miscellaneous Topics on l/f Noise in Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. l/f Noise in Solid-state Devices . . . . . . A. l/f Noise in Tunneling Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Flicker Noise in Schottky Barrier Diodes C. 1 Noise in Junction Devices . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Flicker Noise in JFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Flicker Noise in MOSFETs

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

VI. Miscellaneous Problems A. Formal Explanation

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

225 225 226 23 1 23 1 233 234 235 231 238 239 243 244 245 256 258 268 269 269 212 214 219 28 1 289 290 290 29 1 292 292

I. INTROPUCTION A . Statement of the Problem. Hiktory of the Problem

The name “flicker noise” refers to noise phenomena with a spectrum of the form AZB/p, where A is a constant, Z is the current, f is the frequency, 225

Copyright 0 1979 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-014649-5

226

A.. VAN DER ZIEL

and the q o n e n t s j? and y are also constants. In its most restricted form one requires y = 1 and speaks of true l/fnoise, but often this requirement is relaxed; for example, flicker noise in saturated vacuum diodes with a tungsten filament has y N 2. The exponent j? often has a value of about 2, though deviations are not uncommon. For example, flicker noise in saturated vacuum diodes with a tungsten filament has p N 2, which is in contrast with the shot noise in these diodes, whose spectra show a linear current dependence. Flicker noise in vacuum tubes was discovered by Johnson (1) in 1925, whereas Schottky (2) gave the first interpretation in 1926. In 1937 Schottky (3) showed that flicker noise was more strongly suppressed by space charge than shot noise. Flicker noise in carbon microphones and in carbon resistors was first studied in detail by Christenson and Pearson (4). Here the noise shows roughly a quadratic voltage (or current) dependence, and y IT 1. Later it was found that flicker noise in one form or the other occurred in most electronic devices. This almost universal occurrence (with a few notable exceptions, however!) of flicker noise has raised the question whether there might be a universal flicker noise mechanism behind these various flicker noise manifestations. In the opinion of this reviewer, such a universal physical mechanism is rather unlikely. The flicker noise processes have in common related spectra of the form l/fy with y 21 1; this relationship is mathematical rather than physical. In order to make progress in the understanding of flicker noise, one must try to understand the physics behind it and so comprehend the reasons why different physical mechanisms can give rise to the same type of spectrum. The remainder of this section deals with divergent integrals occurring in l/f spectra. Section I1 discusses two processes that have a llfspectrum and so qualify as llfnoise; but they are usually not recognized as such. Section I11 deals with flicker noise in vacuum tubes; while these devices have been almost completely replaced by solid-state devices, their flicker noise mechanisms are of interest in that they shed light on the general problem of flicker noise. Section IV discusses flicker noise in resistors and photoconductors, whereas Section V deals with flicker noise in solid-state devices. Finally, Section VI discusses deviations from y = 1 and a fundamental noise mechanism that may give rise to llfspectra. B. Divergent Integrals

Spectra of the form l/fy for 0 1) or the upper limit (for y < l), or at both limits (for y = 1, true l/fnoise). We shall

227

FLICKER NOISE IN ELECTRONIC DEVICES

see that this violates some commonsense notions about fluctuating qualities. To demonstrate this, we start with the discussion of the Wiener-Khintchine theorem, which can be stated as follows (5) : Let the autocorrelation function X(u)X(u+ s) of a stationary random variable X(t),having x(t)= 0, be absolutely integrable for the interval 0 Is I 00, and let the spectrum S , ( f ) of X(t)be defined as m

S,(f) = 4

[ Jo

X(u)X(u+ s) cos ws ds

Then S , ( f ) exists and the inverse transformation is equal to X(u)X(u+ s); that is

X(U)X(U + S) =

r

S , ( f ) cos osdf

(la)

+

The condition of absolute integrability of X(t)X(t s) is a sufficient condition for the simultaneous validity of the pair of equations (1) and (la). If the condition of absolute integrability is not satisfied, then the pair of equations (1) and (la) may not be simultaneously valid; that is, even if the integral in (1) exists, Eq. (la) may not be valid. Let us now investigate the convergence of the integral

in more detail. A sufficient condition for convergence at the upper limit (sl + 00) is that for large s IX(u)X(u+ s)l goes to zero faster than l/d with 6 > 1. A sufficient condition for convergence at the lower limit (so+O) is that for small s IX(u)X(u s)l goes to infinity more slowly than l/se with E < 1. Note that for 0 < E < 1, we have p = 00 ; this is mathematically allowed since the integral (2) converges at the lower limit. If the integral (2) diverges at the lower limit (so 0), then the definition (1) is invalid since the right-hand side of (1) diverges at the lower limit for all w. If the integral (2) diverges only at the upper limit (sl co),then the integral of the right-hand side of (1) exists for o # 0 as long as X(u)X(u+ s) goes to zero for s + 00, so that Eq. (1) can be used as a definition for S , ( f ) . If then X(u)X(u s) 2 0 for all s, S,(O) = co. This may in itself be allowed, but there is now no absolute guarantee that the transform (la) of S , ( f ) exists and is equal to X(u)X(u s); that needs to be investigated in each individual case. Let us now investigate what we find in practice. Actually, X(u)is always quite small, so that p is finite and X(u)X(u+ s) is of bounded variation.

+

-+

-+

+

+

228

A. VAN DER ZIEL

There can be, therefore, no doubt that the integral (2) will converge at the lower limit. Moreover, all physical systems are finite and hence they have finite time constants, including a largest one. There can therefore be no doubt that IX(u)X(u + s)J will go to zero faster than 1/sd with 6 > 1 for large s. Only in idealized models, such as diffusion into a half space, can the latter rule be violated. In practice, therefore, the integral (2) will always converge at both limits. An infinite value of in itself does not cause mathematical difficulties, provided that the integral (2) converges at the lower limit. As an example, consider the case where X(u)X(u + s) = a d(s), where a is a constant and 6(s) is the Dirac delta function. We then have' that S,(f) = 2 4 and Eq. (1) is valid because X(U)X(U+ S ) = 2a

cos osdf = u 6(s)

(3)

according to a well-known definition of 6(s). Let us now investigate the integral

j;

Sx(f)df

(4)

If it converges at both limits, we can use Eq. (la) as definition for X(u)X(u s); X(u)X(u + s) then exists and its transform (1) is equal to S,(f). If the integral diverges at one or both limits, difficulties may arise. Divergence of (4)at the upper limit may in itself not be harmful. As an example, consider the case X(u)X(u + s) = a 6(s); then S,(f) = 2u and the integral (4)diverges at the upper limit. Nevertheless Eq. (3) is valid, so that the inverse transform of S,(f) is equal to a 6(s), as required by (la). Divergence of (4)at the lower limit is much more serious, for in that case the integral on the right-hand side of (la) diverges at the lower limit for all s and therefore (la) cannot be used to define X(u)X(u + s). This should be avoided. While the divergence of (4) at the upper limit may not always cause difficulty, it violates the commonsense rule that must be finite. In reality, therefore, integral (4)should converge at both limits. Hence in all noise spectra encountered in practice, there will be a lowfrequency f,, however small but not zero, below which the spectrum S(f) varies more slowly than l/f, so that the integral (4)converges at the lower limit (fo +0). There must also be a high-frequency fb, however large but

+

' 4s:

6(s) ds = 2jTm 6(s)ds = 2, according to the definition of 6(s).

FLICKER NOISE IN ELECTRONIC DEVICES

229

not infinite, above which the spectrum varies faster than l/f; so that the integral (4)converges at the upper limit (fi + 00). The integral

then exists, the Wiener-Khintchine theorem is certainly valid, and our commonsense notions about noise quantities are satisfied. We can arrive at the same result from the point of view of measurements. Suppose we have a spectrum of the form a/f. At very high frequencies the noise is so small that it drowns in the noise background of either the device itself or the measuring equipment. Hence the existence of an a/f spectrum cannot be proved above a high-frequency fd that is a characteristic for either the device or for the measuring equipment. At very low frequencies, when recording the noise and using fast Fourier transform techniques, a measurement at the frequency f needs a time of the order l/J Hence it takes 28 hr to record and measure down to lo-’ Hz; measuring down to zero frequency would require an infinite measuring time. There is, therefore, a lower frequency f,’ below which the existence of a l/f spectrum cannot be proved; this frequency depends on the patience of the investigator and on the stability of the device and the measuring equipment. Beyond the range f,’ If < fd we know nothing about the spectrum; statements like “I believe that l/f noise persists for 0 If I00” are nonverifiable and hence do not belong to the realm of science. It is perfectly compatible with the measurements to assume that S x ( f ) has a frequency dependence for f 4 f,’ and f % fd such that the integral (4)converges at both limits. Another important theorem used in noise calculations is Carson’s theorem (51 which can be formulated as follows. Let a stationary random variable Y(t) be written as Y(t) =

c F(t i

ti)

(5)

where the F(t - ti)% are pulses of the same shape occurring at random instants ti, and F(t - t i ) = 0 for t < ti. Let F(u) be absolutely integrable for the interval 0 5 u I; 00 and let the pulses occur at the average rate 1; then the spectrum of Y(t) may be written as S,(f) = Ul$(f)12

where

(54

230

A. VAN DER ZIEL

The condition of the absolute integrability of F(u) is again a sufficient condition for the validity of the theorem, If that condition is not satisfied, Eq. (5a) might not be valid, even if the integral (5b) exists. We illustrate this with an example. Schonfeld has observed that the pulse (6) F(u) = l/u112 for u > 0

and

F(u) = 0 for u < 0

(6)

gives l/f noise. For if we put u = mu, @(f) becomes

where the integral I exists. Applying Carson’s theorem yields S,(f) = 2Jl@(f)l2= (2I/w)11l2

(74

But since F(u) is not absolutely integrable for the interval 0 Iu I 00, the last step is in doubt, and hence the conclusion is in doubt. We may also illustrate this as follows. A series of random pulses of the form (6) should have an autocorrelation function (5)

+

Y(t)Y(t s) - (p>2 =

u”*(u

+ s)1/2

Unfortunately this integral diverges at the upper limit for all s and at the lower limit for s = 0, i.e., Y(t)has no well-defined autocorrelation function. We can eliminate the difficulty by modifying F(u). If

F(u) = 1/u1I2 for 0 Iu I ul,

F(u) = 0 otherwise

(9)

then for large but finite ul, the function F(u) is absolutely integrable and S,(f) varies as l/f for f % fl, where fl is related to u1 and is very small if u1 is sufficiently large. Moreover, S,(O) is finite. The new model violates only the commonsense rule that Y 2 - (p>’ must be finite. We can eliminate the latter difficulty by further modifying F(u). If

F(u) =

for 0 Iu Iuo,

F(u) = 1/u’I2 for uo Iu Iul,

F(u) = 0 otherwise

(10)

then F(u) is absolutely integrable, the integral over the spectrum exists, and - (Y)’is finite. For uo Q ul there is now a large frequency range l/u, Q f 4 l/uo for which the spectrum viries as l/f.

231

FLICKER NOISE IN ELECTRONIC DEVICES OF 1/ f NOISE 11. UNUSUAL EXAMPLES

A . The Open-Circuit Noise Voltage of a Lossy Capacitor

We shall show that the open-circuit noise voltage of a lossy capacitor shows a llfspectrum. This is rather interesting, for we know that in this case the noise is thermal noise of the dielectric losses. Let E = E' - jd' be the complex dielectric constant of the dielectric used in the capacitor; then its loss tangent is defined as tan 6 = t n / d

(1 1)

The capacitor can now be represented by a capacitance C = dc,A/d and a resistance r = tan 6/wC in series; here E~ is the permittivity of free space, A the electrode area of the capacitor, and d the thickness of the dielectric. Hence, since the noise is thermal noise of the resistance r, the open-circuit noise voltage has a spectrum

S,y) = 4kTr

=

4kT(tan6/wC)

(12)

Since in normal dielectrics C and tan 6 are practically independent of frequency, S,y) represents a 1/f spectrum. This is a case of 1/f noise in which no dc voltage needs to be applied to the device to detect it. In order to explain why tan 6 is practically independent of frequency, one usually introduces a dielectric aftereffect with a continuous distribution g(z) dz in relaxation times such that

jr

g(z)dz = (normalization)

For an aftereffect with a single relaxation time z, one finds

where 4 is the low-frequency dielectric constant (at w = 0) and E, the highfrequency dielectric constant (for wz % 1). One finds this behavior for polar molecules dissolved in nonpolar dielectric liquids. In 1907 von Schweidler (7) proposed to introduce a distribution in time constants in order to explain dielectric properties of solids. In this case Eq. (14) must be written El = E m

+ (E, -

Em)

1

+ w2z2'

Eff

= (E,

- Em)

wzg(z)dz 1 w2z2 (15)

+

232

A. VAN DEX ZIEL

For the case of small losses

[(E,

- 8,)

< E,]

In the particular case that

we obtain

For l/z,

Q w

4 l/t1 this reduces to

which is independent of frequency. For w % 1/z, and w -4 l/z,, tan6 goes to zero. It is not quite clear who first introduced this distribution in z ; for details see Gever's papers (8).Van der Ziel(9) and du Pr6 (10) applied this distribution to explain l/f noise in various devices and materials. One can understand Eq. (16) when one attributes the losses to impurity ions trapped in a potential well, such that it takes an activation energy En to bring it out of the well. In that case one can write for the time constant at the absolute temperature T z = toexp(E,/kT)

(17)

since the ion has a probability B exp( - E , / k T ) to have an energy E,; here B is a constant and zo is a time constant of the order of sec. Converting (16) to a distribution function g(E,) dE, in activation energies, the equation may be written

otherwise, corresponding to a uniform distribution in E, for En, < Ea < E,, . In view of the fact that in most noncrystalline dielectrics there will be a large number of different potential wells in which the ions can be trapped, this is a very reasonable distribution function for En. In Eq. (17a) E,, and E n , are the activation energies of z, and zl, respectively: En, = kT ln(zz/zo),

En, = kT ln(zl/zo)

(17b)

FLICKER NOISE IN ELECTRONIC DEVICES

233

Gevers (8) has applied this model very successfully to the interpretation of dielectric loss measurements as a function of frequency and temperature in various noncrystalline dielectrics. He obtained a number of interesting results of which we mention two. (1) The temperature coefficient of the capacitance C of a dielectric capacitor of loss tangent tan6 is

1 dC

2

where a, is the linear expansion coefficient of the material. Gevers found that the coefficient in front of tan 6 depended only very slightly on the nature of the dielectric. This is not so surprising, for zo does not differ too much for different materials, and the coefficient only varies as In zo. It should be observed that wzo Q 1 for all frequencies of practical interest. Equation (18) agreed quite well with experiment in a number of cases. (2) Tan 6 / T is a function of Tln(l/wz,) only. By measuring tan 6 as a function of T and as a function of w, Gevers was able to evaluate the time sec for glass. constant z,. He found zo N The results obtained by Gevers can be considered as an experimental verification of the validity of the distribution function g(z) dz. The application of this model to the derivation of the lifcharacter of S,y) thus has an experimental basis, and there is no need to invoke a universal l/f noise mechanism in this case. B. Flux Flow Noise in Type II Superconductors

It will now be shown that flux flow noise in superconductors has a spectrum that is constant at low frequencies and varies as l/fat higher frequencies, and that this spectrum comes about by an appropriate superposition of independent random pulses. In a flux flow noise experiment dc current is passed through a type I1 superconducting foil, a magnetic field is directed perpendicular to the face of the foil, and the noise is measured between two probe point contacts made to the foil. In the superconductor the magnetic field breaks up into elementary fluxoids, and due to the flow of current, a Lorentz force acts on the fluxoids, causing them to move across the foil with a uniform velocity v , thereby inducing voltage pulses between the probe contacts. Since the fluxoids are generated at random on the one side of the foil and disappear at random on the other side of the foil, the motion of fluxoids produces a kind of shot noise. It was first assumed that all pulses had the same shape (II), but a more

234

A. VAN DER ZIEL

careful examination showed that the pulse shape depended strongly on the path taken by the fluxoids, so that the noise is a superposition of independent pulses of different shape (12-14). Actually, the problem is somewhat more complicated than that. Usually the fluxoids do not occur as single flux quanta c$o but as bundles of fluxoids of size nc$O.This is caused by the fact that fluxoids can be pinned at pinning centers and are only released when a sufficiently large force is applied. For an infinite foil of width w with probes at a distance 2 4 the spectrum is of the form

{: ( A) +

S ( f ) = S(0) -exp -2n-

[1+

4n;/jx

-27+)]}

(19)

where f, = u/2u and S(0) is the spectral intensity at low frequencies: S(o) = (2nc$0/c)vdc

(20)

Here vd, is the dc voltage generated between the probes, c the velocity of light, and b0 the fluxoid quantum. This equation takes into account that the fluxoids occur in bundles of size nc$o. The calculation applies Carson’s theorem to fluxoids moving through the foil at x and integrates the resulting expression from x = -a to x = +a. We see that the spectrum is constant at low frequencies but varies as l/f at higher frequencies. This agrees with experiment (15, 16). At the higher frequencies Eq. (19) becomes invalid and S ( f ) decreases much faster than l/f because the probes are not “point probes,” as assumed in the derivation of (19), but have a finite size [probe size effect (16)]. Consequently S ( f ) can be integrated for the interval 0 I f I00. A phenomenon that is caused by helium bubbles on the sample due to the boiling of helium, and that results in small temperature fluctuations of the sample, gives rise to spurious noise with a l/f2 spectrum (17).Somewhat confusingly, this is called flicker noise. The importance of this process is that it gives rise to l/f noise as a consequence of the superposition of differently shaped independent pulses.

111. FLICKER NOISE IN VACUUM TUBES

We discuss here 1/f noise in various vacuum tubes, not because these devices are so important in themselves, but because the discussion sheds light on various l/f noise mechanisms.

FLICKER NOISE IN ELECTRONIC DEVICES

235

A. Flicker Noise in Field Emission Diodes

Noise studies in these diodes were performed by Kleint and his coworkers (18), by Timm and van der Ziel(19), and by Shen and van der Ziel (20). Kleint and his co-workers found l/f noise and shot noise; the l/f noise increased strongly with residual gas pressure. Timm and van der Ziel found shot noise and const/fY noise with y N 3 for clean tungsten surfaces ; they attributed this to diffusion noise (see below). Shen and van der Ziel found in some cases y N 1. The current I of a point emitter is of the form

where A and B are constants, F is the field strength at the field emitter, and is the work function. The only parameter that can possibly fluctuate is the work function 4. Hence’

4

But 4 depends on the density n of foreign atoms on the surface. Let the active surface be represented by a circular patch of radius a (the atoms diffuse in and out of this patch), and let ii be the average density of atoms on the surface; then the average number R of atoms on the patch is R = (naz)ii and

But according to Burgess (21) and van Vliet and Chenette (22) a2 ber,(u’/’) kei,(u”’)

SN(f)= 8 var N-

D

+ bei,(u’/’) ker,(u’/’) -U

(24)

where D is the diffusion constant of atoms along the surface, u = oa’/D, and ber,, bei,, ker,, and kei, are Kelvin functions of the first order. Substituting (24) into (23), one finds for the low-frequency limit of the spectrum Sn(o+0) =

-(Pfi/nD)ln(oa’/D)

(244

We assume here for local fluctuations that 6 d/q5 6 1 and 3(Bd1’2/F) 64 6 1 so that the Taylor expansion of (21) is valid. We may then average over the active region of the point and obtain Eq. (22).

236

A. VAN DER ZIEL

and for the high-frequency limit of the spectrum S,(O

8pi

--+

c

co) = nD (oa2/D)3/2

where C is a constant that does not depend on ii, a, and D and is defined by var N = N. The turnover frequency fl occurs at about ln(wa2/D) = 1. Experimentally Timm and van der Ziel found a turnover frequency of the cm2/sec order of 1 Hz This would agree with theory for D = 2 x cm (100 A). The high-frequency form (24b) gives indeed the and a = experimental f - 3 / 2 spectrum. If one defines the equivalent saturated diode current of the noise by equating

SAf) = 2qIeq

(25)

where q is the electron charge, Timm and van der Ziel found various expressions for clean tungsten surfaces. Sometimes I,, could be represented as

I,, = I + (254 for frequencies above 30 Hz.The first term is a shot noise term, the second term a diffusion noise term. Sometimes, however, the following puzzling expression was found

I,, = ( 1 1 / 2 + A1/2f

- 9 2

(25b)

It is difficult to explain the latter theoretically. For cathodes contaminated by residual gases they found

I,,

=

I

+ B/f

(254

where y has a value slightly above 1 at low current levels, increasing to about $ at higher current levels. When a tungsten point was cleaned by flash-heating and then left in contact with the residual gases (gas pressure N lo-'' torr), the noise increased strongly with time, going through a rather sharp maximum after about 5000 min. When the point was contaminated by barium, the noise was again of the form (2%) with y N 1.2 at a low current and y N 1.4 at a higher current. The noise was about two orders of magnitude larger than for an uncontaminated point. Noise of the form (25c) could be caused by superposition of spectra of the form (24) with different work functions and diffusion constants. This could result from patchy surfaces or from faces on the point with different work functions and diffusion constants. What is not quite clear is how y can now depend on the current.

FLICKER NOISE IN ELECTRONIC DEVICES

237

Separate experiments on clean tungsten surfaces indicated that the noise varied as the square of the current, as expected from Eq. (22). Kleint found that the field emission noise increased rather strongly with increasing temperature. Timm and van der Ziel explained this in terms of the temperature dependence of the diffusion constant D in Eq. (24). Taking Kleint’s data for a Ba-covered surface at 295 and 410°K they obtained an activation energy of the diffusion process slightly above 0.5 eV, which is not an unreasonable value. Kleint assumed that the fluctuation 6N of the number of surface atoms was caused by the random arrival and evaporation of impurity atoms at the emitter. If this process has a time constant z, then

S , ( f ) = 4 var N[z/(l

+ w’z’)]

He then introduced a distribution of time constants g(z)dz that was normalized and finally assumed

as could be expected from a distribution in activation energies. This yields

which varies as l/f for l/z2 4 w 4 l/zl. In view of what was said before, this may be only a formal exercise without physical meaning. Timm and van der Ziel have argued that the diffusion mechanism should be the predominant noise source. The measurements are performed in so short a time interval that uery f a v atoms will leave or arrive at the surface during that time. But all the atoms present on the surface will diffuse over the surface and so contribute to the noise. Finally, the model does not readily explain the l/fy noise spectra with 1 < y < $. According to Eq. (23) the noise would be zero if a+/an = 0. Such an effect has never been found. Apparently then, there is still something missing in both theories.

B. Flicker Noise in Photoemission Work function fluctuations such as encountered in field emission diodes might also occur in photocathodes (23) and give rise to l/f7 noise with l$4$. Let Zp be the photoemission current and 4 the work function of the emitting surface. We may then write, if n is the number of photons arriving

A. VAN DER ZIEL

238

per second, n, the number of photoelectrons emitted per second, and q the quantum efficiency of the photocathode,

I, = qn,

= qnq

Fluctuations An, in n, may occur for three reasons: (a) Fluctuations An in n: If they obey a Poisson process, they give a contribution q2q2 = q2q2ii to and hence a contribution

z2

2q2q2ii

(2W

to the spectral intensity Sr( f ) of AZ,. (b) Spontaneous fluctuations in n, : According to partition noise theory they give a contribution q2iiq(1 - q) to and hence the contribution to S r ( f ) is

A T

2q2W(1(c) Induced fluctuations due to fluctuations A$ in qii(dq/d$) A$, the contribution to Sr(f) is

(28b)

4:

Since AI, =

where S,( f ) is the spectral intensity of A$. Adding all expressions yields

The first term gives the well-known shot noise term of I,, whereas the second term is a flicker noise term. Since it varies as I;, it should not show up at low levels of I,, as, e.g., encountered in photomultipliers. Smit et al. (24) found flicker noise in the dark current of photomultipliers by counting techniques; Young discusses the problem of the occurrence of flicker noise in the photocurrent of these devices (25). C. Flicker Noise in Secondary Emission (23,26)

In secondary emitters the anode current I, may be written as I , = 61,

(30)

where I, is the primary current and 6 the secondary emission factor. Now 6 depends on the work function of the secondary emitter; a fluctuation

FLICKER NOISE IN ELECTRONIC DEVICES

239

A$ in 4 gives rise to a fluctuation A6 in 6 such that A6 = (a6/a4)A+. Consequently, if both I , and 6 fluctuate, AI, = 6 AI,

+I

as A4 a4

-

or

Here the last term represents the secondary emission flicker noise. This effect was observed in a Philips EFP 60 tube, which was a tetrode with one stage of secondary emission multiplication. The experiment was most easily performed by inserting a resistance R, between cathode and ground and HF connecting the screen grid to the cathode. The cathode shot noise, cathode flicker noise, partition noise, and partition flicker noise were then all suppressed by negative feedback, whereas the secondary emission flicker noise, which flows from the secondary emitter to the anode, was not affected. By measuring the noise as a function of the resistance value R, , it could be conclusively determined that secondary emission flicker noise did exist and that it had a l/f' spectrum with y N 1. D. Flicker Noise in Vucuum Diodes and Triodes (27) 1. Schottky's Theory of the Effect

Schottky (2)ascribed the flicker noise observed by Johnson (I)to foreign atoms arriving at and departing from the surface at random. He assumed that these atoms spent an average time zo on the surface and calculated the noise spectrum as follows. A single foreign atom, arriving at a point of the cathode and adsorbed as an ion, produces an electric moment p and causes a change 6' in potential ; the effect will extend nearly uniformly over a circle of area CJ around the , co = 8.85 x lo-'' F/m is the dielectric point, so that 6' = P / C J E ~where constant of free space. Let the current flow be temperature limited (saturated diode) and let Jo be the average current density; then the current density in the area CJ changes by a factor exp(qO/kT,) N (1 + qO/kT,), where T, is the cathode temperature. The atom thus produces a current pulse of amplitude i , = aJoq6'/kT, = FJo,

I; = qpc/kT,e,

(31)

of average duration z. If N foreign ions are located on the surface, then the fluctuating current is i(t) = ( N -

mi,

(32)

We deviate here from Schottky's derivation in order to simplify the calculation. The Wiener-Khintchine theorem was not yet known in 1926.

240

A. VAN DEX ZIEL

where N is the average value of N . Assuming a Poisson distribution for N, we have (N - N)’ = N,and the autocorrelation function is [ ~ ( t-) iTJ[N(t

+ s) - iTJ = Nexp(-s/zo)

(33)

Hence the spectrum, as given by the Wiener-Khintchine theorem, is

sr(f)= 4NFZJ;To/(l 4- 0’7;)

(34)

The spectrum thus varies as J;, in reasonable agreement with Johnson’s experiments for o’z; >> 1. The frequency dependence agreed quite well with Johnson’s data, and for oxide-coated cathodes the agreement was thought to be reasonable, since Johnson’s data gave some indication of leveling off at low frequencies. Later it was found that the spectrum usually did not level off and that it was of the form l/f with y close to unity. This can be explained by a wide distribution in time constants, as discussed earlier.

2. Space-Charge Suppression of Flicker Noise Flicker noise in space-charge-limited vacuum diodes was usually thought of as being caused by fluctuations in the emission current I,. As a consequence

Here SI, is the fluctuation in I,, and 61, the resulting fluctuation in the anode current I,. According to Langmuir’s theory of the diode (28)

aI,/ar, = kT,g/qI,

for I,

+ I,

(354

where T, is again the cathode temperature and g is the small-signal conductance. If the emission current fluctuation is interpreted as a fluctuation 8 4 in the work function (Schottky’s microscopic model is easily transformed into this macroscopic model !), then, since 1 s = AoSTZexp(-qWT,),

S r . ( f ) = (I,/kT,)’S+(f)

(36)

Here A. is Richardson’s constant S in the cathode area and S , ( f ) the spectrum of &$. Then S r , ( f ) = g2S,(f)

(364

or, by introducing an equivalent noise resistance Rn with the help of the definition, 4kT0Rn = S,,(f)/g2 = S,(f)

(36b)

FLICKER NOISE IN ELECTRONIC DEVICES

24 1

where To is room temperature. Measurement of R, as a function off thus gives S , ( f ) directly. We can compare flicker effect and shot effect in the case of space-charge limitation. The shot noise then has a spectrum (29)

where 9 = 3(1 - 4 4 ) = 0.644. Since the shot noise in the emission current has a spectrum

we see that

Therefore, the space-charge suppression factor for shot noise is 29 aIJaIs, whereas for flicker noise it is (aIJaI,)2. Since usually aIJaI, < 1, flicker noise is much more strongly suppressed by space charge than shot noise, as Schottky (3) already suggested in 1937. 3. Flicker Noise in Tubes with Tungsten Cathodes, Thoriated Tungsten Cathodes, and Thin-Film Cathodes In good vacuum diodes with tungsten cathodes, the noise under saturated conditions is shot noise down to about 30 Hz and varies as P /f below that frequency. In poorer tubes the noise can be improved considerably by flashheating the filament and thereby driving off foreign atoms from the surface. This indicates that the noise is caused by the arrival at and the departure from the cathode surface of foreign atoms. It cannot be caused by a diffusion process, since in contrast with the field emission cathode, where only the tip of the field emitter was active, the whole cathode is an active region, so that foreign atoms cannot diffuse out of the active region but can only evaporate from the active region. In vacuum diodes with thoriated tungsten filaments ( 2 3 , the noise can be quite low and is of the l/fy type with y N 1. Here the noise can be caused by a diffusion mechanism, since the thorium is contained in pores and thorium ions can diffuse over the surface from pore to pore, covering the surface with a near monolayer of thorium ions on tungsten. This problem has not been solved. The same is true for thin Ba-film cathodes (27). Here barium atoms come out of pores and diffuse over the surface as ions, forming a near monolayer of barium on oxygen on tungsten.

242

A. VAN DER ZIEL

Under space-charge-limited conditions diodes with tungsten cathodes show anomalous Bicker effect having a constant zo/(l + W’T;) spectrum. The effect is caused by the emission of bursts of positive ions from pockets of dirt in the filament. These ions are trapped in the space-charge region, raise the potential minimum, and so increase the noise. As a consequence the effect gives rise to pulses of the form (30) Z(t) = 0

for t < to,

i(t) = io exp[-(t - to)/zo]

for t > to (38)

where zo is the lifetime of the ions in the space charge. The effect disappears when an electrode with negative potential with respect to the cathode is inserted in the tube, indicating that the effect is indeed caused by positive ions. This means that the effect is absent in triodes with a negative grid.

4. Flicker Noise in Tubes with Oxide-Coated Cathodes (27, 30a) Lindemann’s experiments (31) on planar diodes and triodes indicate that SI,f) at constant anode current I, is independent of the cathode-grid distance d. This has three important consequences : (a) Other things being equal, tubes with small cathode-grid distance have the lowest noise resistance R, . The reason is simple, since R, is defined as 4kTRn =

~Ilf)/s:

(39)

and g, increases strongly with decreasing d (approximately as l/d4/3 at constant Za). This rule gives a good guide for selecting low-noise tubes. (b) The theory of flicker noise as suppressed emission fluctuations cannot be valid, for in that case R, would be independent of d. Replacing g by gm in (35a) and substituting (35) and (35a) into (39) gives that R, is a constant, independent of d, whereas in fact SIaf)at constant I, is independent of d. This also excludes Schottky’s flicker noise mechanism as a direct cause of the noise. (c) Neither can the noise be generated deep in the coating as resistance noise. For in that case the noise can be represented by an emf e in series with the coating;2 may now depend on I, but not on the cathode-grid distance d. Therefore, since in this model -

e2 = 4kTRn Af

(394

R, at constant Z, should be independent of d, in contrast with experiment. The latter effect does indeed exist. If the cathode nickel contains a few percent of silicon, the silicon reduces part of the coating under the formation of barium, leaving a high-resistance layer of Ba,SiO, behind at the interface

FLICKER NOISE IN ELECTRONIC DEVICES

243

between nickel and coating. The current flowing through this high-resistance layer generates a huge amount of flicker noise (32). What then is the source of the noise? Further experiments by Lindemann (31,33)suggested that the noise was generated in a thin surface layer and that this surface layer contained an appreciable voltage drop. Fluctuations in this voltage drop could be the source of flicker noise. This brings us to a theory of flicker noise proposed by Johnson and van Vliet and backed up by experimental evidence provided by Johnson (34). The basic assumptions of the theory are as follows. 1. A thin surface layer exists with possibly an appreciable voltage drop in that layer. 2. Donor centers (oxygen vacancies) can diffuse through this layer. 3. Barium ions arrive at random at the surface of the grains, resulting in an increase of donor sites in the area adjacent to the surface. Various kinds of spectra can be obtained : (a) Difusion spectra, which are flat at very low frequencies and vary as l/pIz at higher frequencies; they occur if the field in the surface layer is relatively small. (b) Low field spectra, which are somewhat different in shape but also vary as l/f3I2 at higher frequencies; they occur if the field in the surface layer is relatively small. (c) Strong field spectra, which are similar but vary as l/f” at higher frequencies; they occur if the field in the surface layer is sufficiently large. These predictions were verified for particular diodes. Note that none of these spectra show l/fnoise; neither did the experiments. What is not quite clear is how l/p spectra with y N 1 can be explained. Probably a nonuniformity in the thickness of the surface layer can give rise to an appropriate distribution of time constants ; this might explain these spectra. E. Flicker Partition Noise in Positive Grid Triodes and Pentodes

In 1954 Tomlinson (35) observed a l/f component in partition noise in pentodes. Schwantes and van der Ziel further studied the effect both in positive grid triodes (36) and in pentodes (37). In pentodes one can reduce the cathode current fluctuations by negative feedback by inserting a resistance R, in the cathode lead; the true partition noise, which flows from screen grid to anode, is not affected by the feedback (37). In positive grid triodes one finds shot noise, partition noise, cathode flicker noise, and flicker partition noise. Flicker partition noise indicates that the rate at which the current divides

244

A. VAN DER ZIEL

between the positive electrodes is randomly modulated by a l/f-type process. We can look at this as follows. If I,, I,,, and I , are the cathode, anode, and screen grid currents, respectively, and A is the partition parameter, then I, = I,A,

I , = Z,(1

-

A)

(40)

so that for fluctuations in a small frequency interval AJ Ai, = A Ai, -k I, 81,

Ai2 = (1 -

A> Ai, - I, AA

(404

and Ai, Ai, = A(1 - A)

@ - I:

where @ = S,,m AL = SAWAJ and S I E mand SAY>are the spectra of I, and A. Experiments indicate that S A Y )has a l/fspectrum and that hi, and Ai2 are positively correlated; the latter means that A(1 - A) @ > I:

Iv. FLICKER NOISE IN RESISTORS When dc current is passed through a carbon microphone, a carbon resistor, a thin-film metal resistor, or a semiconductor resistor, noise is generated that is proportional to the square of the current and has a l/f-type spectrum (4).When ac current of frequency& is passed through these devices, noise is generated that is proportional to the square of the rms current and that shows two sideband noise spectra around fo with a spectral intensity varying as l/lfo (IlAfnoise) (38, 39). Since the current is kept constant, the only parameter of the system that can fluctuate is the resistance of the device. One can therefore distinguish between contact noise, in which the fluctuating parameter is the contact resistance, and true resistance noise, in which the fluctuating parameter is the bulk resistance. Fluctuations 6R in the resistance R can readily explain the current dependence of the noise. Let I be the dc current; then this produces a fluctuating voltage 6 V = I 6 R , and hence the spectrum is

-fl

=

IZsRY)

(41)

so that SOY)vanes indeed as P , as found experimentally. Small deviations from the quadratic law can be expected when the resistance R depends slightly on the current, e.g., because current heats the sample, or because R is slightly nonlinear. Strong deviations may be expected when the resistance is strongly nonlinear (40). The problem of explaining the spectral dependence of S u mhas now been moved one step back; it now consists in explaining the spectrum of SRY). The response to ac currents is also easily understood. Let I . cos mot be

245

FLICKER NOISE IN ELECTRONIC DEVICES

the ac current and let 6R(t)be represented by a series representation

c m

6R(t) =

a, COS(U,,~

+ 4")

n=O

-

where %a,* is proportional to l/fn; then

c ancos(ont+ 4") m

W ( t )= I , cos o o t

n=O

c m

=$Io

{a,, cos[(oo

+ on)t+ 4,,] + a,,cos[(wo - on)t-

(42a)

n=O

so that two noise sidebands of frequencies fo k fn and carrier frequency = w0/2x are generated having a spectrum varying as l/lfo - f).

fo

A . Fundamental Experiments

1. Clarke and Voss's Fundamental Experiments (41-43)

Clarke and Voss have demonstrated that resistance fluctuations were indeed responsible for the observed noise. They did this by detecting such resistance fluctuations at zero current. The circuit used is the RC circuit of Fig. 1 and C is so chosen that C R = z has a certain value. The amplified noise is now filtered by a square filter of passband fl - fo so that noise over a frequency band fo < f < fl is passed; usually f, % fo . The filtered signal is then squared. The output of the squaring device may now be written as

Here the first term represents the subensemble average of P(t) for a given T and a given R, and the second term represents the residual fluctuation in each element of the subensemble. If now T and (or) R slowly fluctuates with time, and P(t) is Fourier analyzed for very low frequencies, the contribution of Po(t)becomes negligibly small. So for 2nfz 4 1 we see the slow fluctuations in the first term only.

-R

==c

FILTER fo< f < f,

-

AMPL

2

SQUARING DEVICE

'('I

LOW FREQUENCY FOURIER ANALYZER

-

FIG.1. Schematic diagram for the experiment of Clarke and Voss.

-

246

A. VAN DER ZIEL

We now have three possibilities. (a) 4n2f:z2 g 1 (bandpass completely below the knee): P(t) = 4kTR

jf:

df = 4kTRCf, - fo)

(44)

If T and R both fluctuate (fluctuations AT and AR, respectively) and the resistance R has a temperature coefficient B = (1/R) dR/dT, then if Ro and To are equilibrium values, A(TR) = Ro(l

+ BT0)AT + ToAR

or

A(TN (1 TOR0

AR + BTo) AT -+ To Ro

so that

(b) 4nzfiz2 3 1 (bandpass completely above knee):

or

so that

) ~f: (passband straddles knee): (c) f2, g 1 / ( 2 n ~4

In cases (a) and (b) one obtains a response from S,(f), as well as from S R ( f ) , whereas in case (c) one only obtains a response from S,(f). Clarke

FLICKER NOISE IN ELECTRONIC DEVICES

247

and Voss measured a 20-Mn InSb bridge and found a l / f noise spectrum. Four methods of measurement were used: ( 1 ) Measurement by dc technique. (2) Measurement by ac square-wave bias and subsequent phase-sensitive detection to measure the noise sidebands of the carrier frequency fo. (3) Measurement by positive pulse current bias to reduce heating. (4) By thermal noise measurement [case (b)].

In all cases the same value of SR(f)/R2was found. In a separate calculation it was shown that the noise in these bridges was a factor lo3 larger than predicted by a thermal fluctuation noise theory (see below). Therefore the noise is indeed resistance noise. A similar conclusion was reached by Beck and Spruit (43). 2. Hooge's Fundamental Formula

Hooge and his co-workers (44-53) have shown that for many thin-film resistors made from different materials, and also for many semiconductor resistors, SR(f)/Rzmay be written as SR(f

)/R2= a / N f

(47)

where a is a universal dimensionless constant that is only a slow function of the temperature T and has a value of about 2 x N is the total number of carriers in the resistor, and f is the frequency. This fundamental formula provided the first breakthrough in the characterization and understanding of flicker noise. It is usually interpreted as a noise mechanism due to a volume effect. It also holds for point contacts (54-59). There is no difference to speak of between a for solid and molten gallium (60). Clarke and Voss (42,42) observed, however, that thin manganin films, which had a practically zero temperature coefficient of the resistance, did not show a measurable l/f noise; that is, a is much smaller in that case. They felt that this indicated that the l / f noise could be caused by spontaneous temperature fluctuations of the resistor. This is easily seen as follows. Let R be the resistance of the sample at the temperature T, and let 6T be a spontaneous fluctuation in temperature ; then, since R = f ( T ) , dR 6R=-aT, dT

6R R

-=--

1 dR 6T = /36T RdT

where /3 = (1/R)dR/dT is the temperature coefficient of the resistance of the material. Consequently SR(f)/RZ= B"S,(f)

(484

248

A. VAN DER ZIEL

where S,( f)is the spectral intensity of the temperature fluctuations. Hence for small p, SR(f ) / R Zshould be negligible if the noise was caused by spontaneous temperature fluctuations. Any other mechanism would not give zero noise in manganin. Clarke and Voss (42) also observed that for thin bismuth films the relative noise spectrum was about the same as for thin metal films, despite the fact that the carrier density in bismuth was about a factor lo5 smaller than in metals. According to them, this indicates that the noise was inversely proportional to the volume of the resistor rather than to the number of carriers. This can readily be explained by the temperature fluctuation mechanism, since S,( f)should be inversely proportional to the heat capacity C, (see next section). Hooge’s formula (47) is valid for those metal films for which the conductivity is close to the conductivity of the bulk metal. For very thin films the electrical conduction is probably practically via a hopping process ; in that case the films exhibit much more noise than expected from Eq. (47) (61). The discussion of flicker noise in semiconductor materials is given in Section IV,C. One can now ask how resistance fluctuations can occur. The resistance R of a sample of length L is given by R = Lz/qpN

(49)

where p is the carrier mobility and N the number of carriers. Fluctuations may thus occur either in N or in p. (a) Fluctuations in N : In this case dR/R = - d N / N ,

SR( f ) / R 2 =

sN(f)/N2

(494

s,(f )/pz

(49b)

(b) Fluctuations in p : In this case dR/R = -sp//cl,

SR(f

)IR2

=

Fluctuations in N would be compatible with Eq. (47) if S,( f)were proportional to N . However, in that case there would be no guarantee that a would be a universal constant. We come back to that problem in the discussion of flicker noise in semiconductors. Fluctuations in p would not be so easily compatible with Eq. (47) unless S,( f)/p2 were independent of p and proportional to 1/N. There is no good general argument why this should be the case. Nevertheless, Hooge, for noise in ionic cells (44, and Kleinpenning, for noise in the thermoelectric emf of intrinsic and extrinsic semiconductors (53),both conclude that the

FLICKER NOISE IN ELECTRONIC DEVICES

249

noise arises from mobility fluctuations, and they therefore postulated S,(f)/P2 = d f N

(49c)

We come back to this problem in Section IV,C. Fluctuations in mobility would be easily understandable if the primary fluctuation were a spontaneous fluctuation in temperature. Consider, for example, the case that p = CTY

(50)

Then Sp/p = yST/T and hence

where y N 1 for metals and y that in the next section.

1:

3 for semiconductors. We come back to

3. Clarke and Voss’s Temperature Fluctuation Mechanism (41,42)

It has sometimes been tried to invoke diffusion mechanisms as a source of l/fy noise (62-64). This diffusion may either be particle diffusion or heat diffusion. For example a three-dimensional particle flow out of a small active sphere into a much larger inactive region gives a constant spectrum at low frequencies, and a l/f3I2 spectrum at high frequencies. For other three-dimensional geometries there might be an intermediate region with a l/f spectrum For example, in a thin film of width w and length L the characteristic time constants are z1 = w 2 / 2 n and z2 = c/2D,where D is the diffusion constant. Clarke and Voss (41,42) have shown that there will then be a region 1/z2 < w < l/z, for which the spectrum is about l/f; for w L this can be quite a wide region. Now the question is “What is diffusing?” Carrier diffusion is out of the question, since the time constants are far too short. Diffusion of heat would be much more likely, since it gives time constants that are the right order of magnitude. But it should also be taken into account that the films are usually deposited on a substrate. This means that there is not only heat exchange with the other parts of the film but also with the substrate. This will have the tendency of changing the temperature fluctuations. In addition, it will give rise to much longer time constants, which, in turn, will extend the l/f spectrum to lower frequencies. It might therefore be perfectly feasible that the temperature fluctuation spectrum could be approximated by the piecewise linear graph on a logar-

+

A. VAN DER ZIEL

250 ithmic scale.

In that case

where k is Boltzmann’s constant and C, is the effective heat capacity of the sample. Consequently, for the intermediate frequency region Clarke and Voss (41,42) propose a temperature fluctuation spectrum of the form

For a free-bearing thin film this can be modified to

since f2/f1 = (L/w)’. Clarke and Hsiang (65) found excellent agreement with experiment for the l/f noise of free-bearing tin and lead films at the superconducting transition. Ketchen and Clarke (66) found equally good agreement between theory and experiment and even were able to detect the flattening of the spectrum at low frequencies and the change in slope at higher frequencies expected from Eq. (51). Table I sums up the results obtained by Voss and Clarke (41)for various TABLE I MEASURED b, MEASUREDS , ( f ) / V 2 ,AND CALCULATED S , ( f ) / V 2 FOR VARIOUSMATERIALS AT 10 Hz Measured Measured Material

B

cu Ag Au Sn Bi Manganin

0.0038 0.0035 0.0012 0.0036 -0.0029

l p ~ <10-4

v2

S”(f)/

(Hz-’)

6.4 10-14 6.4 x 0.6 x 7.7 x 13 x <7 x 10-19

Calculated S”(f)/vz (Hz-’) 16 x 2x 0.76 x 7.7 x 9.3 x <3.5 x

10-19

FLICKER NOISE IN ELECTRONIC DEVICES

251

materials. The agreement is quite good, indicating that spontaneous temperature fluctuations give a plausible explanation of l/f noise. However, in InSb layers the measured noise was several orders of magnitude larger than that calculated from (521 indicating that in this case the l/f noise was of a different origin. If the noise is due to spontaneous temperature fluctuations, there should be a correlation between the noise in different parts of the film at low frequencies. Voss and Clarke (42) and Clark and Hsiang (65) demonstrated that this was the case with the help of the arrangement of Fig. 2. Let Vl(t) and V&) be the two noise sources; then the arrangement allows for rneasuring the spectra S+(f) of Vl(t) V,(t) and S-(f) and K ( t ) - Vz(t).The correlation coefficient c between Vl(t) may then be expressed as

+

Good agreement was obtained, indicating that the heat coupling between the films (1) and (2) was the cause of the correlation. However, in InSb layers no such correlation could be observed, indicating that spontaneous temperature fluctuations are not the cause of the l/f noise in this material. For a further theoretical discussion of temperature fluctuation noise see Kleinpenning (67).

POP-II

.j

so

~

0.25 0

0.25

f (Hz)

( b)

FIG.2. (a) Experimental configuration for correlation measurement; (b) fractional correlation for two samples (R. F. Voss and J. Clarke, Phys. Rev. E 13,556 (1976)).

252

A. VAN DER ZIEL

We shall now demonstrate that Eq. (52) is formally equivalent to Hooge’s equation (47)and that it yields the right order of magnitude for Hooge’s constant a. To that end we observe that C, is proportional to the effective number N, of atoms in the sample. According to Dulong and Petit’s law, the heat content Q = 3N,kTy and hence C, = 3N,k, since each atom has 6 degrees of freedom. Therefore T Z

and hence, according to Eq. (50a)

where n is the number of conduction electrons donated per atom. This is of the form (47). For metal films y N 1 ; taking fz/fl = lo5 and n = 1 yields a = 25 x which is the right order of magnitude. Therefore Clarke and Voss’s temperature fluctuation mechanism is compatible with Hooge’s formula. Improved agreement would have been obtained if we had taken n somewhat smaller. For example one would expect n < 1 in Ag, Au, and Cu. More important, due to the effect of the substrate, N, might be larger than the number of atoms in the film, and that would also result in R lower n (68). The latter possibility finds support from measurements discussed in the next section. Since spontaneous temperature fluctuations give rise to fluctuations in the mobility p, the temperature fluctuation noise mechanism is compatible with the mobility fluctuation mechanism, and the latter finds here a natural explanation. One can now also understand why bismuth (41,42)behaved anomalously, in that it had a value for the parameter a that was several orders of magnitude smaller than 2 x According to Clarke and Voss, their bismuth film had N = lO-’N,; nevertheless, the noise was of the same order of magnitude as for the other metal films. This is understandable, since the first half of Eq. (55) should be valid and hence a should be quite small because n is so small. Since N, is proportional to V, one can now see (41, 42) why S,y)is proportional to 1/ V rather than to 1/N. 4. Experiments by the Chicago Group (69-72)

Dutta, Eberhard, and Horn (69) measured l/fnoise in copper whiskers (diameter = 3 pm) ranging in length from 0.15 to 1.20 cm. They found the noise to be proportional to the square of the applied dc voltage, and the noise

KICKER NOISE IN ELECTRONIC DEVICES

253

had a power spectrum of the form constlj”, with y = 1.05 L- 0.05. But the noise power was lo2 to lo3 times larger than predicted by Hooge’s formula (47). They did not find any low-frequency turnover, even for the shortest samples, as would have been expected from Clarke and Voss’s theory (42, 42). While the temperature fluctuation should be somewhat larger than for a thin film on a substrate, they find it hard to explain the large difference in noise magnitude. They also found a rather strong temperature dependence. Gold contact wires were connected to the whisker by silver conducting paint, and the driving dc current was about 20 mA. Since silver paint contacts are sometimes noisy, it was deemed necessary to measure the noise as a function of the sample length. They found indeed that S v y ) / V 2 varied as 1/N as expected from Hooge’s and from Clarke and Voss’s formulas. Eberhard and Horn (70- 72) measured the temperature dependence of l/f noise in silver, gold, copper, and nickel films on sapphire substrates having conductivities comparable to those in bulk material. In contrast with Hooge’s and Clarke and Voss’s formulas, they found a rather strong temperature dependence of S,cf)/V2. They expressed this dependence over a wide temperature range by

S,V)N/ V 2 N A

+ A’ exp( - E,/kT)

(56)

where A and A‘ are constants and E, is a kind of activation energy, which is 1750°K for Ag, 1400°K for Au, and 1250°K for Cu. The noise showed a peak at a higher temperature (400°K in Cu, 500°K in Ag, no peak in Au below 500”K,no primary peak in Ni but a slight secondary peak at the Curie temperature of 625°K). According to these authors there are two types of noise : Type A : Noise that is weakly temperature dependent and strongly substrate dependent, possibly caused by the temperature fluctuation mechanism proposed by Clarke and Voss. Type B: Noise that is strongly temperature dependent and only very weakly substrate dependent, whose origin is unknown. The free-bearing copper whiskers had noise of this type.

These statements are demonstrated in Figs. 3 and 4. Figure 3 shows S;cf)N/V2 at 20 Hz for 800-A Ag films on sapphire and on fused silica substrates as functions of temperature. Also shown is the temperature fluctuation noise deduced from the Clarke and Voss formula, which is found from (52a), and the temperature fluctuation noise as expected from measured temperature fluctuations on a fused silica substrate and on a sapphire substrate (72). There is limited agreement at low temperatures for a fused silica substrate, but the temperature fluctuation noise for the sapphire substrate was too low to show up in the graphs. Figure 4 shows similar data

A. VAN D W ZIEL

254

- -I

g f l

200

n-x

I

__--

100

200

300

400

I

500

T (K)

FIG.3. (i) Voltage noise S,(20)N/V2 versus temperature for 800-A Ag films on sapphire (open circles) and fused silica (open triangles) substrates. Error bars do not reflect calibration errors due to uncertainty of N. (ii) Temperature-fluctuation-induced noise Su,,,(20)N/ V2 versus temperature from direct measurement of temperature fluctuations in 800-A Ag films for sapphire (filled circles) and fused silica (filled triangles) substrates according to Dutta et al. (J. B. Eberhard and P. M. Horn, to be published).

for 800-A Cu films; here the measured noise shows reasonable agreement with the measured temperature fluctuations at low temperatures, both for the fused silica substrate and for the sapphire substrate. The “measured” temperature fluctuation noise is decreased from the Clarke-Voss formula by about a factor of 8 for metals on fused silica and by a factor of 200 for metals on sapphire, independent of the metal. This suggests that the substrate lowers the spontaneous temperature fluctuations, which is not unreasonable, and that the difference between substrates is due to differences in the heat properties of the substrates (68). The fact that the Clarke-Voss formula agrees reasonably well at room temperature is probably accidental, since the measured and predicted temperature dependences vary widely (see Figs. 3 and 4). Eberhard and Horn (71) speculate about the temperature-dependent l/fnoise that is independent of the substrate. They assume primary spectra of the form z/(l + 02z2) with a distribution g(z) dz, as introduced in Eq.

FLICKER NOISE IN ELECTRONIC DEVICES

300

200

100

400

255

500

T (K)

FIG. 4. (i) Voltage noise Su(20)N/V2versus temperature for 800-A Cu films on sapphire (open circles) and fused silica (open triangles) substrates. Errors do not reflect calibration errors due to uncertainty in N. (ii)Temperature-fluctuation-induced noise Su,,,(20)N/ V 2 versus temperature from direct measurement of temperature fluctuations in 800-A Cu films for sapphire (filled circles) and fused silica (filled triangles) substrates according to Dutta et al. (From J. W. Eberhard and P. M. Horn, to be published).

(6), which is caused by a uniform distribution in activation energies. They assume that vacancies are created at the surface or at any crystal deformation, with the equilibrium number of vacancies given by n

=

const exp( -E , / k T )

(57)

where E, is the energy required to create a vacancy. Once created, these vacancies diffuse through the crystal in a process characterized by a diffusion constant D given by

D

-ED/kT] (58) where ED is the energy barrier presented to the diffusing vacancy. If ED is uniformly distributed in energy and 0.15 5 ED 5 1.1 eV, the observed l/f spectrum results. The temperature dependence of the magnitude of the noise would then be governed by E,. The measured values of E, are a factor 8 smaller than in bulk material, but since they increase with increasing film thickness, this mechanism cannot be ruled out. Eberhard and Horn also investigated the voltage dependence of the noise. They corrected for the heating effects by measuring the resistance R versus T a t zero current; then they measured R at the driving current and deduced the device temperature from it. Strange enough, they found that S u m ,even = D,[exp(

A. VAN DER ZIEL

256

after this correction, varied as Vo with b > 2 (typically p = 2.26 k 0.13). They offer no satisfactory explanation for this result. The important conclusions to be drawn from these experiments are the existence of two noise mechanisms: type A and type B, and the effect of the substrate on the noise of type A but not on the noise of type B. B. Noise in Metal-Metal and Semiconductor-Semiconductor Contacts

The work on this problem was mainly performed by Hooge’s group (54-59). A very good summary is given in Vandamme’s Ph.D. thesis (73).

The resistance and the noise have been calculated for the contact resistance R of a half constriction (point contact of radius a on conducting metal plate) and for a symmetric constriction of radius a. They found in the first case R = p/2na, and in the second case

S,y)

=

ap2/40n3na5f

(59)

R = plna, S R y ) = ap2/20n3na5f (60) Here p is the resistivity, n is the carrier density, and a is Hooge’s constant. These formulas were derived under the assumption that Eq. (47) S R y ) / R 2 = a/Nf

is valid for any spherical sheet between two equipotential surfaces a distance dx apart. This leads to a resistance dR between the surface and a noise SdRO

dR = p dx/2nx2,

SdRy)= up2 d ~ l ( 2 n x ) ~ n f

(61)

from which Eqs. (59) and (60) follow by simple integration. It is common practice to eliminate a, which is not so easily determined, by expressing a in terms of the total resistance R, evaluating S R ( f ) / R 2 and , comparing it with the experimental data. This yields for the case of Eq. (60)

Here R was varied by varying the pressure with which the two pieces were pressed together. Reasonable agreement was obtained, indicating that Eq. (47) forms a reasonable basis for noise in point contacts. One should, of course, not expect perfect agreement, for a point contact is not as stable a configuration as a metal thin-film resistor. Moreover, the surface may be covered with a thin oxide layer and that can have an in-

257

FLICKER NOISE IN ELECTRONIC DEVICES

fluence. For example, Vandamme finds that manganin point contacts satisfied Eq. (62), whereas Clarke and Voss found no measurable flicker noise in thin manganin films. It could be that the noise in Vandamme’s case was either due to an oxide film or due to the fact that his manganin material showed noise of type B (in the notation of the Chicago group). This could be decided by an experiment of the type described below. Equations (59) and (60) also hold for semiconductor-semiconductor contacts (73,74). Here it is especially important to take into account the effect of a possible oxide film between the materials. To that end a parameter 4 W a s introduced : 4 = R f i d R c o n s t r i c t i o n . Since Rconstriction = Pbuik/na, and Rfilm= tpfilm/na2,where pfih is the resistivity of the film, t the thickness of the film, and a the contact radius,

4

=

tPfilm/aPbulk

(63)

Generally t 4 a and the total resistance R is = Pbulk(l

+ 4)/na = P b u l d n a p

+

where a,, = a/(l 4) is the “apparent” contact radius (ap c a). For the parameter C is

(634

4

=0

fMf) - m 2 R 3 C=-fS”(f) =-VZ RZ 20nbulk PkIk as follows directly from Eq. (62). For film + oxide, Vandamme finds

Here nfilmand pfilmrefer to the carrier density and the resistivity of the oxide film. It is assumed that the oxide satisfies the relation

CIf

(65) where Nfilmis the effective number of carriers in the oxide film. If the resistance and the noise is film dominated, then since Rfilmis proportional to l/Nfilm, the constant C is proportional to R. These predictions were well varified by experiments on Ge contacts (73, 74) and InSb contacts at 77 and 300°K (74, 75). In particular, the factor C was found to be proportional to the contact resistance when the properties of the contacts were film dominated. To lower the flicker noise in carbon resistors, which consist of carbon grains with the resistance mainly in the contacts between the grains, one can put n resistors R in series and n strings of such resistors in parallel. The total SuY)IV2 =

aiflfilm

=

258

A. VAN DER ZIEL

resistance is then the same, but the noise has been reduced as l/n2, since the volume V of the resistor is n2 times as large. To prove this, consider that a total current I passes through the sample. The current in each string is then l/n times as large, and hence S,y) for each string is l/n times the Soy) as for the single-resistor case. Since the resistance of each string is nR, S r y ) for a single string is l/n3 times the Sly) for the single-resistor case. But there are n such strings, and hence the total value of S r y ) is l/n2 times the value for the single-resistor case; since R has not changed, Soy) varies as l/n2 also. If we shrink all the dimensions of the grains, including the contact radii, by a factor p while leaving the total volume V intact, then there are p 2 as many contact strings and p times as many contacts per string. Then the total resistance R has not changed, and the noise for a given current I has not either. For the current through each string is p 2 as small, and hence S,y) for each contact is p s / p 4 = p times as large, so that Soy) for each string is p 2 times as large. But since the resistance per string is p 2 times as large, S,y) for each string is p 2 / p 4 = lip2 times as large. Since there are p 2 as many strings, the total S,y) has not changed; neither has the total S,y). For a study of l/f noise in HgCdTe devices with current-carrying nonohmic contacts, see Hanafi and van der Ziel(40).

C. Flicker Noise in Semiconductors 1. Validity of Eq. (47) According to Hooge’s compilation of the literature ( 4 3 , Eq. (47) is also This seems to correct for semiconductors with CI of the order of 2 x qualify it as a bulk effect. One has to be somewhat careful about this conclusion. On the basis of a suggestion by Klaassen (76), van der Ziel (77) showed that Eq. (47) could also be derived from the McWhorter model (78), that is, from a surfacecontrolled density fluctuation model. The basic assumptions are

~

1. The electrons interact with oxide traps via surface states with a time constant z. 2. There is a distribution g(z) dz in time constants of the form (6). 3. If 6N is the fluctuation in the number N of carriers in the sample, then 6N2 = PN, where P is proportional to the surface-state density at the interface between semiconductor and oxide. In that case one obtains Eq. (47) (see next section), with = P/ln(~Z/zl)

(66)

FLICKER NOISE IN ELECTRONIC DEVICES

259

For samples in which no special measures are taken, it is quite possible that a lies in the range 1-10 x in fair agreement with CI 2: 2 x 10-j. Hanafi and van der Zief (79), for example, found that for many samples of CdHgTe the value of a lay between 2-9 x I0-j. Those samples that had undergone a surface treatment, however, had values of a that were more than a factor of I0 smaller (1 -2 x 1 0-4) and Broudy (80) even produced a sample that had CI N This speaks in favor of a surface-controlled density fluctuation mechanism. For silicon and germanium samples no such measurements have been reported, but McWhorter’s model was introduced to explain his noise data in germanium filaments; in those filaments he found sound evidence for the existence of a wide distribution in surface time constants (see Section IV,C,2). Moreover, the measurements of flicker noise in p-n junctions and in transistors (Section V) strongly suggest that the noise is caused by fluctuations in the surface recombination velocity s, which parameter is directly proportional to the surface-state density N,, of the semiconductor-oxide interface. Finally, noise measurements in MOSFETs strongly suggest that the noise is proportional to the surface-state density N,, (Section V). All this evidence points again to a surface-controlled density fluctuation mechanism. Nevertheless, there is also strong evidence that the noise is caused by a mobility fluctuation mechanism, in which the mobility in subbands of width AE and Energy E fluctuates independently. We come back to that problem in Section IV-C3. Hanafi and van der Ziel (40, 81) made two current-carrying contacts 1 and 2 and several noise-measuring contacts a, b, c, . . . on a CdHgTe sample and measured the l/f noise between the contact pairs ab, bc. and ac. If S&Cf),s b , y ) , and S,,Cf) are the spectra, they found (67) s ~ ~ C f=) s a b Y ) -k s b c C f ) This remained true even if contact b was a noisy contact when carrying current (40). In other words the noise voltages generated in the adjacent regions ab and bc were independent, in agreement with the Clarke-Voss experiments on InSb (41,42). The same was true if one of the regions ab or bc contained a grain boundary, but the noise in the grain boundary region was about 4 times larger than in the region not containing a grain boundary (81). We can, therefore, draw the conclusion that grain boundaries are a source of llfnoise. This is understandable, for a grain boundary is a conglomeration of dislocation lines containing l/f generating centers with a distribution g(z) dz in time constants of the form (6). Bess (82) has developed a detailed theory for the effect. It should be noted that Montgomery found a different effect (83). He

260

A. VAN DER ZIEL

measured llfnoise generated between adjacent contact pairs ab and bc on germanium and found correlation between the noise in sections ab and bc; in other words, Eq. (67) was not valid. This effect can be explained by assuming that the surface states not only interact with free electrons and with oxide traps, but also generate hole-electron pairs (see next section). The carriers trapped in the oxide then randomly modulate the hole-electron pair emission of the surface states ; this random modulation contributes to the l / f noise. The randomly generated hole-electron pairs were swept through the device by the current and so produced the correlation. Apparently this effect did not take place in the CdHgTe samples. The most clear-cut indication that Eq. (47) was not valid for HgCdTe came from a field effect experiment (79). When providing the surface of a thin HgCdTe sample, epoxied onto a germanium substrate, with an insulating layer and a field plate and then applying a variable dc voltage to the field plate, one can bring the HgCdTe surface from accumulation to strong inversion. The sample was so thin that the inversion layer reduced the effective height of the conducting channel and so increased the resistance R of the sample by about 25%. According to Eq. (47), S,(f) is proportional to R3 and hence should change by about a factor of 2 when going from accumulation to strong inversion. Hanafi and van der Ziel (79) observed, however, that S,(f) was independent of the dc bias on the field plate. This rules out the validity of Eq. (47). Unfortunately, not enough was known about the sample to decide whether a surface effect could fully explain the observations. 2. Mc Whorter’s Surface Model (78,84) McWhorter’s surface model was developed for germanium. Figure 5a shows the energy band structure of a weakly n-type germanium sample and a weaklyp-type inversion layer caused by interface states at or near the semiconductor-oxide interface. There are two kinds of states: fast states at the interface and slow states 0-40 A inside the oxide. A metal layer was deposited on the oxide and an ac voltage was applied between metal and semiconductor; this modulates the conductivity of the sample. Experiments indicated that the conductivity modulation response could be approximated as S,(w) = a In bw

for w 4 wmaXr

S,(w) = S,,

for w % w,,,

(68) This can be explained as follows. Consider first that there is only one type of slow surface state with a time constant z. These surface states store charge and thus can be represented by a capacitance C,/cm2; the excess charge leaks away and this can be represented by a resistance R,/cm2 such that z = C,R, (Fig. 5b). The ac charge SQ, on the capacitance C,,due to the

FLICKER NOISE IN ELECTRONIC DEVICES

26 1

(b)

FIG.5 . (a) Energy band structure at the surface of a semiconductor; (b) equivalent circuit of the field effect.

application of the ac voltage 6 V between metal and interface via the oxide capacitance C/cm2, can for C, B C be written as SQ, = C,SV, = CSV

jwCsR, = C 6 V - j w z 1 joC,R, 1 jwz

+

+

The charge SQs induces an excess charge -SQs in the germanium near the interface, and this gives rise to a sample admittance 6Y = ps,6Q,E/6VL, where is the surface mobility, which is a function of the surface voltage depicted in Fig. 5a; E is the field strength parallel to the surface and L is the device length. Therefore

SY

jwz jwz

= const ___

1

+

262

A. VAN DER ZIEL

This does not resemble (68) at all. Agreement with (68) is obtained by introducing a distribution of time constants : dz/z g(7) d z = ___ W,/z,)

for z1 < z < z,

g(z)dz

=0

This yields 6Y = const

r2

j o z g ( z ) d z = -In( const

(1 + joz) ln(z,/zl)

+

otherwise (71)

)

1 jwz, 1 + jwz,

(72)

or

for ozl% 1

6Y = const

(7W

so that om, = l/zl. But that is exactly the distribution that is needed to produce a noise spectrum with a frequency dependence of the form constlf. McWhorter assumed that the distribution (71) in time constants, needed to explain l/f noise, was due to tunneling from the surface to oxide traps at a depth y. Assuming a uniform distribution in traps for 0 < y < y l , this yields B(Y) dY = dY/Yl

for 0 < Y < Y l

dY)dY = 0

otherwise

(73)

But z = zo exp(ay)

(734

where a N 10’ cm-l; this yields immediately Eq. (71), and hence, if we put 6N2 = PN

This agrees with Eq. (47) with a = fi/[ln(z,/z,)], as stated earlier. McWhorter introduced the following refinement into the theory. If the carrier is trapped by an oxide trap, the surface recombination centers adjacent to the trap begin to emit hole-electron pairs; this increases the conductance, whereas the trapping of an electron reduces the conductance. If M hole-electron pairs are emitted, the noise must be multiplied by the factor [-pn M(pp pn)l2/pi. The effect is usually ignored in other

+

+

FLICKER NOISE IN ELECTRONIC DEVICES

263

semiconductors, but it explains the correlation found by Montgomery for l/f noise in his germanium samples.

3. 'Ihe Case for Mobility Fluctuations The assumption of mobility fluctuations as an explanation of l/f noise was first introduced by Hooge (44)in order to explain a peculiar result found by Hooge and Gaal (85) in electrolytic concentration cells. They measured S,(f)/G2 in electrolytic resistors, where G represents the conductance of the electrolytic device, and they determined S,(f)/V', where V is the terminal voltage, for open-circuit noise in electrolytic concentration cells, and they found

where c2 is the lowest concentration of the cell. Hooge showed that the latter relationship could not be explained by fully correlated ion density fluctuations (that is, full correlation for the positive and negative ion density fluctuations, or 6n = 6 p ) ; such a correlation would be expected from space-charge neutrality considerations. However, if the mobilities of the positive and negative ions fluctuated independently so that Eq. (49c) was satisfied for each, then Eq. (75) could be explained. Van der Ziel (86) has shown, however, that this relationship could also be proved if the density fluctuations of the positive and negative ions were partially correlated. The space-charge neutrality argument can here be overcome if the ions are temporarily adsorbed at the walls, each immobilized positive ion surrounding itself with mobile negative ions, and vice versa. The experiments seem to require that positive and negative ions be adsorbed in bunches. Hence density fluctuations are still a viable option for explaining Hooge and Gaal's results. In semiconductor resistors the situation is similar. Equation (49c) shows how Hooge's formula (47) can be explained in terms of mobility fluctuations. But Section V,C,2 shows how the McWhorter theory can explain the same data. Again this does not allow the conclusion that one or the other explanation must be excluded. The situation is different, however, in thermoelectric cells. Experiments by Brophy (87) showed that the thermoelectric emf showed 1/f noise. This was interpreted as follows. The thermoelectric emf per degree dB/dT, set up by a temperature gradient in the semiconductor, is

264

A. VAN DER ZIEL

when lattice scattering predominates. Here AE, is the distance between the Fermi level and the band edge, T is the absolute temperature, k is Boltzmann's constant, and q is the electron charge; the plus sign holds for n-type and the minus sign for p-type material. Fluctuations in the carrier density would show up as fluctuations in the Fermi level (or in AE,), and this shows up as thermoelectric noise. Kleinpenning (53)repeated the experiments and found a somewhat different result. He found that the current noise and the thermoelectric noise in thermoelectric cells were practically uncorrelated, as required by the mobility fluctuation model, whereas according to the carrier density fluctuation hypothesis they should be fully correlated. In addition, the thermoelectric noise data in intrinsic materials could not be explained by carrier density fluctuations. In this case mobility fluctuations seem to offer the only explanation. But now the hypothesis had to be extended. The conduction band had to be divided up into a large number of subbands, and in each subband the mobility fluctuated independently such that

where c1 N 2 x and the subscript s refers to the subband. This does not involve fluctuations in E,. A similar situation occurred in single-injection space-charge-limited solid-state diodes (88).At low voltages the characteristic is linear (ohmic regime) and at high voltages it is quadratic (space-charge regime). For plane-parallel geometry the mobility fluctuation theory gives

Sdf) = a V 2 / N f ,

I = (aA/L)V

(78)

for the ohmic regime, whereas for the space-charge regime aqLV

Wf)= S A E E O f Y

9 AV2 I = - &&& 8 L3

(79)

Here a N 2 x N is the number of carriers for low voltage, o the lowvoltage conductance, L the device length, A the cross-sectional area, V the F/m, and the applied voltage, q the electron charge, E~ = 8.85 x relative dielectric constant. Under the assumption of density fluctuations, similar noise expressions were found, but the breakpoint in the noise curve occurred at a different voltage than for mobility fluctuation noise. Here again the latter assumption seems to be required. Equations (78) and (79) seem to agree reasonably well with experiment. We now turn to the problem of l/f Hall noise. Brophy (89), Brophy and Rostoker (90), and Bess (92) found that the open-circuit Hall voltage showed l/f noise. This was easily explained in terms of carrier density

FLICKER NOISE IN ELECTRONIC DEVICES

265

fluctuations. If a current I flows in the X direction and a magnetic field B, is applied in the 2 direction, then the Hall voltage across terminals in the Y direction is

for an n-type bar of carrier density n and dimensions w,, w,,, w,. The noise is thus generated by majority carriers, except in intrinsic semiconductors where both types of carriers contribute. In the latter case the experimental data fitted best with the idea of fully correlated hole and electron density fluctuations ; i.e.

An/n = -ApJp

(81)

This would be expected for slow fluctuations where equilibrium is maintained at all times so that p n = n’ = const. However, in the frequency region where generation-recombination noise predominated, they found that the Hall noise was due to correlated carrier density fluctuations with An = Ap. Vaes and Kleinpenning (92)and Kleinpenning and Bell (93) have shown, however, that mobility fluctuations can explain the Hall noise equally well. Hence Hall experiments cannot discriminate between the mobility and carrier density fluctuation theories. Kleinpenning (88) has summed up the mobility fluctuation theory in the following formula for the conduction fluctuation spectrum :

Here CI is Hooge’s constant, E the kinetic energy of the carriers, n k , E ) the density of carriers of energy E at the position r, and ak, E ) the conductivity of carriers of energy E at the position z. The delta function Sk - r’) indicates that fluctuations at r and r’ are independent, whereas the delta function 6(E - E’) indicates that fluctuations at the energies E and E’ are independent. Kleinpenning (88) has also integrated Eq. (82) with respect to E and E’ and obtained S,k,

i,f) = [ ~ ~ a ~ k ) / f n6kk )-] r’)

(83)

This result was obtained by using pb, E ) 1: E-’/’, and n k , E ) N x exp(-E/kT). He used this expression to explain flicker noise in spacecharge-limited diodes. Zijlstra used Eq. (83) directly in order to prove Eq. (79) (934.

266

A. VAN DER ZIEL

It should be emphasized that for space-charge-limited diodes Kleinpenning used Eq. (83) in order to derive the formula

where A is the cross-sectional area of the device. For the space-charge regime this yields Eq. (79). The mobility fluctuation hypothesis is thus needed for explaining the noise data in thermoelectric cells and in single-injection space-chargelimited solid-state diodes. In all other cases the density fluctuation hypothesis explains the data equally well. There is one further piece of evidence in favor of mobility fluctuations. Hooge and Vandamme (94) found from contact measurements on heavily doped Ge and GaAs that the parameter a decreased with increasing doping and could be expressed as a =2 x

lo-3(~/~latt)2

(85)

Here p is the total mobility, phttthe lattice mobility, and pimpthe mobility due to impurity scattering, so that

1 / = ~ l/PIatt

+ VPimp

(85a)

Such a relationship would be expected if lattice scattering were the source of l / f noise and impurity scattering were noiseless. This could be the most direct evidence so far in favor of the mobility fluctuation hypothesis, and could point to phonon scattering as the source of noise. Compare, however, van der Ziel (94a). 4. Integrated and Zon-Implanted Resistors Hsieh has investigated l / f noise in integrated surface resistors and in ion-implanted resistors (95). The results were published in two papers (96,97).A third paper (98) gives the theory. Hsieh et al. expressed the l / f noise in terms of the noise ratio n, defined as

where S;(f) = S,(f) - 4kT/R is the l/f noise generated in the resistor, S,(f) the total noise spectrum, and 4kT/R the thermal noise spectrum. Their data do not allow the evaluation of S R ( f ) / R zand hence of Hooge’s noise parameter a. They found for integrated surface resistors (95) that (n - 1) varied as

FLICKER NOISE IN ELECTRONIC DEVICES

267

12/fw2, where I is the current, f the frequency, and w the device width; also, (n - 1) was independent of the device length L. This is compatible with Eq. (47). noise at low doping In ion-implanted resistors (97) they found l/f levels and l/f noise at higher doping levels. Bilger et al. found l/f noise (99). The latter changed the value of the resistance R by changing the voltage of the epitaxial layer surrounding the resistor. Hooge found that in the expression (100) sR(f)

Af/ R 2 = Af/f

(87)

the constant C varied as R2,whereas Eq. (47) would give C = a/N= (aqp/L?)R. This discrepancy is easily explained by the McWhorter model. We saw in Section IV,C,2 that Eq. (47) resulted if 6 p = /IN, with a = /I/[ln(z2/zl)]. If instead the device satisfied the relationship

6NZ= PN,,

(88)

where N,, is independent of N, then

-sR(f) R2

/I

.%

1n(z2/zl)f N 2

which varies as R 2 ; now Eq. (47) is numerically valid for N = Neq.This agrees with Hooge's plot of their data shown in Fig. 6. Assumption (88) is not an ad hoc assumption made in order to force agreement with experiment but must also be introduced in n-channel MOSFETs to explain the flicker noise data in those devices. Equation (88) says that if N is varied by varying the bias of the epitaxy, then the meansquare value of the fluctuation 6 N is independent of N.

FIG.6. llfnoise of ion-implanted resistor. The resistance R is changed by the depletion voltage. Points R2: experimental results; line R : calculated according to Hooge's relation (47) (F. N. Hooge, Proc. Symp.IlfFluctuations, 1977 Conf. Rep., p. 88 (1977)).

-

-

268

A. VAN DER ZIEL

5 . Noise in Conductors by the Four-Probe Method Hawkins and Bloodworth (101) measured l/f noise in thick-film resistors. They found experimentally that 1/f noise arises between probes placed both at right angles to the current flow on opposite sides of the film and parallel to the current flow at one side of the film. In both cases the l/f noise was of the same magnitude. The noise between the probes on opposite sides of the film is called transverse noise. While the measurements were performed on thick-film resistors, it should be clear that the experiment works equally well for semiconductor resistors. The noise is caused by conductivity fluctuations. A theory of the transverse as well as the longitudinal effect was given by Kleinpenning (102), who also gives a list of references on the subject. D . Miscellaneous Topics on I /f Noise in Resistors

There have been many reports on the excess l /f noise generated in carbon resistors by ac excitation. The noise intensity is proportional to V,., with 2 < n < 4 (103-110). When Kc increases, n decreases from 4 to 2. The intensity is several percent of the intensity of the l/Af spectrum (i.e., the noise sidebands a t f k f , ) . The effect can be explained if one assumes that the contacts between the grains are not perfectly ohmic. The exciting sinusoidal voltage will then lead to a sinusoidal current with a very small extra dc current through each contact. Averaged over all contacts, the dc currents will more or less compensate each other so that the I- V characteristic of the resistor is fairly ohmic. Since the l/f noise generated in a contact does not depend on the direction of the current, the l/f noises of all contacts will add. In addition, noise is generated at the second harmonic at frequencies (2f ff,). The correlation between the noises at f, ,f +f, , and (2f ff,) has been investigated by Jones and Francis (110). The statistical properties of l/f noise have been extensively studied (111-121). The overall amplitude distribution was Gaussian, but the variance noise was not. With variance noise one means the fact that in some devices the variance of one time sample of duration T may be different from the variance of another time sample of duration z. One would expect such a phenomenon, e.g., if the sample gave noise at a high level at some times and at a low level at other times with random transitions from the one level to the other. Of course, more complicated models are possible. Voss (122) measured (V(t)l V(0) = Vo) for various devices showing l/f noise; this is the average behavior of the noise before and after a fluctuation of amplitude Vo. He found that some noise sources were linear and others

FLICKER NOISE IN ELECTRONIC DEVICES

269

slightly nonlinear, whereas systems like p-n junctions showing burst noise revealed a nonlinear mechanism.

V. l/f NOISEIN SOLID-STATE DEVICES A . l/fNoise in Tunneling Devices

The most important tunneling devices are the Josephson junction, the metal-oxide-metal diode, and the semiconductor tunnel diode.

1. l/f Noise in the Josephson Junction

As suggested by Clarke and his co-workers (123,124), the l/f noise in these devices is caused by spontaneous temperature fluctuations at the junction. The device voltage V depends on the critical current I,, such that at a current I > I,, according to the Stewart-McCumber model (125),

where R is a constant equal to the resistance at high current. A temperature fluctuation 6T produces fluctuations in I,, and this, in turn, produces fluctuations in V according to the relation

But according to Voss and Clarke (42) [our Eq. (52a)]

where C, is the heat capacity of the active volume of the junction, and w1 and w2 the width of the film making up the junction (in their case Nb and Pb), so that

In view of Eq. (89), S , ( f ) decreases with increasing current I . As Fig. 7 shows, the agreement between theory and experiment is reasonably good, which is somewhat surprising since Eq. (52a) gives a

270

A. VAN DER ZIEL

-2

-19

-

\

N . ,

-=8 -20-

c

CS

-21-

-

01

-22

BACKGROUND I

I

log f

I

I

(Hrl

FIG.7. Voltage spectra for three values of Josephson junction bias current I. Note that the fluctuations decrease with increasing I (J.Clarke and G . Hawkins, IEEE Trans. Magne. M A G 11, 841 (1975)).

1/f noise regime only if w1 and w 2 differ appreciably (in these experiments w1 and w 2 were of the same order of magnitude). This l/f noise sets a lower limit to the low-frequency application of Josephson junctions (126-128). l/f Noise Noise inin Metal-Oxide-Metal Metal-Oxide-Metal Diodes Diodes 2.2. l/f Flicker noise measurements have been reported by Zijlstra (129) on Ta-Ta,O,-Ta structures and by Liu and van der Ziel(130) on A1-A1,O3-AI structures. The noise was shot noise at high frequencies, as expected, and l/f noise at low frequencies with S,(nvarying as P . In the Ta-Ta,O,-Ta structures the shot noise was suppressed because the electron moving from metal to metal hops from metal to trap to trap * * to metal, and at high frequencies each movement gives rise to independent pulses carrying a fractional charge (131). In the A1-A1,O-Al3 structures the oxide was much thinner (1.20 A), and the shot noise was not suppressed because multiple hopping is a very rare event. The noise was attributed to temporarily trapped carriers. The trapped carriers have two effects :

(a) They modulate the barrier height E,, and this gives rise to noise (see also Section V,B). (b) The random distribution of traps gives rise to a wide distribution in time constants.

FLICKER NOISE IN ELECTRONIC DEVICES

27 1

We now investigate this model. Since the current density J is given by the Fowler-Nordheim equation,

where Eb is the potential barrier height, F the electric field strength, h = h/2n (where h is Planck’s constant), q the electron charge, m* the effective mass, f ( y ) Nordheim’s elliptic function, y = (qFo/n&&o)’/2/Eb, E~ the permittivity of free space, and E the relative dielectric constant of the oxide. Differentiating (92) with respect to E,, assuming that f ( y ) is slow function of E,, yields

6J = - J[l

6Eb + $BE;/’f(y)] Eb

with B =

4(2m*/h2)l12 (4F)

so that S,( f ) = 2qDJ2

where D =

[l

+ %BE;’2f(y)]2&,(f1 2qEb’

where S , , ( f ) has a l/f spectrum because of a wide distribution in time constants. The two types of devices measured gave about the same value of D at low frequencies. For an alternate theory see Kleinpenning ( 1 3 2 ~ ) . 3 . The Semiconductor Tunnel Diode

The (Z- V) characteristic of a tunnel diode has three distinct regimes : (a) the low-voltage regime with positive derivative; (b) the intermediate-voltage regime with negative derivative; and (c) the high-voltage regime with positive derivative. In regimes (a) and (b) the device is a majority carrier device due to tunneling through the barrier. In regime (c) the device is a minority carrier device due to diffusion or thermionic emission over the barrier; this corresponds to what one finds in normal p-n junctions. In regime (a) Agouridis (232) found no flicker noise above 100 kHz; he did no measurements in regime (b) and found a considerable amount of flicker noise in regime (c), all for germanium tunnel diodes. Yajima and Esaki (233) found no flicker noise in regimes (a) and (b) down to 10 Hz for Ge tunnel diodes, whereas they also found a large amount of flicker noise in region (c). The llfnoise mechanism in part (c) of the characteristic should be similar to the noise in p-n junction diodes, except for the fact that the p

272

A. VAN DER ZIEL

and n regions are much more heavily doped. We refer to Section V,C,l for details. The llfnoise in regions (a) and (b) is difficult to measure because of the low impedance levels involved. For example, in the negative conductance region the device may have to be shunted by a small load resistance R, to prevent oscillations. Any possible flicker noise may then drown in the thermal noise background of R , . Since the device is a majority carrier device in regimes (a) and (b), one would expect only a relatively small amount of flicker noise, if any. The basic tunneling mechanism should be noiseless, but some l/f noise might occur due to tunneling via traps in the forbidden gap. See Section V,B,l for details. It would be worthwhile to verify the absence of flicker noise in regimes (a) and (b) more carefully, since it could have important theoretical consequences. B. Flicker Noise in Schottky Barrier Diodes

The Schottky barrier diode is a majority carrier device. Zettler and Cowley (134) have demonstrated that by using a p-n junction guard ring structure it is possible to obtain Schottky diodes with forward currentvoltage characteristics that approximately follow the expected Richardson equation. The guard ring structure also lowers the low-frequency noise because it reduces the current flow across the surface. In the ideal case the llfnoise should be caused by the current flow from bulk to contact and should be inversely proportional to the contact area. This surface effect was corroborated by Wall (135) and by Hovatter (136). Wall first measured the noise in a planar structure and then changed the structure to a mesa structure without changing the contact area and remeasured the noise. He found a decrease in noise by more than one order of magnitude. He concluded from this result that the llfnoise was generated by edge currents. Hovatter found for point contact Schottky barrier diodes made on heavily doped material that the flicker noise at a given current was inversely proportional to the current diameter rather than to the contact area. Hsu (137) measured Schottky barrier diodes with a gate structure. He found less noise at flatband conditions under the gate than when the region under the gate was strongly accumulated. This puts the main source of the noise in the edge (or surface) current. The carriers contributing to this current interact with the surface oxide via interface states. If that were the case, S,y) should indeed vary as l/J would be proportional to the surface state density N,, and could be reduced by appropriate surface treatment.

FLICKER NOISE IN ELECTRONIC DEVICES

273

Hsu (138) has given the theory for the case in which surface effects are absent. The mechanism discussed is a two-step tunneling process in which the electron tunnels from the conduction band of the semiconductor to a trap state and from the trap state to the metal. The fluctuating trap occupancy gives rise to a fluctuation in barrier height, and this produces noise. The tunneling distances depend on the energy of the carriers and decrease with increasing carrier energy, so that a wide range of tunneling times is present. He found for the case of a parabolic band structure and a triangular potential profile,

where

(934 Here Nt is the trap density, T~ the tunneling time constant at the top of the barrier, L, the width of the lower limit of the potential barrier where twostep tunneling occurs, L, the width of the upper limit of the potential barrier where two-step tunneling occurs, w the width of the space-charge region, A the contact area, T the device temperature, Nd the donor density, and E the relative dielectric constant of the material. The spectrum is constant at low frequencies ( 0 7 , < 1) and varies as l/f at intermediate frequencies (l/z2 < w < l/zJ and as l/f2 at high frequencies (wrl > 1). According to this equation, S r ( f ) is proportional to 1/A, as expected. The corner frequency f,, defined as the frequency for which the flicker noise equals the shot noise, is given by

But

is independent of N, so that the corner frequency at constant current density I/A is practically independent of the donor concentration N,. Here vd, is the diffusion potential. According to Grant (139) this flicker noise model is operating in InP Schottky barrier diodes.

A. V A N DER ZIEL

274

C. l/f Noise in Junction Devices

1. l/f Noise in p-n Junctions

If we consider a p+-n junction, then the current flowing in the forward direction is due to holes. In long junctions it comes about because the holes recombine with electrons at the surface of the n region (in germanium and silicon) or at the surface of the space-charge region (in silicon). The noise comes about because the surface recombination velocity s shows fluctuations 6s with a spectrum S,(f). Such a fluctuation would be expected because the surface recombination velocity is modulated by the trapping and detrapping of carriers in traps in the oxide adjacent to the oxide-semiconductor interface (McWhorter model). One would expect 6s to be proportional to SNt, and hence S,(f) should be proportional to S N , ( f ) . But, S N , ( f ) is proportional to q, and hence, since Tt is proportional to the surface-state density N,,, proportional to N,. Since s is also proportional to (or to N,,), we may thus write S J f ) = CS/f

(94)

where C may be a slow function of the surface current density J,. This is essentially Fonger’s result (140). We now turn to the case where the recombination occurs at the surface of the n region. If p’ is the excess hole density at a surface element dA, J, =

6 J , = 4P’dSY

qp’s,

SJp(f)= (qP’)2S,(f)

(95)

whereas the junction current I, =

s s J, dA =

qsp‘ d A = qsp’(O)A,,

(96)

when the n region is so long that practically no holes reach the ohmic contact to the n region. Here p‘(0) is the hole concentration at the beginning of the n region and A,, is an effective recombination area. Moreover, since all surface elements fluctuate independently,

f

cs CZ; A:, p ” d A = q2-[p‘(0)12A:, = -f fs A&

(97)

where AiE is another effective area. We thus see that S,(f) is proportional to If. This agrees reasonably well in some germanium p-n junctions at low currents, but there is often a large dip in the noise (sometimes an actual zero) at higher currents (Z4Z). Fonger attributed it to the current dependence of the series resistance r, of the n region.

FLICKER NOISE IN ELECTRONIC DEVICES

27 5

OHMIC RING CONTACT

OHMIC CONTACT

FIG. 8. Watkins’ arrangement for studying flicker noise in Ge p-n junctions.

Watkins (142) had a different explanation. He used a geometry shown in Fig. 8. The surface to the n region parallel to the junction was so treated that a high recombination velocity was obtained and the width w was so chosen that most recombination occurred at the treated surface. He then expressed the junction voltage V in terms of the hole concentrations p(0) at x = 0 and p(w) at x = w. The zero in the noise occurred if.dV/ds = 0. Van der Ziel (143) calculated dV/ds more accurately and showed that the condition for a zero value in dV/ds was 2p(w)

+ Nd(2 - In 2) = (sw/D,)p(w)

(98)

where D, is the hole diffusion constant. This gives a positive solution for p(w) when sw/Dp > 2, which is the case if s is sufficiently large. As shown from Eq. (981 this can only occur at relatively high injection. For small s the zero should not occur. Gutkov (144) found for germanium p-n diodes that S , ( f ) = const (Z2/szf)

(99)

and concluded that this contradicted Fonger’s equation (94), since it contradicts Eq. (97).Van der Ziel(143) was able to show, however, that the same treatment given for the Watkins experiment yielded for his geometry S,(f) = const

I2

sf(1

+ SW/D,)~

If one now takes the n region sufficiently long, w changes along the surface and one must average over w. This can give a wide range of s for which S , ( f ) varies as l/s2, especially if s is sufficiently large. Hsu et al. (145) found for silicon diodes that the noise spectrum S , ( f ) at constant current was proportional to the surface recombination velocity s, in contradiction with Eq. (97). By assuming that in this case the noise is generated in the junction space-charge region, van der Ziel (143) reconciled this result with Eq. (94). This recombination occurs in a well-defined part of the space-charge region, characterized by the coordinate xl. Let for an applied voltage

276

A. VAN DER ZIEL

V the potential at x1 change by an amount V, and let p(xl) and p1 be the hole concentrations at x1 for applied voltages V and zero, respectively; then the recombination current I, is given by 1, = qSp(x1)Aefi = qsp1 exP(eVl/kT)Ae,

(101)

where Aeff is the effective area of the recombination region. If s now fluctuates by an amount as, then the fluctuation 81, in I, and the fluctuation 61 in I is 81 = 61, = qp1 exp(qI/,/kTMe,ds,

S d f ) = (qP1Ae,)* exp(2qVdkT)Ss(f)

(102) Substituting (94) for Ss(f) and bearing in mind that I = I. exp(qV/mkT), we have S,(f) = const(I)zmY1/Ys/f

(103)

in agreement with Hsu et al.’s data. If Vl N- 1/2V and rn N 1, Sz(f) would be proportional to 1. This current dependence of S,(f) agrees reasonably well with the results obtained by Plumb and Chenette (146) for noise generated in the emitter space-charge region of silicon transistors. For Vl/V > 3, Sl(f) goes faster than linear with the current I. The final conclusion of this discussion is that the noise is due to fluctuations in the surface recombination velocity s, and that there is direct experimental evidence in favor of it. North (147) had a similar theory. He assumed that the noise is caused by fluctuations in the surface potential 4s.These fluctuations are thermal and the fluctuations in the surface recombination velocity s follow from

where Re, is the real part of the equivalent impedance into which 4, looks. To calculate Req, North developed an equivalent network for the surface traps similar to the one shown in Fig. 5b. This is an alternate way of looking at the noise. Hsu (148) has developed a theory for the l/f noise observed in a gatecontrolled diode in terms of surface-state effects. He finds that the noise is proportional to the density of these states and the square of the transconductance gm = aZ/aV of the device, where I is the device current and V the gate voltage. For diodes without gate gm must be replaced by aI/a4,, where 4sis the surface potential. There is then a similarity with North’s theory.

FLICKER NOISE IN ELECTRONIC DEVICES

277

2. Flicker Noise in Transistors

The theory of flicker noise in p-n junctions can be directly applied to flicker noise in transistors. That means that in silicon transistors most of the flicker noise comes from recombination in the emitter-base spacecharge region. In general the flicker noise can be represented by two current generators, i f , and if2,in parallel to the emitter-base junction and the collector-base junction, respectively. In view of what was just said, one would expect if, and ifz to be fully correlated, since they come from the same noise source, and i f , should far predominate over i f z , since the noise is a fluctuation in the emitter-base recombination current IR that is not transmitted to the collector. This was demonstrated by Chenette (149) and further substantiated by Gibbons (150). Probably the clearest demonstration came in a classical paper by Plumb and Chenette (146). Their experimental setup is shown in Fig. 9a. Here a large resistance RE was inserted into the emitter lead and a variable resistor RB was inserted between base and ground. The noise was measured between emitter and ground, and RB was so adjusted that the measured noise was a minimum. The noise voltage u appearing across the emitter terminal is

FIG.9. (a) Plumb and Chenette'scircuitfor studying flicker noise in transistors (IEEE Trans. Electron Devices 4-10, 304 (1963)). (b) Equivalent circuit of flicker noise in transistors, incorporating surface noise and dislocation noise.

278

A. VAN DER ZIEL

where reO = kT/& and if, has been split into two parts, i;z, fully correlated with ifl, and i;,, uncorrelated with ifl, respectively. We see that 3 will go through a deep minimum if

(105a)

By plotting (RB),,,in versus reo/ao = kT/qI,, one should obtain a straight line intercepting the vertical axis at -rb, whereas the slope gives [l c2/aoifl)]. This agreed very well with the experimental data and the values found for i;, were only a few percent of i f l , so that if, can be neglected. Measurements of i& were obtained for small values of Z E , by omitting RB altogether; they indicated that izl varied as Zt,with B somewhat smaller than unity in some units. This is compatible with Eq. (103). Measurements by Viner (151) on silicon transistors operated at elevated temperatures showed another interesting feature. At elevated temperatures ZB = -ICBo + 4 where -ICBois the base current for zero emitter current and rBthe injected base current. Viner showed that the currents ICBoand Zh fluctuated independently, that each showed flicker noise, and that the in his units (TI483). flicker noise spectrum of TBvaried as (4)1.5 One might now ask the question whether the flicker noise in transistors can be completely eliminated by fully passifying the surface. The answer is that this is not the case (252, 253). After fully removing the source of noise at the surface, one is left with llfnoise due to dislocations. The dislocations then act as flicker noise generating centers just as the surface traps did, and the mechanism is probably similar. Since the dislocations are distributed through the bulk, this is a true bulk effect. The flicker noise can be further reduced by reducing the dislocation density. This can be done in two ways

+

(153) :

(a) One starts from dislocation-free material (perfect crystal technology). (b) In n-p-n transistors one deposits P/As mixed doped oxide onto Si using PH, and ASH, in a flow ratio of 4: I and one then diffuses in. This reduces stress in the emitter and so prevents the introduction of dislocations by the diffusion process. The two noise sources, surface l/f noise and dislocation l/f noise, are located somewhat differently in the equivalent circuit. This is shown in Fig. 9b, where ifl represents the surface flicker noise and iil represents the dis-

279

FLICKER NOISE IN ELECTRONIC DEVICES

location flicker noise, and rb = R,, + R,, is the total base resistance. Brodersen et al. (154) have the two noise sources interchanged, however. Mueller (155) has suggested that the flicker noise in p-n junctions and transistors is caused by temperature fluctuations described by a thermal circuit with a wide distribution in time constants (62); this transforms the shot noise into llfnoise. Van der Ziel(156) was able to show that this could only occur when the device was operating close to a thermal instability. Shacter et al. (157) demonstrated that such a situation did occur in second breakdown in power transistors and that it transformed the normal llfnoise into l/f’ noise as expected theoretically. 3. Burst Noise (158)

Burst noise has been observed in planar silicon and germanium diodes and transistors. The phenomenon consists of a random turning off and on of a current pulse of 10-8-10-9 8.It can be described by a random telegraph signal approach (159), and leads to a spectrum const/(l 02r2), where z is the time constant associated with the pulse. The noise can be practically eliminated by using a low source resistance in common base amplifiers. This indicates that the current generator describing the burst noise must be located much closer to the external base contact than the current generator describing the flicker noise generated at the surface. Since burst noise is not a flicker noise phenomenon, we refrain from a more detailed discussion.

+

D . Flicker Noise in JFETs

Good silicon JFETs at room temperature do not show any flicker noise to speak of (160), whereas GaAs FETs show a relatively large amount of it (161). We shall see that this hangs together with the structure of these devices. We shall also see that the absence of l/fnoise in silicon JFETs has important consequences for our understanding of semiconductor noise in general and that it can help pinpoint the source of this noise (162). Let us first assume that flicker noise is a bulk effect and that Hooge’s formula

Mf )/NZ= ./fN

(106)

is valid. Following Klaassen’s theory of flicker noise in MOSFETs (76), we then obtain for the spectrum of the equivalent flicker noise emf 6V in series with the gate

280

A. V A N DER ZIEL

for V, V,, where V, is the pinch-off voltage, Zo the current, V, the drain voltage, gm the transconductance, q the electron charge, p the mobility, and L the device length. We substitute Hooge's constant a = 2 x and further put q = 1.6 x lo-'' C, p = 1.4 x cm2/v sec, k = 1.38 x J/"K, T = 30O0K,L = 10-3~m,Z= 5mA, V, = 2Vygm= 5 x mho, which are very reasonable values. We then obtain for the noise resistance at room temperature R, N lO'O/f ohms. In fact one finds (160) for good JFETs at room temperature R , N 105/(1 + 02z2) ohms. We thus conclude that the noise at room temperature is generation-recombination (8-r) noise and that flicker noise is absent. This means that flicker noise in semconductors cannot be a bulk effect, for if it were, it would be present in JFETs also. Conversely, we may say that if Eq. (106) were valid, then a < 2 x lo-*, in clear violation of the bulk noise hypothesis. But we can go one step further. The only difference between good silicon JFETs and all other semiconductor devices is that the former have no semiconductor-oxide interface, whereas all other semiconductor devices do. This points to the semiconductor-oxide interface as the source of the noise. This leads to the following model. Carriers interact with traps in the oxide near the interface and this produces three effects : 1. The fluctuating occupancy of the traps modulates the surface recombination velociiy s of injected carriers in structures involving minority carrier flow; this explains the l/f noise in p-n junctions and transistors (Fonger model) (140). 2. The fluctuating occupancy of the traps gives rise to fluctuations in the carrier density of the semiconductor material or device; this explains the McWhorter model of flicker noise (78). 3. The fluctuating occupancy of the traps modulates the surface potential of the interface and so gives rise to fluctuations in the (surface) mobility of the carriers in the semiconductor. This explains the Kleinpenning model of flicker noise (53,88).

We wish to point out here that we have now, in principle, unified all the important models of flicker noise in semiconductor materials and devices. We shall come back to this problem in Section V,F. We can now also understand why there is a large amount of flicker noise in GaAs FETs. These devices have a gate width that is much smaller than the source-drain length. Hence there is a large semiconductor-oxide interface area between gate and source as well as between gate and drain. This gives a full contribution to flicker noise, since one would expect Eq. (106) to be approximately valid in this case with a value of a determined by the interaction between the carriers and the oxide traps via surface states.

FLICKER NOISE IN ELECTRONIC DEVICES

28 1

The ramifications of this insight for the theory of flicker noise in MOSFETs will be discussed in the next section. E. Flicker Noise in MOSFETs 1. General Theory

Flicker noise in MOSFETs is usually thought to be caused by carrier density fluctuations, brought about by interaction of free carriers with oxide traps via interface states (76,163-172). In order to clarify the problem, we start here the theory from first principles, rather than applying previously developed formulas. Let ( x , y , z ) be the coordinate system; here x is in the direction of the length L of the sample and y is the direction of the width w , whereas z is perpendicular to the oxide and pointing into it. We consider a surface element AxAy and a volume element AxAyAz in the oxide. Let AN, =n,(E)AEAxAyAz by the number of traps in the volume element AxAyAx with an energy between E and E + AE. Let ANs = n,(E)AEAxAy be the number of surface states in the surface element AxAy with an energy between E and E + AE. Let ANt and AN, be the number of carriers in those states, respectively, and let the interaction occur by tunneling at constant energy E ; then (170) dAN,/dt = g(AN,) - r(AN,),

g(AN,) = aAN,(AN, - AN,),

r(AN,) = PAN,(ANs - AN,)

( 108)

The interaction between the channel carriers and the surface states is so fast that we can ignore it. In equilibrium ANs = ANsa = ANsfi,

AN, = ANta = AN,fi

(108a)

wherefi is the fractional occupancy of the surface states and the traps, and dAN,/dt = 0. Substituting (108a) into this equation yields a = P. Now we look for slow fluctuations GAN, in AN, and slow fluctuations GAN, in AN, that are driven by the fluctuations in g(AN,) - r(AN,). Substituting AN, = ANsa + GAN, and AN, = AN,, + GAN, yields dGAN,/dt = uAN,GAN, - aANsSAN,

+ Ag(t) - Ar(t)

(109)

We now substitute JAN, = -yGAN, and bear in mind that AN, is proportional to AxAyAz and ANs to AxAy. In a subsequent integration process with respect to z, we let Az go to zero, so that the first term in (109) becomes negligible. We may thus put as dominant lifetime t =

l/(aANs)

(109a)

A. VAN DER ZIEL

282

According to the theory of carrier density fluctuations (172)

46AN;z S d f ) = 1 + mzz2

6AN: = g(ANto)t= ANTft(1

and

- ft)

(110)

Hence

where

f, = ( 1

+ exp[(E - E,)/kT]}-'

(1 12)

and E, is the Fermi level, so that ft(l - fd has a very sharp peak at E = E,. We now integrate with respect to z. To that end we bear in mind that in a tunneling model z = zo exp(ez)

(113)

where E is of the order of lo8 cm- '. We now assume a uniform trap distribution for 0 c z c z1 and zero traps outside; this is allowed if z1 is so chosen that traps for z > z1 have so long a time constant z that their effect on the noise cannot be measured. We then have a normalized distribution

for zo < z < z1 and zero outside that interval. Replacing Az by z,g(z)Az, and integrating (111) with respect to z between the limits 70 and zl yields

We next integrate this expression with respect to the energy E. Since f ( l - ft) has a sharp peak at E = E,, we introduce the parameter

J-

W

nr(Ef)eff =

n,(~)ft(l- ft) d~

(115a)

If we also integrate with respect to y between the limits 0 and w, we obtain, since ln(zl/zo) = E Z ~ ,

where S,,(f) is defined for unit area. Hence, z1 has disappeared.

283

FLICKER NOISE IN ELECTRONIC DEVICES

We next illustrate the meaning of Eq. (115a). Assuming &(E) for E not too far from Ef , nT(Ef)eff

=

N

n@f)

(115b)

nT(Ef)kT

For the case in which %(E) depends strongly on E near E = E,, the full definition (115a) must be used. We now bear in mind that %(Ef)is usually thought of as being proportional to the surface-state density ns(Ef) so that nT(Ef)e& is proportional to the effective surface-state density N,, eff. If C is the proportionality factor, we obtain (116a)

It will be shown shortly that for very strong inversion the fluctuation SAn in the number of free carriers AN for area wAx is equal to -SANt, or = -SANt,

S,(f)

= S,,(f),

S"(f)= S"Jf)

(117)

where S , ( f ) is again defined per unit area. Next we consider the case that the device is operated at a very low drain voltage Vd(Vd 4 5 - b).The device is then nearly uniform and the spectrum S N ( f )of the fluctuation in the N carriers in the sample is 'N(f)

(118)

= Sn(f)wL

Since the charge fluctuation SQ in the channel may be written SQ = q S N = C,,wLSV,,,

or

SVeq 0 =-

'

COXWL

SN

where SV,,, is the equivalent emf in series with the gate at near zero drain bias, the spectrum of SV,, , is S"@(f) = (q2/c:;ww"(f)

(1 19)

The equivalent noise at the input is thus a direct measure for S,,(f). This is essentially Katto's result (268, 169). If S , , ( f ) is independent of the applied voltage (V, - b),where V, is the gate voltage and VT the turn-on voltage, as seems to be the case in many is independent of (V, - b);this is Katto's n-channel MOSFETs, SVeqO(f) result (168,272). If, however, S,,(f) is proportional to (V, - b),as seems to be the case in many p-channel MOSFETs, SVeq,(f)is proportional to (V, - VT); this is essentially Klaassen's result (76, 272). It should be borne in mind, however, that Klaassen assumed that S , ( f )

284

A. VAN DER ZIEL

was proportional to n, rather than to (V, - V,) only. He put S,,(f) = an/f, where a is Hooge's parameter. Since qn = Cox(&- V,), this yields

(119a) This gives a slight difference in the dependence of the equivalent input noise on the oxide thickness t in both cases. Since Cox= E E , / ~ , Eq. (119) varies as t2 if S,,(f) does not contain the parameter Cox;whereas Eq. (119) varies as t if S,,(f) is proportional to Cox.Equation (119a) varies as t also. Both cases seem to have been found; for the first see Katto (169),for the latter case see Kaassen (76) and Berz (165). See also a paper by Vandamme and de Kuijper ( 1 6 9 ~ ) . We now want to evaluate Sv0,(V,, V, - V,,f) for arbitrary drain voltage V, (170).According to the Klaassen-Prim approach, as formulated by van der Ziel (173),

S,(f) =

$J

0

where F ( x , f ) = S r ( x , f ) Ax

F ( u , f )du

(120)

and Sr(x, f ) is the spectrum of the primary current fluctuation in the section Ax at x. But dx

where o(V0) = pwCox(Q- V, - V,) is the conductance for unit length at x, p the carrier mobility, V, the gate voltage, and V, the turn-on voltage of the channel. Hence

):();(

~ I ( x t), =

-

- 6AN(x, t )

and

so that

where we have switched to V, as a new variable.

FLICKER NOISE IN ELECTRONIC DEVICES

285

We now define

Now

and (123b) Substituting into (123) yields

- V, -

h - Vd/2Jo"d V,

sue,~(V,

- - VO,f)dVO V,-v,-V, (124)

The first half corresponds essentially to Christensson et al.'s Eq. (9) (163) and the second half to van der Ziel's Eq. (10) (170). If Sueqo( V, - 5,f ) is proportional to V, - V,, we have

which decreases from Sucq ,JV, - V,, f ) at V, = 0 to half this value at saturation (Vd = V, - G).This agrees with experiments for many p-type samples. If Svoq0(V, - h,f) is independent of (V, - V,), the integral in Eq. (124) diverges logarithmically at saturation. This is overcome because the condition of strong inversion is violated when the channel is at or near cutoff; consequently Sueqo(V, - V,, f ) must go to zero when V, - V,;0' we shall see that this comes about because (- 6AN/6ANt)-+ 0 at pinchoff. Both results presuppose, of course, that the channel has uniform doping and a uniform surface. 2. The Correction for Arbitrary Inversion Up to here we assumed that 6AN = -6ANt. Jindal and van der Ziel (174) have shown that this is true in the region of very strong inversion. If this condition is not satisfied, the above results must be multiplied by the

286

A. VAN DER ZIEL

rI

'2

FIG.10. Equivalent circuit of an MOS structure.

factor R2, where R = -6AN/6ANt; this parameter decreases from unity to a very small quantity when the gate voltage changes from the condition of very strong inversion to the condition of very weak inversion. They found from first principles that

This has a very simple equivalent circuit interpretation. Q, is the total charge density, Qinvthe inversion layer charge density, and Qsd = Q, - Qinv the depletion layer charge density; whereas us = q+,/kT, the normalized surface potential. We now introduce the capacitances per unit area : &&

cox= 0 t '

cN -

q2

aQinv

kT au, '

c

W

= - - -q2

aQsd

dT au,

(125a)

where is the relative dielectric constant of the material and t the thickness. This leads to the equivalent circuit of Fig. 10. Since the induced fluctuation is shared by Cox,C,, CNin parallel and only the charge on C, is effective (174, =

cN/(cN

+ cw + cox)

(125b)

For a numerical evaluation of R these capacitances must be evaluated numerically (174). 3. The Effect of Mobility Fluctuations The fluctuations in the occupancy of oxide t r a p not only produce density fluctuations, they also modulate the surface potential and so produce mobility fluctuations. We investigate this problem for a MOSFET at low-drain bias. Let N , be the number of trapped electrons, N the number of free carriers, and 6Nt and 6N the respective fluctuations such that 6N/6Nt = -R. Since

FLICKER NOISE IN ELECTRONIC DEVICES

287

the fluctuation SNi drives the fluctuation 6p in p, Sp and SN will be fully correlated. The relative current fluctuation SZ/Z may then be written

so that the noise calculated previously must be multiplied by the factor (1

+;$

(126a)

a factor that was first proposed by Berz (165).But dp/dN = (dp/d4,)(d$,/dNi) (dNJdN) and SN = SN, (dN/dN,).If we now put dN/dN, = - R, we have (126b) so that the noise calculated in Section KE,l must be multiplied by the factor

(R-fg.2) 2

(126c)

Since d$,/dNi < 0, the two terms add if dp/d$, > 0. Katto (169) essentially added the two terms in (126b) quadratically rather than linearly. This does not make much difference. When the one term predominates over the other, the error is negligible, and the maximum error (of a factor 2) occurs when the two terms are equal. For weak inversion R is extremely small, but the second term can still be quite significant; in that case the mobility fluctuations will predominate. The same conclusion was drawn by Katto.

4. Experimental Data Klaassen (76) and Berz (165) find the noise resistance R,, at low drain voltage to be proportional to the oxide thickness t, whereas Katto (169), Mantena and Lucas (176),Hsu (166) and Christensson and Lundstrom (177)find a t 2 dependence.We have already seen how that can be explained; the t 2 dependence is probably more fundamental. Klaassen (76) finds RnOproportional to - V, forp-channel MOSFETs, whereas Katto (169) and Pai (171) find R,, to be practically independent of - & for n-channel MOSFETs. We saw already how that can be interpreted. Fu and Sah (167) find a more complicated behavior in especially constructed units.

<

<

288

A. VAN DER ZIEL

Takagi and van der Ziel (178) find R, to be rather independent of temperature for p-channel devices and more strongly dependent of temperature for n-channel devices. Christensson and Lundstrom (177) also report temperature data, whereas Rogers (179) has measured down to even lower temperatures. This must be interpreted in terms of the temperature dependence of N,, Klaassen (76) and van der Ziel (180) have shown that R,, is inversely proportional to wL, but the L dependence may no longer be true for the noise resistance R, of short-channel devices at saturation because of the occurrence of hot electron effects (180). Various authors (76, 169, 181) report R, to be proportional to the surface-state density N,, (Fig. 11). Leuenberger (182) finds an N: dependence, but the evidence for the latter is not clear, since it seems that a linear plot would fit the data equally well. D-MOS devices were investigated by Huang and van der Ziel (183) and

lijla

/

7

-

/

I 3

I

L

I

1

I

I I

I

1

I

I

FLICKER NOISE IN ELECTRONIC DEVICES

289

by Takagi and van der Ziel(184). They find a behavior different from other devices. Not only is the current dependence of R, different, but also the spectrum is different: n+-p-p--n+ structures, with the gate over p - p - , have a 1iY.O dependence at lower frequencies and a 1/y.6 dependence at higher frequencies, whereas n+-p-n--n+ structures, with the gate over p-n-, have a l/f'.6 spectrum throughout. This indicates that the 1/f'.6 spectrum comes from the p region and the l/fnoise from the p - region, whereas the n- region gives no measurable noise. Various papers have been written about surface treatment. Yau and Sah (185) used phosphorus gettering, Cheng et al. (186) used the HCl process, whereas Leuenberger (187) claims to have found a process that gives only g-r noise. Hielscher and End (188) have studied the effects of oxygen-reduction treatment on l/f noise. For further experimental papers see the list of references (189-202). For further theories see also references (203-205). Van der Ziel (206) has suggested a limiting flicker noise in MOSFETs due to the dielectric losses of the oxide; in modern devices this limit is not yet reached, however. F. Semiconductor Noise Revisited

The discussion in Section V,D has indicated the possibility of surfacecontrolled mobility fluctuations, and the discussion of Section V,C has indicated that in certain MOSFET cases, especially those involving weak inversion, the mobility fluctuations may predominate. This verifies Kleinpenning's model to a certain extent. Now, it should be taken into account that the MOSFET is a very particular device in which an inverted layer forms a conducting path between source and drain. In most other devices, e.g., in semiconductor resistors, this is not the case; rather, majority carriers in the bulk interact with carriers in a somewhat accumulated or depleted surface, and the carriers near the surface interact with oxide traps. It may well be that in this case the mobility fluctuations also predominate; that needs to be proved by a detailed calculation. There are two experiments for which the above mobility fluctuation model must be carefully scrutinized : (a) Can the model explain the experiment by Hooge and Vandamme (94)? (b) Can the model explain the thermoelectric experiment (53)? For one would, in this case, expect the carrier fluctuations in each subband to be fully correlated with the fluctuating occupancy of the traps.

A. VAN DER ZIBL

290

The space-charge-limitedflow experiment (88) is much less critical, since it only needs predominance of the mobility fluctuations. VI. MISCELLANEOUS PROBLEMS A . Formal Explanation of llfy Spectra with y # 1

In most of the theories developed so far, the spectrum was exactly l/f over a wide frequency range. What one finds experimentally, however, is a l/pspectrum with y close to unity, but not exactly equal to unity. We shall see that an appropriately chosen distribution function in z can formally explain this behavior. Consider a number fluctuation with a single time constant z and a spectrum -

S,(f) = 46N2 [2/(1

+ o"z")]

where 6Ni does not depend on z. Let now the following distribution function be introduced : g(z) dz = A

=0

dz

for zo < z c z1

zfl

otherwise

(128)

with j? # 1, where A is so chosen that

["g(z) dz = 1

J To Then for l/zl c o c l/zo

which yields

(128a)

For zo 4 z1 this gives a l/f2-fl spectrum over a wide frequency range. Butz (207) has given an extension of this model using very rigorous mathematical methods. We now transform back to the y domain by putting 7/70

This yields

= e"y

(130)

FLICKER NOISE IN ELECTRONIC DEVICES

29 1

We then see that the distribution function in y is no longer uniform. It thus seems that a nonuniform distribution function in y can explain deviations from the exact l / f spectrum. Since there is no particularly plausible argument in favor of an exactly uniform distribution in y , a nonuniform distribution seems quite reasonable. In our example we considered fluctuations in the number N of free carriers. Actually the same spectrum would be found if we had considered carriers trapped in the oxide. We could then have explained l/y spectra in the fluctuating surface recombination velocity, in the fluctuating number of free carriers, or in the fluctuating free-carrier mobility. That covers most of the practical cases of l/y noise in semiconductor devices.

B. Handel’s Theory of l / f Noise (208-211)

Handel has proposed a quantum theory of l / f noise caused by a selfinterference effect. In a noise experiment charge carriers pass from one terminal of the circuit element to the other. The corresponding quantum transition amplitude will contain a large term corresponding to the transition without emission of any photon above the low-frequency threshold of the noise measurement. But it will also contain a small term which is the amplitude of the same transition with emission of a bremsstrahlung energy between E and E d&. Consequently, the wave function that describes the coherent propagation of the charge carriers through the circuit element will contain a small part whose frequency is reduced by an amount between E/h and ( E dE)/h. The interference of this small part with the large part corresponding to the absence of bremsstrahlung represents beats with frequencies between E/h and ( E d )/h. This interference appears as a noise current having a quantum expectation value of the spectrum that is nonzero. This spectrum duplicates the number spectrum of the emitted photons. Since the bremsstrahlung radiation power per unit frequency interval is constant at low frequencies, the number spectrum is proportional to l/o. The exact shape of the spectrum is made convergent by radiative corrections, but the exponent of the frequency in the spectrum remains very close to - 1 . In contrast with the theories presented so far, Handel’s theory is a fundamental theory based directly on general physical principles. If correct, it should therefore set an absolute lower limit to the l / f noise process occurring in physical systems. Other l / f noise processes may be present, and we have seen what kind of processes they are, but they will give noise over and above the limit set by Handel’s theory. Tremblay has failed to verify Handel’s theory (211) in detail. The latest information seems to be, however, that this controversy is being resolved by a set of joint papers (212).

+

+

+

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C . Other l / f Noise Theories

There have been a large number of formal theories of l / f noise in which the main purpose is to produce a l / f spectrum without much regard to physical reality. It is beyond the scope of this paper to discuss them.

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Zie&“Noise, Sources, Characterization, Measurement,” p. 114. Prentice-Hall, Englewood Cliffs, New Jersey, 1970. 152. M. Conti, Solid-State Electron. 13, 1461 (1970). 153. S. Yamamoto, M. Wanatabe, N. Fuse, T. Yonezawa, and M. Nakamura, Colloq. Int. C.N.R.S. 204,87 (1972). 154. A. J. Broderson, K. B. Cook, and E. R. Chenette, Colloq. Int. C.N.R.S. 204, 91 (1972). 155. 0. Mueller, IEEE Trans. Electron Deiiices ed-21,539 (1974); Arch. Elektr. Uebertr. 28,492 (1974). 156. A. van der Ziel, IEEE Trqns. Electron Devices ed-22,964 (1975). 157. S. B. Shacter, A. van der Ziel, E. R. Chenette, and A. D. Sutherland, Solid-State Electron. 21,599 (1978). 158. W. H. Card and P. K. Chaudhary, Proc. IEEE 53, 652 (1965); G. Giralt, J. C. Martin, and F. X . Mateu-Perez, C. R. Hebd. Seances Acad. Sci. 261, 5350 (1965); Electron Lett. 2, 228 ( I 966). 159. S. Machlup, J. Appl. Phys. 25, 341 (1954). 160. C. F. Hiatt, A. van der Ziel, and K. M. van Vliet, IEEE Trans. Electron Devices ed-22, 614 (1975). 161. Cf., e.g., K. Takagi and A. van der Ziel, Solid Slate Electron. 22,285 (1979). 162. A. van der Ziel, Appl. Phys. Lett. 33, 883 (1978). 163. S. Christensson, I. Lundstrom, and C. Svensson, SolidState Electron. 11,797 (1968). 164. E. A. Leventhal, Solid State Electron. 11,621 (1968). 165. F. Berz, Solid-State Electron. 13,621 (1970). 166. S. T. Hsu, Solid-State Electron. 13,1451 (1970). 167. H. S. Fu and C. T. Sah, IEEE Trans. Electron Devices ed-19,273 (1972). 168. H. Katto, Y. Kamagaki, and Y. Itoh, Proc. Conf Solid State Devices, 6th, 1974 Supplement to J. Jpn. Soc. Appl. Phys. 44, 243 (1975). 169. H. Katto, M. Aoki, and E. Yamada, Proc. Symp. llf Fluctuations, 1977 Conf. Rep., p. 148 (1977); M. Aoki, H. Katto, and E. Yamada, J. Appl. Phys. 48,5135 (1977). 169a. L. K.J. Vandamme and A. H. de Kuijper, in “Noise in Physical Systems” (D. Wolf, ed.), p. 152. Springer-Verlag, New York, 1978. 170. A. van der Ziel, Solid-state Electron. 21,623 (1978). 171. S. Y. Pai, Ph.D. Thesis, University of Minnesota, Minneapolis (1978). 172. A. van der Ziel, “Noise in Measurements,” Sect. 5.2f. Wiley (Interscience), New York, 1976. 173. F. M. Klaassen and J. Prins, Philips Res. Rep. 22, 505 (1967); see also A. van der Ziel, (172, Sect. 7.2). 174. R. P. Jindal and A. van der Ziel, Solid-State Electron. 21,901 (1978). 175. C. T. Sah, Proc. IEEE 55,654 and 672 (1967). 176. N. R. Mantena and R. C. Lucas, Electron. Lett. 5,605 (1969). 177. S. Christensson and I. Lundstrom, Solid-State Electron. 11,813 (1968). 178. K. Takagi and A. van der Ziel, Solid-state Electron. 22,289 (1979). 179. C. J. Rogers, Solid-state Electron. 11, 1099 (1968). 180. A. van der Ziel, Solid-State Electron. 20,267 (1977). 181. G . Abowitz, E. Arnold, and E. A. Leventhal, IEEE Trans. Electron Devices ed-14, 775 (1967). 182. F. Leuenberger, Helv. Phys. Acta 41,448 (1968). 183. C. Huang and A. van der Ziel, Solid-State Electron. 18,885 (1975).

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184. K. Takagi and A. van der Ziel, Solid-State Electron. 22, 1 (1979); 22 in press (1979). 185. L. D. Yau and C. T. Sah, Solid-State Electron. 12,927 (1969). 186. Y. C. Cheng, J. W. Haslett, E. J. M. Kendall, R. J. Kriegler, and F. J. Scholz, Proc. IEEE 62,859 (1974). 187. F. Leuenberger, Electron. Lett. 7, 561 (1970). 188. F. H. Hielscher and J. H. End 111, Appl. Phys. Letf. 24, 27 (1974). 189. H. G. Dill, Electron. Lett. 3, 341 (1967). 190. T. Tanaka, K. Nagano, and N. Nakemi, Jpn. J. Appl. Phys. 8, 1020 (1969). 191. M. Nakahara, H. Iwasawa, and K. Yasatake, Proc. IEEE 57,2177 (1969). 192. F. Berz and G. Prior, Electron. Lett. 6, 595 (1970). 193. T. A. R. Bhat, Arch. Elektr. Uebertr. 26, 343 (1972). 194. R. J. Hawkins and G. G. Bloodworth, Solid-state Electron. 14,929 (1971). 195. F. Leuenberger, Phys. Status Solid, A 8,545 (1971). 196. V. V. Potemkin and P. K. Kasharow, Radio Eng. Electron. Phys. (Engl. Transl.) 17, 518 (1972). 197. I. Kabayashi, M. Nakahara, and M. Atsumi, Proc. IEEE 61, 1145 (1973). 198. R. S . Ronen, RCA Rev. 34,280 (1973). 199. M. B. Das and J. R. Moore, IEEE Trans. Electron Devices ed-21,247 (1974). 200. G. Broux, R. van Overstraeten, and G. de Clerck, Electron. Lett. 11, 97 (1975). 201. K. Nakamura, 0. Kudoh, and M. Kamashita, J. Appl. Phys. 46. 3189 (1 975). 202. K. L. Wang, IEEE Trans. Electron Devices ed-25,478 (1978). 203. E. H. Nicollian and H. Melchior, Bell Syst. Tech. J . 46,2019 (1967). 204. C. G. Bloodworth and R. J. Hawkins, Radio Electron Eng. 38,17 (1969). 205. J. W. Haslett and F. N. Trofimenkoff, Solid-State Electron. 15, 117 (1972). 206. A. van der Ziel, Solid-State Electron. 18, 1031 (1975). 207. A. R. Butz, J. Statist. Phys. 4, 199 (1972). 208. P. H. Handel, Phys. Rev. Lett. 34, 1492 and 1495 (1975); ibid. Phys. Lett. A 53,438 (1975); Z . Naturforsch., Teil A , 30, 1201 (1975). 209. P. H. Handel, Proc. Symp. l / f Fluctuations, 1977 Conf. Rep., pp. 12, 26. 183, and 206 (1977). 210. P. H. Handel, in “Linear and Nonlinear Electron Transport in Solids” (J. T. Devreese and V. E. van Doren, eds.), p. 515. Plenum, New York, 1976. 21 1. A. Tremblay, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge (1978). 212. P. H. Handel and A. Tremblay (to be published).

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS, VOL. 49

Recent Advances in Electron Beam Deflection EDWARD F. RITZ, JR. Tektronix, Inc. Beaverton, Oregon I. Introduction. . . . . . . . . . . , . , . . . . A. Scope and Organization B. Background.. . . . . . . . . . . . . . . . C. Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Magnetostatic Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Background.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. .

299 299 300 302 303 303 304 307 322 111. Electrostatic Deflection ........................... 323 323 324 C. Magnetically Immersed Electrostatic Yokes. . . . 336 D. Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . 344 IV. Traveling-Wave Deflection . . . . . . . . . . . . . . . . . . . 345 .......... 345 341 348 349 349 349 350 351 354 354 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

I. INTRODUCTION A. Scope and Organization

This review aims to describe and assess the ferment that has overtaken the field of electron beam deflection in the last ten or fifteen years. This agitation is largely due to the impact of the high-speed digital computer on theoretical studies of deflection structures. The computer methods, together with theoretical developments in a more classical tradition and the invention of some new deflectors, have revitalized a field formerly handicapped by a 299

Copyright Q 1919 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-014649-5

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serious lack of theoretical methods that are both powerful and practical. Consequently,this review emphasizes recent advances in methods of analysis, simulation, and design. It might be thought that such an account of progress stemming primarily from the application of computer techniques would be merely a dreary tale of recipes for computing algorithms; such is not the case. Without physical and mathematical insight, the use of computers is sterile and misleading; what is gained in calculating power is more than lost in understanding. Thus, the true significance of the high-speed digital computer is not in its vast calculating power, but rather in the innovative theoretical methods now made practical by the computer. Consequently, in this review I have tried to give a summary of the physical and mathematical bases of the work reviewed sufficient to acquaint the reader with the principles involved. Details must be obtained from the original papers. It is worth remarking that some of the theoretical formulations discussed were described many years ago and have only lately been reinvented or recovered from the literature. Hutter (1974) has recently described the progress in electron beam deflection during the twenty years ending in 1973, and has given an outline of the fundamentals of the subject. Hutter’s review is strongly recommended as preliminary reading; it will be cited frequently to avoid needless repetition of already published material. The present review supplements Hutter’s by including much work prior to 1973 that was not discussed by him, and goes on to describe and assess the many new developments that were just beginning to appear in 1973. Hutter reviewed a period of development that was drawing to a close; as he predicted, the new methods made possible by computers were opening a new era. I shall attempt to review here the beginnings of that era and the current state of progress, as well as to point out possible lines of future development where these can be divined. The body of the present review is divided into four sections: magnetostatic deflection, electrostatic deflection, traveling-wave deflection, and scan magnification. Unfortunately, the sprawling growth of the subject makes logical organization difficult ; consequently, these sections are of grossly unequal length owing to the differing rates of progress in these fields during the period discussed here. Each section begins with a brief introduction containing background information and a plan of that section. The remainder of the section is a synthesis and assessment of the published work. B. Background Electron beam deflection is a venerable topic of study, its continuing practical importance having inspired several generations of investigators. There has been a steady flow of improvements stimulated by the constantly

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increasing stringency of requirements for practical applications. Even so, in comparison with a field such as semiconductor electronics, the pace of progress in electron beam deflection has been rather sedate. Until quite recently most improvements were achieved using empirical methods guided by simple physical models. Theoretical methods suited for practical use have been largely lacking. Theories of deflection have suffered from two principal defects : Realistic theoretical models are too unwieldy and usable models too unrealistic. This circumstance has seriously impeded further progress. In the last ten or fifteen years, the situation has altered dramatically with the application of high-speed digital computers to the perennial theoretical problems of electron beam deflection. The fundamental problem is to calculate the electric or magnetic field produced by a given configuration of conductors, currents, and dielectric or permeable materials, and then to trace the trajectories of the electrons that form the beam. Analytic solutions have been found only in a few cases of limited utility. Consequently, theorists have traditionally resorted to various orders of perturbation expansions which require extensive numerical calculations. The computer has now assumed this heavy burden of calculation, thereby making possible the practical use of these long-established theories. Unfortunately, the convergence properties of the perturbation expansions limit their use absolutely to angles of deflection less than 45" and practically to angles less than 40" at most. Furthermore, in most cases of practical interest the field distribution must be determined by experiment for use in these expansions, as it cannot be calculated analytically. A more fundamental change brought about by the introduction of computer methods is the development of theoretical techniques especially suited for use with a computer. Finite-difference and finite-element calculations of the fields, combined with numerical solutions of the equations of electron motion, have made the old perturbation techniques almost unnecessary. The new methods permit field calculations independent of experiment and trajectory calculations regardless of deflection angle. However, the old methods are still valuable for qualitative understanding of the deflection aberrations. The impact of computer methods has been greatest in the field of magnetostatic deflection. In the study of electrostatic deflection and scan magnification, the new methods have not been employed as yet to full advantage, although some advances have been made. Traveling-wave deflection, on the other hand, has benefited very little from application of computer techniques. Thus, the potential for substantial future improvements in these latter fields certainly exists. Whether or not the potential improvements actually occur will depend largely on the urgency of commercial demands for progress in these relatively undeveloped fields. For example, the recent progress in

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magnetostatic deflection is due to the stringent requirements of wide-angle, large-screen color television, requirements that led to very extensive research. Similar pressures in other fields may well result in commensurate progress based on computer techniques. Recent advances in electron beam deflection are not limited to those brought about by the application of computer methods. In the last thirteen years, there have been substantial improvements in electrostatic deflectors and in mixed-field deflectors employing both electric and magnetic fields. These improvements have been due, in more traditional fashion, to inventions, empirical investigations, and theoretical calculations based on simplified models. Progress in the same vein has also occurred in scan magnification and, to a lesser extent, in traveling-wave deflection. However, full exploitation of the possibilities of these developments hinges on the employment of digital computers for more realistic modeling of these deflection systems. C. Preliminary Definitions

To save space and avoid tiresome repetition, some definitions regarding units, methods, and notation will be made here. The rationalized MKS system of units will be employed except as noted specifically in the text. The standard system of coordinates used is the right-handed rectangular system (x, y , z). The electron beam is assumed always to have a positive velocity component in the z direction. The usual cylindrical coordinates (r, 4, 2) are also employed, with x = r cos $J and y = r sin 4. The rest mass and charge of the electron are denoted by m, and -e, respectively. The ratio elm, is represented by q . As is customary, E, and p, are the permittivity and permeability of the vacuum. It is customary to neglect relativistic corrections in the study of electron beam deflection, even though there are actually a number of systems with beam potentials high enough to show observable relativistic effects. Therefore, only nonrelativistic electron motion will be discussed in this review. The vector differential equation governing the nonrelativistic evolution of the position vector r with time t in the fields of electric field strength E and magnetic induction B is

P = - q ( E + i x B) (1) which represents three coupled scalar equations in x , y , and z. Except in the case of traveling-wave deflection, it will be assumed that the electric and magnetic fields discussed are quasi-static. This assumption places two restrictions on the rate of variation of these fields wih time. First, the variation with time during the transit of a single electron must be neg-

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ligible. Second, the variation must be slow enough that the electric fields generated by magnetic induction and the magnetic fields generated by displacement currents are negligible. The term static deflection is used in this review with the understanding that these two restrictions are satisfied. In mathematical terms, we have

VXE=O

(2)

VXH=J (3) where J is the electric current density. Finally, wherever possible the notation standard in each subfield has been followed. However, this rule has been violated occasionally to avoid use of the same symbol for different quantities or to achieve coherence and consistency in fields where such standards are not established.

11. MAGNETOSTATIC DEFLECTION

A . Background Investigations of magnetostatic deflection have been hampered until recently by two deficienciesof available theories. First, there were no methods for accurately calculating the magnetic fields of practical deflection yokes. Second, even had the fields been known, the amount of calculation required to determine by numerical methods the electron trajectories through the fields was prohibitive before the availability of high-speed digital computers. There were approximate perturbation theories for small angles of deflection, but these also required voluminous numerical calculations. Consequently, designers proceeded empirically with guidance from simple physical models. With the advent of high-speed computers, the burden of calculation was assumed by the machine. This made possible the belated use of the smallangle theory for practical design work. At the same time, the widening availability of large computers made feasible the use of numerical techniques to solve the partial differential equations governing the magnetic fields of many practical deflection systems. Once the fields were obtained, the computer could calculate trajectories numerically for any angle of deflection. The last ten years have produced a series of remarkable improvements in the theoretical modeling of magnetostatic deflection systems. We shall examine this progress by beginning with a discussion of the use of computers in the small-angle theory, following with a presentation of the more accurate and powerful wide-angle techniques, and closing with an assessment of the state of the art.

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EDWARD F. RITZ, JR.

It is assumed that the reader is familiar with the basic types of magnetic deflection yokes and with the fundamental ideas of magnetostatic deflection as discussed in Hutter (1974). Yokes presently in use fall into two basic categories: toroid and saddle. Hybrid yokes with one toroidal and one saddle winding are common. The so-called stator yoke is actually a variant of the saddle type. The function of all types is to provide a magnetic field transverse to the undeflected beam. Pure magnetic deflection then results as described by (1) with E = 0. B. Small-Angle Theory 1 . Summary o j Theory

Limitations of space permit only a brief summary of methods and results to be given here. The original articles should be consulted for details, particularly the papers by Haantjes and Lubben (1957, 1959) and Kaashoek (1968). The basic method of the small-angle theory is first to write the Lagrangian function F(x, y , x', y', z) for an electron moving in a magnetic field and then to expand F in powers of x, y , x', and y', where primes signify partial differentiation with respect to z. This requires writing the magnetic field strength H in the power series

+ )V;)x2 + 2H2xy + &y2 + H y = Ho - (H2 + )&)y2 + 2&xy + H ~ x ' + H, = VAX + Hby - .

H, = Vo - (&

* *

(4)

* * *

(5)

(6) where the horizontal field parameters H,,and the vertical field parameters V , are given by *

*

(7)

Ho = ( H , ) x = y = o

~ , 1=anv,z ( v ),

n>o

x=y=o

In this expansion, the H,,, V,, and their derivatives are considered on a par with x, y, x', and y' in determining the order of each term. With these results, the Lagrangian function is then expanded in a similar power series that contains only even terms :

F(x,y, x', y', Z) = Fo where Fo = 1.

+ F2 + + F4

* * *

(1 1)

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The trajectories are calculated from the Euler-Lagrange equations which F must satisfy:

in which Fo makes no contribution because it is constant. If we now assume that the trajectory remains close to the axis and maintains a small slope, then the higher order terms in F can be neglected in the lowest approximation F = F z . This is called the first-order approximation because only terms of first order or less appear in (12) and (13). If we now consider a deflection system in which the undeflected beam strikes a flat screen located at z = z, in the point (x,,ys) with slope (xi,yi), then the first-order or Gaussian trajectory strikes the screen at (xg,y,) given by xg = x, Yg

= Ys

+ x,

(14)

+ r,

(15)

where

Here Usis the constant beam potential and zo is outside the deflection field on the entrance side. The third-order approximation is obtained by treating F4 as a perturbation of F z , giving terms up to third order in Eqs. (12) and (13). The resulting deviation Ax3(zs)from the Gaussian values (14) and (15) is

A x ~ ( z ,= ) A

3 0 i X

+ (A302 + B303)Xse

+ B306 + B306c)XsY86 + (A309Xf + B 3 1 0 e ) x s + (A312 + B 3 1 1 ) X s ~ y s + A307XsXi2 + A308X8i2 + 2(B308 + B 3 0 8 c ) x x b : + A313XsXs2 + A 3 1 4 X 8 f + B 3 1 5 x x 8 s + A316Xsxsx~ + B317xxiys + A 3 1 8 X 8 8 6 + ( A 3 0 4 g + B305et)xi +

+ ( B 3 1 8 + B31t3c)%x86

(1%

The corresponding expression for Ay,(z,) is obtained from (18) by interchanging the letters A and B, X and Y, and x and y. There are 21 each of the

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EDWARD F. RITZ, JR.

A and B coefficients, which are sums of definite integrals involving the quantities X,, g; their slopes Xi, q ;the functions Ho and Vo; and the fwnctions H , and V . . The explicit expressions are given by Kaashoek (1968). These errors are classified as raster or pattern distortion, curvature of field, astigmatism and astigmatismlike errors, and coma and comalike errors, according to their dependence on X,, x, , ys , x: , and y: . Details are again found in Kaashoek (1968). Yet another approximation can be made by adding F 6 , giving terms of fifth order at most in (12) and (13). The resulting fifth-order deviations involve an unmanageable number of coefficients that can only be dealt with by using simplifying approximations and numerical methods, according to Kaashoek (1968). The theory outlined above requires for its application the functions Ho , V, ,H 2 , V, ,etc., which describe the magnetic field distribution of the deflection coils. These functions of z must be obtained either by calculation or from measurement. Before the recent developments described in the subsequent section on large-angle theory, measurement was required in practice because very few deflection systems had fields that could be calculated with the methods then available. Such measurements are described by Kaashoek (1968) and Vonk (1971). It appears that the third-order theory is reasonably accurate for angles of deflection up to 20 or 25"; the fifth-order theory is useful perhaps to 40". Since the series expansions of the fields and trajectories diverge for angles above 45" and converge poorly near that value, the small-angle theory fails completely at 45".

<,

2. Use of High-speed Computers From the foregoing summary, it is quite clear that even the third-order small-angle theory requires a very large amount of numerical integration to obtain the required aberration coefficients. This fact effectively prevented any but qualitative uses of the third-order theory in practical problems until quite recently. However, the use of high-speed digital computers to evaluate the aberration coefficients has rendered the third-order theory usable. The calculation by computer of third-order aberrations from the smallangle theory has been described by several workers. Wang (1967a) calculates these aberrations by computer and uses the results to eliminate astigmatism and minimize coma in a field distribution for which Ho and H2 are written as sums of Gaussian distributions with various widths and centers. Only one axis of deflection is studied, however. In another paper (Wang, 1967b) he uses a computer to calculate numerically the current density of the spot at the screen by means of the Jacobian of the transformation connecting

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the parameters of the undeflected electron with those of the landing point at the screen. The result is a series of novel and instructive computer-drawn plots of current density in the spot for beams with initially uniform or Gaussian current densities. Kaashoek (1968), using a more general theoretical framework than Wang's, also calculates the third-order aberrations by computer and extends the calculation to fifth-order errors. He relies on field profiles H o , H, , H4, Vo, V, , and V4 determined by experiment. As an example, he discusses the design of a 90" shadow-mask color tube, where 90" is the angle subtended at the plane of deflection by the screen diagonal; that is, a deflection angle of 45". C. Large-Angle Theory

1. General Considerations We have just seen the difficulties that overtake the small-angle perturbation theory when the angle of deflection approaches 45". These difficulties arise from the approximation of the magnetic fields and the electron tiajectories by series expansions about the z axis of the system. Since exact analytic solutions are unknown for most practical magnetic deflection systems, we must resort to numerical techniques in correcting the deficiencies of the small-angle theory. The fundamental requirements for a wide-angle theory are two: first, a method for calculating the magnetic field everywhere in the volume through which electron trajectories will pass, and second, a method of calculating the trajectories themselves. Both of these tasks can be performed by computer using numerical techniques; however, calculating power by itself is not sufficient. The theory of magnetic deflection must be cast into a form that facilitates both the computer calculations and the ultimate practical application of the theoretical results. The calculation of trajectories presents few fundamental difficulties other than obtaining sufficient accuracy at acceptable cost. However, in calculating the magnetic field, the theoretical formulation should be chosen to reduce the very formidable amount of calculation required for threedimensional field distributions. At the same time, this formulation must facilitate both synthesis and analysis. That is, the designer must be able to synthesize a suitable configuration of the deflection system, as well as to analyze the performance of a given structure. Consequently, the greater part of the following discussion is devoted to recently developed methods of calculating the magnetic fields for practical deflection systems.

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EDWARD F. RITZ, JR.

All of the methods of field calculation to be discussed here rely on the use of the magnetic scalar potential Q, defined by

H = -VQ, The use of Q, depends on the vanishing of the current density outside the coil windings and magnetic core of a deflection yoke. From Eq. (3) we then havc

VxH=O where J = 0, which implies (19). However, care is required to ensure that no conductor is ever encircled by any closed contour of integration; otherwise, the line integral of H around such a closed contour would not vanish, and no scalar potential could be defined. If necessary, a suitable barrier surface can be introduced to prevent encirclement of a current filament. In such a case the scalar potential will be discontinuous across the barrier. The magnetic scalar potential must satisfy the Laplace equation

V2@ = 0 as a consequence of (19) and the divergence equation

V * B= p V * H = 0 provided that the permeability p is constant. Thus all the techniques available for the solution of the electrostatic Laplace equation can be applied here as well. To do so requires knowledge of the boundary values of Q,. These values are calculated on the assumption that the permeability of all cores or shields is very high; that assumption justifies the treatment of the surfaces of these magnetic components as equipotentials of Q, if there are no windings on the surface, and permits calculation of the surface distribution of potential should there be surface windings. These results follow from the vanishing of H within the highly permeable members as p/p0 becomes infinite and from the known distribution of current in the windings. Once the boundary potentials are obtained, Eq. (21) can be solved for Q, and the magnetic field H found from (19). For a more thorough discussion of the magnetic scalar potential and its uses, see the books by Jackson (1975) and Stratton (1941). 2 . Calculation of Electron Trajectories

The calculation of the magnetic scalar potential by numerical methods in a computer usually produces a matrix of potential values on a large array of discrete points covering the volume of interest. The calculation of electron trajectories requires a method of numerical integration for the trajectory equation (1) and a method of numerical differentiation and interpolation

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for obtaining the magnetic field H between the points of the original array. The methods used for trajectory calculations are not described very thoroughly in the published studies on wide-angle magnetic deflection. The papers of Bloomsburgh et al. (1965) and Bernstein et al. (1978) offer a few details. Bloomsburgh and his associates divide the volume of the yoke into small cubes in which average values of the potential and magnetic field are taken. However, their numerical procedure for solving the equations of motion is not entirely clear. Bernstein and his colleagues use cubic splines for interpolation and numerical differentiation of the fields. For solution of the equations of motion, they employ a classical fourth-order RungeKutta procedure. The solution of the trajectory equations is a standard problem in numerical analysis; suitable algorithms can be found in textbooks. On the other hand, numerical differentiation of the potential to obtain the magnetic field requires some care to avoid accumulation of errors in the trajectory integration. In the calculation of several trajectories differing only slightly in their initial conditions, careless interpolation and differentiation may introduce errors of the same size as the spacing between trajectories. Thus, a method of smooth interpolation such as spline functions is essential if conclusions regarding deflection aberrations are to be drawn. Hutter (1970) has proposed a solution to this problem: He first calculates a central trajectory numerically and then expands the magnetic field in a power series around this trajectory. Then he determines the paths of adjacent electrons using a perturbation technique based on a curved-optical-axis formalism. Unfortunately, the theory is very complex and requires lengthy numerical calculations for its application, although some useful qualitative conclusions can be drawn regarding the shape of the deflected spot. It appears from the work of Carpenter et al. (1977) and Bernstein et al. (1978) that the use of cubic splines with the Runge-Kutta method is sufficiently accurate for the simulation of convergence errors in color tubes. Here the half-angle of the cone on which lie the three beams of a delta gun or the two outer beams of an in-line gun is roughly 1". However, this reviewer has found no reported attempts to calculate the spot profile for an individual beam of the triad by direct numerical integration of the trajectory equations. Consequently, it is not known if the method is adequate for this task. Such a determination of the spot profile for a narrow beam is a severe test of the numerical accuracy of the field and trajectory calculations. 3. Three-Dimensional Solution of the Laplace Equation

A straightforward method of solving the Laplace equation (21) for the magnetic scalar potential Q, is to carry out a full three-dimensional solution

310

EDWARD F. RITZ, JR.

by replacing (21) with its finite-difference equivalent, which is then solved by standard relaxation techniques using a computer. This reviewer has found only one paper (Bloomsburgh et al., 1965) describing investigations employing this method; in fact, it is the earliest study found in which a computer is used to investigate magnetic deflection. The results are applied to the design of the shadow mask and creen for a 90" color tube. Bloomsburgh and his colleagues employ a hybrid method in which the finite-difference formulation of the Laplace equation is solved iteratively by computer to obtain the potential CD, but the boundary values of CD are found from measurements of the field of the particular magnetic yoke being simulated. The Laplace equation is solved in a quadrant of the volume of the yoke. The boundary values of CD on the surface of this quadrant are found by calculating the appropriate line integrals of the surface components of B as determined experimentally with a Hall-effect probe. The finite-difference equation is solved by first defining a three-dimensional lattice of points to fill the volume of the yoke quadrant. The finitedifference equation relates the potential at each point to the potentials of its six nearest neighbors. Initially, estimated potentials are assigned to the internal lattice points and boundary potentials to the surface points. Then a new potential for each internal point is calculated using the finite-difference equation and the six neighboring potentials. The residual, or difference between new and old values, is used to correct the potential at each point. Finally, the entire lattice is traversed repeatedly in this fashion until each residual is less than a predetermined value. The last array of potentials is the solution of the finite-difference equation and hence an approximation to the solution 0 of the Laplace equation (21). The accuracy of this approximation can be improved in principle by reducing the interval between lattice points. However, the amount of calculation required then increases because of the increased number of points. For a discussion of the finitedifference solution of the Laplace equation, see Weber (1967). The technique of Bloomsburgh and his co-workers is a substantial improvement in the simulation of magnetic deflection yokes. However, there are several disadvantages. First, the experimental determination of boundary potentials is time consuming, difficult, and subject to errors that affect the calculated internal potentials. Second, three-dimensional field calculations need large amounts of storage and much calculating time in the computer because attainment of acceptable accuracy requires a large number of lattice points (9000 in this study). Third, it is difficult to use this method of field calculation in the design of yokes rather than of shadow mask and screen, as originally intended, because there is no explicit connection between the

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

31 1

winding distribution and permeable core on the one hand and the magnetic field on the other. This last deficiency compels the designer to resort to trial and error, with each tentative design requiringa new experimental determination of boundary potentials. In the next section, we consider techniques that reduce or eliminate these difficulties. 4. Multipole Field Expansion a. Reduced Laplace equation. One method of decreasing the amount of numerical calculation required to solve the three-dimensional Laplace equation using finite-difference methods is to reduce the problem to the r-z plane by separation of variables. This is easily done in cylindrical coordinates by writing the scalar potential o(r, 4, z) as a Fourier series in the azimuthal angle : m

@(r,4, z) =

m=l

[A,(r, z) cos rn4

+ Bm(r,z) sin rn4]

(23)

where m is a positive integer, The term in rn = 0 is omitted for reasons explained below. The functions A, and B, are called multipole potentials or functions. Substitution of (23) into the Laplace equation (21) gives a set of equations in r and z only:

with a similar set obtained by replacing A with B. Thus, in place of the full three-dimensional Laplace equation, we have an infinite set of reduced Laplace equations in only two dimensions. In modern magnetic deflection yokes, the current-carrying conductors are usually arranged on a surface of revolution about the z axis. This surface is either a cylinder or a flared horn; the shape of the surface is defined by the winding form and the ferromagnetic core (if one is used). Consequently, the cylindrical coordinate system (r, 4, z) is a natural choice for magnetic yokes, and the Fourier decomposition (23) is well suited to the description of the scalar potential. This reduction to a two-dimensional problem would offer little comfort were it necessary to solve the entire infinite set of equations (24).Fortunately, as we shall see shortly, only a limited number of values for rn is needed for an adequate description of the magnetic field in practice.

312

EDWARD F. RITZ, JR.

The solutions of (24) can be expressed as power-series expansions in r (Glaser, 1952) : m

Am(r,

Z)

=

C n=O

Amn(Z)rZn

The coefficients A,,(z) are found by substituting (25) into (24) :

where the superscript (2n) indicates the 2nth derivative with respect to z. For small r, A, is approximately AAr, Z) = &o(z)r"

(27)

Exactly similar results hold for B,. The terms in m = 1 prevail near the axis. The higher order terms vanish on the axis but act as edge corrections at larger radii. Thus, relatively few terms of (23) are needed in practice; the highest multipole reported in the literature is m = 9 (Carpenter et al., 1977). The above Fourier decomposition of the scalar potential was given by Glaser (1952) for the analogous electrostatic potential. Nomura (1971), Barkow and Gross (1974, 1975), Schwertfeger and Kasper (1974), Munro (1975), Kasper (1976), Hutter et al. (1977), Carpenter et al. (1977), and Bernstein et al. (1978) have all recently applied this analysis to magnetic deflection yokes. Ohiwa (1977) used a similar field analysis in the context of the small-angle theory. The results of some of these investigators are discussed below in more detail. b. Boundary conditions for the reduced Laplace equation. Appropriate boundary conditions on A, and B, are required in solving the reduced Laplace equation (24). Below is a brief discussion of the calculation of these boundary conditions based primarily on the fundamental papers of Nomura (1971) and Schwertfeger and Kasper (1974). For simplicity, we first consider magnetic yokes with highly permeable cores and windings laid directly on the surface of the core with no gap between core and windings. The cross section of the core in the r-z plane has the boundary r = R(z), as shown in Fig. 1. The values of A, and B, on R(z) from (23) are simply the Fourier coefficients of the surface potential :

The behavior of A,,, and B, at large distances and along the axis must also be specified by giving either the potential or its normal derivative. The boundary condition at large distances is conveniently specified by placing

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

I"."

313

SHIELD

z FIG. 1. Longitudinal section of the ferromagnetic Core R(z) of a magnetic deflection yoke. The shield S(z) serves as a computational boundary for determining the magnetic scalar potential Q.

Y

t

FIG. 2. Transverse section of the ferromagnetic core of a magnetic deflection yoke; the angular surface current density is k,(R, 4, z),

a large, azimuthally symmetric, highly permeable shield [S(z) in Fig. 11 around the yoke. It is shown below that the potential vanishes on both the shield and the axis. If the relative permeability p/po of the core is effectively infinite, then the distribution of current on the core determines its surface potential very simply. With that assumption, the magnetic field strength H vanishes inside the core. Therefore, the Ampkre circuital law applied to the circuit C in the transverse plane shown in Fig. 2 gives

-

H dl *

=o' 'J

ka(R,

$9

2) d+

(29)

where ka is the current per radian passing through C. The left-hand side

314

EDWARD F. RITZ, JR.

of (29) is just the potential difference between Po and P:

@(R4, Z) - @(R,4 0 , Z) =

14:

ka(R

$3

Z) d$

(30)

Differentiation with respect to 4 gives

k,(R 494

= d@(R,4, z)/a4

(31)

The surface-current density K(R, 4, z) is conveniently resolved into components K, in the 4 direction and K , tangent to R(z) in the r-z plane. The angular current density k,(R, 4, z) is thus ka(R, 4, Z) = K r z N z )

(32)

Since V * K vanishes by continuity, K, and Krz are related by 1 aK, aK,, +-=O R a+ as

--

(33)

where s is the path length along r = R(z). The surface-current density can be expanded in a Fourier series in the same manner as the magnetic scalar potential. However, the windings of magnetic yokes have symmetry properties that affect the number of harmonics appearing in this expansion. The deflection-producing windings have an axis of symmetry, say = 0, such that ka(-4) = kk,(+) and ka(4 n) = -k,(+). Therefore, only odd rn can appear. The quadrupole windings sometimes used for convergence in color yokes have an axis of symmetry, again taken as 4 = 0, such that k J - 4 ) = kk,(+)and ka(+ + n) = ka(+). Consequently, only even m can occur. The Fourier expansion containing both types of windings is

+

+

OD

ka(R, 4, Z) =

C1 [am(R, Z) cos m+ + Bm(R, Z) sin m43

(34)

m=

in which the term for rn = 0 has been omitted; the coefficient a. for this term vanishes identically because of the symmetry of k,. The coefficients in (34) are

By substituting (23) into (31) and comparing coefficients with (34), we obtain relations connecting the current and potential coefficients on the core : Am(R, Z) = - B m ( R z)/m

(36)

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

315

Thus, each harmonic m of the current distribution produces the same harmonic m of the potential distribution and no other, but with a 90" phase shift in 4 that interchanges sines and cosines. Finally, it remains to specify the potential on the axis and on the shield. Since there is no term m = 0 in (27), the potential is zero on the z axis. The potential also vanishes on the shield with use of (35)-(37) because there are no currents on the shield. With these last boundary values, the solutions of (24) are now completely determined. The multipole theory can also be extended to calculation of the potential for yokes in which the windings are at some distance from the core, a type of construction common in hybrid yokes having a toroidal winding in one axis and a saddle in the other. Details of this extension are given by Nomura (1971); lack of space prevents their inclusion here. In this case the displaced windings form a barrier surface of discontinuity in the potential and the tangential component of H. These discontinuities are determined by the surface-current density I<, of which the K4 component is also written in a Fourier series with the aid of (32)-(34). Finally, Nomura notes that thick windings can be simulated with several current-bearing surfaces of infinitesimal thickness. c. Calculation of the multipole potentials. Two techniques are presently used for calculating the multipole solutions A, and B, of the reduced Laplace equation (24) : the finite-difference and finite-element methods. The former is favored by the majority of the authors listed at the end of Section II,C,3,a; only Munro (1975) describes the use of the latter. A brief summary is given below; for details, the original referencesjust mentioned should be consulted. The finite-difference method relies on the replacement of the original partial differential equation (24) with its finite-difference equivalent, which is then solved by relaxation methods on a lattice of points in the r-z plane within the boundaries given. Several variants of the basic relaxation procedure are available; perhaps the most commonly used is the successive overrelaxation (SOR) method. Some authors transform the original equation (24) into another before carrying out the finite-difference solution (Schwertfeger and Kasper, 1974; Kasper, 1976; Janse, 1971). However, the advantages of these more elaborate methods are not clearly established in the literature. Schwertfeger and Kasper (1 974) assert that the original method converges faster than the modified method; in fact, the latter introduces convergence difficulties.It is worth noting that the authors who have published explicit potential distributions calculated by computer have uniformly used the original formulation (24) (See Nomura, 1971; Barkow and Gross, 1974, 1975; Carpenter et al., 1977; Hutter et al., 1977; Bernstein

316

EDWARD F. RITZ, JR.

et af., 1978). A possible exception is the paper of Van Alphen (1972) discussed in Section II1,B. In the finite-element method described by Munro (1973,1975), the energy stored in the magnetic field of each multipole potential is minimized by an iterative numerical procedure on a mesh of small triangles that covers the r-z plane within the boundaries specified, thereby determining the value of Q, at the vertex of each triangle. This formulation of the problem is completely equivalent to the partial differential equation (24), which is obtained by applying the Euler-Lagrange equations of the calculus of variations to the functional that is minimized in the finite-element method. The finiteelement approach has two advantages : It accommodates irregular boundaries easily and readily includes the effects of spatial variations of permeability. d. Applications of multipole field analysis. Toroidal deflection yokes are the most easily analyzed with the multipole Fourier technique because each turn of the winding lies nearly in a plane that also contains the z axis. Consequently, the surface-current density vector also lies in a plane containing the z axis, thereby ensuring that K4 = 0 and K,.,(R, 4, z) = ka(4)/R(z), where the angular current density k,(4) depends only on 4 and not on z. Thus, the Fourier coefficients a, and p, of the current distributions are constants given by (35). On the core cross section, the multipole potential A, and B, given by (36) and (37) are constants independent of z. Therefore, A,@, z ) and B,(r, z ) can only differ by a constant factor. They are conveniently written in the form A,@, z ) = a,U,(r,

z)

(38)

B,,,(r, 2) = b,U,(r,

Z)

(39)

where a, and b, are constants given by am

(40)

= - pmlm

(41)

b, = u,/m

and UJr, z ) is a solution of the reduced Laplace equation (24) with the boundary condition at the core U,(R,

Z)

(42)

= 1

Thus, the complete potential is m

@(r, 4, z ) =

m= 1

'Um(r,z)[a, cos m+

+ b, sin m$]

(43)

Figure 3 shows the multipole character of the fields produced by the individual multipole potentials.

F UNDAMNTAL

3RD HARMONIC

5TH HARMONIC

7ni

HiRMONIC

FIG.3. Magnetic field patterns in a transverse plane for odd harmonics of the multipole potential (43) in a toroidal magnetic yoke. In the circular diagrams, continuous lines are field lines; the short arrows indicate the force on the electron beam. The rectangular diagrams show beam displacements at the screen (Carpenter el al., 1977).

318

EDWARD F. RITZ, JR.

This is the simplifying characteristic of toroidal yokes : The functions Urnare completely independent of the current distribution. They need only be determined once for a given core and shield geometry. The effects of altering the current distribution k,($) are calculated from the simple summation (43). Furthermore, a desirable potential distribution can be found by tracing

A5 k,zl

(C)

FIG.4. Multipole functions Am@,z) or Um(r,z) for a toroidal magnetic yoke. (a) Fundamental; (b) third; (c) fifth. Closed curves are equipotentials at intervals of 10% of the core potential; orthogonal curves are field lines (Barkow and Gross, 1974).

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

319

trajectories through the potential (43) while adjusting the coefficients a, and b, without reference to the windings. When suitable values for a, and b, are determined, the current distribution that will give these values is immediately known from (35), (40), and (41). Several authors have applied the multipole analysis to the design of toroidal yokb for shadow-mask color tubes with in-line guns in which the three undeflected beams lie in the x-z plane with one beam on the axis and the other two on either side. Barkow and Gross (1974, 1975) describe a yoke for a 67-cm, 110" color tube (- 55" deflection angle) that perfectly converges horizontal lines while leaving a small separation between beams at the top and bottom of the screen. This misconvergence is then removed by an auxiliary quadrupole winding driven dynamically at the slow vertical scan rate. A similar 19-in. (48-cm), 90" system has been studied by Carpenter et al. (1977), Hutter et al. (1977), and Bernstein et al. (1978). Barkow and Gross achieve their design with the use of the harmonics rn = 1,3, and 5. They show very instructive computer plots of the multipole functions Urn@,z), which are reproduced in Fig. 4. Note particularly the concentration of the equipotentials in the vicinity of the core for the higher harmonics. The self-convergence of horizontal lines is obtained by adding the m = 3 term with opposite signs in the horizontal and vertical windings, thereby producing a controlled astigmatism in each axis (Schlesinger, 1949). The horizontal field consequently is pincushion shaped, while the vertical is barrel shaped. Carpenter, Hutter, Bernstein, and their respective colleagues find that odd terms through m = 9 are sufficient to characterize their deflection system; Fig. 5 illustrates the effects of successive harmonics on convergence. Their theoretical studies show excellent agreement with experiment for both magnetic field and trajectory calculations. For example, Fig. 6 shows the measured and computed profile of the transverse magnetic field on the z axis (Hutter el al., 1977); for both vertical and horizontal coils the agree-

t

FUNDRMLNTRL

v

ONLY

WRRW

l a 3 , S

nRRw

x

6.8,

'I

I

I

1,sr

7,

a

s.m

FIG.5. Effect of harmonic content on convergence in a color tube with a toroidal yoke; dimensions are in inches (Carpenter et al., 1977).

320

EDWARD F. RITZ, JR. I

1.0-

n 0.8-

J

g

IA

; w

0.6

.. 5 a 2

n W

0.4-

3

-1

4

B0 0.2-

2 0

VERTICAL COIL

0 t 4

t 3

t 2

t I

+INCHES

0

-+ -I

-2

-3

-4

FIG.6. Comparison of transverse magnetic field along the axis computed by the multipole technique (solid curve) with measured values for a toroidal yoke (Hutter et al., 1977).

ment is very good. Similarly, for a diagonal radius of 9 in. (23 cm), Carpenter et af. (1977) report differences between predicted and measured final deflection in the range of +0.3 to - 1.2%.They also claim good agreement between calculated and measured landing patterns at the screen for the triad of beams, as shown in Fig. 7. As an aid in obtaining a first estimate of the coefficient of the m = 3 term for a specific application, Carpenter, Hutter, Bernstein, and their colleagues use a modified third-order narrow-angle theory. The third-order theory is completely determined by multipole fields with m I 3 and can be inverted algebraically to obtain the ratio of the third- and first-harmonic amplitudes, given the desired amounts of astigmatism and coma. This first estimate is then refined by adding other harmonics and by making trial-anderror adjustments based on wide-angle trajectories. The most complete account of this technique is given by Bernstein et al. (1978). The multipole formulation has also been used to simulate the effects of various disturbing influences on toroidal yokes. Electrostatic lens fields are often present near the entrance to the yoke; these can be included in the trajectory calculation via the term in E in Eq. (1). External magnetic fields such as the terrestrial field can be added into B. Mechanical misalignments are easily studied. Carpenter et al. (1977) give numerous examples of this kind. Barkow and Gross (1974, 1975) also consider some of these disturbing factors.

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

x

7.2s

Y

32 1

-L.

IN 8OTW CR6ES

+ x

COMPUTED VRLUE MERCURED VRLUE

FIG.7. Computed and measured landing patterns for a three-beam color CRT; dimensions are in inches (Carpenter et al., 1977).

Saddle yokes are more difficult to analyze using multipole fields than are toroidal yokes because the current distribution depends on both 4 and z in general. This means that the coefficients a, and p, given by (35) depend on z , and therefore that the surface potential on the core cross section varies with z for each multipole function A , or B,. Thus the boundary conditions for the solution of the reduced Laplace equation (24) depend on the current distribution, in contrast to the toroidal case. Consequently, in principle, both solutions &(r, z ) and B,(r, z ) are required, since the vertical and horizontal current distributions may differ. Furthermore, these solutions of (24) must be recomputed each time the windings are altered because the boundary conditions will also change. The principal result of these added complications is this: Although a given saddle yoke can be simulated accurately at the expense of additional computation time for the Fourier coefficients of the current at many values of z, it becomes very difficult to find a suitable field distribution and to determine the corresponding current distribution during the design process. The design of saddle yokes is therefore much more complicated and indirect than the design of toroidal yokes. This reviewer has found no examples in the literature of saddle yokes designed using the multipole method. However, two papers describe the simulation of the fields of existing saddle yokes. The most complete account is that of Nomura (1971) who analyzes one axis of deflection for a saddle yoke with windings displaced from the surface of the ferromagnetic core. He calculates the Fourier coefficients of the current distribution for m = 1, 3, and 5 at various points along the winding in the z direction. These coefficients provide the boundary conditions for calculating the corresponding multipole potentials, for which Nomura gives equipotential plots. Substantial agreement is obtained between predicted and measured magnetic fields,

EDWARD F. RITZ, JR.

322 1.0-

20AX

COIL

HORIZONTAL WINDING 0.8 -

NONRADIAL SADDLE I (I

W 2

0.6 -ti 0 -I

w 0.4-3 I

B 0.2

SEREEH-

-

0 -

4.

2'

0

2.

4.

t

6'

FIG.8. Computed and measured transverse magnetic field distribution on axis in a saddle deflection yoke (Carpenter et al., 1977).

despite the use of only three harmonics. The paper of Carpenter et al. (1977) also mentions the simulation of a saddle yoke and gives curves (shown here in Fig. 8) of the predicted and measured transverse field strength along the axis, although these workers deal primarily with toroidal yokes. D . Assessment The use of the high-speed digital computer in conjunction with the smallangle theory of magnetostatic deflection has certainly provided a considerable improvement in both the qualitative and quantitative understanding of magnetic systems. Nevertheless, certain weaknesses remain, the fundamental defect being the limitation of the method to deflection angles less than 45". Furthermore, the small-angle theory provides no means to calculate the axial profiles H,, H, , and so forth, for practical saddle and toroidal magnetic yokes with permeable cores. Consequently, rather difficult, expensive, and time-consuming measurements of these quantities must be made. Finally, the extreme complexity of the theory makes design work difficult, even with the aid of computers. The mathematical connection of the winding distribution in the coils with the deflection of the beam is very indirect. It is not always easy to see how to change H,, H, , etc., in order to achieve a required improvement in the aberration coefficients, and it is

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

323

even more difficult to determine the correct winding distribution to achieve the desired values of the H and V parameters. The development of the multipole theory of fields in magnetic deflection yokes has led to great progress in the simulation and design of magnetic deflection systems. The use of the multipole analysis requires the computational power of high-speed digital computers. It is now possible to make very accurate simulations of toroidal or saddle yokes and to trace electron trajectories accurately through their fields by numerical integration of the equations of motion. In the design of toroidal yokes, the multipole formulation greatly aids in synthesizing a suitable winding distribution for a given design. Saddle yokes are not similarly favored and also require more elaborate numerical calculations to simulate the magnetic fields. It is unclear as yet from the published studies whether or not the direct numerical integration of the trajectory equations is sufficiently accurate to permit calculation of spot profiles for narrow beams. However, present accuracy is certainly sufficient for calculating misconvergence patterns in shadow-mask color tubes. Direct solution of the three-dimensional Laplace equation by the finitedifference method has proved much less successful. Even if the correct boundary conditions were calculated using the arguments of Section II,C,4,b, the amount of numerical calculation is still much greater for the three-dimensional potentials than for the two-dimensional multipole potentials. Furthermore, whenever the current distribution is changed, the three-dimensional potential must be completely recalculated. It thus appears that at present, the multipole field analysis, combined with numerical integration of the trajectories, is the best available theory of large-angle magnetic deflection. Great improvements have already been made over the small-angle theory, and further refinements can be expected with increasing use of these techniques.

111. ELECTROSTATIC DEFLECTION A . Background

Papers on conventional electrostatic deflection plates no longer appear with any regularity in the technical journals. In the last dozen years the significant work on electrostatic deflection has dealt with electrostatic yokes, a class of deflectors defined by Hutter and Ritterman (1972) as having a common center of deflection for both the vertical and horizontal axes. A more accurate description would be this : a class of deflectors having

324

EDWARD F. RITZ, JR.

simultaneous deflection in both axes rather than the usual sequential deflection. A center of deflection cannot be defined for certain electrostatic yokes, particularly those that are twisted or magnetically immersed, because the projection of the deflected ray is skewed relative to the axis. Hutter and Ritterman (1972) have proposed a classification of electrostatic yokes which we follow here: resistance yokes, multipole yokes, and pattern yokes. Familiarity with the treatment in Hutter (1974) of the fundamentals of electrostatic yokes is assumed in the following discussion. In addition to these three basic types, we also consider here electrostatic yokes immersed in a static longitudinal magnetic field, although in the strict sense these do not constitute purely electrostatic deflection systems. This permits consolidation of two closely related topics. Recent applications of electrostatic yokes include vidicons, scan converters, and computer memories addressed by electron beams. Resistance and multipole yokes have been little used as yet, but pattern yokes in the form of the so-called deflectron have been increasingly employed commercially and have received substantial attention from researchers. However, the theory of these devices has not yet attained the sophistication of the theory of magnetic deflection yokes in the use of computer methods. The present discussion of electrostatic deflection yokes begins with a summary of the recent work on purely electrostatic yokes, followed by an account of magnetically immersed electrostatic yokes, and ends with an assessment of the recent progress in this field. Electrostatic yokes have been thoroughly reviewed by Hutter (1974); consequently, a rather brief treatment is given here, with the addition of some recent material. On the other hand, magnetically immersed electrostatic yokes have not been recently reviewed in depth; therefore, they will be considered in more detail. B. Electrostatic Yokes 1. General Considerations An electrostatic yoke must deflect the beam simultaneously in both the horizontal and the vertical directions. In an ideal deflector, deflection in one direction is independent of the deflection in the other and is linear in the applied deflection potentials. A simple example of a deflection field with these properties is an electric field E whose magnitude is independent of position and whose direction is orthogonal to the z axis at an angle 4o in the x-y plane, given by

tan

4o = EJE,

(44)

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

325

The potential corresponding to this electric field can be written as V(X,y , Z ) = -E,x - E,J

(45)

V(r, 4, z ) = -E,r cos 4 - Eyr sin 4

(46)

or as The magnitude of the electric field strength is

[El = (E:

+ E;)’’’

(47)

The design of an electrostatic yoke thus amounts to the determination of an arrangement of electrodes that will produce such an electrostatic field. A perfectly ideal deflection field cannot be realized. Practical yokes must have a finite length; this introduces disturbances in the field at the ends of the yoke. Furthermore, practical arrangements of the electrodes usually have additional fringing fields that also produce departures from the ideal uniform fields. Consequently, practical yokes only approximate the properties (44)-(47) of the ideal deflecting field. One method of producing the ideal field is to provide an infinite cylinder of radius Rd with the following potential distribution on its surface: V(Rd,4, z )

=

(V, cos 4

where the deflection potentials V, and

+ Vy sin 4 ) / 2

(48)

V, are defined by

This approach is taken for multipole and pattern yokes, in which the surface on which the electrodes are disposed is circular in cross section. Similar equations hold for conical yokes, for which R, becomes a function of z : Rd(Z).

Another method is employed in resistance and pattern yokes, for which a square tube of infinite length is used. Let the corners of the square cross section in the x-y plane be (Rd, 0), (0, R,), ( - R, , 0),and (0, -&); then the corresponding potentials at these corners will be V,/2, Vy/2,- V,/2, - V,/2. Since the potential (45) or (46) produces evenly spaced, plane-parallel equipotentials inclined to the x axis at an angle 4,, + n/2, the variation of potential along the straight edges of the square is linear. Consequently, were the sides of the square tube made of uniformly resistive material, the correct distribution of potential would be produced within the box. The electrode surfaces so far mentioned have had constant cross sections independent of z. One can conceive of other surfaces, tapered in analogy

326

EDWARD F. RITZ, JR.

with conventional flared or bent deflection plates, for which the cross section would be small at the entrance and large at the exit. This has the effect of increasing the field strength at the entrance, where the deflection is still small, and thereby increasing the deflection obtained for given deflection potentials. In practice, conical pattern yokes are commonly used for this purpose, and a tapered resistance yoke has been reported; these will be discussed shortly. However, tapered multipole yokes seem not to have been used as yet. The theoretical analysis of electrostatic yokes is not so far advanced as that of magnetostatic deflectors. End effects and fringing fields are usually neglected. This approximation often suffices for the determination of beam deflection, but not for the study of spot aberrations and raster distortion. Several authors apply the small-angle deflection theory already outlined for magnetic fields to electrostatic yokes. A very limited use of computers has been reported in calculating fields and third-order aberrations; nothing comparable to the recent work in magnetostatic deflection has yet been attempted, with the probable exception of the work of Van Alphen (1972). These and other matters are discussed more fully in the sections that follow. In the field of resistance yokes, this reviewer has discovered no work reported since the paper of Catchpole and Ceckowski (1969), who described both tapered and untapered structures made from reduced lead glass. Since Hutter (1974) has already reviewed this work, and since Catchpole and Ceckowski did not describe any theoretical techniques of special interest to us here, the reader is referred to Hutter’s review for a fuller account of resistance yokes. There has been much more activity in the study of multipole yokes and pattern yokes. The new work on these topics will therefore be discussed in some detail, and certain aspects of older work not stressed by Hutter will also be mentioned. The main emphasis will be on pattern yokes, as these have been more extensively studied and applied in practice. 2. Multipole Yokes Multipole yokes employ longitudinal electrodes arranged on a cylindrical surface; each electrode extends the full length of the cylinder. The cross section of the yoke does not depend on z. With suitable positions and potentials, these electrodes will produce a field that approximates that of (45) and (46) near the axis, neglecting end effects. Hutter (1974) discusses a number of configurations that have been described in the literature. We shall consider mostly the octupole form (Fig. 9) favored in the recent work reported by Heynick (1970) and Van Alphen (1972); there are apparently no later studies of multipole yokes, although the theoretical paper of Wang

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

327

(1971) is directly applicable. [A confusion of terminology must be pointed out here: The term octupole deflector refers only to the number of electrodes and not to the multipole character of the fields, which in fact are composed of the odd m terms in (23).] Kelly and Wang have both extended the third-order theory described in Section II,B to electrostatic deflection (Heynick, 1970; Kelly, 1977; Wang, 1971). Both workers have given extensive discussions of the influence of the electrostatic field distribution on the errors of deflection. Wang has again used a computer to produce plots of current density in the deflected spot (cf. Wang, 1967b). The work of Kelly was directed toward electrostatic deflectors for electron-beam-addressed memories; that of Wang was primarily an analysis of simultaneous deflectors, although he provided an appendix extending his results to the sequential configuration. Apart from the inconvenience of calculating and using the complicated aberration coefficients, the principal obstacle to the use of the third-order theory to analyze electrostatic yokes is the difficulty of calculating accurate electric fields for practical deflectors. The narrow-angle restriction also must be considered, remembering that the radial displacement of the beam, as well as its slope, must be small if the third-order expansions are to be valid. Since the deflection angle is usually quite modest in most of the recent applications of electrostatic yokes, excessive slopes generally do not occur. However, the beam displacement is frequently comparable to the radius of the yoke, which reduces the accuracy of the third-order theory. Kelly has calculated the electric field for the octupole yoke of Fig. 9 under the assumption that its length is effectively infinite (see Heynick, 1970; Kelly, 1977). The potential then depends only on x and y and is readily determined, in this case by a relaxation method in a computer. Figure 10 shows the resulting electric field E, for vertical deflection alone plotted against the transverse distance x. This quantity and its second derivative dZE,/dx2,evaluated on the axis, determine the electron trajectories in the third-order theory. The form of E,(x) depends on the value chosen for the ratio parameter a defined in Figs. 9 and 10. The most uniform field is obtained for a E $- 1, the value derived by requiring that deflection be the same in the diagonal direction as along the x and y axes. The field is uniform to 1% accuracy within a radius half that of the octupole yoke itself. However, this value of a is not necessarily the optimum for the control of deflection aberrations. Such an optimum would have to be selected with reference to the integrals that give the aberration coefficients. Van Alphen (1972) described a novel deflector in which an octupole similar to Kelly’s is sandwiched between two coaxial cylinders having the same diameter as the octupole. Were all segments of the octupole operated at the same potential with the end cylinders at another common potential,

328

EDWARD F. RITZ, JR.

this structure would be an electron lens. However, with deflection potentials applied to the octupole, a deflection field is superimposed on the focusing field so that simultaneous focusing and deflection are obtained. Note that the applied potentials are different from those used by Kelly. Although Van Alphen reports acceptable raster distortion for an experimental tube with such a deflector, this concept of electrostatic deflection apparently has not passed into commercial use. However, in this reviewer's opinion, the true significance of Van Alphen's work is not the practicality of his deflector, but rather the method apparently used to calculate the deflection field. Regrettably, Van Alphen does not discuss in detail his calculation of the electric field but instead refers to computer programs described by Weber

FIG.9. Octupole deflector (Heynick, 1970).

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION 1.8

1.6

329

1

-

1.4

1.2

t

l’O

t

z

YI

A:;1 a = 0.0

0 1 OXlS

0.2

0.4

0.6

0.8

1.o

1.2

TRANSVERSE DISTANCE

FIG.10. Variation of deflection field E, with transverse distance x in the octupole deflector of Fig. 9 (Heynick, 1970).

330

EDWARD F. RITZ, JR.

(1967) and Janse (1971). However, a reading of these two papers is very revealing. The paper of Weber was cited earlier in this review in connection with the finite-difference solution of the Laplace equation, while the paper of Janse was mentioned in the context of multipole analysis of magnetic yokes. Janse describes the application of the multipole decomposition of the potential to electrostatic lenses with deviations from axial symmetry. The conclusion seems inescapable that Van Alphen has applied this same multipole decomposition to the calculation of the fields in his octupole deflector and then used the finite-difference methods of Weber to determine the multipole functions for the appropriate (even) values of rn (see Section II,C of this review). This application of computerized harmonic analysis to a multipole yoke is particularly important. As in the study of magnetic deflection yokes, the multipole technique permits accurate calculation of the field, including end effects. It seems clear that this same technique can be applied to pattern yokes as well. The result would be a very great improvement in the simulation of pattern yokes. However, this has yet to be done, although a close approach has been reported by Hutter and Ritterman (1972), as we shall see shortly.

3. Pattern Yokes a. Definitions. Pattern yokes consist of a set of electrodes disposed on a suitable surface, usually rotationally symmetric, in such a way that the surface potential given by (48) is approximated. The cross section of a pattern yoke is not independent of z, as it is in multipole yokes. Consequently, Eq. (48) is true of the surface potential only on the average. Several different types of pattern yokes have been proposed in the literature. However, only the deflectron described by Schlesinger (1952, 1956) has been widely used in practice, especially in recent years. Consequently, in this review the discussion is confined to this kind of pattern yoke. Hutter (1974) has given a recent review of various other types. A deflectron consists of four conducting zigzag electrodes, two of which are driven by the horizontal signal V , and two by the vertical signal 5. As with other electrostatic deflectors, it is preferable to operate each pair of plates with push-pull potentials to minimize the deflection errors. A typical cylindrical deflectron is pictured in Fig. 11; conical deflectrons are also employed. Two deflectron patterns are shown unrolled in Fig. 12. One is a standard deflectron, as discussed in Schlesinger (1952, 1956); the other is a twisted deflectron, which will be described shortly. The potential applied to each x electrode is k V,/2 and to each y electrode k 5 / 2 .

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

33 1

FIG.11. Typical deflectron: Tabs at left are for connections. (Ritz, 1974).

beam direction b’b

a

FIG.12. Top: unrolled pattern for an untwisted deflectron. Bottom: unrolled pattern for a twisted deflectron (after Ritz, 1973).

332

EDWARD F. RITZ, JR.

The edge of each conductor is a sinusoidal curve z = (42) sin 4, suitably displaced in 4 and in z ; here 4 runs from -n/2 to + n/2 for a given segment between the inflection points. Note that the cycle length 1is not the unit cell of the pattern; if all cycles had I constant, the repetitive cell would have a length 2A. Some workers (Hutter and Ritterman, 1972; Hutter, 1974) prefer to define a cycle length A = 21. However, since in practice 1varies in length from one end of the yoke to the other in order to control fringing fields, we shall use the original definition 1 of Schlesinger (1952, 1956). In a crosssectional plane, each electrode occupies one-quarter of the circumference of the cylinder or cone, except for small gaps that insulate adjacent electrodes from each other. b. Electricfield. Since the distribution of potential on the surface of a deflectron varies with z and furthermore consists of electrodes having constant potentials that change abruptly from one electrode to the next, Eq. (48) cannot represent the surface potential correctly. However, if 1is much smaller than the radius R, for a cylinder or than Rd(z)for a cone, then the fluctuating fields near the surface of the yoke will not be felt by the beam traveling near the axis, and the true surface potential can therefore be replaced by the average potential taken along the line r = &&), 4 = constant. This average is V(& ,4, z ) = ( V, cos 4

+ V, sin +)/2 Jz

(51)

which is the same as (48) except for the factor 1/$, which represents a loss of deflection sensitivity intrinsic to the deflectron structure. The averaging method was introduced by Schlesinger (1952) on physical grounds; a more rigorous proof was given much later by Hutter and Ritterman (1972). As long as the electron beam does not travel through the rapidly changing field near the inner surface of the yoke, the use of (51) gives surprisingly accurate results. The radial extent of these fringing fields can be controlled by reducing 1where the beam closely approaches the surface of the deflectron (Schlesinger, 1959). Schlesinger (1 972) has recently modified the standard deflectron pattern by twisting it about the z axis through an angle n/2 from one end to the other. Thus each point of the original pattern is displaced along a circumference through an angle nz/2L, where L is the length of the deflectron, and the origin of coordinates is at the entrance to the deflectron. The z coordinate of each point is unchanged. Figure 12 shows an unrolled twisted deflectron. Schlesinger found that twisting improved resolution and reduced raster distortion and collimation or beam-landing errors in deflectrons immersed in a uniform longitudinal magnetic field. Ritz (1973) extended twisting to arbitrary angles and showed that immersed, twisted, cylindrical deflectrons have certain modes of operation in which beam-landing errors vanish. These

333

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

modes are analogous to similar untwisted modes reported by Schlesinger and Wagner (1965). Such immersed deflectrons will be discussed shortly. It seems obvious that one effect of twisting the deflectron pattern will be a corresponding rotation of the electric field vector as z increases; a calculation of the electric field verifies this supposition (Ritz, 1973). The calculation assumes that the fields for an infinite deflectron apply approximately to a long narrow deflectron and that the average surface potential (51), suitably modified for twisted yokes, can be used. Let the electric twist angle be 8,; then the coordinate 4 of a point on the untwisted deflectron becomes 4 - B,z/L and the surface potential (51) for a cylindrical deflectron of radius R, becomes V(&,

4, Z) = [ V, cos(4 - e , l / L ) + V, sin(4

-

e e Z / ~ ) $2 ]/2

(52)

For simplicity, we shall consider only vertical deflection 2 0 and set V, = 0 from now on. The electrostatic potential inside the deflection is obtained by solving the Laplace equation subject to the boundary condition (52) with V, = 0:

c

where I l ( c ) is a modified Bessel function of the first kind. For 4 1, the function Zl(r> is approximately c/2; consequently, if 8,Rd/L is small, (53) is approximately

V ( X y, , Z) = E,,[y cos(B,z/L) - x sin(8,z/L)]

(54)

where E,, = & / 2 , / 2 R d . This potential leads to an electric field vector with components Ex = E,, sin(8, z/L)

(55)

E, = -E,, cos(e,z/L)

(56)

E, = (8,r/L)E,, COS(+- e,z/L)

(57)

The effects of Ez can be neglected under certain conditions discussed by Ritz (1973). Thus, there remains a vector (Ex,E,, 0) of magnitude E,, which rotates linearly with axial position z through an angle 8, from z = 0 to z = L. Note that setting the untwisted field as a special case - 8.- = 0 gives of (55)-(57). The field inside a twisted conical deflectron under similar assumptions was given by Ritz (1976) : Ex =

aE,, sin(8,zlL) 1 (a - l)z/L

+

(58)

3 34

EDWARD F. RITZ, JR.

E, = 0

(60)

where the aperture ratio a = R,/R, is the ratio of the final or exit radius R f of the cone to the initial or entrance radius Ri, and Eoy = 5 / 2 f i 4 . In effect, the electric field is inversely proportional to the local radius of the cone. A similar result for untwisted deflections was first given by Schlesinger (1956). The field analysis just given is approximate and applies only to yokes with large values of L/& and modest values of the electric twist angle Be. It does not include end fields or fringing fields at all. Hutter and Ritterman (1972) have given a more rigorous analysis for a cylindrical deflectron of finite length bounded at each end by cylinders that extend to infinity. Their method is worthy of note because it is closely related to the multipole analysis of the magnetic scalar potential described in Section II,Cy4 for magnetic yokes. They expand the electrostatic potential V(r,4, z) in azimuthal harmonics, but then write the functions Am(r,z) and Bm(r,z) of Eq. (23) as Fourier integrals :

:1

Am(ryz) = -

&(y)

&@cos yz dy

Bm(ryz ) = where d,(y) and hm(y) are the Fourier cosine transforms of am(z)and b,(z) from the Fourier expansion of the surface potential: 00

V(Rd,

4, z,

=

[am(z)cos m4 + bm(z) sin m4]

(63)

m=l

Thus, the Fourier transforms b,(y) and 6Jy) are given by h,(y) = JoNA a,(z) ms yz dz

h,(y) = JoNA b,(z) cos yz dz where A = 2A is the modified cycle length defined prevlouLly and 2N is the total number of cycles of length A. As in magnetic yokes, only the odd values of m occur in the expansion of deflection fields because of the symmetries of the boundary conditions.

RECENT AD'IANCES IN ELECTRON BEAM DEFLECTION

335

The analysis of Hutter and Ritterman is too lengthy to discuss further here; the original paper should be consulted for details. However, these authors reach two important conclusions that can be summarized here. First, average fields of the form (46) actually exist and therefore can be used legitimately for calculations where the fringing fields near the deflectron surface can be neglected. Second, in general, the vertical deflection field has horizontal components, and the horizontal field likewise has vertical components. (Hutter and Ritterman describe a modified deflectron pattern which they call the quilt pattern; this new version has pure vertical and horizontal fields. Unfortunately, the quilt pattern has numerous isolated electrode patches that would be difficult to fabricate and to energize with the proper potentials.) c. Electron trajectories. Electron trajectories for untwisted cylindrical and conical deflectrons can be found analytically for the approximate fields given by (55)-(56) and (58)-(59) with E, = 0 and 8, = 0. Since these trajectories are special cases of the trajectories for arbitrary twist angles with the deflectron immersed in a uniform axial magnetic field, we shall defer discussion of the untwisted results to the following section on immersed yokes. Hutter and Ritterman (1972) used the field calculations just discussed to calculate trajectories and deflection aberrations with the third-order narrow-angle theory. Few general results were given. However, these workers found that spot distortion tends to be smaller in electrostatic yokes than in conventional sequential parallel-plate deflectors. Also, they found that as A/% approaches zero, the electron trajectories are smoother because the oscillatory fringing fields are reduced. This supports the earlier assertion of Schlesinger (1959) that the value of Iz (or A) should be reduced as the beam approaches the deflectron surface. d. Applications. Most of the recent applications of pattern yokes involve immersion in a uniform axial magnetic field; these applications are discussed in the section on immersed yokes. However, there have been two purely electrostatic applications reported recently. Lemmond et al. (1 974) have described a deflection system for addressing a fly's eye or matrix lens in an electron beam artwork camera. Hughes et al. (1 975) have discussed a similar deflector for addressing a semiconductor memory by electron beam. For these applications, the beam must leave the deflector traveling parallel to the axis. Both of these papers describe deflection systems having two deflectrons arranged sequentially along a common axis, as shown in Fig. 13. The electrodes of the second deflectron are connected to the driving potentials with polarity opposite to that of the first deflectron. Thus the second deflectron tends to reduce the radial velocity imparted by the first to the electron beam.

336

EDWARD F. RITZ, JR.

FIG.13. Double deflectron for collimated deflection (Lemmond et al., 1974).

With a proper choice of length and diameter for the two deflectrons, the radial velocity can be completely removed at the exit of the second deflectron so that the beam leaves traveling parallel to the axis. Of course, deflection sensitivity is sacrificed to achieve this collimated deflection. Either cylindrical or conical deflectrons can be adapted to this scheme; Lemmond et al. employ cylindrical yokes; Hughes et al. prefer conical yokes. It appears that the use of the conical deflectrons may reduce the problem of accurate alignment of the two yokes, since both cones can be imprinted on a single conical form. C. Magnetically Immersed Electrostatic Yokes 1. Introduction

In many applications it would be very convenient to shorten the electron beam deflection system by focusing and deflecting the beam simultaneously,

RECENT ADVANCB IN ELECTRON BEAM DEFLECTION

337

as is done in all-magnetic television camera tubes and in the all-electrostatic camera tube described by Van Alphen (1972), for example. Deflectrons immersed in static axial magnetic fields have been much used in such applications. This configuration was first proposed by Schlesinger and Wagner (1 965), who coined the phrasefocus projection and scanning (FPS) to describe its function and provide a convenient label for easy reference. The initial papers on FPS systems dealt with untwisted cylindrical or conical deflectrons inside a solenoid and presented a theoretical analysis based on the infinite deflectron, average-field approximations just described in Section II1,C. Early camera tubes based on this analysis showed excellent resolution, uniform focus, freedom from raster distortion, and low beamlanding error. (Beam-landing error results from nonperpendicular incidence at the target and leads to shading, a harmful variation of signal amplitude across the target of a camera tube.) Beam-landing errors vanish altogether in certain modes of operation described by Schlesinger and Wagner (1965, 1967). The use of electrostatic deflection permitted higher deflection speeds at lower power and reduced the bulk, weight, and sensitivity to mechanical shock of camera tubes. Schlesinger later invented a deflection with 90" of twist to improve further the resolution, raster distortion, and beam-landing characteristics of FPS systems (Schlesinger, 1972). Ritz (1973) extended the concept of twist to arbitrary angles and discovered new shading-freemodes of operation analogous to the previous untwisted shading-free modes. Ritz also gave a general theoretical analysis of the operation of twisted cylindrical FPS systems, an analysis later extended to twisted conical systems (Ritz, 1976).These analyses were based on the infinite-deflectron, average-field approximations. The remainder of Section II1,C contains a summary of the theory of FPS deflection systems and a discussion of recent applications. 2. Theoretical Model A typical FPS deflection system appears in Fig. 14, which schematically represents a television camera tube. A conical deflectron is surrounded by a long solenoid, with a field-forming mesh and a light-sensitive target at the exit and a shield at the entrance. Before traveling through the deflectron and solenoid, the beam passes through a small limiting aperture and an electrostatic prefocus lens; the latter is omitted in some applications. A cylindrical deflectron can be used in place of the conical one. This FPS system is the basis of the theoretical analysis summarized below. The deflectron of length L is placed with its entrance at z = 0 and its exit at z = L. For simplicity, only vertical deflection is considered; the potentials +&/2 are applied in push-pull fashion around the average deflectron potential V,. The anode, deflector shield, and mesh are also set

338

EDWARD F. RITZ, JR.

FIG. 14. Typical FPS deflection system with vacuum envelope not shown. H, heater; K, cathode; G, control grid cup; A, anode cup; ap, beam-limiting aperture; P, electrostatic prefocus lens barrel; S, deflector shield; D. deflectron; M, mesh; T, target; F, focusing solenoid; 5, grid voltage; 6 ;prefocus lens voltage; K, anode voltage; F, target voltage; is, signal current (Ritz, 1976).

at V,. The prefocus lens can be operated as a field-free drift space at V , or as a convergent lens at some other potential. The mesh is taken to be the effective target at z = L. The beam-limiting aperture is placed at a distance A in front of the entrance to the deflectron. The electric field in cylindrical and conical yokes is given by (55), (56), (58), and (59) with E, = 0 in the infinite-yoke, average-field approximation. The magnetic induction is idealized as a uniform axial field inside the yoke and a null field outside; for 0 I z I L the magnetic induction is B, = By = 0

(66)

B, = B, (67) where B, is a constant. Thus, the end fields of both deflectron and solenoid are neglected. 3. Electron Trajectories The following account is based on the analysis of Ritz (1973, 1976) be-

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

339

cause of its generality. The special case of untwisted cylindrical deflectrons was first studied by Schlesinger and Wagner (1965, 1967). The solution of the equation of motion (1) is simplified by the use of the following reduced or normalized variables : (xr

3

Yr 9 2,) =

(x, Y , ~ ) / R o

t, = t/To The normalization constants R, and To are

(68) (69)

Ro = 1Eo,Tm.)2 To = L/io in which the initial axial velocity i, is

i, = (2qv,)”2 (72) The time To is the transit time in the deflectron because the asial velocity is a constant of the motion due to the vanishing of E, + (i x B), in this approximation. This circumstance permits the elimination of z in the electric field (55), (56), (58), and (59) by the substitution t = z / i 0 .Finally, the magnetic field is described by the magnetic dwell angle 8, defined by

em= OT,

(73)

in which w = yB, is the cyclotron frequency. With these preliminary definitions, the equations of motion for a twisted conical yoke become 2, =

8,t, -emj, - ( 2 4 2 1 +a sin (a - l)t,

j;, = emir+ (242

a cos eetr

1

+ (a - l)t,

(74) (7 5)

in which dotted reduced variables denote differentiation with respect to the reduced time t,. Let x, and y, be the solutions for an electron that enters the deflection along the axis; then the general solutions of (74) and (75) can be written x, and xrf and y, y,,. Here xrf and y,, satisfy arbitrary initial conditions and the equations

+

which describe the focusing properties of the solenoid. Thus, the functions

340

EDWARD F. RITZ, JR.

of deflection and focus are mathematically separable, although both are performed simultaneously by the FPS system. Analytic solutions for both the conical and cylindrical cases were giver, by Ritz (1973, 1976). The cylindrical case a = 1 is relatively simple:

These expressions describe a general epi- or hypocycloidal plane curve generated by a point on the circumference of a circle that rolls on the circumference of a second circle. In the untwisted case Oe = 0, the curve is a cycloid, which is the degenerate case where the radius of the second circle is infinite (straight line). The solutions for the conical case are too complicated to reproduce here; they involve the sine and cosine integrals and can be found in Ritz (1976). The complete trajectory is obtained by combining the uniform drift along the z axis at velocity i, with the x-y projection given by (78) and (79); this trajectory is helical in shape. A similar helix is found in the conical case. The focusing action of the solenoid is described by the solutions of (76) and (77). The operation of a long solenoid lens has been analyzed by Schlesinger and Wagner (1965, 1967). They conclude that an image of the beamlimiting aperture is formed at the target if

The magnification M is given by M = cos(8"42)

(81)

Thus, in general, a demagnified image of the limiting aperture is formed at the target. The ratio AIL can be negative for certain values of ;8, this indicates that the prefocus lens must be used to form a real image of the limiting aperture inside the solenoid or perhaps even beyond the target to serve as object for the solenoid lens. Note that the special case M = 0 is not physically attainable ; space-charge effects and aberrations will intervene. These aberrations cannot be calculated accurately from the present model because of the simplifying approximations made.

4. Operating Characteristics It is impossible to discuss all the characteristics of the many possible

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

34 1

FPS systems within the confines of this review. However, some useful general comments can be made. First, the essential features of deflection in FPS systems are determined by the choice of the three parameters O,, Om, and a ; they fix the shapes of the electron trajectories. Consequently, the notation {Oe, Om, a}, or {Oe, Om} for cylindrical deflectrons, serves to specify a particular FPS system or mode. Each mode represents an infinite number of possible embodiments, each differing in dimensions, magnetic field, and operating potentials, but all having the same projected trajectories (78) and (79) Three quantities serve to describe the relative operational properties of the various possible modes. The deflection sensitivity or scan compression k defined by k = [ ~ ~ ( +l ~) ,~( 1 ) ~ ] ~ ’ ~ / 2 n ~ is the final deflection at the target (t, = 1) normalized by the deflection 2n2 for the untwisted cylinder with Om = 0. The scan rotational angle Or defined by tan 0, = -X1(1)/Yr(1)

(83)

gives the angular orientation of the final deflection at the target relative to the deflection for the untwisted cylinder with Om = 0. The shading index s defined by

measures the beam landing or shading error of the system. All three of these parameters are determined completely by O,, Om, and a. The operating characteristics in conventional units are the deflection factor F, or ratio of V, to the final deflection; the magnetic induction Bo; and the beam-landing angle p between the trajectory and the normal to the target. These quantities are given by F = 8JZVaRf/kLz BO

= J27tJOmJva/L

tan p = S [ X , ( ~ )+~ Y , ( ~ ) ~ ] ” ~ / L

(85) (86) (87)

The values of A and M can be found from (80) and (81). Schlesinger and Wagner (1965, 1967) have reported two particularly useful modes (0, &2n, 1}, an untwisted cylinder with +360” of magnetic dwell in which the shading index s vanishes identically. These are the first and most useful of the series of shading-free modes with 8, = 0, a = 1 for

EDWARD F. RITZ, JR.

342

which 8, = 2nn,where n is a nonzero integer. Ritz (1973) discovered twisted shading-free modes characterized by the more general relation - 8, =

8,

2nn

when he extended the theory to twisted cylindrical deflectrons. In general, only the modes with In1 = 1 have sufficient deflection sensitivity to be useful in practice. There are no analogous shading-free modes in conical FPS systems, as was shown by Ritz (1976). Although increasing the aperture ratio gives higher deflection sensitivity, it also spoils the shading cancellation, as shown in Figs. 15 and 16. The fidelity to reality of the theory just summarized can be judged from the experimental curves of Figs. 17 and 18, which show the measured values obtained by Ritz (1973) for 8, and k as functions of the magnetic field (Om). The experiments were carried out with both a long and a short solenoid to estimate the effects of the magnetic end fields. For the longer solenoid (smaller end fields), agreement with the theory was quite good. (Note that the scan rotation 8, is expressed as 8, relative to its value at Om = 0.) Unfortunately, few experimental results suitable for checking other aspects of the theory have been published, except for some data on resolution in Schlesinger and Wagner (1965, 1967), Decker and Knoll (1968), Saldi and Schlesinger (1970), and Ritz (1973). These results also seem compatible with theory. 1.4

1

I

I

1

4

5

1.2 2-

k

2

1.0

-

$

0.8

-

k m z z

0

+ 0.6 U

I!LL

W 0.4

n

0.2

'

1

1

I

2

3

I

APERTURE RATIO a FIG. 15. Increase of deflection sensitivity k with aperture ratio a in conical FPS systems where 8, - Be = -2x (Ritz, 1976).

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

343

U

I

m

0 1

2

3

4

APERTURE RATIO

em

5

a

FIG. 16. Increase of shading index s with aperture ratio a in conical FPS systems where - o. = -2n (fitz, 1976).

4rt 2

L

FIG.17. Comparison of theory and experiment for a cylindrical FPS system with 8. = 4 2 : scan rotation angle versus magnetic dwell angle. Solid line: theory; circles: experiment, long solenoid; triangles: experiment, short solenoid. Dotted lines show shading-free modes (Ritz, 1973).

EDWARD F. RITZ, JR.

344

k

I

I

-3T

l

-2T

!

I

-T

10

I I

T

I

2.7

MAGNETIC D W E L L A N G L E Om

l

l

3T

(rad)

FIG.18. Comparison of theory and experiment for a cylindrical FPS system with 0, = n/2: deflection sensitivity k versus magnetic dwell angle. Solid line : theory; circles : experiment, long solenoid; triangles: experiment, short solenoid. Dotted lines show shading-free modes (Ritz, 1973).

5. Applications There have been numerous applications of FPS systems reported in the recent literature. Much of the original work on these devices was directed toward vidicon camera tubes (Schlesinger and Wagner, 1965, 1967; Decker and Knoll, 1968; Saldi and Schlesinger, 1970; Lockwood and Noble, 1970). Marrs (1972) discusses the application of an FPS vidicon to rapid-scanning optical spectroscopy. Kubota et al. (1977) describe a single-beam color pickup tube that uses the untwisted shading-free mode just discussed. An FPS system with 90" of twist operated in a shading-free mode has been used as the reading gun of a silicon diode array scan converter for high-speed transient record (Hayes, 1975). D. Assessment In the last dozen or so years most, if not all, of the reported research activity in electrostatic deflection has been in the field of electrostatic yokes. Although much work was done early in that period on multipole yokes, in recent years pattern yokes, and particularly the deflectron, have taken the lead. When immersed in a magnetic field, these devices provide many design possibilities. Steady progress has been reported in theoretical studies of electrostatic yokes. Computer simulations have been employed in research on multipole yokes, probably including the multipole analysis used in the study of magnetostatic deflection. However, the theory of deflectrons is generally carried

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

345

out in a framework analogous to the elementary theory of conventional electrostatic deflection plates, although the calculations themselves are much more complex. A limited amount of work has been done using a somewhat more realistic theoretical model, but as yet there is no satisfactory theory of deflection aberrations for deflectron-based systems. The study of conventional electrostatic plate deflection systems has stagnated. Some authors (Moss, 1968; Hutter, 1974) have expressed the opinion that the best configurations probably have already been found by trial and error without benefit of theory. However, this writer is not convinced that the application of computer simulation to the determination of fields and trajectories is fruitless in this context. New methods of analysis often reveal new possibilities. IV. TRAVELING-WAVE DEFLECTION A . Background

When a conventional parallel-plate electrostatic deflector is driven at high frequencies, the variation of the electric field during the transit of a single electron can no longer be neglected. For the case of vertical deflection by a sinusoidal potential of frequencyf, let the quasi-static deflection y,(f, t) at the screen for very low frequencies be

y 0 ( L t ) = Yo exp(i2nft)

(89)

where Yo is the true static deflection for f = 0. At higher frequencies, the deflection y(f, t ) is given by the well-known expression

where Y is the transit angle 2nfT, and To is the transit time of an electron in the deflector. Thus the amplitude of the sinusoidal response is reduced and its phase is shifted. Consequently, the response to a general waveform compounded of many different high frequencies will be distorted by such a deflector. Convenient treatments of this topic are found in Spangenberg (1948), Hollmann (1950), and Andrews (1970). The traveling-wave deflector circumvents the difficulty just described by providing a traveling electromagnetic wave that moves ideally at the axial velocity of the electron beam so that each electron is subjected to a fixed phase of the signal while passing through the deflector. The traveling-wave deflector is thus a slow-wave structure similar to those of traveling-wave

346

EDWARD F. RITZ, JR.

amplifier tubes but designed to use the transverse component of the electric field rather than the longitudinal component. The phase velocity of the slow wave is typically on the order of 10% of the velocity of light in vacuum. In practice, the matching of the wave and electron velocities cannot be maintained at high frequencies because dispersion in the slow-wave structure causes the wave velocity to depart from that of the beam. Consequently, the response of traveling-wave deflectors also falls, although at much higher frequencies than in parallel-plate deflectors with the same length and beam velocity. Of all the subjects discussed in this review, traveling-wave deflection presents the greatest obstacles to theoretical anslysis. Not only is the transit time of the electrons no longer negligible, but the static field equations (2) and (3) fail. Consequently, the motion of an electron must be calculated in time-varying fields that satisfy

V

x E = -aB/at

v

x

(91)

H = aD/at

plus the boundary conditions imposed by the structure of the deflector. As the solution of these equations in practical cases is generally very difficult or impossible, much if not most of the development of traveling-wave deflectors has been accomplished with rather rudimentary theories heavily supplemented with trial-and-error experimentation guided by experience with waveguides, delay lines, and traveling-wave amplifiers. However, there have recently been several significant theoretical contributions, although computer-based methods have yet to realize their probable potential. Hutter (1974) recently reviewed traveling-wave deflection, described the basic types of slow-wave structures used, and gave an account of the principles of their operation. Familiarity with these fundamental ideas is assumed in the following discussion, in which we shall examine some recent contributions to the theory and practice of traveling-wave deflection. Some earlier papers omitted by Hutter will also be mentioned. The complexity of the subject unfortunately limits the present review to a brief outline of the principles involved; for details the original literature must be consulted. Theories of traveling-wave deflection fall into two broad categories : simplified wave theories and true wave theories. In simplified wave theories no attempt is made to solve the field equations (91) and (92) explicitly for the particular structure used. Instead, the bare facts of wave propagation are combined with electric fields derived by approximate static methods. On the other hand, in true wave theories the field equations are actually solved, although often in rather crude approximations. Both classes of theory have

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proved useful in their proper spheres. The remainder of Section IV is organized under the headings of simplified and true wave theories, followed by an assessment. B. SimpliJied Wave Theories

To avoid the very considerable difficulties in actually calculating the electromagnetic fields of slow-wave structures, many authors simplify the problem by using the approximate static field of parallel deflection plates energized by the instantaneous value of the applied signal-wave amplitude at the instantaneous position of the electron being deflected. The spacing of the plates is adjusted to suit the structure being studied. This approximation thus makes use of only the most basic property of an undamped propagating wave: its phase shift with time and axial position. Furthermore, it is impossible to calculate aberrations and raster distortion with this method because the longitudinal electric field components and the variation of the transverse components with position are neglected. Typical of this approach are the efforts of Pierce (1 949), Honey (1954), Chernushenko (1959), Andrews (1970), and Silzars and Knight (1972). In spite of the crudeness of this simplified approach, it does give an intuitive picture of the operation of traveling-wave deflectors, playing a role analogous to that of the elementary parallel-plate theory in electrostatic deflection. It is particularly useful in understanding lumped-element delay lines (e.g., Pierce, 1949) in which cascaded deflection plates (capacitors) connected by small inductors form a structure for which the propagation velocity and phase shift per section are readily calculated. It can also be used to understand meander lines and helical lines (see Hutter, 1974), either paired or over ground planes. However, with these more complicated structures it is difficult to determine the variation of propagation velocity with frequency by calculation. Silzars and Knight (1972) have recently described a hybrid method in which the deflection in each segment of a space-periodic deflector, such as a lumped-element line, a meander line, or a helical line, is calculated by the methods that led to Eq. (90), using phase-shift information from experiments on models of the deflection structure. This permits determination of the deflection sensitivity versus frequency for tentative designs without the expense and difficulty of constructing and measuring complete vacuum tubes. These authors reported good agreement between their calculations and experiments for a complete tube employing a meander line structure. (Silzars and Knight used a computer to perform the calculations just outlined. This

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EDWARD F. RITZ, JR.

is the sole example of computer use in the theory of traveling-wave deflection known to this reviewer.) C . True Wave Theories An accurate picture of the fields in slow-wave structures can only be obtained by solving Eqs. (91) and (92), or equivalently, the wave equation

where vp is the phase velocity of the wave. The wave equation is very hard to solve for most practical structures, but this can be done in a few cases of simple geometry. A system studied with some success is the cylindrical helix, in which the actual spiral structure is replaced by a cylinder having anisotropic conductance. Current can flow on the surface of the cylinder only in a helical fashion; when the surface is unrolled onto a plane, the lines of current flow make an angle with the z axis. Pierce (1950), in a review of traveling-wave amplifiers, gave the wave solutions for such an idealized isolated helix. Recently, Vaynoris et af. (1969) extended this analysis to a helix with concentric cylindrical conducting shields inside and outside. The two annular regions formed may have differing values of the permittivity E . The resulting field components show clearly that in addition to the transverse electric field primarily responsible for beam deflection, a longitudinal field exists that will defocus the beam. The radial position of the beam must be selected carefully to minimize the variation of the transverse field with frequency because at high frequencies the field collapses toward the helix. The azimuthal component of the electric field leads to raster distortion. Finally, the reduction of the transverse dimensions of the system reduces the frequency dependence and increases the magnitude of the deflection. Nishino and Maeda (1972) have given a more elaborate treatment of helical slow-wave lines, in which they include the effects of the gaps between turns of the helix, a clear advance over the method of Pierce (1950) and Vaynoris et a f . (1969). Unfortunately, the results of Nishino and Maeda are very complex, although their calculations show rather good agreement with experiment for a deflector with a bandwidth of 5 GHz. Vaynoris et al. (1 970a,b) have also studied a deflector consisting of one or two rectangular helices inside a rectangular shield. They have approximated these systems with a set of infinite-plane-parallel electrodes ; the electrodes simulating the helices conduct anisotropically as in the cylindrical case. The single helix has an inner plane shield. Their analysis shows that the

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

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single-helix deflector behaves similarly to the cylindrical case previously discussed. The double-spiral system has 30% less deflection at low frequencies and an essential dependence of the deflection field on frequency, which cannot be removed by repositioning the beam. Yamada and Takagi (1972) have analyzed a meander line deflector partially loaded with dielectric to decrease the phase velocity. They used a space-harmonic analysis described by Fletcher (1952) originally and later elaborated by others, and concluded that the dispersion deficiencies of this structure are due to interactions between the adjacent segments of the line. They inserted a shield between segments, producing what they have named a trough-type deflector, and found good agreement between their calculations and experimental measurements. D . Assessment Progress in traveling-wave deflection during the last ten years has been slow paced, although there have been several significant advances in theory using classical techniques of field analysis. Only a single example of computer use was found, and there have been no reported efforts to employ computer techniques to calculate the electromagnetic fields of slow-wave structures for deflection. There appear to have been no radical new inventions, but only modest practical improvements in existing devices according to known principles. It seems to this reviewer that further progress in this field must await the unpredictable invention and the systematic application of the powerful computer techniques now available to calculation of field distributions in traveling-wave deflectors. These methods seem essential for further improvements, since it appears unlikely that the classical methods of analysis will be able to cope with the structures actually used in practice. In this respect, the situation in traveling-wave deflection resembles that in magnetostatic deflection just prior to the introduction of the computer-based harmonic field analysis of Section 11. V. SCANMAGNIFICATION

A . Background Deflection sensitivity is a fundamental constraint in the design of electron beam deflection systems. Limitations imposed on this parameter by the requirements of beam energy and the capabilities of the driving circuitry have far-reaching effects on the dimensions and aberrations of the deflection

350

EDWARD F. RITZ, JR.

system. Consequently, the vision of higher deflection sensitivity has been the holy grail of designers, and numerous methods have been proposed to realize this goal. These schemes fall into two basic categories : postdeflection acceleration (PDA) and scan magnification. In postdeflection acceleration the beam is deflected at a relatively low beam potential and is then accelerated after deflection to a higher potential before reaching the screen or target. In scan magnification an electron optical lens is used to increase the deflection angle produced by the deflector. Some methods of scan magnification incorporate means for postdeflection acceleration also. Systems for scan magnification employ electrostatic or magnetostatic fields to produce either azimuthally symmetric lenses, in which equal lens strengths are achieved in both axes of deflection, or else lenses with two orthogonal planes of symmetry, in which these two lens strengths may differ. When the horizontal and vertical deflectors are sequential, the scan magnification lenses are often placed between the deflectors so that only one axis of deflection will be magnified (usually the vertical). In assessing scan magnification systems, it is not enough to consider only the deflection sensitivity; the effects on resolution must also be examined. That is, the deflection sensibility, or reciprocal of the deflection signal (current or potential) required to move the beam one spot diameter, is the important parameter. If the sensibility decreases with scan magnification, then resolution will suffer in the magnified scan. Pierce (1941) showed that it is impossible to increase the sensibility of magnetic deflectors by means of postdeflection. Gundert and Lotsch (1962) later demonstrated that scan magnification, with or without postdeflection acceleration, necessarily causes a reduction of resolution in magnetic deflection systems. Consequently, recent work is concerned only with electrostatic deflection. Hutter (1974) has recently reviewed research in postdeflection acceleration and scan magnification through 1973. Consequently, in the present review we shall emphasize work done since 1973, together with some significant earlier papers not mentioned by Hutter. This body of work deals almost exclusively with scan magnification rather than postdeflection acceleration, and with electrostatic rather than magnetostatic deflection, although both electric and magnetic magnifying lenses are considered. The remainder of this discussion is organized under the headings, “Lenses with Rotational Symmetry” and “Lenses with Twofold Symmetry.” Section V ends with a brief assessment. B. Lenses with Rotational Symmetry Two types of rotationally symmetric electrostatic lenses have been proposed: a field-forming mesh placed after the deflectors at the beginning of the PDA field, and a conventional lens with multiple rotationally sym-

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

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metric elements. The mesh lens is widely used in practice, especially for oscilloscope cathode-ray tubes. Significant papers on this topic more recent than Hutter’s review of 1974 have not come to light. Only one example of the conventional lens structure has appeared recently : the paper by Schackert (1971). This paper was also mentioned by Hutter but is reviewed again here because of the theoretical methods employed. The lens of Schackert consists of three concentric elements placed after both electrostatic deflectors of an oscilloscope cathode-ray tube. The element nearest the deflectors is operated at the average deflection potential (1500 V), the second at about 200 V, and the third at 20 kV; thus, PDA is also present. The second electrode serves as a control on raster distortion. The lens is convergent and thus causes a reversal of the scan direction. The beam crossover in the electron gun is brought to a focus at magnification M , by a prefocus lens; this image is then focused by the scan magnifier onto the screen with total magnification M , M , ,where M 2is the magnification of the expansion lens. The spot magnification M , is not necessarily equal to the scan expansion factor because the centers of deflection are not imaged at the screen. Schackert analyzed this scan magnification lens with the aid of a computer program that calculates the electron trajectories and apparently also the fields in rotationally symmetric lenses. By this means he evaluated numerous alternative designs in which the potentials and positions of the deflection centers were varied. It was always possible to find a potential for the second lens element that produced acceptable raster geometry. The scan magnification at the potentials given above was 2.8 in the vertical and 2.2 in the horizontal. Raster distortion was satisfactory on a 60- by 100-mm screen. Performance would have been improved with unequal lens strengths in the vertical and horizontal because of the axial displacement of the deflection centers, that is, by using a rotationally asymmetric lens. In principle, one could also use a magnetic scan magnification lens with rotational symmetry. However, no reports of research on such a lens were found. This circumstance no doubt stems from the practical difficulties of this kind of magnifying system. The diameter of the CRT neck imposes limits on the size of externally mounted magnetic lenses, which makes difficult the achievement of sufficient strength without also causing the lens fields to penetrate the deflection region. Furthermore, unless specially wound, a magnetic lens rotates the deflection axes away from their original directions (Gundert and Lotsch, 1962).

C. Lenses with Twofold Symmetry Most workers have devoted their recent efforts to scan magnification lenses with two orthogonal symmetry planes that intersect along the z axis.

3 52

EDWARD F. RITZ, JR.

These lenses, of which the quadrupole is the prototypical example, are convergent in one plane and divergent in the other. This behavior is quite distinct frcim that of the hypothetical asymmetric lens proposed by Schackert (1971), which would be convergent in both axes but with different strengths. Himmelbauer (1969) gave a useful account of the use of electric quadrupole lenses in cathode-ray tubes, which serves as a good background for the discussion below. Both electric and magnetic quadrupole lenses have been studied by various other authors cited below. The use of quadrupolar scan magnification lenses stigmatizes the beam. To remedy this defect, it is common to use additional quadrupoles to compensate by prefocusing the beam; this allows the overall spot magnification to be made equal in the two symmetry planes. Himmelbauer (1969) showed that two such electric quadrupoles are needed for prefocusing. Very similar results have been reported by Lyubchik et al. (1971) and Ovsyannikova et al. (1972). Various locations of the three lenses are possible with respect to the deflectors.

FIG.19. Box lens for scan magnification (Odenthal, 1977).

RECENT ADVANCES IN ELECTRON BEAM DEFLECTION

353

Himmelbauer (1969) and Lyubchik et al. (1971) conclude that although a triplet of quadrupoles produces scan magnification with a round spot, a doublet does not. Martin and Deschamps (1971) describe a CRT in which two electric quadrupoles are used in conjunction with PDA. However, they also require an astigmatism electrode preceding the deflectors and a slot lens in the PDA field-forming electrode to provide the necessary degrees of freedom for the correction of astigmatism. Blazo et d.(1977) describe another CRT employing an electrostatic quadrupole triplet based on Himmelbauer's scheme; they also provide a fourth, weak quadrupole with axes at 45" to the others to compensate for errors of fabrication. This compensator was described by Himmelbauer in the work cited. Odenthal (1977) has described another type of lens with two planes of symmetry. It consists of the boxlike electrode structure shown in Fig. 19 and is placed between the last deflector and the screen in a CRT. The author has calculated the potential within this lens by using a computer program that solves the finite-difference equivalent of the Laplace equation in three

FIG.20. Equipotentials and trajectories in the vertical symmetry plane of the box lens of Fig. 19 (Odenthal, 1977).

FIG.21. Equipotentials and trajectories in the horizontal symmetry plane of the box lens of Fig. 19 (Odenthal, 1977).

3 54

EDWARD F. RITZ, JR.

dimensions; about 28,000 mesh points arranged in a cubic array were used. Electron trajectories were also calculated by computer. Figures 20 and 21 show the equipotentials and trajectories in the two symmetry planes. Magnification of 4.5in the vertical and 4.0 in the horizontal were achieved. The lens is similar to a quadrupole in that one axis converges while the other diverges. Magnetic quadrupoles have not received much recent attention. Only two papers have been seen by this reviewer: Johnson (1964) and Kartashev et al. (1968). It seems that various practical difficulties associated with increased bulk, difficulty of fabrication, and power supply limitations make magnetostatic quadrupoles less attractive than electrostatic units. D . Assessmen1 It appears from the reported applications that lenses with two planes of symmetry are proving more suitable for scan magnification than rotationally symmetric lenses. Considerable progress has been made in the use of electrostatic quadrupoles, especially in triplet configurations. However, practical problems with magnetic quadrupoles make them less useful, and to date little work has been reported. Theoretical studies of scan magnification systems have begun to benefit from the power of computer-based methods of field calculation, although not yet to the extent seen in magnetostatic deflection. No doubt further refinements of present devices can be expected from the resulting improvement in understanding of these systems. VI. CONCLUSION The recent advances described in the foregoing pages bear witness to the improvements in our understanding of electron beam deflection brought about by the application of the high-speed digital computer and suitable theoretical formulations to the perennial problems of field calculation and trajectory tracing. The effects have been greatest in the field of magnetostatic deflection and have been felt to a substantial extent in electrostatic deflection and scan magnification. Only traveling-wave deflection has received little benefit as yet. In electrostatic deflection, some relatively new deflectors have provided new fields for more classical theoretical methods, and useful results have been obtained. There are good prospects for more advances stemming from further use of computer techniques. Similar remarks apply to the study of scan magnification.

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However, it is traveling-wave deflection that has the greatest future potential for progress resulting from computer methods. Since recent work has been confined almost entirely to classical methods of analysis that are incapable of dealing with many of the most useful practical deflectors, we have yet to see even the earliest fruits of the new computer techniques in this field.

REFERENCES Andrews, J. (1970). Ph.D. Thesis, University of Kansas, Lawrence. Barkow, W. H., and Gross, J. (1974). Report ST-5015. RCA Entertainment Tube Division, Lancaster, Pennsylvania. (Read at 2nd Annual Convention, Fernseh und Kinotechnische Gesellschaft, Munich, October 15, 1974.) Barkow, W. H., and Gross, J. (1975). Nachrichtentech. Z. 28, K96. Bernstein, G., Dressel, H., Fye, D., Hutter, R., and Vassell, M. (1978). Sylvaniu Videon 23,27. Blazo, S., Perkins, P., and Hawken, K. (1977). Tech. Dig.-Int. Electron Devices Meet., 1977 p. 81. Bloomsburgh, R., Jones, R., King, J., and Pietrolewicz, J. (1965). IEEE Trans. Broadcast Telev. Receivers BTR-11, 50. Carpenter, M. E., Momberger, R. A., and Schultz, T. W. (1977). IEEE Trans. Consumer Electron. ce-23,22. Catchpole, C. E., and Ceckowski, D. H. (1969). Rev. Sci. Instrum. 40, 1549. Chernushenko, A. M. (1959). Radio Eng. Electron. 4, 120. Decker, R. W., and Knoll, J. S. (1968). Program, Int. Electron Devices Meet., 1968 Abstracts, p. 40. Fletcher, R. C. (1952). Proc. IRE40,951. Glaser, W. (1952). “Grundlagen der Elektronenoptik.” Springer-Verlag. Vienna. Gundert, E., and Lotsch, H. (1962). IRE Trans. Electron Devices ed-9, 197. Haantjes, J., and Lubben, G. J. (1957). Philips Res. Rep. 12, 46. Haantjes, J., and Lubben, G. J. (1959). Philips Res. Rep. 14,65. Hayes, R. (1975). IEEE Trans. Electron Devices ed-22,930. Heynick, L. N., ed. (1970). “High-Information-Density Storage Surfaces.” Final Rep., ECOM 01261-F, Contract DA28-043 AMC 01261(E). Stanford Res. Inst., Menlo Park, California. Himmelbauer, E. E. (1969). Philips Res. Rep., Suppl. 1, 1. Hollmann, H. E. (1950). Proc. IRE 38,32. Honey, R. C. (1954). IEEE Trans., Microwave Theory Tech. 2,2. Hughes, W. C., Lemmond, C. Q., Parks, H. G., Ellis, G. W., Possin, G. E., and Wilson, R. H. (1975). Proc. IEEE 63, 1230. Hutter, R. G. E. (1970). IEEE Trans. Electron Devices 4-17, 1022. Hutter, R. G. E. (1974). In “Advances in Image Pickup and Display” (B. Kazan, ed.), Vol. 1, pp. 163-224. Academic Press, New York. Hutter, R. G. E., and Ritterman, M. B. (1972). IEEE Trans. Electron Devices ed-19,731. Hutter, R. G. E., Bernstein, G., Dressel, H., Fye, D., and Vassell, M. (1977). In “1977 SID International Symposium Digest of Technical Papers,” p. 136. Lewis Winner, New York; copyright, Society for Information Display. Jackson, J. D. (1975). “Classical Electrodynamics,” 2nd ed. Wiley, New York. Janse, J. (1971). Oprik (Stuttgart)33, 270. Johnson, K. E. (1964). Radio Electron. Eng. 28, 115. Kaashoek, J. (1968). Philips Res. Rep., Suppl. 11,l.

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Kartashev, V. P., Kotov, V. I., and Utochkin, B. A. (1968). Radio Eng. Electron. Phys. (Engl. Transl.) 13, 1621. Kasper, E. (1976). Optik (Stuttgart)46,271. Kelly, J. (1977). Ado. Electron. Electron Phys. 43,43. Kubota, Y., Tagawa, S., Sawai, M., Namba, K., and Kakizaki, T. (1977). IEEE Trans. Consumer Electron. ce-24, 114. Lemmond, C. Q., Buschmann, E. C., Klotz, T. H., Jr., and White, G. M. (1974). ZEEE Trans. Electron Devices ed-21, 598. Lockwood, L. W., and Noble, M. L. (1970). J. SMPTE 79,317. Lyubchik, Ya. G., Savina, N. V., Fishkova, T. Ya., and Shkunov, V. A. (1971). Radio Eng. Electron. Phys. (Engl. Transl.) 16, 1733. Marrs, J. M. (1972). Am. Lab. 4, 57. Martin, A., and Deschamps, J. (1971). Proc. SID 12, 16. Moss, H. (1968). “Narrow Angle Electron Guns and Cathode Ray Tubes.” Academic Press, New York. Munro, E. (1973). In “Image Processing and Computer-Aided Design in Electron Optics” (P. W. Hawkes, ed.), p. 284. Academic Press, New York. Munro, E. (1975). J . Vac. Sci. Techno[.12, 1146. Nishino, T., and Maeda, H. (1972). Electron. Commun. Jpn. 55-B,45. Nomura, T. (1971). Elec. Eng. Jpn. 91, 147. Odenthal, C. (1977). In “1977 SID International Symposium Digest of Technical Papers,” p. 134. Lewis Winner, New York; copyright, Society for Information Display. Ohiwa, H. (1977). J. Phys. D . 10, 1437. Ovsyannikova, L. P., Utochkin, B. A., Fishkova, T. Ya., and Yavor, S. Ya. (1972). Radio Eng. Electron. Phys. (Engl. Transl.) 17, 825. Pierce, J. R. (1941). Proc. IRE 29,28. Pierce, J. R. (1949). Electronics 22,97. Pierce, J. R. (1950). “Traveling-Wave Tubes.” Van Nostrand, Princeton, New Jersey. Ritz, E. F., Jr. (1973). IEEE Trans. Electron Devices ed-20,1042; correction: ed-21,314 (1974). Ritz, E. F., Jr. (1974). U.S. Patent 3,796,910. Ritz, E.F., Jr. (1976). IEEE Trans. Electron Devices ed-23,1325; correction: ed-24,775 (1977). Saldi, I. T., and Schlesinger, K. (1970). Opt. Spectra 4, 53. Schackert, P. H. (1971). IEEE Trans. Electron Devices ed-18, 521. Schlesinger, K. (1949). Electronics 22, 102. Schlesinger, K. (1952). Electronics 25, 105. Schlesinger, K. (1956). Proc. IRE44,659. Schlesinger, K. (1959). U.S.Patent 2,904,712. Schlesinger, K. (1972). U.S.Patent 3,666,985. Schlesinger, K., and Wagner, R. (1965). Proc. Electron Laser Beam Symp., 1965 p. 471. Schlesinger, K., and Wagner, R. (1967). IEEE Trans. Electron Devices ed-14, 163. Schwertfeger, W., and Kasper, E. (1974). Optik (Stuttgart)41, 160. Silzars, A., and Knight, R. (1972). IEEE Trans. Electron Devices ed-19, 1222. Spangenberg, K. R. (1948). “Vacuum Tubes.” McGraw-Hill, New York. Stratton, J. A. (1941). “Electromagnetic Theory.” McGraw-Hill, New York. Van Alphen, W. M. (1972). Ado. Electron. Electron Phys. 33A, 511. Vaynoris, Z. A., Matseyka, K. Yu., and Shtaras, S. S. (1969). Radio Eng. Electron. Phys. (Engl. Transl.) 14,622. Vaynoris, Z. A., Matseyka, K. Yu., and Shtaras, S. S. (1970a). Radio Eng. Electron. Phys. (Engl. Transl.) 15, 1122.

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Vaynoris, Z. A,, Matseyka, K. Yu., arid Shtaras, S. S. (1970b). Radio Eng. Electron Phys. (Engl. Trawl.) 15, 1125. Vonk, R. (1971). Philips Tech. Rev. 32,61. Wang, C. C. T. (1967a). IEEE Trans1 Electron Devices ed-14,357. Wang, C . C. T. (1967b). J. Appl. Phys. 38,4938. Wang, C . C. T. (1971). IEEE Trans. Electron Devices ed-18,258. Weber, C . (1967). Philips Res. Rep., Suppl. 6, 1. Yamada, I., and Takagi, T. (1972). IEEE Trans. Electron Devices ed-19,204.

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Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although the name is not mentioned in the text.

A

Bassel, R. H., 135(22, 23), 149(59), 167(95), 169(59), 219,220,221 Batavin, V. V., 54, 79 Bates, D. R., 134(4), 219 Beattie, J. R., 76, 77, 79 Beaty, E. C., 178, 180(128), 222 Beck, H. G. E., 245(43), 247, 268(109), 293, 294 Bell, D. A., 265, 294, 268(107, 115), 294, 295 Bknt, B., 113, 131 Bknk, G.-J., 113, 114, 128, 131 Bennett, H. E., 76, 80 Bennett, L. H., 126, 132 Beresneva, L. A., 60, 76, 79 Berglund, C. N., 60,76,84 Bernstein, G., 309,312,315,316,319,320,355 Berrington, K. A., 179(132), 180(132), 222 Berz, F., 281(165), 284,287,289(192), 296,297 Bess, L., 259,264, 294 Bethe, H. A., 154(74), 220 Bhadra, K., 149(61), 169,220 Bhat, T. A. R., 289(193), 297 Bhattacharyya, P. K., 216,217,224 Bilger, H. R., 267,268(119), 294,295 Birman, A., 135, 148(32), 213, 214, 216(194), 219,223 Blankenbecler, R., 205,223 Blasquez, G., 266(98), 294 Blazo, S., 353,355 Blicharska, B., 113, 131 Bloembergen, N., 101, 131 Bloodworth, G. G., 268, 289(194, 204), 294 29 7 Bloom, A. L., 104, 130,131 B Bloomsburgh, R., 309,310,355 Baklanov, M. P., 38,39,40,64,72, 79 Blumkina, Y. A., 38,39,40,64, 70,71, 72, 74, Baklanov, M. R., 65,66,69,71, 77, 79 76, 77, 79 Barker, A. S.,60,76,84 Bockshtein, M. F., 15, 79 Barkow, W. H., 312, 315, 318,319, 320,355 Bogdanov, A. V., 140(40), 219 Bashara, N. M., 15, 29, 69, 70, 71, 73, 77, 78, Bohm, D., 135(15), 143(52), 163(52), 165(52), 78, 79,80, 81 219,220 359 Abeles, F., 6, 12, 13, 14, 15, 19, 77, 78 Abowitz, G., 288(181), 296 Abragam, A., 87,130 Adams, A. C., 77, 78 Adams, J. R., 77, 78 Agajanian, A. H., 76, 78 Agouridis, D. C., 271, 295 Aguado Bombin, R. M., 77, 78 Alexandrov, E. B., 36, 79 Algazin, Y. B., 38, 39, 40, 64, 65, 66, 70, 71, 72, 74, 76, 77, 79 Alkemade, C. T. J., 238(24), 292 Allen, T. H., 76, 79 Ambrozy, A., 268(120), 295 Anderson, R. J., 180,222 Anderson, R. L., 274(141), 295 Andrasko, J., 106, 131 Andrew, E. R., 121, 123, 126, 130, 131 Andrews, J., 345, 347, 355 Antziferov, A. P., 40,59, 79 Aoki, M., 281(169), 283(169), 284(169), 287 (l69), 288(169), 296 Archer, R. J., 19, 34,40, 63, 69, 70, 76, 77, 79 Archipenko, A. V., 38, 39,40, 64, 72, 79 Arnold, E., 288(181), 2% Aspnes, D. E., 28, 36, 39,40,69,78, 79 Atsumi, M., 289(197), 297 Avdeenko, A. A., 77,83 Avery, D. G., 77, 79 Azzam, R. M. A., 15,29,36,39,40,78,79,80

360

AUTHOR INDEX

Boksenberg, A,, 190,191,222 Bonham, R. A,, 151, 152, 153, 154(75), 155, 163, 164,220 Bootsma, G. A., 64, 69, 70, 75, 76, 77,80,82 Borcard, B., 113, 114, 128, 129, 131 Born, M., 22, 80, 134, 135(11), 136, 145, 146, 147, 150, 153, 154, 155, 156, 157, 161, 162, 163, 164, 165, 166, 167, 168, 169, 172, 173, 178, 179, 180, 181, 186, 187, 188, 191, 193, 194, 197, 199, 204(8), 205 (11),208,209,211,212,214,218,219,221 Bottomley, P. A., 121, 123, 126, 130, 131 Boutry, G.-A., 40,82 Bovbe, W. M. M. J., 116,131 Boyd, R. L. F., 190, 191,222 Bozic, S. M., 244(38), 293 Bozinis, D., 183, 222 Brackmann, R. T.,190, 191, 193(163),222 Braenstein, M., 76,82 Bransden, B. H., 136, 179(132), 180(132), 183, 185(157), 187(157), 219, 222 Brattsev, V. F., 180(139),222 Brittin, W. E., 103, 132 Broderson, A. J., 279,2% Brophy, J. J., 263,264,268(111, 121), 294,295 Broudy, R. M., 259,294 Brow, G., 289(200), 297 Brunner, P., 133 Bull, C. S., 244(38), 293 Burdett, R. K., 248(61), 293 Burge, D. K., 28, 39,40, 76,80, 81 Burgess, A., 204(177), 223 Burgess, R. E., 234(12), 235, 292 Burk, D.,126,132 Burke, P. G., 136, 171,217(205),219,224 Burykin, I. G., 15,80 Buschmann, E. C., 331,335, 336,356 Buttlaire, D. H., 132 Butz, A. R., 290,297 Bychkov, Y. A., 54, 79 Byron, Jr., F. W., 135(28, 30), 136, 140(38), 147(53, 54, 55), 158, 161, 163(53), 164 (53), 167(96), 168, 169, 177, 178, 180 (122), 181,214, 216,219,220, 221,223 C

Cahan, B. D., 28,40,80 Callender, A. B., 39,84 Card, W. H., 279(158), 296

Carpenter, M. E., 309,312,315,317,319,320, 321, 322,355 Carroll, J. J., 77,80 Cartiaux, L., 152(71), 220 Cartwright, D. C., 178(131), 180(131), 183 (153), 222 Catchpole, C. E., 326,355 Ceckowski, D. H., 326,355 Chamberlain, G. E., 177, 178, 179, 180(126), 222 Chan, F. T., 149(63), 150, 152(63, 73), 153, 154, 155, 156, 158, 169, 171, 172, 173, 175, 176, 177, 178, 179, 180, 181, 182 (152), 183, 198, 206, 216(202), 217, 220, 222,224 Chang, C. H., 150, 152(73), 153, 154(73), 155 (73), 156(64, 73), 158, 169, 171, 172, 173, 176(120, 121), 177(120), 180, 181, 182 (152), 183, 216,220,221,222,223 Chang, D. C., 115,131 Chaudhary, P. K., 279(158), 296 Chen, J. C. Y.,135(27), 163(90), 201(174), 202, 203, 204( 174), 206, 207, 208, 209, 210,212,213,216,219,221,223 Chen, S.T.,176(115, 116, 117, 118, 119), 177 (115, 118, 119), 178, 179, 180(115, 118, 119), 221 Chennette, E. R., 235, 266(96, 97, 98), 267 (97), 276, 277, 279(154, 157), 292, 294, 295,2% Cheng, Y.C., 289,297 Chernushenko, A. M., 347,355 Chin, F. K., 76, 80 Choe, H.M., 234(16), 292 Christenson, C. J., 226,244(4), 292 Christensson, S., 281(163), 285, 287, 288, 296 Christy, R. W., 76, 77,81 Chung, K. T., 206(185), 207(185), 221, 223 Chutjian, A., 183(153), 222 Clarke, J., 245, 246, 247, 248, 249, 250, 25 1, 252,253,254,257,259, 269,270,293 Clarke, R. A., 69, 70, 71,81 Claussen, B. H., 76,80 Clayton, H. C., 15,80 Clegg, P. L., 77, 79 Clem, J. R., 234(13, 14), 292 Cole, R. H., 105, 113,131 Coleman, J. P., 179(132), 180(132), 222 Coles, B. A., 117, 131 Conn, G. K. T., 76, 77, 79

361

AUTHOR INDEX

Conti, M., 278(152), 2% Cook, K. B., 279(154), 2% Comell, B. A., 101,131 Coulter, J. K., 77,80 Cowley, A. M., 272,295 Cox, D. M., 208(188), 209(188), 223 Csaki, A., 109,131 Czyzak, S. J., 134(5), 219

D Dagman, E. E., 15,56,60,80 Damadian, R., 116, 119, 125, 126, 129, 131 Das, M. B., 289(199), 297 Dash, W. C., 69,70,80 Daszkiewicz, 0. K., 103, 113, 118,131 Davidson, D. W., 105, 113,131 Decker, R. W., 342,344,355 de Clerck, G., 289(200), 297 de Heer, F. T., 179(136), 180(136), 181(136), 222 de Jongh, J. P., 179, 180(138), 222 Dekluizenaar, E. E., 15,69,70,75,76,77,82 de Kuijper, A. H., 281(169a), 284,2% Dell, Jr., R. A., 268(114), 295 Dell’Oca, C. J., 77, 80 Demichelis, F., 40,80 DeNicola, R. O., 40, 51, 53, 80 Denton, R. E.,77,80 Deschamps, J., 353,356 DeSmet, D. J., 15,80 Devyatova, S. F., 60,76, 79 Dewangan, D. P., 142(49), 143(49), 151, 152, 153(49),220 Diegel, J. G., 117, 131 Diegel, J. L.,115,131 Digman, M. J., 40,83 Dill, F. H., 39,40,81 Dill, H. G., 297 Dillon, M. A., 178, 180(130), 222 Dissanayake, S. P. B., 268(1IS), 295 Dixon, A., 194,223 Dmitriev, J. I., 113, I31 Dokuchaev, Y. P., 40,80 Dolder, K. T.,205,223 Doolen, G. D., 188(160, 161), 189(160, 161), 198(161),222 Donaldson, F. G., 179, 180(137),222 Dom, R., 75,81 Dressel, H., 309, 312, 315, 316, 319, 320, 355

Drude, P., 2, 19,64,80 Dubrovskin, G. V., 140(40), 219 Ducommun, E., 114,131 Dumont, M., 113,131 du Prb, F. K.. 232,292 Dutta, P., 252,253(72), 254,255,293

E Eberhard, J. W., 252,253,254,255,293 Edmonds, A. R., 156, 157(81), 220 Edwards, N. P., 40,80 Egorova, G. A., 49,50,69,70,80 Ellis, G. W., 335, 336,355 El-Shazly, A. F. A., 50,83 Elshazly-Zaghloul, M., 15,80 Eminyan, M., 157,221 End. J. H., 111, 289, 297 Engel, G., 103, 131 Engelsen. D., 15,80,82 Epstein, M., 268( 114), 295 Ernst, R. R., 125,132, 133 Esaki, L., 271, 295

F Fadeev, N. F., 40,80 Fane, R.W., 76,80 Fano, U.,156,220 Fassett, J. R., 249(64), 293 Feoktistov, A. M., 40, 51,83 Feucht, D. L., 15,81 Feuerbacher, B., 77,80 Filatova, E. S.,75,83 Finch, E. D., 104,131 Fisher, R. W., 132 Fishkova, T. Y., 352, 353,356 Fite, W. L., 190, 191, 193(163),222 Fitzgerald, D. J., 275(145), 295 Flaten, C. J., 77,80 Fletcher, R. C., 349,355 Flower, M., 76,80 Fonger, W. H., 274,280,295 Foster, G., 142(50), 143(50), 158, 161(84, 85), 162, 163(50, 89), 165, 167(92), 168, 181 (143, 202(175), 203, 204(175), 220, 221, 222,223 Francis, J. D., 268,294,295

AUTHOR INDEX

362

Franco, V., 135, 148, 149, 150, 167(95), 169 (59), 174, 175, 177, 180(112, 113), 182, 183, 201(65), 202(65), 203, 204(176), 212, 219,220,221,223 Frazee, R. E., 51, 53,80 Frey, H. E., 116,131 Fu, H. S.,281(167), 287,296 Fundaminsky, A., 134(4), 219 Fuse, N., 278(153), 2% Fye, D., 309,312,315,316, 319,320,355 G Gaal, J. L. M., 263, 294 Gabillard, R., 119, 131 Gailitis, M. K., 204(179), 223 Gallagher, A., 171(101), 172(101), 221 Gardner, E., 122, 123, 130, 131 Garwin, R. L., 107,131 Gasse, H. J., 235(18), 292 Gau, J. N., 156, 158, 167(93), 168, 169, 181, 182(79), 220,221 Cell-Mann, M., 211(193), 223 Geltman, S., 156(78), 179, 180(135), 183, 187, 220,222 Genshaw, M. A., 76,77,80,83 George, E. P., 96,97, 130, 132 Gerchikov, A. S.,77, 84 Gerjuoy, E., 135(26, 29), 136, 143, 148(29), 149, 150, 151, 156(29, 80), 157(62), 158 (80), 161(29), 165, 167(95), 169, 175, 176, 182, 189,206, 207,216,219,220,221 Gertreuer, K. W., 116,131 Gevers, M., 232,233,292 Ghosh, A. S. 147(58), 149(60, 61), 164(58), 168(58), 169, 177(124), 216,217,220,221, 224 Gibbons, J. F., 277, 296 Gien, T. T., 143(52), 147(56,57), 163(52), 164 (56), 165, 167(94, 9 9 , 168(98), 220, 221 Giralt, G., 279(158), 296 Glasel, J. A., 107,131 Glaser, W., 312,355 Glauber, R. J., 135, 136, 137, 140, 141, 142, 143, 144, 146, 147, 148, 150, 151, 152, 153, 155, 156, 157, 158, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 186, 188, 191, 193, 194, 195, 197,

199, 201, 202, 205, 206, 207, 208, 209, 210,211,212,213,216,217,218,219 Click, R. E., 98, 131 Gobeli, G. W., 19, 34, 63, 79 Godwin, R. P., 77,80 Goldberger, M. L., 205,211(193), 223 Golden, J. E., 190, 191, 192, 194, 195, 196, 197, 198, 199,222,223 Goldsmith, M., 125, 126, 131 Goldstein, H., 135(10), 219 Goodell, W. V., 77,80 Gorshkov, M. M., 78,80 Gottfried, K., 135(12), 219 Goubau, W. M., 270(126, 127, 128), 295 Graf, V., 109, 131 Graffunder, W., 242(30a), 292 Grannel, P. K., 89,132 Grant, A. J., 273, 295 Gray, D. T., 122, 123, 130,131 Grebnev, N. I., 65,66, 70,71, 72, 74,76, 77, 79 Greef, R., 40, 80 Greenstein, L. J., 268(121), 295 Gross, J., 312, 315, 318, 320, 355 Grove, A. S.,275(145), 295 Grushetzky, V. V., 15,80 Gundermann, R., 40,49,8J Gundert, E., 350,351,355 Gupta, A., 21 1(191), 212(191), 213(191), 214 (191), 215(191), 223 Gutkov, I. D., 275,295

H Haantjes, J., 304, 355 Hacsaylo, M., 36,83 Halford, J. H., 76,80 Hallenga, K., 105, 130, 131 Hambro, C., 206(185), 207(185, 186), 208, 209,223 Hambro, L., 221 Hamilton, Sir W. R., 135, 143(52), 163(52), 165(52), 219,220 Hanafi, H. I., 244(40), 258,259,260,293,294 Handel, P. H., 291,297 Hannam, H. J., 237(23), 238(23), 292 Hanne, F., 154(75), 220 Harkness, A. C., 77,80 Harrison, M. F. A., 194(169), 223 Haslett, J. W., 289(186, 205), 297

363

AUTHOR INDEX

Hass, H., 76,80 Hauge, P. S.,39,40, 79,80,81 Hausser, R., 99, 131 Hawken, K., 353,355 Hawkins, G., 269(123, 124), 270,295 Hawkins, R. J., 268,289(194,204), 294,297 Hayes, R.,344,355 Hazebroek, H. F., 40,81 Hazelwood, C. F., 115,131 Heddle, D. W. O., 134(2), 219 Hender, M. A., 179(137), 180(137), 222 Hennel, J. W., 103, 113, 118, 131 Henrion, W.,15,81 Henty, D. N., 40,81 Hertz, H. G., 103, 131 Heynick, L. N., 326,327,328, 329,355 Hiatt, C. F., 279(160), 280(160), 296 Hidalgo, M. B., 156(78), 179, 180(135), 188, 189(160, 161), 198(161), 220,222 Hielscher, F. H., 289, 297 Hien, N. C., 40, 69, 70, 71, 81 Hiltbrand, E., 113, 114, 128, 129, 131 Hilton, A. L., 40, 51, 81 Himmelbauer, E. E., 352, 353,355 Hinshaw, W. S.,121, 123, 130,131 Ho, T. S., 149(63), 152(63), 153(63), 220 Holland, G. N., 121, 123, 130,131 Hollmann, E. E., 345,355 Holmes, D. A,, 15,8I Holscher, A. A., 40,81 Holt, A. R., 179, 180(133), 222 Homer, L. D., 104,131 Honey, R. C., 347,355 Hooge, F. N., 247, 248, 253, 256, 258, 263, 265, 266, 261, 268, 279, 280, 284, 289, 293,294,295 Hoppenbrouwers, A. M., 244(39), 293 Hoppenbrouwers, A. M. H., 247(46, 48, 54, 55), 256(54, 55), 268(112), 293, 295 Hopper, M. A., 69, 70, 71,81 Horn, K., 75,81 Horn, P. M., 252,253,254,255,293 Houdard, J., 40,82 Hoult, D. I., 132 Hovatter, M. J., 272, 295 Hsiang, T. Y., 250, 251,293 Hsieh, K. C., 266, 267(97), 294 Hsu, S.T., 272, 273, 275, 276, 281(166), 287, 295,296

Huang, C., 288,296 Huang, J. T. J., 2 16(202), 2 17,224 Hudson, D. I., 56,58,81 Hughes, R. H., 180,222 Hughes, W. C., 335,336,355 Hummer, D. G., 204(177), 223 Hunderi, O., 39,40,81 Hunt, J., 179(133), 180(133), 222 Hutchison, J. M. S.,126, 132 Hutter, R.,309, 312, 315, 316, 319, 320, 355 Hutter, R. G. E., 300, 304, 309, 312, 315, 319, 320, 323, 324, 326, 330, 332, 334, 335, 345, 346, 347, 350,351,355 1

Ibach, H., 75,81 Ibrahim, M. M., 69, 70, 71,73,81 Icenogle, H. W., 76, 77,81 Ichiji, K., 15, 81 Idezak, E., 50,83 Inch, W. R., 117,132 Ishihara, T., 201(174), 202, 203, 204(174), 207 (186), 208, 209, 210, 212, 213, 223 Itoh, Y.,281(168), 283(168), 296 Ivanova, N. S.,49, 50,69,70,80 Iwasawa, H., 289(191), 297

J Jackson, J. D., 308,355 Jain, A., 192, 194, 198,223 James, T. L., 91,92, 106, 113, 130, 132 Jamieson, M. J., 171, 221 Janse, J., 315, 330,355 Jasperson, S.N., 28, 39,40,81 Jedrychowski, A., 113,131 Jerrard, H. G., 36,40,49,81 Jindal, R. P., 285,286(174), 2% Joachain, C. J., 135(30), 136,137(37), 140(39), 147(53, 54, 55), 152(71), 161, 163(53), 164(53), 177, 178, 180(122), 181, 200(37), 211(192), 214, 219,220,223 Johnson, D. L., 76,81 Johnson, J. B., 226,239,240,292 Johnson, K. E., 354,355 Johnson, P. B., 76, 77,80,81 Johnson, R. R., 243,292 Jones, B. K., 268,294,295

AUTHOR INDEX

Jones, G. E., 40, 51,81 Jones, G. P., 100,132 Jones, M. L., 77,81 Jones, R., 309, 310, 355 Joshi, S. K., 182(146, 147, 148), 222

K Kaashoek, J., 304, 306,307,355 Kabayashi, I., 289(197), 297 Kakizaki, T., 344, 356 Kamagaki, Y., 281(168), 283(168), 296 Kamal, A. N., 205,206,223 Kamashita, M., 289(201), 297 Kannenwurf, C. R., 268(114), 295 Kartashev, V. P.,354,356 Kasai, T., 40,81 Kasharow, P. K., 289(196), 297 Kasper, E., 312, 315,356 Katto, H., 281(168, 169), 283, 284, 287, 288 (169), 296 Kawabata, S., 15, 81 Kedzia, J., 247(60), 293 Kelly, J., 327, 328, 356 Kelsey, E. J., 181, 204(180), 222, 223 Kendall, E. J. M., 289(186), 297 Kessing, R. G. W., 134(2), 219 Kessler, J., 154(75), 220 Ketchen, M. B., 250, 270(126, 127, 128), 293, 295 Khayrallah, G. A., 143(51), 151, 152, 153(51), 164(51), 165, 176(116, 117), 180(51), 220, 221 Kieffer, L. J., 134(3), 219 Kim, Y.-K., 152(73), 153, 154(73), 155(73), 156(73), 220 Kinbara, A., 15, 77,84 King, J., 309, 310,355 Kinosita, K., 34,81 Kirchmayer, S., 113, 131 Kirillova, M. M., 77, 81 Klaassen, F. M., 258, 279, 281(76), 283, 284, 287,288,293 Kleinpenning, T. G. M., 247(50, 51, 52, 53), 248,251,264,265,266,267,271,280,289, 290(88), 293,295 Kleinpoppen, H., 157(83), 171, 172,221 Kleint, C., 235,237,292 Klotz, T. H., Jr., 331, 335, 336,356

Knausenberger, W. H., 76,84 Knight, R., 347, 356 Knispel, R. R., 115, 116, 117, 118, 130, 231, 132 Knoll, J. S., 342, 344,355 Koenig, S. H., 105, 107, 109, 112, 118, 130, 131, 132 Kontzevoy, Y.A., 40, 80 Kostuk, V. P., 76, 77,83 Kotegawa, H., 194,223 Kotov, V. I., 354, 356 Koutcher, J., 126, 131 Kraiss, E., 171, 172,221 Kriegler, R. J., 289(186), 297 Krotkov, R. V., 216(196), 223 Krueger, H., 171(99), 172(99), 221 Kruger, J., 77, 84 Kruuv, J., 116, 131 Krynicki, K., 98, 99, 130, 132 Kubota, Y.,344,356 Kucirek, J., 76, 81 Kudin, V. D., 40, 80 Kudo, K., 40,81 Kudoh, O., 289(201), 297 Kudryavtzev, E. N., 40,80 Kumar, A., 125,132 Kumar, S., 182, 183(150, 151), 211, 212, 213, 214,215,222,223 Kuntz, I. D., 106, 113, 130,132 Kuppermann, A., 178(131), 180(131), 222 Kurdiani, N. J., 77, 81 Kurepa, M. V., 177,221 Kushkova, A. S., 64,84 Kuyatt, C. E., 177(126, 127), 178(126, 127), 179(126, 127), 180(126, 127), 222 Kuzjmin, V. L., 15, 81 Kuznetsov, Y.N., 54, 79 Kwong, R., 202(175), 203,204(175), 223

L Lamy, P. L., 76,81 Landau, L. D., 217(204), 224 Lassettre, E. N., 178, 180(130), 222 Latour, L. J., 167(96), 168(96), 221 Lauterbur,P. C., 119, 120, 129, 130,132 Lee, J. S., 154(75), 220 Leech, J. W.,134(4), 219 Leluk, L. G., 76, 77,81,83

AUTHOR INDEX

Lemmond, C. Q., 331, 335, 336,355,356 Leuenberger, F., 288, 289,296,297 Leventhal, E. A., 281(164), 288(181), 296 Levy, M., 140(46), 220 Lewis, B. R., 212(86), 221 Lieber, M., 152(73), 153, 154(73), 155(73), 156(73), 220 Lifshitz, E. M., 217(204), 224 Lihle, F., 40,81 Lin, C. C., 180(141), 222 Lindeman, J., 116,131 Lindemann, W. W., 242,243,292 Lindstrom, T. R., 107, 112, 118,132 Lisitza, M. P., 76, 77,81 Liu, A. J. D., 71,81 Liu, S. T., 270,295 Lloyd, C. R., 163(86), 165(86), 213, 217(86), 221 Lluesma, E. G., 40,81 Lockwood, L. W., 344,356 Long, R. L., 208(188), 209,223 Lorteye, J. E. J., 244(39), 293 Lotsch, H., 350, 351, 355 Lubas,B., 103, 113, 118,131 Lubben, G. J., 304,355 Lubinskaya, R. I., 15,80 Lucas, R. C., 287,296 Liith, H., 75.81 Lukes, F., 76, 77, 81,84 Lundstrom, I., 281(163), 285(163), 287, 288, 296 Lyubchik, T. G., 352, 353,356

M MacAdam, K. B., 157(83), 221 McClure, D. E., 40,81 McConkey, J. W., 179(137), 180(137), 222 McConnel, H. M., 103,132 McCrakin, F. L., 34,81 McCredie, J. A., 117, 132 McCumber, D. E., 269,295 McDowell, M. R. C., 136, 171, 179, 180, 204 (106), 219, 221, 222 Macek, J., 156, 158, 167(93). 168, 169, 181, 182(79), 204( 180), 220, 221, 222, 223 McGuire, J. H., 188, 189, 190, 191, 192, 194, 195, 196, 197, 198, 199,222,223 Machlup, S., 279(159), 2%

365

McNamara, D. A., 266(97), 267(97), 294 McWhorter, A. L., 258, 259, 260, 262, 267, 274,280,294 Madan, R. N., 142(47, 48), 151, 152, 153(48), 158,220 Maeda, H., 348,356 Magnin, P., 113, 114, 128, 129,131 Mahan, H., 171, 172,221 Maja, M., 40,80 Malenka, B. I., 140(43),220 Malin, M., 55,81 Malm, W. G., 77,81 Mansfield, P., 89, 121, 122, 123, 124, 125, 130, 132 Mantena, N. R., 287,296 Mardezhov, A. S., 20, 36, 40,42,48,81,82 Marrs, J. M., 344, 356 Martin, A., 353,356 Martin, J. C., 279(158), 2% Massey, H. S. W., 134(4), 154(74), 174(110), 219,220,221 Maten-Perez, F. X.,279(158), 2% Matheson, C. C., 40,49,81 Mathieu, H. J., 40, 81 Mathur, K. C., 163(90), 182,221,222 Matseyka, K. Y.,348,356, 357 Matson, G. B., 132 Maudsley, A. A., 123, 124, 125, 130, 132 Maxwell, L. R., Sr., 126, 132 May, E. J. P., 268(105, 108), 294 Meadows, A,, 40,82 Medeiros, J. A., 216(196), 223 Melchior, H., 289(203), 297 Melmed, A. J., 77,80 Mendez-Morena, R. M., 152(71), 220 Mertens, F. P., 76, 77,81 Meulen, Y.J., 40, 69, 70, 71,81 Meyer, F., 15, 64, 69, 70, 75, 76, 77,80,81,82 Michel, D., 107, 132 Mielczarek, S. R., 177(126, 127), 178(126, 127), 179(126, 127), 180(126, 127), 222 Milaslavsky, V. K., 76,83 Miller, F. T., 180(141), 222 Minkoff, L., 125, 126,131 Mironov, F. S., 15, 42, 82 Mishnev, V. I., 82 Mitra, C., 204(178), 223 Mittleman, M. H., 140(39), 219 Miyamoto, A., 192(166), 193,223

366

AUTHOR INDEX

Mizgireova, L. P., 76, 82 Mohan, M., 197, 198,223 Moiseiwitsch, B. L., 134(1), 179(133), 180 (133), 219, 222 Molibre, G., 135, 219 Momberger, R. A., 309, 312, 315, 317, 319, 320, 321, 322,355 Monin, J., 40,82 Montagnon, N. B., 268(103), 294 Montgomery, H. C., 259,263,294 Moore, J. R., 289(199), 297 Moore, W. J., 268(117), 295 Moore, W. S., 121, 123, 130, 131 Morgan, H. G., 268(105, 108), 294 Morgan, L. A., 171(106), 204(106), 221 Morozov, V. N., 77,84 Morrison, D. J. T., 171,221 Moss, H., 345, 356 Moss, T. S., 52,82 Mott, N. F., 154(74), 174(110), 220,221 Motulevich, G. P., 77,82 Moustafa Moussa, H. R., 179, 180, 181,222 Mueller, O., 279, 296 Muller, R. H., 40,81 Munro, E., 312,315,316,356 Muntjewerff, W. F., 238(24), 292 Myerscough, V.P., 171(106), 204(106), 221

N Nagano, K., 289(190), 297 Nakahara, M., 289(191, 197), 297 Nakamura, K., 289(201), 297 Nakamura, M., 278(153), 296 Nakemi, N., 289(190), 297 Namba, K., 344,356 Narumi, H., 192, 193, 194(168), 201(173), 202, 203, 204(173), 223 Neal, W. E., 77, 78 Neal, W. E. J., 76,80 Newman, R., 69, 70,80 Nichols, B. L., 115, 131 Nicolet, M. A,, 267(99), 294 Nicollian, E. H., 289(203), 297 Nishino, T., 348,356 Noak, F., 109, 131 Noble, M. L., 344,356 Nomerovannaya, L. V., 77,81 Nomura, T., 312, 315,321,356 Norman, J. E., 76,80

Norris, H., 40, 49, 81 North, D. O., 241(29), 276,292,295 Norton, T. G., 222 Noskov, M. M., 77,81 Nuttall, J., 188(161), 189(161), 198(161), 222 Nyce, A. C., 16,82 0

Ochkur, V. I., 143, 147,151,152,153,163,164, 165,180,181,202,212,213,218,220,222 Odenthal, C., 353,356 Oh, S. D., 204(180), 223 OHandley, R. C., 28, 39,40,81 Ohiwa, H., 312, 356 Okamoto, G., 40,81 Oldham, W. G., 55,82 Oljshanetzky, B. Z., 64, 66, 72,82 Opal, C. B., 178, 180(128), 222 Oppenheimer, J. R., 134,219 ORahilly, R., 122, 123, 130,131 Ord, J. L., 36,40,82 Ortmans, L. H. F., 247(59), 256(59), 293 Osswald, R., 15, 81 Outhred, R. K., 96,97, 130,132 Ovsyannikova, L. P., 352,356

P Packard, M., 110, 132 Packer, K. J., 99, 100, 107, 108, 130, 132 Padalka, V. G., 76, 77, 82, 83 Pai, S. Y.,281(171), 283(171), 287, 296 Panjkin, V. G., 40,56,60,76, 79,80 Pankov, J. I., 50,82 Parks, H. G., 335,336,355 Passaglia, E., 34, 81 Pearson, G. L., 226,244(4), 292 Peart, B., 205, 223 Pedinoff, M. E., 76, 82 Pela, C. A., 40,81 Percival, I. C., 157, 158,220 Perkins, P., 353, 355 Peterkop, R. K., 184,222 Pfeifer, H., 107, 132 Philipp, H. R., 69, 70, 76, 71,82 Phinarev, M. S.,54,78,82 Pierce, J. R.,347, 348, 350, 356 Pietrolewicz, J., 309, 310, 355 Pintar,M. M., 115,116,117,118,130,131,132

367

AUTHOR INDEX

Platt, B. C., 76, 77, 81 Plumb, J. L., 276, 277,295 Plumb, R. C., 76, 77,81 Poe, R. T., 216(199), 223 Pope, J. M., 101, 131 Possin, G. E., 335, 336,355 Potapov, E. V., 49, 50,69,70, 76,80,82 Potemkin, V. V., 289(196), 297 Potter, R. F., 69, 70, 82 Primak, W., 76,82 Prins, J., 284, 296 Prior, G., 289(192), 297 Prishivalko, A. P., 22, 23, 82 Pritam, R., 216(199), 223 Pruniaux, B. R., 77, 78 Pullan, B. R., 86, 132 Purcell, W. E., 268(113), 295 Pushpavati, P. J., 270(131), 295 Pykett, I. L., 89, 125, 130, 132

Q Quigg, C., 136,219

R Rack, A. J., 241(29), 292 Rakov, A. V., 49, 50, 69, 70, 76, 80, 82 Rao, B., 40, 83 Rao, B. S., 77,81 Rau, A. R. P., 156,220 Raudrant, D., 113,131 Reich, H. A., 107,131 Repinsky, S. M., 64, 65, 66, 72, 79, 82,84 Rezvyi, R. R., 40,54,78,80,82 Rice, J. K., 178(131), 180(131), 217(203), 222, 224 Rice, S.O., 227(5), 229(5), 230(5), 292 Richardson, J. E., 205, 206(183), 223 Richardson, J. M., 240,249(62), 272, 279(62), 293 Ritterman, M. B., 323,324,330,332,334,335, 355 Ritz, E. F., Jr., 331, 332, 333, 337, 338, 340, 342, 343, 344,356 Roberts, E. F. J., 40, 82 Roberts, S., 76, 77,82 Rodberg, L. S., 210(190), 21 1(192),223 Rogers, C. J., 288,296 Ronen, R. S., 289(198), 297

Rosendorff, S., 135, 148(32), 213, 214, 216 (194), 219, 223 Rostoker, N., 264,294 Rothen, A., 40,82 Rudge, M. R. H., 171, 183, 185(156), 187,221, 222 Rumble, Jr., J. R., 176(117), 221 Ryberg, R., 39,40,81 Rzhanov, A. V., 7, 15, 26, 29, 30, 64, 66, 70, 72, 79, 82 S

Sah, C. T., 281(167), 286(175), 287, 289, 296, 297 Saha, B. C., 177,221 Saifi, M. A., 51, 53, 80 St. John, R. M., 171, 180,221,222 Saldi, I. T., 342, 344,356 Salzberg, C., 76, 77, 82 Saprykina, G. A., 15,80 Sarkar, K., 177(124), 221 Sasaki, T., 77, 80 Sato, N., 40,81 Savina, N. V., 352, 353,356 Sawai, M., 344,356 Saxena, A. N., 19,77,82 Saxon, D. S., 140(45), 220 Schackert, P. H., 351,352,356 Schey, H. M., 171(105), 221 Schiff, L. I., 135(16), 140(44, 45), 174(109), 219,220,221 Schillinger, W. E., 109, 132 Schlesinger, K., 319, 330, 332, 333, 334, 335, 337, 339,340,341, 342,344,356 Schmidt, E., 77,82 Schnatterly, S. E., 39, 81,84 Schonfeld, H., 230, 292 Scholz, F. J., 289(186), 297 Schottky, W., 226, 239, 240, 241, 242, 272, 273,292 Schueler, D. G., 49,82 Schultz, T. W., 309, 312, 315, 317, 319, 320, 321, 322,355 Schutten, J., 179(136), 180(136), 181(136), 222 Schwantes, R. C., 237(23), 238(23, 26), 243 (37), 292, 293 Schwertfeger, W., 312, 315, 356 Seaton, M. J., 134(6), 137, 157, 158, 205, 219, 220

368

AUTHOR INDEX

Stchehaye, R., 113, 129,131 Sellars, W. D., 268(105, 108), 294 Semenenko, A. I., 7, 15,20, 26, 29, 30, 36, 38, 39, 40, 42, 46, 48, 49, 51, 54, 56, 60, 64, 70, 72, 75, 79,80, 81, 82, 83, 84 Semenenko, L. V., 7, 15, 20, 26, 29, 30, 36, 40, 42, 46, 48, 49, 51, 54, 56, 59, 60, 64, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 82,83,84 Shacter, S. B., 279, 296 Sharp, A. R., 116,131 Sharshunov, A. G., 40, 51,83 Shashkin, V. V., 40, 79 Shchigolev, B. M., 73,83 Shen, S. W., 234(15), 235,292 Sheorey, V. B., 156(80), 158(80), 169(80), 182 (80), 220 Shklyarevsky, I. N., 50, 76, 77, 81, 82,83,84 Shkunov, V. A., 352,353,356 Shtaras, S. S., 348, 356, 357 Shubin, A. A., 77,82 Shurkliff, W. A., 22,83 Shwartz, N. L., 36, 56, 60, 75, 76, 79,80, 83 Shwartz, N. P., 40, 79 Sil, N. C., 149(60), 204(178), 220,223 Silzars, A., 347, 356 Sinfailam, A. L., 206(185), 207(185), 221, 223 Sinha, P., 149(60), 220 Sirohi, R. S., 76, 77,80,83 Skidowski, M., 77,80 Skolnick, L. P., 76,82 Slater, I. C., 64, 83 Slevin, J., 157(83), 221 Smidt, J., 116, 131 Smit, C., 238,292 Smith, K., 171(105), 221 Smith, L. E., 77,83 Smith, N. V., 76,83 Smith, R. C., 36,83 Smith, S. J., 134(1), 171(101), 172(101), 196, 208(188), 209(188), 219, 221,223 Smith, T., 34,40, 76, 77,83 So, S. S., 55, 76,83 Sokolov, A. V., 17,83 Sokolov, V. K., 7, 15, 26, 29, 30, 36, 40, 42, 46, 49, 51, 54, 59, 64,69, 70, 71, 75, 77, 79,81,82,83,84 Solomon, I., 101,132 Soonpaa, H. H., 15, 77, 78,81 Spangenberg, K. R., 345,356

Spenke, E., 241(29), 292 Spruit, W. P., 245(43), 247,293 Srivastava, M. K., 182, 183(150, 151), 192, 194, 198, 211(191), 212(191), 213(191), 214(191), 215(191), 222,223 Stafsudd, 0. M., 76,82 Standage, M. C., 157(83), 221 Stanford, M., 126,131 Starunov, N. G., 77,83 Steinberg, H. L., 34, 81 Stepanov, S. A., 38, 39,40, 64, 72, 79 Stern, E. A., 77,80 Stewart, W. C., 269, 295 Stobie, R. W., 40,83 Stoffel, A., 40,81 Stohrer, M., 109, 131 Stoll, M. P., 76,83 Stone, I. L., 248(61), 293 Storsiek, M., 268(118), 295 Strasilla, U.J., 268(116), 295 Stratton, J. A., 308,356 Strnat, K., 40,81 Strocken, J. T. M., 247(52), 251(52), 293 Stromberg, R. R., 34,77,81,83 Strutt, J. W., 2, 83 Strutt, M. J. O., 268(116), 295 Studna, A. A., 40,77, 79,83 Sucher, J., 140(46), 220 Suchorukov, 0. G., 49,51,82,83 Sutcliffe, H., 268(104), 294 Sutherland, A. D., 279( 157), 296 Suzuki, H., 178, 180(129), 222 Svensson, C., 281(163), 285(163), 296 Svitashev, K. K., 7, 15, 20, 26, 29, 30, 36, 38, 39, 40, 42, 46, 48, 49, 51, 54, 56, 59, 60, 64,65, 66, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80,81, 82, 83,84 Swartz, H. M., 118,132 Szczepkowski, T. W., 103, 113, 118,131

T Taft, E. A,, 69,70, 76, 77,82 Tagawa, S., 344,356 Tai, H., 149(59), 167(95), 169,220 Takagi, K., 279(161), 288,289,296,297 Takagi, T., 349,357 Takasaki, H., 40,84 Takayanagi, T., 178, 180(129), 222 Tanaka, T., 289(190), 297

369

AUTHOR INDEX

Tandon, J. L., 267(99), 268(119), 294,295 Tao, L. C., 76,81 Tenney, A., 151, 174, 177, 178(111), 180(111), 216,220,221,223 Teshima, R., 205,206(183), 223 Teubner, P. J. O., 163(86), 165, 212, 213(86), 217(86), 221 Tewari, K. C., 98,131 Thaler, R. M., 210(190), 211(192), 223 Thkroux, P., 76,81 Thomas, B. K.,135(29), 136, 143, 148(29), 149, 150, 151, 156(29, 80), 157(62), 158 (80), 161(29), 165, 169(80), 175, 176, 177, 182(80), 183, 189, 198, 201(67), 202(65), 203,204(176), 206,207,216,219,220,223 Thompson,R.T., 115,116,117,118,130,131, 132 Thomson, J. J., 134(7), 219 Thutupally, G. K. M., 76,84 Tijburg, R. P., 247(58), 256(58), 293 Timm, G. W., 235,236,292 Tomar, M. S., 15,84 Tomlin, S. G., 76, 77,80,84 Tomlinson, T. B., 243,293 Torrey, H. C., 128,132 Toshima, N., 150,220 Trajmar, S.,178(131), 180(131), 183,217(203), 222,224 Tremblay, A., 291,297 Treu, J. I., 40,84 Tripathi, A. N., 182,222 Trofimenkoff, F. N., 289(205), 297 True, J. I., 39,84 Truhlar, D. G., 178, 180(131), 217(203), 222, 224 Tsuji, A., 192(166), 193, 194(168), 201(173), 202,203,204(173), 223 Tully, J. A., 204(177), 223 Tzwelych, N. G., 76, 77,BI

U Ulmer, R., 171(99), 172(99), 221 Uryvsky, Y.O., 78,84 Usoskin, A. I., 77,84 Utochkin, B. A., 352,354,356

V Vaes, H.J. M., 265,294

Vainshtein, L. A., 171,221 Valabrega, P., 40,80 Van Alphen, W. M., 316, 326, 327, 328, 330, 337,356 van Blerkom, J., 152(72), 220 Vandamme, L. K. J., 247(49, 56, 57, 58, 59, 60, 256, 257, 266, 281(169a), 284, 289, 293,294,296 Van de Hulst, M. C., 22, 23,84 van der Ziel, A., 232, 234(15, 16), 235, 236, 237(23), 238(23, 26), 239(27), 241(27), 242(27, 31), 243(37), 244(40), 249(63), 252(68), 254(68), 258, 259, 260, 263, 266, 267(97), 270, 274(141), 275, 278(151), 279, 280(160), 281(170), 282, 284, 285, 286(174), 288, 289, 292, 293, 294, 295, 296,297 van Dijk, H. J. A., 247(48), 293 Van Eck, J., 179, 180(138), 222 Van Gurp, G. J., 233(1 l), 234(17), 292 van Helvoort, G. J. M., 268(109), 294 van Ooyen, D. J., 233(1 l), 292 van Overstraeten, R.,289(200), 297 van Wet, K. M., 235, 240(28), 249(63, 64), 271(132), 279(160), 280(160), 292, 293, 295,296 van Wijngaarden, J. G., 240(28), 292 Varian, R., 110,132 Vadjeva, L. L., 46,60,64,66,72,76, 79,82,84 Vassell, M., 309, 312, 315, 316, 319, 320, 355 Vaynoris, Z. A., 348,356,357 Vedam, K., 55,76,81,83,84 Verleur, H. W., 60, 76,84 Verminsky, Y.S.,77,84 Vidhani, T., 197, 198,223 Villa, J. J., 76, 77, 82 Viner, J., 278,296 von Engel, A., 194(169), 223 Vonk, R., 306,357 von Schweidler, E., 231,292 Vorobjeva, P. L., 15,80 Voss, R. F., 245, 247, 248, 249, 250, 251, 252, 253,254,257,259,268,269,293,295 Vriens, L., 177, 178, 179, 180(127), 222 Vuskovic, I., 177, 221

W Wagner, R.,333,337, 339,340,341,342,344, 356

370

AUTHOR INDEX

Walker, Jr., J. D., 171, 221 Wall, E. L., 272,295 Wallace, S.J., 140(42),220 Walters, H. R. J., 182,222 Wang, C. C. T., 306,326,327,357 Wang, C.-S., 206(184), 223 Wang, K. L., 289(202), 297 Washburn, H. A,, 15,77, 78 Watkins, T. B., 275,295 Watson, K. M., 206(184), 223 Watters, R. L., 266(97), 267(97), 294 Waylonis, J. E., 76, 80 Weber, C., 310,328, 330,357 Weigold, E., 212(86), 221 Weisman, I. D., 126, 132 Welti, D., 125, 132 White, G. M., 331, 335, 336, 356 Whittaker, E. R., 135(14), 143(52), 163(52), 165(52),219,220 Wilkin, C., 135(22, 23), 219 Williams, J. F., 136, 162, 163(87), 168, 169 (97), 219,221 Williams, J. L., 248(61), 293 Williams, J. R., 163, 164, 212,221 Williams, K. G., 208(187), 223 Williams, W., 183, 222 Williamson, Jr., W., 140(41), 142(50), 143 (50), 158, 161(84, 85), 162, 163(50, 89), 165, 167(92), 168, 181(145), 202(175), 203,204,220,221,222,223 Willis, B. A., 168, 169(97), 221 Wills, B. L., 36, 40, 82 Wilmans, I., 40, 81,84 Wilson, R. H., 335, 336,355 Winterbottom, A. B., 36, 37, 84 Winters, K. H., 152(71), 220 Woessner, D. E., 115, 131 Wolf, D., 268(118), 295 Wolf, E., 22,80, 135(11), 205(11), 219 Wolf, W. L., 76,77,81 Wong, T. C., 154(75), 220 Woods, M. M., 126,132

Woollings, M. J., 179, 180, 222 Worthington, B. S., 121, 123, 130, 131 Wright, J. G., 40, 49, 81 Wyatt, P. W., 15, 84

Y Yablontzeva, T. M., 70, 71, 72, 74, 76, 77, 79 Yajima, T., 271,295 Yamada, E., 281(169), 283(169), 284(169), 287(169), 288(169), 296 Yamada, I., 349,357 Yamaguchi, T., 15,77,84 Yamamoto, M., 34,40,81,84 Yamamoto, S.,278(153), 296 Yan, G., 77,80 Yarovaya, R. G., 76,77,81,83 Yasatake, K., 289(191), 297 Yates, A. C., 151, 174, 177, 178(111), 180 (lll), 216, 220,221, 223 Yau, L. D., 289,297 Yavor, S. Y., 352,356 Yolken, H. T., 77,84 Yonezawa, T., 278(153), 296 Yoshida, S.,15,84 Young, A. T., 238(25), 292 Young, L., 69, 70, 71, 77, 80, 81

2 Zaner, K. S., 132 Zopassky, V. S., 36, 79 Zarowin, C. G., 40,80 Zeidler, J. R., 77, 78 Zettler, R. A., 272, 295 Zijlstra, R. J. J., 265, 270, 294, 295 Zimmerman, J. R., 103, 132 Zipp, A., 106, 113, 130,132 Zolotarev, V. M., 76, 77,84 Zudkov, N. M., 54,79

Subject Index A

Abeles’ matrix, homogeneous layer, 10- 15 Absorbing film fundamental equation of ellipsometry, 55, 61 parameters, 54-63 Adsorption of bromine, 66-68 of krypton, 64 of methane, 64 of oxygen, 63-64 Adsorption-desorption ellipsometry study, 63-68 partial reversibility, 64-68 Aiming function, 56-58 level lines of, 57 vs. parameters, 58,60 Amniotic fluid, and nuclear magnetic resonance, 113-1 14 Ampkre circuited law, 313-314 Anisotropic media, ellipsometry, bibliography on, 15 Anomalous ilicker effect, 242 Aqueous fluids, and nuclear magnetic resonance, 112-1 14 Atomic collision, 133-224 parameters, 156-158, 172-173 qualitative description, 134-135 review papers, 135-137 Atomic density, 88-89 Atomic excitation cross section, 134- 135 Azimuth angle, of compensator, 25

Bonham-Ochkur method, 151-153 Born approximation in ionization, 186 second, 145 vs. Glauber approximation, 197-199 Born-Bonham-Ochkur exchange amplitude, 153 Born direct scattering amplitude, 150 Born series, scattering amplitude, 145 Bound state wave function, 159 Burst noise, 279

C

Cancer in situ measurement, 119- 129 relaxation time, 116 tumor growth, rate, 116-118 Capacitance, temperature coefficient, 233 Carrier density fluctuation, 281-285 Carson’s theorem, 229-230 Cathode ray tube, 319, 321 Chicago group experiment, 252-256 Clarke and Voss experiment for resistors, 245-247 temperature fluctuation mechanism, 249252 Compensator azimuth angle, 25 Stokes parameter, 26 Contact noise, 244 Correlation time agarose gel, 106 continuous distribution, 105-106 B discontinuous distribution, 104 Biopsis, 112-1 19 tissue, 115-116 Blankenbecler-Goldberger approximation, vs. distribution function, 106 205-206 Coulomb-eikonal scattering amplitude, 200Blood 20 1 in situ diagnosis, 127 Coulomb-Glauber approximation, 201-203 measurement, cow, 113 total cross section, 203 and nuclear magnetic resonance, 112- 113 Coulomb-Glauber-Ochkur amplitude, 202relaxation time, 99 204 371

372

SUBJECT INDEX

Coulomb potential, 185-186 Coulomb wave function, 195, 200 D

Damadian’s Fonar, 125 Dark current, 65 Debye-Stokes theory, 92 Deflection, 330-332 Deflectron double, 336 electron trajectories, 335, 338-340 electrostatic potential, 333 twisted vs. untwisted, 332-333 Density of atoms, 88-89 of velocity spectrum, 92 Desorption, thermal, 64-65 Dielectric film, measurement, superthin, 48 Dielectric film on semiconductor substrate, see Substrate-dielectric film system Dielectric loss, thermal noise of, 231-233 Dielectric properties, time distribution, 231 232 Diode current, saturated, 236, see also specific diode Dipolar magnetic interaction, 106-108 in aqueous solutions, 100-106 in biological substances, 95-108 relaxation time, 95-96 in water, 95-100 Dislocation density, reduction of, 278 Distorted-wave approach, 21 1 Divergence equation, 308 Drude classical theory, 64 Dynamic constants, evaluation, 94-95

E Eikonal approximation, 133-224 from Green’s function, 140-142 from Lippmann-Schwinger equation, 137140

review papers, 135-137 for scattering amplitudes, 137- 147 second-order corrections, 144- 147 vs. Glauber approximation, 158-160 Eikonal-Born series, 147 compapson of results, 164

Eikonal exchange amplitude, 142 Eikonal scattering amplitude, 141 Eikonal-type approximation, 205-216 Electric field multilayer reflecting system, 7-9 multiple yoke, 327-329 pattern yoke, 332-335 of protons, local, 89-90 tangential component, 9, 14 Electron beam deflection, 299-357 background, 300-302 definitions, 302-303 Electron scattering from ions, 199-205 from neutral atoms, 148-183 Electron-helium scattering, 174- 182 differential cross section, 176-180 total cross section, 179-180 Electron-hydrogen ionization differential cross section, 193 Eikonal approximation, 187-194 integrated cross section, 194 total cross section, 190-193 Electron-hydrogen scattering, 148-173 differential cross section, 162-163, 166-168 excitation cross section, 170-171 excited state, 152-156 experimental data, 160-173 Glauber theory, 148-158 total cross section, 161-162, 164-166, 168169 Electron-lithium scattering, 182-183 Electron-molecule collision, and Glauber approximation, 216-217 Electrostatic deflection, 323-345 background, 323-324 Electrostatic deflection yoke, 324-336 magnetically immersed, 336-344 multipole, 326-330 octupole deflector, 328 pattern, 330-336 theoretical analysis, 324-326 types, 324 Electrostatic potential, deflection, 333 Ellipsometer angle-of-incidence, 35 configuration bibliography, 40 light beam intensity, 22-28 LPE-3M, 41

373

SUBJECT INDEX

schematic, 21 threshold sensitivity, 35-36 Winterbottom automatic, 36-38 Ellipsometric measurement automation, 36-39 infrared, 51-52 zones, 31, 33 Ellipsometric angle and bromine adsorption, 66-67 experimental techniques, 20-21 values, 59-60 Ellipsometry anisotropic media, bibliography on, 15 experimental studies, 43-75 fundamental equation of, 4,45-46, 55,61 techniques, 1-84 Emission, secondary flicker noise in, 238-239 Epitaxial film-substrate system silicon, 53 measurement of thickness, 50-54 Error in ellipsometry, 49-50 in optical constant determination, 73-74 statistical, 58 Euler-Lagrange equation, 305 Extinction coefficient, 73 vs. refractive index, 70-71 vs. temperature, 71

F Field emission diode, flicker noise, 235-237 Fixed-scatterer approximation, 147 Flicker noise, 225-297 background, 225-226 defined, 225-226 examples, 231-234 field emission diode, 235-237 flux flow noise, 233-234 from grain boundaries, 259 Hall noise, 264-265 JFET, 279-281 Josephson junction, 269-270 junction device, 274-279 Lossy capacitor, 231-233 metal-metal contacts, 256-258 metal-oxide-metal diode, 270-271 from mobility fluctuation, 263-266 photoemission, 237-238

p-n junction, 274-276 problems, 290-292 resistor, 244-269 Schottky barrier diode, 272-273 secondary emission, 238-239 semiconductor-semiconductor contacts, 256-258 semiconductor tunnel diode, 271-272 semiconductor, 289-290 solid-state device, 269-290 space-charge suppression, 240-241 spectrum integral, divergent, 226-230 thermoelectric cell, 263-264 transistor, 277-279 tunneling device, 269-272 types, 253 vacuum diode, 241-243 vacuum pentode, 243-244 vacuum triode, 242-244 vacuum tube, 234-244 7 # 1, 290-291 Flux flow noise, 233-234 Focus projection and scanning deflection system, 337-338 applications, 344 deflection sensitivity, 341-342 motion, equation of, 339-340 operating characteristics, 340-344 scan rotational angle, 341, 343 shading index, 341, 343 theory vs. experiment, 344 Fonger model, 280 Fonger’s equation, 274-275 Fourier decomposition, scalar potential, 312 Four-probe method, conductors, 268 Fowler-Nordheim equation, 27 1 Free precession, 110-1 12 apparatus diagram, 110 in weak magnetic field, 128

G

Gaussian trajectory, 305 Generalized oscillator strength, 154-1 55 maxima, 155 minima, 154-155 Germanium opttcal characteristics, 69-75 6ptical constants, 70-71, 73

374

SUBJECT INDEX

Glauber amplitude for helium scattering, 174 for multielectron atoms, 195 Glauber-angle approximation, 206-209 excitation cross section, 208-209 Glauber approximation, 133-244 derivation, 142-144 electron-molecule collisions, 216-21 7 electron-lithium scattering, 182-183 multielectron atoms, 194-195 review papers, 135-137 second-order corrections, 144- 147 unrestricted, see Eikonal approximation vs. Born approximation, 197, 199 wave function, 188 Glauber-Bonham-Ochkur approximation, 151

Glauber-Bonham-Ochkur exchange amplitude, 152-153 Glauber exchange amplitude, unrestricted,

Ions, electron scattering from, 199-205 Isotope, identification, 87-88

J JFET, flicker noise, 279-281 Josephson junction flicker noise, 269-270 voltage spectra, 270 Junction device flicker noise, 274-279 p-n junction, 274-276 K

Klaasen-Prins approach, 284 Klaasen’s theory, 279 Kleinpenning model, 280

L

181-182

Glauber-Ochkur amplitude, 143- 144 Glauber path, 148 Glauber scattering amplitude, 182 in electron-hydrogen scattering, 148- 158 Grain boundaries, source, flicker noise, 259 Green’s function, 140-141

Lagrangian function, 304 Langmuir’s theory, 240 Laplace equation, 308 reduced, 311-315 three-demensional, 309-31 1, 323 Large-angle theory, magnetostatic deflection, 307-322

H

Hall noise, 264-265 Handel’s theory, 29 1 Hooge’s formula, 247-249 for JFET, 279 and temperature fluctuation mechanism, 252

validity of, 258-260 Hylleraas wave function, 174 I

Index of refraction, complex, 6 Inhomogeneous substances dipolar magnetic interaction, 106- 108 with suspended cells, 106-107 Interferometry, infrared, 53-54 Ionization Eikonal approximation, 187-194 by electron collision, 183-199 of multielectron atoms, 194-199 total cross section, 186-187

Larmor pulsation, 88 Lens with rotational symmetry, 350-351 with twofold symmetry, 351-354 Light beam convergent, 51 intensity, ellipsometer, 22-28 and linear optical system, 22 output intensity, 28, 35, 37 Line scanning, 123-1 24 Lippmann-Schwinger equation, 137-139 Lorenz-Lorentz relation, 64 Lossy capacitor flicker noise, 231-233 temperature coefficient, 233

M McWhorter surface model, 258-263,280 energy band structure, 261 field effect, 261 and Hooge’s formula, 258-259 and resistor, 267

375

SUBJECT INDEX

Magnetic deflection yoke, see also Saddle yoke: Toroidal yoke longitudinal section, 313 transverse section, 3 13 types, 304 Magnetic dipole interaction, see Dipolar magnetic interaction Magnetic field in situ diagnosis, 127-129 multilayer reflecting system, 7-9 pattern, toroidal yoke, 3 17 pattern, toroidal yoke, 317 spatial variation, 121 tangential component, 9, 14 transverse, 320, 322 Magnetic scalar potential, 308 Magnetostatic deflection, 303-323 angular current density, 314 background, 303-304 electron trajectories, 308-309 Laplace equation, 309-3 1 1 large-angle theory, 307-322 multipole field expansion, 31 1-322 multipole potential, 315-316 reduced Laplace equation, 31 1-315 saddle yoke, 321-322 small-angle theory, 304-307 Mansfield’s techniques, 123-125 Maxwell macroscopic equations, 6 Medical diagnosis new method, justification, 86 by nuclear magnetism, 85- 132 physical methods, 86 Metal-metal contacts, flicker noise, 256-258 Metal-oxide-metal diode, flicker noise, 27027 1 Mobility fluctuation, 263-266 in MOSFET, 286-287 in semiconductor, 289-290 in solid-state diode, 264-266 theory, 264-265 Modified-Glauber approximation, 147, 213216 comparison of results, 164 Molecular exchange, 102-104 Monte Carlo integration method, 181 MOSFET carrier density fluctuation, 281-285 equivalent circuit, 286 experimental data, 287-289 flicker noise, 259, 267, 281-289

inversion correction, 285-286 mobility fluctuation, 286-287 noise vs. interface state density, 288 Multielectron atom ionization, 194-199 total cross section, 194, 196-198 Multilayer reflecting system future research, 19-20 with inhomogeneous layer, 6- 15 optical properties, 6 Multiple field expansion, 31 1-322 Multipole field analysis, 316-322 Multipole function, 318 Multipole potential, 315-316 finite-difference method, 3 15 finite-element method, 3 16 Multipole theory, 323 Multisensitive point method, 121-123 Muller matrix, linear optical system, 24

N NMR, see Nuclear magnetic resonance Nuclear magnetic resonance and amniotic fluid, 113-114 application to tissue, 109- 130 and blood, 112- 113 for diagnosis of cancer, 118-1 19 environmental dynamics, 91-95 environmental structure, 89-91 future, 129- 130 imaging, see Zeumatography information provided by, 87-95 in medical diagnosis, 85-132 of ordered systems, 89 and pathological tissues, 94, 116-1 18 and pleural fluid, 114 techniques, 109- 1 12 and tissues, 114-1 19 Null technique, automation of, 36-38 0

Optical characteristics bibliography, 75-77 determination, 68-75 germanium, 69-75 silicon, 69-75 Optical constants germanium, 70-71 silicon, 70-71 Oxide film, change in thickness, 48-49

376

SUBJECT INDEX

P Pathology, nuclear magnetic resonance, 94 Pattern yoke applications of, 335-336 defined, 330-332 electric field, 332-335 Phase nomogram, silicon substrate-dielectric film, 50 Phenomenological model, nuclear magnetic resonance, 93-94 Photoemission, flicker noise, 237-238 Photometric technique, automation, 38-39 Planar and multiplanar imaging method, 124125 Plane homogeneous reflecting system, reflectivity coefficients, 4 Pleural fluid, nuclear magnetic resonance, 114 Polarization, state of, 22 Polarization fraction, 156-158 comparison of results, 169 helium, 181 vs. electron impact energy, 172-173 Polarizer, Stokes parameter, 25 Polarizing angle automated photometric technique, 38-39 automation, 36-38 estimation, 28-36 with fixed compensator, 29-31 generalized, 42 null technique, 28-34 and oxide film, 48 photometric technique, 34-35 Positron-hydrogen scattering, 160- 161 Postdeflection acceleration, 350-351, 353 Projection reconstruction method, 119-120 Protein correlation time, 104 relaxation time, 103-105 Proton Hamiltonian interaction, 95 spin interaction, 96 Proton-helium scattering, 176

R Reflecting system, Stokes parameter, 25, see also specific system Refractive index, 49-50,73 vs. extinction coefficient, 70-71 vs. temperature, 71

Relative reflectivity coefficient, 4 experimental technique, 20-43 future research, 39-43 generalized, 42 matrix procedure, 7-10 multilayer reflecting system, 6-20 plane homogeneous reflecting system, 4 problems, 39-43 thin-fdm system, 15-19 Relaxation time amniotic fluid, 114 cancerous tissue, 116 dielectric, 105 dipolar magnetic interaction, 95-96 dispersion, 98 evaluation, 93 molecular exchange, 102- 104 nonparamagnetic ionic solution, 103 nonresonant rotating frame, 101 nuclear magnetic resonance, 90-91 paramagnetic system, 101-102 polarizing field, 111 protein, 103-104 pure water, 98-100 rotating frame, 100-101 sensitivity, 91 soft biological tissue, 107-108 spin-echo technique, 109-1 10 spin-lattice, 90, 99 spin-spin, 90-91 spleen tissue, 118 tissue, 115-116 vs. correlation time, 96-98 vs. frequency, 96-98 vs. Larmor frequency, 104-106 vs. tissue water content, 117 Resistance noise, true, 244 Resistor Chicago group experiments, 252-256 Clarke and Voss experiments, 245-257 flicker noise, 244-269 Hooge’s formula, 247-249 integrated, 266-267 ion-implanted, 266-267 Rotational motion, tissue, 91-92 Runge-Kutta procedure, 309 S

Saddle yoke, 321-322 Scan magnification, 349-354

377

SUBJECT INDEX

background, 349-350 box lens, 352-354 deflection sensitivity, 349-350 Schottky barrier diode, 272-273 Schottky’s theory, 239-240 Search program, 56-62 incorrect parameter determination, 62 Semiconductor flicker noise, 258-268,289-290 Hooge’s formula, 258-260 mobility fluctuation, 263-266 Semiconductor-semiconductor contacts, flicker noise, 256-258 Shot noise field emission diode, 235-236 metal-oxide-metal diode, 270 photoemission, 238 positive grid triode, 244 Schottky barrier diode, 273 superconductor, 233 transistor, 279 vacuum diode, 24 1 Silicon optical characteristics, 69-75 optical constants, 70-71, 73 Single-point spin mapping technique, 121-123 Singlet scattering amplitude, 187 Small-angle theory, 322-323 errors, 306 magnetostatic deflection, 304-307 use of computers, 306-307 Solid-state devices, flicker noise, 269-290, see also specific device Solid-state diode, mobility fluctuation, 264266 Spectral density, 92 Spectrum diffusion, 243 divergent integral, 226-230 low field, 243 strong field, 243 Static deflection, defined, 303 Stewart-McCumber model, 269 Sticking coefficient, 67-68 Stokes parameter, 22-28 for compensator, 26 for polarizer, 25 for reflecting system, 25 Substrate and very thin transparent film relating system, 18-19 Substrate-dielectric film system

germanium, 44-47 parameter measurement, 44-50 phase nomogram, 50 silicon, 44-45, 47 Substrate-inhomogeneous layer-homogeneous layer reflecting system, 17-18 Substrate-inhomogeneous layer reflecting system, 17 Substrate-inhomogeneous layer-two homogeneous layers reflecting system, 15-17 Substrate-two homogeneous layers reflecting system, 17 Successive overrelaxation method, 3 15 Superconductor, flux flow noise, 233-234 Surface-controlled density fluctuation model, 258-259 Surface potential, Fourier expansion, 334

T Temperature fluctuation mechanism Clarke and Voss, 249-254 Hooge’s formula, 252 measurement vs. calculation, 250 noise correlation, 251 Thermodesorption, 64-65 Thermoelectric cell, flicker noise, 263-264 Thin fi1m parameter, 43-63 relative reflectivity coefficient, 15-19 Thomas-Chan reduction method, 175-176 Thomas-Gerjuoy integral representation, 175- 176 Thomas-Gerjuoy reduction method, 150151, 175, 189 Tissue, 114-116 application of nuclear magnetic resonance, 109-130 correlation time, 115-1 16 in situ, 119- 129 inuitro, 112-119 large-scale heterogeneity, 107 microscopic structure, 91-93 pathological, 116- 118 relaxation time, 107-108, 115-116 rotational motion, 91-92 small-scale heterogeneity, 108 translational motion, 92 water content, 89 Toroidal yoke, 316-320 magnetic field pattern, 317

-

378

SUBJECT INDEX

Trajectory, electron calculation of, 308-309 deflectron, 338-340 pattern yoke, 335 scan magnification, 353 Transistor burst noise, 279 flicker noise, 277-279 noise voltage, 277 Plumb and Chenette circuit, 277 Translational motion, tissue, 92 Transverse noise, 268 Traveling-wave deflection, 345-349 background, 345-347 simplified wave theory, 347-348 true wave theory, 348-349 Triplet scattering amplitude, 187 Tumor, see Cancer Tunneling device, flicker noise, 269-272 Tunnel diode, 271-272 Two-potential eikonal approximation, 209215 differential cross section, 212-215

U

with oxide-coated cathode, 242-243 with thin-film cathode, 241-242 with thoriated tungsten cathode, 241-242 with tungsten cathode, 241-242 Vacuum pentode, positive grid, 243-244 Vacuum triode with oxide-coated cathode, 242-243 flicker noise, 239-243 Vacuum tube, flicker noise, 234-244, see also specific tube Vanadium dioxide film, ellipsometric angle, 60-61 Variance noise, defined, 268 Voltage noise vs. temperature, 254-255 Volterra integral equation, 141 W

Water dynamic processes, 100 magnetic dipole interaction, 95-100 Watkins experiment, 275 Wiener-Khintchine theorem, 227-229, 240 Winterbottom automatic ellipsometer, 36-38 Wrist, nuclear magnetic resonance image, 122

Urine, in situ diagnosis, 128 Z

V

Vacuum diode flicker noise, 239-243

Zeumatography in medical diagnosis, 126 Fourier transform, 125-126