ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 39
CONTRIBUTORS TO THISVOLUME J. Boulmer J.-F. Delpech Wolfgang Franzen D. C. Lain6 John H. Porter J. Stevefelt J. L. Teszner S. Teszner J. te Winkel M. J. Whelan
Advances in
Electronics and Electron Physics EDITEDBY L. MARTON Smithsonian Institution, Washington, D.C. Assistant Editor CLAIREMARTON
EDITORIAL BOARD E. R. Piore T. E. Allibone M. Ponte H. B. G. Casimir W. G. Dow A. Rose L. P. Smith A. 0. C. Nier F. K. Willenbrock
VOLUME 39
1975
ACADEMIC PRESS
New York San Francisco London
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT 0 1975, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED on TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC on MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
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LIBRARY OF CONGRESS CATALOG CARDNUMBER:49-7504
ISBN 0-12-014539-1 PRINTED IN THE UNITED STATES O F AMERICA
CONTENTS CONTRIBUTORSTO VOLUME 39 FOREWORD. . . . .
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vii ix
Electron Diffraction Theory and Its Application to the Interpretation of Electron Microscope Images of Crystalline Materials M . J . WHELAN I . Introduction . . . . . . . . . . . . . . . . . I1. Theories of Electron Diffraction . . . . . . . . . . . 111. Applications to Electron-Microscope Image Contrast of Crystalline Materials . . . . . . . . . . . . . . . . . . IV . Concluding Remark . . . . . . . . . . . . . . . Appendix: General Solution of the Dispersion Equations . . . . References . . . . . . . . . . . . . . . . . .
1 5 52 68 68 70
Energy Spectrum of Electrons Emitted by a Hot Cathode WOLFGANG FRANZEN AND JOHN H. PORTER
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I Introduction . . . . . . . . . . . . . 11. Effect of Surface Barrier on Electron Energy Distribution
. . 111. Physical Situation Outside Spherical Cathode . . . . IV. Space-Charge Density for Spherical Cathode . . . . . V . Solution of Poisson’s Equation . . . . . . . . . VI . Self-consistent Solution for Space Charge and Potential .
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73 75 80 83 87 90 93 95 99 103 109 114 115 117
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121 122 123 132 140 160 177 178
VII . Effect of Space-Charge Barrier on Electron Energy Spectrum VIII . Design Principles for a Spherically Symmetric Gun: Field Matching IX . Construction of Electron Gun . . . . . . . . . . . . X . Electron Energy Analysis . . . . . . . . . . . . . XI . Results of Experimental Study of Electron Energy Distribution . . . . . XI1. Relation Between Results Reported Here and Earlier Work Glossary of Symbols Used in Text . . . . . . . . . . References . . . . . . . . . . . . . . . . . .
Low-Temperature Rare-Gas Stationary Afterglows J.-F. DELPECH.J . BOULMER. AND J . STEVEFELT Introduction . . . . . . . . . . . Units and Notations . . . . . . . . . I . General Description of the Stationary Afterglow . 11. Microwave Diagnostics Techniques . . . . . I11. Ionic Population . . . . . . . . . . IV . Excited States Populations . . . . . . . V . Conclusion . . . . . . . . . . . References . . . . . . . . . . . . V
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vi
CONTENTS
Advances in Molecular Beam Masers
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D C L A I ~
I. Introduction . . . . I1 Principles and Techniques I11. Beam-Maser Spectroscopy IV . Beam-Maser Amplifiers . V Beam-Maser Oscillators . VI . Other Properties . . . References . . . . .
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183 184 199
219 221 241 244
Past and Present of the Charge-Control Concept in the Characterization of the Bipolar Transistor J . TE WINKEL I . Introduction . . . . . . . . . . . . . . . . . I1. First Years of the Transistor Era in Brief . . . . . . . . . 111 Origin of the Charge-Control Concept . . . . . . . . . IV. Physical Background of the Charge-Control Concept . . . . . V Hybrid-Pi Equivalent Circuit as a Small-Signal Version of a ChargeControl Representation . . . . . . . . . . . . . . VI . Ebers-Moll Model Derived from the Charge-Control Concept . . . VII . Large-Signal Charge-Control Model . . . . . . . . . . VIII. Concluding Remarks Regarding Charge-Control Models in General IX . Linvill Lumped Model . . . . . . . . . . . . . . X . Miscellaneous Applications . . . . . . . . . . . . . XI . Charge-Control Concept in the Computer-Aided Analysis of Transistors and Circuits . . . . . . . . . . . . . . . . . XI1. Conclusion . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .
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253 254 258 263 266 269 271 274 275 277 284 287 288
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Microwave Power Semiconductor Devices I
s. TESZNER AND J . L. TESZNER Introduction . . . . . . . . . . . . Two-Terminal Devices . . . . . . . . . I . Bulk Diodes . . . . . . . . . . . . References for Section I . . . . . . . . . I1. Junction Diodes . . . . . . . . . . . References for Section 11. A . . . . . . . References for Section 11. B . . . . . . . . References for Section 11. C . . . . . . . . AUTHOR INDEX . SUBJECT INDEX .
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291 293 293 318 320 350 364 377
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391
CONTRIBUTORS TO VOLUME 39 Numbers in parentheses indicate the pages on which the author’s contributions begin.
J. BOULMER, Groupe d’Electronique dans les Gaz, Institut d’Electronique Fondamentale, Facultt des Sciences, Universitt Paris-XI-Campus d’Orsay, Orsay, France (12 I ) J.-F. DELPECH,Group d’Electronique dans les Gaz, Institut d’Electronique Fondamentale, Faculte des Sciences, Universitt Faris-XI-Campus d’Orsay, Orsay, France (1 21)
FRANZEN, Department of Physics, Boston University, Boston, WOLFGANG Massachusetts (73) D. C. LAIN& Department of Physics, University of Keele, Keele, Staffordshire, United Kingdom (1 83) JOHNH . PORTER,Department of Physics, Boston University, Boston, Massachusetts (73) J. STEVEFELT, Groupe d’Electronique dans les Gaz, Institut d’Electronique Fondamentale, Facultt des Sciences, Universitt Paris-XI-Campus d’Orsay, Orsay, France (121) J. L. TESZNER, Direction des Recherches et Moyens d’Essais, Paris, France (291)
S TESZNER, Centre National d’Etudes des Ttlecommunications, France (291) J.
WINKEL, Philips Research Laboratories, Eindhoven, The Netherlands (253)
TE
M. J. WHELAN, Department of Metallurgy and Science of Materials, University of Oxford, Park Road, Oxford, England (1)
vii
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FOREWORD Six critical reviews on widely different subjects constitute our present volume. In the first review M. J. Whelan starts by pointing out what an important tool the electron microscope has become for the study of the solid state. The interpretation of electron microscope images, based on scattering considerations, was the subject of a review in Volume 32 of these Advances. Whelan’s review contributes a very necessary counterpart to the earlier one by examining the theory of electron diffraction and its application to the image interpretation of crystalline materials. Very often the practitioners of electron microscopy and of electron diffraction assume tacitly that their instrumentation provides a monochromatic electron beam. Twice before, in Volumes 13 and 29, we published reviews examining the energy distribution of electrons emitted by a hot cathode. In this volume W. Franzen and J. H. Porter bring us up to date on this important topic. The reactions of excited or ionized atoms and molecules at thermal energies are best studied in low-temperature afterglows. J.-F. Delpech, J. Boulmer, and J. Stevefelt discuss in their review such afterglows in rare gases. This field is growing rapidly, and we may find it useful to come back to the same subject in a few years. D. C . Lain6 points out in the next chapter how unjustified is the apparent neglect in the literature of molecular beam maser research. This important device is now studied more often in other countries than in the United States. In view of the abundance of undeveloped suggestions and ideas, it is hoped that this review will contribute to new efforts. Our fifth review concerns the use of the charge-control concept in the characterization of the bipolar transistor. J. te Winkel shows how a new concept allows new insights into the operation of a well-known device. The volume ends with the first part of a review, by S. Teszner and J. L. Teszner, on microwave power semiconductor devices. F o r a long time microwave power generation and reception seemed to remain the last holdout for vacuum tube devices. In recent years semiconductor device development succeeded in invading this area and, although total replacement of vacuum tubes by semiconductors is not yet in sight, progress in this field is so remarkable that an extensive review is more than justified. We expect to publish in forthcoming volumes the following critical reviews. ix
X
FOREWORD
Time Measurements on Radiation Detector Signals The Excitation and Ionization of Ions by Electron Impact Nonlinear Electron Acoustic Waves. I1 The Photovoltaic Effect I n Situ Electron Microscopy of Thin Films Physics and Technologies of Polycrystalline Si in Semiconductor Devices Charged Particles as a Tool for Surface Research Electron Micrograph Analysis by Optical Transform Electron Beam Microanalysis Electron Polarization in Solids X-Ray lmage Intensifiers Electron Bombardment Semiconductor Devices Thermistors High Power Electronic Devices Atomic Photoelectron Spectroscopy. I and I I Electron Spectroscopy for Chemical Analysis Laboratory Isotope Separators and Their Application Recent Advances in Electron Beam-Addressed Memories Nonvolatile Semiconductor Memory Devices Light Emitting Diodes, Methods and Applications. I and I1 Generation of Images by Means of Two-Dimensional Spatial Electric Filters Mass Spectroscopy Nonlinear Atomic Processes High Injection in a Two-Dimensional Transistor Semiconductor Microwave Power Devices. I1 Plasma Instabilities Basic Concepts of Minicomputers Physics of Ion Beams from a Discharge Source Physics of Ion Source Discharges Auger Electron Spectroscopy High Power Electron Beams as Power Tools Terminology and Classification of Particle Beams On Teaching of Electronics Wave Propagation and Instability in Thin Film Semiconductor Structures The Gunn-Hilson Effect A Review of the Application of Superconductivity
S.Cova John W.Hooper and R. K. Feeney R. G. Fowler Joseph J. Loferski A. Barna, P. B. Barna, J. P. Pkza, and I. Pozsgai J. Kobayashi J. Vennik G. Donelli and L. Paoletti D. R. Beaman M. Campagna, D. T. Pierce, K. Sattler, and H.C. Siegmann J. Houston D. J. Bates, R. Knight, and S. Spinella G. H.Jonker G. Karady S. T. Manson D. Berknyi S. B. Karmohapatro J. Kelly J. F. Vervey H. F. Matad
H. F. Harmuth F. E. Saalfeld, J. J. Decorpo, and J. R. Wyatt J. Bakos W. L. Engl S. Teszner and J. L. Teszner A. Garscadden L. Kusak Gautherin and C. Lejeune Gautherin and C. Lejeune N. C. Macdonald and P. W. Palmberg B. W. Schumacher B. W. Schumacher and J. H.Fink H. E. Bergeson and G. Cassidy A. A. Barybin M. P. Shaw W. B. Fowler
FOREWORD
xi
Minicomputer Technology Digital Filters Physical Electronics and Modeling of MOS Devices
C. W. Rose S. A. White J. N. Churchill, T. W. Collins, and F. E. Holmstrom
Supplementary Volumes Sequency Theory Computer Techniques for Image Processing in Electron Microscopy
H. F. Harmuth
W. 0. Saxton
Almost simultaneously with the appearance of this volume we are fortunate enough to publish, as a supplementary volume to Advances in Electronics and Electron Physics, a monograph ‘‘ Charge Transfer Devices ” by C. H. SCquin and M . F. Tompsett. We wish to express our best thanks to the many friends whose help makes it possible to produce these volumes. As in the past we would be very grateful for further advice and constructive criticism.
L. MARTON C. MARTON
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Electron Diffraction Theory and Its Application to the Interpretation of Electron Microscope Images of Crystalline Materials M. J. WHELAN Department of Metallurgy and Science of Materials, University of Oxford, Parks Road, Oxford, England
I. Introduction................................................................................ 11. Theories of Electron Diffraction .. .................................................... A. Kinematical Theory of Electron Diffraction ......................................... B. Dynamical Theory of Electron Diffraction........................................... C. Absorption due to Inelastic Scattering ............................................... 111. Applications to Electron-Microscope Image Contrast of Crystalline Materials ....... A. Perfect-Crystal Contrast Effects-Extinction Contours ............................. B. Images of Planar Defects.. ............................................................ C. Images of Dislocation Lines.. ......................................................... IV. Concluding Remark ....................................................................... Appendix: General Solution of the Dispersion Equations .............................. References ................. .............................................................
1 5 5 24 43 52 52 57 61 68 68 70
I. INTRODUCTION
In recent years the electron microscope has become a tool of considerable importance in the study of the solid state. In its conventional form (the transmission electron microscope or TEM) it enables the interior structure of thin films of solid materials to be examined directly and crystal defects such as dislocation lines, responsible for physical, mechanical, and chemical properties, to be observed. In the form of the scanning electron microscope (SEM) it enables the surface of a bulk specimen to be studied. In both types of instrument the image of the specimen is formed by causing an electron beam to be scattered by the specimen and by using the scattered electrons to form an image. In this review we have in mind primarily the case of the TEM, where real electron images are formed by a sequence of electron lenses in a manner analogous to the working of an optical microscope, as shown schematically in Fig. la. This shows a microscope employing three magnifying lenses (objective, intermediate, and projector lenses) and a single condenser lens for illuminating the object. The electron beam incident on the 1
M. J. WHELAN
2
E
Electron source
Condenser lens
Focal plans of objective
Intermediate
lens 2nd. Intermediate ~
[tor
Im Microscopy
(a)
Diffraction
(bl
FIG.1. Image formation in the electron microscope. After P. B. Hirsch et al. (16).Courtesy of Butterworths.
specimen is an almost parallel beam and the situation pertaining in the region of the objective lens is shown in Fig. 2. The incident beam in passing through the specimen is scattered and if the specimen is thin and crystalline, the scattering consists of a number of discrete diffracted beams, known as Bragg reflections, together with diffuse background scattering (Fig. 3) which increases with increasing specimen thickness. The Bragg beams are often thought of as arising from elastic scattering (scattering of the incident beam without change of energy), while the diffuse scattering is attributed to inelastic processes in which the incident electron transfers energy to the specimen. Such a classification, while useful, is too restrictive, and it is now known that both quasi-elastic and inelastic processes may
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
/I
3
Electron Beam
Objective Lens
Objective Aperture
FIG.2. Diagram illustrating the mechanism of contrast production in the electron microscope for the bright-field image. Scattered beams are prevented from reaching the image by the objective aperture.
FIG.3. Electron diffraction pattern of a single crystal of aluminum. Discrete spots are Bragg reflections. The central beam is marked. Note the diffuse scattering and the dark and light lines, known as Kikuchi lines.
4
M. J. WHELAN
contribute to the background, while some inelastic processes contribute intensity close to the directions of Bragg beams. The image is formed by those electrons which pass through a small aperture in the objective lens known as the objective aperture. This aperture is moveable during operation of the microscope and it can be used to select either the directly transmitted beam (as shown in Fig. 2) or one of the diffracted beams. Such images are known as bright-field and dark-field images,,respectively. The objective aperture acts as an angular filter, allowing only those electrons in a range of solid angle to contribute to the image. Evidently then, the interpretation of image contrast in crystalline specimens will depend on an understanding of the scattering of the incident electron beam by the specimen, and the theoretical apparatus for this constitutes what is known as the dynamical theory of electron diffraction. Section I1 of this review is an account of this theory, the initial impetus for the development of which ( 1 ) was the discovery of electron diffraction (2, 3) nearly 50 years ago, and its subsequent uses in the study of the structure of molecules and crystals. In Section I11 we then briefly review some of the applications of the theory to the study of defects in crystals. The number of applications is now so large that it is not possible in a short account to give complete coverage. The applications discussed are therefore intended to be illustrative rather than exhaustive. As mentioned above, we have primarily in mind the applications of the theory to transmission electron microscope images. Nevertheless, it should be pointed out that a variant of the theory has been applied with success to the study of certain contrast effects known as channeling patterns and to images of stacking faults and dislocations observed with crystalline materials in the SEM (4, 5), and because of the reciprocity principle (6) the theory is also applicable to image contrast in the scanning transmission electron microscope. In this review we will not in the space available be able to discuss applications in these fields, but, nevertheless, their existence should be noted. It should also be pointed out, before embarking on the theory, that from the experimentalviewpoint the recording of electron diffraction patterns and images go hand-in-hand in modern transmission electron microscopes, which are able to record electron diffraction patterns from small areas (- 1 pm diam) of the specimen. This technique is known as selected-area electron diffraction and was first developed by Boersch (7,8)and later by Le Poole (9). The principle of the method is shown in Fig. lb. A selector aperture is inserted in the image plane of the objective lens to select an area of the first intermediate image. The strength of the intermediate lens is then reduced to image the focal plane of the objective on the final viewing screen. In this plane the diffraction pattern of the specimen is formed just as in Abbe's theory of the optical microscope. After removing the objective aper-
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
5
ture the diffraction pattern of the area of specimen selected is visible on the final screen. This technique is very useful in the study of crystalline materials, where the diffraction pattern enables information about the orientation of the specimen relative to the incident beam to be obtained, as well as information on the important Bragg reflections excited. Analogous techniques (10-12) for recording selected-area “channeling patterns ” in the SEM have also been developed in recent years. 11. THEORIES OF ELECTRON DIFFRACTION
The range of electron energy of interest in electron microscopy is from a few thousand electron volts up to several hundred thousand electron volts, and electrons in this range are usually described as “fast.” Electrons of energy less than about a kilovolt belong to the “1ow”energy region and are of interest for surface diffraction studies in lowenergy electron diffraction (LEED) (2). The elastic scattering of fast electrons by single atoms is sufficiently strong to necessitate a multiple scattering (dynamical)treatment in crystals, and a simple physical way of approaching multiple scattering is by methods similar to those originally used by Darwin (13) for the case of X-ray diffraction. This approach which was developed by Howie and Whelan (14, 1 5 ) [see (16) for review], uses wave-optical methods and gives results identical with a more formal wave-mechanical treatment formulated by Bethe (1). We therefore first discuss scattering by single atoms and then the so-called “kinematical” theory of diffraction. The latter is the theory normally used in X-ray diffraction where multiple scattering is neglected. It is not a good approximation in fast electron diffraction, except for very thin crystals or for weak Bragg reflections. Nevertheless an account of this is useful as groundwork before proceeding to the general theory, and also because in the wave-optical approach the scattering in the crystal is calculated from the scattering by thin layers which are considered to scatter kinematically themselves. However, the latter is not a necessary assumption. Wave-optical methods have also been formulated for LEED (17), where scattering by single layers of atoms cannot be treated accurately by kinematical theory. A. Kinematical Theory of Electron Diffraction 1. Elastic Scattering by Single Atoms
The incident fast electron of momentum p has a de Broglie wavelength
1 = h/p. Expressing p in terms of the accelerating potential E we obtain
[
1 = h 2moeE 1 + _ _
(
2mpOc2
41
-
M. J. WHELAN
6
TABLE I ELECTRON WAVELENGTHS A N D OTHeR
DATAFOR
ELECTRONS OF VARIOUS ENERGlEs
1 10 50 80 100 loo0
0.388 0.122 0.054 0.042 0.037 0.0087
2.580 8.194 18.67 23.95 27.02 114.7
0.062 0.195 0.41 0.50 0.55 0.94
1.002 1.020 1.098 1.157 1.196 2.957
where m, is the electron rest mass, e is the electronic charge, and c is the velocity of light. The term in E2 in Eq. (1) is a relativistic correction amounting to about 5 % at 100 kV.Typical electron wavelengths are shown in Table I, together with the velocity relative to c and the relativistic mass m relative to mo.If the incident plane wave exp (2nik r) with wave vector k (k = 1-') has unit amplitude, the wave scattered by an atom situated at the origin, in the direction k' to a large distance r, is given by the asymptotic expression
f(28) exp (2nikr)/r, (2) where for elastic scattering k' = k, and 28 is the scattering angle between k and k;f(28) is the atomic scattering amplitude and has the dimensions of length;f(20) is related to the differential cross section for elastic scattering into the solid angle dn by
For fast electrons the first Born approximation gives a reasonable estimate offas the Fourier transform of the atomic potential V(r)
f(28) =
j V(r) exp (-2niK - r) dr,
(4)
where K = k - k and K = 2 sin 0/A. The potential V(r) is related to the atomic charge density ep(r') by V(r) = e
j
P ( 4 dr'. r-r'I
Equation ( 5 ) states that V is the convolution of p with r-'. Hence from convolution theory the Fourier transform (4) is the product of the Fourier
ELECTRON DIFFRACTION THEORY A N D ELECTRON MICROSCOPY
7
transforms of p and r-’. Since p(r) = Z6(r) - p,(r) where the 6 function describes the nucleus of charge + Ze and pe(r)describes the electron density in the atom, the Fourier transform of p is Z - F ( K ) , where F is the X-ray scattering factor given by F(28) = p,(r) exp (- 2niK * r) dr. Also the Fourier transform of r - is l/nK2. Thus Eq. (4) may be transformed to
’
S
In Eq. (6) a. is Bohr’s radius h2/me2(0.527 di for m = mo).To take account of relativistic effects a. should be calculated using the relativistic mass. Equation (6) can be used to estimatef from known values of F. Bothfand F are tabulated in the International Crystallographic T a k s . At small angles (K,8 -,0) Eq. (6) gives f ( 0 )= Z(r2)/3ao
where (r2) =Z-’
s
9
4nr4p,(r) dr
(7)
(8)
is the mean-square radius of the atom. For hydrogenf(0) = ao, while for most atoms f ( 0 ) is in the 1-10 di range. This is to be compared with the atomic scattering amplitude for X-rays which is of the order lo-” cm at low angles for elements of medium Z . It is evident that electron scatteringis about lo4 times as strong as X-ray scatteringand this explains why multiple scattering effects are much more important for electrons than for X rays. It is also clear from Eq. (6)that the electron scattering amplitude derived from the first Born approximation is real and positive. That this cannot be strictly correct is seen from the “optical theorem” which states that
where cr is the total atomic scattering cross-section for all processes, elastic as well as inelastic. It follows therefore thatf(0) at least will have an imaginary part. In general f(28) should include a phase term (ie., f ( 2 8 ) = I f(28) I exp [iq(28)]).The calculation of the phase 428) requires a more sophisticated analysis either by taking account of higher Born approximations or by using the partial-wave solution (18). Discussion of this and its effect on the analysis of gas diffraction patterns where neglect of q can lead to erroneous bond lengths in molecules has been given by several workers (1921). The phase q for medium Z can be of order 0.1 rad and increases with increasing Z and 8. However we wish to emphasize that in our development of the theory we obtain the correct results for dynamical scattering in a
8
M. J. WHELAN
crystal by assuming the first Born approximation to correctly describe atomic scattering. If a scattering amplitude with an imaginary part is substituted in the wave-optical theory it leads to absorption effects (22,23),as discussed in Section II,C,2,a and such “absorption ” is physically untenable if only elastic scattering is considered (24, 25). In fact in the wavemechanical development of the dynamical theory the fundamental quantity entering the theory is the Fourier coefficient of the periodic lattice potential and this is given (apart from constants) exactly by the Fourier transform [Eq. (411. Before concluding this section we examine the total cross-section for elastic scattering by single atoms obtained by integration of Eq. (3). We have, since the scattering is independent of azimuth,
d o = 2a sin 28 d(28) = 2aK dK/k2.
(10)
Equation (3) then gives
211
o,, = - J” I f ( K )I2K dK.
k2
Equation (11) can be evaluated numerically from tabulated values off(K), or analytically by using Wentzel’s formula
f = 2Z/ao(41t2K2+ R - 2 ) .
(12)
If RZ = (r2)/6 = aof(0)/2Z, Eq. (12) satisfies (7). Equations (1 1) and (12) give oe]= znzf(0)/2aao.
(13) For an aluminum atom with 100 keV electrons, a. = 0.442 K , f ( O ) = 7.29 A (relativistically corrected). This gives oel = 4.7 x cm’ which may be slightly too large. Misell (26) quotes a figure of 659 A for the mean free path for elastic scattering in aluminum which leads to o,, = 2.5 x lo-’’ an2. Data for carbon, aluminum, and gold are given by Misell (26).The angular distribution of elastic scattering is also wide compared with the acceptance angle of a typical objective aperture. For a 20 p diam aperture in an objective of focal length 3 mm, 28,, ‘Y 3 x rad. Integration of Eq. (11)using (12) shows that only 3% of the elastic scattering from an aluminum atom passes through the objective aperture. While the above refers to scattering by a single atom, the same would approximately describe the scattering per atom in an amorphous solid or gas, where the phase relationship between scattering from different atoms is approximately random. However, as we shall see next, the regular periodic
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
9
arrangements of atoms in a crystal leads to well-defined phase relationships and the angular distribution of the scattering from the crystal is quite different from the sum of that from each atom treated independently. 2. Kinematical Scattering by a Perfect Crystal The atoms in a crystal are arranged in a regularly repeating pattern in space, the repeat unit being called the unit cell. The edges of this cell are the basic vectors of the direct lattice. The reciprocal lattice is defined by the following basic vectors: a* = (b x c)/V,, b* = (c x a)/K, c* = (a x b)/V,, where V , = a * (b x c) is the volume of the cell in the direct lattice. A vector g = ha* + kb* + k* (h, k, 1 integers) in the reciprocal lattice has the following useful properties :
n, (14) where dhk, is the spacing between planes in the direct lattice with Miller indices hkl, and n is a unit vector normal to these planes; g=
-
g r, = integer,
(15) where r, is a vector to the nth lattice site in the direct lattice. Denote the scattering amplitude of the contents of the nth unit cell byf, and consider the scattering of an incident plane wave of unit amplitude by the crystal. The phase difference between scattering from the cell at the origin 0 and that at r, is (Fig. 4) 2a - (r, cos /3 A
- r,
cos a ) = 2n(k’ - k) * r,
= 2nK
- r, .
The total amplitude scattered by the crystal to a point at large distance r is A = r-
n
f, exp (-2aiK
*
r,),
(17)
9 0
FIG. 4. Diagram illustrating the diffraction of the primary wave to a point r at large distance.
M. J. WHELAN
10
where it is assumed that r is much larger than the crystal dimensions. The factor r - arises in the same way as in Eq. (2). If all unit cells are identical, Eq. (17) becomes
'
A =J C exp (- 2niK * rn). r
r
n
The sum in (18) is maximized if K r, is an integer for all n. This is the condition for a diffracted beam to occur and is known as the Laue condition. From (15) it is clear that this condition is satisfied if K = g (a vector in the reciprocal lattice). The geometrical interpretation of this condition is called the Ewald sphere construction and is shown in Fig. 5a. A sphere with radius
(aI
(bl
FIG.5. Ewald sphere construction for determining the direction of a diffracted wave. (a) The Bragg condition is exactly satisfied when the reciprocallattice point g lies on the sphere; (b) a deviation from the Bragg condition is indicated by g lying off the sphere.
k = A-' is envisaged and the origin of the reciprocal lattice is placed at the extremity 0 of a radius vector in the direction of the incident wave. If a reciprocal lattice point g coincides with the sphere a diffracted wave is established in the direction k joining the center C of the sphere to g. This geometrical illustration of the h u e condition is very useful, and is easily seen to be equivalent to the Bragg law (2dhk,sin 0 = A) from the geometry of the triangle Cog and by using Eq. (14). For fast electron diffraction the radius of the Ewald sphere is large compared with Igl. For example in Table I k = A- = 27 A-' at 100 keV while 181for a low-order reflection is about 0.5 A- '. The Ewald sphere in the vicinity of 0 in Fig. 5a is practically a plane and the diffraction pattern is then effectively the projection of a section through the reciprocal lattice. For k close to a prominent zone axis in the crystal the diffraction pattern is a cross grating of spots (Fig. 3). One other consequence of the large radius of the Ewald sphere is that the Bragg rad for low-order reflections at 100 keV. angle 0 is small, about The Ewald sphere construction is useful for visualizing the situation when the h u e or Bragg condition is not exactly satisfied. For an infinitely
'
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
11
large crystal the reciprocal lattice points are perfect points, and Bragg reflection occurs only for one particular angle of incidence. However, for a crystal of finite size some diffracted intensity occurs even when g lies off the sphere as shown in Fig. 5b. Consider the scattering in direction k defined by the small vector s in Fig. 5b, where K = k - k = g + s. The phase term in Eq. (18) is then 2nK r, = 2n(g + s) r,, and since g r, is an integer Eq. (18) becomes
The argument of the exponential varies slowly from cell to cell in the lattice and the sum can therefore be replaced by an integral over the crystal volume
f exp (-2nis * r) dr. (20) rV, crystal Equation (20) shows that each reciprocal lattice point can be thought of as having a finite extension given by the amplitude distribution A(s),which is simply the Fourier transform of the function which is unity inside the crystal and zero outside. This is called the shape transform. Each reciprocal lattice point is broadened to the same extent regardless of the indices hkl. This broadening is known as particle-size broadening in X-ray diffraction. Equation (20) can be evaluated for a crystal with a rectangular parallelepiped shape, and by allowing two dimensions to become large, we obtain the following for a plate crystal of area y and thickness t: A(S)= -
j
where s, u, w are orthogonal components of s, the component s being normal to the plate (Fig. 8). Equation (21) shows that the intensity distribution around the reciprocal lattice point is in the form of a spike normal to the crystal plate. This is shown schematically in Fig. 6a, and is the well-known diffraction function of a slit of width t in physical optics. The central maximum has a width 2/t which broadens as the thickness decreases. Equation (21) can be used to calculate the absolute intensity scattered by a crystal and hence the intensity of the diffracted wave emerging from the crystal. We consider the symmetrical Laue case shown in Fig. 7a where the Bragg planes are perpendicular to the crystal surface so that incident and diffracted beams enter and leave the crystal symmetrically at angle 8 to the planes. The spike in Fig. 8 then bisects the angle between k and k.The total intensity scattered over a sphere S of large radius r is 1 IA(s, u, w ) 1’ dS.
M. J. WHELAN
12
EO
FIG. 6. Intensity distribution (schematic) around a reciprocal lattice point (a) on the kinematical theory, (b) on the dynamical theory. After M. J. Whelan. Courtesy of the Institute of Metals.
Using Eq. (21) and the fact that dS/r2 = dodw/k2 cos 8, the total intensity is
The area of the crystal projected along k is y cos 8. Therefore the intensity per unit area of the diffracted wave emerging from the crystal is
We put
g,
= n k cos ~ e~(2e).
(24) 5, has the dimensions of length and is called the extinction distance for reasons which will become clear later. In generalfdepends on the scattering amplitudes of the atoms in the unit cell through a structure-factor term f(2e) = Cfi(2e) exp (-2nig i
T~),
(25)
wherefi is the scattering amplitude of the atomj at zi in the unit cell. Some typical values of extinction distances computed from (24)and (25) are shown in Table 11. It is noted that extinction distances are of the order of a few
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
13
Incident beam Reflecting planes
Diffracted beam
Transmitted beam
(a)
Diffracted beam
Transmitted beam (b)
FIG.7. Diagrams illustrating geometrical situations in diffraction. (a) The symmetrical Laue case; (b) the symmetrical Bragg case. After M. J. Whelan. Courtesy of Pergamon Press.
/
V E w a l d sphere
FIG. 8. Ewald sphere construction showing the intensity distribution along the spike through the reciprocal lattice point for a plate crystal.
M. J. WHELAN
14
TABLE I1 EXTINCTION DISTANCES t., I N A
IN
SOME
CRYSTALS
Reflection g
A1
Cu
Au
111 200 220 311 222
556 673 1057 1300 1377
242 281 416 505 535
159 179 248 292 307
Reflection g
Mt3
Zn
Tl00 1120 2200 TlOl
1509 1405 3348 1001
553 497 1180 35 1
Reflection g -
Diamond
111 220 311
400
Si _ _ _ _ _ _ ~
______
476 665 1245 1215
602 757 1349 1268
hundred angstroms for low-order reflections. Equation (23) can then be written
IW)l2 =
(;)IS.
Some important contrast effects in crystals can now be understood qualitatively in terms of Eq. (26) and Fig. 8. First we have the case of bend contours, an example of which is shown in Fig. 9. If the thin foil has uniform thickness but is buckled, some areas of the foil will be locally at the Bragg reflecting position. In terms of Fig. 8 the intensity distribution around the reciprocal lattice point has a fixed width but brushes through the Ewald sphere as the foil bends. Thus the diffracted beam will have a maximum corresponding to s = 0, with subsidiary maximum on either side, while the direct beam will exhibit complementary behavior. A bright-field image will therefore show a dark fringe at the positions in the foil where s = 0, together with subsidiary dark fringes. These are known as bend extinction
ELECTRON DIFFRACTION THEORY A N D ELECTRON MICROSCOPY
15
FIG. 9. Bend extinction contours in aluminum due to 311-type reflections. Pairs of parallel contours visible correspond to reflections from opposite sides of the Bragg planes. After Hirsch et al. (16).Courtesy of Butterworths.
contours (27) and Fig. 9 shows such contours in the image of a buckled aluminum foil. A number of contours caused by reflections of the type 220 and 311 are visible. Since the foil is thin and the reflections are high order with large extinction distances (Table II), the kinematical theory gives a fairly good description of the contours. If the contours are due to a loworder reflection however, the kinematical theory gives a poor description, and dynamical theory is needed to account quantitatively for the fringes (see Section 111,A). Second, there is the case of thickness extinction contours (27) occurring in a foil of varying thickness, an example of which is shown in Fig. 10 for the dark-field image of a dimple in an aluminum foil. In this case the orientation of the foil is fixed, that is, s is fixed in Fig. 8, while the thickness t varies along the wedge. Therefore the width of the distribution in Fig. 8 around g varies along the wedge causing subsidiary maxima to brush through the Ewald sphere. This gives rise to “wedge fringes” in the image as seen in Fig. 10. These are fringes of constant thickness of the foil. According to Eq. (23) the fringes have a spacing corresponding to a depth periodicity in the foil of At = s-l and an intensity which varies as s-’. The kinematical theory is therefore inapplicable at s = 0 when dynamical theory must be employed. The form of the depth oscillations of direct and diffracted waves predicted by kinematical theory is shown schematically in Fig. lla.
re
16
M. J. WHELAN
FIG. 10. Thickness extinction contours around a dimple in an aluminum foil (111 reflection; dark field image). After P. B. Hirsch er al. (29). Courtesy of the Royal Society.
3. Kinematical Scattering by an Imperfect Crystal
By an imperfect crystal we mean a crystal containing defects such as dislocation lines, stacking faults, or strains due to inclusions or precipitates. The intensities of diffracted beams are appreciably affected by elastic displacements caused by the presence of such defects. A dislocation line causes a displacement R of an atom from its perfect crystal position, the function R depending on the distance from the dislocation as well as on its character. The method of calculating the contrast in the image of the dislocation uses the so-called “column approximation” shown in Fig. 12 for the case where a dislocation line parallel to the surface is situated at 0.It is required to calculate the amplitudes &(t) and &,(t) of direct and diffracted waves at a variable point Bat the bottom of the foil. This is done by carrying out the calculation for the hatched column in Fig. 12 as though it were a crystal of infinite lateral extent. The displacement R(z) in the column has a functional form which depends on the position of the column. As the position varies the intensities I&,(t)12 and I+,(t)lZ vary and give the forms of the bright-field and dark-field images of the dislocation. The column approximation was first introduced by Whelan and Hirsch (28) in the calculation of stacking-fault images, where it was shown that the approximation gave the same result as a complete wave-matching calculation. It was later used to treat kinematical scattering from
Direct wave
I
Diffracted wave I
Direct wave
I
Diffracled wave
(b)
FIG.11. Intensity oscillations (schematic) of direct and diffracted waves in a crystal. (a) Kinematical region where the diffracted beam is weak; (b) dynamical region where s = 0. The depth periodicity of the intensity oscillations in (b) is given by the extinction distance T e . After M. J. Whelan and P. B. Hirsch (28). Courtesy of the Philosophical Magazine. I
x*-
Incident wave
\
\ \
\
’
z
Bottom
FIG.12. Diagram illustrating the concept of the column approximation, and the coordinates used to describe a dislocation in a foil. r is a radial coordinate measured from 0.After A. Howie and M. J. Whelan (15). Courtesy of the Royal Society.
M. J. WHELAN
18
dislocations (29), and subsequently in the dynamical theory of defect scattering (14, 15). The approximation is valid in the case of electron diffraction where Bragg angles are usually small. It has been discussed by Howie and Basinski (30) and by Takagi (31). An atom at r, in the perfect crystal is displaced to r, + R, in the imperfect crystal. Equation (18) of the previous section should therefore be replaced by 4-
A= r
C exp [ -274g + s) n
(r,
+ R,)].
Now s is of the order of t - ’ while R, is of the order of an interatomic distance. Therefore s * R, in Eq. (27) is negligible and using (15) we may write
f A =“
C exp (- 2xig n
R, - 2nis * r,).
(28)
Proceeding as for Eq. (20) we obtain
f exp (-2xig R(r) - 27th * r) dr, A ( s )= (29) rV, crystal where R(r) is a continuous function of c such that R(r,) = R, . Equation (29) shows that the amplitude distribution around the reciprocal lattice point is the Fourier transform of a function whose modulus is unity inside the crystal and zero outside, and whose phase [ -2ng R(r)] depends on the displacement field. As an example consider the case of a stacking fault on an inclined plane as shown in Fig, 13. A stacking fault produces a constant shear R between the two halves of the crystal separated by the fault. Putting
P
FIG. 13. Diagram illustrating a stacking fault running across a thin crystal on an inclined plane. Parts of the crystal above and below the fault are displaced relative to each other by a constant vector R,which is not a lattice translation vector. After M. J. Whelan. Courtesy of the Institute of Metals.
ELECTRON DIFFRACTION THEORY A N D ELECTRON MICROSCOPY
19
a = 2ng R(z), we have in Fig. 13 a = 0 for the part of the column above the fault and a = 2ng R for the part below the fault. The part of the integral in Eq. (29) which depends on z' is therefore Jco,"m"
=
exp ( -2aisz') dz' + exp ( - ia - 2nisz') dz', ,I*, iZ+'
(30)
where z is the depth of the fault below the center of the foil. Evaluating this we obtain = (ns)-l
exp (-$a)[sin
(nts + &) - sin $ exp (-2nisz)I. (31)
column
The intensity of the diffracted beam at the bottom of the column in Fig. (13) is then
+
sin2 (nts $a) + sin2 &a- 2 sin &asin (nts + i a ) cos 2nsz
(42
(32)
For an inclined fault Eq. (32) predicts contrast in the form of fringes. The fringes arise from the cos 2nsz term and their depth periodicity is Az = si.e., the same as for wedge fringes in the kinematical theory. We note that Eq. (32) reduces to the perfect-crystal expression [Eq. (2611 when a = 2nn, where n is an integer. This occurs when g R = n and the stacking fault is then invisible. This rule enables R to be determined if a systematic examination is made of the vanishing conditions for various Bragg reflections. The method has been applied to stacking faults in face-centered cubic materials (28), to the hexagonal AlN structure by Drum (32), and to other materials by numerous authors. An instructive geometrical interpretation of the column integral [Eq. (30)] is the amplitude-phase diagram, which for a stacking fault with phase angle a = -3. is shown in Fig. 14. The first integral in Eq. (30) is represented by a chord PQ of a circle of radius (271s)- ',while the second integral in Eq. (30) is represented by a chord QP" of a second circle of the same radius intersecting the first circle at Q at an angle a. The arc POQ of the first circle is equal to the length $t + z of the column above the fault (Fig. 13), while the arc QP" is equal to the length it - z of the column below the fault. The point 0 in Fig. 14 corresponds to the center of the foil in Fig. 13. If a = 0 or 2nn, the first and second circles become coincident and the integral in Eq. (30) is given by the chord P P . This is the amplitude diffracted by the perfect crystal. When a = - 120"as shown, the amplitude is PP" and it is clear that this can be considerably different from the perfect
',
.
M. J. WHELAN
20
FIG. 14. Amplitude-phase diagram for a face-centered cubic crystal containing a stacking fault of phase angle -2n/3. After P. B. Hirsch et al. (29). Courtesy of the Royal Society.
crystal amplitude. Moreover, as z varies (i.e., as Q varies in Fig. 14) the amplitude PP" oscillates with a depth periodicity Az = s- '. The amplitude-phase diagram approach is also useful for evaluating the column integral of Eq. (30) when a is a continuous function of z as for a dislocation. The situation is as shown in Fig. 15. A screw dislocation line A B parallel to the surface causes the column CD in the perfect crystal to deform to the shape EF. For a screw dislocation
R = (b/2n)tan-'
(33)
(z/x),
where b is the Burgers vector of the dislocation and
a = 2xg * R = n tan-' (z/x);
(34) n = g b is an integer for a perfect dislocation where b is a lattice vector. The column integral is then
-
+1 2
L"m"I-=, =
'
exp (- in tan- (z/x)- 2nisz) dz,
(35)
where the variables are defined in Fig. 15. Amplitude-phase diagrams representing Eq. (35)are shown in Fig. 16 for the cases n = 1, 2nsx = 1. The integral Eq. (35)has a large value when the two phase terms in the integrand have opposite signs, whereas it has a small value when the phase terms have the same sign. In the former case the amplitude-phase diagram is as shown in Fig. 16a while in the latter case it is as shown in Fig. 16b. Hirsch, Howie,
*
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
21
FIG. 15. Crystal containing a screw dislocation AE parallel to the plane of the foil. After P. B. Hirsch et 01. (29).Courtesy of the Royal Society.
la)
(b)
FIG. 16. Amplitude-phase diagrams for a column of crystal close to a screw dislocation as in Fig. 15. The radius of the broken circle is ( 2 m - I . The diagram is an unwound spiral (a), or wound-up spiral (b) depending on the signs of the phases of the terms of the kinematical integral [Eq. (35)’J. After P. B. Hirsch et al. (29).Courtesy of the Royal Society.
22
M. J. WHELAN
and Whelan (29) took (AB)' in Fig. 16a as a measure of the dislocation image profile where A and B are the centers of the initial and final circles, analogous to those discussed for a stacking fault. This procedure removes the depth oscillations arising from the positions of P and P' near the circles and gives the image profiles shown in Fig. 17 for a screw dislocation for various values of n = g b.
/I= 27rsx
FIG. 17. Intensity profiles of images of a screw dislocation for various values of n. After P. B. Hirsch et al. (29). Courtesy of the Royal Society.
We note from Fig. 17 that the image profile has a peak which lies to one side of the core of the dislocation and that the profile becomes narrower for increasing s. For n = 2 the peak occurs at x = (2ns)- and the half-width of the peak is Ax N (ns)-'. The side on which the peak lies is determined by the behavior of the phase terms in the integrand as mentioned above. The physical interpretation of this is that the peak lies on the side where the Bragg planes are tilted locally toward the reflecting position by the displacement field of the dislocation. This can be seen easily for the case of an edge dislocation (Fig. 18) where the opposite sense of tilt of the planes on either side of the dislocation is clear. The one-sided nature of the image of a dislocation when the foil is tilted away from the Bragg position is useful for
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
23
I I
0
0
0
0
~
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
FIG. 18. Arrangement of atoms around an edge dislocation in a simple cubic lattice. Note the opposite sense of tilt of lattice planes on opposite sides of the dislocation. After A. H. Cottrell. Courtesy of the Clarendon Press.
determining the Burgers vector of a dislocation (see Section III,C,2), while the narrow image profile at large s is used in “weak-beam” technique (33)of high-resolution imaging of defects (see Section 111,C,3).For further details of kinematical theory the reader should consult references (16) and (29).
4. Limitations of Kinematical Theory
Equation (26) shows that at the Bragg position (s = 0) I $g = n2t2/ri, i.e., the intensity of the diffracted beam increases quadratically with foil thickness t . However, it cannot become greater than the incident beam intensity (unity), and hence an upper estimate to the foil thickness is t = l g / nN 5,/3. The foil will have to be appreciably thinner than this for the kinematical theory to be a reasonable approximation at s = 0. For loworder reflections Table I1 shows that this usually requires the thickness to be less than 100 di. Most foils examined in the TEM are thicker than this, so evidently the kinematical theory is a poor approximation at s = 0. However, as noted for Fig. 9, the theory is reasonable for higher order reflections in moderately thin crystals. The kinematical theory is also a good approximation even for thick crystals provided the deviation from the Bragg conditions is sufficiently large. The requirement here is that t g s % 1. For example, in the weak-beam technique (Section 111,C,3) a value of res N 5 usually suffices to
-
M. J. WHELAN
24
obtain narrow kinematic images of dislocations (34). This means that the intensity in the tails of the curve of Fig. 6a approximates to that of the more exact dynamical distribution in Fig. 6b. B. Dynamical Theory of Electron Difraction 1. Wave-Optical (Darwin) Theory
In the theory a distinction is made between the two geometrical situations illustrated in Figs. 7a and 7b, which are known as the h u e and Bragg cases, respectively. The solutions of the theory have quite different forms in these cases. The Laue case is relevant to transmission diffraction, whereas the Bragg case is relevant to reflection diffraction, and historically was the case first studied by Darwin (13) using the wave-optical approach for X-ray diffraction. Bethe ( 1 ) subsequently developed a general theory based on the Schrodinger equation and this was applied by MacGillavry (35) to the Laue case. Heidenreich and Sturkey (36) and Heidenreich (27) used MacGillavry’s theory to explain thickness extinction contours. At a much later date the wave-optical method was revived by Howie and Whelan (24, 25) and applied to the two-beam Laue case. Sturkey (37) extended the method to the n-beam case. a. Equations of dynamical equilibrium. In the wave-optical method the crystal is divided into a number of slabs or layers parallel to the surface and the equations describing dynamical equilibrium between waves in a layer are formulated. In the Bragg case (Fig. 7b) these layers may be taken as the Bragg planes, while in the Laue case (Fig. 7a) the layers are perpendicular to the Bragg planes. We consider first the two-beam symmetrical Laue case where only the incident and one diffracted wave are appreciable. Let their amplitudes by 40(z)and +,(z) as a function of depth z in the crystal (Fig. 12). As the incident wave propagates into the crystal its amplitude is depleted by scattering into the diffracted beam and vice-versa and the equilibrium can be described by the following equations:
*
in
= - 4o
to 3 h = -in4 0 dz
+ in 4, -
exp (2nisz),
tg
exp(-2nisz)+-4,, in
dz 5, to where 5, is the extinction distance given by Eq. (24) and tois given by a similar equation with f (20) replaced by f (0).Equation (36a) states that in a layer of thickness dz in Fig. 12, c$o changes through forward scattering (first term on right-hand side) and through Bragg scattering from the diffracted
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
25
beam (second term on right-hand side). The factor i occurring in these equations represents a phase change of n/2 on scattering, which arises through the reconstruction of a plane wave from waves scattered by single atoms in a layer taking due account of phase. Such phase changes are well known in problems in physical optics. s is the parameter denoting deviation from the Bragg condition introduced in the kinematical theory. That these equations are plausible can be seen by integrating Eq. (36b) under kinematical conditions (40N 1, 4, small). For a foil of thickness t we obtain the result of Eq. (26). A variety of equations like Eqs. (36) may be obtained by making suitable phase transformations. Put
4bb) = 40(4 exp ( - inz/50)9 & J z ) = 4#(z)exp (2nisz - niz/t0).
(374 (37b)
Equations (36a) and (36b) then become
The solutions of Eqs. (36) and (38) differ only by phase factors and since these have no effect on intensities both systems of equations give the same results. The transformation (37a) is equivalent to correcting for the mean refractive index of the crystal because the directly transmitted wave 4,(z) exp (2nix * r) is re-expressed as &(z) exp [2ni(x * r + z/2t0)],where x is the wave vector of the electron wave in vucuo. The phase of the latter expression shows that the major part of the z dependence of q50 in the former expression is accounted for by a change of wave vector to K where K, = xz ( 1/2t0).K is the wave vector of the electron wave inside the mean inner potential of the crystal. The corresponding refractive index is
+
where N = V; is the number of “atoms” per unit volume. Typically p - 1 is of order for 100 kV electrons. b. Perfect-crystal solution. We solve Eqs. (38a) and (38b), but for convenience we drop the primes on &, &, Eliminating either & or c$# from Eqs. (38) leads to the following equation satisfied by both 4o and 4,:
26
M. J. WHELAN
A solution of Eq. (40) of the form exp (2niyz) exists where 2
y - sy - ( 1/2t,)z = 0,
(41)
i.e., there are two values of y:
JW), + JW).
$1)
= +(s -
(42)
y‘2’
= +(s
(43)
Consider first the root y(l). We have
where
w is a dimensionless parameter which denotes deviation from the Bragg reflecting position. It is usual to choose C, and C, so that IC,(’ + 1 C,1’ = 1. From Eq. (45)we then find
cp=
[:( -
l+J-)]
W
112 Y
l+w
Similarly for the root y(’),
cp= cp=
-J-)]
[:( [:( -
-
1
1 +J-2)]
W
112
l+w W
112
l+w
For each root y(i) the wave propagating in the crystal is of the form
Pi)(’)= c$g) exp (2niK r) + 4:) exp (2ni(K + g) r) = Cg)exp (2nikg) r) + Ct)exp (2ni(kg)+ g) * r),
(47)
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
27
where
(p) - K I + Y(0. 0 2 -
(53) There are two functions B(’)and B(’) corresponding to the two values of y. The functions B(’)are two-beam approximations to Bloch functions. c. The dispersion surface. Since by Eqs. (42) and (43) 7“) is a function of orientation of the wave vector K in the crystal, Eq. (53) shows that as the orientation varies, kg)will trace out a surface in k space which is known as the dispersion surface. There are two branches of this surface for the twobeam case corresponding to y ( ’ ) and y(’), and these are labeled (1) and ( 2 ) in Fig. 19. The vector A 0 represents the incident wave vector K. The locus of K is a sphere of radius K centered at 0. y(i) is the distance of A parallel to the z direction from the two branches. The Brillouin-zone boundary is a plane bisecting the reciprocal lattice vector g. In practice K z 50 g so that the sphere of radius K is practically a plane in the vicinity of the Brillouin-zone boundary. The two branches of the dispersion surface are then a hyperbolic surface of rotation about g asymptotic to the two spheres in Fig. 19. The dispersion surface is useful as a geometrical aid for describing Bloch wave vectors in the crystal. If the dispersion surface is known the crystal wave vectors kg)can be found simply by constructing a normal to the crystal surface through the end of K at A in Fig. 19. The intersections of the normal with the branches of the surface give the “wave-points” of the vectors kg). The concept of the dispersion surface can be generalized to the n-beam case, where more than one strong Bragg reflection is excited. The surface then consists of n branches asymptotic to spheres of radius K centered at the n reciprocal lattice points of the Bragg reflections. The form of the n-beam surface usually needs to be obtained by numerical computation. d. Solution in the h u e case. We introduce a useful notation due to S. Takagi for describing deviation from the Bragg reflecting condition. In place of w we introduce p defined by
w
= cot
p.
(54)
In Fig. 19, w = + co (p = 0) corresponds to A being far to the right of the Brillouin-zone boundary. w = 0 (p = ~ / 2 in ) the Bragg reflecting position when A is at the zone boundary. w = - 03 (/3 = A) occurs when A is far to the left of the zone boundary. s is taken as positive if the reciprocal lattice point g is inside the Ewald sphere. In terms of p
Cb” = C”’= costp,
(554
cb’) = -c(’) e = sin 4s.
(55b)
28
M. J. WHELAN Rrillouin zone g
r x/
IY\
(center 0 )
FIG. 19. Diagram illustrating the construction of the dispersion surface. The Ewald (reflecting)sphere is also shown. After A. Howie and M. J. Whelan (14). Courtesy of the Royal Society.
We now express the total crystal wave as $(r) = $(')W)(r) + $ ( ' ) P ( r ) ,
where B") is given by Eq. (52); i.e., $(r) = $(')Cb') exp (27rikg) r)
+ $(')CL2) exp (2nikb') . r)
+ $(')Cr) exp [27ri(kb') + g) + $(2)Ck') exp [2ni(kbz)+ g)
*
r] r].
(57)
Choose r = 0 at the top surface of the crystal. The first two terms on the right represent the direct wave; the second two terms represent the diffracted wave. Equating the first to unity and the second to zero at the top surface we obtain $'"C"' + p""= 1 (584 $(lql)
+ $(Z,C'Z' 9 = 0,
(58b)
29
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
which gives +(I) =
cos +p,
(59a)
+(2) = sin i p . (59b) The amplitudes of the direct and diffracted waves at the bottom surface of a crystal of thickness t are
Apart from an unimportant phase term, Eqs. (60a) and (60b) simplify with the aid of Eqs. (42), (43), (54),(55), and (59) to
The intensity of the diffracted beam obtained from Eq. (61b) is
where s is an effective value of s given by
s = JW Equation (62) is similar to the kinematical expression in Eq. (26) except that s is replaced by s. This has the effect of removing the divergence in the depth periodicity of the diffracted wave predicted by kinematical theory when s + 0 noted in Section II,A,2. At s = 0 the depth periodicity of (62) is 5 , and the form of the intensity oscillations of direct and diffracted waves is as shown in Fig. 1lb. This demonstrates the physical significance of the extinction distance t, as a characteristic oscillation distance arising from the coupling of direct and diffracted waves. The term “extinction” denotes the fact the direct wave is extinguished by scattering to the that in a depth diffracted wave. The expression (62) gives the form of the “rocking curve of the crystal on the two-beam dynamical theory. Figure 6b shows this curve schematically for a crystal of moderate thickness compared with the kinematical rocking curve of Fig. 6a. For t 6 <,, Fig. 6b approaches the kinematical curve. For t % t,, Fig. 6b consists of an envelope of width 2/t, containing rapid oscillations of periodicity As 2 t - ’ .
+<,
”
M. J. WHELAN
30
e. Solution in the Bragg case. We mention this for the sake of completeness and of comparison with the Laue solution. The geometry is shown in Fig. 7b. Let the crystal be semi-infinite and let x denote depth below the surface. Equilibrium equations corresponding to Eqs. (38a) and (38b) are
d40 in cot 0 dx
50
40
+
in cot ~
5,
e
4iJ '
(644
Proceeding in a manner similar to the Laue case we find that exp (- nisx cot
e) exp
[ ye J-] -
is a solution of these equations, where w = &(s + 5; '). The exact Bragg condition w = 0 now occurs at s = - 1/t0,and this is a result of refraction in the Bragg case. The amplitude of the wave reflected from the crystal is We see that for I w 1 < 1 the crystal wave decreases exponentially with depth and the incident wave is totally reflected. The rocking curve in the Bragg case is shown in Fig. 20. The width of the region of total reflection is 2/5,. This curve was first obtained by Darwin (13) for the Bragg case in X-ray diffraction. The physical reason for the difference between the solutions in the Bragg and Laue cases is easily seen from the dispersion surface of Fig. 19. In the
FIG.20. Rocking curve in the Bragg case of diffraction. The ordinate is the intensity of the reflected wave.
ELECTRON DIFFRACTION THEORY A N D ELECTRON MICROSCOPY
31
symmetrical Laue case the normal to the crystal surface through the end A of the wave vector K is parallel to the zone boundary, and real intersections with the two branches of the dispersion surface occur. Thus the wave vectors of the crystal waves are real giving rise‘to propagating waves. However, in the symmetrical Bragg case the normal to the crystal surface through A is perpendicular to the zone boundary. Hence there is a range of angle around the Bragg position where no real intersections with the two branches of the dispersion surface occur. The normal components of crystal wave vectors then have imaginary parts giving rise to evanescent crystal waves. The phenomenon is not unlike that occurring for total internal reflection of light at an interface. f: Equations for an imperfect crystal. Equations (36a) and (36b) can be generalized to the case of an imperfect crystal by use of the column approximation as in the kinematical theory. In place of Eqs. (36)-(38) we obtain
% = -40 in dz
5,
exp (-2nisz - 2nig * R ) + -in4 , , 50
where
Equations (66) and (68) are equivalent for the calculation of intensities. Equations (66a) and (66b) demonstrate that the displacement R ( z )enters the dynamical theory in the same way as in kinematical theory, while in Eqs. (68a) and (68b) the term p’(z), which represents tilting of the Bragg planes, acts to modify s locally. In the numerical calculation of intensities convenience determines which of Eqs. (66) and (68) should be used. If p’ is easily calculated Eqs. (68a) and (68b) are most convenient since these equations d o not involve the calculation of exponentials at each step of an integration.
M. I. WHELAN
32
2. Wave-Mechanical (Bethe) Theory a. Basic formulation. The starting point in this development of the theory is the Schrijdinger equation for the wave function $(r) describing the motion of the fast electron of energy eE in the crystal:
where -eV(r) is the crystal lattice potential. Put
where x is the wave vector of the fast electron in vacuum. Equation (70) then becomes V'$(r) 4n2[x2 U(r)]$(r) = 0. (73) Equation (73) also describes the relativistic case provided that x is a relativistic wave vector [i.e., I is calculated from Eq. (l)] and that m in Eq. (72)is the relativistic mass of the electron. V(r) and V(r) have the periodicity of the lattice and can be expanded as a Fourier series
+
+
Uh exp (27cih r) =
U(r) = h
2me h
1 V, exp (27cih - r),
(74)
h
where g and h are reciprocal lattice vectors. In the absence of absorption V(r) is real. Hence
u, = u!h
9
(75)
while if the origin of coordinates is at a center of symmetry, uh=
u-h= U t ,
(76)
so that uh is real. Frequently the center of symmetry coincides with an atom where the potential energy is a minimum, i.e., where V(r) has a maximum. In this case the important Fourier coefficients are positive. If the lattice potential is assumed to be a superposition of free-atom potentials (i.e., the effects of bonding is assumed to be negligible), Uh can be estimated from the formula
where I h I
=2
sin $ / I , V, is the unit cell volume, andfis given by Eq. (25).
ELECTRON DIFFRACTION THEORY A N D ELECTRON MICROSCOPY
33
The term exp ( - M h ) is the Debye-Waller factor which takes account of the smearing of the lattice potential caused by thermal vibrations. As noted in Section II,A,l, the atomic scattering amplitudesfj need to be calculated from the first Born approximation. Sincefin Eq. (4) contains the relativistic mass rn, Uh will increase with energy as rn/rno (Table I). This is also clear from the definition (72) since V(r) is independent of E. Solutions of Eq. (73) exist in the form of Bloch functions
-
IC/(r) = C C,(ko) exp (2nikg r), Q
(78)
where k, = k, + g (see Fig. 19). Substitution of (78) and (74) in (73) gives after manipulation
Equating the coefficient of each exponential in this sum to zero we obtain the dispersion equations
This infinite set of equations (one for each g ) can be regarded from two viewpoints. The first (adopted by the solid-state theorist) regards k, as given and solves (79) as an eigenvalue equation for E, which from (71) is determined by x 2 . The coefficients C,(k,) in Eq. (78) are obtained as eigenvectors of Eq. (79). The second regards E or x 2 as predetermined and calculates the wave vectors k, of Bloch waves with the given energy E. This viewpoint is the one adopted in dynamical theory where the energy of the fast electron inside the crystal is determined by the accelerating potential, while interference effects occur between various Bloch wave vectors kg’ characteristic of this energy. It is customary to rewrite Eqs. (79) as
where
This amounts to changing the vacuum wave vector x to K, the crystal wave vector corrected for mean refraction. The refractive index obtained from (81) is (for U , small in comparison with x 2 )
Using Eq. (77), Eq. (82) is seen to be equivalent to Eq. (39).
34
M. J. WHELAN
Although Eqs. (80) are infinite in number, each containing an infinite number of terms C , , in practice an approximate solution with a finite number of terms must be employed. If n equations each with n terms C , are used, this is referred to as an n-beam case, where, apart from the direct wave, n - 1 diffracted waves will be taken into account. b. Boundary conditions. Let the normal to the crystal surface be the z direction. The incident vacuum wave vector x has components (xZ,x,) as shown in Fig. 21. A necessary boundary condition is that all direct crystal wave vectors have the same component tangential to the surface as that ofx.
FIG.21. Matching of wave vectors at the crystal-vacuum interface.
Thus K,, kor are equal to x, and these wave vectors differ only in their normal components. Equations (80) are quadratic in k& (see the Appendix). Hence for the n-beam case there will be 2n values of k O L .Their physical significance can be seen from Fig. 22 which refers to the two-beam case. The normal along z through the end of x will not only intersect the upper branches 1 and 2 as discussed in Section II,B,l,c, but it will also make intersections with branches 1’ and 2’ which correspond to Bloch waves which propagate backwards. To a good approximation backwardpropagating waves can be neglected in highenergy transmission diffraction (see below), so there will be only two forward-propagating waves. Similarly in the n-beam case only n forward-propagating waves need be considered.
35
ELECTRON DIFFRACTION THEORY A N D ELECTRON MICROSCOPY
FIG.22. Schematic diagram illustrating the branches of the dispersion surface in the twobeam case. Branches 1 and 2 correspond to forward-propagating Bloch waves, while branches 1' and 2' correspond to backward-propagating waves.
This is a considerable simplification in the highenergy case, since the neglect of back-scattered waves enables the boundary conditions to be applied at successive plane interfaces consecutively and independently. The boundary conditions are also simplified. For the complete system of 2n waves there must be 2n boundary conditions. These are the usual conditions arising from the continuity of current at an interface, namely and d+/dz must be continuous. There will be n conditions arising from continuity of one for each plane wave, and similarly n conditions arising from continuity of d+/dz. These are the necessary 2n conditions. However when back-scattered waves are negligible, only the n conditions arising from the continuity of t+h need be considered to determine the n forward-propagating Bloch waves. We illustrate this with reference to the two-beam case for the crystal-vacuum interface. Consider the situation shown in Fig. 23. Let the amplitude of the incident wave be unity. The specularly reflected wave has amplitude R , and the reflected diffracted wave has amplitude R,. The crystal Bloch waves corresponding to branches 1 and 2 of the dispersion surface in Fig. 19 have amplitudes + ( I ) and @'). Continuity of gives
+
+,
+
36
M. J. WHELAN
FIG.23. Diagram illustrating the waves generated at a plane interface by an incident wave of unit amplitude. (a) Real-space diagram showing the incident wave of wave vector x and unit amplitude, the surface reflected waves R,, R , and the transmitted crystal Bloch waves I//’) and +(2’. (b) Ewald sphere construction for determining the wave vectors of vacuum waves in (a).
where k‘”, k@’arethe Bloch wave vectors and -x: is the z-component of the wave vector of the reflected diffracted wave R e . Since we have four equations and four unknowns ($(”, ,!()’, R o , R B )the equations are soluble. At high energies however, g is small compared with x. Hence the z components of all the wave vectors in Eqs. (84a) and (84b) are very nearly equal. Equations (84a) and (84b) then reduce approximately to ,,!,(l)chl) + ,,!,(2)C‘2)= 1 - R 01 (85a) ,,!,(l)ChI) + ,,!,(2)C(2) =
-R
(85b) Comparison of Eqs. (85a) and (85b) with Eqs. (83a) and (83b) shows that Ro = 0; Re = 0. Thus reflected waves are negligible and the boundary conditions 8
,,!,(l)Chl) + $(Z)C&2’= 19
$(l)cy) + $WC‘2’ 9 =0
8 ’
(86a)
(86b) derived from continuity of $ alone suffice to determine ,!,(’) and $(’). Equations (86a) and (86b) appeared previously in the wave-optical approach [Eqs. (58)l. The above discussion can be generalized to the n-beam case [see Pendry (38)].
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
37
c. Solution in the n-beam h u e case. The general method of solution of the equations in expression (80) is given in the Appendix, although in practice considerable simplification occurs at high energy in the case where all important g vectors are parallel to the surface of the crystal (i.e.,g2 = 0). This case can be considered as the n-beam generalization of the two-beam symmetrical Laue case. If the important g points lie on a single row of the reciprocal lattice, we have the case of “systematic” reflections. They may also form a two-dimensional net known as the first Laue zone. In the following the normal components of vectors and their components tangential to the crystal surface have subscripts z and t , respectively. We drop the subscript on k,. Using the facts that g, = 0 and that k, = K,, it is easily shown that in Eq. (80) K’ - k,Z = Kf - kf - (2K, * g + g’). (87 ) Since K , , K, are fixed by the incident beam direction, Eq. (80)can be written as an eigenvalue equation
where
The diagonal elements of the matrix a are functions of orientation of the incident beam. From the condition (75) a is an Hermitian matrix and has real eigenvalues (i.e., kf is real), while if condition (76) is satisfied a is a real symmetric matrix. If there are n important g vectors, the matrix a is n x n, there are n eigenvalues k;)’ - K f ( i = 1 ... n), and n eigenvectors Cr). The eigenvalues give 2n Bloch wave vectors k k:) corresponding to forward- and backward-propagating waves. We also have the result
arising from the orthogonality and normalization of eigenvectors belonging to different eigenvalues. Further simplification occurs at high energy where the radius of curvature of the Ewald sphere is large and the Bragg angles are small. The geometry of Fig. 19 then gives - (2K,
- g + g’) * 2K,sg .
(91)
38
M. J. WHELAN
We also have k: - K : '2K,y,
(92)
where y=k,-K,. (93) These approximations hold when y and s are small in comparison with K , which implies that the Bragg angles are small and that K , on the right of Eqs. (91) and (92) can be replaced by K. In place of Eq. (88) we then have
where
The two-beam extinction distance is given by
5,
= K cos O,/V,,
(96) where 0, is the Bragg angle for the reflection g. In view of Eq. (77), this definition is seen to be equivalent to that of Eq. (24) iff(20) is modified by exp ( - M e ) to take account of thermal vibrations. The matrix A is also Hermitian and its eigenvalues y") give the n forward-propagating Bloch waves with wave vectors k!) = K 2 + (97) As the orientation of the incident beam varies (i.e., K, and sg varies) the vectors (k?), K,) will trace out the n branches of the dispersion surface. The formal solution in the Laue case is easily obtained if y") and Cjli) are known. These may be calculated from the matrix A,, by digital computer using standard matrixdiiagonalization procedures. Consider a crystal slab as shown in Fig. 24. Let the amplitudes of waves incident on the top surface and transmitted through the bottom surface be 4; and 4,, respectively. If the amplitude of the ith Bloch wave is i,bcn,the boundary conditions at the top and bottom surfaces give
C cjli)i,b(O= ,#,;,
(98a)
i
1 Cjli) exp (27~iy(~)t)i,b(~) = 4, . i
(98b)
Equations (98a) and (98b) give the following matrix relation between the n-column vectors Q and Q' representing the amplitudes 4, and @:,
Q = C[exp (2niy")t)],C- '4'
= PQ',
(99)
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
39
FIG.24. Illustrating the amplitudes ## and da of waves above and below a slab crystal of thickness t.
where Cgi(= C:)) is a unitary matrix and the subscript d denotes a diagonal matrix. With 4' = [l 0 0 ...I, Eq. (99) is the formal solution for the diffracted beam amplitudes for an incident beam of unit amplitude. The matrix P is known as the scattering matrix. d. The two-beam symmetrical h u e case. In this case only the direct and one diffracted beam g are important and from Eqs. (95a) and (95b) the matrix equation (94) is
where 5, is given by Eq. (96). In the two-beam symmetrical h u e case this equation is not restricted to small Bragg angles. For Eq. (100) to have nonzero solutions for C , and C , , the determinant of the matrix must vanish. This gives y2 - sy - (1/25$ = 0. We have already met this equation in the wave-optical treatment, and the eigenvalues y(l) and y'2) are given by Eqs. (42) and (43). The first of Eqs. (100)gives C$)/C$)= 2t, y") [cf. Eqs. (45) and (4911. Proceeding as in the wave-optical treatment and using the notation (54), the matrix C is given by
40
M. J. WHELAN
It is easily shown from Eqs. (42), (43), (W), and (101) that for unit incident beam amplitude (4' = [ LO]) the amplitudes 4o and c$8 transmitted through a slab crystal of thickness t are given by Eqs. (61a) and (61b). The wavemechanical treatment therefore gives results identical with the wave-optical treatment. e. Imperfect crystal. The simplest case to consider is that of a stacking fault which is the planar interface between two slabs of crystal as in Fig. 25, which have a relative displacement R which is not a lattice vector. As thus
FIG. 25. Illustrating two superposed slab crystals 1 and 2. If the relative shear R between them is not a lattice translation vector a stacking fault is formed. After P. B. Hirsch et al. (16). Courtesy of Butterworths.
envisaged the fault is parallel to the surface, but the results for this case can also be applied from point to point of an inclined fault by use of the column approximation (Section 11,,4,3). Let the upper slab 1 be fixed and the lower slab 2 be displaced by R. If Bloch functions in slab 1 are represented by Eq. (78), those in slab 2 are IC/(r) =
c C,(ko) exp ( W k O + g)
*
(r - R))
8
= exp ( - 2niko
*
R)
C C8(ko)exp ( - 2nig * R ) 8
x exp (2nik8 * r).
(102)
The phase factor exp (-2xik0 * R) can be removed since it is independent of g, and we see that the C i s in slab 2 are obtained by multiplying those in slab 1 by exp ( - 2nig R). The matrix C2 for slab 2 is therefore Q; 'C, where C is
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
41
the matrix for slab 1 and Q2 is a diagonal matrix whose elements are exp (2nig * R). The scattering matrix for slab 1 is, from Eq. (99),
P, = C[exp (2niy"'t,)]dC-1,
(103)
while that for slab 2 is
P2 = Q; 'C[exp (2niy"'t2)],C- 'Q2 ,
(104) where tl and t 2 are the thicknesses of the slabs. The scattering matrix for the pair of crystals is p = P2(t2)P1(11). (105) Evidently this procedure can be generalized to an assembly of slabs, the scattering matrix of slab n being
P,
= Q;
'C[exp (2niy(i)t,)],C- 'Q, .
(106) If the thickness t , approaches an infinitesimal quantity 6z, Eq. (106) can be expanded to first order in 6 z :
P, = I 4-2niQ;
lc($i))dc'Q, 62.
(107) Now c(y"')dc- = A, where A is given by Eq. (95) for the undisplaced slab 1. From Eq. (107) we obtain the rate of change of the wave-amplitude column vector
+:
dz
= 2niA(zM,
where A(z) = Q-'(z)AQ(z). (109) Equation (108) generalizes the two-beam equations obtained in the waveoptical theory [Section II,B,lf]. Defining +' = @, Eqs. (108) and (109) give
3 = 2ni(A + (&)&', dz where (&), is a diagonal matrix whose elements are given by Eq. (69). Equation (110) is the generalization to the n-beam case of Eqs. (68a) and (68b). The transformation from to does not affect intensities 1 +g l2 since the elements of Q have unit modulus. The transformation is analogous to the transformation of Eqs. (67a) and (67b). Equation (108) or (110) can be used to calculate the diffracted n-beam intensities near a dislocation line where R(z) and hence p,(z) vary continuously along the column in Fig. 12. It is instructive to evaluate the wave amplitudes in the two-beam case for
+ +'
M. J. WHELAN
42
a stacking fault using the scattering matrix (105). The pair of crystal slabs is shown in Fig. 25 together with the coordinates used. Using Eq. (101), for slab 1 we have
For slab 2 we have
)(
0 sin +/3 exp (2;iy(')t2) -sin +/3 e-'" cos +/3 e - i a exp (2niy(2)t2)
fa cia)(
cos +/3
- sin
sin +/3
cos +/3 eia
40(f1)
&,(tl)
)
, (112)
where a = 2 ~ gR. Combining Eqs. (111) and (112) and putting 40(0)= 1, +,(O) = 0, corresponding to an incident wave at the top, we can multiply out the matrices. The following result is obtained-omitting an unimportant phase term exp [ai(y(') + y ( 2 ) ) t ) ] :
4,(t) = cos (n Akt) - i cos /3 sin (A Akt) + 4 sin2 /3(eia- 1) cos (n Akt) - 4 sin' /3(eia- 1) cos (271 Akt'),
4,(t) = i sin /3 sin (A Akt) + 4 sin /3(1 - e - i a ) [cos /3 cos (n Akt) - i sin (n Akt)] - 3 sin B(1 - e-iu) [cos /3 cos (ZA Akt') - i sin (2n Akt')],
(113a)
(113b)
where t' = t l - i t is the distance of the fault from the center (Fig. 25) and where
Ak = ,/W/c&.
(114)
For an inclined fault, t' in Eqs. (113a) and (113b) varies across the fault giving rise to intensity oscillations in the form of fringes of constant thickness. The depth periodicity of the fringes is At' = Ak- = <= /,. Furthermore if a = 0 or 2nn, Eqs. (113a) and (113b) reduce to the perfect crystal result [Eqs. (61a) and (61b)], i.e., the stacking fault is invisible. Applications of Eqs. (113a) and (113b) ®iven in Section II1,B.
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
43
C. Absorption Due to Inelastic Scattering Inelastic scattering involves a transfer of energy between the fast electron and the specimen, the total energy and momentum of the system being conserved. The specimen can be considered as a single atom or as an assembly of atoms. In the latter case some inelastic processes are essentially characteristic of the constituent atoms, while others are characteristic of the collective behavior of the assembly. It is therefore useful to discuss first the types of inelastic scattering processes which occur in single atoms and crystals. Inelastic scattering gives rise to absorption of the primary wave by virtue of the fact that electrons may be scattered outside the objective aperture (Section I) by inelastic processes, and are thus lost from the image. If ginis the total atomic inelastic cross section for scattering outside the aperture, the contribution to the absorption coefficient for the primary beam is where N is the number of atoms per unit volume. If the solid is truly amorphous, the elastic scattering cross section g,, for scattering outside the aperture as given by Eq. (11) also contributes to absorption of the primary beam so that However, in a crystal the elastic scattering interferes to produce Bragg reflections, and while these cause intensity variations in the primary beam (e.g., Figs. 9 and 10) such intensity changes are referred to as “extinction” rather than “absorption ” effects. For extinction the intensity removed from a beam by scattering to other Bragg beams is mainly returned after a certain propagation distance so that no true absorption occurs. Inelastic processes, on the other hand, scatter intensity away from the Bragg beams and since such intensity does not generally return to the Bragg beams, this is referred to as “absorption.” In this case the mean absorption coefficient is given by Eq. (115) only. The crystal periodicity also gives rise to additional special effects whereby absorption depends on the particular Bloch wave symmetry. The mean absorption coefficient is then insufficient to describe such “anomalous” transmission effects, which are reviewed in Section II,C,2. 1. Inelastic-Scattering Processes
a. Single-electron excitation. The fast electron excites the specimen system (either a single atom or a solid) to a higher energy state by raising an electron of the system to a higher energy level. For a single atom the possible
44
M. J. WHELAN
transitions are between occupied and unoccupied single-electron states of the atom, transitions to occupied states being forbidden by the exclusion principle. Thus a hole is created in an inner shell, the ejected electronappearing in an unoccupied higher energy state or in a continuum state corresponding to ejection from the atom. Such transitions are referred to as interband transitions. In a solid the transitions take place between states in various energy bands. The deeper lying bands will be narrow corresponding to the inner energy levels of the constituent atoms whereas the higher bands are broad. For metals where there is a partially filled band of conduction electrons, intraband transitions can occur in addition to interband transitions. These correspond to excitations between states below and above the Fermi level in the conduction band. In the Born approximation the differential scattering cross section for an inelastic process in which an atom (or solid) is raised from its ground state 0 to an excited state n is given by
where qno = k - k is the momentum change of the fast electron in the process k k', a. is Bohr's radius, and
The $'s are normalized one-electron wave functions of the atom (or solid). Equation (117) is a generalization to inelastic scattering of the results of Eqs. (3) and (6) for elastic scattering. Hartree-Fock type wave functions are available for a number of atoms which enable estimates of inelastic cross sections to be made. For a solid, detailed calculations are difficult because of lack of knowledge of the wave functions in the solid. However, singleatom results can be applied to a solid if a tight-binding model is assumed applicable. While this is of course a crude approximation when applied to the wave functions of valence and conduction electrons, it is often used in the absence of a better approximation. For single atoms the total cross section ginfor inelastic scattering can be obtained by summing Eq. (117) over all n. If qnois approximated by an average wave vector change q, independent of n, the sum can be evaluated using the fact that the $,'s form a complete set. We obtain the Morse formula
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
45
where
In Eq. (120) the sum is over occupied states only. ij is the average wavevector change for inelastic scattering at a scattering angle 28 equal to zero. Values ofJj and S for a number of elements are provided by the work of Freeman (39-41) on Compton scattering. These may be used to estimate inelastic cross sections (42). A useful estimate of S ( q ) is the Raman-Compton formula S ( q ) = Z - F z / Z , where F is the X-ray scattering factor (Section II,A,l). Using a screened Coulomb model for F, integration of Eq. (119) gives in the notation of Section II,A,l
Since the logarithm term varies slowly with Z, binwill vary likef(O), which on the Fermi-Thomas model of the atom varies as Z1I3.For an aluminum atom using the figures given in Section II,A,l and assuming q = 1.6 x A-’, corresponding to an average loss of about 10 eV, we find that bin‘Y 8 x 10cm2, i.e., almost twice the total elastic scattering cross section estimated in Section II,A,l. Comparison of Eqs. (13) and (122) shows that apart from the logarithm term, the ratio be,/binvaries as 2. The ratio is about unity for an atom of medium Z such as copper (44). We therefore see that inelastic scattering predominates over elastic scattering for light elements and vice-versa for heavy elements. The angular distribution of single-atom inelastic scattering has a halfrad. Inelastic scattering is therefore very sharply peaked in width q/k the incident beam direction. For example, using the data of Section II,A,l, we find that about 80% of the inelastic scattering of an aluminum atom passes through a 20 pm diam. objective aperture. This compares with about 3% for elastic scattering (Section II,A,l). b. Plasmon excitation. It is well known from the work of Bohm and Pines (45) that the electrons in a solid can undergo collective or plasma oscillations of fairly definite frequency. For a free-electron plasma the frequency is given by
’*
-
where n is the number density of electrons in the plasma and m and e are the
46
M. J. WHELAN
electron mass and charge. An incident fast electron is able to excite quanta of these oscillations, known as plasmons, of energy
E, = ha,. (124) Typically the plasmon energy E , is a few tens of volts, and certain materials cause very sharp plasmon losses in the energy-loss spectrum of a beam of transmitted electrons. For example, aluminum has a loss at about 15 eV 1 eV. Other examples of materials with with a half-width of the peak sharp losses are magnesium (10.5 eV) and silicon (17.2 eV). On the other hand, carbon exhibits a broad plasmon peak in the energy-loss spectrum at about 24 eV, the broadening being attributed to the short plasmon lifetime. In thicker specimens multiple losses due to repeated plasmon excitation occur. The intensity of the energy loss nE, in a specimen of thickness t is given by a Poisson distribution [Marton et al. (46)]:
-
where 1 is the mean free path for plasmon excitation. Comparison with experiment (46) enables 1 to be estimated. For aluminum 1 is about 1500 A for 100 keV electrons. The differential mean free path (or differential cross section per unit volume) for scattering through an angle 20 is [Ferrell (47)]
where 4E= E,/2E is the half-width of the angular distribution and E is the incident beam energy. Since E , is of the order of 10 eV, 4E is of order rad at 100 keV. The narrow angular distribution arises essentially from the long-range nature of the plasmon interaction. In this respect the scattering is similar to that caused by singleelectron excitations. In a crystalline specimen, where Bragg reflections occur, each Bragg beam is subject to small-angle inelastic scattering due to plasmon and singleelectron excitations. c. Phonon scattering. An incident fast electron may also be scattered inelastically by a solid through the creation or destruction of quanta of lattice vibrations or phonons. Again energy and momentum must be conserved, but the energy transfer is very small in comparison with the incident beam energy. Typical phonon energies are less than kOD, where k is Boltzmann’s constant and OD is the Debye temperature, i.e., less than 0.05 eV in many materials. Thus the energy loss is negligible and such scattering is referred to as “ quasi-elastic.” In fact, in X-ray diffraction, phonon scattering
ELECTRON DIFFRACTION THEORY A N D ELECTRON MICROSCOPY
47
is usually treated by ignoring the lattice dynamics and calculating the timeaveraged elastic scattering from the lattice with frozen-in ” lattice displacements caused by thermal vibrations. This is referred to as thermal diffuse scattering, and the terms thermal diffuse scattering and phonon scattering are synonymous. The former description, however, illustrates the point that phonon scattering usually occurs through large angles. This is due to the facts that phonons with large wave vectors in the Brillouin zone are more easily created or destroyed and that in crystals Umklapp processes can occur where the fast electron is simultaneously Bragg reflected. Electrons that have suffered phonon scattering therefore appear in the diffraction pattern as diffuse scattering between Bragg beams. The diffuse scattering itself can also exhibit diffraction effects such as Kikuchi lines and bands (see Fig. 3). The diffuse nature of phonon scattering implies that such scattering is normally outside the angular range of acceptance of the objective aperture in an electron microscope and such scattering therefore gives rise to absorption. Theoretical treatments of phonon scattering have been given by Yoshioka and Kainuma (48),Whelan (43), and Hall and Hirsch (49). “
2. Inelastic Scattering by Crystals a. The complex lattice potential. Yoshioka (50)first showed in a formal quantum-mechanical treatment that the effect of inelastic-scattering processes on the elastic scattering by crystals could be represented by the addition of a small imaginary part V(r) to the crystal lattice potential V(r), which gives rise to absorption of the elastic wave in the same way as a complex refractive index gives rise to absorption of an electromagnetic wave. A similar mechanism was postulated by Moliere (51) for the Bragg case and a similar theory was developed for X-ray diffraction by von Laue (52) to explain the experiments of Borrmann (53).The imaginary part of the lattice potential in general has the periodicity of the lattice. This leads to certain special effects for crystals which we shall now outline. The effects are known as “anomalous” absorption or transmission, or in the case of X-ray diffraction as Borrmann transmission. The theory to be outlined should be regarded as phenomenological and as a first approximation only, since it does not treat in detail the effect of small-angle inelastic scattering which passes through the aperture of the electron microscope. Such small-angle scattering may produce image contrast which has to be added to that due to elastic scattering. For example, electrons scattered through small angles by plasmons and singleelectron excitations are expected to produce contrast similar to elastically scattered electrons [Howie (54), Humphreys and Whelan (55)], whereas those scattered through small angles by phonons
48
M. J. WHELAN
may produce little or no contrast. Clearly the situation is complicated and the absorption itself can be a function of the objective aperture size. The imaginary part V(r) is typically of magnitude & to fb of the real part V(r). A scaled potential U'(r) can be defined by a relation similar to and Vi Eq. (72)and the imaginary potential will have Fourier coefficients V;, given by an expression like Eq. (74). Absorption parameters are defined by relations like Eq. (96), i.e.,
-
re= K cos i9,/Un .
(127)
It is also clear from a relation like Eq. (77) for the imaginary part of the potential that absorption can also be described in the wave-optical approach by allowing the scattering factorf(20) to have an imaginary part. However, it should be emphasized that this description is valid only when the imaginary part is attributed to inelastic processes and not to phase-shift effects due to higher Born approximations as mentioned in Section II,A,l. b. Anomalous transmission. The absorption parameter t:, determined from Eq. (127) by the Fourier coefficient U:, is responsible for an average absorption coefficient which is independent of direction of propagation and affects all Bloch waves equally. This is easily seen from the wave-optical approach (Section II,B,l,a) where it was shown that refraction arises through a change in the z component of the wave vector by l/2to.If l/to becomes complex via the substitution l l -+-+;, to to
i to
the z component of the Bloch wave vector develops an imaginary part i/2&, and this leads to an exponential decay of wave amplitude, with an absorption coefficient for amplitude given by
In a crystal, however, the periodic complex potential will exhibit peaks in those regions of the unit cell where absorption occurs, i.e., in the vicinity of atomic centers, and such peaks give rise to special absorption effects which are dependent on the particular Bloch wave symmetry and on the direction of the wave vector of the Bloch wave. Since the extent of excitation of a particular Bloch wave depends on the direction of the incident wave, the effective absorption coefficient of the incident wave depends on orientation. The effect is referred to as anomalous absorption or anomalous transmission, and the physical explanation of the phenomenon in the two-wave case is illustrated in Fig. 26, which shows the symmetry of the two Bloch waves at
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY type !wave
49
type 2 wave
.A
reflecting planes
FIG,26. Schematic diagram illustrating the nature of Bloch waves in the two-beam case at the Bragg reflecting position. The current flow is parallel to the reflecting planes. Absorbing regions at atoms are shaded. The type-2 wave is absorbed more than the type-1 wave. After H. Hashimoto et a/. (59). Courtesy of the Royal Society.
the reflecting position (w = 0, B = 42). From Eqs. (47), (48) and (50)-(52) we find that the Bloch functions B(') and B(') can be written as
B") = - i@ exp (2ni I kill + fg 1 z) sin ngx, 8'') = @ exp (2ni I k;')
+ Q Iz) cos ngx,
(130)
(131) where x is the coordinate normal to the reflecting planes in Fig. 26 and z is the coordinate parallel to these planes. The current flow corresponding to Eqs. (130) and (131) is in the direction of kt)+ ig, parallel to the reflecting planes, and is modulated sinusoidally in the x direction as shown in Fig. 26. It is clear that if the absorption is localized near the atoms (shaded regions in Fig. 26) the type-1 Bloch wave will be absorbed less than the type-2 Bloch wave, since the former has its peaks localized in regions of the lattice where the absorptive potential is small, whereas the latter has its peaks coincident with the regions of strong absorption.
M. J. WHELAN
50
c. Dynamical theory including absorption. A simple approach to the formal theory of anomalous transmission is via the wave-optical theory outlined in Section 11,BJ. The basic equations are obtained from Eqs. (36a) and (36b) by the substitution (128) and a corresponding substitution for
mg:
l l i -+-+--.
5,
(9
r;
With these substitutions Eqs. (37), (38), and (40) become
( 134a) (134b)
The last equation is satisfied by both Faand @, . As before, we try a solution of Eq. (135) of the form exp (2niyz) and find that y") = f [ s - (1 -t w z fi
+ 2i(e/t;)1'z/&J,
(w - JFii7)/25, - i/(2cgJCL7),
(136a)
+ JCi7)/29, + i / ( 2 5 ; J W ) (136b) We have neglected ti-' and assumed that (,/cg is small compared with y") = (w
unity. Thus from Eqs. (133a) and (133b)we find that the amplitude absorption coefficients for the two Bloch waves for branches 1 and 2 of the dispersion surface (Fig. 19) are given by (137a) (137b) Providing is positive, which is the case when peaks in the imaginary potential coincide with atomic centers of symmetry, we see that the Bloch wave on branch 1 of the dispersion surface is absorbed less than that on branch 2. An extreme effect occurs when = <,; for then the absorption of wave 1 becomes zero at w = 0. This corresponds to the complex potential
ELECTRON DIFFRACTION THEORY A N D ELECTRON MICROSCOPY
51
varying like 6 functions at the atomic centers. In this case the absorption is localized very close to the atomic centers in Fig. 26, and it is clear that transmission of wave 1 with zero absorption can occur. The situation is closely approached in X-ray diffraction where the absorption is due mainly to photoelectric processes in the K shell. For electrons it has often been assumed in calculations that 5: = tb and that = 105%.Absorption parameters for electrons have been estimated by Hashimoto (56) and by Metherell and Whelan (57) using experimental profiles of thickness extinction contours. For aluminum t: 15%is about 30. Theoretical calculations of for various elements as a function of temperature and g have been made by Humphreys and Hirsch (58). The n-beam formulation of dynamical theory for the Laue case given in Section II,B,2,c is easily generalized to cover absorption if the latter is assumed to be a small perturbation. In this case the matrix A,, in Eq. (94) is replaced by
+ iAbh
7
(138)
where Abh = Ub-h/2K. (139) If the absorption is small Abh is small compared with A g h ,and the eigenvector matrix Cgi(= Ct)) obtained by diagonalizing the matrix (138) hardly differs from that obtained in the absence of absorption. Thus to a good approximation C is obtained by diagonalizing the matrix A,, and has the property
C ' A C = (f))d, (140) where d denotes a diagonal matrix. However, absorption does affect the eigenvalues y") which develop imaginary parts $). q(') may be calculated approximately, by performing the operation (140) on the matrix A of Eq. (139). The diagonal elements of the resulting matrix are identified with q(". We obtain the following expression:
This equation can be used to calculate the amplitude absorption coefficients 2mfi) for the various Bloch waves in the n-beam case. Equation (141) may be evaluated in the two-beam case using Eqs. (55a) and (55b) for Ct).The result is easily shown to be identical with that of Eqs. (137a) and (137b). d. The h u e solution with absorption. In the two-beam theory the only modification of the previous development (Section II,B,l,d) required is the introduction of exponential attenuation factors in the Bloch functions of
52
M. J.
WHELAN
Eq. (52) corresponding to the absorption coefficients of Eqs. (137a) and (137b), since, as discussed in the previous section, the Ci)are approximately unchanged by the introduction of absorption. It is easy to show that Eqs. (61a) and (61b) are replaced by $,(t) = exp (-nt/&)
iw
( 142a) (142b)
where
Similarly the n-beam matrix theory of Sections II,B,2,c and e is appliwhere q(’) is given by cable provided y“’ in Eq. (99) is replaced by y(” + Eq. (141). In particular Eqs. (1 13a) and (1 13b) for a stacking fault are applicable if the arguments of the sine and cosine functions are made complex by the substitution
and by including a mean-absorption term exp (-nt/tb). Examples of calculated fringe profiles are given in Section II1,B. 111. APPLICATIONSTO ELECTRON-MICROSCOPE IMAGE CONTRAST OF CRYSTALLINE MATERIALS The mechanism of image contrast in crystalline materials examined in the TEM was outlined in Section I. We now proceed to give some examples of how image contrast can be computed from the theory developed in Section 11, and how it may be used to interpret TEM images of crystalline materials. A. Perfect-Crystal Contrast Effects-Extinction Contours
By the term “perfect crystal” we mean a crystal containing no lattice defects such as dislocations or stacking faults, but which may be subject to elastic bending and may be of wedge-shaped cross section. In this case the crystal exhibits extinction contours due to thickness variation or bending, and these have been discussed on the basis of the kinematical theory in Section II,A,2. We now examine the profiles of extinction contours predicted
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
53
by the dynamical theory as outlined in Section II,B,l,d, with the modification to include anomalous absorption as outlined in Section II,C,2,d. The basic equations for the calculation of bright-field and dark-field images are Eqs. (142) and (143), and Fig. 27 shows examples of the profiles of bend extinction contours computed from these equations. The ordinates are the intensities of the bright-field and dark-field images, 1 &, l2 and 1 4, 12, shown by full and broken lines, respectively, as functions of w, the deviation parameter in Eqs. (142a) and (142b) for a crystal thickness t = 45,. The broken curves are symmetrical about the intensity axis and are not shown for negative w. Figure 27a is for the case of no absorption while Figs. 27b and 27c show the effect of increasing absorption represented by the parameter &. In all curves = (;, corresponding to an extreme anomalous absorption effect as discussed in Section II,C,2,c, where the type-1 Bloch wave has zero absorption at w = 0. In Fig. 27a we see that the bright-field and dark-field images are complementary, as expected for zero absorption. With increasing absorption the bright-field profile becomes asymmetrical about w = 0. The dark-field profile, however, remains symmetrical. We also note that the amplitude of the oscillations of the curves decreases with increasing absorption. Since the oscillations are caused by interference between waves on branches 1 and 2 of the dispersion surface in Fig. 19, the decreasing amplitude reflects the fact that the type-2 wave is being removed preferentially by anomalous absorption. The width of the reflecting curve (rocking curve) of the dark-field image also differs considerably from that predicted by kinematical theory, which we saw in Section II,A,2 was given by 2/t (Fig. 6a). The reflecting curve on the dynamical theory consists of oscillations of approximate spacing lit modulated by an envelope of half-width 2/<, as shown schematically in Fig. 6b. When absorption occurs the oscillations in Fig. 27 are reduced in amplitude, and the width of the reflecting curve is given roughly by 2/<,. In contrast to the result of kinematical theory, this width is independent of crystal thickness. It is as though the crystal, when thicker than a n extinction distance <,, behaves in the dynamical case as though it were a crystal of thickness 5,. Figure 28 shows an example of a bright-field image of a bend extinction contour due to the 111 Bragg reflection in an aluminum foil whose thickness is increasing from left to right. Since the Bragg angle for a 111 reflection is small, such contours are due to pairs of reflections 111 and ITT, corresponding to reflections from opposite sides of the Bragg planes. Unlike the case for high-order reflections shown in Fig. 9, where the pairs of contours are well separated, for low-order reflections the bright-field images of each contour merge. The resulting bright-field image profile therefore consists of a pair of
<,
<:
M. J. WHELAN
54
(a
,
I
-3
-2
-I
0
-3
-2
-I
0
I
2
3
2
3
w
FIG.27. Rocking curves computed by the two-beam theory for a crystal thickness t = 4((,. Full curves refer to bright field; broken curves refer to dark field. (a) No absorption; (b) = 0.05; (c) to/(;= 0.1, = (;. After H. Hashimoto et al. (59). Courtesy of the Royal Society.
(#/re
rb
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
55
curves such as in Fig. 27b placed back-to-back and merging around w = -2. This gives a bend contour in the form of a dark band with subsidiary fringes visible in thin regions (Xin Fig. 28) but not in thick regions ( Y in Fig. 28). It is clear from Fig. 28 that good transmission in a thick crystal is obtained on either side of a dark bend extinction contour. This good transmission is due essentially to the fact that for this orientation, mainly the
FIG.28. Bright-field image of a bend extinction contour in a region ofvarying thickness of an aluminum foil. A : principal contour corresponding to 111 and TTT reflections. X :thin region (t = 1.55,). Y: thick region. Note the reduced intensity inside the contour compared with that outside, particularly in thick regions. After H. Hashimoto et al. (59). Courtesy of the Royal Society.
type-I Bloch wave is excited and that this wave is easily transmitted because of the anomalous absorption effect. It is true to say that, because of anomalous absorption, it is possible to obtain transmission through much greater thicknesses of crystalline material than would be possible in the absence of the effect. Figure 29 shows computed intensity profiles of bright-field and darkfield images as a function of crystal thickness for different amounts of
M. J. WHELAN
56
(0)
0
\
1
2
3
4
I
2
3
4
5
6
I.
6
FIG.29. Profiles of thickness fringes computed from the two-beam theory at w = 0. (a) No absorption; (b) e,/ca= 0.05;(c) &,/fa = 0.10. co= f;.After H. Hashimotoetal. (59). Courtesy of the Royal Society.
absorption. The curves give the intensities in images of wedge-shaped crystals, the so-called thickness. extinction contours or Pendellosung fringes. Figures 10 and 30 are typical examples. For zero absorption the fringes persist indefinitely with increasing thickness, whereas in practice the fringes fade out as shown in Fig. 30, so that only about five fringes are clearly visible. We see from Fig. 30 that although the fringe visibility is reduced in the thick region, the transmission is still good and dislocation lines are clearly visible. The decrease in fringe visibility is again a result of the type-2 Bloch wave being removed preferentially by anomalous absorption. Comparison of observed profiles of thickness extinction contours with theory enables absorption parameters to be estimated (56, 57). For further details reference should be made to Hashimoto et al. (59).
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
57
FIG. 30. Thickness extinction contours in a bright-field image of a wedge crystal of Cu + 7 wt.% Al alloy. 11 1 Bragg reflection. A number of dislocations is visible as well as some faulted regions showing fringes. Note that the thickness contours fade out in thick regions. Note also the “dotted” and “zig-zag” behavior of dislocation images. After A. Howie and M. J. Whelan (15). Courtesy of the Royal Society.
B. Images of Planar Defects 1. Stacking-Fault Images
The case of a stacking fault was discussed in Section II,A,3 on the kinematical theory, where Eq. (32) predicted fringes in the region of overlap of the two parts of the faulted crystal (Fig. 13).The fringes are loci of constant z in Fig. 13, and run parallel to the intersection of the fault plane with the foil surface. Stacking faults were first observed with the TEM by Bollmann (60). Figures 31 and 32 show typical examples of fault fringes where the phase angle a = 2ng R has magnitude 2n/3, typical of stacking faults in closepacked structures. Figure 31 refers to faults in hexagonal AlN crystals, a
58
M. J. WHELAN
FIG. 31. Fringes at stacking faults on the basal plane of AIN. la1
= 2x/3.
After C. M.
Drum and M. J. Whelan. Courtesy of the Philosophical Magazine.
material with the wurtzite structure, while Fig. 32 refers to a stacking fault on a (111) plane in a facecentered cubic Cu + 7 wt% A1 alloy. Equations (113a) and (113b) [with the substitution of Eq. (144) to take account of absorption] are used to calculate the profiles of bright-field and dark-field images on the two-beam dynamical theory. For faults with phase angle 2n/3 the following results have been noted: a. For zero absorption the contrast consists of fringes of constant z, while away from the B r a g reflecting position the fringes are alternately strong and weak. b. The fringe profile is uniform across the rojected width of the fault, the depth periodicity of the fringes being &,/ For J large k deviation ?. from the Bragg condition the weak fringes vanish and the periodicity tends to s- as predicted by kinematical theory. c. At the Bragg reflecting position, the strong and weak fringes become
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
59
FIG. 32. (a) Bright-field and (b) dark-field images of a stacking fault in Cu + 7 wt.% A1 alloy. G( = + 2 a / 3 . After H. Hashimoto et al. (59). Courtesy of the Royal Society.
indistinguishable, and the depth periodicity is then *tg.This is understandable in terms of the depth oscillation of direct and diffracted waves shown in Fig. Ilb. At points along the fault differing in depth by itg,one or other of these waves has zero amplitude and only one plane wave is incident on the fault. I t therefore propagates into the lower crystal as though no fault were present. The periodicity itgis in contrast to the periodicity 5, of wedge fringes which would be observed in either part of the faulted crystal in Fig. 11b alone. d. In the absence of anomalous absorption, bright-field and dark-field images are complementary, and are both symmetrical about the center of the fault, i.e., the fringes near the intersections of the fault with top and bottom surfaces of the crystal appear similar. e. When anomalous absorption is present the fringe profile is no longer uniform across the fault, the visibility of the fringes being greater near the edges of the fault as shown in Fig. 32. f. For faults with phase angle 2a/3, the bright-field image remains symmetrical when absorption is introduced, whereas the dark-field image becomes asymmetrical as shown in Fig. 32, one edge of the dark-field image being bordered by a light fringe and the other by a dark fringe. Figure 33 shows that theoretical calculations predict the observed effect of asymmetry of the dark-field image.
M. J. WHELAN
1 '4,
FIG.33. Computed profiles of the stacking fault images in Fig. 32. The full line is the = bright-field image; the broken line is the dark-field image. a = + 2 n / 3 ; t / ( , = 7.25; (,,/to 0.075; 5: = w = -0.2. After P. B. Hirsch et al. (16). Courtesy of Butterworths.
rb;
2. Determination of the Type of Stacking Fault
In face-centered cubic materials stacking faults can be of two main types known as intrinsic and extrinsic faults. An intrinsic fault is produced by removal of a layer of close-packed atoms (or by simple shear on a {Ill} plane), while an extrinsic fault is produced by inserting a layer of closepacking (or by shear in different directions on two adjacent planes). In materials science it is important to be able to distinguish these two types, which have opposite shear vectors R and hence, for a given g, have phase angles a (= 2ng * R) of modulus 2n/3, but differing in sign. Computations show that with a suitable convention for defining R, when a is positive the edge fringe is light on the bright-field image. For example, a is positive in Fig. 32. Moreover, the sense of inclination of the fault can be determined from the fringes in the dark-field image. The edge of the fault where the fringes in the dark-field and bright-field images are similar corresponds to the intersection of the fault with the top surface of the crystal (where the electron beam enters). We can therefore say that P is the top surface and Q is the bottom surface for the fault in Fig. 32. Thus the sense of inclination of the fault is determined, and since g is known, it is possible from a knowledge of the sign of a to determine the sense of R,and hence to determine whether the fault is intrinsic or extrinsic. The method was first described by Hashimoto et al. (59), and has been further developed by Gevers et al. (61), who pointed out that the type of fault can be determined from the dark-field image alone, the sense of inclination of the fault not being required. They gave the following simple rule. The diffraction vector g is marked on the electron micro-
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
61
graph as in Fig. 32. If on the dark-field image g points from the light edge fringe at one intersection towards the dark edge fringe at the other and if the the fault is intrinsic. A Bragg reflection is in the class {111}, {220}, {W}, converse observation would imply that the fault is extrinsic. A second class of reflections including (200): {222}, (440)is covered by a converse rule. It is clear that in Fig. 32 the fault is intrinsic since the Bragg reflection is (111).
C. Images of Dislocation Lines The application of kinematical theory to dislocation image contrast was outlined in Section II,A,3, where the column approximation (Figs. 12 and 15) proved to be a useful concept in evaluating wave intensities from point to point at the exit surface of a crystal where the lattice displacements vary continuously near a dislocation line. The same approximation is made in the dynamical theory. The basic situation is therefore as illustrated in Fig. 12. A dislocation line is situated at 0 and causes a displacement R(z) of the atoms in the shaded column of crystal. R(z) is calculated from dislocation theory, and is inserted in Eqs. (66a) and (66b) [or in the equivalent Eqs. (68a) and (68b)l. Generalization to the n-beam situation and to include absorption is via Eqs. (108)-(110) and (132). For a screw dislocation R(z) is given by Eq. (33), while for a general dislocation lying parallel to the surface in Fig. 12 isotropic elasticity theory gives 2n
sin 2 0 4(1 - V)
(1 - 2v) ----In 2(1 - v)
Irl
+-
(145) where v is Poisson’s ratio, CD = 4 - y (Fig. 12), b is the total Burgers vector, be is the edge component of b, and u is a unit vector along the dislocation. If g lies parallel to the surface of the crystal (which is usually approximately true), g (b x u) = (g be) tan y and putting v = 4 we find
-
-
2 z g - R = ( g . b ) 4 + g . b e [ ~ s i n 2 4 + t a n y ( $ I nIrl - i c o s 2 4 ) ] . (146) The quantity 2ng * R to be inserted in Eq. (66) therefore depends not only on (g * b) but also on (g be) for a dislocation of mixed character. From the point of view of materials science it is important to have methods of determining the character of a dislocation line observed with the electron microscope, that is, of determining its Burgers vector. It is therefore necessary to examine the features of computed images of dislocations and to compare them with observations. In general the solution of the dynamical
-
M. J. WHELAN
62
Eqs. (66) and (68) cannot be obtained by analytical methods for an arbitrary displacement R(z), and resort is made to numerical integration of the equations by computer. We now discuss briefly some results and applications. 1. images of Dislocation Lines of Mixed Character
-
The image of a dislocation depends on the quantity g b in Eq. (146), which for whole dislocations is an integer. In practice the most important values of g * b are 0, & 1, +2. The case g b = 0 is discussed in the next section. Since changing the sign of x in Fig. 12 is equivalent to changing the sign of g b, it is not necessary to discuss positive and negative values of g * b separately. Figure 34 shows image profiles for g b = 1 of mixed dislocations determined by a parameter p = g b,/g b. The crystal is relatively thick ( t = 8r,) and is set at the Bragg position w = 0. The curve for p = 0 is for a pure screw dislocation; p = 1 is for a pure edge dislocation. All curves in Fig. 34 except the broken curve refer to y = 0 in Fig. 12. It is seen that the image of an edge dislocation is wider than that of a screw dislocation by a factor of about two. The broken curve in Fig. 34 shows the effect of tilting an edge dislocation so that y = 45".
I
- 0.6
-0.4
-0.2
0
0.4
0.2
-
I
0.6
FIG.34. Computed bright-field images of mixed dislocations. g b = 1, t/& = 8, y / t , = 4, co=(b,
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
63
Examination of computed images of dislocations leads to the following main conclusions. a. For g * b = 1 near w = 0, the image of a screw dislocation in the interior of a thick crystal, where anomalous absorption is important, consists essentiallyof a single dark peak, whose position and width are insensitive to crystal thickness and position of the dislocation. The half-width of the peak is about it, for a screw dislocation and about twice this value for an edge dislocation. Bright-field and dark-field images appear similar under these conditions rather than complementary as would be the case for zero absorption. b. Where the dislocation line approaches the surface, oscillatory effects are visible which are similar in nature to the enhanced fringe contrast near the surfaces observed at stacking faults (Fig. 32). For sufficiently thin crystals oscillatory effects are observed along the entire length of an inclined dislocation. Examples of oscillatory contrast are visible in thin and thick regions of Fig. 30. These effects give the image of an inclined dislocation a zig-zag or dotted appearance depending on the local thickness. c. For g * b = 1 the image peak is localized near to the core of the dislocation for w = 0, but is deviated to one side for I w I > 0, in agreement with the kinematical prediction (Section II,A,3 and Fig. 17). d. Images for g b = 2 show more complicated profiles with double peaks and a line of no contrast at the dislocation core.
2. Determination of the Burgers Vector of a Dislocation
-
When g b = 0 we see from Eqs. (33) and (146) that a screw dislocation will not produce any contrast, i.e., the dislocation is invisible. The simple explanation of this is that the displacements (which are parallel to b) lie in the Bragg reflecting planes. These may be thought of as mirrors reflecting the electron beam, and it is clear that when g b = 0 the reflecting planes are not distorted by the strain field of the dislocation. The g * b = 0 criterion is a useful method for determining the direction of the Burgers vector b. The specimen is mounted in a goniometer holder and tilted so that images are formed with various Bragg reflections excited in turn. From the observed vanishing of the dislocation contrast in certain reflections the direction of the Burgers vector can be determined. The g b = 0 criterion of invisibility is valid exactly only for screw dislocations to which isotropic elasticity theory is applicable. It is only approximately valid for edge and mixed dislocations because of the term in g * be in Eq. (146). Usually, however, the residual contrast due to this term is weak. Figure 35 is an example of the vanishing of a dislocation B in A l + 4 wt% Cu alloy. The specimen contains long screw dislocations which
-
-
64
M. J. WHELAN
FIG.35. Electron micrographs illustrating the vanishing of a dislocation B when g * b = 0. After P. B. Hirsch et a/. (29). Courtesy of the Royal Society.
have become slightly helical by vacancy climb, so that it is known that the Burgers vector of dislocations A and B are along their lengths. In Fig. 35a the 020 reflection is excited, both dislocations have I g * b I = 1, and both are visible. In Fig. 35b the 220 reflection is excited as shown by the diffraction pattern. For this reflection g b = 0 for B leading to invisibility. The magnitude and sense of the Burgers vector can be obtained by comparison of the observed images with computed images. A most useful comparison method uses the technique of computer simulation of images first.devised by Head (62),whereby a line-printer is used to produce a scale of gray by overprinting type symbols. The intensities computed from Eqs. (66a) and (66b) are displayed as “dots” of varying densities, and the image is constructed as a two-dimensional array of dots. Figure 36, taken from Head‘s paper (62), shows an example for dislocations in ordered B brass. In this material, elastic anisotropy is important, and screw dislocations for which g b = 0 are nevertheless visible as a pair of closely spaced images with a line of no contrast at the core (Fig. 36a). Figures 36a-d show experimental images, Figs. 36e-h are computer simulated images for b = [Tl I], while Figs. 36i-1 are simulated images for b = 4[T11]. Direct comparison shows that the Burgers vector b = [ T l l ] fits best the experimental images. Refinements of this technique now use cathode ray tube devices for constructing the image (see Fig. 37). For a review of the computer simulation technique the reader should consult reference (63).
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
65
FIG.36. Comparison of experimental and computer simulated images in /? brass. (a)-(d), experimental images. (e)-(h), computer simulated images for b = [Tll]. (i)-(1). computer simulated images for b = 4[1ll]. (i), TTO reflection; (ii), 121 reflection; (iii), 020 reflection; (iv), 110 reflection. After A. K. Head (62). Courtesy of the Australian Journal of Physics.
66
M. J. WHELAN
FIG.37. Experimental images (lower pair) and computer simulated images (upper pair) of dislocation loops. Upper pair is due to Bullough, Maher, and Perrin (65).The lower pair is due to Mazey et a / . (66). Note the change in size of the images with change in diffracting condition. Courtesy of the Philosophical Magazine.
An important application is the determination of the Burgers vectors of closed loops of dislocation line. Loops are produced by clustering of point defects (vacant lattice sites, interstitial atoms) introduced by processes such as quenching or radiation damage by fast nuclear particles, and they affect the mechanical and physical properties of the material. It is often required to know whether a given loop is of vacancy type or interstitial type. These two types of loop (analogous to intrinsic and extrinsic faults) differ in the sign of the Burgers vector, and hence a determination of the sign serves to distinguish the type of loop. In the previous section and in Section II,A,3 it was stated that the image peak of a dislocation observed in a deviated condition ( I w I > 0) lies to one side of the dislocation core. The side on which the peak lies is determined by the sign of the quantity (g b)s, where s is the deviation parameter (Fig. 5b). For a dislocation loop, the image peak therefore lies entirely inside or entirely outside the position of the core. Hence by observation of the size of the image for opposite signs of the deviation parameter s (or alternatively of g) it is possible to determine b and hence the type of loop. Details of the method are given by Groves and Kelly (64) and in reference (16). We illustrate the method by showing in Fig. 37 a comparison of experimental images of loops in aluminum irradiated with a particles, with computer simulated images due to Bullough, Maher, and Perrin (65). The
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
67
change in size of the dislocation loops with tilting is clearly visible in both experimental and computed images. In this case the loops in aluminum were found to be of interstitial character (66).
3. Weak-Beam Images We saw in Section III,C that at the Bragg reflecting position (w = 0) the half-width of the image of a screw dislocation is about $&, i.e., about 100 A for a typical low-order reflection in a metal. This limits the dimensions of detail which can be resolved in images formed using strong beams. For example, the images of pairs of dislocations would merge if they were closer 100 A and they would not be resolved as separate. It is important to than be able to make observations at higher resolution in order to distinguish closely spaced dislocations. Examples of such applications are the resolution of pairs of partial dislocations resulting from dislocation dissociation and the imaging of very small dislocation loops produced by clustering of point defects. In the former case the dissociation depends on the stacking-fault energy of the material and in many metals the dissociation is less than 50 A. It is therefore important to improve the resolution. A method known as the weak-beam technique was developed by Cockayne, Ray, and Whelan (33), and the basis of the method can be understood by reference to the images of a screw dislocation calculated by kinematical theory (Fig. 17). The image profiles are plotted as a function of the dimensionless parameter 2mx. It follows that if s is increased the half-width Ax of the dislocation image is decreased. For a screw dislocation when g b = 2, Ax 5: ( R S ) - ' . Hence if s 5: 2 x lo-' A-', Ax 'v 15 A. The image width is therefore very narrow. Figure 17 refers to the dark-field image, the overall intensity level of which varies as s - 2 for large s [Eq. (23)]. It is therefore difficult to make observations in the bright-field image since the contrast is low. However, if the dark-field image is used, the contrast of the dislocation peak is high even though the overall intensity is low. The technique therefore uses the dark-field image with large s (w = 5, s 5). To reduce aberrations the diffracted beam must be aligned on the axis of the electron microscope. This requires the specimen and the incident beam to be tilted. In recent years this has become a practical possibility by the development of reliable goniometer stages and electromagnetic beam tilting facilities. Since .the image is weak, exposures of the order of minutes are required. This places rather stringent conditions on the amount of stage drift which can be tolerated. The development of reliable (lowdrift) specimen stages has therefore contributed to the realization of the weak-beam technique. Figure 38 is an example showing the resolution of partial dislocations in Cu 10 wt% A1 alloy. The dislocation is close to edge orientation and the
-
-
+
68
M. J. WHELAN
FIG. 38. Weak-beam image of a dissociated edge dislocation in Cu + 10 wt.% A1 alloy. After Cockayne et al. (33). Courtesy of the Philosophical Magazine.
weak-beam image is formed with the 220 Bragg reflection indicated on the diffraction pattern. From measurements of the separation of the partial dislocations the stacking-fault energies of various materials have been estimated (67-69).
IV. CONCLUDING REMARK From what has been mentioned in this review it will be apparent that quantitative information concerning crystal defects is now readily obtainable with the electron microscope. Such information has become available through our greatly improved understanding of the mechanisms of scattering of electrons in crystalline materials, particularly in imperfect crystals, and the electron microscope is now an observational tool of central importance in the study of materials.
APPENDIX : GENERAL SOLUTION OF THE DISPERSION EQUATIONS Using the fact that K, = k, it can be shown that in the dispersion equations (80)
K2 - ki = (K: - 2K,
. & - g’)
- k i - 2g,k,,
(A.1)
where the subscripts z and t denote the components of vectors normal and tangential to the crystal surface. If gz = 0, Eq. (A.l) reduces to Eq. (87).
ELECTRON DIFFRACTION THEORY A N D ELECTRON MICROSCOPY
69
Define n x n matrices B and D by
The dispersion equations (80) may then be written (D - k,B - kf1)C = 0
(A.4)
where I is the unit matrix and C is the column vector [C,]. For a nontrivial solution.
(D-k,B-ktI( =O.
(‘4.5 1
Equation (AS) is an n x n matrix quadratic equation and has 2n values of k,. n of these correspond to forward-propagating Bloch waves, while the remainder describe backward-propagating Bloch waves. For a particular k , , the n-column vector [C,]gives the corresponding Bloch function of Eq. (78). Rather than solve (AS) the following procedure (70) gives the 2n values of k, as eigenvalues of a 2n x 2n matrix. Put
(B + k,I)C = X.
(A4
Equations (A.4) and (A.6) can then be combined as
The k, are now given by the eigenvalue equation
-B - k, D
0-k,
Expanding (A.8) gives (AS). The above procedure is related to the “mixed representation” method of Marcus and Jepsen ( 7 1 ) for calculating LEED intensities [see also Tournarie (72)], where the wave function is expanded as
and where the g,‘s are parallel to the surface and form the reciprocal lattice net of the surface periodicity. The lattice potential U ( r )is expanded as (A. 10)
70
M. J. WHELAN
Substituting (A.9) and (A.lO) in the Schrodinger equation (23) gives for the column vector JI = [$,,I
c!!! =
dz2
-4n2E(z)JI,
(A. 11)
where E,, = Kt - 2K,
*
g, - g:,
Egh = ug-h(Z).
(A.12)
It is readily shown from Eq. ( A . l l ) that the 2n column vector [JI, dJl/dz] satisfies
Equations ( A . l l ) and (A.13) apply to a case where the potential is not periodic in the z direction and they take account of forward-and backwardpropagating waves. Equations (A. 13) can be regarded as a generalization of the Darwin method to take account of backward-propagatingwaves. If only periodicity parallel to the surface is important as in the Laue case (ie., 92
= 0)
J/,(z) = C , exp (27rik,z), uh,(z) = uh *
(A. 14) (A.15)
If (A.14) and (A.15) are substituted in (A.13) we obtain the result (A.7) for gz = 0.
ACKNOWLEDGMENTS The author is grateful to numerous colleagues for stimulating discussions and for kindly supplying photographs.
REFERENCES H. A. Bethe, Ann. Phys. (Leipzig) 87, 55 (1928). C. Davisson and L. H. Germer, Phys. Rev. 30,707 (1927). G. P. Thomson, Proc. Roy. SOC.Ser. A 117, 600 (1928). G. R. Booker, A. M. B. Shaw, M. J. Whelan, and P. B. Hirsch, Phil. Mag. 16, 1185 (1967). J. P. Spencer, C. J. Humphreys, and P. B. Hirsch, Phil. Mag. 26, 193 (1972). J. M. Cowley, Appl. Phys. Lett. 15, 58 (1969). 7. H. Boersch, Ann. Phys. (Lpipzig) 26, 631 (1936). 8. H. Boersch, Ann. Phys. (Leipzig) 27, 75 (1936).
I. 2. 3. 4. 5. 6.
ELECTRON DIFFRACTION THEORY AND ELECTRON MICROSCOPY
71
9. J. B. Le Poole, Philips Tech. Rev. 9, 33 (1947). 10. C. G. van Essen, E. M. Schulson, and R. H. Donaghay, Nature (London) 225,847 (1970). I t . C. G. van Essen, E. M. Schulson, and R. H. Donaghay, J. Mater. Sci. 6, 213 (1971). 12. R. J. Woolf, D. C. Joy, and D. W. Tansley, J. Sci. Instrum. 5,230 (1972). 13. C. G. Darwin, Phil. Mag. 27, 315,675 (1914). 14. A. Howie and M. J. Whelan, Proc. Roy. SOC.Ser. A 263, 217 (1961). 15. A. Howie and M. J. Whelan, Proc. Roy. SOC.Ser. A 267,206 (1962).
16. P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan, “Electron Microscopy of Thin Crystals.” Butterworths, London, 1965. 17. E. G. McRae, Surface Sci. 11,479 (1968). 18. N. F. Mott and H. S. W. Massey, “The Theory of Atomic Collisions,” 3rd ed., Chapter 2. Oxford Univ. Press (Clarendon), London and New York, 1965. 19. R. Glauber and V . Schomaker, Phys. Rev. 89, 667 (1953). 20. J. A. Hoerni and J. A. Ibers, Phys. Rev. 91, 1182 (1953). 21. J. A. Ibers and J. A. Hoerni, Acta Crystallogr. 7, 405 (1954). 22. H. Boersch, 0. Bostanjoglo, and H. Raith, 2.Phys. 180, 407 (1964). 23. H. Boersch, G. Jeschke, and H. Raith, Z. Phys. 181, 436 (1965). 24. A. Fukuhara, Proc. Phys. SOC.London 86, 1031 (1965). 25. M. J. Goringe, Phil. Mag. 14,93 (1966). 26. D. L. Misell, Advan. Electron. Electron Phys. 32, 63 (1973). 27. R. D. Heidenreich, J. Appl. Phys. 20, 993 (1949). 28. M. J. Whelan, and P. B. Hirsch, Phil. Mag. 21, 1121, 1303 (1957). 29. P. B. Hirsch, A. Howie, and M. J. Whelan, Phil. Trans. Roy. SOC. London 252, 499 (1960). 30. A. Howie and Z . S. Basinski, Phil. Mag. 17, 1039 (1968). 31. S. Takagi, J. Phys. SOC.Jap. 26, 1239 (1969). 32. C. M. Drum, Phil. Mag. 11, 313 (1965). 33. D. J. H. Cockayne, I. L.F. Ray, and M. J. Whelan, Phil. Mag. 20, 1265 (1969). 34. D. J. H. Cockayne, 2.Naturforsch. A 27, 452 (1972). 35. C. H. MacGillavry, Physica (Hague) 7, 329 (1940). 36. R. D. Heidenreich and L. Sturkey, J. Appl. Phys. 16, 97 (1945). 37. L. Sturkey, Proc. Phys. SOC.London 80, 321 (1962). 38. J. B. Pendry, J. Phys. C 2, 2273 (1969). 39. A. J. Freeman, Phys. Rev. 113, 176 (1959). 40. A. J. Freeman, Acta Crystallogr. 12, 274, 929 (1959). 41. A. J. Freeman, Acta Crystallogr. 13, 190, 618 (1960). 42. M. J. Whelan, J. Appl. Phys. 36, 2099 (1965). 43. M. J. Whelan, J . Appl. Phys. 36, 2103 (1965). 44. F. Len& 2.Naturforsch. A9, 185 (1954). 45. D. Bohm and D. Pines, Phys. Rev. 82,625 (1950); 85, 338 (1951); 92,609 (1953). 46. L. Marton, J. A. Simpson, H. A. Fowler, and N. Swanson, Phys. Reu. 126, 182 (1962). 47. R. A. Ferrell, Phys. Rev. 101, 554 (1956). 48. H. Yoshioka and Y. Kainuma, J. Phys. SOC.Jap. 17, Suppl. B-11, 134 (1962). 49. C. R. Hall and P. B. Hirsch, Proc. Roy. SOC. Ser. A 286, 158 (1965). 50. H. Yoshioka, J. Phys. SOC.Jap. 12,628 (1957). 51. K. Moliere, Ann. Phys. (Leipzig) 34, 461 (1939). 52. M. von h u e , Acta Crystallogr. 2, 106 (1949). 53. G. Borrmann, Phys. 2.42, 157 (1941). 54. A. Howie, Proc. Roy. SOC.Ser. A 271, 268 (1963). 55. C. J. Humphreys and M. J. Whelan, Phil. Mag. 20, 163 (1969). 56. H. Hashimoto, J. Appl. Phys. 35, 277 (1964).
72
M. J. WHELAN
A. J. F. Metherell and M. J. Whelan, Phil. Mag. 15, 755 (1967). C. J. Humphreys and P. B. Hirsch, Phil. Mag. 18, 115 (1968). H. Hashimoto, A. Howie, and M. J. Whelan, Proc. R o y . Soc. Ser. A 269, 80 (1962). W. Bollmann, Proc. Eur. Reg. Con5 Electron Microsc., 1st p. 316 (1956). R. Gevers, A. Art, and S . Amelinckx, Phys. Status Solidi 3, 1563 (1963). 4. K. Head, Aust. J . Phys. 22, 43 (1969). A. K. Head, P. Humble, L. M. Clarebrough, A. J. Morton, and C. T. Forwood, “Computed Electron Micrographs and Defect Identification.” North-Holland Publ., Amsterdam, 1973. 64. G. W. Groves, and A. Kelly, Phil. Mag. 6, 1527 (1961); 7, 892 (1962). 65. R. Bullough, D. M.Maher, and R. C. Perrin, Phys. Status Solidi €3 43,689 (1971). 66. D. J. Mazey, R. S. Barnes, and A. Howie, Phil. Mag. 7, 1861 (1962). 67. D. J. H. Cockayne, M. L. Jenkins, and 1. L. F. Ray, Phil. M a g . 24, 1383 (1971). 68. 1. L. F. Ray, and D. J. H. Cockayne, Proc. R o y . Soc. Ser.A 325, 543 (1971). 69. M. L. Jenkins, Phil. Mag. 26, 747 (1972). 70. R. Colella, Acta Crystallogr. 28A, 11 (1972). 71. P. M. Marcus and D. W. Jepsen, Phys. Rev. Lett. 20, 925 (1968). 72. M. Tournarie, J . Phys. Soc. Jap. 17, Suppl. B-11, 98 (1962).
57. 58. 59. 60. 61. 62. 63.
Energy Spectrum of Electrons Emitted by a Hot Cathode WOLFGANG FRANZEN
AND
JOHN H. PORTER
Department of Physics, Boston University, Boston, Massachusetts
I. Introduction .................................................................. 11. Effect of Surface Barrier on Electron Energy Distribution ............................ 111. Physical Situation Outside Spherical Cathode.. .......
........ IV. Space-Charge Density for Sphe V. Solution of Poisson’s Equation ........................................ VI. Self-consistent Solution for Spa Potential ... .................... VII. Effect of Space-Charge Barrier on Electron Energy Distr VIII. Design Principles for Spherically Symmetric Gun: Field Matching.. ................ IX. Construction of Electron Gun ................................. ... X. Electron Energy Analysis ................................................... XI. Results of Experimental S ctron Energy Distribution.. .................... XII. Relation Between Results ... ....................... Glossary of Symbols Use References ........................................ .......................
75
87 90 93 95 99 103 109 114 115
117
I. INTRODUCTION The energy distribution of electrons emitted by a thermionic cathode has been a controversial subject for over twenty years (1-19). As is well known, and as will be shown in Section I1 of this paper, an elementary argument leads to the conclusion that the thermoelectrons emerging from a hot cathode should be described by a half-Maxwellian distribution in initial kinetic energy and angle of emission, that is, a Maxwell-Boltzmann distribution multiplied by the component of velocity perpendicular to the surface (20). When an intense accelerating electric field is applied to the cathode, electron emission is enhanced by tunnelling through the surface barrier, and the electron spectrum is modified in a manner described by the FowlerNordheim theory (21). In the absence of a space-charge barrier outside the cathode, that is, in the emission-limited (low electric field) case, several careful experiments have shown that the energy distribution of the electrons is indeed halfMaxwellian (22,23).In the case of field emission, theory and experiment are also in good agreement (24, 25). It is only when the emission current be73
74
WOLFGANG FRANZEN AND JOHN H. PORTER
comes large enough to cause the formation of a repulsive space-charge barrier in front of the cathode, in other words, in the space-charge-limited case, that theory and experiment have been reported to disagree. Further reference to the literature on this subject will be made in Section XI1 of this review. It should be noted that when a space-charge cloud surrounds the cathode, the electron energy spectrum is filtered twice as the electron leaves the metal, first by the surface barrier (or Schottky barrier) and then again by the space-charge barrier. The surface barrier converts the Fermi distribution inside the metal to a half-Maxwellian distribution just outside, as mentioned earlier, but the effect of the space-charge barrier cannot be described so simply. This second barrier selectively reflects the outward-movingelectrons in a manner that depends on the geometry of the electron gun, as Hasker has pointed out in an important series of papers (26-29), and as we shall demonstrate in our analysis of this problem. Because of the mathematical difficulties encountered in the self-consistent computation of the spacecharge potential distribution, the filtering effect of the space-charge barrier on the energy spectrum has not been taken into account in most previous discussions of this subject; Hasker’s articles (26-29) provide the only exception. A comparison between experiment and theory is meaningful only when the theoretical prediction is based on an accurate description of the physical environment of the cathode. For many electron guns, such a description is difficult because of a lack of symmetry in the configuration of the electrodes. For example, it is likely that a hairpin cathode surrounded by a Wehnelt cylinder, as in several early experiments on energy distribution ( I , 13), has a nonuniform space-charge barrier which would cause the energy spectrum of electrons passing through it to be distorted. The extent of this filtering effect of the barrier on the energy distribution cannot be ascertained in this case, because the space-charge-limited potential in such an electron gun has not been computed. In view of this situation, we decided to re-investigate this subject, both experimentally and theoretically, using a geometrical configuration of electrodes amenable to analysis, namely a hemispherical cathode surrounded by a concentric hemispherical anode, with boundary condition so chosen that the electric field between the two electrodes is spherically symmetric (32).In this geometry, an exact self-consistent solution of the spacecharge potential problem can be found, using the full initial energy and angle distribution of the thermoelectrons (33). This solution can then be applied to a computation of the filtered energy spectrum (19).An experimental realization of such an electron gun was used as an electron source in a Purcell-type electron monochromator which selected slices of the initial energy spectrum for sub-
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
75
sequent acceleration. The acceleration potential needed to raise the energy of electrons so selected to the energy of the 19.3 eV scattering resonance of helium (34, 35) then became a measure of their initial kinetic energy. As will be demonstrated in this review, the energy spectra obtained by this novel technique agree well with the predicted energy distributions, obtained by filtering the initial half-Maxwellian distribution through the computed space-charge potential barrier. Only one parameter, namely the value of Richardson’s constant A for our (dispenser) cathode, was regarded as adjustable when comparing theory and experiment. The principal purpose of this review is to present a complete account of this work, so that its results can be evaluated in the light of the controversy mentioned earlier. It has been pointed out on a number of occasions (30,31)that a cathode with a microscopically smooth surface and a uniform work function is an idealization not realized in practice, except under very unusual circumstances, as in Shelton’s experiment (22). Why then can these surface irregularities be neglected in our prediction of the energy and angle distributions? The answer to this question is that the irregularities can be neglected if, from the point of view of their linear dimensions, they are much smaller than characteristic dimensions of the electrostatic field, such as the distance between the cathode and the virtual cathode. At a distance from the cathode several times larger than the size of surface bumps or work function patches, the equipotentials of the electrostatic field become smooth, and reflect the overall shape of the cathode surface, in accordance with a well-known property of such a field. All electrons passing through such a smooth equipotential surface have surmounted the same surface barrier, although the shape of the barrier, as a function of distance measured in the outward direction, varies from one location to another. As far as the energy and angle distribution of the transmitted electrons is concerned, the effect of replacing the actual surface barrier by the effective barrier just described will be small if the dimensional criterion mentioned earlier is satisfied. In the case of dispenser cathodes, as used in our experimental work, surface irregularities are known (36) to have dimensions of the order of 1 pm, as compared to a cathode radius of 125 pm, and a cathode-virtual cathode distance of about 25 pm in our apparatus (see Section VI of this paper). From this point of view, our dispenser cathode is effectively “smooth,” a condition that would not be likely to be fulfilled by an oxide-coated cathode, for example. 11. EFFECTOF SURFACE BARRIERON ELECTRON ENERGY DISTRIBUTION
We assume that the state of the electrons inside the emitter may be described to a good approximation by the free-electron theory of metals. This implies that the emitter has a nearly spherical Fermi surface in a partly
76
WOLFGANG FRANZEN AND JOHN H. PORTER
filled conduction band. The arbitrarily chosen zero of potential energy is the bottom of the conduction band. Phase space for the (nearly free) electrons inside the metal is then six dimensional, an element of volume of this space being do = dp, dp, dp, d x d y dz in Cartesian coordinates, where p, ,p, ,and pz are the three components of the linear momentum. It is convenient to choose the xy plane as the surface from which the thermoelectrons emerge, and to let the positive z axis be the outward normal. For the components of the momentum, cylindrical coordinates are more appropriate than Cartesian coordinates. Thus we define pT = (p: + pi)''* as the component of momentum parallel to the surface of the metal, pz is the component perpendicular to the surface, and 4 = tan-'(p,/px) is the azimuthal angle in the x y plane. An element of volume of phase space then becomes
do = PT
dpT dpz d 4 d x dy dz.
But we know that there are 2/h3 states available per unit volume of phase space, the factor of 2 arising from the two possible spin orientations that are allowed for each energy state by the Pauli principle. Hence in do there are
available states, each of which may be occupied by at most one electron. The probability that a state of energy E is occupied is {exp [(E - E,)/kT]
+ I}- ',
at an absolute temperature T, where EF is the Fermi energy, also measured positively upward from the bottom of the conduction band. Then the number of electrons in the specified volume of phase space do is
In view of our choice of zero of the potential energy, E is just the kinetic energy of the electrons inside the metal:
E
=
(1/24(P-4+ Pf).
The distribution (1) describes a state of thermal equilibrium inside the metal, assumed to remain undisturbed by the small outward flow of thermoelectrons over the surface potential barrier. The barrier itself is described as a sudden rise in potential energy, from zero inside the metal to U , outside. When an electron approaches the surface with a positive z component of momentum p.- such that pf/2m 2 U , , it will pass over the barrier and leave
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
77
the metal. Tunnelling through the barrier is neglected in this analysis, so that the exact shape of the barrier is unimportant. Of principal interest for our purposes is the distribution in components of momentum, at the instant of emission, of the group of electrgns with sufficient energy to pass over the barrier. The process of energy selection of the thermoelectrons at the surface of the metal, and their subsequent acceleration by external electric fields, imply that the emitted electrons are not in thermal equilibrium, either with one another, or with their surroundings. Nevertheless, we can treat the portion of the distribution ( 1 ) that arises from passage over the surface barrier, upon normalization, as a probability distribution that can be propagated to the outside of the metal, subject to the known dynamics of electron motion. The constants of motion during the crossing of the barrier are the total energy E, the transverse component of momentum pro, and the azimuthal angle 4, which is measured in the plane of the surface (the x y plane). Just outside the surface the constant total electron energy can then be written
E = (1/2m)(pt,, + p - f , ) + U o = c0 + U o , where is the initial kinetic energy. Then E - EF = ( 1/2m)(ple
+ P ; ~ )+ W, = E , + W, ,
(2) where p t o / 2 m = p l / 2 m - U,, p T , = p T , and W, = U , - E , is the work function of the cathode. In these, as in subsequent expressions, the subscript o denotes conditions at the surface of the cathode. Electrons that pass through an element of area d A = dx d y of the surface between times t = 0 and t = dt must have come from a (hypothetical) depth dz = vzo dt = ( p Z , / m )dt inside the metal. At t = 0, the group of electrons that will pass through d A in time dt thus fills an element of volume of configuration space dV = d x d y dz = uZ, d A dt. (3)
Substituting (2) and (3) in ( l ) , and replacing p z , and pT, by muzo and muT,, respectively, we obtain
This expression describes the number of electrons emerging from area d A in time dt, and having an outward component of velocity between vzo and vz, duzo, a transverse component between uTo and vT0 + dvTo,in an azimuthal direction between 4, and 4, + d 4 , .
+
78
WOLFGANG FRANZEN AND JOHN H. PORTER
At the temperatures employed in thermionic emission, generally W,/kT % 1, so that unity in the denominator of ( 4 ) can be neglected in comparison with the much larger exponential function. Furthermore, it is desirable to divide by d A dt, to obtain the number of electrons emitted from unit area of the surface per unit time, in the specified ranges of velocities and azimuthal angle:
with the restriction uzo2 0. The total number of electrons emerging from unit area per unit time, regardless of velocity or direction of motion is then N,=2n
(
'm3)
exp
(- 2): 1
1
m
d u z o uTouzo
kT
=
(
4nmk'
exp
(- $1.
The outward surface current density J , is just e times as large, where e is the electronic charge: J , = ( 4nmek' 1)T'
exp
(- 2).
(7)
This is Richardson's law, and the constant 4nmek' A=-----= 120 x lo4 A/m'("K)' h3
is known as Richardson's constant. On dividing ( 5 ) by (6),we obtain the probability that an electron emerging from the emitter will have an outward component of velocity between or, and uzo+ dozo,a transverse component between uTo and uT0+ doTo,and an azimuthal angle of motion between 4oand 4, + d4,:
The restriction uzo 2 0 applies here.
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
79
The probability distribution (9) is known as a half-Maxwellian distribution, since it can also be used to describe the velocity distribution of molecules of a gas escaping from a container into a vacuum through a small hole, when the gas obeys Maxwell-Boltzmann statistics, and the molecules are in thermal equilibrium at a temperature T before escaping (37). In order to make use of important invariants of the electron motion outside the cathode, it is convenient to express (9) in two other sets of generalized coordinates. One such set consists of the initial kinetic energy, E, = f m ( v - f , of,), the azimuthal angle 40,and the angle made by the initial velocity vector with the normal to the surface of the metal, namely 8, = tan- l(~T,/~z,). The Jacobian for this transformation of coordinates is
+
so that
= m dozo doTo d 4 0 ,
and the probability distribution becomes
Q&,
0,) d ~ do, , d4, = n-l(kT)-’ exp ( - E , / ~ T ) E de,, sin 6, cos 6, do, d 4 , ,
(10)
with the restriction 0 I 8, I4 2 . When the cathode is spherical, another useful .set of coordinates consists of the initial orbital angular momentum Lo = mvT,ro about the center of the cathode of radius ror the initial radial kinetic energy er0 = E*, = 4moIo, and 4,. The Jacobian of the transformation to these variables is
so that the desired distribution has the form
80
WOLFGANG FRANZEN AND JOHN H. PORTER
The surface barrier of the cathode thus has the effect of filtering out of the Fermi-Dirac distribution (1) inside the metal a half-Maxwellian portion defined by any one of the functions (9)-( 11). It is important to emphasize that (9)-(11) define not only the energy spectrum of the emerging thermoelectrons at the instant of emission, but also their angular distribution. This aspect of their motion influences the formation of the space-charge potential barrier outside the cathode in a geometry-dependent manner.
111. PHYSICAL SITUATION OUTSIDE SPHERICAL CATHODE
As discussed in the Introduction, the energy spectrum of the electrons emitted by a hot cathode is filtered twice, once by the surface potential barrier of the metal, and again by the space-charge barrier that surrounds the cathode. The space-charge potential distribution can be computed readily only for certain simple geometrical configurations, such as a spherical cathode surrounded by a concentric spherical anode, a cylindrical cathode surrounded by a concentric cylindrical anode, and a plane cathode parallel to a plane anode. Approximate solutions in these cases have been known for many years (18,31,38-41), but in most cases these solutions are based on somewhat unsatisfactory simplifying assumptions, such as the neglect of the initial energy and angle distribution of the emitted electrons. Analytical solutions for a plane geometry without simplifying assumptions have been discussed by Lindsay (42) and Amboss (43). Exact numerical solutions for the spherical and cylindrical case have been found recently by Porter et al. (33).These solutions are exact in the sense that the physical situation is described exactly, and the only approximations are those necessary in numerical integration. The solution of the spherical case is needed for the computation of the filtering effect of the space charge on the electron energy spectrum and will be discussed in detail in the next three sections. If cathode and anode are concentric spheres, a system of spherical polar coordinates is appropriate, as shown in Fig. 1. It is clear that regardless of the space charge, symmetry requires that the electric field between the two electrodes depend on r only. From this fact it follows that the orbital angular momentum L = Lo of the electrons about the origin must be conserved. The other constants of motion are the total energy co, and the azimuthal angle
40 -
Now let E, be the radial kinetic energy at any radius r ; E,. is its value at the surface of the cathode, as in Eq. (11). Since the electric field, and therefore the electrostatic potential U(r), are functions of I only, the potential
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
81
FIG.1. Diagram of the system of coordinates used to specify the position and velocity of an electron outside the spherical cathode. One octant of the cathode sphere of radius ro is shown, with the origin 0 of the coordinates at the center of the sphere. Electrons emerge from the cathode at the point P on the z axis, moving in the direction of the initial velocity vector v, which makes an angle 0. with the z axis. The xy plane is tangent to the cathode at P, and the initial transverse component of velocity vTo= vxo + vyo is the projection of v, on the tangent plane.
energy of the electrons -eU(r) will also depend on r alone. [Note that U ( r ) as defined here is zero at the surface of the cathode.] The constancy of the total energy implies that E,
L2
= E , ~i- 2= 2
W
O
E,
L2 +7 - eU(r), 2mr
where ro is the radius of the cathode. The radial kinetic energy at any radius r can then be written 8,
= E,.
+c 42m( ro
+ etr(r).
82
WOLFGANG FRANZEN AND JOHN H. PORTER
Thus from the point of view of the radial motion, -1 times the effective potential energy of the electrons is
The quantity W(r,L) is just e times the effective radial electrostatic potential seen by an emitted electron. Evidently, this potential depends not only on the actual electrostatic potential U ( r )as modified by the space charge, but also on the constant orbital angular momentum L Under space-charge-limited conditions, U ( r ) has a negative minimum U(r,) at i d , the radius of the virtual cathode. The effective radial potential e - W(r, L), however, contains an additional term e-'(L2/2m)(r; - r-') which increases from zero at r, to the asymptotic value e-'(L2/2mrf) for large I , as illustrated in Fig. 2. One would expect that for large values of the orbital angular momentum L the effective radial potential first rises to a
I
Radius (r) -+
Region Region I
II
'a
Region 111
FIG,2. Plot of eU(r), the term (L2/2m)(r;2- r - z ) , and the sum of these two quantities, W(r, L), which is e times the effective radial electrostatic potential, as functions of the radial distance r from the center of the cathode, for different values L, > L, > 0 of the orbital angular momentum L For L = L,, the three regions I, 11, and 111 discussed in the text are indicated.
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
83
positive maximum and then decreases through zero at r. to a negative minimum e - W(r,,,L) at r b , which may be different from rd . For smaller values of L, the initial positive rise of e-'W(r, L ) is absent. The likelihood that this centrifugal barrier effect will indeed give rise to a positive maximum in the effective radial potential, as described above, may be judged from the following argument. The effective radial electric field at the cathode is evidently
'
d - - [e- 'W(r, ,!,)Iro dr
=
-
If we express the initial angular kinetic energy L2/2mr; in units of kT, where T is the cathode temperature, so that L2/2mr: = vkT, where v is diminsionless, the absolute value of the first term on the right-hand side of (14) becomes L2/emr: = 2vkT/er,. For our cathode, ro = 1.25 x lo-' cm, and T 1 2 W K , so that kT/e 0.10 V. Then the contribution of the centrifugal barrier to the effective radial electric field at the cathode becomes
-
-
r 2
L -~
-
emr,"=
- 16.0~V/cm.
The positive (repulsive) electric field of the space-charge cloud represented by the second term of (14),on the other hand, is of the order of 100 V/cm for our cathode, as we shall see later on. Hence, for our cathode, a value of v s 6, i.e., an initial angular kinetic energy of 6kT, would be required for the negative effective centrifugal barrier field to cancel the actual (positive)electric field of the space-charge barrier at the cathode. The population of electrons with such a large angular kinetic energy is very small. Hence, although the centrifugal barrier distorts the effective radial potential barrier, it does not give rise to a positive maximum for most electrons in our geometry. Iv.
SPACE-CHARGE
DENSITY FOR
SPHERICAL CATHODE
We shall describe a self-consistent procedure for computing both the space-charge density and the electrostatic potential outside a spherical cathode surrounded by a concentric spherical anode. Because of the spherical symmetry of the electric field between these two electrodes, the angular momentum of an electron about the center of symmetry (the origin, that is, the common center of cathode and anode spheres) is constant, as was pointed out above. Therefore, the transverse, or angular, component of velocity ZIT is known at any radius, once its value at the cathode has been specified, independent of the form of the potential function U(r).The other constants of motion are the total energy E, and the azimuthal angle 4.
84
WOLFGANG FRANZEN A N D JOHN H. PORTER
Let us suppose that u = (u.f + u ; ) l / ’ is the velocity of an electron in the field space at a radial distance r, where
+
u, = (2~,/m)’/*= {(2/m)[.zrO W(r, L)]}’/’
(15) is the radial component of velocity. The function W(r,L) is defined by ~ the radial kinetic energies at r and r,, as in (12a). Eq. (13), and E, and E ? are If there are p(u, R) du dR electrons per unit volume moving in the direction of the solid angle enclosed between R and R + dR, and of velocity between u and u du, then the space-charge density is evidently given by
+
o =e
J J p(u, R) du dR,
where the integral extends over all values of both variables. An expression for p(u, R) du dR may be found by relating it to the function Qd(&,,L, q5) dEr0dL dq5 defined by (11). It is clear that the number of electrons with velocity between u and u + du, and moving in the direction dR, that cross the surface of a sphere of radius r concentric with the cathode in unit time is just 4nr2p(u,R)u, du dR. This quantity should be equal to the number of electrons leaving the cathode in unit time, with radial kinetic energy between E,. and E,. + d.cr0, orbital angular momentum between L and L + dL, and azimuthal angle of motion between q5 and q5 + dq5: 47cr2p(u,Q)u, du dR = 4 n r ~ N o Q d ( ~Lr)odsr0L , dL dq5.
(16)
Here, of course, dQ = sin 0 d0 dq5; u = u,/cos 8; cr0 + W(r,L ) = (m/2)u,2, and L = muTr = mru, tan 8. As before, 0 is the angle between a radius and the vector v (see Fig. l), N o is given by (6),and Q&,,, L) by (1 1). Hence a differential of the charge density becomes do = ep(u, R) du dR
L) dEr,,L dL dq5 - eriNoQd(&r,, ~ ~ _ _ _ . . _ 9
~~~~~
r’v,
J , exp [ - (era + L2/2mr:)/kT] dE, L dL dq5 --
+
2n(kT)’r’{%[~,~ W ( r ,LzTlr
’
(17)
where we set J, = e N , , as in Richardson’s law [Eq. (7)]. In this expression, the potential function U ( r )is contained in W(r, L),as given by (13). Thus we can integrate (17) over the variables E,. and q5 without involving the form of V ( r ) .The integration over E,, requires some care, as reference to Fig. 2 will
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
85
make clear. The diagram shows that for fixed L the field space can be divided into three distinct regions. Region I extends from the cathode radius ro to the second zero of the effective radial potential at r,, region I1 from r, to the minimum at r b , and region I11 from rb to the anode. As stated earlier, for certain ranges of values of L, one or more of these regions may be absent. In region I, the energy distribution of the forward component of the electron current is not changed by reflection from a potential barrier in that region, but there is also a backward current of electrons reflected in region 11. In region 11, the radial energy distribution changes systematically as r increases, because of reflection of the lowest energy electrons. Furthermore, a changing backward current is superimposed on the forward current. I n region 111, there is no backward current, and no further potential barrier intervenes to inhibit the forward (outward) flow of electrons. I n regions of space where forward and backward currents flow, both contribute to the space charge. In the integration of Eq. (17) over E ~ we~ must , keep in mind that croitself and E, =
cr,
+ W(r,L )
must both always be positive quantities. For the forward current, it follows from this requirement that if W(r, L) > 0 for points of radius r and any smaller radius, the lower limit on croiszero. If W(r,L) < 0 for radius r, or for some radius smaller than r, the lower limit on E , ~is - W(rmin, L), where rmin is the radius of the point of lowest (most negative) effective radial potential between r and r o . For the backward current, the upper limit on cro is always - W(rb,L), where rb is the radius at which the function W(r, L) attains its minimum value. The lower limit on cro is zero in region I, and - W(r,L) in region 11. Keeping these considerations in mind, we obtain for the differential of charge density integrated over cro and 4 the expression
[
do’ = K F ( r , L) exp W ( r ,L)k-TL2f2mr,Z] L dL, ~
~ _ _ _ _ _ _ _ _ _ _
where the function F(r, L) has the following forms in the three regions, which are distinguished from one another by the subscripts I, 11, and 111:
86
WOLFGANG FRANZEN AND JOHN H. PORTER
Here we have set I W, I = - W(rb,L) and K = Jon1~’/[(2rn)’~’(kT)’/ZrZ]. The symbol erf refers to the error function defined by erf 3, =
2
A
exp ( -x’) dx.
A further integration of the differential do‘ over L can now be performed, using either an assumed form of U(r),as in the first step of the calculation, or the form obtained in the previous iteration, as in all subsequent steps. [Note that the potential function U ( r )is contained in W(r,L) by Eq. (13).] Then
where we have set t = L2/2rnr?kTand G(r, L) = F(r, L) exp [W(r,L)/kT].
(21) U ( r )is known at a series of radii r j , as discussed below. For each value of U(rj),the integral of Eq. (20)has been evaluated numerically by use of the Gauss-Laguerre quadrature formula which may be written (44) j o m e - y ( x ) dx = i = 1w if ( x i )
+ R, .
(22)
Here x i is the ith zero of the Laguerre polynomial I,,(x), and the weights w i are given by
The remainder R, has the value (2n)! Tabulated values of x i and wi are available (44). In terms of the expansion (22), the expression for the charge density then becomes
where now
f(5) = G[r, (2rnri$kTt)’/’]. The different forms of F(r, L) [related to G(r, L) through Eq. (21)] for the three regions that should be used in this expression are given by Eqs. (19a)-( 19c).
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
87
V. SOLUTIONOF POISSON’S EQUATION In the case of spherical symmetry, as assumed here, the electrostatic potential U ( r )between cathode and anode depends on the radial coordinate r only. Hence it suffices to consider the radial part of Poisson’s equation:
d 2 U ( r )+------. 2 d U ( r ) - o(r) dr2 r dr K, Here x0 is the permittivity of free space. This equation must be solved for V ( r )subject to the boundary conditions U ( r o )= 0 and V ( i A ) = U A ,where ro and r A are the radii of cathode and anode, when the charge density o(r)is assumed known. In order to express U ( r )and o(r)numerically, it is convenient to choose a discrete set of radii between ro and rA, and to define corresponding sets of discrete values for U ( r ) and o(r) at these points. We thus have three onedimensional arrays,
r j , U j , and
aj;
j = 1, 2,
..., N .
Here rl = To, r N = r A , r j + > r j , and N is the number of points in the mesh. The actual radii r j may be chosen in any suitable way; the simplest choice is to make them equally spaced. However, for economy in computer time and machine storage capacity, N should be as small as allowed by the amount of detail desired in the result. In the case of a small spherical cathode surrounded by a large concentric spherical anode, the potential varies rapidly with radius near the cathode, but much more slowly near the anode. Consequently, it is desirable to have a high density of mesh points when the radius is small, and a low density when the radius is large. To this end, we define a set of mesh steps dj by
6j=rj+l - r j ;
j = 1,2,
..., N ,
(27 1
j = 1, 2,
..., N - 1.
(28)
where we let
6j+l = Saj = Sjd,;
Here the letter j used as a superscript denotes an exponent, whereas the subscriptj serves as an index. Thus we will have equally spaced mesh points if we let S = 1, but mesh points of increasing separation for S > 1. We also impose the condition rl = ro and rN = r A , as stated earlier, so that N- 1
rA=
r,+
C
j= 1
N- 1
dj= r,+
C
j= 1
SJ-ld1.
88
WOLFGANG FRANZEN AND JOHN H. PORTER
Thus if S, N, r o , and rA are specified, the value of 6, is determined:
The radius of the jth mesh point will then be given by
In our computation, the values S = 1.05 and N = 160 were chosen. To illustrate the geometrical meaning of this choice, we may define an effective mesh size Ti by
rjthus measures the ratio of the cathode-anode radial distance to the size of the mesh step dj. For the values of S and N just given, Tj varies from about 20 near the anode to about 47,000 near the cathode. For j > 60, the distance scale defined by (31) is approximately logarithmic¶as may be shown by considering that for S = 1.05, SS9= (1.05)” z 18.
Hence we can let S j - - 1 z S j - for j > 60, with an error of about 5 % for j = 60, but a much smaller error as j increases. We can also set
SN- - 1 z SN- to a high degree of approximation. With these approximations¶ (31) becomes r j - ro z ( r A - ro)Sj-N,
(33)
which may be solved for the index j to yield
a relation that will prove to be useful later. To find a numerical method for determining a set of values of the potential U, from a given set of values of the space-charge density aj in this unequally-spaced array, let a = 6,-
= rj
- r j - 1,
/3=aj=rj+, -rj,
(35)
where of course a and /3 are both functions of the indexj. Then the potentials
89
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
U j p 1 , U , , and U j + l at the three neighboring radii r j - l , r j , and r j + l are related to each other by the truncated Taylor expansions
(Y)j+ -ip ’(”b,”Ij uj-a (Y), i (2’1,’
uj+l=uj+p
2
~
Uj-I =
~
+-az
,
(37)
~
These equations may be solved for the two derivatives, with the result u 2 ~ j +-l p2Uj-l
+ (/I-’a 2 ) U j
ffbb+ P ) U U j + ]+ pu,-, - (ff + P ) U j
f
1-
(39)
When these expressions are substituted in Poisson’s equation (26), the difference equation
uJ.= a( 1 + u/rj)Uj+ + p( 1 - p / r j ) U j - - @(a + p)oj/2lcO a + p + (a’ - p 2 ) / r j ~~
~
~
f
(40) is obtained to provide a value for the potential U j at the point rj in terms of the space-charge density ojat r,, and the potentials U j + at r j + 1, and U j at r j In order to compute the potentials at each of the mesh points, the method of successive overrelaxation (38) has been employed. At the beginning of the computation, an arbitrary set of values of the potential U j and the space charge 0, at each of the mesh points rj is assumed. A sweep over the mesh from one end to the other, with fixed potentials at the mesh boundaries, U = 0 at r l = ror and U , = U , at r, = rA, then yields a set of corrections to each U j based on (40).The actual correction applied at each point, however, is made larger than the computed value by a constant factor, the overrelaxation constant a. The optimum value of a is determined empirically so as to achieve rapid convergence of the solution obtained by successive sweeps over the mesh. In our case, a value of a = 1.9395 was chosen. The absolute value of the largest change in any one U j produced in one pass was used as a measure of convergence. When this quantity had decreased to some preset limiting value (usually one part in lOI4 attained with an IBM 360 model 50 computer), the computation was terminated. To check the accuracy of the difference equation (40) as a replacement for Poisson’s differential equation (26), a computation was carried out for the
90
WOLFGANG FRANZEN AND JOHN H. PORTER
case of zero space charge (aj = 0 for all r j ) for which an exact solution (the solution of Laplace’s equation for the potential between two concentric spheres) is known. The two solutions differed at most by one part in lo3, which was considered satisfactory. VI. SELF-CONSISTENT SOLUTION FOR SPACE CHARGE AND POTENTIAL Two related calculations have been described above. In the first place, the space charge aj is computed at a set of radii r j when the corresponding potentials Ujare assumed known; the current density J , at the surface of the cathode is also specified. In the second place, the set of potentials U jis in turn calculated from the values aj of the space charge. In order to find a self-consistent solution for both quantities, both types of calculations must be carried out alternately until repetition of the procedure produces no further change, that is, until both solutions converge. As in the computation of the potential, however, an adjustable relaxation factor is needed in order to achieve a convergent solution without excessive oscillation of the calculated values. When the charge density is computed from a new set of potentials, the predicted changes in oj are usually larger than necessary in this case, so that underrelaxation must be employed. To illustrate this procedure, let a>be the space charge at r j calculated from the newest set of potentials Uj,and let aj be the previous value of the same quantity. Then a; rather than a>is used to compute the next set of potentials, where a; = aj
+ b(oJ-
(41) The underrelaxation factor b used here was normally in the range 0.2-0.3. Uj).
Figure 3 illustrates the effect of the underrelaxation constant on the damping of oscillations observed in successive iterations of the self-consistent solution. The formulas and the sequence of operations described above have been incorporated in a computer program written in IBM Fortran IV(G) language, which was executed in double precision on an IBM 360 model 50 computer. A detailed account of the solutions thus obtained for the spherical geometry described above, as well as for the case of cylindrical symmetry (cylindrical cathode surrounded by concentric cylindrical anode) will be found in Porter et al. (33). This reference also contains a demonstration of the influence of the initial angular distribution of the thermoelectrons on the space-charge-limited potential, a plot of the changes in radius and depth of the potential minimum as functions of cathode temperature, and a comparison of the computed potential variation with the predictions of the classical Langmuir-Blodgett model (45, 46). This model neglects the initial energy of
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
91
20.0 Nlok
2.01 10 ~ ' ~ " " ' ~ ~ " " " ' " ' " ~ ' ' ' ~ " ~ ' ~ ' " ' 1 ' ' ' " ~ 5 10 15 20 25 30 35 40 45 50 Iteration Number
FIG. 3. Example of the variation of the computed anode current with major iteration number for insufficient damping of the solution, i.e., an excessively large value of the underrelaxation constant b.
the electrons entirely, so that an unlimited supply of zeroenergy electrons is available at the surface of the cathode, and there is no potential minimum in front of the cathode. Of principal interest to a study of the energy distribution of spacecharge-filtered thermoelectrons is the form of the solution applicable to the configuration of our experimental study of this problem, as presented in graphical form in Fig. 4. In this case, the hemispherical cathode was a Philips dispenser cathode (47), the ratio of anode radius to cathode radius rJr0 was 138, and the effective anode-cathode potential was 4.80 V at a cathode temperature of 1315"K, after a correction for the contact potential had been applied by a method that will be described later. The work function of the = cathode was assumed to vary with temperature according to Wo(T) (1.67 + 3.2 x 10-4T) eV (47, 48), and Richardson's constant (the only adjustable constant in the solution, as explained later) was given a value of A = 7.0 x lo4 A/rnZ("K)'. A special technique was employed in the construction of the electron gun in order to simulate a spherically symmetric electric field despite the use of hemispherical electrodes, as described in SectionVIII. The change in shape of the negative potential region near the cathode as a function of cathode temperature is influenced by three factors, all of which have entered into the calculations outlined above: (i) The emission current is a function of temperature in accordance with Richardson's law [Eq. (711;(ii) the anode-cathode potential difference seen by an electron depends on temperature, because the work function of the cathode and therefore the
WOLFGANG FRANZEN AND JOHN H. PORTER
UA = 4.00 Volts h / r o = 136
0
0
L
I
Mesh Point Index Number
1 I I I I
I I Ill 1.1
1.2
1.5
2.0
3.0
5.0
I 10
1 I l l 20
50
I 100
Distance from Origin in Cathode Radii
FIG. 4. Plot of the potential distribution between two concentric spheres of the same dimensions as the cathode and anode of the electron gun used in this experiment, as computed by the method described in the text. Four different cathode temperatures are indicated. Mesh point index number refers to the nonlinear scale used in the solution of Poisson’s equation. The effective anode-tocathode potential was assumed to be 4.80 V at 1315°K.At other cathode temperatures, the potential is slightly different because of the variation of the cathode work function with temperature.
anode-cathode contact potential vary with temperature; (iii) the energyand-angle distributions of the emitted electrons, which enter into the calculation of the space-charge density, are functions of temperature. For the highest cathode temperature (1412°K) the second zero of the potential occurs near r r 2r,, corresponding to a mesh-point index number j r 60. Beyond that point, the variation of potential with radius is approximated reasonably well by a straight line, on the nonlinear scale employed for the abscissa in Fig. 4. If we call u ( j ) = U , = U(r,) the potential as a function of the index j, this relation may be expressed by u ( j )s
U A
~
100 ( j - 60);
j 2 60,
(42)
where U Ais the anode potential. We recall that N = 160 is the total number
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
93
of mesh points, so that ~ (1 6 0= ) U A = U(rA).In view of the approximate relation (34) between j and rj we can write
which holds for j 2 60, when S = 1.05 and rA/ro= 138, as in our experiment. (42) may then be written in the useful form U ( r ) z 0.2UA log
(43)
VII. EFFECTOF SPACE-CHARGE BARRIER ON ELECTRON ENERGYSPECTRUM
To compute the effect of the space-charge barrier in front of the cathode on the energy spectrum of the thermoelectrons, we consider the two forms of the half-Maxwellian distribution (10) and (11) that describe the energy and angle distribution at the instant of emission, after integration over the ignorable coordinate 4: q,(E,,
2
(
8,) dEo d8, = (kT)Zexp - $ ) E , ~
dE, sin 8, cos 8, d8,, (loa)
Here we have dropped the subscript o on L = Lo. In these equations, we recall that E, and E , ~are the total energy and the radial part of the kinetic energy at r, , respectively, 8, is the angle made by the velocity vector of the electron with respect to a radius (i.e., a normal to the surface of the cathode), and L is the constant angular momentum about the origin. The condition that a given electron be able to pass through the potential barrier in front of the cathode is simply that its outward radial velocity, given, for example, by (15), remain positive for all values of r. We may formulate the passage condition equally well in terms of the radial kinetic energy E, given by (12b): E,
= E,
L2 - -+
2mr2
eU(r) > 0,
(12c)
94
WOLFGANG FRANZEN AND JOHN H. PORTER
for all r. Here E , = cr0+ L2/2mr:, as in (12a), and U ( r ) is the electrostatic potential between cathode and anode, as computed in the last section and plotted, for example, in Fig. 4. Equation (12c) implies that for a given value of E,, the angular momentum L must not exceed a maximum value given by where the subscript “min” on the right-hand side indicates that the quantity in braces is to be evaluated at its smallest value between r = ro and r = r A . A subsidiary condition E, + eU(r) 2 0 for all r is implied here. We may then formulate (12c) as the condition 0 < L < La,.
(45 1
Now since L = Lo = muT,ro = ro sin 8 , ( 2 m ~ , ) ~where / ~ , uTo = u, sin 8, is the initial transverse velocity, the maximum value of L defined by (44) corresponds to a maximum value of the angle 8, that the initial direction of motion may make with the surface normal:
The passage condition then becomes
0 I 8, < (eo)max
*
(471
It is implied here that (8,),ax I 4 2 , and that the quantity in braces is to be evaluated at its smallest value in the range ro < r < r A . We now consider the form (loa) of the initial energy spectrum of the thermoelectrons. When this equation is integrated over 8, from 8, = 0 to 8, = ,8 ,(), while E, remains fixed, what remains is just the distribution in energy E, of the electrons that have surmounted the space-charge barrier:
where the quantity in braces is evaluated at its minimum value for each value of E , , subject to the condition E, 2 - e U ( r ) for all r. Since U ( r )has a negative minimum u d = U(rd)= - I U d 1 at the radius of the virtual cathode, this condition may be written E, 2 e I Ud 1, so that the range of energies for which (48) is valid is defined by epdI
IE,
(49)
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
95
The solution of the potential problem discussed earlier has provided us with a set of values U j of the potential at a corresponding set of radii r j . It is then a simple matter to program a computer to locate the smallest value of the quantity in braces in Eq. (a), for every value of E,. This result should be compared with a half-Maxwellian distribution in energy, independent of angle of emission, that can be obtained by integrating Eq. (10a) over all values of 8,:
On comparing (48) and (49) with (50), it is clear that the filtered energy spectrum is not simply a half-Maxwellian distribution cut off by the minimum U d= U(r,) of the potential outside the cathode, but that a more complex distortion of the spectrum has taken place. This distortion is evidently caused by the dependence of the passage condition (45) or (47) on the angle of emission at the surface of the cathode. The particular manner in which a space-charge minimum will distort an electron energy distribution clearly depends on the geometry of the electrodes. Our result is valid only for concentric spheres. VIII. DESIGN PRINCIPLES FOR A SPHERICALLY S ~ M E T RGUN: IC FIELDMATCHING A spherically symmetric electron gun has several important advantages. Since the angular momentum of an emitted electron is a constant of the motion as a consequence of the spherical symmetry, its transverse velocity is inversely proportional to the radial distance from the center of the cathode. If the anode sphere is much larger than the cathode sphere, the transverse velocity can be made negligibly small near the anode. The electron beam beyond the anode will then be largely free of crossfire, a desirable condition for the operation of an electron monochromator. Furthermore, since the electron orbits near the anode are then almost completely radial, the effective source (defined by the region of intersection of the tangents to the orbits extrapolated backward toward the cathode) may be very small, smaller than the cathode sphere itself. In practice, however, it is not possible to construct a completely spherical cathode that is supported mechanically and heated electrically. Instead, segments of spheres must be employed as electrodes, as, for example, two concentric hemispheres. As illustrated in Fig. 5, the hemispheres necessarily possess a boundary plane. The field over the surface of this plane must then be adjusted in such a way that an electron moving from one hemisphere to another does not sense the presence of a boundary, or in other words, so that
96
WOLFGANG FRANZEN AND JOHN H. PORTER
I
I
I
I
\
,
I / I
FIG.5. Diagram of a spherical cathode C and a concentric spherical anode A divided into hemispheres by the plane P. Electron beam is considered to exist in the upper hemisphere only.
the electric field between the hemispheres is purely radial, as it would be between two complete spheres. This objective can be accomplished in principle with Pierce’s method (49), widely employed in many devices (37-41). Pierce’s approach is to consider the field space (in the vacuum outside the cathode) as divided into a region filled with space charge in the form of streaming electrons, and a charge-free region. The division between these two regions is maintained by a suitable configuration of equipotential electrodes in the charge-free region. We have chosen instead to use a solid (material) boundary in the form of an annular electrode. A potential distribution is maintained over the surface of this electrode so as to match the radial variation of the space-chargelimited potential between two concentric spheres, as obtained from our solutions of Poisson’s equation. The principle of the gun structure is illustrated in Fig. 6. Our objective is to maintain a potential U ( r ) over the surface of the electrode E such that U ( r ) varies with radius just as in the Poisson solution shown in Fig. 4 for a particular cathode temperature. It is clear. that there are two distinct matching problems, one for the radial region I , Ir IrE between the radius of the cathode r0 and the radius rE of the circular hole in the electrode E, and a second one over the surface of E, from rE to the anode radius r A . As far as the outer region is concerned, we observed earlier that the space-charge-limited potential between two concentric spheres is approximately logarithmic with radius for I 2 2r0, as expressed, for example, by Eq. (43):
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
FIG. 6. Diagram of a practical hemispherical structure. C-Cathode; A-anode; electrode. The plane P of Fig. l lies on the surface of the electrode E.
97
E-
This holds for rA/ro= 138, as in our experiment. U Ais the potential of the anode, including the anode-cathode contact potential, with respect to a zero of potential at the cathode: U(r,) = 0. Such a variation of potential with radius can be achieved readily over the surface of a circular (annular) disk by allowing a constant radial current to flow through a layer of conductive material of uniform resistivity, from an inner circular hole of radius rEto an outer rim of radius rA. The second region over which fields must be matched is in the vacuum between the cathode and the inner hole of the electrode E; it extends from r = r, to r = rE in the (extrapolated) plane of the surface of E. This plane intersects the cathode just where the hemispherical tip of the cathode is joined to its cylindrical stem. Our procedure has been to choose that value of the hole radius rE for which the field in the cylindrical region, below the plane just referred to, best approximates the solution for concentric spheres illustrated by Fig. 4 for radii between r, and rE. To this end, the potential as a function of radius between two concentric cylinders of radii ro and rE,and held at fixed potemials V(r,) and U(rE),respectively, must be compared with the variation of the potential between two concentric spheres of the same radii and held at the same fixed potentials. A solution of the space-charge problem for two concentric cylinders is available (33). A comparison between the two potential functions, as plotted in Fig. 7 for various values of r E , shows that the cylindrical potential always lies below the corresponding spherical one, and that the best fit will be obtained when the radius rE is made as small as allowed by the extent of the logarithmic region. ) the outer cylindrical electrode In the plot of Fig. 7 the potential U ( r E of of radius rE has been fixed at the same value as the potential of a sphere of the same radius. It is of interest to investigate whether the match of spherical
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WOLFGANG FRANZEN AND JOHN H. PORTER
-1.0
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FIG. 7. Example of the potential variation between concentric cylinders (dotted and dashed lines) compared with that between concentric spheres (solid line), when the inner cylinder (cathode) has the same radius as the inner sphere, and the outer cylinder is given several different radii. For each radius, the outer cylinder has a potential equal to the concentric sphere's potential at that radius. In this example, the cathode temperature was chosen to be 1300"K,the cathode-to-anode potential 4.4 V, and Richardson's constant 10 x lo4 A/m2 deg'.
and cylindrical potentials can be improved by giving the cylindrical electrode a somewhat higher potential than the spherical potential at rE. For this purpose, we have plotted in Fig. 8 the potential as a function of radius between two cylinders of radii ro and rE = 4r0, respectively, when U(rE)is varied, while V(r,) = 0 remains fixed, as always. Furthermore, at r = rE,the plot of V ( r )so obtained is joined to a logarithmic potential between rE and rA, with constant U(rA),Superimposed on this graph is a plot of the potential function between two spheres of radii ro and rA, respectively. It is clear that for the best overall match between the cylindrical and the spherical potential functions, U(rE) for the cylindrical electrode should be somewhat larger than the spherical potential for the same radius. Such a procedure was followed in our experiment. However, at best only an approximate match of potential functions can be achieved by this technique, as is clear from Fig. 8.
ENERGY SPECTRUM OF THERMIONIC ELECTRONS 50
-
4.5
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99
-Solution
for complete spherical geometry U(r ) = 1 15 Volts
...............
FIG. 8. Composite plot of the concentric cylinder potential between the cathode and rE joined to a logarithmic potential from rE to the anode, compared with the potential that would exist between concentric spheres. Dimensions are the same as those in the gun actually constructed.
IX. CONSTRUCTION OF ELECTRON GUN On the basis of the design principles just outlined, we have constructed the electron gun shown in Fig. 9a and b. The cathode, which has a hemispherical tip of radius 0.13 mm (0.005 in.), is a typeB Philips dispenser cathode encased in a carburized molybdenum body (47,50). In order to define the position of the cathode accurately, the cathode body is supported by two precision-ground fused quartz spacers. The dimensions of the cathode body and the supporting structure are chosen so that play between the pieces is eliminated at operating temperatures by thermal expansion. The hemispherical anode has a radius of 1.75 cm, so that the ratio of anode to cathode radii is 138, as mentioned earlier. The electrode E (part number 2 in Fig. 9b) consists of an annular alumina disk, 0.25 mm thick and having an outside diameter of 3.60 cm, which is coated on the side facing the anode with a thin layer of graphite of
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WOLFGANG FRANZEN A N D JOHN
H.PORTER
t 0.295Cm
FIG. 9a. Diagram of cathode and cathode body. Body ( 1 ) is constructed of carburized molybdenum, whereas the insert (2) consists of impregnated tungsten. Hemispherical tip of the insert juts through the hole in the intermediate electrode shown in (b) and serves as an electron source for the gun. When the gun is assembled, a double-spiral heater coil is inserted in the cathode body from the left.
resistance 200 R, measured from the central hole to the outer rim. The coating extends through the hole of 0.50 mm radius to the back face of the disk, which is covered with a much thicker layer of graphite. Thus the front and rear surfaces of the disk are connected electrically. Contact is made to the outer rim on both sides, so that a conventional (positive) current can flow radially inward over the front surface toward the central hole, pass through the hole, and then flow radially outward to the outer edge on the back side. This pattern of current flow, designed to provide a logarithmically varying potential on the front face, has the advantage of approximately cancelling the magnetic field generated by the current sheets on the two sides of the disk. In order to achieve the correct logarithmic potential distribution, the graphite layer on the front face of the disk must be uniformly thick. For this purpose, the alumina disk was mounted on a heated rotating table during the coating process. Graphite in the form of Aquadag (51) dissolved in distilled water was then sprayed on the spinning disk with the aid of an airbrush. To check the uniformity of the surface layer deposited by this technique, the potential distribution over the front face of a coated disk mounted on the gun was measured, with the result shown in Fig. 10.
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
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FIG.9b. Diagram of electron gun used in the experiment reported here: (1) cathode, (2) intermediate electrode, (3) anode, (4) outer deflecting sphere, and (5) inner deflecting sphere.
The cathode is heated by means of a heater coil inserted into the hollow part of the carburized molybdenum body. The coil is in the shape of a double spiral, so that the current flowing through it generates at most a magnetic quadrupole field. Nevertheless, it was considered desirable to eliminate this residual magnetic field as well by pulsing the heater current on and off.When the heater current is on, the electron flow from the cathode is cut off by application of a synchronous negative voltage pulse to the intermediate electrode. On the other hand, when the heater current is pulsed off, the electrode is maintained at the correct operating potential, as described earlier. A balanced ac bridge technique is used to eliminate switching transients from the potentials applied to the other electrodes of the gun. Thus all measurements of electron energy spectra were carried out in the absence of heater current.
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WOLFGANG FRANZEN AND JOHN H. PORTER
“i
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FIG. 10. Plot of the potential measured on the graphite-coated intermediate electrode with a constant radial current flowing (points) compared with a logarithmic variation of potential (solid line).
The circular exit slit in the anode has a width of 1 mm and transmits an electron beam in the form of a hollow cone of vertex angle 120°, as shown in Fig. 9b. The solid angle subtended by the anode aperture at the cathode is a/10 sr, so that of a typical current of 50 pA flowing toward the hemispherical anode (one-half the value calculated for a complete sphere), 2.5 pA could be expected to emerge from the exit slit (32). With the exception of the cathode, all surfaces of the gun (and all the conducting surfaces of the monochromator described below) are coated with graphite. Thus despite the use of several different metals in the construction of our apparatus (copper, molybdenum, and titanium), a uniform contact potential with respect to the heated cathode is maintained everywhere (52). As far as our experimental measurement of the energy spectrum of emitted thermoelectrons is concerned, the principal feature of the electron gun just described is the radial symmetry of the space-charge cloud that can be expected to form in front of the hemispherical tip of the cathode. This radial symmetry, which simulates the space-charge distribution around a complete sphere, has made it possible for us to predict the filtering effect of the space charge on the energy spectrum.
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
103
X. ELECTRON ENERGYANALYSIS In order to analyze the energy spectrum of thermoelectrons, an electron monochromator is needed. It suffices to have an energy resolution several times smaller than the width (typically about 0.25 eV) of the energy distribution to be analyzed. O n the other hand, a reliable technique is needed to measure the relative energy of electrons selected from different portions of the spectrum. For these reasons, a Purcell-type electron spectrometer ( 5 3 ) was employed under operating conditions that provided an energy resolution of about 80 meV. Different parts of the emitted-electron energy spectrum could be selected by changing the electric field applied to the spherical-sector prism which serves as an energy-dispersing device, as explained below. The relative energy of the electrons emerging from the final (energy-selecting) aperture of the spectrometer was then measured by first accelerating the electrons with a potential that varied linearly with time in saw-tooth fashion, then scattering them through 90" from an atomic beam of helium, and recording the phase of the saw-tooth potential sweep at which the 19.3 eV elastic scattering resonance (34, 3 5 ) appeared. Although the principle of the Purcell monochromator is well-known ( 5 3 ) a brief description emphasizing particular features of our instrument may be helpful. In this connection, we refer to the diagram of Fig. 11. Here C denotes the cathode, H the (circular slit) aperture in the hemispherical anode, S ,
FIG.11. Schematic diagram illustrating the principle of the Purcell electron monochromator, as explained in the text. C is the cathode, H the anode slit, S, and S, the deflectingspheres of radii R and R , , respectively, and Band B' are focal points for two different electron energies. An energydefining aperture at B is indicated. Line CDB is the optical axis of the instrument.
,
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WOLFGANG FRANZEN AND JOHN H. PORTER
is a sector of angle 0 of an equipotential sphere of outer radius R,, and S2 refers to a sector of a larger hollow equipotential sphere of radius R2.The common center D of the two spheres lies on the axis of symmetry C D B of the instrument. An electric field is maintained between S1and S2. For a particular electron energy and normal (perpendicular) passage through the field edge near H, the electron orbit between S1 and S2 is a circle of radius K = i ( R , + R2).Electrons following this circular orbit leave the field region at H and return to the axis at B. The "normal" orbit just described is indicated by the solid curve in Fig. 11. Electrons of the same energy, but leaving the cathode in a slightly different direction (indicated by the angle with respect to the axis), follow an elliptical path between S, and S2,as shown by the dashed curve. To first order, this orbit when continued into the field-free space at right will also cross the axis of symmetry at B, which thus represents a first-order focal point for electrons of a particular energy. On the other hand, electrons of lower energy will experience a greater deflection by the electric field between the spheres and therefore cross the axis at a point B' closer to the prism than B (dotdashed curve). Thus the electrostatic prism represented by the electric field between S, and S2 has the effect of forming a chromatic (i.e., energydispersed) image of the cathode on the axis of symmetry. A particular portion of the initial energy spectrum can then be selected by placing an aperture at B. In operation, the aperture remains in a fixed position, but the potential between the spheres is changed in order to pick out electrons of other energies. In our monochromator, the prism angle is 0 = 70",and q = 30",so that y = - (0+ q ) = 80". (See Fig. 11 for a definition of these angles.) Under these conditions, the angle of inclination of the conical electron beam with respect to the axis is 60" on the object (cathode) side and 10" on the image side. This choice of dimensions results in a linear electron-optical magnification by a factor of about five. In other words, an image of the cathode formed by monoenergetic electrons on the right side of the prism is five times larger than the cathode itself. This degree of magnification has the effect of reducing space-charge effects on the image side, particularly in the crossover region near the axis. The effective energy of electrons inside the monochromator is smaller than their initial energy added to the anodecathode potential (6.7 V)by an amount equal to the contact potential between the cathode and the graphitecoated anode. (The anode in turn is at the same potential as the body of the monochromator, all conducting parts of which are also coated with graphite.) Two different methods were employed to measure this contact potential. In the first place, as the temperature of the cathode is lowered, the observed electron energy spectrum shifts toward lower electron energies, as we would expect from the shrinking of the potential barrier surrounding the
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
105
cathode. At a sufficiently low temperature (1235°K in our case), the repulsive potential barrier disappears entirely. No further shift of the energy distribution occurs as the temperature is lowered still further, as observed experimentally. The temperature just mentioned thus marks the boundary between emission-limited and space-charge-limited electron flow. It is clear that the extrapolated lowenergy foot of the energy distribution recorded at the emission limit corresponds to electrons leaving the cathode with zero initial energy. The overall potential needed to give such electrons an energy equal to the helium resonance energy was measured to be 21.3 V. The difference between this quantity and l/e times the known helium resonance energy of 19.3 eV (34) is the cathode-to-graphite contact potential of 2.0 V at a cathode temperature of 1235°K. The energy of the electrons leaving the monochromator through its exit aperture can also be deduced from the geometry of the deflecting spheres and the potential difference maintained between them. If we use the symbol VAto denote the effective anode-cathode accelerating potential, i.e., the actual potential U Acorrected by the cathode-to-graphite contact potential, the total energy of electrons following the normal path (defined above) through the prism may be written
where E, is the initial energy of the electrons on leaving the cathode, R = f(R, + R,) is the mean radius of the two deflection spheres, e is the electronic charge, and X(R) is the (radial) electric field between the two spheres:
Here AU denotes the potential difference between the two spheres. Combining (51) and (52) we obtain
For electrons of zero initial energy, E, = 0, so that
where AU, is the potential difference between the deflecting spheres when electrons of zero initial energy are focussed on the exit aperture. In our case, (R2/R1) - (R, /Rz) = 0.459 and AU, = 2.17 V so that (53b) yields V:,= 4.73 V. The difference between this value and the applied potential
106
WOLFGANG FRANZEN AND JOHN H. PORTER
U A= 6.7 V again equals the anode-cathode contact potential at a cathode temperature of 1235"K, in approximate agreement with our earlier argument. As stated earlier, the work function of the cathode was assumed to vary with temperature in accordance with ( 4 7 ) :
W,(T)= (1.67 + 3.2 x 10-4T) eV.
(54)
Since the contact potential discussed above is equal to the difference between the work functions of graphite and of the cathode, the temperature variation (54) implies that the effective anode-cathode potential UX will vary with temperature in accordance with UX(T)= UX(1235"K) + 3.2 x 10-4(T - 1235) V.
(55)
As discussed in Section VI, this temperature dependence was assumed in the computation of the space-charge-limited potential inside the electron gun. The experimental determination of the temperature that marks the transition between emission-limited and space-charge-limited current flow also makes it possible to assign a value to Richardson's constant A for our cathode. It is well known (40) that the value of A for practical cathodes is usually smaller than the theoretical value A = 120 x 104A/m2("K)Zgiven by Eq. (8). Only under exceptional circumstances (22) is the theoretical value approached. For these reasons, it is appropriate to regard Richardson's constant as an adjustable parameter, indeed the only such parameter in our analysis. We have chosen A so that in a gun of spherical geometry in which cathode and anode have the same dimensions as in our gun, and under the influence of an anode-cathode potential varying with temperature as discussed above, the predicted transition temperature has the experimentally determined value of 1235°K. The value of A so defined is A = 7.0 x 104A/m2("K)2. After passing through the exit aperture of the monochromator, the energy-analyzed electron beam enters an asymmetrical three-element cylinder lens which serves to accelerate the electrons, and at the same time to focus them on an atomic beam of helium crossing the electron path at right angles, as illustrated in Fig. 12. For a final electron energy of about 20 eV, the convergence half-angle of the conical electron beam is reduced from 10" to about 2.5" of arc by the action of the lens. Helium gas used to form the atomic beam is first purified outside the vacuum system by passage through liquid-nitrogen-cooled activated charcoal, and is then collimated with the aid of a mosaic collimator consisting of an array of a large number of 2 pm diameter holes in a glass disk 1.5 mm thick. Electrons scattered by the helium beam through 90"are collected in a
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
107
FIG. 12. Diagram that illustrates the acceleration, focussing and scattering of energyanalyzed electrons. Electrons leave the electron monochromator by passing through the energydefining aperture 1, in order to enter a three-element accelerating-focussing cylinder lens. Three electrodes of the lens are identified by the numerals (2), (3), and (4). Electron beam is focussed on a helium beam that emerges from the mosaic collimator (5). Transmitted-current Faraday cup is shown as (6), whereas (7) identifies three Faraday cups, placed at 54", W, and 126"with respect to the incident electron beam, respectively. (8) refers to one of four collimating copper blocks which serve to define the angular spread of the scattered electron beam. In this experiment, only electrons scattered through 90" were recorded.
Faraday cup. The current from the cup is amplified by means of a vibratingreed electrometer, the output of which drives the vertical ( y ) axis of an x-y recorder. A linearly rising potential is used both to drive the horizontal ( x ) axis of the recorder and to sweep the final accelerating potential through a range of 1.00 V at a rate of 2 mV/sec. Thus the x coordinate of the pen of the recorder is a direct measure of the accelerating potential applied to the electrons. For example, if the fixed accelerating potential is set at 21.0 V, and the sweep is adjusted so as to go through zero potential at its midpoint, the overall accelerating potential varies from 21.5 to 20.5 V as the recorder pen moves from left to right. Because of the cathode-to-graphite contact potential of about 2.0 V, the actual potential seen by the electron beam varies from 19.5 to 18.5 V in the course of the sweep. Let us now assume for the purpose of illustration that the deflecting potential AU between the spheres S , and S2 of Fig. 11 is set so that electrons of initial energy 0.1 eV pass through the monochromator exit aperture. Such electrons need an additional energy of 19.2 eV in order to attain the helium resonance energy of 19.3 eV. The resonance will therefore appear on the recorder paper at a value of x that corresponds to an applied potential of 19.2 + 2.0 = 21.2 V, that is, 3/10 of the total sweep distance from the left-hand edge of the recorder paper.
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WOLFGANG FRANZEN AND JOHN H. PORTER
If the deflecting potential is set to select electrons of higher initial energy, less accelerating potential will be required to give them a total energy of 19.3 eV, and the resonance will appear on the recorder paper to the right of the resonance discussed in the previous example. Hence the position of the helium resonance on the recorder paper is a direct measure of the initial electron energy. The initial energy spectrum of the electrons can then be obtained by recording helium resonances at different deflection potentials and measuring the monochromator output current in each case, as illustrated by Fig. 13. All fixed potentials in the apparatus are derived from batteries, which are kept in shielded enclosures outside the vacuum system, and are connected to the apparatus through two-stage rf filters.
FIG.13. Set of helium resonances recorded successively on a single piece of recorder paper. Pen of the recorder moves from left to right as the energy of the accelerated electrons decreases. Vertical position of the pen is proportional to the number of electrons scattered through 90"by the helium beam. As the electron energy passes through 19.3 eV, the helium resonance energy, the scattered current experiences a sudden dip. Different resonances recorded on this diagram correspond to different initial electron energies, selected by varying the potential difference between the two deflecting spheres. Horizontal positions of the resonance minima are direct measures of the initial electron energy in each case, as explained in the text. Relative vertical positions of the different tracings have no significance.
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
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The monochromator and scattering chamber are housed in a stainlesssteel vacuum chamber evacuated with a 6 in. diam mercury diffusion pump. A Freon-cooled baffle and a liquid-nitrogen trap separate the vacuum enclosure from the pump. The foreline contains two additional Freon-cooled baffles and a Zeolite trap to prevent contamination of the mercury. When assembled with a copper gasket and baked at 120°C for 24 h, an ultimate vacuum of 1.0 x torr was achieved in this system. In the experiment reported here, a Viton gasket was used in the jaws of the Wheeler flange of the main vacuum system, and the monochromator alone was baked by heating it with an internal heater winding. Under these conditions, a vacuum of about 1.2 x lo-' torr could be obtained without gas input. With helium flowing into the system, the pressure rose to several times 10- torr, causing a drop of about 25% in the current caught in the Faraday cup below the scattering region (see Fig. 12), as compared to the transmitted current (typically about 4 x lo-' A) without gas input. The monochromator and scattering chamber are surrounded inside the vacuum system by a vertically mounted cylindrical mu-metal shield, 12 in. diam and equipped with end caps. The top cap has a 1.5 in. diam hole to permit entrance of the coaxial filament leads, whereas the bottom cap has a 6 in. diam pumping hole. To limit the effect of these holes on the residual magnetic field inside the shield, a pair of 36 in. diam Helmholtz coils outside the vacuum system is used to compensate the vertical component of the Earth's magnetic field. The Helmholtz coils are energized when demagnetizing the mu-metal cylinder in situ by passing a slowly diminishing ac current through a winding wrapped around it. After such a demagnetization, the residual magnetic field inside the monochromator did not exceed 0.2 mG in any direction, as measured with a flux-gate magnetometer.
XI. RESULTSOF EXPERIMENTAL STUDY OF ELECTRON ENERGY DISTRIBUTION In order to compare with experiment the shape of the energy spectrum predicted by E q . (48), the cathode temperature must be known. An optical pyrometer focused on the tip of the cathode is a device suitable for measuring this temperature. Since the cathode tip is ordinarily hidden and cannot be viewed optically, the electron gun (minus anode) was taken out of the monochromator and mounted so as to expose the tip to view. The stainlesssteel bell jar normally used as a vacuum envelope was replaced by a glass bell jar. The system was then evacuated and the cathode heated in the same steps of heating power that were used in the experimental study of the energy spectrum. For each setting, the tip was viewed through a calibrated optical pyrometer. Brightness temperature readings for tungsten obtained in this
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WOLFGANG FRANZEN AND JOHN H. PORTER
manner were converted to absolute temperature with the aid of a table of emissivities (54). Four different power settings were chosen, at which the temperature of the cathode was found to be 1253", 1315", 1358",and 1412"K, respectively. It is believed that these temperatures are accurate to f 15°K. In order to measure the shifts that take place in the electron energy distribution as the temperature of the cathode is changed, an absolute scale is needed for the overall potential applied between the cathode and the scattering region (ground potential in our apparatus). It is desirable to measure this potential without drawing current from the cathode circuit. We were able to satisfy this condition approximately by connecting the emf to be measured in series opposition to the potential generated by a set of batteries. The connection between the two circuits was completed through the input impedance (lo8 Q) of a vacuum-tube voltmeter. The batteries were so chosen that the difference between the two opposing potentials did not exceed 0.1 V, as read on the voltmeter. It follows that the current drawn from the cathode circuit by the measuring circuit was less than lo-' A. With the two circuits connected in opposition, as described, a divider constructed of 0.025% precision resistors was connected across the batteries in such a way that a known fraction of the potential drop could be measured with a potentiometer that had a standard-cell reference. It is estimated that this technique provides values of the overall potential to an accuracy of k 6 mV. After every such measurement, and prior to recording a set of helium resonances, a fiducial mark corresponding to the measured cathode potential was inscribed on the recorder paper. In this fashion, the same absolute voltage scale was established on every sheet of recorder paper. As we know from our earlier discussion, during a run at a fixed cathode temperature the accelerating potential applied to electrons emerging from the exit slit of the monochromator was swept linearly over a 1 V range. Somewhere during a sweep, the vertical position of the recorder pen would experience a sudden dip, because of a drop in the 90" scattered electron current provoked by the 19.3 eV helium resonance. From the calibration of the recorder and the position of the fiducial mark mentioned above, the accelerating potential corresponding to the dip was known precisely on the absolute scale described earlier. On going from one sweep to the next, all experimental conditions remained the same, except for the intersphere potential AU, which was changed in small steps, thereby selecting successively different slices of the initial electron energy spectrum. A change in AU causes the resonance pattern to appear in a different place on the recorder paper. The initial energy distribution is then obtained by plotting the electron current emerging from the monochromator (and caught in the transmitted-current Faraday cup) as a function of resonancedip position, expressed on the absolute voltage scale previously established.
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The results of runs at the four measured cathode temperatures have been superimposed in Fig. 14. Experimental points are identified by discrete symbols. We note that as the cathode temperature is lowered, the pattern shifts to the left, toward higher accelerating potentials, or what amounts to the same thing, toward lower initial electron energies, as we expect from the shrinking of the potential barrier of the virtual cathode. By reducing the cathode temperature below the lowest measured temperature (1253"K), we were able to show that the shift of the pattern to the left ceases, indicating the vanishing of the repulsive barrier. In this way, the absolute shift of Acceleratlng Potentla1 (VOIIS)
1.Ooo-
21.200
21.100
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I
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600
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FIG. 14. Comparison between experimental energy distribution (discrete points) and theoretical distribution (curves) computed by the method described in the text, at four different cathode temperatures.
the pattern recorded a t 1253°K was shown to be 14 mV. By extrapolating a plot of shift versus cathode temperature to zero shift, as illustrated by Fig. 15, it was found that the transition from space-charge-limited current flow t o emission-limited flow takes place at 1235°K for our cathode. As explained in Section X, this argument enables us to deduce the cathode-graphite contact potential by comparing the extrapolated foot of the 1235°K energy spectrum (obtained by shifting the 1253°K spectrum
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WOLFGANG FRANZEN AND JOHN H. PORTER
Cathode Temperature
\OK)
FIG. 15. Plot of the position of the extrapolated low-energy foot of each of the energy distributions shown in Fig. 14, as a function of cathode temperature. Dots indicate the positions of the feet; a straight line has been drawn through them. Intercept of this line with the temperature axis corresponds to the temperature at which the spacecharge barrier in front of the cathode disappears, as explained in the text.
14 meV to the left), namely 21.3 eV, with the known (34) helium resonance energy of 19.3 eV. Furthermore, we are then able to establish an initial electron energy scale: The initial electron energy at any point on Fig. 14 is equal to 21.3 eV minus e times the accelerating potential at the point in question, as indicated at the bottom of the graph. Finally, the correct value of Richardson's constant for our cathode is then determined so that in the solution of Poisson's equation for our gun geometry, the negative minimum in the potential will disappear at 1235"K, as discussed earlier. Using this value of Richardson's constant (A = 7 x 104A/mZ("K)2),the energy spectra were computed numerically from Eq. (48), using the potentials of Fig. 4, but then convoluted with an assumed Gaussian resolution function having a full width at half maximum of 82 meV, which corresponds to the measured resolution of the spectrometer. The convolution was computed numerically with the aid of the Gauss-Hermite quadrature formula (44). The resulting distribution functions at the four measured temperatures are represented by the continuous curves superimposed on the
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experimental points of Fig. 14. It is clear that the experimental and theoretical spectra agree well with one another, particularly at the highest and the lowest temperatures employed in the experiment. To investigate the agreement between the predicted and the observed shapes of one of the energy distributions more closely, a spectrum with a large number of experimental points was recorded at the highest operating temperature (1412"K), where deviations between theory and experiment should be most marked, according to the views of some investigators. The results of this experiment are plotted in Fig. 16. (The measurements of Figs. Accelerating Potential (Volts)-
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FIG. 16. Detailed study of the electron energy distribution at a cathode temperature of 1412°K. Discrete points are experimental,but the continuous curve is a theoretical distribution calculated on the basis of an initially half-Maxwellian energy spectrum, filtered by the space charge of Fig. 4, and convoluted with the resolution function of the spectrometer.
14 and 16 were carried out about one week apart.) As in the earlier experiment, there is no evidence of any significant distortion or shift of the experimental spectrum relative to the spectrum calculated on the basis of a half-Maxwellian initial energy distribution filtered by an isotropic spacecharge barrier surrounding the cathode.
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WOLFGANG FRANZEN AND JOHN H. PORTER
XII. RELATIONBETWEEN RESULTSREPORTEDHEREAND EARLIER WORK Many experimental results have been published on the subject of the energy distribution of thermoelectrons (6, 9-15), following Boersch’s discovery ( 1 ) of the so-called anomalous energy spread. In most cases, energy spectra much wider than a half-Maxwellian distribution and energy shifts depending on some power of the current density were reported. Explanations advanced for these observations include the interaction of the electrons with longitudinal space-charge oscillations (4,statistical fluctuations of the current density ( 5 9 transverse instabilities in the electron beam (7), fluctuations of the electric microfield (56), noise fed into the electron beam by the electron gun (1 7), an energydependent reflection of electrons approaching the surface barrier (57), and Coulomb interactions between individual electrons (58), leading to broadening by a relaxation process (9, 18). The entire subject of broadened energy distributions in electron beams has been reviewed recently by Zimmermann in another article in these Advances (18), and it would serve no purpose to review the subject again at this point. However, it is appropriate to discuss the mechanism of energy broadening by relaxation in relation to our experimental results. Zimmermann (18) and others have shown that in a drifting dense electron beam, a relaxation process is operative in the course of which the thermal agitation energies stored in the longitudinal (beam motion) direction, and in transverse directions, tend toward equipartition. The necessary exchange of energy is brought about by collisions between the electrons in the beam. When viewed from the point of view of a coordinate system moving with the average beam velocity, one may regard this process as an approach to thermal equilibrium among the different degrees of freedom, or in other words, an increase in the entropy of the beam. A relaxation temperature TI may then be defined, which characterizes the longitudinal thermal velocity distribution after the attainment of equilibrium. If the longitudinal thermal energy is initially small compared to the transverse thermal energy, such a relaxation process will tend to broaden the energy spread in the direction of motion. These theoretical predictions are not in conflict with the results of our experiment because beam relaxation before energy dispersion is not possible in our monochromator. Immediately after passing through the anode slit, our electron beam is dispersed in energy as it passes between the deflecting spheres. Furthermore, in contrwt to the high transverse and low longitudinal electron temperatures assumed by Zimmermann and characteristic of electron guns with converging beams, conservation of angular momentum in
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
115
the diverging beam leaving the cathode of our electron gun causes the transverse thermal velocity to become very small compared to the longitudinal thermal velocity. If relaxation were present in our apparatus, its effect, therefore, would be the opposite of the effect envisioned by Zimmermann: The transverse thermal energy would increase at the expense of the longitudinal thermal energy. It is also important to emphasize, as Zimmermann has done (59), that a relaxation process may broaden an energy distribution, but cannot bring about an overall shift. The large energy shifts observed in the early experiments referred to above suggest the presence of substantial space-charge barriers. A correlation between the amount of the shift and the observed broadening of the spectrum pointed out in many cases indicates that both effects proceed from the same cause, namely the filtering of the electron energy distribution by the repulsive space-charge barrier surrounding the cathode. If this barrier is not uniform, as seen by electrons following various possible orbits from cathode to anode, then the energy distribution will be both broadened and shifted. Unless special precautions are taken in the design of the gun, the spacecharge barrier will usually be nonuniform. The burden of proof that uniformity has indeed been attained is on the experimenter. In the absence of such proof, it appears that no valid conclusions can be drawn concerning the energy spectrum of the electrons leaving the cathode.
GLOSSARY OF SYMBOLS USEDIN TEXT Cartesian coordinates Cartesian components of linear momentum Element of volume of phase space Transverse component of linear momentum Azimuthal angle in the x, y plane Electron energy state inside metal Fermi energy inside metal Planck's constant Absolute temperature of cathode Number of electrons in d@ inside metal Electron rest mass Potential step at the surface of the cathode Initial kinetic energy of emitted electron Work function of cathode Element of area of cathode Element of volume Number of electrons emerging from area d A of cathode in time dt, with specified range of velocity components Initial components of electron velocity parallel and perpendicular to surface of cathode, respectively
WOLFGANG FRANZEN AND JOHN H. PORTER
116
r Jo
A
'1
U j = U(rj)
J'
N
' uJ
Number of electrons emitted per unit area per unit time, with specified ranges of velocity components Number of electrons emitted per unit area per unit time, independent of velocity or direction of motion Electronic charge Surface current density Richardson's constant Initial velocity distribution of electrons Initial polar angle Sacobian of transformation from one system of generalized coordinates to another Initial energy and angle distribution of electrons Radius of spherical cathode Initial radial kinetic energy Angular momentum about center of cathode Initial distribution in radial kinetic energy and angular momentum Radial kinetic energy Electrostatic potential between cathode and anode Radial distance from center of cathode e times the effective radial potential Radius of virtual cathode Radius at second zero of W(r, L) Radius at minimum of W(r, L) Ratio of initial angular kinetic energy to kT Radial electron velocity Space-charge density Number density of electrons with specified range of velocity and solid angle Element of solid angle Radius of point of lowest radial potential between ro and r Differential of charge density integrated over 8," and 4 Function defined by Eqs. (19a)-( 19c) Constant used in (19) and defined there Dummy variable Variable used in (20) and defined there Function defined by Eq. (20) Mesh-point index Radius at the jth mesh point Potential at jth mesh point Weight factor defined by Eq. (23) ith zero of Laguerre polynomial L,(x) Remainder in Gauss-Laguerre quadrature formula Function used to illustrate use of Gauss-Laguerre formula Permittivity of free space Radius of spherical anode Anode potential Potential and space charge at j t h mesh point, respectively Total number of mesh points Mesh step defined by Eq. (27) Mesh step expansion factor Effective mesh size defined by Eq. (32)
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
117
Parameters defined by Eqs. (35) Overrelaxation constant Underrelaxation constant Potential expressed as a function of the mesh-point index j The distribution QJE,, 0,) integrated over 4 The distribution Q , , ( E ~ ~L) , integrated over r#~ Maximum angular momentum defined by Eq. (44) Maximum polar angle defined by Eq. (47) Distribution in energy of electrons that have passed through the space-charge barrier Potential at the radius r,, of the virtual cathode Half-Maxwellian distribution in energy, independent of angle of emission Intermediate (field-matching) electrode Radius of hole in intermediate electrode Inner and outer deflecting spheres, respectively Radii of S, and S, , respectively Sector angle of deflecting spheres Cathode Common center of S, and S, Aperture in hemispherical anode Focal points of monochromator for particular electron energies Mean radius of deflecting spheres Angle between electron path on the object side and axis of symmetry Angles characteristic of monochromator defined in Fig. 11 Electric field between deflecting spheres as a function of R Radial distance from common center of the two deflecting spheres Potential difference between the two deflecting spheres Anode-cathode potential corrected for contact potential Value A U when electrons of zero initial energy are focussed on exit aperture Relaxation temperature of electron beam in Zimmermann's theory
ACKNOWLEDGMENTS We would like to thank Dr. Rajendra Gupta, who first discovered the shifting of the position of the helium resonance when the deflecting spheres were tuned, Dr. Robert Wenstrup, who made significant contributions to the experimental work reported here, and the Boston University Computing Center for the use of its facilities. We are also indebted to the National Science Foundation for the award of several research grants, in the course of which much of this work was carried out.
REFERENCES I . H. Boersch, Z. Phys. 139, 115 (1954). 2. A. R. Hutson, Phys. Rev. 98, 889 (1955). 3. G. F. Smith, Phys. Rev. 100, 1115 (1955). 4. H. Fack, Physik. Verhandl. 6, 6 (1955). 5. W. Veith, Z. Angew. Phys. 7 , 437 (1955). 6 . W. Dietrich. Z. Phys. 152. 306 (1958). 7. B. Epszstein, C . R. Acad. Sci. (Paris) 246, 586 (1958).
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WOLFGANG FRANZEN AND JOHN H. PORTER
8. D. Hartwig and K. Ulmer, Z. Angew. Phys. 15, 309 (1963). 9. D. Hartwig and K. Ulmer, Z. Phys. 173, 294 (1963). 10. K. Ulmer and B. Zimmermann, 2. Phys. 182, 194 (1964). 11. J. A. Simpson and C. E. Kuyatt, J. Appl. Phys. 37, 3805 (1966). 12. W. A. M. Hartl, Z. Phys. 191,487 (1966). 13. W. H. J. Anderson, Brit. J . Appl. Phys. 18, 1573 (1967). 14. J. E. Collin and A. Magnee, Bull. SOC.Roy. Sci. Liege 9-10. 522 (1967). 15. R. Speidel and K. H. Gaukler, Z. Phys. 208, 419 (1968). 16. K.-J. Hanszen and R. Lauer, Z. Naturforsch. A 24, 214 (1969). 17. M. Fisher, J. Appl. Phys. 41, 3615 (1970). 18. B. Zimmermann, Adv. Electron. Electron Phys. 29, 257 (1970).
19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 3 7.
W. Franzen, R. Gupta, J. H. Porter, and R. S. Wenstrup, J. Appl. Phys. 44, 145 (1973). P. A. Lindsay, Adv. Electron. Electron Phys. 13, 181 (1960). R. D. Young, Phys. Rev. 113, 110 (1959). H. Shelton, Phys. Rev. 107, 1553 (1957). A. H. Beck and C. E. Maloney, Brit. J. Appl. Phys. 18, 845 (1967). R. D. Young and E. W. Miiller, Phys. Rev. 113, 115 (1959). A. V. Crewe, M. Isaacson, and D. Johnson, Rev. Sci. Instrum. 42, 411 (1971). J. Hasker, Philips Res. Rep. 20, 34 (1965). J. Hasker, Philips Res. Rep. 21, 122 (1966). J. Hasker, Philips Res. Rep. 24, 23 1 (1969). J. Hasker. Philips Res. Rep. 24, 263 (1969). See Zimmermann (18), pp. 298-299. K. Amboss, Adv. Electron. Electron Phys. 26, 42 (1969). J. H. Porter and W. Franzen, Rev. Sci. Instrum. 44, 868 (1973). J. H. Porter, W. Franzen, and R. S . Wenstrup, J. Appl. Phys. 43, 344 (1972). G. J. Schulz, Phys. Rev. 10, 104 (1963). C. E. Kuyatt, J. A. Simpson, and S . R. Mielczarek, Phys. Rev. A 138, 385 (1965). H. Ahmed and A. H. W. Beck, J . Appl. Phys. 34, 997 (1963). N. Ramsey. “Molecular Beams,” p. 20. Oxford Univ. Press, London and New York
(1956). 38. C. Weber, In “Focussing of Charged Particles” (A. Septier, ed.), Vol. I, Chap. 1.2, pp. 45-100. Academic Press, New York (1967). 39. G. R. Brewer, In “Focussing of Charged Particles” (A. Septier, Ed.), Vol. 11, Chap. 3.2, pp. 23-72. Academic Press, New York (1967). 40. W. B. Nottingham, In “Encyclopedia of Physics” (S. Fliigge, ed.), Vol. XXI, pp. 1-175. Springer-Verlag, Berlin and New York (1956).
41. J. R. Pierce, “Theory and Design of Electron Beams.” Van Nostrand-Reinhold, Princeton, New Jersey (1954). 42. See Lindsay (20), pp. 243-245. 43. See Amboss (31), pp. 34-42. 44. P. J. Davis and I. Polonsky, In “ Handbook of Mathematical Functions” (M. Abramowitz and I. A. Stegun, eds.), Chap. 25. p. 890. Nat. Bur. Std., Washington, 1964. 45. I. Langmuir and K. Blodgett, Phys. Rev. 24, 49 (1924). 46. 1. Langmuir and K. Blodgett, Phys. Rev. 22, 347 (1923). 47. R. Levi, J. Appl. Phys. 26, 639 (1955). 48. G. A. Haas, In “Methods of Experimental Physics,’’ (L. Marton, V. W. Hughes, and L. Schultz, eds.), Vol. 4, Part A, Chap. 1. Academic Press, New York (1967). 49. J. R. Pierce, J . Appl. Phys. 11, 548 (1940). 50. E. S. Rittner and R. Levi, J . Appl. Phys. 33, 2336 (1962).
ENERGY SPECTRUM OF THERMIONIC ELECTRONS
51. 52. 53. 54. 55. 56.
Acheson Colloids Company, Port Huron. Michigan. J. H. Parker and R. W . Warren. Rru. Sci. Insrrum. 33, 948 (1962). E. M. Purcell. Phys. R P U .54, 818 (1938). G . A. W. Rutgers and J. C. DeVos, Physira 20, 715 (1954). W. Veith. Z . Angew. Phys. 152. 306 (1955). P. Schiske. Physik. Verhundl. 12, 143 (1961). 57. See Nottingham (40). pp. 22-24. 58. K . H. Loeffler. 2. Anyew. Phys. 27. 145 (1969). 59. See Zimmermann (18). p. 291.
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Low-Temperature Rare-Gas Stationary Afterglows J.-F. DELPECH, J. BOULMER,
AND
J. STEVEFELT
Groupe d'Electronique dans les Gaz, Institut d'Electronique Fondamentale,. Faculte des Sciences, Universitk Paris-XI-Campus d'orsay, Orsay, France
I.
11.
111.
IV.
V.
Introduction.. .............................................................................. Units and Notations.. ... General Description of th ........................................ A. Electron-Neutral collisions B. Coulomb Collisions ................................................................... C. Particle Diffusion to the Walls D. Electron Energy Balance. ............................................................. Microwave Diagnostic Techniques ....................................................... A. Microwave Propagation in a B. Electron Density and Collisio C. Electron Temperature Measurements Ionic Population .................. ........................ A. Ion Diagnostics by Mass Spectrometry ................ B. Ionic Mobilities, Atomic-to-Molecular Ion Conversions.. .......................... C. Electron-Ion Recombination . ............... ......................... Excited States Populations.. ..... ............... ......................... A. Energy Levels of Rare-Gas Atoms and Molecules .................................. B. Afterglow Spectroscopy .................. ....................................... C. Spectroscopic Diagnostics Techniques . . D. Metastable Population ................... ............... Conclusion .................................................................................
...............
.........................
121 123 126 130 132
140 142 147
160 160 168 177 178
INTRODUCTION Afterglows are particularly well suited to the study of reactions involving excited or ionized atoms or molecules at thermal energies. A laboratory afterglow is the mixture of electrons, ions, and excited molecules present in a neutral gas, either after cessation of the active discharge (stationary afterglow, where time is the main parameter) or far enough from the active discharge region, in a flow of gas (flowingafterglow, where space is the main * Laboratoire associb au CNRS. 121
122
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
parameter). In this review, we shall be particularly concerned with stationary afterglows in rare gases at low temperatures, from cryogenic temperatures of a few degrees kelvin to room temperature around 300°K. In an afterglow, the energy deposited during the discharge in the ionization of the neutral gas and in the excitation of long-lived metastable species relaxes slowly through radiative and collisional processes (typical relaxation times range from a few microseconds to a few milliseconds). In many cases, though macroscopic rates are slow, microscopic rates are fast and a quasiequilibrium is maintained. Because of the essentially total lack of chemical reactivity of the rare gases with themselves in their ground electronic state, the final state of a pure rare-gas afterglow is undistinguishable from its initial state, before the discharge-relaxation is complete. However, as we shall see, the details of the energy transfer processes leading to total relaxation are rather intricate. While their general features are now beginning to be understood, particularly in helium, much remains to be done. Atomic and molecular processes leading to energy relaxation in a plasma are very interesting from a fundamental point of view; there is an increasing realization that their practical importance may also be large in various fields, from thermonuclear fusion generators to magnetohydrodynamics and to far UV highefficiency rare-gas lasers. Such a brief review of this extensive and fast-growing subject is bound to be far from exhaustive. Excellent reviews have been written of late on related subjects, and are referred to in the text, which they complement. Whenever possible, we have deliberately avoided historical exposition; we are acutely aware of having thus omitted a number of important references, but we have been careful to refer indirectly to them through the bibliographies of some of the recent references we have cited.
UNITSAND NOTATIONS S.I. (Systeme International-MKSA) units are used in literal formulas throughout the text, with accepted notations and values of the fundamentals constants (Taylor et al., 1969). For example, h is Planck's constant, h = h/27c, E~ = 1O7/4ncZis the vacuum permittivity (note that po c2 = EO '). Total normalization is, however, not practical in numerical expressions, because of widespread usage (angstroms for optical wavelengths) and also because there are natural yardsticks (Bohr's radius a. = 5.29177 x 10- l 1 meter for atomic and molecular phenomena). In numerical expressions, particle number densities (electronic density n, , neutral density no, ...) are quoted in cm- 3, temperatures in "K, frequencies in Hz,pressures in torr (1 torr corresponds to a neutral particle density no = 3.536 x 10l6 cm-3 at 0°C; in S.I. units, 1 torr is 1.33322 N m-').
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
123
I. GENERAL DESCRIPTION OF THE STATIONARY AFTERGLOW A . Electron-Neutral Collisions A collision between particles is said to be inelastic when the internal energy of at least one of the collision partners is changed. It is said to be elastic if there is no change in internal energy. 1. Elastic Electron-Neutral Collisions
a. Cross sections. In a classical picture, an atom of radius a,, (Bohr radius) would present to an incident electron a velocity independent cross section nai 10- l6 cm2. In fact, atomic dimensions are so small that the size of the quantum-mechanical wave packet associated with the incident electron is comparable to atomic size even at moderate energies, and quantum interference effects have to be taken into account (McDaniel, 1964). Except for helium, cross sections turn out to be strongly energy dependent in the rare gases at low energies. In the case of argon, krypton, and xenon, the cross-section versus energy curves exhibit a very marked minimum: This is the well-known Ramsauer effect. Electron elastic collision cross sections for momentum transfer on raregas neutral atoms are given in Table I. This table has been compiled for nearly equidistant points on a logarithmic energy scale; linear interpolation between adjacent points introduces no significant loss of accuracy. A critical evaluation of measurement techniques is given by Bederson and Kieffer (1971); see also Sol et al. (1973) for comments on the microwave method. Most available experimental results have been discussed by Gilardini (1972). In the case of helium, we have used below 0.1 eV data obtained by Sol et al. (1973) through microwave absorptivity measurements with selective electron heating. Above 0.1 eV, we have used the cross sections deduced from swarm measurements by Crompton et al. ( 1970).Expected systematic errors should be lower than & 5 % over the quoted range. For neon, low-energy cross sections have also been deduced by Sol et al. (1975) from microwave measurements. Above 0.1 eV we have used the results deduced by Robertson ( 1972) from his swarm measurements. Overlap between these two sets of measurements is quite satisfactory over their common range of validity (between 0.03 and 0.3 eV), and expected systematic errors over the quoted range should be lower than f7%. For argon, krypton, and xenon, we reproduce data deduced by Frost and Phelps (1964) from swarm measurements. Systematic errors may be below + l o % far from the minimum, but they might exceed &30% at the minimum. b. Macroscopic rates. In a plasma, individual collisions manifest themselves as macroscopic (average) rates. It is easy to see that on the average
-
124
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
TABLE I
ELECTRON-NEUTRAL ELASTIC COLLISION CROSS SECTIONS FOR crn') MOMENTUMTRANSFER IN RARE GASES (IN UNITSOF Electron energy (ev) 1.8 x 3.0 x 6.0 x 1.0 x 1.8 x 3.0 x 6.0 x 1.0 x 1.8 x 3-0 x 6.0 x 1.0 x 1.8 x 3.0 x 6.0 x 1.0 1.8 3.0 6.0 10.0
10-4 10-4 10-4 10-3 10-3 10-3 10-3 10-2 lo-' lo-' lo-' lo-' lo-'
Mass (au) Mass ratio 2m/M
Helium
5.14 5.19 5.26 5.32 5.38 5.44 5.51 5.57 5.63 5.69 5.77 5.87 6.11 6.35 6.66 6.85 6.98 6.89 6.0 1
Neon
Krypton
Xenon
6.1 4.2 2.8 1.5 0.45 0.20 0.15 0.4 1.1 2.2 4.1 8.7 13.8
26.0 21.0 16.0 10.0 6.8 3.2 0.93 0.49 0.62 1.7 4.8 12.6 19.3
116 87 61 33.5 20.5 10.2 3.2 1.3 1.6 6.9 17 32 32
40 2.72 x 10-5
1.30 x 10-5
-
0.28 0.30 0.32 0.36 0.40
0.46 0.52
0.64 0.73 0.89 1.09 1.40 1.62 1.79 1.91 2.14
-
4 2.72 x 10-4
Argon
20 5.45
10-5
84
-
131 8.32 x 10-6
there is total electron-neutral momentum transfer during a collision: After a collision the electron loses all memory of the direction of its former velocity. However, elementary considerations show that the modulus of its velocity is almost unchanged, electronic mass m being much smaller than atomic mass M: The electron loses only on the average a fraction 2m/M of its kinetic energy during the collision. The mass ratios 2m/M of the rare gases are also shown in Table I. The monokinetic electron-neutral collision rate for momentum transfer at velocity u and at neutral density no is (1) which is usually a function of velocity. When the electron velocity distribution function is Maxwellian-and we shall see later (Section 1,B) that this is hT(V)
= nOgMT(u)U,
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
125
usually the case-the resulting average electron-neutral momentum transfer rate (or collision frequency) at electron temperature T, is
where the bar denotes an average over velocity space. In view of its fundamental importance, this average is often simply called the electron-neutral collision frequency, and is denoted veo (subscript e0 denotes electron neutral). is nearly energy independent (this is In the particular case where aMT nearly true in helium, and may be used with caution for order-of-magnitude estimates in the other rare gases),
The electron-neutral collision rate for energy transfer BENis related to vMT as follows :
-
vEN
1 torr, 273”K, vMT = veo = 2.1 x lo8 sec-I lo4 sec-I. The corresponding energy relaxation time constant is
In
helium
at
= (2m/M)veo= 5.7 x
and
The mean-free-path concept is sometimes useful. The electron-neutral mean free path for momentum transfer is dMT
2
(6)
(nO0MT)-
Electronic motion is randomly directed; an electron at point xo at time is at point x after N mean free paths, with
t =0
(x
- x0)’
= Nd;,
.
(7)
Since it takes M/2m collisions to relax electron energy to l/e of its initial value, the electron-neutral energy relaxation mean free path is
126
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
At 10 torr, U)O"K, A,, = 0.34 cm in helium. Expressions (7) and (8) are fair approximations in helium at low electron density. In the other rare gases, or when electron-ion collisions become significant, they may be used for order-of-magnitude estimates. A more careful evaluation is a complex problem in plasma kinetic theory (Shkarofsky et al., 1966). 2. Inelastic Electron-Neutral Collisions Inelastic excitation or ionization collisions of electrons with ground-state atoms are negligible in an afterglow, where only an extremely small fraction of the electron population has an energy above ground-state excitation or ionization thresholds. The discussion of some particular phenomena involving p = 2 metastable states will be deferred until Section IV,D. The situation is quite different for highly excited states near the continuum limit, where the energy difference between levels becomes of order kT, or lower. These processes play an important role in the study of electron-ion recombination quasiequilibrium and will be discussed together with the closely related problem of collisional-radiative recombination (Section 111,C). A review on electron impact excitation of atoms has been published by Moiseiwitsch and Smith (1968).
B. Coulomb Collisions 1. Coulomb Collision Rates
Charged particles in a plasma interact with each other through a screened Coulomb potential (Shkarofsky et al., 1966).In an afterglow, where ions are usually singly charged, momentum transfer between electrons and ions proceeds at a rate
or numerically in practical units (see Introduction) v,,(sec-
I)
2
n 3.63 2In (AE), T,3/2
where In (Ac), the so-called Coulomb logarithm, is a slowly varying function of electron temperature T, and density n,:
The Coulomb logarithm is equal to 6.5 under fairly typical afterglow conditions (n, = 10'' cmP3, T, = 300°K).
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
127
The electron-ion rate, or electron-ion collision frequency for momentum transfer, v e i ,is of the same nature as the electron-neutral collision frequency ve0 defined previously. The electron-ion rate for energy transfer is similarly (2m/M,)vei,where Mi is the ionic mass. Note that the above rates are defined under the assumption of quasineutrality, i.e., n, n, , which usually holds in a plasma.. Generalization of the electron-ion rate for energy transfer is straightforward in the case of a mixture of ions of different masses. Energy is exchanged between electrons at a rate v,, = v,,, where v,, is the electron-ion collision frequency for momentum transfer given by relation (9). Because of the mass ratio, interelectronic energy exchange is thus orders of magnitude faster than electron-ion energy exchange. Ion-ion and ion-neutral energy relaxation is, on the contrary, very efficient, since the mass ratio is of order unity in this case. When total discharge energy is low enough, it is thus generally fair to assume that = To E Twalls. The situation becomes much more complicated when these near-equalities are not true (Poukey et al., 1969). Electron-heavy particles’ energy relaxation being quite inefficient, the average electron energy may be very different from To. The electron energy balance in an afterglow will be discussed in Section I.D.
-
2. Electron Energy Distribution Since electrons interact strongly with each other, and much more slowly with heavy particles, it is usually reasonable to assume that they have a Maxwellian energy distribution at temperature T, . More precisely, the dimensionless electron-electron to electron-neu tral energy relaxation ratio Z is
or numerically, assuming In (A,)
‘v
6.5,
In an afterglow, where the tail of the electron energy distribution does not play an important role, since ionization or excitation by electronic impact is normally negligible, the condition Z 2 1 is usually sufficient to ensure that the central part of the electron energy distribution is reasonably Maxwellian for most applications. For example, in helium at pressure 10 torr, with n, = lo9 cmW3and T, = 300”K, 2 N 9.
128
J.-F. DELPECH, J. BOULMER, A N D J. STEVEFELT
When electron temperature is substantially elevated above 300°K by microwave heating, the above condition does not always hold. However, it may be shown by solving Boltmann’s transport equation that the electron energy distribution still tends to remain Maxwellian in the presence of microwave heating (Frommhold et al., 1968). C . Particle Diffusion to the Walls
1. Diffusion Equations
Let us consider a closed vessel containing particles of an arbitrary nature which are immediately destroyed upon colliding with a wall (for example, metastable atoms which are deexcited by collision, or ions which are neutralized by the electrons present on the walls). Their equation of continuity (McDaniel, 1964) is an(r’ ‘)- DV2n(r, t). -at
The particle number density n(r, t ) is a function of position r and of time is the diffusion coefficient and V2 the Laplacian operator acting on the space variable. Given the initial distribution n(r, t = 0 ) and the boundary condition n = 0 on the walls, the partial differential equation (12) can be solved, assuming constant and uniform particle temperature, to yield the particle density at any time t > 0 and at any position. Equivalently,Eq. (12) is related to the eigenvalue equation t;D
where n is an eigenfunction and - l/A2 the corresponding eigenvalue; this defines the diffusion time constant A2 D
TD=-.
(14)
Whatever the initial spatial distribution, it is easy to see that after a time long enough compared to rD the dominant diffusion mode corresponds to the eigenvalue - l/A2 of largest argument: The lowest-order diffusion mode becomes dominant. In afterglow studies, the vessel often has the shape of a cylinder of length L and radius R such that L? 9 R2. The lowest-order diffusion length A is then given by
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
129
and the corresponding timedependent solution of Eq. (12) is
where no is a constant related to the space-averaged particle density at time t = 0. J o is the zeroth-order Bessel function, and 2.405 its first root. Initial longitudinal homogeneity is maintained over most of the length of the cylinder if L2B R2. The following relation holds in this case between the space average H of the density and its central value n(r = 0): = 0.432n(r = 0).
(17)
In actual afterglow work, nonlinear loss terms or even source terms should in many cases be added to Eq. (12) to describe electron or metastable atom density changes with time, and temperatures are not always constant or independent of position (Poukey et al., 1969). However, even when diffusion losses are small compared with nonlinear loss terms, numerical analysis (Gray and Kerr, 1959) shows that the actual spatial particle distribution remains quite close to its limiting shape, i.e., to a zero-order Bessel function in the case of a long cylinder. 2. Ambipolar Diffusion
If electrons and ions diffused independently to the walls, the electrons being much lighter and having a much larger average thermal velocity, would diffuse much more rapidly: This would violate charge neutrality. Coulomb interaction thus sets up a space-charge field which accelerates the ions and slows down the electrons; the energy associated with this space charge is of order kT, . The corresponding diffusion regime, where electrons and ions diffuse at the same rate and where n, z ni is conserved, is called ambipolar diffusion. The ambipolar diffusion coefficient D, (Oskam, 1958) is directly proportional to ionic mobility pi:
D. LS measured in cm2 sec- ;it varies inversely with pressure, provided ionic composition remains constant. In practical units, since ionic mobility is usually tabulated in cm2 V- sec- at NTP gas density (no = 2.69 x 10’’ cm- 3), the following relations hold:
130
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
and if To = T, = 300"K, D,p(cm2 sec- torr) = 4 3 . 2 3 ~, ~ (20) where p is the measured gas pressure, in torr. The following table of D , p at 300°K was deduced from experimental mobilities (Table IV) using relation (20). TABLE 11
REDUCED AMBIPOLAR DIFFUSION COEFFICIENT D , p FOR RARE GAS IONS AT 300°K GASTEMPERATURE Ionicspecies
He'
He:
Ne+
Net
Ar+
Ar:
Kr+
Kr:
Xe'
Xe:
D d cm2 sec-' torr
450
722
173
266
67.0
116.7
38.9
43.2
26.8
34.2
Oskam (1958) has treated in great detail the ambipolar diffusion problem in a mixture of ions. Deviations from ambipolar theory occur at very low electron densities, where charge neutrality cannot be maintained (Gerber and Gerardo, 1973). They are negligible in most laboratory afterglows. Diffusion may be a significant electron cooling mechanism in lowpressure afterglows when the diffusion time constant q, is not large compared to the electron-heavy particle energy relaxation time (2rnveO/M)- This effect is particularly important in the heavier rare gases near their cross-section minima (Boulmer et al., 1972).
'.
D. Electron Energy Balance Electron temperature is fairly high during the discharge: A few electron volts are needed to sustain ionization and to balance out diffusion losses. During the afterglow, electron temperature relaxes through electron-heavy particle collisions, with a characteristic time constant defined in Section 1,A. However, atoms (or molecules) excited on a metastable state constitute an energy reservoir (Goldstein, 1955) which may be coupled to the electron gas through various mechanisms. Experimentally, electron temperature often relaxes quite slowly and this should be attributed to long-lived excited species (Ingraham and Brown, 1965). Various processes are possible, some of which are discussed in Section IV,D. In this section, we shall assume that there are p ( t ) ion-fast electron pairs created per unit time and volume. Extension to other cases is straightforward. We shall also assume that the initial fast-electron kinetic energy #,
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
131
is large (15-20 eV). It is usually quite reasonable to assume that the fastelectron number density is negligible compared to thermalelectron number density. Fast electrons first relax their kinetic energy on the neutral gas, while at the same time diffusing freely to the walls: Their energy loss due to the ambipolar space charge field, of order kT, , where T, is the bulk electron temperature, is negligible compared to their large kinetic energy. The fraction of fast electrons which is not lost by diffusion reaches finally a characteristic energy u* where electron-neutral and electron-electron energy relaxations proceed at the same rate: They now become “ thermal” electrons, interacting strongly with the bulk of the Maxwellian electron distribution function. When discharge energy is small enough not to perturb appreciably the neutral gas, this problem can easily be formulated and solved in terms of Boltzmann’s kinetic equation. Characteristic energy u* is such that
In helium, u*(eV) z 3.1 x 103(n,/no)1/2. The fraction of fast electrons which is not lost by diffusion is exp ( -<’/A2), where A is the fundamental diffusion length of the container and where
is the square of the distance it takes for a fast electron to relax from urnto u* (0, is the velocity associated with kinetic energy urn).ve0 should be evaluated at some energy between urnand u * ; exact results can be obtained from a solution of Boltzmann’s equation. In helium at 10 torr and 300”K, = 0.24 cm. The number of thermal electrons created by unit time and volume (i.e., the apparent electron creation rate) is
<
s ( t ) = p ( t ) exp
(- t2/A2),
(23)
and the corresponding electron energy source term is
(1
3 2 n, kT,) = s(t)u,
.
(24)
In the presence of an inhomogeneous electron heating term, thermal gradients may appear in the electron population. They become nonnegligible on a distance of order A,, [Eq. (8)J and it can be seen that this distance may easily become small compared to the tube radius. However, it should be
132
J.-F.
DELPECH, J. BOULMER, AND J.
STEVEFELT
noted that there is no particular boundary condition for the electron temperature on the walls: Sheath potentials, which develop in the vicinity of the plasma boundary, are normally sufficient to ensure negligible electronenergy transport between the bulk of the electron gas and the walls (Dougal and Goldstein, 1958). 11. MICROWAVE DIAGNOSTICS TECHNIQUES
A. Microwave Propagation in a Plasma
1. Plasma Dielectric Constant In the absence of a magnetic field, the relative complex dielectric constant E of a plasma at frequency o/2a is
where v is an average electron-heavy particle collision frequency related to the collision frequency for momentum transfer. The plasma frequency o p / 2 a =fp is related to the electron density n, by
or numerically, in practical units, fp(Hz) = 8979nf’’. At an electron density of 10” cm-3 there corresponds, for example, a frequency of about 9 GHz, or an in uacuo wavelength of 3 cm. In what follows we shall only consider the case of a high-frequency wave (0’ b o;)propagating in an unmagnetized low-loss plasma (v’ 4 0’). These restrictions can be lifted with some additional algebra (Heald and Wharton, 1965). If we further assume that electron-ion collisions are negligible and that the electron energy distribution is Maxwellian at temperature T,, the average electron-heavy particle collision frequency v to be used in Eq. (25) becomes
Equation (27) is similar to Eq. (2), but it involves a different power of velocity. If the momentum transfer cross section cMT is independent of energy, Eq. (27) reduces to 4
VHF = 3Veo
(28) where v,, is the electron-neutral collision frequency introduced in Section LA. 9
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
133
When electron-ion collisions are not negligible, the formulation of the problem becomes much more complicated. However, it turns out (Shkarofsky et al., 1966) that v,,(ions
+ neutrals) = v,,(neutrals) + vei ,
(29)
where vei is given by Eq. (9).
2. Microwave Perturbation of a Plasma The complex part of the plasma dielectric constant corresponds to plasma absorptivity. As a result, a fraction of the energy propagated by a high-frequency electromagnetic wave through the plasma is dissipated in the electron gas, and increases its temperature. Assuming an electron thermal conductivity large enough to ensure thermal homogeneity (see Section 1,D) a steady-state electromagnetic field at radian frequency o,, of timeaveraged value E;, (the bar denotes suitable averaging over plasma volume) raises electron temperature by
As will be seen below (Section II,B), it is not always easy to relate the electric field inside the plasma E,, to the incident microwave power. In deriving Eq. (30), electron energy relaxation was assumed to be mainly due to collision with heavy particles. During microwave heating, the electron energy distribution remains normally Maxwellian (Ginzburg and Gurevitch, i960-see also Section 1,B). Selective heating of the electron gas by a microwave electromagnetic field is an extremely powerful technique which has found wide applicability since its introduction by Goldstein et al. (1953). In addition to electron temperature control, it has been used to induce precisely controlled electron density variations through the electron energy dependence of ionization loss terms (Delpech and Gauthier, 1972). B. Electron Density and Collision Frequency Measurements
1. General Principles Measurements of' the real part E' and of the imaginary part E" of the relative complex dielectric constant of a homogeneous plasma yield the plasma electron density and collision frequency by inversion of Eq. (25):
EN
V=W-
1
- & I '
134
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
We shall restrict ourselves, as before, to microwave diagnostics of a low-loss plasma, i.e., w 2 % wp’ and v z 4 u2. Microwave diagnostics of a real, nonhomogeneous plasma reduces to the measurement of an average of the relative complex dielectric constant of the plasma; average values of n, and possibly v obtain through Eqs. (31) and (32), considering the actual microwave field distribution within the plasma and the plasma-microwave interaction geometry. Interaction may take place, either in nonresonant structures (free space, waveguide) or in resonant structures. Despite the complexity of the general problem stated above, it usually turns out that relatively simple approximation techniques are available. Microwave methods are generally applicable and are very reliable; they are particularly well suited to precision measurements in lowtemperature afterglow studies. Microwave power should of course be low enough so as not to heat significantly the electron gas [i.e., AT, Q T,, where AT, is given by Eq. (30)].
2. Typical Experimental Systems As mentioned before, diagnostics can be made either in a resonant system or in a nonresonant, propagating system. a. Resonant systems. The high-performance microwave cavity system used at the University of Pittsburgh was described by Frommhold et al. (1968) (see Fig. 1). This system has been used for some of the electronmolecular ion recombination studies described below (Section 111,C). The cavity operates on three modes: The first is used to ionize the gas, the second wavemeter
,/>
FIG. I . Simplified block diagram of a three-mode microwave cavity used for microwave diagnostics with electron heating. From Frommhold el a/. (1968).
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
135
is used to measure the detuning of the cavity by the electrons and so determine the electron density, and the third is, in fact, a nonresonant waveguide mode for application of the microwave heating field. The cavity radius is 3.65 cm,and the distance between the resonant irises is 29.2 cm; resonant frequencies are in the 3 GHz range. Density is measured by a low-power ( < 2 pW) probing signal; its frequency is measured with a high-Q cavity wavemeter, and the reflected signal is detected by a sensitive superheterodyne receiver and displayed on an oscilloscope screen. The cavity resonant frequency corresponds to a minimum in the reflected power. Curves of resonant frequency shift versus time can be obtained by successively using different frequencies and noting the times of minimum reflection. Higher frequencies are necessary for measuring higher electron densities ; a high-Q closed cavity would then become impractical because it could not be large enough to accommodate an afterglow cell. A confocal spherical Fabry-Perot has been used near 32 GHz by Collins et al. (1972b). The mirrors were separated by a distance equal to their radius of curvature (22.9 cm) and had a diameter of 9.9 cm. Such a structure can accommodate large plasma cells and can be calibrated for high-precision dielectric coefficient measurements. b. Nonresonant systems. Resonant systems are not well suited to automated data acquisition techniques which become more and more necessary, as precision and signal-to-noise ratios are pushed further. Because of their resonant nature, they have good sensitivity but they can only measure one density point at a time. Furthermore, transient cavity quality factor measurements are not easy; meaningful measurements of the imaginary part of the plasma dielectric coefficient are in practice also very difficult. In the system shown in Fig. 2, the plasma is confined in a cylindrical tube located inside a standard X-band waveguide (1.02 x 2.29 cm inside dimensions). In conjunction with a fast data acquisition system, this system yields reliable electron density measurements in the 106-10'2 cm-3 range, and is well suited to collision frequency measurements, especially in cryogenic afterglows (Sol et al., 1973, 1975). Sensing power is below 1 pW near 8.77 GHz. Phase is measured at 45 MHz IF, after heterodyning to improve the signal-to-noise ratio and to insure a high rejection between the heating wave and the microwave receiver; complex plasma dielectric coefficients can still be measured during comparatively high-power microwave heating. With simple additions to the microwave circuitry the same setup is well suited to radiometric measurements (see Section 11,C). It has been used for electron-He: recombination measurements at 4.2"K (Delpech and Gauthier, 1972) as well as for a study of electron-neutral interactions in helium and neon at cryogenic temperatures (Sol et al., 1973, 1975).
136
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
CW klystron reference
precision hase shifter
FIG. 2. Travelling-wave microwave afterglow diagnostics system used by the Gaseous Electronics Group at Orsay. Simple replacement of the sensing wave klystron by a calibrated noise source transforms this interferometric system into a high-sensitivity radiometer.
3. Data Interpretation
Whatever experimental system is used, the actual plasma dielectric constant and even possibly its space variation is to be deduced from a measured global dielectric coefficient (Thomassen, 1963). In practice, it is often possible to deal with electron density inhomogeneities by assuming that the spatial electron density distribution corresponds to a fundamental diffusion mode (Section 1,C). Afterglow plasmas are often created inside glass containers which themselves affect the field distribution in the propagating (or resonant) mode (Kinderdijk and Hagebeuk, 1971). To be specific, we shall refer in what follows to a waveguide propagation experiment. Generalization to other cases involves no fundamental difficulty. The exact solution to the problem of a nonuniform plasma in a circular container located inside a rectangular waveguide can in general only obtain through machine computations (Baier, 1970). However, an approximate solution by variational or by perturbation methods (Harrington, 1961) is often precise enough to be of great value. It is well known that in a waveguide the fields E,, at z = 0 and E ( z ) at z # 0, where z is measured along the waveguide, are related by
E ( z ) = Eo exp (-az -#z),
(33)
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
137
where a and B are, respectively, the attenuation and phase constants characteristic of the corresponding mode of the guide. The time-varying term in Eq. (33) has been left out; we assume in what follows that the microwave frequency is such that only one mode propagates in the guide. The free-space propagation constant is (34)
YO
In the waveguide with the empty glass container, it is Yc
=ac
+a I
(35)
and finally, with the plasma present inside its container y = yc
+ Ay = (ac+ Aa) + j ( B c + AP).
(36) The complex propagation constant difference with and without plasma, Ay, is the quantity that is actually measured experimentally. The problem now is to relate its real (Aa) and imaginary (AB) parts to the plasma relative complex dielectric constant 6. In the high-frequency low-loss case, it may reasonably be assumed that the plasma does not perturb significantly the electric field. Then (Harrington, 1961)
The index X refers to integration over the waveguide cross section with surface element da z is the unit vector along Oz and Po is the free-space propagation constant. In fact, the plasma is not uniform across the guide (i.e., E is a function of position) and a spatial average is measured (Frommhold and Biondi, 1968):
-
The symbol (. (37) now becomes
stands for “microwave spatial average.” Equation
The problem has now reduced to the evaluation of the ratio G of two integrals :
138
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
Note that the electric field which appears in Eqs. (37)-(40) is unperturbed by the plasma, but not by the container. We shall see below that the effect of the container on the ratio G may be large. In most cases, u,’ 4 3/: and our initial hypothesis of high-frequency (02%- a :) diagnostics of a low-loss (v’ Q 02) plasma implies fl, %- A/3 > Au. One may then write
Comparison with relations (31) and (32)shows that a collision frequency measurement will not be affected by the value of G while a density measurement will. If one is only interested in relutioe variations of n, it should furthermore be noted that in a low-loss plasma, (ne)pW is linearly proportional to the phase shift A/3 of a high-frequency microwave propagating through the plasma. However, a precise absolute value of n, is usually needed, and G should be evaluated carefully. To this purpose, a first, simple approximation may be made. Assume that the electric field is perturbed neither by the plasma nor by the glass container. Computation of G is then quite simple; it is evaluated by replacing the electric field E in Eq. (40) by its expression for the corresponding unperturbed mode. This approximation may be reasonable in some configurations, but it is clear that the glass tube cannot usually be treated as a small perturbation. A better approximation obtains if one evaluates the field inside the glass tube by a quasi-static method. The electric field inside a glass container of inner and outer radii R , and R, small compared to wavelength of relative dielectric constant E, located inside a uniform electric field perpendicular to the tube axis is also uniform, and its amplitude E,, is related to the outside field amplitude Eo by
This may be used as an approximation to the actual situation in a mode where electric field lines are parallel. In this case, this approximation to the electric field inside the plasma E,, should also be used in Eq. (30) to compute electron temperature variations with microwave heating. This correction factor C may be substantially lower than unity; it is equal to 0.80 under typical conditions (glass tube with E, = 4.5, inside radius 0.4 cm, wall thickness 0.1 cm). It approaches unity with large tubes and very thin walls.
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
139
Once the ratio G has been evaluated through substitution of CEOas given by Eq. (43), it remains to evaluate p, in Eq. (41). p, is the wavenumber in the guide with the container present but without plasma. Even an approximate evaluation is not straightforward, because the container can rarely be treated as a small perturbation. The simplest solution is a direct measurement in a standing wave, either with a microwave probe, or by measuring the spatial frequency of the light modulation induced by standing wave microwave heating of the plasma. Using Po instead of p, in Eq. (41) may easily lead to errors of 3&40% in waveguide electron density measurements. Another method is direct system calibration against a known lowpermittivity dielectric like CO, gas (Collins er al., 1972b). Collision frequency measurements are less sensitive to the actual choice of C or /I, They . are nevertheless quite difficult, plasma absorptivity being low. Interferences between waves scattered at the two ends of the plasma tube are particularly difficult to take into account.
C. Electron Temperature Measurements The effective power P radiated in a frequency band Aw/2.n at center frequency 0/2n by a Maxwellian electron gas of temperature T, is given by Kirchhoffs law, which may be written in the limit ho 4 k T :
AU P(o)= A(w)kT - . 2w
(44)
Plasma absorptivity A ( o ) varies between 0 and 1. It is of course directly related to the imaginary part of the plasma dielectric coefficient defined by Eq. (25). The problem of plasma radiation and of temperature measurements at microwave frequencies has been reviewed by Bekefi (1966). In a low-loss afterglow, and particularly at low temperature, the radiated noise power to be measured becomes very small, in general much smaller than the noise temperature of available receivers. Synchronous detection with long integration time constants becomes necessary. Furthermore, an afterglow plasma being essentially transient, sampling techniques are necessary to sort out the desired time periods, in connection with multichannel analysis. Measurements are made more difficult by the fact that there are perturbing thermal sources around the plasma (container or wall absorptivity at different temperatures) as well as plasma-waveguide mismatches. These problems have been solved in the often-used standing-wave radiometer design of Ingraham and Brown (1965). It is also often very useful to measure electron temperature during microwave heating. This poses a severe rejection problem, since heating
140
J.-F.
DELPECH, J. BOULMER, AND J.
STEVEFELT
powers may be of the order of a watt, i.e., from 10 to 14 orders of magnitude larger than the thermal power to be measured. The use of a travelling-wave radiometer becomes necessary to ensure uniform plasma heating. This problem was solved by Delpech and Gauthier (1971).Their radiometer has a bandpass of 250 MHz around 9.5 GHz, and its practical sensitivity is better than 1°K.Heating-wave frequency is 8.6 GHz, and the necessary rejection is obtained through a combination of filters and circulators. The implementation of such a radiometer is not particularly complicated, its design being directly compatible with the travelling-wave diagnostics system shown in Fig. 2, by simply replacing the klystron (probing wave generator) by a variable-temperature noise source. 111. IONICPOPULATION
A. Ion Diagnostics by Mass Spectrometry A mass spectrometer selects and detects ions according to their chargeto-mass ratio; its use is often essential for reliable afterglow studies, since the measurement of an ionic rate is of little value if the corresponding ionic species is not well identified. Afterglow production and loss processes of helium ions have been studied by Phelps and Brown (1952)with a magnetic deflection mass spectrometer. However, afterglow plasma mass spectroscopy now relies almost universally on the use of quadrupole-type mass filters followed by detection by secondary electron emission and electron multiplication. Their many advantages were initially pointed out by Mosharaffa and Oskam (1966). In addition to small dimensions and relative ease of coupling to plasma containers, quadrupole mass filters allow some trade-off between sensitivity and resolution through simple electronic adjustments. While resolution may usually be kept to a relatively modest value in rare-gas afterglows, where ion complexity is limited, sensitivity should in general be as high as possible. Under practical afterglow conditions, transmission efficiency of a quadrupole mass filter may approach 100%. The general features of the quadrupole mass filter have been reviewed by Oskam (1969). Figure 3 shows a typical experimental setup used by Bhattacharya (1970b)in his studies of ion conversion in the heavier rare gases. The ion population is sampled during the afterglow through a small aperture (50 jim diameter) in the flange at one end of the tube. Pressure in the filter-detector assembly should be kept below 10- torr to minimize the effect of collisions between the ions and residual gases; fast differential pumping is thus necessary, while a leak valve simultaneously maintains constant pressure in the discharge region. For ultimate purity, the whole system should be thoroughly cleaned and baked out at high temperature.
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
TO VACUUM PUMP
141
E L E C T R O N MULTIPLIER
FIG. 3. Typical experimental coupling of a discharge tube to a quadrupole mass filter. From Bhattacharya (1970b).
Figure 4 shows the very large dynamic range and sensitivity of the system used by Gerber and Gerardo (1973) to study the transition between ambipolar and free diffusion in the late afterglow of a helium discharge. The range over which they were able to follow He: ion density is larger than 10'. The curve labeled He" corresponds to the background count rate, which they ascribe to secondary electrons produced by metastables striking the surfaces. In this case, the mass filter was used to focus the electrons onto the electron multiplier. This background was effectively used by Gusinow and Gerber (1970) to study the afterglow decay of a metastable species which was most probably the molecular triplet He,(3C: ) (see Section IV,A). Mass spectrometry cannot by itself yield the absolute number density of ions inside the discharge tube, since it samples an ion current to a wall. When a given ionic species is dominant, this problem can be circumvented by simultaneous independent microwave measurement of electron density, under conditions where the quasi-neutrality condition is known to hold in the plasma. This is the case at early times ( t < 10 msec) under the experimental conditions of Fig. 4. However, in the case of a mixture of ions, spatial ionic density distributions may not be homothetical, and detection sensitivity may not be independent of ion mass. Furthermore, ion dissociation may take place near the sampling hole, where electric potential gradients may be large, under the influence of extracting fields or even of ambipolar fields (MartiSovitS, 1970). This is at present one of the main limitations of this otherwise extremely useful technique.
142
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
FIG.4. Time dependence of the average electron density and the various ion wall currents in helium at moderate pressure. From Gerber and Gerard0 (1973).
B. Ionic Mobilities, Atomic-to-Molecular Ion Conversions 1. Theory
a. Mobility. The motion of an ion in a gas under the influence of an electric field depends on the detail of the ion-neutral interaction. Velocity is linearly proportional to the electric field felt by the ion; the proportionality constant is called the mobility. In the low-field, low-temperature limit valid
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
143
for afterglow plasmas, interaction is of the induced dipole type ( l/r4 potential), to which one should add an exchange interaction in the case of atomic rare-gas ions moving in their parent gas (McDaniel, 1964). The pure induced dipole case results in the well-known Langevin formula p(cm2 V - ' s- I ) = 35.9(crmr)- '/',
(45 )
which relates the mobility p to the atomic polarizability a (in units of ui) and to the reduced atom-ion mass rn, = mimo/(mi + m,) also in atomic units (unit proton mass). In this approximation, mobility is independent of temperature and of the nature of the ion; it only depends on the masses and on the nature of the gas. Charge exchange reactions of the type
R + + R + R iR+
(46) increase significantly the ion-neutral collision cross section, as compared to pure induced dipole interaction, and thus decrease ionic mobilities. Low-field mobility should in this case depend on gas temperature and on details of the electronic wave functions. Holstein (1952) has derived a general method for calculating the mobility of an ion in its parent gas. TABLE Ill ZEROFIELDTHEORETICAL IONIC MOBILITIES (in cm2 V - sec- ') REDUCED TO NTP GASDENSITY (no = 2.69 x lOI9 cm-')
Polarizability (au) Atomic mass Molecular ion mobility (Langevin theory) Atomic ion mobility (Langevin theory) Atomic ion mobility at 300°K including charge transfer"
Ar
Kr
Xe
2.65
11.0
16.7
27.1
4 18.7
20 6.03
40 2.09
84 1.17
131 0.74
21.6
6.96
2.42
1.35
0.85
11.1
4.2
1.62
1.0
0.66
He
Ne
1.38
As quoted by Biondi and Chanin (1954)
Table 111 lists theoretical low-field mobilities. The Langevin formula [Eq. (4511 was used for molecular ions; results with and without charge exchange (at 300°K)are compared for atomic ions. Note that charge exchange is important, notably in the case of helium. Mobilities are universally
144
J.-F. DELPECH, J. BOULMER, A N D J. STEVEFELT
quoted in units of cmz V- sec- I , and reduced to NTP gas density, i.e., no = 2.69 x 10'' ~ m - ~ . b. Zon conversion. The existence of diatomic rare-gas ions was first demonstrated by Tiixen (1936).The diatomic ions of all the rare gases were identified in a mass spectrometer by Hornbeck and Molnar (1951). At these low pressures, they were formed by the reaction R*
+R +
Ri +e
+ KE,
(47) where R* is a highly excited atomic state, within at most 2 eV from the ionization continuum. At higher pressures, molecular ion formation is governed by the termolecular process R+
+ R + R + R i + R + KE.
(48)
Mahan (1965) has shown that experimental results are reasonably well accounted for by assuming that this reaction proceeds through resonant ionatom charge transfer during grazing collisions, with, therefore, negligible kinetic-energy exchange, in the proximity of a third body of arbitrary nature. c. Gas mixtures. Heteronuclear rare-gas molecular ions have also been identified in mixtures of gases, and some of their properties have been studied, notably in the case of (He Ne)+ (Veatch and Oskam, 1970). In the absence of ion conversion reactions, the reduced mobility p of an ion in a mixture of gases is given by Blanc's law (Biondi and Chanin, 1961), 1 _--fl fz (49) P P1 Pz where f l and fz are the fractional concentrations of gases 1 and 2, respectively, i.e.,fl +fz = 1 ; p1 and p z are the reduced mobilities of the ion in each of the pure gases. + - 9
2. Experimental Results The first modern measurements of ionic mobilities in carefully purified gases, in a near-zero electric field situation at room temperature, are due to Biondi and Chanin (1954).The same authors later published results at temperatures ranging from 77" to 300°K (Chanin and Biondi, 1957). The next major advance was made when positive identification of the ion species was made with an ion mass filter coupled to a drift tube (Madson et al., 1965). Subsequent measurements (see, for example, Bhattacharya, 1970a) showed that earlier mobility results were indeed correctly ascribed to the corresponding ions. Ion mass analysis of a cryogenic afterglow plasma at
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
145
82"K, in an isotopic mixture of helium (90% 4He and 10% 3He), enabled DeVries and Oskam (1969) to prove the existence of triatomic and even of (minority) tetratomic helium ions (see Fig. 5).
p. = 5.7
torr
Tg = 8 Z 0 K
II
7%
"HI;
1
G
n
lo
'
1
I0
1
1
12
14
L
1
16
Mars (omu 1
FIG. 5. Mass scan of ions with masses between 8 and 17 amu present during the decay period of a plasma produced in 4He containing 10% 'He. From DeVries and Oskam (1969).
The tetratomic ion He: was also eventually detected as a very low concentration minority ion at room temperature by Gerber and Gusinow (1971). However, no data are available concerning this ion. Finally, the coupling of an ion mass filter to an afterglow plasma cell enabled precise measurement of the rates of termolecular ion-neutral association reactions (Vitols and Oskam, 1972, 1973a,b). Gas purity is of paramount importance in such studies, as demonstrated by these authors; large variations in the observed reaction rates should be expected for extremely low concentrations of impurities. Accepted experimental values of mobilities at 77", 195", and 300"K, and of atomic molecular ion reaction rates at 300"K, are shown in Table IV.
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
146
TABLE IV ACCEPTEDEXPERIMENTAL VALUES OF IONICMOBILITIES AND IONIC CONVERSION RATESIN THE LOW-FIELD LIMITAT NTP GAS DENSITY (no = 2.69 x 10''
'
Zero-field mobilities p(cm2 sec- V- ') Ionic Species He+ He: He:
77°K 16.2 f 0.3'
195°K
W K
Termolecular reaction rates K(10-32 cm6 sec-') at W K
11.8 f 0.25' 16.1 f 0.3'
10.4 & 0.1' 16.7 k 0.17'
6.T [14.4 at 77"KId large'
18.0 f 0.2'
Ne+ Net
5.21 6.7,
4.3' 7.3f
4.0 f O.lh 6.15'
Ar+ Ar:
2.2f 2.71
1.951 2.9'
1.55' 2.7f
19,
0.90k 1.O'
23"
Kr+
fi:
(1.0 at 90°K)'
-
1.O'
4.4 f O.Sh
0.62 f 0.03" 0.79'
Xe Xei
+
20 f 2n
a Unless otherwise quoted, systematic experimental errors should be within 5 % on mobility measurement but may reach 20% on ion conversion rates. Beaty and Patterson (1968). Patterson (1970). Gerber et al. (1966). Bhattacharya (1970b). * Gerber and Gusinow (1971). Biondi and Chanin (1954). ' Patterson (1968). Beaty (1956). f Chanin and Biondi (1957). " Bhattacharya (1970~). Hackam (1966). " Vitols and Oskam (1973a). I, Vitols and Oskam (1972).
'
'
Comparisonsbetween Tables I11 and IV show that theories based on induced dipole interaction, with the addition of charge exchange reactions for atomic ions, are in reasonable agreement with experimental results. However, disagreement with Langevin's theory is quite marked in the case of the diatomic helium ion He:. This may be due (Beaty et al., 1966; Patterson, 1970) to reactions of the type Hei('Z:)
+ He(l'S)+
(He:)*
unstable
He(1'S) + He:('Z:).
The lifetime of the reaction complex (He:)* would depend on the available kinetic energy and on the details of the He:-He interaction. (He:)*
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
147
-
may either stabilize by releasing its 0.17 eV of energy to the neutral gas, or dissociate back to He: + He, at higher temperature and/or lower pressure. Process (50)would account for the reduced He: mobility and also for He: + He: conversion, which is known to be fast below 100°K (see Table IV). There is no direct evidence for this process, but it may be substantiated by the fact that a “fast” mass 8 ion having a room temperature mobility around 20 cm2 V-’ sec-’ has been identified (Madson et al., 1965)in very high purity helium gas. Beaty et al. (1966)suggested an explanation by showing that a metastable He: (“C:) ion had a bound state. Such an excited ion should not be liable to charge exchange reactions of the type (46),and should have a mobility near the Langevin limit. Its presence would be dependent on the ion source condition, as it could not simply be produced by reaction (48).
C . Electron-Ion Recomb inat ion
To a good approximation, a cold rare gas in its normal state consists of neutral atoms only. Ionization energy present at the end of the discharge must decay to zero during the afterglow. Charges which diffuse out of the plasma eventually recombine on the walls, essentially through nonradiative processes, and rates are fast enough to maintain a near-zero charged particle density condition near the walls. In this section, we shall be concerned with the mechanisms and rates of volume recombination, which play a very important role in many afterglow plasmas (the very name of the afterglow refers to the light emission due to electronic recombination of ions during the postdischarge period). The capture of an electron by a positive ion requires that the electron go from a free positive-energy state to a bound negative-energy state. The electron-ion complex should thus in some manner give up energy, either by itself (radiation, dissociation) or by interaction with a third body (electron, neutral atom). It seems thus reasonable to classify recombination mechanisms according to the main process by which the energy difference between the initial electron-ion pair and the final neutral species is carried off. It is now generally believed that the two main recombination mechanisms in afterglow plasmas are collisional-radiative recombination, for atomic ions, and dissociative recombination, for molecular ions, with the possible exception of the diatomic helium ion He:, whose recombination processes are not yet completely understood. The macroscopic recombination rate is usually defined as a two-body
148
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
rate describing the decay of free electron density n, in the absence of any other process,:
where plasma neutrality and a dominant ionic species have been assumed. When recombination is a three-body process, it is still traditionally measured by a two-body rate which now becomes a function of third-body density. A general survey of recombination phenomena has been recently published by Bates (1974). 1. Collisional-Radiative Recombination
The energy levels of excited atoms of high principal quantum number p may be expected to be nearly hydrogenic, so that the ionization energy E, of state p is
where R , is the ground-state hydrogen atom ionization energy (13.6 eV in practical units) and a. = 5.29 x lo-” rn is Bohr’s radius. Bound electrons of high principal quantum number are thus hundreds of a. away from the nucleus. While this is a large range on an atomic scale, it is a very short range on a plasma scale, since the screening length for the Coulomb potential is usually orders of magnitude larger than ao:Even with large principal quantum number, a bound electron “feels ” an unscreened Coulomb potential. a. Colliswnal processes. Consider now the following reversible reactions : R*(p) + e s R*(q) e, (53)
+ R + + e + e$R*(q) + e.
(54) When dealing with states of high principal quantum number, the correspondence principle holds and Coulomb collisions between free and/or bound electrons may be treated classically. For an energy transfer collision to be efficient, collision time should not be substantially larger than bound-electron orbiting time. This means that the two interacting electrons should also be within a distance no larger than the radius of the electronic orbit, i.e., of order a. p 2 . This is again a very close collision on a Coulomb potential scale: Long-range plasma effects should not be expected to play a role in this process. It is thus finally sufficient to treat classically the three-body processes in a Coulomb potential described by reactions (53) and (54).
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
149
This has been done in detail by Mansbach and Keck (1969).They found that contrary to what had often been assumed the classical impulse approximation (Gryzinski, 1959) was only valid for energy transfers greater than a few k T , . Their approach involves a minimum of hypotheses and seems to describe well collisional transitions between highly excited Rydberg states. Rate coefficients K are easily derived from their work. For transitions between bound states of principal quantum numbers p < q [reaction (53)], K(4, p)[cm3 sec- ‘3 =
2.86 x
TY7
p6.66 q5 ’
(55)
and for transitions between state p and the continuum [reaction (54)], 3.75 x 1 0 - ~ 7 ~ , _ _ P6.667 ~
K(c, p)[cm3 sec- ‘1 = _
TY’
in practical units (see Introduction). Consider now an equilibrium situation where the electrons are at temperature T, and the population and depopulation rates of different energy states are dominated by electronic collision processes which are the inverse of each other. The relative populations N in different energy states characterized by principal quantum numbers p and q are then related by Boltzmann’s law
where the subscripts E indicate equilibrium, and the g’s are the statistical weights. The number of free electron states available per ion of degeneracy g i is (2gi/ni)(271.mkT,/h2)3/2,where ni is the ionic density. Substitution into (57) yields Saha’s law, which is written, taking the ionization energy as energy zero,
The excited states-electron equilibrium described by Saha’s law cannot extend down to the ground state (p = l), since an afterglow plasma is essentially out of equilibrium; Saha’s law is valid only for highly excited levels, and reflects the quasi-equilibrium condition of an afterglow plasma. There is always a level of principal quantum number p below which collisional deexcitation becomes more probable than collisional excitation: This results in a global loss of ionization, and this is the basic phenomenon of collisional recombination.
150
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
When collisional processes are dominant, Mansbach and Keck (1969) compute the recombination rate coefficient n 2 a,(cm3 sec- ') = 3.8 x (59) TY' where the subscript c stands for collisional. Capture and stabilization by neutrals R + e R e R * ( p ) R, (60) R*(p) R e R * ( q ) R, (61)
+ + +
+ +
should also play a role in high-pressure plasmas where neutral density is much larger than electron density. However, because of the mass ratio 2m/M, electron-neutral atom energy exchange is very inefficient compared to electronelectron exchange. The corresponding collisional recombination rate aN was computed by Pitaevskii (1962):
where oMTis the electron-neutral cross section for momentum transfer (Table I) and no is the neutral density. Although not negligible, the correcm3 sec-' in helium at sponding rate is small; it amounts to 7 x 100 torr, 300°K. b. Radiatioe processes. An interacting electron-ion pair may stabilize on a level p by emitting a photon of energy hv. This is the inverse of the process of photoionization. For electron energies lower than a few eV, summation of this process over all available bound states (Seaton, 1959) yields a recombination coefficient 1.55 x lo-'' a,(cm3 sec-') = 7Y3 ' where the subscript R stands for radiative. Radiative deexcitation of the bound state
'
R*(P) R*(q) + hv (P 4 ) (64) by means of spontaneous light emission may also occur; the inverse radiative lifetime of a hydrogenic level p is given with sufficient accuracy by the well-known expression +
(651 The density of excited states on high quantum levels being extremely small, the inverse processes of photoionization and photoexcitation are negligible in low-temperature afterglows. A ( ~ ) [ ~ -=' I1.6 x
1010p-4.5.
151
LOW-TEMPERATURE RAREGAS STATIONARY AFTERGLOWS
c. Collisional-radiative recombination. Radiative and collisional processes combine in a complex way. At very low electron densities and relatively high electron temperatures, recombination is essentially radiative, while in the opposite domain, it is essentially collisional. In low-temperature afterglow plasmas, collision processes usually play a dominant role, but radiative processes cannot be neglected. The general problem of interacting radiative and electronic collisional processes is extremely complicated. In their classical paper, Bates et al. (1962)made the important remark that relaxation times of the excited atoms are extremely fast compared to plasma relaxation times under most conditions. Rates may be found by solving the set of linear equations which express equality between production and destruction rates. In their computation, they used collisional rates calculated in the impulse approximation (Gryzinski, 1959) which lead to overestimating by a large factor the probability of small energy transfers. Stevefelt et al. (1975)have used the same technique together with the presumably more correct rates (55) and (56)to numerically derive the collisional-radiative recombination coefficient at electron temperatures below 4000°K.It is well approximated by aCR(cm3sec- ’) =
1.55 x
lo-’’
Ty3
+
6.0 x
lo-’
n:.3’
+
3.8 x 10-9 Q.5
ne.
(66)
t.1 loo
1 I
1010
I
10”
n. (cm-3)
FIG.6. Collisional-radiative recombination coefficient for three temperaturesas a function of electron density.
152
J.-F. DELPECH, J. BOULMER, A N D J. STEVEFELT
It can be seen that the collisional and radiative limits are recovered but that &R 2 ac + aR. The inequality is due to the complex interaction between collisional and radiative processes at intermediate electron densities and temperatures. Expression (66) has been plotted on Fig. 6. Note that collisionalradiative recombination is a slow process, even at low temperature (300°K) and high electron density (10' cm- ') and that it can easily be swamped out by competing processes.
'
2. Dissociative Recombination The processes leading to dissociative recombination of a molecular ion are illustrated on Fig. 7. If the repulsive curve (A-B)of an excited state of the molecule crosses the electron-ion energy curve at a suitable point, the system
~
Internuclear separation
FIG.7. Schematic representation of dissociative recombination process.
may transfer to this state. After this capture step, the system may either autoionize again or begin to dissociate, the nuclei moving apart down the potential slope to (B). This stabilization step yields two atoms, one of which is excited, with a kinetic energy uK equal to the energy difference between the initial molecular and the final atomic state. The final step is radiative deexcitation of the atomic state. This can be summarized by the following sequence of reactions : capture, R: stabilization,
+ e * (Rf)""stable;
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
153
deexcitation, R * ( p ) + R*(q)
+ hv.
(69)
Dissociative recombination processes have been recently reviewed by Bardsley and Biondi (1970), Oskam (1969), and Bates (1974). In what follows, we shall only summarize important resdts needed for an understanding of dissociative recombination processes in rare gases. The cross section for dissociative recombination may be written as the product of the cross section for electron capture, B , , ~ ~ ( U where ), u is the electron energy, and a survival factor S which may be expressed in terms of the stabilization probability A, and the autoionization probability A, (Bates and Dalgarno, 1962)
For a Maxwellian distribution of incident electron velocities, the dissociative recombination coefficient is given by
One should not infer from Eq. (71) that the dissociative recombination rate has a T i 3/2 dependence; in fact, it turns out that the integral should in many cases be linearly dependent on electron temperature, at a given gas temperature, and that uDR should approximately vary as T i There is at present no theory available for a priori exact calculation of uDRfor a given molecular ion. However, order-of-magnitude estimates of the various quantities involved in Eq. (71) show that dissociative recombination rates should be quite large-much larger than collisional-radiative recombination rates-in typical afterglow plasmas where molecular ions with suitable curve crossings are present. If the potential curves for R i and (R;)unsl,cross close to the equilibrium separation for R l , recombination rates should decrease rapidly with increasing vibrational quantum number of the ion, mainly because the survival factor S should decrease exponentially. If vibrational energy levels are assumed to be in equilibrium at gas temperature To,O’Malley (1969) found by simple considerations that
where hw, is the vibrational energy spacing of the molecular ion (Table VII).
154
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
3. Experimental Results
Low-temperature recombination rates are usually deduced from the time evolution of the ion or electron density in an afterglow, provided the ionic composition is accurately known. This measurement is not straightforward, because various loss and source terms may combine in the equation giving the time evolution of the electron density. Combining Eqs. (51) and (12) yields, assuming no source term,
an, = D,V2n, - an,’, at
(73)
This nonlinear partial differential equation cannot be integrated in terms of simple functions in the general case. It has been solved numerically by Gray and Kerr (1959) and by Frommhold and Biondi (1968). This equation should only be used after ascertaining the absence of significant ionization processes during the afterglow. a. Molecular ion recombination. Careful microwave measurements of electron density variations in the afterglow of highly purified rare gases have been made by Oskam and Mittelstadt (1963) under conditions where molecular ions were known to be dominant. The electron density variation as a function of time was compared to the microwave averaged (see Section II,B,3) machine solution of Eq. (73). These authors obtained recombination coefficients of all rare-gas diatomic homonuclear ions ( R i ) except helium, where significant measurement could not be made. These ions were expected to recombine following a dissociative process; it had been shown earlier (Biondi and Holstein, 1951) that the recombination spectrum of neon was atomic and that the observed lines originated from levels more than 0.85 eV from the ionization potential. Consideration of Fig. 7 shows that this is exactly what should be expected in the case of dissociative recombination. It was reasonable to assume that the heavier rare-gas diatomic ions behaved similarly to neon. However, in helium, the same authors have shown that visible recombination radiation includes lines originating from high lying states within 0.3-1.5 eV of the atomic ionization potential. If they were due to dissociative recombination, the ion would have to be in highly excited vibrational states. Relatively abundant molecular helium light had also been frequently observed in helium afterglows. This made the process of dissociative recombination rather less likely for He:. Connor and Biondi (1965) and Frommhold and Biondi (1969) have confirmed the essentially dissociative nature of N e i and A r i electronic recombination; they were able to detect the kinetic energy uK of the dissociation products (Ar* or Ne*) through the broadening of the lines emitted
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
155
during optical deexcitation (B to C transition on Fig. 7). The measured kinetic energy uK is in reasonable agreement with what may be expected from spectroscopic data (Table VII) on neon molecules. The diatomic nature of the recombining ion was confirmed by Biondi and his co-workers who used mass spectrometric detection techniques; data analysis was made through a computer fit of the ion continuity equation (73) under conditions where ionization sources in the afterglow were negligible. Recombination rates were measured at constant gas temperatures (300"K), at electron temperatures between 300" and 4600°K in neon (Philbrick et al., 1969), and between 300" and 10,000"K in argon (Mehr and Biondi, 1968). Finally, in a series of experiments with a shock tube, Cunningham and his co-workers (O'Malley et al., 1972; Cunningham and Hobson, 1972) were able to vary simultaneously the electron and the ion vibrational temperatures. Electron density was measured by double-probe techniques and the recombination rate was deduced from an approximate solution of the electron continuity equation. Experimental data are in good agreement with each other over their common range of validity. Furthermore, they agree well with qualitative theoretical models, both at constant gas temperature and varying electron temperature and at covariant electron and gas temperatures (see Fig. 8).
FIG. 8. Electron temperature dependence of e - N e i recombination rates: open circles fixed 300°K gas temperature, varying electron temperature (Frommhold et al., 1968); solid circles covariant electron and gas temperatures (OMalley et al., 1972). Solid line corresponds to Eq. (72) and the dashed line is a simple power approximation, both with the data of Table V.
Table V summarizes experimental data concerning e - R: dissociative recombination rates. Note that the vibrational spacings h o e deduced from Table V for Ne; and Ar: ions are in very reasonable agreement with independent spectroscopic data (see Table VII). The rate and the general features of the electronic recombination of He: are also in qualitative agreement with a dissociative model, although the
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
156
TABLE V DISSOCIATIVE ELECTRONIC RECOMBINATIONRATESOF DIATOMIC RARE-GAS MOLECULAR IONS Gas temperature To = 300°K
a=ao
Nei Ari Kri Xei
(3 G K ) - Y ~
Value at 300°K ao(IO-' cm3 sec-')
Exponent
1.73 f 0.2 8.5 k 0.8
0.49" 0.67'
12
*
1'
14f 1'
Covariant electron and gas temperatures ( T = T. = To)
Y
no(10-
' cm3 sec- ' )
-
hw,(eV)
b,
2.0 9.6 13
0.086 0.054 0.069
0.43" 0.67' 0.5'
-
-
-
Valid from 300 to 4600°K; Philbrick et al. (1969).
'Valid from 300 to 10,OOO"K; Mehr and Biondi (1968). Oskam and Mittlestadt (1963). " From 500 to 3000°K; OMalley et a!. (1972). From 900 to 2700°K; Cunningham and Hobson (1972). The quoted value of hw, fits the available data well but should not be taken too seriously in view of the restricted temperature range.
actual processes are not known. Measurements have been made around 77°K (Gerard0 and Gusinow, 1971) and 4.2"K (Delpech and Gauthier, 1972). The rates have also been measured at various gas temperatures between 4.2"and 77°K (Devos, 1973).The electron temperature was varied by microwave heating between gas temperature and 200°K. The results are well approximated by - 1.08
a(cm3 sec-1) = 2.2 x lo-'(-)77°K
-0.32
(A) 77°K .
(74)
Since He: vibrational levels cannot possibly be excited at such low temperatures if they are in thermal equilibrium with gas (see Table VII) the electron and neutral temperature dependences may point to the importance of rotational processes. At a given gas temperature, the recombination rate shows no measurable dependence on gas pressure; this should rule out any contribution of ion conversion phenomena to the neutral temperature dependence of the globally measured electron-ion recombination rate. However, no mass spectroscopic analysis has been possible below liquid nitrogen temperature. b. Electronic recombination of He:. The diatomic helium ion He: is
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
157
known to be the majority ion in helium afterglow plasmas near room temperature at pressures above a few torr (see Section 111,B). Its electronic recombination processes are not well understood. Experimenters do not even agree on its recombination rate and on its functional dependence (or lack of dependence) on T,, To, nerand no! Ferguson et al. (1965) give a thorough discussion of the evidence available ten years ago. Despite some experimental observations (Chen et al., 1961 ; Rogers and Biondi, 1964) they conclude, essentially from spectroscopic evidence on the behavior of the atomic and molecular lines in afterglows (Collins et al., 1963), that dissociative recombination was unlikely and that collisional-radiative recombination was likely. Further studies (summarized, for example, by Collins and Hurt, 1969) partly supported this conclusion, but could not rule out the presence of a small amount of dissociation. It should also be mentioned that the theoretical evidence (Mulliken, 1964) makes dissociative recombination rather unlikely, since there is apparently no curve crossing such as the one illustrated on Fig. 7, at least in the u = 0 state. Let us now discuss very briefly some recent experimental results in roomtemperature helium afterglows. Berlande et al. (1970) have analyzed the time dependence of the electron density in an afterglow. Assuming that losses were only due to diffusion and recombination, and that the electron-ion source term was negligible, they found a recombination coefficient in reasonable agreement with the predictions of collisional-radiative (CR) recombination theory, taking into account collisions both with electrons and with neutrals. However, their neglect of the electron-ion source term was possibly not justified, particularly under such broad experimental conditions. Johnson and Gerardo (1971) have used a microwave perturbation technique to show that an electron source term was indeed very important under their experimental conditions. The net rate of decay of free electrons was found to be similar to that observed by Berlande et al. (1970), but the actual recombination coefficient became five to six times higher when the electron-ion source term was taken into account. In later work, Johnson and Gerardo (1972) reported the observation of a strong coupling between electron density and atomic metastable density. This may point to the presence of an important dissociative mechanism somewhere in the recombination of He:. Collins et al. (1970) have made a direct spectroscopic measurement of the rate of electronic recombination of He:, using methods very similar to those used for He+ (see next section). Their results are in general agreement with the CR recombination scheme, although they probably do not rule out the presence of simultaneous dissociative recombination.
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
158
In fact, there are good reasons to believe that He: is continuously produced in highly excited vibrational states (Stevefelt, 1973;Grossheim et al., 1973) during the afterglow, and that favorable curve crossings leading to dissociative recombination do exist for some intermediate vibrational states. This is borne out by the conclusion reached by Robben (1972), in his comments on spectroscopic measurements by Stevefelt and Robben (1972), that dissociative recombination of He: in vibrationally excited states should be in part responsible for the formation of excited helium atoms during an afterglow. Boulmer et al. (1973) have developed a new afterglow analysis technique involving no assumption about the functional dependence (or lack of dependence) of the recombination rate coefficient on experimental parameters such as electron density or temperature. Explicit account is taken of the measured electron source term due to ionization during the afterglow. The electron temperature is also measured, either directly by radiometry, or indirectly through collision frequency measurements. Measurements at 295°K neutral temperature, at pressures from 20 to 30 torr, at electron temperatures from 295" to 5WK, and at electron densities between 8 x 10'' and 2 x 10' cm- are in reasonably good agreement in absolute value and in functional dependence on n, and T, with the predictions of collisionalradiative recombination theory. Finally, in a recent paper, Boulmer et al. (1974) have reported that the measured molecular spectrum of a helium afterglow plasma does not fit the available recombination theories. Excited states of high principal electronic quantum number are in thermal equilibrium at a substantially different temperature than that of free electrons. Furthermore, the characteristic temperature of highly excited states is only marginally affected by controlled microwave heating of the electron gas. The rotational temperature of the He2(3p3/II,)states was also found to be substantially above electron and gas temperature and to be essentially independent of electron heating. The phenomena involved in the electronic recombination of He: cannot be explained simply in term of available recombination theories and rotational-vibrational phenomena should probably be expected to play some role in these processes (Stevefelt, 1973). It is interesting to remember at this point that mobility measurements of He: are also to some extent contradictory (Section 111,B). In conclusion, it may be said that e - He: recombination measurements are difficult because of two facts-the recombination coefficient always turns out to be quite small, and an ionization source term plays a large role under most afterglow conditions in helium, precluding simple use of Eq. (73). c. Atomic ion recombination. Atomic ion recombination rates in rare gases are only known in the case of Hef. Kuckes et al. (1961) and Hinnov
'
LOW-TEMPERATURE RAREGAS STATIONARY AFTERGLOWS
159
and Hischberg (1962) have made detailed microwave and optical studies of the afterglow plasma present in the Princeton B-1 Stellarator after a discharge in helium at. pressures between 0.25 and 100 mtorr. They demonstrated that high principal quantum number levels were indeed in Saha equilibrium [Eq. (58)] with the free electrons and that the density and temperature dependence of the recombination rates were in general agreement with theory. Robben et al. (1963)obtained detailed information on a helium afterglow by a spectroscopic study of an arc jet. They were able to infer from their measurements collisional transition rates between levels. These results and the measured recombination rates are in good agreement with later theoretical values (Mansbach and Keck, 1969; Stevefelt et al., 1975). General spectroscopic features and recombination rates were also found to agree reasonably well with theory by Collins et al. (1972b) and by Stevefelt and Robben (1972).
X
FIG.9. Collisional-radiativeelectronic recombination of He' : (V)Hinnov and Hirschberg (1962); ( x ) Robben et al., (1963); ( 0 )Stevefelt and Robben (1972); ( 0 )range covered by Collmsetal., (1972b).ThecoordinatesX = log [(nJ10'o)-0~2s*~]and Y = log [(nJ10'o)0.'63a] have been chosen because they yield a twodimensional representation of aCR[Eq. (66) and solid line]. The dashed line corresponds to the collisional limit.
These four sets of measurements are summarized on Fig. 9 along with theoretical results of Stevefelt et al. (1975). Note that the coordinates are not simple functions of a, n,, and T,;they were chosen because they allow a universal twodimensional plot of the recombination rate given by Eq. (66). Recombination mechanisms are not known for the heavier rare-gas atomic ions. There is some evidence that they follow the collisional-radiative
160
J.-F.
DELPECH, J.
BOULMER, AND J. STEVEFELT
model (Vitols and Oskam, 1973b), but in view of the complexity of their energy level structures, other mechanisms may well be competitive with collisional-radiative phenomena. IV. EXCITEDSTATESPOPULATIONS A . Energy Levels of Rare-Gas Atoms and Molecules
1. Atoms
Table VI lists a few important energy levels of rare-gas atoms (Moore, 1952) with the ground state as energy zero. Conventional Paschen notation is used, except for helium. Energy level structures are very similar in the heavier rare gases from neon to xenon, because they all have filled outer shells in their electronic ground states and all their known excited states have a p5 core. The structure is quite different in helium, which has only two valence electrons. Energy level diagrams and conversion tables between Paschen and systematic notation will be found in the AIP Handbook (Crosswhite, 1972). 2. Molecules
For low-pressure afterglow purposes, ground-state rare-gas molecules may be regarded as unstable, in the sense that they have no significant potentialenergy minimum lower than the energy of the separated atoms in their ground states. However, they have excited and ionized states which are stable against dissociation. Table VII lists the energy values which correspond to the lowest excimer states (%:) and to the ionic ground states of diatomic homonuclear rare-gas molecules, as well as known data concerning the triatomic helium ion He:. Energies are given relative to the ground state of the atom, and dissociation energies Do are given for vibrational ground state u = 0. The energy separation between the u = 0 and u = 1 vibrational levels is h o e . Energies and wavenumbers are related by 1 eV = 8065.47 cm-'.
(75)
The atomic unit of length a, (Bohr's radius) is
a,
E 5.29
x lo-'' meter.
(76) The rotational energy u,,, corresponding to the rotational quantum state K is u,,, = B K ( K
+ l),
(77) where B is the rotational constant related to the interatomicdistance and the atomic mass.
IMPORTANT ______~
ELECTRONIC ENERGY LEVELS OF RARE-GAS ATOMS
_____~
Resonant line to ground state Upper level
(4
First ionization energy (eV1
24.59
Wavelength
Helium
2'P
584
Neon
1% 1% Is4 1% 1% 1% 1% 1%
744 736 1067 1048 1236 1165 1470 1296
Argon Krypton Xenon
21.56 15.76 14.00 12.13
Metastable states Designation
Energy (ev 1
23s 2's 1% Is3 1% Is3 1% 1% 1% Is3
19.82 20.62 16.62 16.72 11.55 11.72 9.92 10.56 8.32 9.45
First nonmetastable states Designation
Energy (ev1
20.96 21.22 16.67 16.85 11.62 11.83 10.03 10.64 8.44 9.57
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
TABLE VI
161
162
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
TABLE VII
IMPORTANT ENERGY LEVELS OF RARE-GAS MOLECULES
Gas
Molecular state
Energy (ev)
Dissociation energy Do (ev)
Helium
He,(%:) He: He;
17.97 22.22 22.05
1.85 2.36 0.17
Neon
Ne,(?Z:) Ne; Ar2(3E:) Ar: Kr,(’E:) Kr; ~e,(Ti) Xe:
15.5 20.2 10.4 14.5 8.7 12.8 7.3 11.2
1.1 1.1 1.2 1.4 1.2 1.2 1.0 1 .o
Argon Krypton Xenon
Vibrational Interatomic Rotational separation distance rc constant B (au) ( i w 4e v ) (ev) 0.224 0.211 0.104 0.166 0.041 0.077 0.077 0.038 0.054 0.03I 0.03I 0.017 0.020
1.98 2.04 2.34
9.52 8.97 1.70
3.7 3.2 4.5 4.5 5.5 5.5 6.4 6.4
0.55 0.73 0.18 0.18 0.06 0.06 0.03 0.03
Energy levels and spectra have been extensively studied for the helium molecule. The values listed have been taken from Ginter and Ginter (1968), Brown and Ginter (1971), and Liu (1971). The He: data are taken from computations by Vauge and Whitten (1972), except for the dissociation energy taken from Patterson (1968). The three vibrational separations correspond to the symmetric and asymmetric stretching and to the bending modes, in that order. Available information about the other rare-gas molecules is much less precise. The data presented in this table are our estimations using Mulliken’s (1970) suggestions, in combination with experimental results obtained in recent VUV spectroscopic studies. B. Afterglow Spectroscopy 1. Spectroscopic Data
The main features of the rare-gas spectra, conversion tables between Paschen and systematic classification, photoelectric recordings of spectra from the ultraviolet to the infrared, and transition probabilities of important atomic lines will be found in the AIP Handbook (Crosswhite, 1972). Table VIII lists some spectroscopic data taken from this reference for important lines of helium, neon, and argon.
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
163
TABLE V l l l RARE-GAS ATOMIC SPECTROSCOPIC DATA
Transition
Wavelength
Absorption oscillator strength
S",
300°K Doppler I/r width 6 (GH4
Helium Ground-2' P 2lS-3'P 23S-33P 2%3 3 P
584.3 5015.7 3888.7 5875.7
0.276 0.151 0.0645 0.609
19.04 2.22 2.86 1.89
0.2 17 1.02 0.337 4.82
Neon Ground-Is, Ground- Is, IS2-2Pz lS3-2Pz 1s4-2P2 1s5-2P2
735.9 743.7 6599.0 6163.6 6030.0 5881.9
0.161 0.012 0.147 0.241 0.028 0.032
6.76 6.69 0.75 0.8 1 0.83 0.85
0.357 0.026 2.9 4.5 0.5 1 0.56
Argon Ground- Is, Ground- Is, b2P, lS3-2P4 lS,-2P* IS5-2P,
1048.2 1066.7 7503.9 7948.2 8424.7 7635.1
0.252 0.06 1 0.13 0.56 0.4 1 0.24
3.36 3.30 0.47 0.44 0.42 0.46
1.123 0.277 14.8 7.8 18.8 4.3
(A)
k,/N at 300°K (lO-lzcm')
In this section, v will denote an oscillation frequency. This usual notation is more convenient here than the radian frequency o = 2nv which was used in former sections to preclude any confusion with collision frequencies. The width of lines emitted by a plasma is in general much larger than the natural linewidth, because of the presence of several broadening mechanisms-pressure broadening caused by neutral atoms, Stark broadening due to charged perturbers, and Doppler broadening. We shall assume that the lines are broadened by the thermal motion of the radiating atoms or molecules, i.e., that Doppler broadening is dominant. This convenient assumption is, however, by no means always true and should be carefully examined in actual applications (Griem, 1964). Dominant Doppler broadening corresponds to a Gaussian line shape
164
J.-F. DELPECH, J. BOULMER, A N D J. STEVEFELT
normalized to unity, i.e., such that +m
J"
-m
Y V dv) = 1,
(79)
where 6 is the l/e width related to the kinetic temperature T, to the atomic mass M, and to the center frequency v,,:
l / e widths are listed in Table VIII in frequency units (GHz). The full width at half maximum (FWHM) is also often used. It is related to 6 ( l / e ) by FWHM = 2[1n (2)]'12 S ( l / e ) .
(81)
The observed line shape is Gaussian only when this width is considerably greater than the width produced by other broadening mechanisms. Even then, far wings tend to deviate from this shape, because other mechanisms result in a much slower decay of the wing intensities. The dimensionless absorption oscillator strength f,, also listed in Table VIII is directly related to the atomic transition probability Am,:
where v,, is the center frequency, gm the upper-level degeneracy, and g, the lower-level degeneracy. Frequency v,, is related to upper- and lower-level energies Em and En through Planck's constant h : hv,, = Em - E n .
(83)
2. Absorption and Emission of Light b y a Plasma Absorption per unit length of monochromatic light at frequency v is
where N , ,N , are the population densities of lower and upper levels n and m, respectively, and ,J.! the absorption profile normalized to unity [Eq. (79)]. We shall limit ourselves in what follows to the homogeneous case (densities and kinetic temperatures constant along the line of sight) where induced
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
165
emission is negligible (upper-levelpopulation small compared to lower-level population). Absorption per unit length now becomes
Its expression at line center for a Gaussian line shape is often used:
ko =
e2 L m Nn 4 8 4 i .c0 mc
,
Normalized factors ko / N are listed in Table VIII. The dimensionless product ko 1, where 1 is the length of the plasma along the line of sight, plays an important role in plasma spectroscopy. It is called the optical depth. Total plasma absorptivity k(v)l at frequency v may be written
While not negligible, optical depths are rarely very large in afterglow plasmas except for transitions to the ground state. For example, at 300°K in helium for the 5015.7 hi line, with a He (2's)density of 10" cm-3 and a plasma length of 1 cm along the line of sight, ko = 0.10. In a precision study, possible fine structure, pressure broadening, and isotopic shifts should be taken into account. In the neighborhood of a strong spectral line, at frequency vmn,the radiated power per unit volume, per unit solid angle, per unit frequency interval, is
where L,,, is the emission profile normalized to unity. In what follows, we shall assume that the emission and absorption profiles L,, and ,!I have identical Gaussian shapes Y v ) [Eq. (78)] and that the plasma is homogeneous. Emission and absorption may be combined into the equation of radiative transfer which becomes in an homogeneous plasma, neglecting scattering (d/dx)l(v,x ) = j ( v ) - k(v)l(v,x),
(89)
where Z(v, x ) is the radiant power per unit surface, per unit solid angle, and per unit frequency interval at abscissa x along the line of sight. We shall now
166
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
integrate this equation under different assumptions and with a variety of boundary conditions. a. Plasma light emission in the optically thin limit. If the optical depth is small, either because the plasma length 1 along the line of sight is small or because lower level density is small, the second term on the right-hand side of Eq. (89) may be neglected. Integration then yields, with the boundary condition Z(v, 0) = 0 (no external light source), under our assumption of a Gaussian line shape,
This defines j , which may be written to a very close approximation, making use of the fact that the linewidth is small compared to frequency (6 $ v,):
Total power radiated on the mn line by unit surface and by unit solid angle obtains, by integrating over the whole line,
6. Nonoptically thin case. When the absorption term is not neglected, integration of Eq. (89) yields, under the same conditions as before,
Total power radiated on the mn line by unit surface and by unit solid angle is
Using Eq. (87) and the assumption 6 4 v,
which may be written
yields
167
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
or
)}I')?(-[
I,,(nonthin) = I,,(thin)S(ko I).
jirn( 1 - e x p ( - k o I e x p
S(koI) = -~ 6&ko1 1
(97) dv(98)
can be integrated by a series expansion (Mitchell and Zemansky, 1934)
+c a,
S(k0I) = 1
p=1
( - ko (p+l)!JP+l'
(99)
The optically thin limit is correctly recovered. For ko I I2, S(ko I) is reasonably well approximated by
S(ko 1) 2 1 - 0.353ko I
+ 0.065(ko1)'.
(100) c. Plasma light absorption. Let us now integrate Eq. (89) in the case of an external light source imposing a boundary condition I(v, 0). We shall assume that this light source is weak enough not to perturb appreciably the excited state densities. It should then not interfere with plasma light emission as given by Eq. (97). The monochromatic relative attenuation is
This is the attenuation which would be measured with a monochromatic external light source (laser source). The plasma Gaussian lineshape would then be directly recovered. The incident light may also have a nonzero linewidth di around center frequency v i . We shall assume that is well approximated by a Gaussian profile Li(V) =
1
Si&
~
exp
[ (y)2]. -
The observed attenuation integrated over the spectrum is now
x
( 1 -exp ( -koIexp [-("dy..)i]/)dv,
which may be integrated by a series expansion
(103)
168
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
where a is the linewidth ratio a = Si/S.
(105) In the important particular case where vi = v,, (same gas in the source and the plasma, but possibly different excitation conditions), Eq. (104) reduces to the function O3
A,=-C
p=1
(-kOl)’ -
p ! J W ’
tabulated by Mitchell and Zemansky (1934). An approximation to the inverse problem of obtaining ko 1 from a known A, is given below [Eq. (107) and (108)l. C. Spectroscopic Diagnostics Techniques
Plasma spectroscopic techniques are described in detail in reference books, particularly by Griem (1964) and by Lochte-Holtgreven (1968). In this section we shall only describe techniques which have been found to be applicable to afterglow spectroscopy.
1. Emission Spectroscopy Emission spectroscopy aims at deducing absolute or relative populations of a radiating excited state from measurements of the absolute or relative intensity of the light emitted on one (or possibly a few) of the lines originating from that level. In making use of Eqs. (96) and (91), one needs to know from independent measurements the oscillator strength of the transition, as well as the plasma length 1. If the optical depth ko 1 is not negligible it should also be measured or independently known. Note that in many cases, contrary to what had been assumed in deriving Eq. (91), the plasma is not spatially uniform. One should then resort to Abel inversion techniques to know the spatial distribution of the excited species (Griem, 1964). In a typical experimental setup, a known plasma volume is observed through an optical system (mirrors, lenses, apertures) under a known solid angle. Light is then spectrally analyzed, usually through a grating monochromator, and its intensity is measured with a cooled photomultiplier; the photon counting mode is more convenient at very low light levels. Possible fluorescence of the quartz windows should be taken into account. Robben (1971) has analyzed optimal uses of photomultiplier tubes. The whole system should be calibrated against a blackbody source of known temperature (Griem, 1964).
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
169
Absolute optical intensity measurements are beset with many difficulties, and systematic errors may substantially exceed 30%. Geometrical and optical calibrations are not easy, and in many cases oscillator strengths are not even known with good precision (Crosswhite, 1972). Large progress has recently been made in photomultiplier cathode technology, and global efficiencies of a few percent from 3000 to 8000 di are now becoming practical with all losses included (mirrors, windows, grating, phototube quantum efficiency); dark count rates may be well below ten pulses per second. Absolute emission spectroscopy is a unique diagnosticstool, because it is the only direct technique available to probe highly excited species densities. Figure 10 shows data obtained in helium at 11 torr; axial particle densities
En, eV
FIG. 10. Excited helium state densities along plasma column axis in a helium afterglow at 1 1 torr. From Stevefelt and Robben (1972).
were deduced from local measurements by Abel inversion (Stevefelt and Robben, 1972). Highly excited levels are, to a good approximation, in Saha equilibrium with the free electron gas [Eq. (58)]. The same equilibrium temperature can also be found among sublevels of principal quantum number p = 3 and 4. In this example, the whole situation is in agreement with the collisional-radiative recombination model of the He' ion (Section 111,C). Relative intensity measurements are easier, the prime consideration being now dynamic range and signal-to-noise ratio. They may also be extremely useful, particularly in studying time evolution of excited particle densities duripg the afterglow (Fig. 11).
170
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
FIG.11. Afterglow intensities of the 5015 A line, proportional to He(3'P) density, and of the 4650 A band, proportional to He2(3%,) density in helium at 3 torr. From Collins and Hurt (1969).
2. Absorption Spectroscopy Measurement of the plasma optical depth ko 1 for a given transition at frequency v, yields through Eq. (86) the lower-level density, provided the corresponding oscillator strength is known. Consideration of Table VIII shows that for a reasonable path through the plasma, lower-level densities should be relatively large if the attenuation is to be experimentally measurable. Optical attenuations are not easy to measure because an attenuation is essentially a small difference between two large quantities [see Eq. (loll] and because optical quantum noise is quite large, because of the large energy of photons in the visible range. Source intensity cannot be made arbitrarily
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
171
large to minimize excited level perturbations; a suitable rejection filter should be inserted between the source and the plasma to avoid interaction at frequencies other than v, (Fitzsimmons et al., 1968). Absorption spectroscopy may be used for molecular density measurements (Phelps, 1955 and 1968) but absolute measurements become very uncertain in this case, because of the rotational structure of the band and because oscillator strengths of molecular transitions are not well known (Stevefelt and Robben, 1972). In a typical experimental setup (Ellis and Twiddy, 1969), the external light source has an approximately Gaussian lineshape of width di around v,, [Eq. (102) with v i = v,] and the measured quantity is the attenuation A, related to the optical depth by Eq. (106). It is often difficult to know the linewidth ratio a [Eq. (105)l with enough accuracy. It can be measured (Dixon and Grant, 1957) with a singlepass/double-pass method as shown in Fig. 12. It should also be noted that A, becomes nearly independent of a when a’ 6 1. One should thus try to Single poss
A
Double pass
I
1 I
4
I
FIG.12. Principle of the single-pass/double-pass method. From Ellis and Twiddy (1969).
172
J.-F. DELPECH, J. BOULMER, A N D J. STEVEFELT
keep the source linewidth as small as possible, either with an incoherent source and a Fabry-Perot filter or with a high-resolution tunable laser. A numerical inversion is necessary to deduce the optical depth ko I from the measured attenuation A,. A good approximation to this numerical inversion obtains by putting
+ 0.73a2)'/2A,]
(107)
+ 0 . 0 7 5 ~-) ~ax2 10 .
(108)
x = In [l - (1
and computing
kol E -(1
~
Error is below 1 % for ko I I 1 and a I 1, but increases rapidly outside this range. A variation of the single-pass/double-pass method may also be used to measure kol. This method consists in comparing the light intensity ZJI) emitted by a length I of plasma [Eq. (9711 and the light intensity Imn(21) emitted by a length 21; the latter may be obtained by reflecting back the light into the plasma. This is the Ladenburg and Reiche line absorption technique (Mitchell and Zemansky, 1934). The measured absorption is
where S is the function defined by Eq. (99). Under actual experimental conditions, one should take into account parasitic reflections and losses ;the two paths across the plasma also may not be identical. Reasonably accurate numerical inversion of Eq. (109) obtains by using Eqs. (107)and (108) with a = 1. 3. Active Spectroscopy
The development of truly tunable lasers that span the entire visible and near infrared spectrum to more than 8OOO di, as well as the near UV down to 3500 di (down to 2400 di with optical frequency doubling) has opened a whole new range in plasma spectroscopy. Dye lasers pumped by nitrogen lasers (Wallenstein and Hansch, 1974) appear particularly promising for afterglow plasma studies. They are able to generate very fast light pulses (a few nanoseconds) of high power (a few kilowatts); this corresponds to quite reasonable energies (typically more than lOI4 photons per pulse). With a single frequency-selecting grating, linewidth is below 0.2 di from 3500 to 8OOO A. Addition of an intracavity
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
173
etalon permits linewidths and stability in excess of 5 x 10- di (0.6 GHz at 5000 di) while maintaining tunability. Dye lasers may also be pumped with flashlamps or with CW lasers. They then deliver larger energies for longer periods, and this may be useful in some applications. Solid-state diode lasers have also been used (Ku et al., 1972). The first, obvious application of tunable lasers is as light sources in the absorption experiments described in the former section ; they are particularly convenient, since they deliver well collimated light beams with linewidths significantly smaller than plasma Doppler linewidths in many cases. They can also be used in the related technique of excited-state population measurement by optical pumping and fluorescence (Ku et al., 1972). However, their most novel feature is in the field of active spectroscopy. Dimock et al. (1969) were among the first to point out their many possibilities; the laser light source now becomes a means for holding the upper and lower states of a transition equally populated during the laser pulse, and to study subsequent reactions. Population of excited states of atoms with coherent light has been studied by McIlrath and Carlsten (1972).Consideration of Tables VI and VII shows that most energy states of electronic quantum number p 2 3 in rare gases, including the continua, can be: reached from the relatively highly populated atomic or molecular metastable levels ( p = 2) with present day nitrogen-laser-pumped dye lasers. Collins et al. (1972a)have populated the He(53P)level by excitation from He(23S) with a flashlamp-pumped dye laser frequency doubled to 2945 A, and have obtained information on excitation transfer and associative ionization processes involving He(S3P). Collins and Johnson (1972) have used similar techniques to study rotational relaxation in molecular helium.
D . Metastable Population Both p = 2 nonmetastable states of helium, He(2lP) and He(23P), are radiatively coupled to the metastable states of same multiplicity, He(2'S) and He(23S). They both behave essentially like ordinary nonmetastable excited atoms; even though the optical depths of lines terminating on metastable states are not negligible under normal afterglow conditions, the plasma is very far from being optically thick toward them. The situation is quite different for the Isz and Is, levels of neon through xenon (Paschen notation). They are optically coupled to the ground state only, and the corresponding lines are resonance lines: The plasma is optically thick toward them. For example, it can be deduced from Table VIII that in argon at 1 torr, 300"K, optical depth is about unity for a propagation
174
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT
length smaller than 10- cm for the two Ar( 1s)ground-state allowed transitions. Apparent radiative decay (Holstein, 1947 and 1951) is thus extremely slow for these excited states and they behave in some respect like metastable states. It is also important to note that the four 1s levels of all the heavier rare gases are quite close in energy, except for the 1s2 and Is, levels of krypton and xenon; collisional coupling between levels should be expected to determine to a large extent the observed decay. Metastable atoms are principally formed during the discharge but some are produced during the afterglow, since both collisional-radiative and dissociative recombination end up on a metastable level. This has been demonstrated in atomic helium by Johnson and Gerard0 (1972). The situation is more complicated in molecular helium, where collisional couplings between singlet and triplet states are strong; radiative deexcitation of singlet states to ground should be very fast, since there is no significant reabsorption in molecular helium, the ground state being dissociative. At the present, there is no point in writing a general metastable continuity equation, since many processes and still more rates are unknown. Rates quoted in the following section have been measured under experimental conditions suitably chosen to enhance a few selected processes. 1. Metastable Diflusion to the Walls Metastables diffuse to the walls with a diffusion coefficient D M . In the absence of other loss or creation process, their density M would be solution of the continuity equation (12) which becomes now
The general comments of Section I,C apply to this case. Known metastable reduced diffusion coefficients D M p are listed in Table IX. Comparison between Tables I1 and IX shows that atomic-ion and atomic-metastable diffusion coefficients are very close to each other. Precise measurements of the He(2,S) diffusion coefficient have been made by FitzSimmons et al. (1968) down to cryogenic temperatures. 2. Ionizing Collisions between Metastable Atoms These reactions are a particular case of the Penning reaction R*(met.) + R*(met.) + R + + R + e + KE. (111) The kinetic energy is essentially coupled to the electron. This reaction has been shown to be an important ionization source term and contributes to
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
175
TABLE IX REDUCED METASTABLE DIFFUSION COEFFICIENTS AT 300°K
Metastable species
DM P
(cm' sec- I tom) 440 f 50" 470 f 25" 361 f 34' 171' 55.0" 51.6d 38'
21'
' Phelps (1955).
'Gusinow and Gerber (1970). Phelps (1959). Ellis and Twiddy (1969). ' (Theory) Palkina et a/. (1969).
electron heating (see Section 1,D). The ion formed may be a molecular ion on highly excited vibrational states (Garrison et al., 1973). The rate fl of process (111) is known in helium to be around 2 x 10cm3 sec- from 4.2 to 500°K (Phelps and Molnar, 1953; Hurt, 1966). It is not known in other rare gases, but there are reasons to believe that it should have the same order of magnitude, with little temperature dependence (Bates et al., 1967). 3. Atomic-to-Molecular Metastable Conversion
These conversions proceed through reactions of the type R*(met.) + R
+ R -,Rf(met.) + R + KE.
(112)
Corresponding three-body metastable conversion rates k3 are listed in Table X. If no other creation or loss mechanism was present, metastable density M would be solution of aM at
-=
-k3niM.
The sharp decrease of the three-body conversion rate with temperature in
176
J.-F. DELPECH, J. BOULMER, AND J. STEVEFELT TABLE X THREE-BODY METASTABLE CONVERSION COEFFICIENTS
Metastable species
Temperature
Three-body rate k3 (cm6 sec- ')
Phelps (1955). Phelps and Molnar (1953). Phelps (1959). Ellis and Twiddy (1969).
helium (Table X) is consistent with the presence of a hump on the He( 1%) - He(23S) interaction curve and with diffusion coefficient variations with temperature (Fitzsimmons et al., 1968).
4. Metastable Deexcitation by Superelastic Electron Collisions The reaction R*(met.) + e + R + e
+ KE
(114) is the inverse of the inelastic excitation by electron impact reaction. The electron gains a large kinetic energy, essentially equal to the metastable excitation energy. While not contributing to ionization during the afterglow, superelastic reactions of type (1 14) may contribute an important electron energy source term. The rate ksup of this reaction in helium has been recently deduced by Nesbet et al. (1974)from theoretical considerations coupled to known experimental values of the excitation cross section. They find ksup = 2.9 x lo-' cm3 sec- for electron temperatures between 0 and 2000°K. This is a fairly large rate. 5. Elastic Electron Scattering by Metastable Rare-Gas Atoms Electron-metastable interaction is strong in view of the large polarizability of metastable rare-gas atoms. Theoretical cross-section predictions by Robinson (1969)range from a few 10-l4 cm2 for the heavier rare gases to cm2 for helium at 300°K. 4x
LOW-TEMPERATURE RARE-GAS STATIONARY AFTERGLOWS
177
6. Miscellaneous Reactions Involving Metastable Atoms a. Helium.
He(2lS)
+e
-+
He(2%) + e + 0.79 eV,
(115)
with cross section 0 = 3 x l o v i 4cm’ (Phelps, 1955),
+ He
He2(2%:)
He2(2’X:)
+ He - 0.29 eV,
(116) where the singlet helium molecule is subsequently deexcited by allowed radiation to the dissociating ground state. A relatively large cross section (0 10- cm’) has been estimated for this process by Teter and Robertson (1966). +
-
He(2’S) + He + He + He + hv
(117) (collision-induced radiative deexcitation of the singlet metastable helium atom), with cross-section 0 z 3 x lo-’’ cmz (Phelps, 1955). b. Neon. The following reactions have been observed and the cross sections deduced by Phelps (1959). Cross section 0 z cm’: Ne(ls,)
+ e + Ne(ls,) + e + 0.0517 eV.
(118)
Cross section 0 z lo-’, cm’: Ne( 1s3)+ e -+ Ne( Is,)
+ e + 0.0446 eV.
(119)
Cross section 0 z 5 x 10- l9 cm’: Ne(ls,)
+ Ne + Ne(ls,) + Ne + 0.0517 eV.
(120)
Cross section 0 s 6 x lo-’’ cm’: Ne( 1s3)+ Ne -+
[
Ne( Is,) Ne( Is,)
+ Ne + 0.0446 eV + Ne + 0.0963 eV
*
(121)
c. Argon. Ellis and Twiddy (1969) have observed deexcitation of argon metastable atoms by collision with ground-state argon atoms. Cross sections cm’ for the Ar(ls,) level and 0 = 1.8 x lo-’’ cm2 for are 0 = 1.0 x the Ar( Is,) level.
V. CONCLUSION Our understanding of stationary afterglows at temperatures between cryogenic and room temperature has substantially progressed in the last few years. We understand now that energy stored in long-lived excited species is slowly released during the afterglow, producing electron-ion pairs, and that
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J.-F. DELPECH, J. BOULMER, A N D J. STEVEFELT
the large kinetic energy of these electrons plays a large role in heating the bulk electron gas. This is a consequence of the fact that excited species of rare gases are chemically reacting; Penning or Hornbeck-Molnar processes (see Section IV,D) may in some sense be thought of as a form of chemiionization. The electron continuity equation, the energy balance equation, and the metastable continuity equation usually form a set of closely coupled nonlinear, second-order, partial differential equations. This situation is far from simple and it should now be clear that many conclusions which were formerly drawn from elementary considerations of a simplified, uncoupled electron continuity equation are questionable. In addition, many of the processes which have been put forward to account for experimental observations are still not completely elucidated, and much remains to be done in order to understand precisely what really happens in low-temperature rare-gas afterglows. ACKNOWLEDGMENTS It is a pleasure to acknowledge many helpful discussions with Professor L.Goldstein and with members of the Groupe d’Electronique dans les Gaz at Paris-XI University.
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Rogers, W. A., and Biondi, M. A. (1964). Phys. Rev. 134, A1215. Seaton, M. J. (1959). Mon. Nor. Roy. Asrr. SOC. 119, 81. Shkarofsky, 1. P., Johnston, T. W., and Bachynski, M. P. (1966). “The Particle Kinetics of Plasmas.” Addison-Wesley, Reading, Massachusetts. Sol, C., Boulmer, J., and Delpech, J.-F. (1973). Phys. Rev. A 7 , 1023. Sol, C., Devos, F., and Gauthier, J.-C. (1975). Phys. Rev. (to be published). Stevefelt, J. (1973). Phys. Rev. A 8, 2507. Stevefelt, J., and Robben, F. (1972). Phys. Rev. A 5, 1502. Stevefelt, J., Boulmer, J., and Delpech, J.-F. (1975). Phys. Rev. (to be published). Taylor, B. N., Parker, W. H., and Langenberg, D. N. (1969). Rev. Mod. Phys. 41, 375. Teter, M. P., and Robertson, W. W. (1966). J . Chem. Phys. 45, 661. Thomassen. K. I. (1963). J. Appl. Phys. 34, 1622. Tiixen, 0. (1936). Z. Phys. 103, 463. Vauge, C., and Whitten, J. L. (1972). Chem. Phys. k t t . 13, 541. Veatch, G. E., and Oskam, H. J. (1970). Phys. Rev. A 2, 1422. Vitols, A. P., and Oskam, H. J. (1972). Phys. Rev. A 5. 2618. Vitols. A. P., and Oskam, H. J. (1973a). Phys. Rev. A 8, 1860. Vitols, A. P., and Oskam, H. J. (1973b) Phys. Rev. A 8, 32 1I. Wallenstein, R., and Hansch, T. W. (1974). Appl. Opt. 13, 1625.
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Advances in Molecular Beam Masers D. C. LAINE Department of Physics, University of Keele. Keele, Staffordshire, United Kingdom
1. Introduction. ........................................................ 11. Principles and Techniques ......................................... A. Molecular Beam Sources ............................
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I. INTRODUCTION The publication by Gordon, Zeiger, and Townes in July 1954 announcing the operation of a device that could be used as a very-high-resolution spectrometer, microwave amplifier, or oscillator of high-frequency stability inaugurated a new class of devices based on the principle of stimulated emission from a substance in nonthermal equilibrium. The original proposal by Townes’ group to use the J = 2 + 1 rotation inversion transition of ND, at a wavelength of 0.5 mm was given in an internal report of the Columbia Radiation Laboratory, New York, as early as 1951. However, the device eventually constructed was based on the longer wavelength J = 3, K = 3 183
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inversion transition of NH3 at 12.5 mm, using electrostatic state selection of molecules (Gordon et al., 1954). Similar but independent proposals by Basov and Prokhorov at the Lebedev Physical Institute, Moscow, for a molecular-beam spectrometer or oscillator were published in the same year (Basov and Prokhorov, 1954). The early pioneering work of the Columbia and Lebedev groups was to lay the foundations not only of molecular beam maser research, but also of the new subject of quantum electronics.This new discipline, based on the use of quantized systems for the construction of practical amplifiers, oscillators, and other devices expanded very rapidly, particularly in the area of lasers. The importance of the pioneering efforts of Basov, Prokhorov, and Townes was recognized by the international scientific community by the award of the Nobel Prize for Physics in 1964. With all this activity it is perhaps not surprising that molecular beam masers have somewhat faded from the research scene. Accordingly, progress in this area is not particularly well known, a situation which it is hoped the present review will in a small measure go some way to redress. Over the 21-year lifespan of molecular beam maser (MBM) research, a large literature of original work has accumulated which is now well in excess of 400 titles. A substantial fraction of the published work owes its origin to the efforts of researchers in the Soviet Union, but on the whole, the extent of that contribution does not appear to be well known in the Western hemisphere, undoubtedly due to problems of translation. The literature on MBMs is rich in undeveloped suggestions and ideas. Many of these have been discussed in an earlier review (Laine, 1970) which also gives a fairly comprehensive list of references to beam masers and their applications up to 1969. The list of references given in this review is intended to be representative rather than an attempt to be complete. The aim of the present article is to review progress and achievements with molecular beam masers rather than dwell on undeveloped ideas and suggestions. Emphasis will be placed on techniques and applications wherever possible. Representative examples will be given of fundamental MBM research work carried out since 1954. Work carried out over the last five years is given particular attention. 11. PRINCIPLES AND TECHNIQUES
In the early 1950's, gas spectroscopists were seeking new methods of improving the resolution and sensitivity of their spectrometer systems. Of the various suggestions under active investigation the molecular beam approach was perhaps the least complex, although the requirements of large beam intensity for reasons of maximum possible sensitivity and low beam
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divergence for good spectral resolution were in conflict. Indeed, the method did not look particularly promising except for the strongest spectral lines until the principle of selection of molecules in particular quantum states as used in MBMs was introduced. The basic problem of any molecular beam method is that as a consequence of beam formation, the density of molecules in the spectrometer cell is very low at any given time, in comparison with bulk gas spectrometer systems. In a conventional effusive molecular beam, the molecules may be assumed to be in or very near the thermal equilibrium condition. Consequently in the presence of external radiation, absorption and emission processes are nearly in balance with fractionally more molecules in the lower energy state, leading to a small net absorption. If this equilibrium condition is disturbed by some method or other, an improved spectrometer sensitivity can in principle be obtained, either in the form of an enhanced absorption (antimaser action) or stimulated emission (maser action). If all the molecules in the lower state are removed, then the signal is enhanced in magnitude by a factor of kT/hv over the thermal equilibrium value, where T is the absolute temperature and hv the energy between the quantum levels of interest. At microwave frequencies (24 GHz) this factor is about 250, whereas at radio frequencies (1 MHz) approximately 6 x lo6, assuming T = 290K. These figures, although only rough, show quite clearly that enhanced MBM spectrometer sensitivities of two orders of magnitude or more are in principle possible at microwave wavelengths and longer. A useful increase of sensitivity in the millimeter to long submillimeter wavelength region is also obtained. The first MBM spectrometer system, which was operated with electrostatic sorting, showed an improvement of about two orders of magnitude in sensitivity and a factor of about five in resolution in comparison with earlier moleciilar beam work (Strandberg and Dreicer, 1954). Furthermore, since molecules in the beam were state selected to give a net emissive rather than absorptive signal the possibility arose of constructing a quantumelectronic amplifier or oscillator. MBMs have three clearly identifiable component parts: gas source, state separator, and a region in which state selected molecules and radiation interact. These are indicated in Fig. 1. The gas source produces an intense and highly collimated molecular beam. The molecules then enter the electrostatic state separator which selects molecules in particular quantum states. Thus the molecular beam is changed from being potentially weakly absorptive on entry to being potentially strongly emissive after passing through the state selector. The beam then enters a region in which emission from it may be stimulated, which usually takes the form of a tuned resonant structure.
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FIG. 1. Typical two-chamber molecular beam maser system.
A. Molecular Beam Sources For the successful operation of a MBM system it is essential to have an intense and highly directional molecular beam. Typically, the molecular flux will be of the order of lo’* molecules sec- or more which constitutes a severe gas “leak” for the associated vacuum pumps to handle. Since the background pressure in the state separator and maser resonator chamber must be of the order of 10- torr or less, fast pumping is required. Thus for a given pumping speed, the maximum permissible molecular flux is essentially determined by the properties of the beam source. Molecular beam sources for masers are either of the effusive or nozzle type and both have been studied in great detail, both experimentally and theoretically, by many investigators. For a more detailed discussion of such sources than is presented here reference should be made to a recent review by English and Zorn (1974). In the early types of MBMs effusive sources were employed. One method of producing a highly collimated beam was to use a crinkle foil source, prepared by rolling two thin strips of nickel foil on to a spindle. One strip was finely corrugated by passing it through a pair of meshing gear wheels and the other was flat. An aligned stack of photographically etched metal foils has also been investigated. Perhaps the most widely used multichannel effuser has been the extended klystron grid structure. Multichannel effusers are usually operated with gas pressures in the range 0.5-10 torr and have a very favorable ratio of forward intensity to total gas flow in comparison to that of a thin wall orifice. Thus, in instances where gas conservation is important, multichannel effusers are advantageous. A summary of experimental results with several types of effusers has been given by Helmer et al. (1960a). Since multichannel beam sources are
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quite difficult to construct, there has been a tendency to use single-tube effusers,even though directivity for a given gas flow is generally reduced. Typically, these have length to diameter ratios of 10, and a diameter of 1 mm. Since the flow rate of gas is somewhat increased by their use, additional pumping speeds may be required to maintain a given background pressure in the maser cavity chamber. In MBMs it is desired that the greatest fraction of the total flux of molecules should be in the quantum states used for the particular maser transition investigated. Since this fraction is temperature dependent, it follows that room temperature will not generally be the most appropriate if this fraction is to be a maximum. Thus, for most maser gases in the ground vibrational state, the beam is advantageously cooled by operating the effuser at a low temperature. At the same time, the maser linewidth is slightly reduced since the forward velocity of the molecules is decreased. However, for molecules in excited states with energies 2 kT above the ground state, it is more appropriate to heat the effuser. This has been demonstrated by Tucker et al. (1971) who used the l , o + l , , rotational transition of formaldehyde in the first excited vibrational states of both the v6 and v 5 bending modes which lie, respectively, at 1167 and 1280 cm- above the ground state. Single-hole nozzles of small diameter (0.3 mm and below) are now widely used to form intense molecular beams in masers. With them the possibility of dynamic cooling occurs, with an associated enhancement of molecular populations in low-lying rotational states. In MBMs, nozzle sources have improved the sensitivity of spectrometers over that with effusive sources by roughly an order of magnitude. Typically, nozzle sources are driven by pressures ranging from 15 to 250 torr or higher, the upper value depending on pumping speed. Suitable holes have been made in thin metal diaphragms by (i) piercing with a needle (Dymanus, 1975), (ii) the use of a jeweller's drill, (iii) electrical spark erosion, or (iv) focused laser radiation (Dyke et al., 1972). When the maser gas is chemically unstable, as for example with the free radical OH, the gas sources described so far are unsuitable. OH radicals are formed near the exit of the flow tube by the reaction H + NO2 -, OH + NO. The atomic hydrogen is produced by an electrical discharge of moist hydrogen molecules or in water vapor. The beam of radicals is then finally produced by a 2 mm hole in a chemically inert Teflon diaphragm (Dymanus, 1975). When the MBM spectrometer is used with substances which possess a low vapor pressure at room temperature, it is necessary to heat the sample to obtain a more intense beam. Such a procedure has been used by Kukolich and Nelson (1971) with formamide heated to 100°C.
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In MBMs a large flux of molecules is necessary because of the inefficiency of electrostatic state selection and the small fractional population of molecules usually encountered in the particular quantum states used. Consequently, to maintain a low background pressure so as to avoid beam scattering, fast pumping is required. This is easily obtained with condensable gases such as ammonia by the use of liquid nitrogen traps which act as cryogenic pumps. When the MBM oscillator held out great promise as a primary standard of frequency in the late 1950's and early 1960's several attempts were made to make sealed-off ammonia MBM systems in which the only form of pumping was cryogenic, thus eliminating bulky diffusion pumps. While such systems were used with limited success, it was generally necessary to supplement the cryogenic pumping with a small vac-ion pump. A more detailed discussion of sealed-off maser oscillator systems is given in Section V,D,l,b. In general, cryogenic pumping is inconvenient and the sole use of diffusion pumps is preferred. However, to obtain sufficiently fast pumping, very large pumping speeds are required so as to obtain a sufficiently low background pressure in the maser for oscillation or for other studies. In present day MBMs, differential pumping, is usually used with the gas source chamber separately pumped from the state separator and resonator chamber. In the case of MBM oscillators, operation without the assistance of cryopumps is rare without differential pumping, although has been achieved by Skvortsov el al. (1960) with ammonia and by Krupnov and Skvortsov (1963b) with formaldehyde. Differentially pumped maser oscillators operated without cryogenic pumping have been reported by White (1959), Grigor'yants and Zhabotinskii (1961a), Bardo and Laine (1971a), and Maroof and Laine (1974) with the J = 3, K = 3 inversion line of 14NH3, and by these last investigators also with the J = 3, K = 2 inversion line of the same molecule. The use of a nozzle source together with a skimmer and a total pumping speed of 1.2 m'sec-', permitted operation of the maser oscillator with a threshold voltage of 7.2 kV and 25.0 kV with the l4NH, J = 3, K = 3 and J = 3, K = 2 inversion lines, respectively. This result for the latter line is important, since it is now possible to operate the maser as a secondary standard of frequency on a favorable transition without the operational time limit necessarily imposed by cryogenic pumping. Furthermore, changes in oscillation frequency which often follow topping up of the liquid nitrogen traps as a result of changes of background pressure, thermally induced drifts of resonator tuning, loading, and so on are greatly reduced. Similar advantages are gained in MBM spectroscopy. For example, where very weak spectral lines are studied, and thus slow scanning rates employed for maximum sensitivity, slow background pressure changes or thermal drifts which could severely affect the apparent lineshape, are eliminated.
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B. State Separators
It is well known that molecules which possess a large electric dipole moment can be readily deflected by the use of nonuniform electric fields with a gradient perpendicular to the molecular beam axis. If the strength of the applied field changes radially, with the molecular beam forming an axis, a transverse potential well is created for molecules with the appropriate sign of Stark interaction. This not only allows the possibility of spatial state selection but also the focusing of molecules under favorable circumstances. Therefore, the beam intensity at a distant point from the beam source may be increased relative to that for an unfocused beam. The device whereby state selection is effected is often called afocuser or state separator. It should be pointed out that these terms are not always interchangeable, since state separation is not always associated with focusing, but rather may be the result of differential defocusing. The condition for focusing is that the maximum Stark energy must be greater than the radial or transverse kinetic energy of molecules in the beam. For MBM operation, it is necessary to obtain a population excess of molecules in the upper of a pair of energy states between which transitions are desired. If the signs of the Stark interaction are different, for example with energy levels that repel each other in an applied electric field, state separation is a simple matter. All that is required is a transverse potential well for molecules whose Stark energy increases (positive Stark slope) in an applied electric field. Thus with NH, , molecules in upper inversion states are focused and those in the lower states (negative Stark slope) defocused and lost from the beam. This is achieved by the use of an electrode structure in which the static electric field increases in a direction perpendicular to the beam axis. However, for other species of molecules, situations arise where the Stark interactions [henceforth designated here as ( + ) or ( - ) according to Stark slope] may be of the same sign for both upper and lower levels, thereby making state separation difficult. For example with HCN, as shown in Fig. 2, molecules in the states J = 1, F = 1 and J = 0, F = 1 have a ( - ) Stark interaction and thus neither state is focused. Since the upper state has a weaker Stark interaction than the lower, maser action may still be achieved. For pure rotation states, ignoring the effects of hyperfine structure, the Stark energy is greatest for molecules with a small value of rotational constant B, large electric dipole moment p, and low value of rotational quantum number J for which there are few M sublevels in an applied electric field. For linear or symmetric-top molecules with K = 0, focusing is most effective with low values of J . When the rotation levels split on account of K , I, A, or inversion doubling, focusing is again most effective with low values of J .
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F
J.0
1
L \ ,E
I
0
Stark field
FIG.2. Energy level scheme and Stark effect for HCN: emissive and absorptive transitions labeled (e) and (a), respectively. After De Lucia and Gordy (1969).
Moreover, for very closely spaced levels such as those used in radiofrequency MBMs, focusing is generally more effective than in microwave masers, and accordingly transitions between doublets with J values up to 10 have been investigated, for example, with the transitions lo,, -+ lo4, for CH,O at 1.5 MHz (Takami and Shimizu, 1966). However, in recent years beam intensities and the sensitivity of MBM spectrometers have both been increased and molecules with high J values can now be studied. Examples are the 7,, + 7,, transition of HDO at 8.6 GHz (Verhoeven et al., 1968) and the 8,, -+ 881 transition of C4H4O at 2.1 GHz (Tomasevich et al., 1973). In early MBMs, the quadrupole electrode configuration was employed (Gordon, 1955). The electrodes were cylindrical tubes whose inner surfaces were shaped to form hyperbolas. For such a system the electric field is zero on the beam axis, rising linearly to a maximum value between the electrodes. The electric field in such an electrode array may be calculated exactly. In general, however, cylindrical electrodes are employed for convenience of construction. A problem with the quadrupole state selector is the relatively narrow acceptance angle for focusing. It was shown by Helmer et al. (1960a) that a greater solid angle of molecules could be captured from the gas source if the state selector is shaped lengthwise to the form of a parabola. Unfortunately, the construction of a state selector with a cross-sectional area which increases along its length is quite difficult, especially in the parabolic form. A linear taper is a useful compromise. In general, however, quadrupole state selectors are not very transparent to molecules deflected out of the beam and state-selection efficiency may suffer on account of molecular scattering. Multipole rod focusers (Shimoda, 1957) consisting of eight or twelve
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electrodes, whose potential well is better described as square rather than harmonic, as with the quadrupole type, are often favored by virtue of their transparency to molecules deflected out of the beam. The rod electrodes are typically made of a nonmagnetic material, usually stainless steel, between 150 and 350 mm long and 2-20 mm diam. Precise dimensions are not critical, but clearly the higher the frequency of the maser transition, the smaller the resonator and the smaller the diameter of the state-selected beam required. The rods are held at both ends by glass or Teflon rings to form a cylindrical electrode array. Operation is typically at 30 kV or more. For symmetric-top molecules with K = 0 or for linear molecules, there is an optimum value of field for the most efficient state selection. This is due to a reversal of the sign of the Stark interaction from (+ ) to ( - ) at high values of applied electric field for certain quantum states. A modification of the multipole focuser is the double ladder device which ideally focuses molecules to a line, rather than a point as in an electrode system with radial symmetry. This type of state selector is particularly convenient for the excitation of the flat-plate capacitor of an LC tuned circuit of a radio frequency MBM (Shimoda, 1964), rectangular microwave cavities (Becker, 1963a), or open resonators. This last type of resonator has been used by several investigators. For example, Marcuse (1962) used radially arranged electrodes (confocal resonators, HCN); Krupnov and Skvortsov (1965~)used a parallel-ladder state separator (disc resonator, CH,O), and Laine and Smart (1971) have used a nonparallel double ladder, thereby increasing the solid angle of molecules captured by the state selector (disc resonator, NH,). Other types of state selectors have also been used for MBMs. The ring and bifilar helix system was first proposed by Krupnov (1959) and has been investigated by Shcheglov ( 1 9 6 9 Basov et al. (1963b), Becker (1963b), and Mednikov and Parygin (1963). Experimentally, the bifilar helix has the experimental advantage that the electrodes are smooth along their length, so that electrical breakdown is minimized. This type of focuser is generally considered to be one of the most efficient from the point of view of capture angle. However, these focusers are prone to be microphonic unless the helix is supported at points along its length or made up from several short sections arranged in series. Ring focusers are much more rigid, but they are difficult to fabricate without sharp edges on the electrodes which may lead to electrical breakdown. The difficulty of fabrication of ring focusers has been circumvented by Bardo and Laine (1971a)who have used the If turn sprung ends clipped from domestic safety pins. The smooth ring profile decreases the tendency for electrical breakdown, although their thickness inevitably increases scattering of molecules deflected out of the beam. A selection of (+ ) focusers is shown in Fig. 3.
192
D.
......... ......... @
c. LAINB
3 _*$I
-.. -. +. . 2: :
-+-+-+-+(C 1
, +* z: : I
~~
-+ 0 e.0
2:
+*
:
(b)
FIG.3. Selection of positive Stark slope state separators: (a) eight pole; (b) double ladder; (c) ring; (d) safety pin.
Molecular beam focusing has also been obtained using the neutral particle analog of alternate gradient focusing as used in charged particle accelerators. This type of focuser, first discussed by Auerbach et al. (1966) was shown capable of focusing molecules of ammonia by Kakati and Laine (1971) and produces either an enhanced absorption or emission condition depending on the focuser-maser cavity geometry. However, both (+) and (-) molecules are converged and consequently the device is rather unsuitable for spectroscopic applications, since it is only the differential focusing effect which is detected. Focusers for ( - ) molecules only should also be briefly mentioned. Molecules which have hitherto shown an enhanced absorption in (+) focusers, would with a (- ) system give rise to state selection suitable for maser action. Examples of possible molecules are HCN (DeLucia and Gordy, 1969) and OCS (Wang et al., 1973). In general, space focusing of (-) molecules is rather more difficult than with (+) molecules. The experimental problem is that the maximum electric field is required on the axis of the molecular beam without excessive beam scattering by one or more focuser electrodes. Methods investigated include coaxial electrodes (Helmer et al., 1960a,b; Laine and Sweeting, 1971a) and its variant, the single-wire system with the outer electrode effectively removed to infinity; the crossed-wire method (Laine and Sweeting, 1971b); and the alternate gradient focusing method already mentioned. Of all these methods, the crossed-wire focuser appears to be the most promising and in the case of ammonia has produced absorption signal enhancements in excess of 20 times that of the unfocused thermal beam. State separation of molecules may be made semipermanent by using an electret coating on the state-selector electrodes. The first proposal for the utilization of electrets in MBMs was made by Oraevskii (1964). An account
ADVANCES IN MOLECULAR BEAM MASERS
193
of experimental investigations on model focuser systems has been given for a longitudinal rod focuser constructed with four knife-edge electrets (Gubkin and Novak, 1971) and for an electret-coated ring focuser. (Gubkin et al., 1972). The first experimental results with an electret focuser in a working molecular beam maser were reported by Laine and Sweeting (1971~)using the classic electret material, carnauba wax. In the course of these investigations solid ammonia itself was found to possess electret properties (Laine and Sweeting, 1971d). A detailed account of maser operation with electrets has been given by Laine and Sweeting (1974). As molecules pass through the state selector, they experience a timevarying electric field. Fourier components of this field in the moving frame of reference of the molecules may induce transitions between closely spaced levels, thus altering the intensities of the maser hyperfine lines. With rf MBMs, this phenomenon causes nonadiabatic focusing and the maser signal intensity is therefore reduced. A suitable shaping of the electrodes and avoiding the use of focusers with electrodes perpendicular to the beam axis, which produce longitudinal electric fields, can reduce the effect. The close spaced hyperfine components of microwave MBM spectra also undergo nonadiabatic transitions which tend to thermalize state-selected level populations. If the spacing is large, however, focusing is adiabatic, which leads to anomalous hyperfine intensity ratios (Gordon, 1955). A normal population of the hyperfine levels may be restored by state scrambling with crossed static and resonant electric fields of the order of 10 kV m-l placed immediately in front of the maser cavity (Tucker et al., 1971). The effect of a weak static electric (or magnetic) field placed in front of a maser cavity was first studied by Basov et al. (1963a).Detailed studies of this phenomenon, attributed to spatial reorientation of molecules, have been subsequently made by several investigators including Krupnov and Skvortsov (1965b),who showed that the electric field between the end of the focuser and the resonator had a longitudinal character irrespective of whether or not the state separator employed a longitudinal electric field with respect to the molecular beam axis. The maser will be more strongly excited if the microwave field of the resonator is also longitudinal with respect to the beam axis. In the case of CH20, the effect of spatial reorientation has been found to be dependent upon the sign of the weak electric field relative to that of the fringe field of the state separator (Krupnov and Skvortsov, 1966). The effect of electric field symmetry upon spatial reorientation of molecules in the fringe field of a quadrupole state separator and a cylindrical maser cavity has also been studied by Koshurinov (1969). These investigations show the importance of taking the fringe field of the state selector into account in detailed studies of maser behavior.
194
D. C. LAlNk
C. Resonant Systems
One of the simplest systems for obtaining an interaction between a molecular beam and a microwave radiation field is a piece of rectangular waveguide, short circuited at one end (Krupnov et al., 1967~).A small hole in the side of the waveguide allows the molecular beam to pass through. It is more usual, however, to use a resonant interaction region for the maser so that emission intensity is increased and regeneration becomes possible. Such a resonant system may take several different forms, depending essentially on the frequency region employed. At radio frequencies, the resonant system simply consists of a capacitor and inductor in parallel. The molecular beam is stimulated to emit radiation as it passes through the capacitor. The low Q value of such an arrangement may be enhanced by a Q multiplier. Since the capacitor plates can be made accurately parallel, a highly homogeneous electric field can be produced between them. MBM Stark spectroscopy then becomes possible. Several transitions of formaldehyde have been studied with this system (Shimoda et al., 1960; Shimizu, 1963).The method has been used for the lowest frequency MBM spectroscopic measurements yet to be carried out, at 0.6 MHz. For UHF operation the resonant interaction region may take the form of a half-wave parallel line resonator (Radford and Kurtz, 1970). Again Q multiplication is possible. If the parallel lines are constructed in the form of accurately parallel plates, MBM Stark spectroscopy is also feasible. A system of this type has been operated with HCN at 449 MHz. At microwave and higher frequencies, either closed or open resonators may be used. The open interferometer types of resonator are particularly useful for millimeter wave MBMs and the closed types at lower frequencies. In view of their importance for MBM work, both types of resonator will now be reviewed in some detail.
1. Closed Resonators Closed resonators of use in MBMs may be either cylindrical or rectangular. Of the cylindrical types, operation in the TM,,, mode is favored because of the uniform microwave field distribution down its axis, leading to a strong interaction between molecules and radiation field. Moreover, the total linewidth of the MBM signal given by 0.9uO/L(where u, is the mean molecular velocity, L the length of the cavity) is the narrowest, for a given value of L, of all possible cavity modes. The resonant frequency of a cavity operated in this particular mode is independent of length. Therefore a long interaction region between molecules and radiation field may be realized in practice,
195
ADVANCES I N MOLECULAR BEAM MASERS
thus lowering the flux necessary to sustain MBM oscillation or alternatively to reach a given spectrometer sensitivity. The practical limit to length is reached when line broadening due to the transverse Doppler effect becomes dominant over that due to the transit time. The molecular beam enters and leaves the cavity through short metal cylinders push fitted into its two ends, whose internal diameters are well below cutoff for propagation of the MBM radiation. Alternatively, radiation from the ends of the cavity may be prevented by the use of a wire grid or mesh. Measurements of deviations from uniformity of the radiation field in an Eolo mode cavity with cutoff sections of various sizes have been reported by Suchkin and Rakova (1962). The loaded quality factor of the TM,,, mode with a silver-plated cavity is typically in the range 5000-8000 at 24 GHz, and 20,000 at 1.6 GHz. For MBM spectroscopy, a wide tuning range is important if the inconvenience of fabricating a new cavity for spectral lines separated by much more than 0.1 % of the nominal MBM frequency is to be avoided. Therefore, the circular mode TE,, which is dependent on length is often used. For this mode, the emission linewidth is given by 1.2u0/L which is larger, for given values of uo and I, than with a uniform radiation field on account of the single half-wavelength of radiation along the beam axis. In order to obtain a long interaction region, the diameter of this type of cavity is near to cutoff which restricts its tuning range to about 0.2% of its nominal operating frequency. A narrow linewidth is therefore obtained at the expense of tuning range. Furthermore, while a high Q value can be obtained with the TEoll mode, the electric field value is zero on the axis, and the maser will therefore be most effectively operated with a broad beam of molecules. Various other circular cavity modes have also been used, for example, TE,, TM, with an ammonia maser (Bonanomi et al., 1957) and TE030 with a hydrogen cyanide maser (Marcuse, 1961a). A comparison of the efficiency of various modes for producing MBM action has been undertaken by Shimoda et al. (1956). For MBM spectrometers and oscillators it is usual to operate the cavity with a single port. Coupling may be via a thin-walled iris in the side of the cavity, or more conveniently via a loop at low microwave frequencies, for example, with ND, (Basov et al., 1963b). For spectroscopy, maximum sensitivity is obtained when the cavity is overcoupled (reflection coefficient 0.33) (Thaddeus and Krisher, 1961). However, the beam maser oscillator should be undercoupled to give a low loss if a low oscillation threshold flux is desired. Circular cavity modes for which there exists one or more nodes on the axis cannot be used for spectroscopy on account of Doppler line splitting. However, beam maser cavities have been operated with such modes as oscillators, for example, TM,,,, TMo13, TE013 (Bonanomi et al., 1957; Barnes, 1959; Saburi and Kobayashi, 1960; Becker, 1966).
,,
,,
,,
196
D.
c. LAINB
Rectangular cavity modes have also been used in MBMs, for example, TMl mode rectangular resonators have been operated with NH3 (Becker, 1963a; Shimoda et al., 1956), with D,CO (Tucker and Tomasevich, 1973), and with CHzO (Kukolich and Ruben, 1971).A comparison of the efficiency of various rectangular modes by Shimoda et al. (1956)shows that this particular mode is preferable to others. The method of constructing microwave MBM cavities also deserves brief mention. Rectangular cavities may be easily made either from precision waveguide or machined flats. Circular mode cavities are somewhat more difficult to fabricate. At frequencies of 10 GHz or less, lathe machining is sufficient to give satisfactory accuracy and smooth surface finishes to yield a high Q value. At higher frequencies, small tolerances on the dimensions become essential to obtain a high Q value. In such instances lathe machining is usually followed by reaming or lapping. An excellent finish and dimensional accuracy have been obtained by forcing a steel ball of the appropriate size down a slightly undersized cavity. If a quartz cavity is used (Vonbun, 1960) there is little alternative to grinding and polishing prior to chemical deposition of a layer of silver on the inner surface. With all these methods, silver plating is desirable to increase the Q value. For high precision cavities, the method of electroforming is accurate, simple, and convenient. Microwave cavities may be tuned by one of several methods. With the TM,, cavity mode coarse tuning is achieved by a mechanical screw adjustment of the axial position of one of the end cutoff sections. However, in order to avoid frequent readjustments of tuning, temperature stabilization of the cavity is usually necessary. Cavities operated in the TMolo mode may also be tuned mechanically or thermally, or both. If the cavity is made with a slightly undersized diameter, accurate tuning can be accomplished by a flow of heated fluid, thermostated to the required temperature. Temperature control to within 0.01"C by this method has been achieved (Vonbun, 1960). Alternatively, electrical heating can be used. Here, a bifilar winding is essential in order to avoid any possibility of Zeeman splitting of the maser line. A copper heater wire can be used as its own thermostat if placed in one arm of a bridge fed by a power oscillator (Strakhovskii and Tatarenkov, 1963). Alternatively, a temperature-sensor winding can be used. Another method is to mill a single narrow slot along the full length of the resonator and then tune by squeezing, either by a mechanical drive, or by thermal expansion. In the latter method, a tungsten wire may be used which is clamped and held in tension by the natural springiness of the slotted cavity. The length of wire is chosen so that the thermal expansion of the cavity is compensated by the expansion of the wire and the cavity is stabilized against changes of ambient temperature. The wire hay also be heated electrically by inserting it in one arm of a feedback bridge oscillator. By this method, the cavity may be
ADVANCES I N MOLECULAR BEAM MASERS
197
thermally tuned over a range of about 0.05% of its natural frequency. This particular tuning technique (Shimoda, 1961a) has a shorter time constant relative to other thermal tuning methods, since the thermal capacity of the wire is quite small in comparison to that of the cavity itself. The TM,,, mode cavity may be tuned mechanically by variation of the depth of penetration of a dielectric or metal probe projecting into the cavity interior. The probe is usually situated midway along the length of the cavity, opposite the coupling hole. This method is particularly useful in conjunction with cavities of quartz or Invar, so chosen on account of their small or zero temperature coefficient of expansion. Precise tuning of the maser cavity to the center frequency of the maser line is rather important for a maser oscillator used as a frequency standard. The establishment of the exact tuning point in a MBM oscillator is discussed in Section V,B,2. 2. Open Resonators There are several drawbacks to the use of closed-cavity resonators, especially for MBM spectroscopy. These include restricted tuning range and unsuitability for precision Stark spectroscopy. Moreover, at high frequencies, closed resonators have a rather low Q value and small diameter. The effective gas flux in the resonator is severely reduced because of the difficulty of precise focusing and also due to pressure buildup within the cavity itself. While the use of the intense and highly collimated nozzle beam sources would help the high-frequency problem, the difficulties of lack of tunability and impossibility of precision Stark spectroscopy remain. Many of these constraints are lifted by the use of open resonators. Open resonators were first proposed for beam maser operation by Dicke (1958) and Prokhorov (1958). They were first used in ammonia MBMs at 24 GHz (Barchukov et al., 1963a,b). With 1 spacing between plates of 200 mm diameter, a Q value of 7000 was obtained. In later work, again using ammonia, but with 1/2 spacing between plates of 150 mm diam, a diffraction-limited Q value of 2500 was obtained for the lowest order mode of a resonator operated in transmission (Laine and Smart, 1971). Despite such a low Q value, oscillation was readily obtained with an extra high tension (EHT) as low as 10 kV applied to a double-ladder type of focuser. Flat plate resonators have also been used in a formaldehyde maser operated at 72 GHz (Krupnov and Skvortsov, 1963b, 1964a, 1965b). A Q value of 2000 was obtained with 1/2 to 21 spacing. Again, oscillation was readily achieved. The use of this type of resonator as a Stark cell is well illustrated by the formaldehyde MBM which may be tuned by as much as 3 MHz by applying fields of 170 V between resonator plates spaced 2 mm apart (Krupnov and Skvortsov, 1964a). It should be noted in this context
198
D.
c. LAINB
that the lo, + Ooo, 72.8 GHz line of formaldehyde has no Stark components and so line broadening which ultimately causes the oscillation to cease is due to electric field inhomogenities rather than hyperfine structure. It may be wondered why oscillation can be easily obtained with such low Q values. The reason is that this type of resonator has an exceptionally wide acceptance angle for molecules in a plane parallel to the plates, but the usual acceptance angle is in a plane normal to the plates. Thus a larger fraction of the beam flux is utilized than is usual in a cylindrical system which offsets the somewhat inferior filling factor of open resonators relative to cylindrical cavities. Furthermore, the open structure of the cavity leads to a greater gaspumping efficiency. When parallel plate resonators are operated in a mode with a single node along the molecular beam trajectory, maser line splitting occurs. This has been observed in both ammonia and formaldehyde MBMs at 23.87 and 72.8 GHz, respectively (Laine and Smart, 1971; Krupnov and Skvortsov, 1964a). An oscilloscope trace of this type of line splitting with ammonia is shown in Fig. 4.
FIG.4. Oscilloscope trace of emission line split by longitudinal Doppler effect in an ammonia maser operated with a plane-parallel open resonator with two maxima of the radiation field along beam direction normal to effuser gas source. From Laine and Smart (1971) by courtesy of the Institute of Physics.
In the construction of this type of resonator, particular care needs to be taken to obtain flat mirrors. A flatness of the order of 1 pm can be obtained by surface grinding brass or copper mirrors. Surface flatness can be measured by standard optical methods. The mirrors are usually mounted in a very rigid frame to avoid flexing and loss of flatness. Coarse tuning can easily be accomplished by a mechanical drive which alters the spacing between the plates or by the use of quartz spacers. By the former method the resonant frequency may be scanned over the range of several gigahertz. Fine tuning over a range of &lo0 MHz or more, depending on frequency, may be effected by slight pressure on one of the plates. Alternatively thermal tuning may be used, which at the same time can be used to stabilize the resonant frequency of the cavity against changes of ambient temperature. Excitation
ADVANCES IN MOLECULAR BEAM MASERS
199
of this type of resonator may be either via the fringe field (Barchukov et al., 1963a) or by means of the coupling hole method, either in transmission or reflection. Mode patterns may be explored by means of a half-wave dipole probe. Millimeter wave MBMs have also been operated with confocal resonators, both as spectrometers and oscillators. The first MBM to be operated with this type of resonator was the HCN maser which oscillated at 88.6 GHz (Marcuse, 1961b, 1962). Stimulated emission at 23.87 GHz has also been obtained with a half-confocal resonator using NH, (Strauch et al., 1964). Oscillation in the HCN confocal mirror MBM has also been studied at 88.6 and 177.3 GHz (De Lucia and Gordy, 1969). Spectroscopic studies of HCN, DCN, ND,H, and D,O have been carried out with a confocal mirror geometry with Q values in the range 3 x lo5 at 88.6 GHz and 5 x lo5 at 177.2 GHz (De Lucia and Gordy, 1969). The observation of the D,O, l l o -,l o , transition at 316 GHz is the highest frequency MBM transition yet to be observed. A conical rooftop resonator with an apex angle of 179'0 has also been studied in a MBM system (Laine and Smart, 1971). A Q of lo4 was obtained at 24 GHz.
111. BEAM-MASER SPECTROSCOPY
The potentialities of the MBM as a spectrometer of exceptionally high resolution (7 kHz at 24 GHz) were clearly demonstrated in the original and now classic paper of Gordon on the hyperfine structure of 14NH, (Gordon, 1955). The success of that early work doubtless may be attributed to the choice of ammonia as the working maser medium. This gas possesses a particularly intense microwave spectrum and also a strong Stark interaction which permits efficient focusing of molecules in particular quantum states. Although good sensitivity was obtained by Gordon with ammonia, MBM spectroscopy with other molecules looked rather less promising owing to the lack of MBM sensitivity in comparison with other well-established methods. Thus little effort was made at the time to extend the MBM spectroscopic technique to other molecules, although limited success was reported for cyanogen chloride (CICN) (Thaddeus et al., 1958, 1960). The next advances came in the late 1950's with the molecules water (HDO) and formaldehyde (CH20, CHDO) at various microwave frequencies. The molecule C H 2 0 was studied particularly successfully with radio-frequency MBMs. Here, the close-spaced K-type doublet levels which gave rise to these low-frequency transitions were found to be particularly favorable for state selection as discussed in Section 1I.B.
200
D.
c. LAIN&
Other molecules of interest from a spectroscopic point of view were also investigated over a period of nearly a decade, some with exceptionally high resolution of 350 Hz or less. In general, however, the potentialities of MBM spectroscopy remained unrealized until the end of the 1960's when a new motivation for high-resolution spectroscopy arose in connection with the discovery of interstellar molecules. In such work, the search for microwave emission and absorption spectra from cosmic sources, and measurements of their Doppler shifts required precise laboratory measurements of molecular transition frequencies. This need, combined with the improved sensitivity of MBM spectrometers in comparison to earlier efforts, produced an impetus for further work. Moreover, excitation mechanisms of cosmic masers, or antimasers (NH, , CH,O, H,O, OH, etc.) are currently of great interest and laboratory MBMs offer scope for related experiments. Within a period of 3 to 4 years, starting in the late 1960's, the number of molecules studied in MBM spectrometers has more than doubled. It should not be forgotten, however, that this state of affairs took 15 years or more to reach from the date of publication of Gordon's original MBM spectroscopy paper in 1955. In the subsections which follow, the main features of MBM spectroscopy will be considered. A general approach is taken in this review to illustrate the various techniques and type of spectroscopic measurements which can be made. A list of molecules for which data have been published in the open literature is presented in Table I. A thorough discussion of high-resolution MBM spectroscopy of individual molecules is not attempted. For such details, reference should be made to an earlier survey by Laine (1970) or to an extensive review by Dymanus (1975) which is exclusively devoted to beam maser spectroscopy. A. Linewidth
In Section II,C, the importance of the cavity mode in relation to the linewidth observed in a MBM was discussed. The narrowest linewidth is obtained with a uniform radiation field in the interaction region. In this case the linewidth between half-power points is given by Av = 0.9u/L. This value is realized to a good approximation in both radio-frequency MBM spectrometers in which a capacitor interaction region is used, or at microwave frequencies with a cavity operated in the TM,,, mode. When there is a single half-wavelength of radiation along the beam axis, the linewidth increases to 1.2u/L. For higher order modes with two or more halfwavelengths along the cavity axis, line splitting is observed. Such modes are clearly unsuitable for spectroscopy. It is clear that the longer the interaction region and the smaller the molecular velocity, the narrower the linewidth.
ADVANCES I N MOLECULAR BEAM MASERS
-
20 1
In an ammonia MBM operated in the TM,,, mode, the calculated 4 kHz, assuming a cavity of length 120 mm and mean linewidth is molecular velocity of 5 x 10' msec-'. Longer cavities may be used, but the likelihood of exciting higher order resonator modes and the presence of molecule-wall collisions make cavities much longer than 150 mm at 24 GHz rather unattractive. At low frequencies, however, such as with ND, (1.5-1.7 GHz), resonator dimensions are large and quite narrow lines, typically 800 Hz, may be obtained with a single cavity. With a single resonator 1 meter long, a linewidth of about 300 Hz was obtained for the ( J , K, r ) = (3, 1, 3 - ) -,(3, 1, 3+), 5.0 GHz transition of CH,OH (Heuvel and Dymanus, 1973). High-resolution MBM spectroscopy has also been carried out with Ramsey's method of separated oscillating fields. With this well-known technique molecules entering the first of a well-separated pair of cavities undergo a 4 2 pulse excitation and a further n/2 pulse in the second cavity downstream from the first. Thus molecules receive a net II pulse, corresponding to a maximum transition probability for emission after passing through both cavities. The width of the central peak is approximately the inverse of the transit time between cavities. Separations between cavities of 1 meter or more are typical. The maximum separation is limited on account of beam divergence over the large distance between resonators, which greatly reduces the beam flux at the second cavity. Clearly, the use of nozzle sources for separated cavity MBM spectroscopy is advantageous for producing an intense well-collimated molecular beam. Open resonators which possess a greater transverse area perpendicular to the axis of the molecular beam may make useful second cavities in a MBM Ramsey scheme. Resolutions of 350 Hz have been obtained for many of the inversion transitions of ammonia including 14NH3 (Kukolich, 1965; Kukolich and Wofsy, 1970), "NH, (Kukolich, 1968a) and NHzD (Kukolich, 1968b). Details may be found in Table I. The best resolution yet to be reported for 14NH, is 240 Hz (Krupnov et al., 1967~).A linewidth of 350 Hz obtained with the separated cavity method has also been reported for the 1 + 1 transition of H,CO (Kukolich and Ruben, 1971)and the 616 -,5,, transition of H 2 0 (Kukolich, 1969a). Narrow spectral lines in MBM spectrometers are only obtained with weak stimulating signals of lo-', W or less because of saturation broadening at higher powers. In general, therefore, superheterodyne detection is used to detect weak emission spectra, although the submillimeter maser transition l,, + lol of D,O at 317 GHz has been detected by a square-law video system (De Lucia and Gordy, 1970). Saturation of maser emission lines has been studied in radio-frequency masers (Shimizu, 1963).
,
TABLE I LISTOF GASES STUDIEDBY MOLECULAR-BEAM-MASER SPECTROSCOP~ Main line transition(s)
Gas'
Ammonia ( p = 1.5 D) NH3
( J , K)* = (4, 2)+ (3, I ) +
(4, 3)_+ (3, 2)+ (7, 6)+ (LI ) + (1, I ) +
22,924.9 23,098.8 23,694.5
(2. 2)*
23,722.6
0.35
(3. 3)+
23,870.1
0.24
(4.4)k
24,139.4 24,533.0 22,624.9 22,649.8 22.789.4
0.35 0.35 0.35 0.35 0.35
23,046.1 23,422.0 23,922.3 24,553.4 18,807.7 25,023.8 110,153
0.35 0.35 0.35 0.35 7.0 0.35
(2, 2 ) k (3. 3)+ (4, 4)+ (5, 5 ) +
NH,D
J ~ .
Best-known resolution (kHz)
21,703.4 22.234.5 22,688.2 22,834.2
(5, 5)_+ (1, I ) *
"NH,
Center frequency main line(s) (MHz)
(6, 6)+ (7, 7)+ I =3,,+3a3 ~ I 414
-t
4a4
1 1 1 -+ 101
0.35 0.35 7.0 0.35 -
0.35 0.35
-
Principal references'
Kukolich and Wofsy (1970)' Kukolich and Wofsy (1970)' Gordon (1955)' Kondo and Shimoda (1965); Kukolich (1%5)*, (1967). Wang ef al. (1973) Kukolich and Wofsy (1970)' Gordon (1955); Kukolich (1967). Gordon (1955); Kukolich (1967)' Gordon (1955); Kukolich (1967); Krupnov et 01. (1967~)' Kukolich and Wofsy (1970)' Kukolich and Wofsy (1970). Kukolich (1967)' Kukolich (1967). Morozov er a/. (1965); Kukolich (1967)' Kukolich (1968a)* Kukolich (1968a). Kukolich (1968a)' Kukolich (1968a)' Thaddeus er a/.. (1964a)' Kukolich (1968b) Garvey and De Lucia (1972)
p 9
-
7.0
De Lucia et al. (1970)';
1,613.0 1,656.2
0.80 0.80 0.80 0.80
De Lucia and Gordy (1970) Basov and Bashkin (1968). Basov and Bashkin (1968)' Basov and Bashkin (1968)* Basov and Zuev (1961): Zuev (1962).
5.872.2 17.074.5 17.404.8 17,616.2
1.5 6.0 6.0 6.0
Wang Wang Wang Wang
9,098.3
3.0
De Zafra (1971)'
23.885
1.5
Thaddeus er al. (1958):. (1960)
22,204.2
4.0
CDZF2 Fluoroform ( p = 1.6 D) CHF,
22,727.9
6.0
Kukolich and Nelson (1972a)* Nelson et a/. (1974)'
20,697.7
2.5
CDF, Formaldehyde ( p = 2.3 D) H,CO
19,842.2
4.0
NHD,
111
101
57.674.8 1,509.2
ND3
1,560.8
Carbonyl fluoride ( p = 1.0 D) COF,
Cyanoacetylene ( p = 3.6 D) HC3N Cyanogen chloride ( p = 2.8 D) ClCN
2-1
Difluoromethane ( p = 1.9 D) CHZF2
0.656 0.8 19
532
-
533
10
1.485 4.573
7.0 6.5 6.4
18.284
6.5
and and and and
Kukolich Kukolich Kukolich Kukolich
(1973)* (1973)' (1973)' (l973)*
Kukolich et al. (1971a); Reynders et a/. (1972)' Kukolich e r a / . (1971a)' Shimizu (1963); Takami and Shimizu (1966)* Takami and Shimizu (1966)' Takami and Shimizu (1966)* Takuma (1961); Shimizu (1963)*; Takuma et al. (1966) Shimoda et al. (1960); Shimizu (1963)*; Takuma er al. (1966)
(continued)
TABLE I-continued Main line transition(s)
GSb
Center frequency main line(s) (MHz)
54.848 4,236.0 4,829.7
6.4 3.0 0.35
4,968.9
3.0
14,488.5
5.0
28,974.9 72.409.4 72,838.1 ~
~
1
3
~
0
H2C180 HDCO
D,CO Formamide ( p = 3.7 D) H2NCH0 Formic acid (p = 1.4 D) HCOOH HCOOD
Best-known resolution (kHz)
4,593.1 13,778.8 4,388.8 13,166.1 5,346.1 16,038.1 6,096.1
-
-5
15 15 2.0 -
2.0 -
1.5 5.0
1.5
Principal references' Shigenari er a/. (1963) Tucker et a/. (1971)' Shigenari (1967); Tucker et a/. (1971)' Kukolich and Ruben (1971)'; Tucker et a/. (1971)' Thaddeus et a/. (1959) (1964b)'; Tucker e t a / . (1972) Takuma et a/. (1959)' Krupnov and Skvortsov (1963a)' Krupnov and Skvortsov (1%3a)' Tucker et a/. (1971). Tucker et a/. (1972) Tucker et a/. (1971)' Tucker er a/. (1972) Tucker er a/. (1971)' Thaddeus et a/. (1959). Thaddeus et a/. (1964b)' Tucker and Tomasevich (1973).
2 1,207 23,081
8.0 8.0
Kukolich and Nelson (1971). Kukolich and Nelson (1971)'
22,471.2 24,569.0 21,732.5
5.0
5.0 4.0
10,753.3
4.0
Kukolich (1969b)' Kukolich (1969b)' Kukolich (1969b)'; Ruben and Kukolich (1974)' Ruben and Kukolich (1974)'
P
a
DCOOH Formyl fluoride ( p = 2.0 D) HFCO DFCO Furan ( p = 0.7 D) C4H4O
Hydrogen cyanide ( p = 3.0 D ) HCN
2- 1 DCN
1 +o
D"CN
2- 1 1-0
Hydrogen peroxide ( p = 2.1 D) H202 Hydrogen selenide ( p = 0.6 D) HDSe
-
22.01 1.6
5.0
Kukolich (1969b)*
22,156.9 21,703.5
6.0 6.0
Kukolich (1971b)' Kukolich (1971b)*
4,575.9 4,381.9 4,103.1 3,753.8 2,062.3
0.45 0.45 0.45 0.45 0.45
Tomasevich Tomasevich Tomasevich Tomasevich Tomasevich
448.95 88,631
177,260 72,400 144,800 71,175 37.518.3
- 8,771 9.138.5
Hydrogen sulfide ( p = 1.6 D) HDS D2S Hydroxyl ( p = 1.7 D) OH
21T3,2, J
= 312
2nn,l, J = 912
3.0 -5
-
10
20 -5 -
3.0 -
11,283.8
5.0
91,359.1
6.0
1.612- 1.720 (4 hyperfine lines) 23,826.6 23,817.6
2.5 -
er 01.
(1973)*
et al. (1973)* er al.
(1973)'
ef al. (1973). er 01.
(1973).
Radford and Kurtz (1970). Marcuse (1961a); De Lucia and Gordy (1969); Garvey and de Lucia (1974)' De Lucia and Gordy (1969)* De Lucia and Gordy (1970)' De Lucia and Gordy (1969)' Garvey and De Lucia (1974)' Dymanus (1975) Chandra and Dymanus (1972)' Bluyssen et al. (1967a)' Thaddeus et al. (1960)*. (1964a) De Lucia and Cederberg (1971). ter Meulen and Dymanus (1972)' ter Meulen and Dymanus (1971) ter Meulen and Dymanus (1971)
(continued)
P
s
b:
TABLE I-continued ~~
Gasb Isocyanic acid ( p = 1.6 D ) HNCO
Main line transition(s)
Center frequency main line(s) (MHz)
--
JY-,K, = 10, - 0 0 0
101 + 000 Methanol ( p = 1.7 D) CH,OH
CH,OD
Methyl chloride ( p = 1.9 D) CH,CI CH,"CI CH,DCI CD,CI Methyl cyanide (p = 3.9 D) CH,CN CD,CN Methyl isocyanide (p = 3.8 D) CH,NC Pyrrole ( p = 1.8 D) C J W
J=1-0 1-0 1-0
1-0 1-0
1-0 3-2 1-0
21,981 20,393 2,502.8 5,005.3 17.5 13.3 18.957.8
(J,K,1)=(2,1,3-)-(2,1,3+) (37 1. 3 - ) - ( 3 , 1, 3 + ) (6, 1, 3 - ) - + ( 6 , I, 3 + ) K = O - 1; J = 1
Best-known resolution (kHz)
--
6.0 6.0
Kukolich et al. (1971b)' Kukolich et 01. (1971b)'
1.0
Heuvel and Dymanus (1973). Heuvel and Dymanus (1973)' Heuvel and Dvmanus (1973)' Castleton and Kukolich (1973)' Castleton and Kukolich (1973)' Castleton and Kukolich (1973)' Castleton and KukoGch (1973)'
0.3 0.9 6.0
2
18,991.6
6.0
3
19,005.6
6.0
4
18,957.1
6.0
26,585
5.0
26,176
5.0
24,660 2 1.687
5.0
18,397.8 15,716.0 47,148
4.0 4.0
--
- 20.105
Principal references'
5.0
-
6.0
Kukolich and Nelson (197k'. 1973) Kukolich and Nelson (1972a*, 1973) Kukolich (1971~). Kukolich (1971~)' Kukolich et al. (1973a)' Kukolich et al. (1973a)' Garvey and De Lucia 1972) Kukolich (1971d); (1972) Gaines and Tomasevich (1973)
P P
Water (p = 1.9 D)
H2O
22,235.1
HDO
8,577.8 10,278.2
4 0
422,
3i3 +220
44i
+
110 +
532
101
10,374.3
0.35
~
5.0
2.0
10,919.4
-
10,947.1
-
316,800
-
Bluyssen er al. (1967a); Kukolich (1969a)'; Verhoeven and Dymanus (1970) Verhoeven er a/. (1968) Thaddeus and Loubser (1959)'; Thaddeus et al. (1964b); Bluyssen et 01. (1967a) Verhoeven et al. (1969)*; Verhoeven and Dymanus (1970) Bluyssen e l a/. (1967b); Verhoeven and Dymanus (1970) Verhoeven er 01. (1968); Verhoeven and Dymanus (1970) De Lucia and Gordy (1969; 1970)
* 2 5
i2 2
01
x
0
E
5
$lJ W
m Common isotope assumed unless otherwise indicated. Dipole moments in Debye units. Asterisk relates to paper reporting linewidths indicated; approximate linewidths inferred wherever possible if not given explicitly.
208
D.
c. LAINB
In MBMs operated at frequencies of 100 GHz and higher, the Doppler width due to beam divergence tends to dominate over transit-time broadening unless particularly well-collimated beams of molecules are used. Distortion of the maser lineshape may occur due to the presence of electric and magnetic fields within the MBM cavity or by dispersion effects. Both of these may lead to measurement errors of the natural molecular resonance frequency. Screening against stray magnetic fields can be effected by the use of mu-metal. Cavity temperature control using a thermostated fluid flow is preferable to electrical heating which may possibly produce a small unbalanced magnetic field. Even the weak magnetic field of the Earth may be important, especially with the free radical O H which has a large magnetic moment (ter Meulen and Dymanus, 1972). Stray electric fields can also be a serious problem because of the high voltages applied to the focuser and the lack of self-screening at the open ends of the resonator. Their effect can be reduced by earthed grids placed over the ends of the resonator at the expense of reduced beam intensity. Dispersion effects which arise when the MBM resonator is tuned away from the spectral line may be quite severe (Gordon, 1955; ter Meulen and Dymanus, 1972) and can lead to quite large errors in measurements of the precise frequency of the line center. To avoid line distortions of this type, the resonator tuning should be made to track the frequency of the excitation signal. This presents problems with thermal tuning of the resonator, on account of the long thermal time constants. This limits the maximum permissible frequency scanning rate if dispersion effects are to be avoided. Lineshape distortions may also arise with fast frequency scans if the frequency spacing of signal Fourier components is not significantly less than the bandpass of the spectrometer-detector system. The effect of this type of distortion may be eliminated by recording with both increasing and decreasing excitation frequencies and taking the mean frequency of the two amplitude maxima.
B. Sensitivity According to Dymanus (1975), the maximum signal-to-noise ratio (S/N)maxfor a reflection cavity MBM spectrometer is given, in cgs units, by
for a univelocity beam and uniform radiation field along the interaction region. For a thermal beam, the numerical factor must be replaced by 0.8, and u is then taken to represent the most probable velocity of the molecules
ADVANCES IN MOLECULAR BEAM MASERS
209
emerging from an orifice. Here, V and QLare the volume and loaded quality factor of the resonator, respectively, F the noise figure of the receiver assumed independent of the input power, Af the receiver bandwidth, n the population excess of upper-state molecules, and plzthe dipole matrix element for the maser transition. The minimum detectable beam flux nminfor a univelocity beam is obtained from Eq. (1) by setting (S/N)maxto unity. A similar expression for (S/N)maxhas been derived by Thaddeus and Krisher (1961). Equations for nminhave previously been derived by Shimoda et al. (1956) and Shimoda (1962).A value of (S/N)maxof between lo4 and lo5and nmin lo9 sec-' with Af 10 Hz is typical for the ammonia MBM at 24 GHz. Other molecules give rather less favorable values but may yet be sufficient for detailed spectroscopic studies. It has been shown by Beers (1961) that at frequencies between 1.5 and 2.0 GHz, the conventional bulk gas absorption spectrometer and MBM system with an effusive gas source have approximately equal optimum voltage signal-to-noise ratios. At higher and lower frequencies, the conventional absorption spectrometer has, respectively, greater and smaller sensitivities than the MBM spectrometer. However, with the advent of nozzle gas beam sources this conclusion needs revision, since the signal-to-noise ratios of present-day MBMs probably gain an order of magnitude through their use. Thus equality of optimum signalto-noise ratios of the two methods should be reached in a frequency range substantially higher than that given by Beers. At low frequencies the Doppler width, which is proportional to frequency, may be quite narrow in conventional gas spectrometers, given sufficiently large enclosures and sufficiently low gas pressures. Thus any improvement in resolution of the MBM at low frequencies over nonbeam methods may not be worth the effort entailed, in view of the attendant experimental complexity and limited tuning range of the MBM spectrometer. 150 GHz and above, the sensitivity of the molecular At frequencies of beam absorption spectrometer can equal that of the MBM spectrometer and be considerably greater at higher frequencies. At frequencies where sensitivities of thermal beam and MBM spectrometers are about equal, it may be more appropriate to operate the molecular beam system with antimaser type of state selection and focusing, so that the absorption is enhanced. The absorption intensity may also be increased by rotational cooling of the molecular beam by the use of high-pressure nozzle-type gas sources.
-
-
-
C. Spectrometer Systems The maximum stimulated emission signal obtained in a beam maser operated well below the condition for oscillation is given by the condition that molecules should be induced to make a single downward transition
2 10
D.
c. LAINB
from upper to lower maser levels as they pass through the maser cavity. In general, the power input into the MBM cavity to meet this condition is typically W or less. Excitation of the maser with a power level much higher than this not only reduces the strength of the emission, but also power saturates and thus broadens the spectral line. In a practical MBM spectrometer, therefore, the power available for detection is necessarily low and, in general, video methods of detection are unsuitable. The only known MBM spectrometer in which video detection has been employed is that used by De Lucia and Gordy (1970) with the l,, + lol transition of D,O at 3 16.8 GHz. Generally, therefore, superheterodyne detection is used when local oscillators of sufficient power and mixers of sufficiently low noise figure are available. For centimeter and longer wavelength MBMs, klystrons serve as local and signal oscillators and semiconductor diodes as mixers. At short millimeter wavelengths, fundamental frequency klystrons or other types of oscillator are expensive, and so harmonic generation and mixing may be favored. For superheterodyne detection, local oscillator and stimulating sources of power are separated in frequency so that the beat frequency after mixing falls within the passband of the IF amplifier. If harmonic mixing and detection are employed, the frequency separation of the primary sources of power will be the I F divided by the harmonic number. Because of the high resolution of the MBM spectrometer, the stimulating signal must be frequency stable to within a small fraction of the maser linewidth. At radio frequencies this is no problem, since for spectral lines a few thousand hertz wide, a signal frequency stability of 10 Hz is only the order of 1 part in lo6 at 10 MHz. At 10 GHz this frequency stability represents a spectral purity of 1 part in 10’. Therefore for high resolution work at microwave and higher frequencies, stimulating signal frequency stability is important. If a klystron is used as a signal source, frequency stabilization is necessary. This usually takes the form of phase-locking the klystron to a high harmonic of a good quality quartz-crystal oscillator operating in the region 5-10 MHz. The stimulating signal is swept across the emission spectrum by frequency modulation of the quartz oscillator. While it is not essential to frequency stabilize the local oscillator klystron, a narrowband IF amplifier can be used with an associated reduction of noise bandwidth if this klystron is also frequency stabilized. When the signal frequency is scanned through the MBM spectrum, it is desirable that the local oscillator klystron tracks the frequency of the signal klystron. This is easily achieved by using a phase-lock loop to frequency stabilize the local oscillator klystron relative to the signal klystron. It is also possible to use frequency multiplication of a stable quartz-crystal frequency to stimulate emission directly (Shigenari, 1967). However, a spectral purity of a high order is called for, which, in the
211
ADVANCES IN MOLECULAR BEAM MASERS
absence of the flywheel action of a klystron-oscillator signal source, may be insufficient for really high resolution studies. This does not appear to be a problem if a high-grade frequency synthesizer is used. Moreover, currently available synthesizers not only offer an extremely stable and spectrally pure output power, but permit either continuous or digitally controlled frequency sweeping. The digital data obtained by the latter technique is particularly convenient for subsequent computer processing. For example, Ruben and Kukolich (1974) have used a digitally swept source for spectroscopic studies of formic acid, HCOOD, using 200 Hz incremental frequency steps. Frequency determinations of spectral lines may be easily made by measurements of the signal oscillator frequency with a digital counter. Superheterodyne detection is also possible using a single klystron. A small fraction of the output of the klystron of frequencyfis taken to a semiconductor microwave mixer diode which is modulated by a power source at, say, 30 MHz. The power reflected from the diode then contains the frequencies (Jf* 30) MHz. One of the sidebands is then used to excite the maser transition. The signal reflected from the cavity is then mixed with local oscillator power at frequency5 to produce an IF signal at the mixer modulation frequency. A particularly simple MBM spectrometer based on this type of system has been described by Bonanomi and Herrmann (1956), the essentials of which are sketched in Fig. 5. However, it should be noted that the presence of the second sideband which does not contribute to the signal information carries mixer noise and so should be eliminated from the detector by some form of microwave filtering. The single klystron scheme has been used by several investigators (for example, by Takuma el al., 1959 MODULATOR
KLYSTRON
DISPLAY
-
\<
q E p'm c-
L -
MASER CAVITY
i
L _ I
-,.
FIG. 5. Superheterodyne detection scheme using a single klystron.
212
D.
c. LAINB
and Thaddeus and Krisher, 1961)as a practical and simple method of MBM spectroscopy. Frequency tracking problems do not arise with this technique and only a single klystron needs to be frequency stabilized. Signal frequency sweeping can be carried out by frequency modulation of either (i) the klystron via its quartz frequency control loop, keeping the modulator frequency constant, or (ii) the modulator frequency with the klystron at a constant frequency. The single sideband technique does suffer from mixer noise, which places it at a slight disadvantage in terms of spectrometer signal-tonoise ratio relative to schemes that use two independent power sources to provide signal and local oscillator power. Thus, for MBM spectroscopy of the highest sensitivity, the single klystron scheme, despite many experimental advantages, has this one important drawback. The single klystron scheme has been used for MBM spectroscopy up to frequencies of about 200 GHz. For this purpose, the output of a klystron of frequencyf, (65-70 GHz) is divided into two parts, one of which passes into a semiconductor diode harmonic generator together with 30 MHz rf power. This arrangement produces output frequencies cotresponding to harmonics nfk, where n = 2, 3 . .. with sidebands at multiples of 30 MHz. One of the signals of frequency (nfk f 30) MHz is used to excite the MBM. The other portion of the klystron power is applied to a second mixer, thus generating harmonics of frequency nfk which provides local oscillator power. Beats between this harmonic power and the signal used to stimulate emission provides a 30 MHz I F which may be amplified in the usual way. This scheme has been used by De Lucia and Gordy (1969) for studies of HCN and DCN at frequencies of 177 and 145 GHz, respectively. In MBM superheterodyne receivers, noise originating in the detector and receiver itself is important if the maximum signal-to-noise ratio is to be obtained. Thus, the effect of local oscillator noise is reduced to a low level at a frequency for which low-noise operation can be achieved, typically 30 MHz. Intermediate frequency noise figures of 1.5 dB are usual, with a bandwidth of 2 MHz. The noise bandwidth of the superheterodyne spectrometer is typically 10-100 kHz. For maximum sensitivity, it is important that this bandwidth is reduced. This can be achieved using some form of synchronous detection. Here, the signal is modulated by some means and then compared in phase with a reference signal at the modulation frequency. Signals of the same frequency and phase then contribute to the detector output. A noise bandwidth of 1 Hz or less, is usual. Several methods have been employed to effect the necessary modulation. These are briefly discussed as follows: (i) Source modulation. Here the signal is frequency modulated over a small range (200-1000 Hz) and simultaneously frequency swept across the
-
ADVANCES IN MOLECULAR BEAM MASERS
213
MBM emission spectrum. The small periodic frequency modulation is produced by phase modulation of the output power from the crystal oscillator or frequency synthesizer prior to multiplication up to the maser frequency. In this method, the output of the phase-sensitive detector is the derivative of the lineshape. The method has been used extensively by Kukolich (1965) for MBM spectroscopy with single- and two-cavity beam masers. ( i i ) EHT modulation. The beam of molecules entering the MBM cavity may be rapidly switched on and off by modulating the focuser voltage. If an alternating voltage is used, the molecules are focused twice per cycle, so that phase-sensitive detection must be at twice the modulation frequency (Thaddeus and Krisher, 1961). If the focuser supply is simply switched on and off (Shigenari, 1967),or an alternating voltage applied with a large dc bias (De Lucia and Gordy, 1969), the molecules are focused only once per cycle. Modulation frequencies in the range 25-200 Hz are typical. ( i i i ) Stark modulation. This method of modulation has been used with rf masers which employ a capacitor as the MBM interaction region. The maser emission is modulated by applying a low-frequency Stark field across the capacitor plates. A modulation frequency of 175 Hz is typical (Shimoda et al., 1960). This method is also potentially useful for MBM spectrometers that employ open resonators or cylindrical cavities split longitudinally along their length. In contrast to the EHT modulation method, which is prone to induce unwanted signals, modulation voltages of the order of a few tens of volts, or less, are required. ( i u ) Beam chopping. Mechanical modulation of the beam has the advantage that the possibility of electrical pickup is minimized. The beam chopper consists of a circular disk with radial slots of a width equal to the effuser diameter. The chopper wheel is rotated by a synchronous motor, mounted either in the vacuum or via a rotating high-vacuum feedthrough (Heuvel and Dymanus, 1973; Krupnov and Skvortsov, 1964b). Modulation frequencies in the 70-260 Hz range have been employed. A beam of light chopped by the same or duplicate disc and detected by a photodiode or phototransistor can be used to provide a frequency reference for phase-sensitive detection. An example of a MBM spectrum obtained with the beam-chopping technique is shown in Fig. 6. (u) Depolarization modulation. In a two-cavity MBM operated in the Ramsey arrangement, the oscillating polarization carried by the molecular beam from the first cavity to the second may be readily modulated by applying a periodic voltage ( - 50 V) across a pair of capacitor plates placed across the beam, immediately in front of the second cavity (Krupnov et al., 1967~).With this method, a high modulation frequency is possible and thus enhanced sensitivity since the beam traverses only the second cavity after modulation. In contrast, beam chopping, focuser, or source modulation
214
D.
c. LAINB
I
1
10 360.000+...
14.450
1
1
,
1
14.400
1
1
1
1
1
frequency(MHz)
FIG.6. Group of maser hyperfine transitions for H D ” 0 . From Verhoeven et al. (1969) by courtesy of the American Institute of Physics and the authors.
would necessarily need to be at a low frequency on account of the long time of flight of molecules between cavities. The method is shown in Fig. 7. In general slow frequency sweep rates are required to use the longest time constants possible after phase-sensitive detection for signal averaging. The maximum useful time constant is determined for a given spectrometer by instability and drift, but may be as long as 15-20 sec. Longer effective time constants can be obtained by using digital averaging using a computer of average transients (CAT). The advantage of this type of system is that the spectrum is repetitively scanned typically in 10-15 sec, stored, and averaged over an hour or more. The signal can be read out at any time convenient to the operator. By this method effects due to long time instabilities of the spectrometer are averaged out. Integration times of many hours are routinely used. However for optimum results it is necessary that exactly the
FOCUSER
--+{ I
=+--I
NOZZLE & SKIMMER
CAVITY 1
I
_.
CAVITY 2
f
/
CAPACITOR
ATTENUATOR 1ATTENuATORH
~E$E;
/-
FIG.7. Schematic diagram of molecular beam maser system employing Ramsey’s scheme of separated cavities. Adaptation from Krupnov et a/. (1967d).
ADVANCES I N MOLECULAR BEAM MASERS
215
same portion of the spectrum is covered on each repetitive scan. This requires precise frequency control (Reynders et al., 1972). Brief reference should also be made here to the MBM spectrometer system that employs Ramsey’s separated cavity scheme. One form of this spectrometer system is shown in Fig. 7. The power level of the stimulating signal in the first cavity is set for n/2 pulse excitation of molecules. A second n/2 pulse is applied in the second cavity, corresponding to an overall n pulse, i.e., a transition from upper to lower maser levels during the total time of flight through the cavity system. The application of the technique to MBMs has been discussed in some detail by Kukolich (1965). However, as noted in Section III,A, the signal in the second cavity is often rather weak so the method can only be used for strong spectral lines. An example of the MBM Ramsey resonance pattern using source modulation is shown in Fig. 8.
I
-2 0
I
-1 0
I
1
00
I I
10
FIG.8. Ramsey-type resonances for the 1 + 1 transition of formaldehyde (CH,O): (a) slow scan, (b) fast scan. Solid line for slow scan from experimental data. Points calculated values. Frequencies in kHz relative to 4829659.9 kHz. From Kukolich and Ruben (1971) by courtesy of Academic Press and the authors.
216
D. C. LAINh
Finally, the oscillating MBM has been used for studies of the hyperfine structure of 14NH3.This specialized method is discussed in Section V,D,2,c. An important and essential part of MBM spectroscopy is the precise frequency measurement of spectral lines. For this purpose, accurate frequency markers are required, which with slow sweep methods are conveniently made directly on the recordings of spectra. High-quality frequency sources are used which may be checked against standard frequency transmissions, or against laboratory frequency standards. Frequencymeasurement methods depend on the particular system used. A typical scheme is to beat local and signal oscillator reference frequencies together prior to harmonic generation and measuring the frequency differences with an electronic counter. This avoids the need for direct measurement at the maser line frequency itself, provided that both the multiplication factor and accurate value of the local oscillator reference frequency are known. Typical frequency-measurement techniques of strong spectral lines with oscilloscope presentation have been described by Thaddeus and Krisher (1961). D . Molecules Investigated
Molecules of spectroscopic interest are those that possess a hyperfine structure that cannot be resolved by means of conventional absorption waveguide spectroscopy. Such structure may be due to a weak magnetic interaction as in formaldehyde (CH,O) or deuteron quadrupole coupling as in methyl chloride (CH,DCl or CD3Cl). Measurements of the deuteron coupling allow precise determination of molecular field gradients, since accurate values of the deuteron quadrupole moment are known. Experimentally determined values of electric field gradients in a given molecule may then be used as a check for theoretical calculations based on molecular electronic wavefunctions. Other molecules of interest are those which may be found in outer space for which precise values of spectral line frequencies are required for systematic radio-telescopic searches of possible new molecular species such as formamide (NH,CHO), or furan (C4H40), and others; or for precise measurements of Doppler shifts in emission spectra [for example, with ammonia (I4NH3), hydroxyl (OH), or formaldehyde (CH,O)]. In rare cases when the MBM line intensity is sufficient to sustain oscillation, the precise line-center frequency can be established by methods discussed in Section V,B,2, as for example, for the lo, + Ooo transition of CH,O (Krupnov et al., 1969). The particular choice of gas for study in the MBM spectrometer depends upon favorable molecular populations in the quantum states of interest and a sufficiently large Stark effectfor efficient electrostatic focusing. If the maser
ADVANCES IN MOLECULAR BEAM MASERS
217
gas possesses a rich hyperfine structure, then molecules will be distributed over a large number of levels and associated spectral lines will in general be weak. Over the lifetime of the MBM considerable attention has been given to problems of spectrometer sensitivity, efficient state-separation systems, and the creation of favorable molecular populations in quantum states of particular interest. These have been discussed in earlier sections of this review. Probably the greatest single instrumental advance has been the use of nozzle sources which produce an order-of-magnitude enhancement of MBM spectrometer sensitivity.With present-day techniques, MBM spectroscopy is no longer restricted to molecules containing only a few atoms, as exemplified by observations with the ring molecule furan (Tomasevichet al., 1973) which contains nine atoms. In view of the discussion of particular molecules in the context of MBM spectroscopy in an earlier review (Laine, 1970; and more recently Dymanus, 1975), only a few general comments will be made here. In Table I a summary is given of molecules investigated by MBM spectroscopy for which data have been published in the open literature. Approximate frequencies of the main line transitions are given, which in some cases are mean values for a group of lines. The best-known spectral resolution is presented wherever possible. Electric dipole moments are also given in view of their relevance for efficient state separation. Molecules such as cyanoacetylene ( p = 3.6 D), formamide ( p = 3.7 D), methyl isocyanide ( p = 3.8 D) and methyl cyanide ( p = 3.9 D) possess some of the largest dipole moments of molecules studied in MBM spectroscopy, whereas hydrogen selenide ( p = 0.6 D) and furan ( p = 0.7 D) feature among the smallest. The molecules studied generally possess low values of quantum number J for which a small number of M sublevels are obtained. However, examples of high-value J states are the -+ transition of formaldehyde (CH,O) at 1.5 MHz and 880 + 881 transition of furan at 2.1 GHz. The first transition is observed as a consequence of particularly efficient state separation for close lying doublets; the second transition illustrates the advantage associated primarily with the use of a nozzle g a s source. The frequency range covered by MBM spectroscopy goes from 0.66 MHz for the 3 3 0 - + 3 3 1 transition of CH,O (Takami and Shimzu, 1966) to 316.8 GHz for the l l 0 -+ lo, transition of D,O (De Lucia and Gordy, 1970). Molecular weights of molecules investigated range from 17 (OH, NH,) to 68 (C4H40). Studies of hyperfine transitions of molecules with molecular weights of 100 or more now appear to be feasible. These figures give some indication of the present range of MBM spectroscopy. Of the molecules listed in Table I, ammonia, formaldehyde, and water, including deuterated species have been studied in great detail. Where several isotopic substitutions are possible, such as with methyl alcohol (CH, 37C1,
218
D. C. LAINl?
CH2DCl, CD3Cl), small changes in molecular structure are revealed by high-resolution measurements of C1 quadrupole coupling, which reflects changes in electric field gradients and may be related to the chemical properties of the molecule. For example, increased deuterium substitution on the methyl group results in a systematic decrease in ,'Cl quadrupole coupling. Moreover, substitution of ,'Cl for 35Cl shows that the decreased electric field gradient at the C1 atom is not a result of the increased mass of the methyl group when deuterated (Kukolich and Nelson, 1972b).This type of investigation is a good example of the usefulness of MBM spectroscopy for chemical structure investigations. Weak-field Zeeman spectra have been studied in CH,O in the range 0-55 G for the transitions 6,, + 6,, (Shigenari et al., 1963), 43, + 432 and 532 + 5,, (Takuma et al., 1966), and l,, + l,, (Shigenari, 1967), and with 14NH, for the inversion transition J = 3, K = 2 (Shimoda, 1961b; Kondo and Shimoda, 1965). Molecular g factors have been obtained from such measurements. High-field Zeeman MBM spectroscopy is currently attracting attention from which improved data on g values have been obtained. Also, measurements of magnetic susceptibility anisotropies become possible from which molecular quadrupole moments can be calculated. Chemical shift anisotropies also become measurable. Examples of molecules studies are the l,, + l,, transition of CH,O at 17 kG (Kukolich, 1971a); 313+ 2,, and 44, + 523 transitions of D,O, the 220+ 2,, transition of HDO, and 6,, + 532 transition of H 2 0 at 13 kG (Verhoeven and Dymanus, 1970);the 30, + 2,, transition of CH,F2 at 100 kG (Kukolich and Nelson, 1972a);the J = 3, K = 2 inversion transition of I4NH3 at 19 kG (Kukolich, 1970) and at 100 kG (Kukolich and Castleton, 1973). With this last transition, the high-field spectra showed resolved splittings due to a difference in molecular g vaiue for upper and lower inversion states, a fact inferred much earlier from maser oscillation characteristics (Shimoda, 1961~).In summary, it is evident that there is considerable potential for studies of molecular Zeeman effects at high magnetic fields using the high resolution available (4 kHz at 22 GHz and 100 kG in the case of CH2F2)with MBM spectrometers. In contrast to Zeeman studies, MBM Stark spectroscopy is virtually unexploited. Only a few instances are known of MBM Stark studies. These include the C H 2 0transitions 431 -,432 (Shimoda et al., 1960)and 330 + 331 (Takami and Shimizu, 1966),studied in rf beam masers in which a homogeneous Stark field can easily be obtained. Stark effects due to the electric field (D x B) produced by the motion of C H 2 0 have been observed for the I,, -P l I 1transition in high-field Zeeman studies (Kukolich, 1971a). The influence of the Stark effect on the J = 3, K = 2 lines of "NH, has been studied by Bajgar et al., (1969). Clearly, more extensive use of the plane parallel Fabry-Perot resonator for MBM spectroscopy would facilitate
ADVANCES IN MOLECULAR BEAM MASERS
219
Stark-effect studies in spectrometers operated at frequencies of 20 GHz upward. Below this frequency the resonator would be impracticably large, unless low Q values could be tolerated. An example of the use of the FabryPerot resonator for Stark effect studies is the lol+Ooo transition at 72.8 GHz (Krupnov and Skvortsov, 1964a).
IV. BEAM-MASER AMPLIFIERS When a MBM is operated under regenerative conditions just short of oscillation, it can act as a very high gain amplifier over a very narrow bandwidth near the center frequency of the molecular resonance. Early in the development of the MBM, this type of amplifier was recognized to be capable of low noise operation, which was the prime reason so much interest was shown in maser amplification in general in the mid 1950's. The basic theory of the MBM amplifier was first given by Basov and Prokhorov (1955) and was developed in subsequent publications. Gain, linearity saturation, and bandwidth were all considered, although the important feature of low noise operation was not considered in any detail. The theory of the maser amplifier, published independently by Gordon et al. (1955), gave an analysis of the low-noise properties predicted, together with power gain, saturation, and bandwidth. These aspects, together with some applications of the MBM amplifier are briefly discussed in the next two sections. A . Gain, Bandwidth, Saturation, and Noise The MBM amplifier is usually operated under highly regenerative conditions and so power gain is obtained at the expense of amplification bandwidth. The maximum possible bandwidth without regeneration is given by the inverse time of flight of molecules through the maser cavity, typically 5 kHz. However, under regenerative conditions the bandwidth is very much smaller and may be as little as 300 Hz with a gain of 20 dB (Helmer, 1957b). Examples of regenerative narrowing have been given by Zuev and Cheremiskin (1962) who observed narrowing from 4500 to 100 Hz and 750 to 20 Hz with NH, and ND,, respectively. The theoretical dependence of bandwidth on the time of flight of molecules and the excitation parameter (which is defined to equal unity when oscillation sets in) has been investigated by Zuev and Cheremiskin (1962) and Strakhovskii and Cheremiskin (1963). The gain, bandwidth, and relative merits of transmission versus reflection MBM amplifiers have been considered by Gordon and White (1958). Measurements of gain have been made on a C H 2 0 maser by Krupnov and Skvortsov (1965a) who have also studied the dependence of
220
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c. LAINB
excitation parameters on molecular reorientation processes in both ammonia and formaldehyde MBMs (Krupnov and Skvortsov, 1966). A systematic study of the gain, bandwidth, and saturation characteristics of an ammonia MBM has been made by Collier and Wilmshurst (1967). The maximum power emitted by the molecular beam is approximately &Nhvo, where N is the total number of upper-state molecules entering the cavity per second and vo the molecular transition frequency. Typically, this power is of the order of W. Clearly, saturation of a high-gain maser amplifier will occur with an input power substantially less than this value. Thus it is only for input powers of lo-'' W or less that reasonably linear power amplification can be obtained. The low-noise property of the MBM amplifier is a consequence of amplification without charge carriers, since the molecules are uncharged. As a result, an input signal only has amplifier noise added to it from thermal radiation within the cavity and associated waveguides and from spontaneous emission from the molecules. Since the fluctuation in beam flux is very small, typically 1 in lo6, noise due to this source is generally negligible. Furthermore, such noise would only become evident in the presence of an input signal. In comparison to thermal radiation noise sources, spontaneous emission may be neglected at frequencies commonly associated with MBMs. When the maser is operated at high gain, and power losses to the walls of the cavity are small in comparison with the power emitted by the beam, a noise figure close to unity is expected (for example, see Shimoda et al., 1957; Helmer and Muller, 1958). This prediction was confirmed experimentally by several different groups within a few weeks of each other (Alsop et al., 1957; Gordon and White, 1957; Helmer, 1957b). The lowest effective input noise temperature measured was 74 f 16 K (Gordon and White, 1958). This result was in good agreement with the predicted value of 68 3 K. An upper limit of 20 K was given for the noise temperature of the beam itself. Noise measurements of a unilateral two-cavity MBM have also been made by Sher (1958) and fair agreement with theory again obtained. B. Practical Systems As an amplifier the MBM has a very narrow bandwidth, saturates at W or less, and is essentially untunable. It is not surprising therefore that rather few applications have been found for the MBM amplifier. One of the earliest applications was as a high-gain narrow bandwidth buffer amplifier between a maser whose noise figure was to be measured and a balanced crystal detector (Gordon and White, 1958). An additional use has been to decrease the frequency dependence of maser oscillation upon cavity detuning by increasing the effective molecular Q in a two-cavity MBM with
ADVANCES IN MOLECULAR BEAM MASERS
22 1
two head-on molecular beams passing through two cavities in series. In this system, one cavity acted as oscillator and the other as amplifier of the molecular ringing signal (Mukhamedgalieva et al., 1965, 1966). A further application has been as a preamplifier in an electron resonance spectrometer and a useful improvement in signal-to-noise ratio obtained (Gambling and Wilmshurst, 1963; Collier and Wilmshurst, 1966). The use of the MBM as a Q multiplifier was first proposed by Townes (1960). Such a system has been considered theoretically and experimentally for use in conjunction with a paramagnetic resonance spectrometer by Mollier et al., (1973). Different modes of operation were investigated: classical, Q multiplier, and marginal oscillator. With the last two techniques the noise factor of the spectrometer was reduced to a value close to the theoretical limit, F = 1. Spectrometer detector systems of this type have been proposed for studies of paramagnetic samples with long relaxation times. This type of application for the MBM is further discussed in Section V,D,2c.
V. BEAM-MASER OSCILLATORS By virtue of their potential application as standards of frequency, MBM oscillators have been very thoroughly investigated. Yet, despite great efforts to improve upon its frequency reproducibility and stability the molecular beam maser oscillator is, at its best, useful only as a secondary standard of frequency. In contrast the atomic beam maser (ABM), first operated in 1960 by Ramsey and colleagues (Goldenberg et al., 1960), has the accuracy of a primary standard of frequency. The ABM uses the storage box principle to reduce the spectral linewidth to 1 Hz or less. The latter type of maser oscillator, which lies outside the scope of this survey, has recently been reviewed by Audoin et al. (1971) and by Ramsey (1973). Although the usefulness of the MBM oscillator has been somewhat overtaken by events, there are certain applications for which its frequency stability and high output power relative to that of other quantum oscillators can be gainfully employed. These aspects are discussed in Section V,D. Moreover, the MBM oscillator has been, and continues to be, of considerable physical interest. Studies of its properties have revealed many features in common with other quantum-mechanical devices, for example, NMR and laser oscillators. In particular, the dynamic characteristics which are the subject of current investigations, are discussed in some detail in Section V,C. A. Conditionsfor Oscillation If the power emitted from the molecular beam exceeds the power loss from the MBM cavity, the system becomes self-excited and an oscillation builds up from thermal radiation noise within the cavity to a level
222
D.
c. LAINE
determined by saturation effects. The condition for oscillation may be written (Shimoda et al., 1956)
where nminis the threshold flux for oscillation, v the mean molecular velocity, p I 2 the matrix element of dipole moment, and A, I, and Q the crosssectional area, length, and quality factor, respectively, of the MBM cavity. For high-frequency MBMs the cross-sectional area A is small, and owing to beam divergence it becomes difficult to obtain oscillation. Therefore, at short wavelengths, the use of open resonators is preferred, despite a rather low filling factor. The flux requirement is eased by the fact that for open resonators a much greater fraction of the total molecular flux is used in comparison to that employed for closed resonators. This is because widely divergent molecules can be utilized, provided that their trajectories lie approximately in a plane perpendicular to the axis of the open resonator. Molecules with which MBM oscillation has been achieved are listed in Table 11. It is perhaps surprising that so few molecules have been found to be suitable in the 21 years of MBM oscillator studies. The most successful operation has been with ammonia in its various isotopic forms. With the exception of HCN, all the other molecules have been used in MBM oscillators employing closed resonators. MBM oscillation with HCN has only been successful with open resonators (De Lucia and Gordy, 1969). The transitions (3, 3)* of 14NH3 at 23.87 GHz and l o , +Ooo of CH,O at 72.84 GHz have both sustained oscillation in resonators of the open and closed variety. Open resonator oscillators employing ammonia have been described by Barchukov et al. (1963a,b)and Laine and Smart (1971). Oscillation with formaldehyde MBMs has been reported by Krupnov.and Skvortsov (1963a) for a closed resonator and by the same authors employing an open resonator (Krupnov and Skvortsov, 1964a). Improvements in stateselection efficiency have recently led to the observation of oscillation on four components of the weak J = 1, K = 1 inversion line of 14NH,. The use of a nozzle beam source in a formaldehyde (CH20) MBM improved the beam intensity sufficiently to cause simultaneous oscillations on the F = 2 + 2, 1 --* 2, and 1 + 1 hyperfine components (Tucker et al., 1970). It is possible that more extensive use of dynamic cooling with nozzles driven with a high back pressure will lead to additional MBM oscillation frequencies in due course. If a nonuniform radiation field exists in the MBM cavity, the emission linewidth is broadened, or even split, depending on the form of the field variation. If a high-order mode is excited, with more than one maximum of radiation field along the resonator axis, the maser oscillation commences at
223
ADVANCES IN MOLECULAR BEAM MASERS
TABLE I1 LISTOF GASESA N D TRANSITIONS FOR WHICH MOLECULAR-BEAM-MASER OSCILLATION HASBEENOBSERVED' Gas
Ammonia NH 3
Transition
Frequency (MH4
( J , K)+ = (1, I ) +
(2, 2)+
23,694.5 23,722.6 22,834.2 23,870.1
2 1.784.0 22,789.4 23,922.3 1,656.2
"NH,
Formaldehyde CH,O
101 Hydrogen cyanide HCN
J=l+O 2-1
(I
+
000
Principal reference(s)
Krupnov and Shchuko (1969) Gordon et al. (1955); Ito and Yamamoto (1964) Bonanomi et al. (1958); Shimoda (1961a) Gordon et al. (1955); Shimoda et al. (1956); Basov (1956); Bonanomi and Herrmann (1956), etc. Bajgar et a/. (1968) De Prins (1962) Takahashi et a/. (1960) Basov and Zuev (1961); Basov et al. (1963b)
4,829.6 Tucker et al. (1970) (3 hyperfine lines) 72,838.1 Krupnov and Skvortsov (1962, 1963% 1964a); Krupnov et a/. (1969) 88.63 1 88,633 177,260
Marcuse (1961b, 1962); De Lucia and Gordy (1969) De Lucia and Gordy (1969)
Common isotope assumed unless indicated otherwise.
the frequency of one or other of the two components, depending on the precise cavity tuning. A small cavity detuning causes the oscillation frequency to jump from one of the split line components to the other. Under certain circumstances, a biharmonic oscillation can be sustained (Becker, 1966).Clearly, such higher order cavity modes require a larger beam flux to sustain oscillation than with a uniform cavity radiation field. Studies of such MBM oscillators may only be observed with strong maser lines, usually the J = 3, K = 3 inversion line of I4NH3. Biharmonic oscillation has also been observed in a single-cavity ammonia maser (J = 3, K = 3 inversion line of
224
D. C. LAINb
14NH,) operated in a weak transverse magnetic field (Logachev et al., 1968; Lefrere and Laine, 1973). Biharmonic oscillation has also been obtained using an open resonator with 14NH3(Barchukov et al., 1963b). B. Amplitude and Frequency Characteristics The amplitude and frequency characteristics of the MBM oscillator have been the subject of a great number of investigations. Since these topics have been covered in some detail in an earlier review (Laine, 1970), only a brief summary is given here. New work is, however, given closer attention. The essential features of MBM oscillators have been described by Shimoda (1957, 1958), Barnes (1959), Strakhovskii and Cheremiskin (1963), and others. 1. Amplitude Characteristics
The amplitude characteristics of the MBM oscillator may be described in term8 of pressure behind the gas source, the EHT applied to the state separator, and cavity tuning. In a MBM oscillator operated with a single cavity, the precise form of the amplitude characteristics depends on details of construction. The general features for each of the experimental parameters cited are now briefly summarized for a MBM oscillator operated either with a uniform or a single maximum of microwave field ( n = 0 or 1,respectively) along the cavity axis as follows. For given values of cavity tuning and focuser voltage the oscillation amplitude curve passes through a single maximum value as the gas source pressure is varied. The oscillation amplitude rises rapidly from zero once a critical gas source pressure is reached. With effusive sources, the maximum amplitude is reached with gas pressures in the region 2-10 torr and with nozzle sources 20-30 torr, the precise value depending on gas source parameters and particular species of molecule employed. At high gas pressures, molecular beam scattering arises within the focuser and cavity, which reduces the amplitude of oscillation. With a nozzle source, gas pressures of half an atmosphere or more may be used before oscillation finally ceases. With a given source gas pressure and resonator tuning, the amplitude of oscillation as a function of focuser voltage follows a monotonic curve of decreasing slope. The tendency for the maser oscillation amplitude to saturate at large EHT values has been studied theoretically by Shimoda et al. (1956) using univelocity theory and for a velocity distribution by Helmer (1957a) and Shimoda (1957). When the focuser voltage and gas source pressure are held constant, and the cavity tuning is varied, the amplitude characteristic takes the form of a
ADVANCES I N MOLECULAR BEAM MASERS
225
semiellipse with the maximum value at zero detuning. The amplitude is given by
(3’ +
(0- wo)’
= B2,
(3)
where E is the electric component of the radiation field in the cavity, w the angular frequency of oscillation, o,,the molecular angular resonance frequency, and B a constant dependent upon cavity dimensions and beam flux (Helmer, 1957a; Shimoda, 1957). Experimental results are only in rough agreement with theory. An asymmetry, attributed to unresolved hyperfine structure, has been noted for the J = 3, K = 3 inversion line of 14NH3. When n 2 2, and the molecular beam passes through one or more nodes of radiation, the spectral line is split on account of the longitudinal Doppler effect. The amplitude of oscillation is no longer given by Eq. (3) and now possesses two peaks with a central minimum value (Barnes, 1959). Becker (1966) has investigated an amplitude-hysteresis phenomenon found to occur with n = 3, in a TM013 mode cavity. The oscillation amplitude characteristics in ammonia masers operated with cascaded cavities, coupled unilaterally by molecules which pass through them in succession, has evoked considerable interest over a period of many years. In this type of maser, the oscillation amplitude in the second cavity (C,) has been studied as a function of (i) the amplitude of oscillation in the first cavity (C,) or related parameters, such as focuser voltage or beam flux, with both cavities tuned to the maser line center vo, and (ii) detuning of the first cavity with the second cavity accurately tuned to vo. In (i), it has been shown that the amplitude of oscillation in C2 passes through the sequence: zero-maximum-minimum-maximum, as the radiation level in C1 is increased (Basov et al., 1963a). This sequence is readily obtained by increasing the focuser voltage from low to high values, with an optimum value of gas pressure behind the beam source. With a high value of EHT, this sequence is obtained with beam source pressures up to the optimum value corresponding to maximum excitation in C,. For pressures beyond this value, the sequence reverses after the second maximum (Strakhovskii and Tatarenkov, 1962), giving a total of three maximum and two minimum values. At lower EHTs, only a single maximum is obtained as the source gas pressure is varied. In the case of (ii), if C, is tuned through its full frequency range for oscillation, then for low focuser voltages the amplitude of oscillation in the second cavity shows a single maximum. At higher focuser voltages, the intensity of maser oscillation in the second cavity shows two maxima with a minimum in the center (Strakhovskii and Tatarenkov, 1962; Laine and Srivastava, 1963; Basov et al., 1963a).
226
D.
c. LAINB
The theory of the oscillation-amplitude characteristics of two-cavity beam masers has been considered by several authors (Basov et al., 1963a; Smith and Laine, 1968). However, theoretical analyses that can account for the phenomena observed have only recently been given (Krause, 1968; Bardo and Laine, 1971b). Moreover, an explanation based on similar lines has been put forward for the NMR two-coil flow maser analog (Krause, 1969; Krause and Laine, 1973). These analyses are based on the fact that at high levels of oscillation in C,, molecules are given a pulse (using NMR terminology) in excess of 4 2 . The 4 2 condition is then reached by detuning C, and the amplitude of C, radiation field reduced in accordance with Eq. (3).This condition corresponds to maximum polarization of the molecular beam. With further detuning of C,, the amplitude of oscillation in C, gradually falls to zero. If the amplitude of oscillation in C, at zero detuning is initially less than 4 2 , only a single maximum in C, is obtained as C1 is tuned through vo . If C, and C2 are tuned to v o , then a gradual increase of excitation in C, upon an increase of EHT will first cause n/2 excitation of the beam, and later 3n/2 excitation or more at the highest values of applied voltage. This results in the appearance of two amplitude maxima in C2 . The appearance of two or more maxima in C, as the gas flux is varied with high fixed value of EHT may be explained in a similar manner. For larger values of detuning of C,, and a strong state-separated molecular beam, a biharmonic oscillation can be obtained in C2. This phenomenon, first observed by Higa (1957) is a consequence of self-oscillations in C, beating with the oscillation forced by the molecular beam polarization carried from C, to C2.With small values of detuning of C,, the oscillation in C2 is phase-locked to the ringing signal carried by the beam from C, and biharmonic oscillation is suppressed. However, with large values of detuning, oscillation in C, ceases, and strong self-oscillation is obtained in C, . Biharmonic operation is further discussed in Section V,C. The amplitude of MBM oscillation can be considerably modified by the presence of weak electric or magnetic fields. This type of effect was first observed by Basov et al. (1963a) with electric and magnetic fields of the order of lo4 V m- and T, respectively, acting on the molecular beam prior to entering the maser resonator. The phenomenon is a consequence of space quantization of molecules in the state separator which results in a nonisotropic spatial orientation of molecules prior to entering the MBM cavity. Since the interaction between molecules and the radiation field depends upon their relative orientation in space, the amplitude of oscillation will depend upon reorientation effects between focuser and resonator in a single cavity MBM or between cavities in a two-cavity maser. In the latter case, however, the beam polarization is destroyed by the application of a large applied field (electric or magnetic) and, consequently, a forced oscilla-
'
ADVANCES IN MOLECULAR BEAM MASERS
227
tion in the second cavity may be reduced to zero. The spatial reorientation phenomenon, already discussed in Section II,B, has been investigated in some detail by Basov et al. (1963a), Krupnov and Skvortsov (1965b, 1966), and Koshurinov (1969). The phenomenon is of particular importance in connection with frequency-standard applications of the maser (Strakhovskii et al., 1966; Krupnov et al. 1967a; Logachev et al., 1968).
2. Frequency Characteristics Frequency pulling characteristics of the MBM oscillator were intensively studied in view of potential frequency standard applications in the late 1950's and early 1960's. The causes of frequency shifts are complex. In summary these include (i) the effect of cavity tuning; (ii) the shift in the maximum value of the MBM line due to unresolved hyperfine structure and the dependence of this shift on focuser voltage and gas source pressure; (iii) frequency dependence upon beam intensity and focuser voltage; (iv) the effect of nonuniform emission of molecules in the resonator causing unbalanced traveling waves in the MBM cavity; (v) spatial reorientation phenomena by weak electric or magnetic fields. The main source of frequency pulling is associated with cavity tuning. The oscillation frequency v is related to the cavity frequency v, by the expression v - Y O = (Qc/Qm)(v,
vo)f(o), (4) where vo is the center frequency of the molecular resonance, Q, and Q, the quality factors associated with cavity and molecules, respectively, andf(0) a function of the order of unity, dependent upon the amplitude of oscillation (Gordon et al. 1955; Shimoda et a/., 1956). For an ammonia maser, Q, /Q, is about 10- and thus the effect of cavity detuning on MBM oscillation frequency is reduced by the same factor. A large value of Q, improves the frequency stability of the maser, but a limit is set by Doppler broadening and the presence of unresolved hyperfine structure. According to Eq. (4)the effect of variations of Q,, Q,, andf(0) vanish when vc = v o . Thus the use of the MBM oscillator as a frequency standard requires precision tuning of the cavity. Methods used include modulation of Q, or f(0), and a point of cavity tuning sought where the frequency shift is either a minimum or ideally zero. Molecular Q modulation is readily achieved by using a small transverse Zeeman field applied to the cavity. The functionf(0) may be varied by many methods which include the following: modulation of the gas source pressure (Nikitin and Oraevskii, 1962), changes of the voltage applied to the state selector (Helmer, 1957a), spatial reorientation of molecules prior to entering the MBM cavity (Krupnov et -
228
D.
c. LAINB
al., 1967a, 1969), and background-pressure modulation (Strakhovskii and Tatarenkov, 1964). Of these methods magnetic tuning is found to be particularly convenient, although even for the J = 3, K = 2 ammonia inversion line, the field dependence of magnetic tuning can cause errors in cavity tuning. Some of the best results have been given by Hellwig (1966a) for the J = 3, K = 3 inversion line of I4NH3and a frequency reproducibility of 1 in 10" claimed. However, it has been shown by Nikitin and Oraevskii (1962) that modulation of the gas source as a method for setting the maser cavity frequency to vo may be yet more accurate, since the magnetic-tuning method is complicated by spatial reorientation effects (Krupnov et al., 1967a). Other frequency pulling effects are associated with traveling waves in the resonator and unresolved hyperfine structure of the MBM spectral line. These and the elimination of their effects are discussed in Section V,D,la,b. If the maser is operated with more than a half-wavelength along the cavity axis, splitting of the spectral line leads to the possibility of oscillation on one or other of the split line components. In general, as the cavity is tuned about vo, the oscillation frequency jumps by an amount given approximately as Av = nu/L, where n is the number of half-wavelengths along the cavity axis and L the length of the cavity. In practice Av is 10 kHz or more. The maser oscillates stably on either one or the other of the split line components. The mean of the two frequencies at which the frequency jump occurs was originally used to define the maser oscillator frequency (Bonanomi et al., 1957; De Prins and Kartaschoff, 1962). A frequency hysteresis has been found to be associated with this frequency jump (Becker, 1966), which is associated with the amplitude hysteresis mentioned earlier. Thus the frequency jump method for defining the maser oscillator frequency is perhaps not the best. The frequency characteristics of two cavity masers have also been investigated, with the original aim of producing a frequency standard of improved accuracy. If cavities C1and C2are coupled both via the molecular beam and externally, the effective molecular Q is increased and the rate of detuning with respect to frequency pulling of the MBM oscillation decreased. For example, a MBM oscillator operated with opposed beams and externally coupled cavities has been investigated by Holuj et al. (1962) for I4NH3 and Kazachok (1966) for "NH, . Increases in effective molecular Q up to five times that of a single-cavity maser have been secured. External microwave coupling is not, however, essential, as shown by Belenov and Oraevskii (1963, 1964, 1966), if opposed molecular beams are used which couple the two cavities bilaterally. In this case, the frequency range of oscillation synchronization and effective molecular Q depends upon the time of flight of molecules and the decay of polarization between cavities. Investigations on MBMs of this type have been carried out by Mukhamedgalieva and
ADVANCES IN MOLECULAR BEAM MASERS
229
Strakhovskii (1965). Moreover, a frequency jump has been found to occur upon resonator detuning which is not predicted by existing theory. It is clear from investigations to date that two-cavity masers are not yet fully understood. Finally, the frequency fluctuations of the MBM oscillator should be mentioned. The narrow linewidth of the MBM is related to the low-noise properties of the device, and is given by the Blaquiere-Townes equation (Grivet and Blaquiere, 1963) 26v =
471k T(Av)' PO
9
where Av and 6v are, respectively, the half-linewidths at half intensity of the maser emission and oscillation power. The theory of fluctuations in MBM oscillators has been investigated by Gordon et al. (1955), Shimoda et al. (1956), and Wang (1960) using univelocity theory, and by Troitskii (1958a,b) using a velocity distribution. Oscillation-amplitude fluctuations have been observed experimentally by Smith and Laine (1967). Frequency fluctuations have been studied experimentally by Tolnas (1960), Chikin (1962), Allan (1966), and Kleiman et al. (1969). The use of the exceptionally high shortterm frequency stability (- 1 part in 2 x 1013 for 10- W and 1 sec integration time) of the MBM oscillator in practical applications is discussed in Section V,D,2b.
''
C. Dynamic Properties It is only relatively recently that the dynamic or nonsteady-state properties of molecular beam maser oscillators have been investigated in any detail. Such studies depend upon evoking changes in the level of steady-state oscillation. This may be achieved either by external means or may be self-induced. The first example of modulation of the amplitude of a maser oscillation by external means was given in the work of Alsop et al. (1957), where coherence effects in the maser were noted when used as a Stark-switched super-regenerative amplifier-oscillator system. Oscillation modulation phenomena were also noted by Basov et al. (1963a) in experiments on spatial reorientation of molecules in an ammonia maser operated with cavities in series. In the latter investigations, a large oscillation-amplitude modulation was caused by applying a relatively low electric field (- 20 kV m- I ) across a molecular beam between cavities of a MBM operated in series. Modulation of the oscillation in the second cavity of such a two-cavity maser has also been achieved by mechanical impedance modulation of the output
230
D. C . LAINfi
waveguide attached to the first oscillating cavity (Laine and Smith, 1966).In the same work, amplitude modulation of the oscillation in a single-cavity maser was also reported for the method of modulating the voltage applied to the state selector. It was found that a modulation amplitude of 50 mV superimposed upon the 20 kV applied to the state selector could be easily detected using phase sensitive techniques. The response of the maser oscillation to periodic modulation of the state selector at various frequencies up to 2 kHz gave some insight into the velocity distribution of molecules in the MBM oscillator. When the condition for oscillation in a molecular beam maser is suddenly overfilled beyond a certain value, the oscillation amplitude follows a damped oscillatory behavior before settling to a steady state. The phenomenon is related to the periodic transfer of energy between molecules and the radiation field as the molecules shuttle between the upper and lower maser levels during their time of flight through the cavity resonator. The frequency v of the amplitude modulation is given approximately by v = p12E/h, where pI2 is the dipole matrix element for the transition and E the amplitude of the electric component of the maser oscillation radiation field. In a typical MBM system, this frequency ranges from approximately 4-10 kHz or more, depending on the value of E. This phenomenon is quite general and has been observed to occur in various types of maser and laser oscillator systems. The first observations of oscillation settling processes in molecular beam masers were carried out by Grasyuk and Oraevskii (1964). However, they were not successful in observing the oscillation-amplitude transient they predicted, on account of the long time of formation of the active molecular beam by applying a high-voltage pulse to the state selector. A report of the experimental observation of the oscillation-settling transient was first published by Laine and Bardo (1969) who used a type of molecular Q switching. The method relied upon saturation broadening of the maser transition by a strong frequency swept microwave signal injected into the maser cavity. A more flexible method for the observation of transient properties of molecular beam maser oscillators has been employed using Stark broadening of the maser transition within the maser cavity (Shakhov, 1969, and independently by Bardo and Laine, 1971~).This technique is also a type of molecular Q switching and is much more useful than the frequency sweep method described, since boxcar-integration techniques can be employed to improve the signal-to-noise ratio. This approach also permits direct measurement of the time of silence from the instant of switch-off of the Stark voltage to the point in time at which the oscillation first appears above noise. A family of curves of oscillation-amplitude transients obtained by the Starkswitching method is shown in Fig. 9.
ADVANCES IN MOLECULAR BEAM MASERS 3L
c2
n
231
Focuser voltage in 2kV steps
.Q
$
::
c
3 El
+2
E
0 0
0.5 The
1msec
FIG.9. Family of ammonia maser oscillator settling transientsfollowing oscillation quenching by means of an intracavity Stark probe in the cavity: 0 V at origin and -400 V 1 msec later; voltages relative to cavity. (Courtesy P. R. Lefrere.)
Basov et al. (1967b) showed that if one of the oscillation parameters of a strongly oscillating maser could be periodically varied at an appropriate frequency, large pulsations of the oscillation amplitude could be evoked. Evidence for such forced pulsations or spiking behavior has been given by Bardo and Laine (1969) using a slow frequency sweep modification of the method used for demonstrating oscillation-amplitude transients. Oscillation pulsations have also been produced by modulation of the voltage applied to a Stark probe at the natural period of transients and also by impedance modulation of the loading to the first cavity of a two-cavity maser and observing the oscillation pulsations in the second cavity. Self-modulation phenomena in MBM oscillators have been noted by many investigators. When a MBM with two cavities in series is operated as an oscillator, with both cavities tuned to the molecular resonance, oscillation in the first cavity phase-locks the oscillation in the second. However, if the first cavity is detuned to the edge of its range for self-sustained oscillation the phase-lock condition is no longer met as the frequency of oscillation of the first cavity is pulled [Eq. (411. A self-oscillation in the second cavity now arises independently of the forced-oscillation signal. The signal from the second cavity is then found to be amplitude modulation (Higa, 1957) as a consequence of beats between self- and forced oscillations. The amplitude modulation reaches a depth of 100% and is typically in the range 4-10 kHz. It has recently been found that when such a cascaded cavity maser is
232
D.
c. LAINB
operated under conditions for very strong oscillation the amplitude modulation is no longer sinusoidal. Such nonsinusoidal beats have also been observed in an analogous maser system employing a flowing nuclear spin system (Krause, 1969; Krause and Laine, 1973). At the highest amplitude of oscillation in the beam maser, the second cavity is driven into a continuous pulsation regime of operation (Laine and Maroof, 1974). An example of such pulsations is shown in Fig. 10. This behavior is identified with the paramet-
FIG. 10. Oscillation pulsations in the second cavity of a molecular-beam maser operated with two cavities in series and biharmonic oscillation in the second cavity.Time base: 200 p e c between grid lines.
ric excitation of oscillation transients in the second maser cavity by the periodic variation in cavity radiation field resulting from the Higa-type amplitude modulation. Thus, if the beat frequency coincides with the period of the oscillation-amplitude transient, or some multiple of it, continuous pulsations result. Other amplitude modulation effects have been noted in MBMs as a consequence of the fine structure of spectral lines (Laine and Smith, 1966; Krupnov and Shchuko, 1969) or as a consequence of higher order mode operation (Becker, 1966). The operation of ammonia beam masers in a weak applied magnetic field is of especial interest since the beat frequency can be controlled (Logachev et al., 1968). In experiments with a Zeeman maser operated with two cavities in series a magnetic fielddependent phase shift between oscillation beat frequencies in the selfexcited first and passive second cavity has been observed (Lefrere and Laine, 1973). Furthermore, it
ADVANCES IN MOLECULAR BEAM MASERS
233
has been shown that the beat frequency of this type of molecular beam maser may be phase-locked to a periodic modulation of either the molecular or cavity Q. Modulation of the cavity Q was secured by impedance modulation by means of a waveguide switch, and molecular Q modulation by an intracavity Stark probe. In both cases, it was convenient to observe phase-locking by observation of the amplitude modulation of the forced oscillation in the second cavity. When the weak-field Zeeman maser was operated in the almost-phase-locked condition, the Zeeman beat signal was found to be highly nonlinear. This result has been attributed to periodic pulling phenomena (Laine and Lefrere, 1973). Amplitude modulation phenomena have also been observed in beam maser oscillators using a Fabry-Perot resonator (Barchukov et al., 1963b). However, the origin of such results are not yet clear. D. Systems and Applications
Very early in the development of the MBM, it was shown that as an oscillator it possessed a spectral purity and frequency stability of a very high order. Accordingly much effort was expended in attempts to develop the maser as a primary standard of frequency and extend its various uses in this role. Over the years, other applications of the MBM oscillator have also emerged, including spectroscopy, noise measurement, and studies of the physics of quantum-electronic oscillators in general. In the discussion which follows, a summary is given of MBM oscillator development and its instrumentation with particular reference to frequency standards. 1. Maser Development and Instrumentation Several related areas of activity emerged in connection with different aspects of maser construction and related electronic instrumentation. These were (i)work directed toward improved frequency stability and reproducibility of laboratory masers, (ii) the construction and field operation of portable designs of masers, (iii) the practical problems associated with the low power output of the maser-typically W-and the transference of both shortand long-term frequency stability of the maser to more convenient frequency regions. a. The molecular beam maser as a laboratory standard offiequency. It has been seen in Section V,B,2 that the frequency of the maser is influenced by many factors. These may be associated with the lack of precision of maser cavity tuning; the presence of unbalanced traveling waves in the cavity; the existence of the unresolved hyperfine structure of the spectral lines employed; the stability of the gas flux and state-selected beam intensity; the
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stability of the background gas pressure with time; and effects of uncontrolled electric and magnetic fields within or in the vicinity of the maser cavity. All of these features have been studied in great detail and are now sufficiently well understood for associated frequency-pulling effects to be much reduced. Of the various approaches to overcome these adverse influences, certain features are to be found in common. For instance the cavity will be made of low expansion coefficient material such as quartz or Invar and may be critically coupled to a second cavity to give a flat phase response and therefore zero temperature coefficient of pulling at the central frequency of the maser oscillation. Tuning of the cavity to the center frequency of the spectral line will be achieved by either magnetic field modulation of the transition (Hellwig, 1966a) or by varying the gas flux from the effuser or nozzle (Nikitin and Oraevskii, 1962). The maser will be operated with two head-on molecular beams so that traveling-wave effects are eliminated in first order. Either a single cavity or two spatially separated cavities will be used. The maser will be operated on a spectral line with the simplest possible hyperfine structure. In most cases the J = 3, K = 3 inversion line of 15NH3or the J = 3, K = 2 inversion line of 14NH3or 15NH3, all of which have no quadrupole hyperfine structure, will be chosen. The lo, + 0,, line of formaldehyde which has no hyperfine structure is not very satisfactory as a maser medium for frequency standard applications because of poor control over the beam intensity as a consequence of the method of preparation of the gas from the polymer (Krupnov et al., 1969). Ammonia, on the other hand, is readily controlled (Johnson, 1957; Strakhovskii et al., 1965). The state-selected beam intensity in the case of formaldehyde is insensitive to EHT at the point when the Stark energy goes through a maximum, whereas for ammonia, there is no such optimum value of EHT, and voltage stabilization is necessary. To guard against background pressure fluctuations, continuous-flow liquid nitrogen traps should be used. However, it is advantageous not to use liquid nitrogen at all in a laboratory maser, as demonstrated by Krupnov and Skvortsov (1963b) for the lol-+ Ooo transition of formaldehyde,and by Maroof and Laine (1974) for the J = 3, K = 2 transition of ammonia (14NH3).Finally, the effects of external electric and magnetic fields can be reduced by appropriate screening, although spatial reorientation effects will generally persist. Laboratory beam masers intended for use as frequency standards have been built by many investigators. For example, masers built as potential frequency standards and operated with balanced beams have been operated on the transitions J = 3, K = 3 (14NH3)by Mockler et al. (1958)and Barnes et al. (1962); J = 3, K = 3 ("NH,) by De Prins et al. (1961a,b) and De Prins (1962); J = 3, K = 2 ( 14NH3)by Saburi et al. (1962a,b),Nikitin and Oraevskii (1962), and Hellwig (1966a); lol+ Ooo ("CH,O) by Krupnov et al.
ADVANCES IN MOLECULAR BEAM MASERS
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(1969). The best results appear to have been obtained by Hellwig (1966a) who claimed a resettability of 1 part in 10" by the use of sufficiently low beam intensities, modulation of the beam intensity in order to avoid the longitudinal Doppler shift in a balanced beam arrangement, and a localized magnetic field for tuning the cavity. This result is at least a factor of 100 better than achieved with the early MBM oscillators. b. Maser as a portable frequency standard. So far the discussion of maser oscillators has been centered on the evolution of laboratory frequency standards. Brief mention should also be made of the efforts aimed at developing more portable and compact versions of molecular beam masers, with secondary frequency-standard capability and a frequency reproducibility of several parts in 10". The main problem with such a maser was to produce a robust sealed-off device which did not use bulky vacuum pumps. All designs of sealed-off masers were based essentially on cryogenic pumping of ammonia with liquid nitrogen. In most cases this form of pumping was supplemented with a small getter-ion pump to take care of any outgassing or very small leaks of gases which could not be readily frozen out. The ammonia gas could be recycled after condensing on the cooled surfaces of the maser by letting the whole maser come up to room temperature, then recondensing the evaporated ammonia back into the gas reservoir. By such a method Johnson (1958) was able to operate the maser oscillator continuously for 5 days at a time. The recycling of the ammonia gas took about 1 hour. The liquid nitrogen consumption was 0.25 liter per hour. A relative tuning accuracy better than 5 x was achieved. More detailed accounts of the construction and performance of this and other types of sealed-off maser operating on the same principles have been given by Awender (1959), Guttwein and Plotkin (1959), Hopfer (1960), Krupnov et al. (1961), and Taylor (1963). A sealed-off maser made with a glass vacuum envelope and liquid nitrogen cooled activated charcoal used for pumping was reported by Dudenkova (1961) and continuous operation times up to 10 hours obtained. All of these portable maser oscillators had a relatively poor frequency stability, ranging from 1 part in lo8 to 5 parts in lo'', depending on the degree of control of cavity tuning. These masers, operating on the J = 3, K = 3 line of I4NH3 lacked the sophistication of their laboratory counterparts. Clearly, a balanced beam arrangement operated with "NH, could have increased their frequency stability and reproducibility by an order of magnitude. The use of liquid nitrogen was a great disadvantage from the point of view of operational convenience. But even this necessity has been overcome in one instance by operating a maser in an artificial Earth satellite in the vacuum of space (Basov et al., 1967a).The small size of the molecular beam maser and good frequency stability under conditions of intense vibration under satellite launch conditions makes it quite competitive with other types of frequency
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D.
standards. The use of opposed beams and an Invar cavity thermostated to better than O.Ol"C, produced a relative stability during a single run of observations of 1 part in lo", and the maser frequency fluctuated less than 1 part in 10" between runs of measurements. The line J = 3, K = 3 of 14NH3 was used. It should be noted that the stability measurements were made via a two-way Earth-satellite radio link which itself could introduce phase fluctuations as propagation conditions varied. c. Methods offrequency translation. The next area of activity to be discussed is that of translation of the frequency stability of the natural oscillation of the maser to the spectral region of interest, which may be either higher or lower than that of the maser itself. For this purpose it is usual to use a phase-locked system, as shown in Fig. 11. Here it will be noted that the
-
I
I MASER OSCILLATOR
t
us
I I YF! I "0
IF
I&
MUL IPLIER
STABILITY-1 in 10"
REFERENCE
P.S.D.
t
CONTROL
FILTER
I
STABILIZED OSCILLATOR
bus/'
t VARIABLE REACTANCE
FIG. 11. Schematic diagram of phase-lock system for translation of frequency stability of the maser oscillator to a lower frequency oscillator.
reference oscillator can be of much lower frequency stability than the maser, yet not significantly degrade the overall system frequency stability. Moreover, the use of a reference oscillator permits the provision to be made of a frequency controlled output at a standard frequency. With such a phaselocking technique, the stability of the maser oscillator can be transferred to any other higher power oscillator at any other frequency. Thus, by the use of harmonic multiplication, phase stable sources of radiation may be made available up to frequencies of 140 GHz or more (Strauch et al., 1964). At microwave frequencies phase-locking of a klystron has been employed (Vasneva et al., 1957; Grigor'yants and Zhabotinskii, 1961b; Hardin et al., 1964) and at lower frequencies a crystal oscillator has been stabilized using a multiplier chain (Nikitin, 1958; Vasneva et al., 1958a; Murin, 1959;Stitch et
237
ADVANCES I N MOLECULAR BEAM MASERS
al., 1960; Barnes, 1961, and others). Thus the spectral purity of the maser, or a fair approximation to it, is transferred to the auxiliary oscillator. In this way, a molecular frequency standard could be constructed with a standard frequency output of I, 5, and 10 MHz, etc. A more difficult problem is to preserve the stability and original frequency of the maser oscillator and to increase the power output. In the late 1950's and early 1960's effective low-noise amplifiers for this purpose were not readily available, so an alternative solution to this problem was found. Thus, several schemes were developed. One was to use an IF offset phase-locked klystron, mixed with its I F (Stitch et al., 1960). However, such a scheme posed severe instrumental problems in order to avoid regenerative feedback. A more satisfactory method has been developed by Bershteyn et al. (1959) who phase-locked a klystron with an output power of 60 mW, operating at one eighth of the maser frequency. Power at the maser frequency was then obtained by harmonic multiplication of a portion of the klystron output. The scheme employed is shown schematically in Fig. 12. 23.870GHz
h MULTIPLIER
FREQUENCY CONTROL
W)
KLYSTRON 7978.33MHz
MULTIPLIER (x3)
MULTIPLIER o( 8)
ISOLATOR
BALANCED MIXER
~ ~ - L i i q IF AMPLIFIER
FIG. 12. System for milliwatt power output generation at molecular-beam-maser oscillation frequency. Adaptation from Bershteyn et al. (1959).
2. Applications a. Use as a standard of frequency. It has been pointed out in previous sections that the molecular-beam maser is sufficiently well developed to make a very useful frequency standard with both excellent long- and shortterm frequency stabilities. Its potential in these respects was the mainspring
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of a great deal of activity in the first decade of molecular beam maser work. Once the precise frequency of the maser oscillation had been established (for example, Mockler et al., 1958; Saburi and Kobayashi, 1960; Bryzzhev et al., 1960)it could be used as a frequency standard either at the natural frequency of the maser (for example, Vasneva et al., 1958b; Shimoda and Kohno, 1962) or by frequency translation to a few million hertz using a precision quartz oscillator phase-locked to the maser via a harmonic multiplier chain. It has been shown by many investigators (for example, Leikin, 1960; Bryzzhev et al., 1960; De Prins et al., 1960; Saburi and Kobayashi, 1960; Konstantinov, 1961; De Prins and Kartaschoff, 1962) that the maser frequency stability was sufficient to permit measurements of irregularities in the rate of rotation of the Earth with reference to the UT, and UT, astronomical time systems. For this purpose, the MBM frequency standard was only used intermittently for sufficiently long periods of time to effect the comparison with a quartz crystal oscillator used continuously as a frequency reference. Frequency comparisons between cesium atomic-frequency standards and molecular beam masers using standard frequency transmissions were also made in an attempt to prove the consistency of atomic and molecular frequencies produced by completely different devices (Blaser and Bonanomi, 1958; Blaser and De Prins, 1958). Within the limits of accuracy, both standards were in agreement. Similar measurements were also made between cesium standards. These investigations were of importance at that time, since the frequencies adopted for atomic or molecular standards were derived from smoothed universal time UT2, which itself is variable in time. Clearly, this was a very unsatisfactory state of affairs and was only resolved when the cesium resonance at 9.192 GHz came to be used to define the second. The ammonia maser, albeit in a modest way, therefore made a contribution to the adoption of the atomic standard of frequency and time. The frequency stability of MBM oscillators even in early designs was generally superior to the best quartz frequency standards and could be used as an absolute frequency reference which permitted the possibility of resetting the quartz oscillator frequency from time to time, or making measurements of the aging rate of the quartz crystals. Both short-term fluctuations and long-term drifts of a quartz crystal frequency multiplied to the maser frequency have been carried out, for example, by Morgan and Barnes (1959) and Hellwig (1966b). The excellent relative stability of a pair of MBM oscillators excited interest in possible experimental tests of ether drift. Experiments were carried out by Cedarholm and colleagues (Cedarholm et al., 1958; Cedarholm and Townes, 1959) using two masers with their molecular beams pointing in opposite directions, but both parallel to the supposed direction
ADVANCES IN MOLECULAR BEAM MASERS
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of motion through the ether. Rotation of the masers through 180" should, on the basis of a presumed ether drift, produce a change in relative frequency (Mdler, 1957; Cedarholm and Townes, 1959), but experimentally no such shift of significance was found. However, the interpretation of this particular experimental result has been the subject of some debate which is beyond the scope of this review (Carnahan, 1961,1962; Kroll and Lamb, 1962).Another experiment proposed by Strakhovskii (see Basov et aZ., 1961) relied on the measurement of the phase difference between two unsynchronized masers placed several meters apart on a rotating carriage. According to relativity theory, rotation of the carriage about a vertical axis through 180"should not cause a phase shift. As far as is known, the experiment has not yet been carried out. The general problem of investigations of relativistic or cosmological effects with atomic and molecular frequency standards has been discussed at length by Basov et aZ. (1961). However, it is clear that relativistic-type experiments at the present time are best carried out, not with molecular beam masers as in the early experiments, but with the hydrogen maser oscillator because of its superior frequency stability. Such an experiment with a hydrogen maser has been recently discussed by Vessot (1971). b. Applications as a spectrum analyzer. The excellent short-term frequency stability of the MBM has been discussed in Section V,B,2. This property of the maser has been put to good use as a high-resolution spectrum analyzer. The spectrum of an oscillator yields data about linewidth, sidebands, and noise spectrum and in general gives more information than measurements of long- and short-term stability. For this type of investigation, it is essential that the signal to be studied should be heterodyned with as spectrally pure a power source as possible, so that the power spectrum is displaced to lower frequencies where narrowband filters can be used for beat-signal analysis. For example, Barnes and Heim (1961), by phase-locking a crystal oscillator to an ammonia maser, were able to provide an exceptionally phase stable reference frequency at 265.5 MHz which was then used to heterodyne with a frequency-multiplied output based on a 5 MHz quartz oscillator. The power spectrum of the signal under investigation at 265.0 MHz was thus displaced to 0.5 MHz, which was then spectrally analyzed. Other investigations have been carried out by Hellwig (1966b) who investigated the K-band spectrum of a frequency-multipliedsignal derived from a 100 kHz quartz crystal controlled oscillator, with a J = 3, K = 2 ammonia MBM oscillating at 22.83 GHz. Similar measurements were also carried out with the same system on the noise spectrum of a klystron, phase-locked to a high-order harmonic generated from a quartz crystal oscillator.
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c. Spectroscopic applications. When the molecular beam maser is used as a spectrometer, it is usually employed well below the threshold for oscillation. If the maser is capable of oscillation, however, certain types of spectroscopic data can be obtained. For example, a measurement of the maser oscillator frequency tuned precisely to the line center by one of the methods discussed in Section V,B,2 has been used to improve the values for the rotational constants of formaldehyde (Krupnov et al., 1969).The presence of unresolved hyperfine structure may also be inferred from the differences of the maser oscillator frequency when tuned to an apparent center frequency by various methods, for example, by EHT, nozzle pressure, or magnetic modulation. Evidence for the unresolved hyperfine structure of the 14NH3 line J = 3, K = 3 transition was obtained many years before the components were finally resolved in a conventional suboscillation beam-maser spectrometer by Krupnov et al. (1967~)and by Kukolich (1967). Furthermore, evidence for differences in the magnetic coupling constants in the upper and lower inversion states of the J = 3, K = 2 line of 14NH3were adduced from oscillation frequency pulling characteristics (Shimoda, 1961~).The linewidth of the molecular emission may also be determined experimentally by observations of the frequency pulling using Eq. (4). Thus, measurements of the cavity detuning relative to the line center, the associated change of maser frequency, and the cavity Q are sufficient for a value of the spectral linewidth to be calculated. Such a method has been used by Trkal et al. (1969) to obtain the ratio of spectral linewidth of the J = 3, K = 2 relative to the J = 3, K = 3 inversion line of "NH, .This method has also been used to obtain molecular Q values for single-cavity masers relative to that of MBMs operated with two cavities in series and two head-on molecular beams (Mukhamedgalieva and Strakhovskii, 1965; Kazachok, 1966; Krupnov et al., 1967b). If a maser can be made to oscillate simultaneously on a pair of closely spaced lines, the beat frequency should be determined primarily by the frequency interval between oscillating components. Such a maser has been operated on the J = K = 1 inversion line of 14NH3(Krupnov and Shchuko, 1969) on the central doublet and on the high-frequency quadrupole satellite doublet. However, the analysis of the experimental data is quite difficult owing to various frequency pulling effects. Nevertheless, Krupnov and Shchuko were able to determine values for the magnetic coupling constant and the quadrupole coupling constant although agreement with previously published values was poor. Another method of using a maser oscillator in a spectrometer system makes use of the principle of robbing molecules from the oscillation transition by satellite line transitions which have one or more energy levels in common. Thus by noting the reduction of the oscillation amplitude as the
ADVANCES I N MOLECULAR BEAM MASERS
24 1
frequency of an external signal was swept through the hyperfine transitions, weak quadrupole lines have been observed (Shimoda and Wang, 1955). Magnetic satellite lines have also been observed by the same method (Sircar and Hardin, 1964). The J = 3, K = 3 inversion spectrum of I4NH3 was studied in both cases. The method does not appear to have been used for any other transition of ammonia, or with other molecules, despite the simplicity and sensitivity of the technique. Finally, quite a different spectrometer application must be mentioned. This is the method of Hardin and Uebersfeld (1970) and Mollier et al., (1973) who have used the maser as a marginal oscillator type of detector and as a Q multiplier in an electron spin resonance spectrometer. The low-noise properties of the maser, coupled with its good frequency stability, have given excellent spectrometer sensitivity. A second MBM oscillator acts as the radiation source in the Q multiplier case. These methods have already been mentioned in Section IV,B. d. Applications to general studies in quantum electronics. Many properties of the beam maser oscillator are related to analogous phenomena in nuclear spin resonance and laser physics. Several such analogs have already been mentioned in this review. In general, the relative simplicity of the molecular beam maser as a two-level quantum system is attractive for investigations which are obviously related to other areas of quantum electronics. While analogs have been stressed, it is also true to say that some experiments have also been carried out which appear to be unique to molecular beam masers, for example, with two-cavity systems. Examples already discussed elsewhere in this review are the beat mode phase shifts (Lefrere and Laine, 1973) and the detection of molecular state modulation (Laine and Lefrere, 1973), both in a Zeeman beam maser oscillator. Without going into further detail, it is clear that the physics of molecular beam masers is by no means exhausted. This is especially true of molecular beam masers operated under conditions of exceptionally strong oscillation now made possible using fast pumping and nozzle beam techniques. Oscillation envelope transients and pulsations obtained in such maser systems are of particular interest at the present time.
VI. OTHER PROPERTIES A . Population Studies
The saturation properties of a beam maser oscillator may be studied indirectly by observations on the state in which the molecules exist when they emerge from the cavity. The increase in the flux of molecules in the
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lower inversion state of a 14NH3 maser oscillator has been studied by Helmer et al. (1960a,b) by the use of a coaxial lower state focuser and ionization detector. It was first shown by Strakhovskii and Tatarenkov (1962) that by using a second cavity in a spectroscopic mode of operation, the molecular beam emerging from a self-oscillating maser cavity could be highly absorptive, showing that the excess population of the molecules in the beam was transferred to the lower state of the MBM transition. The condition for maximum flux of lower-state molecules is that of a-pulse excitation in the first, self-oscillatingcavity. Further investigations carried out by Laine and Bardo (1970) showed that as the amplitude of oscillation in the first maser cavity was increased from zero to a high value, the spectral line in the second cavity went through the sequence:absorption-emission-absorptionemission. In this experiment, forced oscillation in the second cavity was suppressed by depolarizing the beam with a weak intercavity Stark field. The emission signal obtained under conditions of very strong oscillation in the first maser cavity was close to 2a-pulse excitation of the beam as a whole. Furthermore, it was shown (Laine and Bardo, 1971) that when fast molecules in the molecular beam undergo a-pulse excitation, some slow moleculesin the velocity distribution of the molecular beam undergo 2a-pulse excitation and bum a narrow dip in the wider spectral profile of the fast molecules. In Section V,A it was mentioned that an ammonia maser may be operated in a biharmonic mode by applying a weak transverse magnetic field across the maser cavity. When the magnetic field strength is sufficiently small and the biharmonic oscillation near a mutual phase-lock condition, the oscillation amplitude beat envelope is highly nonsinusoidal. If the spectral state of the molecules is examined by means of a second cavity operated in a spectroscopic mode, it is found that the flux of molecules is highly absorptive and becomes amplitude modulated. This molecular flux modulation is attributed to periodic pulling (snapping beats) of the biharmonic oscillations which causes microwave field amplitude and frequency modulation at the biharmonic beat frequency. The result is that the net downward transition probability of molecules in the beam is modulated at the same frequency, giving the flux modulation observed (Laine and Lefrere, 1973). The population of states in a molecular beam maser will also be affected by scattering of the beam by the background gas, usually the same gas as constitutes the beam. Such scattering effects may be severe in masers operated without cryogenic pumping, when the background pressure may be high, or when the maser gas is not easily trapped, as for formaldehyde. However, it is only recently that scattering cross sections of molecular beams with target molecules have been measured. Such experiments are discussed in Section VI,B,2.
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B. Relaxation Efects
1. Rapid Passage Phenomena Very little work has been carried out on relaxation effects (TI or T,) in molecular beam masers. Some observations were made by Laine (1966) on molecular ringing in a single-cavity ammonia MBM. The form of the experiment was analogous to the well-known wiggles phenomenon of NMR and the transverse relaxation time T, could be inferred from the rate of decay of the coherent spontaneous emission. Such a rapid passage through a single line, with a Lorentzian lineshape yields wiggles with a decay envelope of amplitude A = A, exp ( -t/T,), where T, is the transverse relaxation time for a nonregenerative MBM system. In the presence of regeneration, however, the rate of decay is reduced and may be much longer than the low-gain value of T, , reaching infinity when the condition for oscillation is reached. The further analog of beating ofbeats was observed by Laine and Sweeting (1971e) using a I4NH, beam maser operating on the split main line of the J = 1, K = 1 inversion transition. The inverse of the period between amplitude maxima gave the separation between spectral components of the main line.
2. Relaxat ion Cross-Section Measurements Quite recently, a new use for the beam maser has arisen in connection with measurements of relaxation cross sections of various gases with a beam of molecules in well-defined quantum states (Kukolich et al., 1973b; Wang et al., 1973; Ben-Reuven and Kukolich, 1973). The method uses a sourcemodulated MBM spectrometer operated with cavities in a Ramsey-type configuration, with the addition of a scattering chamber containing various gases placed between the spectrometer cavities. Two basic types of experiment are possible with such a system. The first is that if only the second cavity is used as the spectrometer detector of the molecular beam, molecules initially selected in the upper state of a pair of maser transition connected energy levels will be disturbed from that state by the gas in the scattering chamber. From measurements of the reduction of the maser signal in the spectrometer cavity as a consequence of beam scattering at various scattering chamber gas pressures, the scattering cross section can be obtained from the relation I = I, exp ( - n a L ) , where I and I, are the attenuated and unattenuated signal intensities, respectively, n the density of scattering gas, L the
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length of the scattering region, and cr the scattering cross section. This particular experiment gives relaxation information related to TI-type processes. The second type of experiment uses both resonator cavities. The molecular beam i s first excited by a 4 2 pulse into a coherent superposition state by a resonant radiation field in the first cavity, then passes into the scattering chamber and finally into the second cavity, where the beam is subjected to a further 4 2 pulse, phase coherent with the first. Interference between transition amplitudes in the two cavities produces the usual Ramsey resonance pattern. Any decrease in signal amplitude in the second cavity is then due t o a loss of molecules from the superposition state. The collision cross section related to T,-type processes may then be obtained using the same relation as before. Ben-Reuven and Kukolich (1973) have shown experimentally that with nonpolar scattering molecules, the scattering cross sections for both types of process (TI and T,) are approximately equal, whereas with polar molecules, the scattering cross section for superposition-state molecules is significantly larger than for the unpolarized beam. Thus here Tl c T,, in contrast to the result Tl > T2 normally encountered in magnetic resonance. The experiment has been carried out for a beam of NH3with the scattering gases NH, , OCS, CF,H, CH,F, N, , He, Ar, N,O, CH,CN (Kukolich et al., 1973b), and for a beam of OCS with the first five molecules (Wang et al., 1973). With the OCS beam, focusing enhances the lower state population of the rotational transition J = 1 -,2 and is therefore not a maser system. In this case, however, scattering transitions from these two states which may occur to J = 0 and 4 levels as well as one another cannot be neglected, as in the case of ammonia for which the inversion doublet spacing is much less than the rotational spacing and the dipole transition matrix elements between inversion levels are large. Thus this type of scattering with an OCS beam constitutes a third type of experiment, in which there is scattering of the beam molecules into a distribution of rotational states. It is clear from these scattering experiments that valuable data may be obtained using MBM techniques. This new area of MBM work is one of considerable potential, since data relevant to the study of the formation of interstellar molecules may be obtained. Related experiments not yet carried out are those of beam-beam scattering which may help elucidate possible mechanisms responsible for the excitation of cosmic masers.
REFERENCES Allan, D. W. (1966). Proc. I E E E 54, 221. Alsop, L. E., Giordmaine, J. A., and Townes, C. H. (1957). Phys. Rev. 107, 1450. Audoin, C., Schermann, J. P., and Grivet, P. (1971). Adoan. At. Mol. Phys. 7 , 1. Auerbach, D., Bromberg, E. A., and Wharton, L. (1966). J . Chem. Phys. 45, 2160. Awender, H. (1959). Elektron. Rundsch. 12,458.
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Strakhovskii, G. M., and Cheremiskin, I. V. (1963). Tr. Fiz. Inst. P.N. Lebedeu 21,68. English translation: (1964). In “Soviet Maser Research” (D. V. Skobel’tysn, ed.), p. 56. Consultants Bureau, New York. Strakhovskii, G. M., and Tatarenkov, V. M. (1962). Zh. Eksp. Teor. Fiz. 42, 907. English translation: (1962). Sou. Phys.-JETP 15, 625. Strakhovskii, G. M., and Tatarenkov, V. M. (1963). Izu. Vyssh. Ucheb. Zaued. Radiofz. 6, 1273. Strakhovskii, G. M., and Tatarenkov, V. M. (1964). Izu. Vyssh. Ucheb. Zaued. Radiofz. 7,992. Strakhovskii, G. M., Tatarenkov, V. M., and Sudvilovskii, V. Yu. (1965). Prib. Tekh. Eksp. 3, 245. English translation: (1965). Instrum. Exp. Tech. 3, 730. Strakhovskii, G. M., Tatarenkov, V. M., and Shumyatskii, P. S. (1966). Radiotekh. Elektron. 11, 519. English translation: (1966). Radio Eng. Electron. Phys. 11, 438. Strandberg, M. W. P., and Dreicer, H. (1954). Phys. Rev. 94, 1393. Strauch, R. G., Cupp, R. E., Lichtenstein, M., and Gallagher, J. J. (1964). In ‘‘ Proceedings of the Symposium on Quasi-Optics” (J. Fox, ed.), p. 586. Polytechnic Inst. Brooklyn, New York. Suchkin, G. L., and Rakova, G. K. (1962). Radiotekh. Elektron. 7, 1251. English translation: (1962). Radio Eng. Electron. Phys. 7, 1172.
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Takahashi, I., Hashi, T., Yamano, M., Yamarnoto, M., and Suzuki, S. (1960). J. Phys. SOC.Jap. 15, 531. Takami, M., and Shimizu, T. (1966). J. Phys. SOC.Jap. 21,973. Takuma, H. (1961). J. Phys. SOC.Jap. 16, 309. Takuma, H., Shimizu, T., and Shimoda, K. (1959). J. Phys. SOC.Jap. 14, 1595. Takuma, H., Evenson, K. M., and Shigenari, T. (1966). J. Phys. SOC. Jap. 21, 1622. Taylor, E. R. G. (1963). R A E (Farnborough) Tech. Note No. WE25. ter Meulen, J. J., and Dymanus, A. (1971). Symp. Struct. Spectrosc., 26th. (Abstr.) p. 78. ter Meulen, J. J., and Dymanus, A. (1972). Astrophys. J. 172, L21. Thaddeus, P., and Krisher, L. C. (1961). Rev. Sci. lnstrum. 32, 1083. Thaddeus, P., and Loubser, J. (1959). Nuouo Cimento 13, 1060. Thaddeus, P., Javan, J., and Okaya, A. (1958). Bull. Amer. Phys. SOC. [2] 3, 28. Thaddeus, P., Loubser, J., Krisher, L., and Lecar, H. (1959). J . Chem. Phys. 31, 1677. Thaddeus, P., Loubser, J., Javan, A., Krisher, L., and Lecar, H. (1960). In “Quantum Electronics” (C. H. Townes, ed.), p. 47. Columbia Univ. Press, New York. Thaddeus, P., Krisher, L. C., and Cahill, P. (1964a). J . Chem. Phys. 41, 1542. Thaddeus, P., Krisher, L. C., and Loubser, J. H. N. (1964b). J . Chem. Phys. 40,257. Tolnas, E. L. (1960). Bull. Amer. Phys. SOC.5, 342. Tomasevich, G. R., Tucker, K. D., and Thaddeus, P. (1973). J . Chem. Phys. 59, 131. Townes, C. H. (1960). Phys. Reu. Lett. 5, 428. Trkal, V., Bajgar, V., and Vavra, A. (1969). N a t . Conf Radiospectrosc., 1st p. 143. Troitskii, V. S. (1958a). Radiotekh. Elektron. 3, 1298. English translation: (1958). Radio Eng. Electron. 3, 115. Troitskii, V. S. (1958b). Zh. Eksper. Teor. Fiz. 34, 390. English translation: (1958). Sou. Phys.J E T P 34, 271. Tucker, K. D., and Tomasevich, G . R. (1973). J. Mol. Spectrosc. 48, 475. Tucker, K. D., Tomasevich, G. R., and Thaddeus, P. (1970). Astrophys. J. 161, L153. Tucker, K. D., Tomasevich, G. R., and Thaddeus, P. (1971). Astrophys. J. 169, 429. Tucker, K. D., Tomasevich, G. R., and Thaddeus, P. (1972). Astrophys. J. 174, 463. Vasneva, G. A.. Gaigerov, B. A,, Grigor’yants, V. V., Yelkin, G. A., and Zhabotinskii, M. E. (1957). Radiotekh. Elektron. 2, 1300. English translation: (1957). Radio Eng. Electron. 2, 116. Vasneva, G. A., Grigor’yants, V. V., Zhabotinskii, M. E., Klyshko, D. H., Sverdlov, Yu. L., and Sverchkov, E. I. (1958a). 102. Vyssh. Ucheb. Zaued. Radiojiz. 1, 185. Vasneva, G. A., Grigor’yants, V. V., Zhabotinskii, M. E., Klyshko, D. H., Sverdlov, Yu. L., and Sverchkov, E. 1. (1958b). Radiotekh. Elektron. 3, 569. English translation: (1958). Radio Eng. Electron. 3, 167. Verhoeven, J., and Dymanus, A. (1970). J. Chem. Phys. 52, 3222. Verhoeven, J., Bluyssen, H., and Dyrnanus, A. (1968). Phys. Lett. A X , 424. Verhoeven, J., Dymanus, A,, and Bluyssen, H. (1969). J. Chem. Phys. 50, 3330. Vessot, R. F. C. (1971). Proc. Con5 Exp. Tests Graoitation, California Inst. Tech., p. 54. Vonbun, F. 0. (1960). Rev. Sci. lnstrum. 31, 900. Wang, T. C. (1960). J. Phys. Radium 21, 261. Wang, J. H. S., and Kukolich, S. G. (1973). Bull. Amer. Phys. SOC. [2] 18, 59. Wang, J. H. S., Oates, D. E., Ben-Reuven, A., and Kukolich, S. G. (1973). J . Chem. Phys. 59, 5268. White, L. D. (1959). Proc. Annu. Symp. Freq. Control, 13th p. 578. Zuev, V. S. (1962). Opt. Spektrosk. 12, 641. English translation: (1962). Opt. Spectrosc. 12, 358. Zuev, V. S., and Cheremiskin, I. V. (1962). Radiotekh. Elektron. 7, 918. English translation: (1962). Radio Eng. Electron. Phys. 7, 869.
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Past and Present of the Charge-Control Concept in the Characterization of the Bipolar Transistor J. TE WINKEL Philips Research Laboratories, Eindhoven, The Netherlands
I. Introduction ............
....................... ............................................. ........ .... IV. Physical Background of the Charge-Control Concept ................................
ra in Brief 11. First Years of the Transi 111. Origin of the Charge-Control Concept . .
V. Hybrid-Pi Equivalent Circuit as a Small Signal Version of a Charge-Control Representation ..........................................................................
IX. Linvill Lumped Model.. ... X. Miscellaneous Applications ................................................
253 254 258 263 266 269 271 274 215 211 277 278 28 1
B. Lightly Doped Collector C. Gummel’s Charge-Control Relation ............................................... D. Extensions of the Charge-Control Concept for Improved Time-Dependent 283 Response. ................................................................. XI. Charge-Control Concept in the Computer-Aided Analysis of Transi 284 .......... 284 ....................................................... 286 B. Analysis of Circuits Containing Transistors 287 XII. Conclusion ........................................................................ 288 References ..............................................................................
I. INTRODUCTION A quarter of a century has passed since the invention of the transistor. Consequently a new generation of solid-state physicists, device technologists, and circuit engineers has succeeded the one that took an active part in the early development and application of the device. This then would mark an appropriate point in time to review the efforts that have been made to derive concise representations of the physical and electrical properties of the various transistor types An important function of such characterizations or models is to serve as communication channels between the three classes of workers mentioned above. Ideally the models should be securely founded in solid-state physics, incorporate the main technological parameters, and provide the data necessary for efficient circuit design. Progress in developing the physically oriented electrical models referred to has been most marked in the case of bipolar transistors; it has led to some very specific ways of describing the operation of the device. One of these, 253
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known as the charge-control concept, will be the subject of the present paper. Briefly, the application of the charge-control concept will mean that bipolar transistor modeling is discussed from the viewpoint that one is dealing with a device in which charges control the electrical performance; the nature and the location of those charges being a subject of discussion as well. However, the charge-control approach as understood in this paper will also encompass a number of other approaches that have been presented in the course of time under various other names. As the discussion will show, all approaches are intrinsically the same and represent equal levels of approximation; they therefore lead to electrical models that differ only slightly. In the hope of removing a considerable amount of confusion and to emphasize the intrinsic similarity of approach, all models will be designated by a common name, the charge-control model. If necessary, additional reference to their origin will be made. This name is preferred above others as it refers to the main physical principle involved. This review has been written taking a midway position between device and circuit designers. The author, who has heard each party complain about the lack of meaningful information supplied by the other, hopes that it may help to bridge a communication gap that has been in existence too long. 11. FIRST YEARS OF THE TRANSISTOR ERAIN BRIEF
It is worth recalling that the invention of the transistor resulted from the work of a team of physicists. In the best tradition of their discipline an analysis of the basic physical effects involved was published immediately after the demonstration of the feasibility of the new device (I). The equally thorough treatment of the pn junction (2) led to the concept of a junction or bipolar transistor. Thus, at the time (1950) when realizability was demonstrated, the elements for a consistent theoretical background of the junction transistor were available (3, 4). It does credit to the high standard of the work performed in those early years when it is noted that the basic theory has remained valid in all its essential aspects and that the approximations and approximate reasonings then introduced in dealing with the junction transistor are still in use today. The point may be illustrated by the brief review of the fundamentals of transistor theory given below. The recapitulation has also been included to serve as a reference when dealing with the main topic of the paper. Condensed into a number of separate statements it will run as follows: (i) The electric current flowing in a semiconductor is carried by both electrons and holes. The flow and the mutual interaction of the charge carriers are governed by five equations ; headed under their familiar names they read:
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
255
the continuity equations, V .j ,
=e
an at ~
+ eR(n, p ) ,
the transport equations, j , = ep,nE
+ eD, Vn,
j , = e p p p E - eD, Vp;
(1)
the Poisson equation, e V.E=-(p-n+N; E
-Ni).
Here n and p represent the electron and hole concentrations, j , and j , the electron and hole current densities, and E the electric field. The mechanism by which electron-hole pairs are created or annihilated is represented by the generation-recombination function R(n, p). Electron and hole mobilities and diffusion constants are represented by p, , p, and D, , D,, respectively, e is the electron charge, and E the dielectric constant of the semiconductor material. The densities of the ionized donor and acceptor impurities N & and NA are quantities that can be controlled by the device designer. It is helpful in a general discussion and also often allowed by the geometry of a particular device to assume that the spatial variation of the various quantities is large only in one single direction. Equations (1) then reduce to a set describing scalar quantities that depend on time and a single space coordinate: ax
=e
an at
+ eR(n, p ) ,
j , = ep,nE
an + eD, ax'
ax
= - e - a P - eR(n, p ) ,
at
j , = epppE - eDP
dP
ax ' -
(2)
(ii) To obtain transistors of practical usefulness the impurity concentrations have to be made relatively large and the impurity type should change rather abruptly from dominantly acceptor or p type to donor or n type. Hence a detailed knowledge of the p n junction is a prerequisite for all further work. (iii) For the pn junction region an approximate description can be given which amounts to a division into a number of separate regions each with very different properties. A depletion or space-charge layer will extend over a narrow region at both sides of the metallurgical junction (i.e., the plane where the net impurity changes from p type to n type) and two so-called
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quasi-neutral regions occupying the remainder. In the depletion region a strong electric field will exist determined by the space charge due to the fixed ionized impurity atoms in the absence of mobile carriers. The designation quasi-neutral region stems from an analysis [using Eqs. (1) or (2)] that shows that in a homogeneous semiconductor appreciable deviations from neutrality can only occur over extremely small distances or during very short times. For all practical purposes, therefore, the net charge and the electric field outside the depletion region may be put equal to zero (this statement needs reconsideration for nonhomogeneously doped material, see below). The description will remain valid when a moderate amount of current flows across the junction (the low-level injection situation discussed later). A most important approximation, violating physical reality, is constituted by the assumption that the change from one region to the other is an abrupt one. (iv) Any voltage applied to a pn junction will appear nearly wholly across the depletion layer; the current that flows will be an exponential function of the voltage. Depending on the polarity of the voltage the current in one direction, the forward direction, will be many orders of magnitude larger than in the other, the reverse direction. Exponential relations will exist between the mobile carrier densities in the quasi-neutral regions at the boundaries of the depletion layer and the applied voltage. (v) In thermal equilibrium the principle of detailed balance requires the product pn of hole and electron concentrations to be constant everywhere. Coupled with the charge-neutrality requirement and the practical need for large impurity concentrations the notion of majority and minority carriers arises (in p-type material: holes and electrons). The distinction will remain useful in the normal active operation of a device. (vi) For efficient operation transistors require pn junctions that pass currents consisting nearly exclusively of one kind of charge carriers. The doping levels at the two sides of the junction then should differ by several orders of magnitude. For instance, from a heavily doped n region a current consisting largely of electrons will flow into a lightly doped p region and add minority carriers to the ones already present there. Such a junction is called an injecting junction. A distinction must be made between situations where the injection raises the total minority concentration to levels comparable with the majority concentration and others where the majority-minority relation is maintained. The restriction to the latter category, that is, to what is known as low-level injection, is fundamental to the subject matter of this paper, hence it will be assumed to apply to all subsequent discussions without being constantly referred to. (vii) When technological advance (the advent of the diffusion process) made it possible to have p and n regions with impurity-concentration
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
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profiles varying in magnitude, the quasi-neutrality concept needed reconsideration (5). From an analysis, again using (1) or (2), it follows that in this case an electric field must be present to prevent the majority carrier profile, also of varying magnitude, from leveling out by diffusion. However, the net charge that is needed to sustain the field, proves to be negligibly small so that quasi-neutrality, again, can be assumed. This does not mean that the field itself can be neglected; through the transport equations it will take part in controlling the flow of carriers. Its value will depend solely on the prescribed impurity concentration profile if, again, the current is kept to moderate values so that no significant change in the majority carrier level can occur.
To fit the junction transistor into the mixed pattern ofexact and approximate reasoning that has been presented above, it will be introduced as a semiconductor crystal in which by a suitable doping process an n p and a pn junction have been made. If the distance between the junctions is small enough, interaction is possible; the whole then will constitute a npn transistor.* In principle two modes of operation can be envisaged. In the first, one junction is permanently biased in the forward direction, the other in the reverse direction ; this represents the transistor amplifier. In the second, two permanent situations can occur, both junctions are biased in the same direction either forward or reverse; a temporary changeover situation with one forward and one reverse bias also exists. This is the transistor switch; the forward situation amounting to a near short circuit, the reverse situation to a near open circuit. To study the electrical properties of the junction transistor, one can proceed in a manner that closely follows the pn junction approximations discussed; the transistor is divided into a succession of quasi-neutral and space-charge regions. In each region the set of equations (1) or (2) is solved making full use of the simplifications that the particular nature of the region will allow. Next, the individual solutions are joined by suitable boundary conditions and the final result expressed as relations between the applied voltages and the terminal currents. This procedure has become known as the regional approach. The amplifying mode of operation was the first to be analyzed along the lines described. The year 1952 may be regarded as marking the start of a flow of publications dealing with the detailed electrical characterization of the junction transistor aimed at its use as a circuit component; references (6) and (7) may be cited as examples of the manner in which the information was presented. It cannot be said, however, that the circuit designers reacted * Throughout the paper npn transistors will be dealt with only, the changes to be made with respect to the pnp variety being obvious.
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enthusiastically; in their view the presentation was too mathematical and, above all, it lacked the simplicity of the electron-tube representation they were accustomed to. What was really wanted to describe the small-signal amplifying properties amounted to an equivalent circuit of simple structure, or in other words, to a combination of a few standard circuit elements and a controlled voltage or current source that could be fitted logically into the circuit to be designed. Preferably, the elements of the equivalent circuit should be accessible for measurement. A circuit that proved to be extremely useful for the purpose, though not particularly noted when first published in 1954 (a), is the one that in the course of time has become known as the “ hybrid-pi small-signal equivalent circuit.” Basically it is obtained by further approximating the results of a device characterization analysis as outlined above, making the assumption that small variations are superimposed on constant currents and voltages of much larger value. Further discussion will be deferred to a later section where the circuit is examined in relation to the charge-control approach. Also in the year 1954 a study of the switching mode of operation, reported in two successive publications (9, I O ) , produced an equivalent circuit adapted to this particular situation. Taking its name from the authors it became known as the Ebers-Moll large-signal equivalent circuit.” The designation “ large-signal ” indicates that currents and voltages vary from normal to very small values so that the nonlinear relation between them must be taken fully into account. It does not mean that the restriction to moderate current levels that applies generally to the present discussion has to be lifted. The merits of the Ebers-Moll model were also recognized only slowly; the subject will be returned to in a later section. To conclude this survey of the information that was available to circuit designers in these early years of transistor development, it must be remarked that the “black-box” concept still had the largest number of supporters. One preferred to regard the transistor as a device provided with two pairs of terminals and described by certain relations between the applied voltages and currents. Such relations could be obtained either from one’s own measurement or from the manufacturer; any connection to physical processes inside the device was preferably ignored. “
111. ORIGIN OF THE CHARGE-CONTROL CONCEPT To introduce the basic principles of control-by-charge,” one may start by remarking that the collection of holes or electrons present in any particular region of the transistor can be regarded as a charge. Next, one can investigate whether that charge has a definite relation to some voltage or current elsewhere in the device. Should it happen that a single charge appears to be related to several other quantities, the concept that the charge is in control follows directly. It is not necessary that the charge be discernible “
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259
as such or that it be accessible from the terminals, for example, for measurement; in fact, in many instances the quasi-neutrality requirement will prevent this. Nevertheless, the charge will represent a physical entity. It may be expected, therefore, that the charge-control concept will be useful in establishing a physically oriented model of the transistor. The formulation is sufficiently general to be applicable to other semiconductor devices or electron tubes ( 1 1 ) . However, the question of whether the charge-control approach is a fruitful one may have a different answer in each particular case. The first publication of the charge-control concept applied to a junction transistor, the now classical paper by Beaufoy and Sparkes (12), dates from 1957. Looking back one may wonder why three years had to pass, counting from the Ebers-Moll papers, even when one takes into consideration that it required some imagination to promote a nonaccessible charge to the role of controlling agent. A major reason might well have been the rather abstract mathematical approach that was customary at that time in publications dealing with electrical characterization. The treatment was mainly directed at obtaining voltage-current relations, largely ignoring the hole and electron concentration patterns and the charge that these represented. Strangely enough, such patterns had been given due regard in Shockley’s classic paper on the pn diode (2) and a minority carrier profile had been used to illustrate the base-width modulation effect (13). This good practice was not continued, however; consequently, in the literature up to 1957, hardly any other reference to concentration patterns can be found. To the recollection of the author, concentration patterns nevertheless did remain in favor at several places, mostly for illustrative purposes in informal discussions. They also proved helpful in explaining the basic properties of the transistor to newcomers. It is an additional merit of the EkaufoySparkes paper that it made this practice more widely known, as later publications and textbooks testify. In describing the junction transistor as a charge-controlled device, a distinction between two lines of attack seems useful. The first may be characterized as mainly qualitative, simple, and approximate, the second as quantitative, precise, and more detailed. Generally speaking, the first line was adhered to in the Beaufoy-Sparkes paper and a number of subsequent publications; the second line became of interest when the comparison of the various transistor models was due, a subject to be reviewed in a later section. To conclude the present section a brief description of the qualitative approach will follow. It may be read either as a condensed report of the basic views that emerged from the publications mentioned above or as an introduction to the second line of attack. The amplifying and switching modes of operation will be dealt with separately; in both cases the simplifying assumption will be made that in the base region neither an electric field exists nor recombination takes place.
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In the amplifying mode the electron (minority) concentration profile in the neutral base region will be determined by a specific value at the emitter side and have zero value at the collector side; the first being determined by the applied forward bias voltage, the second by the fact that the full reverse bias sweeps away all electrons arriving at the boundary. The current that flows through the base from emitter to collector does so by diffusion of minority carriers. As it has to be constant in every cross section, the carrier concentration profile should have a constant gradient. The pattern that results is shown in Fig. la. In the figure the area of the triangle will represent the total electron charge in the base region; its value is determined by the gradient and the electron concentration at the emitter side, that is, by the collector current and the base-emitter voltage, respectively. To complete the picture one can lift the restriction mentioned above and allow recombination to occur; the amount is assumed to be so small that the change in
FIG. 1. Illustration of the control function of the base minority charge,
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
26 1
collector current through the loss of electrons is insignificant. The base current that has to furnish the holes destined to recombine with those lost electrons then will be determined by the total number of electrons present, that is, by the base electron charge. The concept that the base minority charge controls the collector and base currents as well as the base-emitter voltage now follows logically. The control function of the base charge can be put to further use in demonstrating in what way the collector-base voltage affects transistor performance (the base-width modulation or Early effect). A decrease in collector-base voltage will narrow the depletion layer and thereby widen the neutral base region. The charge pattern will change as shown in Figs. l b and lc, drawn for constant base-emitter voltage and collector current, respectively. The changes in collector current in the first case and in base-emitter voltage in the second can be interpreted as showing the existence of a finite collector conductance and a voltage feedback from collector to emitter. When the charge-control principle is investigated in connection with time-varying quantities, one starts with the very plausible assumption that the relations between stored charge, collector current, and base-emitter voltage as described above remain unchanged. However, the base minority charge, that now varies in magnitude (Fig. Id), will need a supply of an equal amount of majority charge to maintain quasi-neutrality. This in its turn will cause an additional base-current component, equal to the time derivative of that charge, to flow in the base terminal. Quite generally then, one can state that when time-varying quantities have to be considered, the charge-control concept refers to control by a charge and its time derivative. The control by the time derivative of a charge also provides a convenient way to incorporate junction capacitances in the general concept. In this case the charge is constituted by majority carriers (holes) that flow into or out of the neutral base region proper, in order to compensate for the change in space charge that accompanies a shift in the depletion layer boundary ; the latter, in its turn, is caused by a change in the bias voltage. The junction capacitance, equal to the ratio of the two changes, is the more common measure of the effect. Turning now to the switching mode of operation the external conditions particular to that mode must be recalled. These are: a voltage source to be applied between base and emitter that functions as the control of the switch by providing bias of either reverse or forward polarity, and a source of large reverse bias connected between collector and emitter in series with a suitable load. Very little current will flow when the base-emitter junction is reversely biased; the switch is then said to be open. Changing the polarity and increasing the magnitude of the base-emitter voltage will first establish the condition already described as the amplifying mode, characterized by a reverse
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bias across the collector-basejunction. A further increase of the base-emitter bias and the corresponding increase in collector current will lead to a large voltage drop across the collector load, a situation that leaves only a small fraction of the bias source between collector and emitter. The collector current then will have a constant value (source voltage divided by load resistance) and the collector-base junction will be biased in the forward direction (the remaining collector-emitter voltage has dropped to a value lower than the base-emitter bias). This particular situation of high-current and low-voltage operation is commonly known as saturation; the switch is said to be closed. In saturation the base minority charge pattern will take the form shown in Fig. le. As a consequence of the forward bias the concentration at the collector side is no longer zero, the fixed value of the current implies a concentration gradient of fixed value as well. Using the same arguments as above, one can again view the total base minority charge and its time derivative as quantities that (with other charges) control the base current. To be able to make a similar statement regarding the junction voltages and the collector current a partition of the base charge has to be made; two possibilities exist as shown in Figs. If and lg. The two parts in Fig. If correspond to a forward and reverse amplifying mode; the charge in each pattern then can be assumed to control a junction voltage and a current, the collector current being the difference of the two separate current contributions. In Fig. lg the triangular pattern in itself would represent the situation when saturation has just started (at zero collector-base bias). Recalling that in saturation the collector current is wholly determined by the external circuit, the charge contained in it can be considered as controlling the collector current in all stages of saturation. The other contribution is a specific saturation charge; its magnitude will depend on the excess base current supplied after the onset of saturation. Arguments for a control of the junction voltages by the two charges can be found; however, such a control function is not normally used in connection with this particular representation. One of the objects in proposing the charge-control concept was to provide a simple and easy way to analyze amplifying or switching circuits; in other words, to provide the tools for a charge analysis. A set of equations that summarizes the previous discussion in the form of relations between charges and currents then reads as follows:
ic=
Q -C -, TC
iE
+ iB + ic = 0.
(3)
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
263
An npn transistor has been assumed, making the minority charge Q a negative quantity; currents have been taken to be positive when flowing into the transistor. The summation signs express the fact that the charge Q may be partitioned as one sees fit and that time-varying majority charge packets may be added to account for junction capacitances. The base-current contributions due to recombination are entered to a first-order approximation that is as currents proportional to the excess minority carrier charge. The constant zB has the dimension of time; it may have different values in different parts of the base. A similar remark applies to tC . The equilibrium value of any particular charge is represented by Qo . It may be remarked here that in many instances the three equations will suffice for the analysis of switching circuits, a fact that may well have been decisive in the proposal of the charge-control concept. This is all the more plausible because the relevant parameters appeared to be accessible for measurement. Should the partial charges have to be related to junction voltages, one may observe that these charges are proportional (or at least approximately so) to the minority concentrations at the junction boundaries, which in their turn are exponential functions of the bias voltages applied to the junctions. Then for any junction voltage uj , with uT = kT/e the thermal voltage. It must be noted that Eqs. (3) have been put forward solely as an analytical formulation of Fig. 1. They do not hold at zero bias (Q = Qo). To make the equations formally correct one can follow various authors and present the collector current as dependent on excess charges Q - Qo . For all practical purposes the difference is immaterial.
BACKGROUND OF IV. PHYSICAL
THE CHARGE-CONTROL CONCEPT
The presentation of the junction transistor as a charge-controlled device occurred at a time which, in retrospect, can be considered as marking the beginning of an explosive growth in transistorized digital equipment. It is understandable therefore that subsequent publications, including those by the proposers themselves, elaborated almost exclusively on the representation of the switching properties of the transistors and did so in a straightforward manner, defining switching parameters, calculating switching times, and the like. In view of this state of affairs little or no effort was made to found the new principle securely upon basic transistor theory or investigate what limitations might be involved in its applications. Several years passed, therefore, before these points were taken up; review papers (14, 1 5 ) that appeared in 1964 and 1967 may be mentioned in this respect. However, the
J. TE WINKEL
264
main objective of these studies was a comparison of the various large-signal transistor models in existence and the demonstration of their intrinsic similarity; questions of a fundamental nature were discussed only briefly and under simplifying assumptions. As far as the author is aware no discussion in depth has been presented since. Consequently, he feels committed to provide a concise version of such a treatment, which will occupy the remainder of this section. As a first step one has to take the continuity equation (1) for the base majority carriers (holes in an npn transistor) and integrate both sides over the whole transistor volume. For the right-hand side this will result in the sum of the time derivative of the total hole charge and the change of that charge per unit of time due to the generation-recombination process. If a regional approach is adhered to, both contributions can be evaluated for each region separately and summed. For the left-hand side Gauss's theorem can be applied, which equates the volume integral to the surface integral of the hole current density. Provided that the current at the base terminal consists only of holes and that the other terminal currents are carried by electrons only, the surface integral reduces to the base current. If also, as a first-order approximation, the net generation-recombination rate is assumed to be proportional to the excess hole charge, the following charge-analysis version of the continuity equation results: iB =
Q-Qo
1
T
+ cdQX '
(4)
This equation has the same form as the first of the set (3); it now refers to hole charges, however, which also may be located outside the neutral base region. It is understood that the proportionality constant, the mean lifetime T, may have values that differ locally. If one wishes to establish relations with the collector and emitter currents, as is done below, the excess hole charge in the neutral base can be replaced (with a change of sign) by the excess electron charge. The major point concerning Eq. (4)is to note its general validity; there are no restrictions in geometry or current level. The only provision that had to be made in its derivation is hardly ever significant. The next step confines the discussion to the quasi-neutral base region and invokes the restriction to moderate current levels, that is, to excess carrier concentrations so small that the relative change in the majority (hole) concentration profile is negligible and the electric field can be assumed to have a fixed value. The current flowing through the region from emitter to collector will consist of minority carriers and is thus governed by the electron continuity and transport equations (1) or (2). If, in addition, the generation-recombination term is approximated by one proportional to the
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
265
excess electron concentration, two linear relations between electron current and electron concentration are obtained. Consequently, the base region can be regarded as a linear system for which the principle of superposition applies. Hence, within the current limit imposed, one is free to partition the electron concentration pattern in any way that one sees fit in order to simplify the discussion. When doing so, no restrictions are imposed either by the device geometry or by the presence of an electric field. A further consequence of the linearity of the basic equations is that linear relations must exist between the currents entering the base region from the emitter and collector sides and the electron concentrations a t the boundaries: iE = allnE+ al2nC, i, = a z lnE
+ a22nc.
(5)
In general, these relations will deal with frequency- or time-dependent quantities ; the parameters aij then will be either complex functions of frequency or real functions of the Laplace-transform variable s. The charge-control concept can now be presented as an approximate formulation of the set (5) that has been derived by combining the transport equation from the set (1) with the integrated version of the continuity equation (3), both being restricted to the neutral base region proper. The use of the set (l), while deleting the continuity equation, is equivalent to the assumption that the generation-recombination effect is zero and that all quantities concerned are time invariant. In that case the set (5) will reduce to a single equation describing a transverse current that enters and leaves the base region unchanged in magnitude and that is linearly related to nE and n, by real coefficients. A linear relation then also will exist between the current and the total base minority charge. Turning now to Eq. (3), the integrated continuity equation, one may remark that this equation in fact prescribes the difference between the currents entering the base region from the emitter and leaving it to the collector. Hence Eq. (3) can be viewed as providing the correction that has to be applied to the transverse current described above to make good the deletion of the continuity equation from the set (1). The whole procedure can be formulated analytically as follows : ic = A , , QB
.
nE
+ A,2n,
+
= ~ c i c QBs =
iE
+ iB + i,
~
7B
3
~ Q B
QB-QO
Ig
7
+-9
= 0.
dt
266
J. TE WINKEL
Taking into account that Q B in (6) represents a minority carrier charge and that the discussion has been confined to the quasi-neutral base region, the last three equations of the set (6) are identical to those appearing in (3). Thus it may be concluded that the sketch of the charge-control concept given in section I11 is applicable to a range of situations much wider than the one discussed there; the main limitation appears to be the restriction to moderate current levels. The general linear relation between Q B and i, contained in (6) is demonstrated in Fig. lg. One has to identify the charge contained in the triangle with 7, i, and the additional saturation charge with Q B s . It must be noted that in setting up Eqs. (6), the transverse current has been identified with the collector current and that the emitter current has been obtained by applying the correction. This is arbitrary; within the range of validity of the approximation one could just as well have started with the emitter current or one could have chosen some intermediate value for the transverse current and have divided the correction (the base current) into two parts, branching off one part at the emitter boundary and the other at the collector boundary. The approximation involved in the adoption of the charge-control concept can now be formulated. Basically, one neglects the fact that each volume element of the base region contributes to the base current individually and evaluates instead the sum of all contributions by integration of the continuity equation. When relating the base current that is obtained in this manner to the current traversing the base from emitter to collector, one assumes that it is branched off when the latter just enters or just leaves the base. This leads to a transverse current in the base region proper that has a time-invariant relation to the total minority charge. Thus, in applying the charge-control concept, the time that is necessary to adapt the base minority charge to a change in transverse current is neglected.
V. HYBRID-PI EQUIVALENT CIRCUIT AS A SMALL-SIGNAL VERSION OF CHARGE-CONTROL REPRESENTATION
A
When the transistor is operated in the amplifying mode, the chargecontrol relations for the neutral base region, Eqs. (6), can be simplified, as n, = 0 in this case. Thus, as a first step in developing a charge-control equivalent circuit:
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
267
For the relation between base charge and base-emitter voltage, the exponential relation already quoted applies:
~ X (uBE/uT). P One now can follow standard practice by assuming that one is dealing with small time-varying quantities superimposed on much larger constant quantities. Eliminating the charge and changing from the time domain t o the frequency domain two equations relating small-signal alternating currents and voltage result; in complex notation one has QB
= Qo
& -_ -1, ~ B E UT
with I, representing the steady-state collector current. These equations can be interpreted as representing a rudimentary common-emitter equivalent circuit that consists of a voltage-controlled current source and an input circuit composed of a resistor and a capacitor in parallel (Fig. 4).To be able to present the circuit elements by more conventional symbols the following substitutions from charge-control to smallsignal parameters have been made: = g,,, , r8/zc = Po , IC/UT
l/r,
=wT,
the transconductance. the low-frequency current amplification factor; for a well-designed transistor Po 9 1.
(9)
the transition frequency.
The input elements then will have values g i= g,/po and C i= g , , , / q . It is of interest to note that the current amplification outside the low-frequency range will be represented by
= 1 for o 2 wT. so that Invoking Eq. (3), hole charges located outside the neutral base region have to be considered in the second step. First, one will have to deal with the time-varying charges associated with depletion-layer width variations; as discussed earlier they will be adequately expressed by capacitances across the junctions (Fig. 2). As shown, it is worthwhile to separate the junction capacitance Cej from the input capacitance C i ;the value of the latter will depend on I,, in contrast to the first which depends on the junction voltage.
268
J. TE WINKEL
FIG.2. Hybrid-pi small-signal equivalent circuit.
A second hole charge to consider is the one present in the emitter which will function as a minority charge in that region. In complete analogy to the base minority charge, its circuit representation again would amount to a resistor-capacitor combination. This, however, can be combined with the existing pair by suitably altering Po and w T . A third contribution is constituted by the charge made up by the mobile carriers in the collector-base depletion layer; for narrow-base transistors it may no longer be negligible compared to the base minority charge. That charge (consisting of electrons) has its counterpart in a majority charge (holes) located in the base at the depletion-layer boundary. Time variations would, analogous to the variations of the base minority charge, lead to the representation by a capacitance parallel to C i. As before, it can be incorporated in C iby a further change in w T , leaving the circuit configuration of Fig. 2 unchanged. The procedure is an alternative to the one that interprets a varying mobile charge in the collector-base depletion layer as a transport of charge packets resulting in a signal delay. As the last step, a completion of the circuit with representations of the semiconductor-material bulk resistivity and of the base-width modulation effect is necessary. In many instances single resistors in series with the base and collector terminal connections will fulfill the first requirement and a resistor connected between collector and emitter the second. (It can be demonstrated that the base-width modulation mechanism pictured in Figs. l b and l c leads to this result when applied to present-day transistor types). The final result then, shown in Fig. 2, is an electrical circuit that represents the small-signal behavior of the transistor in the common emitter connection. It has been built up by consistently applying the concept that charges control electrical parameters and eliminating those charges afterwards. By inspection one may establish that the circuit is identical to that known as the hybrid-pi equivalent circuit, a representation generally derived by applying the basic equations (1) to the neutral base, making a number of simplifying assumptions (one-dimensional geometry, absence of an electric field, etc.) and adding extraneous effects separately (8). It must be concluded that the hybrid-pi equivalent circuit is nothing else than a small-signal version of an elaborate charge-control model. As the
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
269
exposition shows, the circuit will be applicable over a much wider range of situations than would follow from the original derivation. The general lowlevel injection or moderate current restriction will remain, of course. The typical charge-control approximation prescribing a time-invariant relation between transverse current and base charge has resulted in a frequencyindependent transconductance, in agreement with the first-order approximation used in the direct derivation.
VI. EBERS-MOLL MODELDERIVED FROM THE CHARGE-CONTROL CONCEPT As has been noted earlier, the Ebers-Moll model was put forward as a concise description of the switching mode of operation. Its essential feature is the superposition of two separate transistors which each operate in the amplifying mode and do so in mutually inverse directions. As discussed in Section IV, the use of the superposition principle is quite legitimate regarding minority carrier concentrations and currents; it is limited only by the general restriction to moderate current levels. More specifically, following Fig. If but discarding the linear shape of the patterns, the partition has to be performed in such a way that two patterns result, each having a given value at one boundary and falling to zero a t the other. For each single pattern, Eqs. (7) will apply and the elimination of the charge and the change to the frequency domain could proceed in the manner indicated in Section V. However, the superposition principle now makes it necessary t o consider the common-base connection and to concentrate on the ratio of collector and emitter currents. Taking first that partial charge that represents a transistor amplifying in the forward (emitter to collector) direction (Fig. 3a), one will have -l'"=aF= aOF ~ E F 1 + ~ O F S / W F'
+ +
= 0,
(10) in view of future inverse transforms, the Laplace-transform variable s has replaced jo.Similarly for the second, the reverse, transistor (Fig. 3b): iEF
i,F
+
iER igR
i,F
+ iCR = 0.
270
J. TE WINKEL
I
1
I
FIG.3. Ebers-Moll large-signal model.
Exponential relations exist between the currents and the junction voltages : i,,
=
i,(exp
vBE/uT
-
I),
iCR= iCs(exp uCE/UT - 1).
(12)
Equations (lob(12) may be interpreted as describing the equivalent circuit shown in Fig. 3, amounting to the superposition of Figs. 3a and 3b. Ideal diodes are used to represent the nonlinear relations (1 1). Junction capacitances and bulk series resistances have been added as before. The circuit may be recognized as the basic form of the Ebers-Moll model (10).It follows from the exposition given here that the model has a broader range of validity than the original treatment would suggest. Several remarks should be made. First, the relations (1 1) differ from what could have been obtained by combining (7) and (4). In fact, using the option discussed in Section IV, the base current has been divided into parts: one part, branched off at the emitter boundary, has been assigned to the forward transistor, the other, branched off at the collector boundary, to the reverse transistor. The second remark concerns the tacit assumption that the reverse transistor behaves just like the forward transistor in the sense that it injects minority carriers in the base ; a symmetrical situation that normally occurs with alloy transistors having a collector more heavily doped than the base. The reverse is true, however, for transistors made by the diffusion process; in saturation, injection into the collector region will take place causing significant changes in the concentration patterns there. The situation deserves a separate discussion which will be given in a later section.
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
27 1
The third remark deals with the common-base current amplification factors ciF and ctR occurring in the model. As shown, they are single-pole functions of frequency and the cut-off frequencies w F and oRcorrespond to the transition frequencies that can be assigned to the common-emitter amplification factors BF and &. This is not in accordance with the results of a direct calculation for a simplified situation, e.g., starting with Eqs. (2) and assuming a uniform field, which would result in ci cut-off frequencies in excess of wF and o, and excess phase factors in ci (16).The discrepancy can be traced back to the main approximation in the charge-control approach: the time-invariant relation between transverse current and base charge. When necessary, corrective changes may be applied, of course.
VII. LARGE-SIGNAL CHARGE-CONTROL MODEL
When having to deal with the charge-control description of the largesignal behavior of the transistor, the reviewer is faced with the difficulty that the content of that particular model has changed considerably in the course of time. It must be recalled first that Beaufoy and Sparkes in their classical paper ( 1 2 ) did present two ways of partitioning the base charge, one according to Fig. lf, the other according to Fig. lg. Only the latter was elaborated on, however, which was also the case in a number of succeeding publications. As was mentioned in section 111, this representation stresses the relations between charges and currents and is therefore not very well suited to be expanded to a circuit model, required to cover relations to the applied voltages as well. It was realized that the first-mentioned way of partitioning did not have this drawback and that on these grounds the charge-control concept could be used effectively in providing a physical background to the Ebers-Moll model (14). The equivalence of both approaches could also be demonstrated (15). The discussion in Section VI has, in fact, summarized these points. A specific charge-control equivalent circuit can be developed if the forward and revere partial charges Q B F and QBR are taken as independent variables. Currents and voltages can then be expressed by two sets of chargecontrol relations analogous to the set (3) ( 1 5 ) :
.
kF
QBF
= __ TCF
7
J. TE WINKEL
272 and
The equivalent circuit representation of (13) is shown in Fig. 4; it needs an additional network element-the store-which is defined as follows: (a) the element represents a stored charge Q, (b) the current through the element
8
FIG. 4. Charge-control large-signal equivalent circuit with charges as independent variables.
is equal to dQ/dt, (c) the voltage across the element is identically zero. The circuit has been completed with bulk resistances and junction capacitances as before. A further development is possible when one again eliminates the charges and takes the transverse currents themselves as independent variables; by definition then: and ii = QBR/ZcR (14) (subscripts N and I refer to normal and inverse operation; note that iN and i, differ from the forward and reverse currents iEF and iCRappearing in Fig. 3). It makes sense to replace the specific charge-control parameters by their small-signal equivalents, using Eqs. (9). The resulting circuit is shown in Fig. 5. The voltage sources can conveniently be related to the currents iN and i I by putting i~ = Q B F / k F
. =QOF zON ZCF
and
QOR iol = -, ZCR
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
273
8. 8
FIG. 5. Charge-control large-signal equivalent circuit with currents as independent variables.
thereby changing the set (13) into iN
= iON(exp VEB/VT
- I)?
i, = iol(expvCB/UT - 1). (15) Pursuing the transformation of the charge-control circuit, one may next incorporate Eqs. (15) in the circuit representation by regarding them as related to a set of ideal diodes characterized by the intercept currents iONand io, (“intercept refers to extrapolated values of iNand il for uEB = uCB = 0). A similar set with intercept values iON /POFand iol /PoRcan represent the recombination currents flowing from emitter to base and from collector to base, respectively. These transformations also allow the replacement of the stores by ordinary capacitances whose values will depend on iN and il . Introducing, for convenience, the characteristic transition frequencies oF= 1 / T c F and wR = l / ~ , -the ~ , capacitance values can be derived from (13)-(15) as ”
c
ed
=2 dQ - i N duEB
+ iON
~ F v T
’
The final result is shown in Fig. 6. It must be noted that the junction capacitances Cejand CCjparallel to Cedand c , d have to be drawn separately since they are voltage dependent. Bulk series resistances have been added as in previous circuits. The equivalent circuit obtained may be regarded as the present-day version of the charge-control description of bipolar transistor behavior (17, 18).
274
J. TE WINKEL
FIG. 6. Charge-control large-signal model for circuit analysis.
VIII.
CONCLUDING
REMARKSREGARDINGCHARGE-CONTROL MODELSIN GENERAL
Summarizing the discussion in the preceding sections one can state that: (i) The known transistor models (the -hybrid-pi and Ebers-Moll equivalent circuits, Figs. 2 and 3) can be developed in a straightforward manner from the basic principles of control by charge. (ii) The consistent application of the charge-control concept will lead to equivalent circuit representations like those in Figs. 4-6; one has the choice either to present the charges explicitly and accept nonstandard circuit elements or to eliminate the charges and obtain a compatible representation. In the contemporary literature (17, 28) one finds some contracted versions of Fig. 6 presented as the nonlinear equivalent circuit, omitting any further specification. In the opinion of the reviewer this is not a commendable practice. Even when recognizing that charges no longer occur in the circuit, Fig. 6 must be presented as a charge-control equivalent circuit or model; adding, if one wishes, the qualifications nonlinear or large signal. In doing so, the physical principles underlying the model are brought forward and the user is alerted to the basic approximations involved. The reviewer might also point out that a much clearer and more unified presentation of the hybrid-pi and Ebers-Moll equivalent circuits could be obtained if these circuits were regarded as modifications of the general charge-control model. It may easily be verified that Fig. 2 is a linearized version of Fig. 6 under the condition that the base-collector junction is reversely biased; and that Fig. 6 reduces to Fig. 3 if the currents injected into the base iEF and iCR are taken as independent variables instead of the transverse currents iN and i, . The change has also been expressed as one from a transport model to an injection model (1 7).
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
275
IX. LINVILL LUMPEDMODEL In 1958 an alternative modeling procedure was introduced by Linvill (19, 20). Having its origin in a university environment, it reflects the need, existing at that time, to teach transistor fundamentals to an ever growing number of engineering students. Basically, then, the lumped transistor model can be considered as an attempt to facilitate the acceptance of new and unfamiliar concepts by pointing out their similarity to wellestablished ones. As a logical consequence the similarity was turned into an analogy and subsequently into an equivalent circuit or model. The brief exposition that follows has to start by writing down the continuity and transport equations related to the electrons (minority carriers) in the transistor base. It is assumed that the carrier flow is one dimensional and that the electric field is absent; as a first-order approximation the generation-recombination contribution is taken as being proportional to the excess electron concentration. To emphasize currents instead of current densities the equations are multiplied by the base cross-sectional area A . From ( 2 ) one then will have
din -
ax
dn n - no =eA -++A-----, at T8
dn in = eAD, - . ax The crucial point is that similar relations also will apply to an R-G-C transmission line. The differential equations applicable in that case may be derived from the elementary section of this line shown in Fig. 7a. For a line of length I having a total series resistance R, total parallel conductance G , and a total parallel capacitance C they read
_ -ai- --- cav+ G 0, ax i at I -
Transmission-line theory will provide an approximate solution to these equations that, in the form of an equivalent circuit, is shown in Fig. 7a. It may be regarded as consisting of two “lumps”, each composed of a resistance, a conductance, and a capacitance. Exploiting now the analogy between the two sets of equations, a similar and also approximate representation can be drawn for the transistor base, Fig. 7b. The terminal quantities are now the currents entering the base from the emitter and the collector and the negative of the excess electron concentrations at the boundaries; the
276
J. TE WINKEL
FIG.7. Illustration of the Linvill-lumped model.
network-like elements will get new names and new symbols: one has the “diffusance” H,, the “combinance” H,, and the “storance” S. Taking the base width as W , it is easily established from the analogy that
H, =
eAD, ~
w ’
eAW Hc=-, 7B
S = eAW.
It is seen that the lumped model prescribes a transverse current that has a time-independent relation to the concentrations at the boundaries as well as storage and recombination currents that are branched off to the base terminal just after entering or just before leaving the base region. Hence, at least for the simplified situation considered here, the lumped model and the charge-control model are identical. The lumping procedure can be extended in various ways. First, also in analogy with a transmission line, several sections like that in Fig. 7b can be cascaded to obtain a more precise representation. Second, the electric field can be accounted for. In the elementary transmission-line element (Fig. 7a), a voltage-controlled current source then has to be added parallel to the resistance R dx/l. Although the transmission line itself will no longer resemble any physical structure, the general theory can be applied and a first-order approximation taken. The resulting two-terminal-pair network will differ from that shown in Fig. 7a in the sense that it is now nonsymmetrical and nonreciprocal. In the transistor analog a fourth network-like element has to be defined: the “driftance” F to be connected in parallel to H , .
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
277
The current through that element is understood to be F times the sum of the excess charges at the terminals, with
F = eAp,E. The other elements in the analog will be constituted by storances and combinances as before; the values on either side must differ, however. Using the same arguments as before it may be seen that also in this more general situation the charge-control and lumped approaches lead to identical models. In conclusion it may remarked that the lumped model has the merit that it clearly shows the nature and the extent of the approximation that is involved and thus, implicitly, also of the approximation in the chargecontrol models. As a means to establish a complete circuit model, the lumped approach must be considered inferior to the charge-control approach ;the derivation is somewhat cumbersome involving newly defined elements. Moreover, these parameters are expressed in terms of physical quantities not readily accessible to measurement, in contrast to the chargecontrol parameters which are electrical quantities.
X. MISCELLANEOUS APPLICATIONS A. Self-Analog or Separation Technique
O n inspecting the analog or circuit model shown in Fig. 6, one may note that the frequency or timedependent behavior is wholly determined by the four capacitances shown. It is then possible to describe its behavior on a contracted frequency scale or on an expanded time scale by multiplying each capacitance by the same given factor. In this situation, provided that the factor is chosen large enough, the actual transistor could serve as a true and complete representation, linear as well as nonlinear, of the time-independent portion of its own model; in other words, it could act as its self-analog. If one could find a suitable scaling method, the scaled capacitances could be added externally and then provide a simulated transistor that can be used for analog-computation purposes (21-23). The process, in fact, separates timedependent and time-invariant behavior, hence the second name. The determination of the added capacitances is simple as far as it concerns the junction capacitances ; ignoring voltage dependence and assuming the series resistances to be sufficiently small, fixed capacitors of the proper value can be connected between the base terminal and the collector and emitter terminals. With respect to the other capacitances, one can invoke the charge-control concept and recall that the charges that they contain will be proportional to direct currents defined in the model. These currents can, in
278
J. TE WINKEL
principle, be determined by measuring or sensing the terminal currents of the actual transistor. Suitably amplified they can next be used to inject properly scaled amounts of charge into capacitors connected across the relevant transistor terminals. A simple example of the technique is shown in Fig. 8. It is assumed that the transistor is described by a model in which the base current is branched off from the emitter current when that current just enters the base region.
FIG.8. Transistor self-analog.
One then has to deal with the total base charge QB = tg iB only. As shown, the base current is sensed across a small resistance r. The amplifier (amplification factor G) provides a voltage source QBrG/tB to charge the external capacitor C , , making the scaling factor equal to
K
= rGC,t,
For the practical realization, one will need an amplifier with a lowimpedance output and an adjustable amplification factor in which the input and output circuits are virtually disconnected. An operational amplifier with voltage feedback will fulfill this requirement. B. Lightly Doped Collector
As has been remarked earlier, modern doublediffused transistors will show a different saturation behavior when compared to the earlier alloy types, because of the fact that the technological process leads to a collector that is more lightly doped than the base. The physical aspects of the changed condition have been discussed in a number of publications (24-26). Bearing in mind that the charge-control and related models were conceived at a time
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
279
when the alloy type was predominantly in use, a reconsideration of the applicability of the concept to saturated double-diffused transistors seems necessary. The discussion of the phenomenon will be based on the sketch of a typical concentration pattern shown in Fig. 9 (derived from a computer simulation). It is seen that the partition of the total charge turns out differently here. The forward charge can be assumed to reside in the base region as before, while the reverse charge will be mainly located in the
-distance
(pm)
(a/ FIG. 9. Carrier concentration patterns (a) and electric field (b) for a doublediffused transistor in saturation.
epitaxial collector region. For the forward charge the known relations to terminal currents and voltages will apply; the reverse charge, however, corresponds to a concentration pattern that differs appreciably from the ones discussed so far, demanding a further investigation into the charge-control relations applicable to this particular case. The patterns shown in Fig. 9 depict the high-level injection situation that generally prevails in the collector region, that is, the minority (hole) and the majority (electron) concentrations are many times larger than the donor concentration N ; . The holes will constitute the main part of the reverse charge as understood in the charge-control concept, the hole concentration is seen to drop from a value p o at the metallurgical junction to zero at some point in the collector region. Quasi-neutrality will again exist in the part of the collector occupied by the injected carriers (n - p E N ; ) , leaving the electric field concentrated in an ohmic conduction region occupying the
280
J. TE WINKEL
remainder. An externally applied voltage between collector and base terminals then will be found nearly wholly across the ohmic region, thereby determining the extent of that region and thus, with the current, controlling the stored charge. The relations between the various quantities have been worked out in Reference (27); for the high-level injection situation described above ( p o B N A ) one has
QRic
= D,(eN,A)2
(
uB'C;,
'.
Referring now to Fig. 4 one may identify Q R with the reverse base charge QBR in the conventional charge-control equivalent circuit. The current i, in (21) represents the current crossing the base-collector junction; referring now also to Fig. 5 one will have
.
1,=1jq-1,=---.
QBF
QBR
TCF
TCR
The quantity uB,,, represents a fictitious internal junction voltage uBC,
= Rcic
- uCB .
Here uCB stands for the externally applied voltage and R, for the collector series resistance pertaining to the whole region, that is, taken from the metallurgical junction to the external connection. Finally uo will represent a fixed built-in voltage, with n, the intrinsic carrier concentration : uo = 2 u T ( 1 n, n 5- N 5 A ).
Turning now to the general charge-control equivalent circuit (Fig. 4),it may be concluded from the above that for the case of the lightly doped collector the circuit configuration can be maintained but that the exponential relation between ucB and QBR has to be replaced by a much more complicated one, involving also Q B F . As a further investigation will show, the proportionality between QeR and i, also becomes questionable, which would necessitate ?cR to be current dependent. A small-signal equivalent circuit that incorporates the findings of the investigation described here has been published (27), as well as a study of the behavior of the reverse charge in the time domain (28). As far as the author is aware, no investigations have yet been made into a modification of the general large-signal equivalent circuit (Fig. 6). However, practical experience has indicated that the original version still can serve as a useful firstorder approximation provided that one chooses the reverse parameters for a best fit and not for physical relevance (17).
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
28 1
It must be noted that the simple division of the collector into a quasineutral and an ohmic region is not generally valid. In specific situations (lightly doped relatively thin epitaxial collector layers) a space-charge region with its associated electric field will occupy the largest part of the collector. It appears that four distinct models are necessary to describe the injection phenomenon completely (29). Each model will lead to a different relation between reverse charge and terminal voltages and currents. The reasoning given above is also applicable to a double-diffused transistor that operates in the reverse mode by changing the functions of collector and emitter. Such a situation occurs in a particular class of switching circuits that has gained importance recently and that is known by the name “integrated injection logic.” The collector-basejunction is then biased in the forward direction and emitter and collector currents flow in a direction opposite to the one considered thus far. A charge-control equivalent circuit in the sense of Fig. 4 is applicable, with the forward charge made up by the holes in the lightly doped collector and the reverse charge by the excess electrons in the base. A calculation of the switching times from the concentration patterns is then possible (30). C . Gummel‘s Charge-Control Relation
The usefulness of the conventional charge-control concept can be increased considerably by the addition of a new general relation between stored charge and terminal voltages and currents first described by Gummel (31). The procedure will remove one of the major restrictions to the charge-control approach, the assumption of a low-level injection situation. In the derivation of the new relation the main assumption of the chargecontrol concept is maintained. The time-invariant and time-dependent contributions to the base-current are considered negligibly small compared to the current traversing the base region so that they will not affect the charge concentration patterns. This leaves a constant time-invariant current traversing the base and adjacent regions. The current density will follow from the transport equation; assuming a onedimensional flow and an npn transistor one will have j,
= ep,E
+ e D , ddn --. x
For constant j , this is a differential equation in n that can be integrated over the whole length of the transistor to yield the new charge-control relation. It is convenient, however, to exclude the ohmic regions in the emitter and
J. TE WINKEL
282
collector from the integration path and represent these by currentdependent additional series resistances. After some manipulation the following relation results: I,, = l o -
. QQ, po(
.
2- “‘i
exp-
- exp-
or
.
Thus, within the assumptions quoted (and applying some other very minor restrictions), a very simple relation is seen to exist between the transverse current i n , the applied voltages corrected for the ohmic drops uEB and uCB, and the charges Q, and QPo. These charges now refer to the total hole (base majority) charge contained in the region covered by the integration (not necessarily the base region proper); QPorepresents the value that Q, takes for zero bias. Equation (22) can also be written in the form
.
1,
.
= i N - i, = io
Qpo ~
Q,
(
)
OEB exp - 1 - io
OT
::( ”u”,”
)
exp - - 1 ,
~
(23)
which shows that the partition of the transverse current into a normal and an inverse component, an essential feature in the charge-control models discussed earlier (Figs. 4 6 ) , can be maintained. However, the normal and inverse currents now contain the ratio QPo/Q, as a multiplying factor. In the low-injection situation this quantity is unity, but high injection levels and base widening will lead to appreciable differences between Q, and QPo. A study of the dependence of Q, on terminal currents and vohages then will make it possible to include these effects in the charge-control approach and to establish what has been called the “ integral charge-control model.” A detailed study of the modeling procedure with the help of the new additional equation (23) has been given in Reference (32). It is shown there that a satisfactory representation of the high-injection phenomena can be obtained if, in spite of the fact that the linearity and superposition arguments no longer apply, the base-charge partitioning method used in the lowinjection case is retained. In particular, the concept of a forward and a reverse excess charge proportional to the normal and inverse current, respectively, remains usable. If, for circuit analysis purposes, one should wish to use (23) in conjunction with an equivalent circuit of the type shown in Fig. 6, it follows from the structure of that equation that one has to replace the ideal diodes by less than ideal ones whose current-voltage relations are (33) iN
= ioNG(exp$
Q,
i, = i,,&(exp$
Q,
-
I),
- I).
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
283
The incorporation of hot carrier effects and collector avalanche multiplication in the integral charge-control model are discussed in References (34) and (35),while a detailed description of the extraction of model parameters from measurements is contained in Reference (36).
D . Extensions of the Charge-Control Concept for Improved Time-Dependent Response One of the basic approximations made in the charge-control approach is the lumping of time-dependent contributions to the base current into two currents branched off at the junction boundaries. In connection with the small-signal representation it has been noted that this would cause nonnegligible discrepancies at high frequencies. A brief discussion of extensions of the charge-control concept aimed at improving the time-dependent behavior would seem to be in order. For one of the proposed extensions, known as the Narud-Meyer model (37-39), one has to return to ( 5 ) and regard these equations as a fourpole-like set linking currents crossing the junctions with carrier concentrations at the junction boundaries. With suitable multiplying factors a similar set would link currents and partial charges. In Section IV the coefficients aij were taken to be constant-a zeroth-order approximation which was corrected later. As an alternative one may recognize their frequency or time dependence and represent them by the first-order terms of power-series expansions as follows :
a21= -,A2 1 1 ST21
+
It is easily shown that the quantities t l l and t 2 2represent the correction refered to, leaving z l 2 and T~~ as additional parameters that can provide the improved time or frequency response sought for. A systematic study of higher-order approximations has been presented in Reference (40). It must be noted that, due to the fourpole type of formulation, it is not possible to represent the extension by a slight modification of an equivalent circuit of the kind shown in Figs. 4 and 6. The Linvill model (Section IX) can offer a second possibility for extension if more than one lumped section is used to represent the transmission-
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J. TE WINKEL
line analog. This again amounts to a fourpole-like representation, a rather cumbersome one, not very well suited for practical application. A third possibility for extending the charge-control concept (41) becomes apparent when it is realized that the fundamental difference between the various models and the actual transistor is that between a lumped and a distributed ladder network. In the language of electrical network theory this could be described as the difference between a minimum-phase and a nonminimum-phase network. Thus the extension envisaged might take the form of an additional phase shift of the transmitted electrical signal. To a first approximation, the shift would be proportional to frequency and therefore, in terms of the time domain, it would amount to a signal delay of fixed value. Turning to the charge-control approach, it appears upon closer investigation that such delays should be incorporated in the relations between controlling and controlled quantities. The charge-control equations for the base region (13) would then change to
and three similar relations for the reverse quantities. It can be demonstrated, by comparison with closed-form analytical solutions of the base transport equations, that (26) indeed provides a much improved description of timedependent behavior ; expressed in terms of the frequency domain, the improvement would amount to an extension of the useful frequency range well beyond the transition frequency wF = l/tCF.It also appears that the values of the delays t l Fand t Z F can be determined from 7CF and the electric field; the field dependence is rather weak, however, so that in many instances the zero-field value &F may be used as a default value for both 7 1 and ~ 7ZF. XI. CHARGE-CONTROL CONCEPT IN THE COMPUTER-AIDED ANALYSIS OF
TRANSISTORS AND
CIRCUITS
A. Transistor Analysis
When digital computers of sufficient size became available and suitable advanced computational techniques were developed, a new and different way for analyzing transistor behavior was opened. The major approxima-
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
285
tion that had prevailed in the classical approach, the division of the transistor into space-charge and neutral regions, could be abandoned in favor of a procedure that considered the device as a single structure completely described by the five basic equations (1) which now were to be solved by direct numerical methods; starting from a given, possibly also computed, donor and acceptor concentration pattern. As might be expected, the new approach evolved in a number of successive steps, the situation handled increasing in complexity each time. Thus the first published analysis concerned steady-state one-dimensional systems only, that is, solutions of the set (2) with time-dependent terms deleted. In later developments time dependence was taken into account as well as refinements like the dope and field dependence of mobility and a detailed representation of the generation-recombination mechanism. The latest achievements in the field of computer simulation concern a two-dimensional analysis of the steady state including the refinements indicated (42). However, considering the computing time and memory space needed in this type of simulation, it would seem highly improbable that, even with refined techniques, two- or three-dimensional time-dependent simulations could become a routine operation. Here the charge-control concept can be extremely useful through its intrinsic property of providing a partial extension from steady-state to time-varying behavior. Following the treatment in Section IV, one regards the time-varying charges, applied voltages, and transverse currents as generated by a succession of steady-state solutions and concentrates the specific time-dependent effects in a base-current contribution that is equal to the time derivative of the charge made up by the carriers that communicate with the base terminal (holes in an npn transistor). That charge and its dependence on the other quantities also can readily be determined from the computation. The method described also allows the determination of the parameters of the small-signal equivalent circuit (Fig. 2) directly from the steady-state simulation. It can easily be shown that the transition frequency that can be assigned to the overall common-emitter current amplification factor can be found from the ratio of small increments of the hole charge referred to above and the collector current:
Similarly, the various capacitances in the equivalent circuit can be determined by taking the ratios of small increments of charge and voltage. A meaningful division of the total charge into parts associated with the junction and quasi-neutral regions must be possible, however.
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J. TE WINKEL
B. Analysis of Circuits Containing Transistors The availability of large digital computers has also led to significant changes in the procedures applied to the design and evaluation of complex electrical networks. It was realized that an advantage in time and effort could be gained if the conventional method of making repeated measurements on a breadboard model were replaced by optimization procedures performed on a simulated network with the help of a computer. The interest in the new technique grew with the breakthrough of a new development in semiconductor device technology : the integrated circuit which, for the present purpose, can be regarded as a functional unit consisting of a large number of active and passive circuit elements concentrated on a single chip of semiconductor material. As the technological development proceeded, it soon appeared that an optimally designed integrated circuit would require a reconsideration of the configuration of the electrical circuit itself. The chip area would be used most economically if the circuit function were performed by a large number of active elements (transistors and diodes) and by the smallest possible number of passive elements (resistors and capacitors). The development sketched above, besides again stressing the need for computer-aided analysis and design, also points to the desirability of a transistor model or equivalent circuit that can be used in conjunction with a general network analysis program. The charge-control concept has proved to be a valuable tool here also; all models that have been proposed for the purpose of computer-aided design belong to the general class of chargecontrol models discussed in the preceeding sections. The proposals made consist of an extended version of the Narud-Meyer approach (Section X,D) (38, 39), the Ebers-Moll or injection model, and the full-scale chargecontrol or transport model (17). Experience has shown that the transport model, here represented in equivabt circuit form by Fig. 6, is to be preferred. The following reasons have governed the choice : (i) The equivalent network is a simple one and fully compatible with existing network-analysis programs ; in particular, it exhibits controlled sources that depend directly on controlling currents without a time or frequency dependence intervening. (ii) The equivalent network has been derived by physical reasoning; it might be expected, therefore, that the parameters that ultimately determine the electrical behavior will be more closely related to technological parameters than otherwise would be the case. That property would also be a valuable one when the parameters are determined by measurement or when a statistical analysis or design procedure is envisaged (17). One might further expect that the dependence of those parameters on temperature and on
CHARGE-CONTROL CONCEPT OF BIPOLAR TRANSISTORS
287
terminal voltages and currents would follow the simplest possible laws. Careful analysis of measured data confirms the expectation (17). (iii) It also follows from the physical nature of the charge-control model that contracted or expanded versions of the basic circuit, as presented here in Fig. 6, can be constructed without much difficulty. In this way, in largescale programs it would be possible to trade accuracy for simplicity (or computing speed and cost). For instance, if very-low and very-high currents can be excluded from the range of operation, the two ideal diodes determining i, and i, (Fig. 6) and the two current sources can be combined into a single source i, - i, (18). On the other hand, a very large range of currents can be accommodated if Gummel's integral charge-control relation (Section X,C) is incorporated in the model and if recombination occurring outside the neutral base region is accounted for. The first object is fulfilled when the ideal diodes determining i, and i, are replaced by nonideal ones behaving according to (24), the second by adding further nonideal diodes between emitter and base and between collector and base, the diode currents now figuring as exponential functions of uEB/nuT and ucB/nuT, 1 < n c 2. The expansion of the basic charge-control circuit can be performed in a systematic manner, making only those additions required for a given current level (43, 44). Summarizing, it can be stated that the consequent application of the charge-control concept has led to the development of a series of efficient bipolar transistor models that are fully compatible with standard networkanalysis programs. Due to the simple topology of the model and the limited number of parameters (about 40 for the most elaborate version, inclusive of the parasitics in an integrated-circuit environment) no excessive demands will be made on the memory capacity of the computer.
XII. CONCLUSION This review has attempted to sketch the growth to maturity of a basic concept in bipolar transistor theory. In doing so, several aspects besides the purely historical one have been given attention. First, the characterization aspect that has led to the charge-control model or equivalent circuit as a concise and efficient way to present the electrical properties of a transistor. Second, the communication aspect in which the charge-control model, just because of its derivation from physical principles, functions as a common ground for discussions between circuit designers and device designers. A similar remark applies to teaching in the field of transistor technology and transistor circuitry. A final remark should be made in connection with the accuracy of the charge-control approach. As has been discussed, frequency- or time-
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dependent behavior generally is represented by way of a first-order approximation only; improvements are possible and have been discussed as well. A full assessment of the validity range of the original and extended versions of the charge-control concept remains to be made, however. Simulation techniques and the help of a digital computer may provide the answer.
REFERENCES* 1. J. Bardeen and W. H. Brattain, Phys. Rev. 75, 1208 (1949). 2. W . Shockley, Bell Syst. Tech. J . 28, 435 (1949). 3. W . van Roosbroeck, Bell Syst. Tech. J . 29, 560 (1950). 4. W. Shockley, M. Sparks, and G. K. Teal, Phys. Rev. 83, 151 (1951). 5. H. Kromer, Arch. Elektr. Uebertragung 8,223-228,363-369, and 499-504 (1954); See also “Transistors I,” p. 202. R.C.A. Labs, Princeton, New Jersey, 1956. 6. J. M. Early, Bell Syst. Tech. J . 32, 1271 (1953). 7. R. L. Pritchard, Proc. IRE 42, 786 (1954). 8. L. J. Giacoletto, R C A Rev. 15, 506 (1954). 9. J. J. Ebers and J. L. Moll, Proc. IRE 42, 1761 (1954). 10. J. L. Moll, Proc. IRE 42, 1773 (1954). 1 1. R. D. Middlebrook, Proc. Inst. Elec. Eng. Part B 106,887 (1959). 12. R. Beaufoy and J. J. Sparkes, Automat. Telephone Eng. Journal 13,310 (1957). 13. J. Zawels, R C A Rev. 16, 360 (1955). 14. D. J. Hamilton, F. A. Lindholm, and J. A. Narud, Proc. IRE 52, 239 (1964). 15. D. Koehler, Bell Syst. Tech. J . 46,523 (1967). 16. D. E. Thomas and J. L. Moll, Proc. IRE 46, 1177 (1958). 17. J. Logan, Bell Syst. Tech. J . 59, 1105 (1971). 18. J. Logan, Proc. IEEE 60, 78 (1972). 19. J. G. Linvill, Proc. IRE 46, 1141 (1958). 20. J. G. Linvill and J. F. Gibbons, “Transistors and Active Circuits.” McGraw-Hill, New York, 1961. 21. H. K. Gummel and B. T. Murphy, Proc. IEEE 55, 1758 (1967). 22. B. T . Murphy, Bell Syst. Tech. J . 47, 487 (1968). 23. E. J. Angelo, J. Logan, and K. W. Sussman, IEEE Trans. Comput. 17, 113 (1968). 24. C. T. Kirk, Jr., IRE Trans. Electron Devices 9, 164 (1962). 25. L. A. Hahn, Proc. IRE 55, 1384 (1967). 26. J. R. A. Beale and J. A. G. Slatter, Solid State Electron. 11, 241 (1968). 27. H. C. de Graaff, Philips Res. Rep. 26, 191 (1971); Electron. Lett. 7, 73 (1971). 28. W . J. Chudobiak, IEEE Trans. Electron Devices 17, 843 (1970). 29. H. C. de Graaff, Solid State Electron. 16, 587 (1973). 30. F. M. Klaassen, IEEE Trans. Electron Devices 22, 145 (1975). 31. H. K. Gummel, Bell Syst. Tech. J . 49, 115 (1970). 32. H. K. Gummel and H. C. Poon, Bell Syst. Tech. J . 49,827 (1970). 33. B. R. Chawla, IEEE J. Solid State Circuits 6,262 (1971). 34. G. Persky, Bell Syst. Tech. J . 51, 455 (1972). 35. H. C. Poon and J. C. Meckwood, IEEE Trans. Electron Devices 19,90 (1972). 36. H. C. Poon, IEEE Trans. Electron Devices 19, 719 (1972).
Note: extensive bibliographies are contained in Refs. 14, 15, and 32.
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37. C. S. Meyer, D. K. Lynn, and D. J. Hamilton, “Analysis and design of integrated circuits,” Chap. 4. McGraw-Hill, New York, 1968. 38. F. A. Lindholm, Solid State Electron. 12, 831 (1969). 39. F. A. Lindholm, 1 E E E Trans. Circuit Theory 18, 122 (1971). 40. W. L. Engl and J. K. Kioustelidis, Solid State Electron. 12, 239 (1969). 41. J. te Winkel, I E E E Trans. Electron. Devices 20, 389 (1973). 42. J. W. Slotboom, I E E E Trans. Electron Devices 20, 669 (1973). 43. F. A. Lindholm, S. W. Director, and D. L. Bowler, I E E E J . Solid State Circuits 6, 213 (1971). 44. J. G. Fossum, 1EEE Trans. Electron. Deuices 20, 582 (1973).
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Microwave Power Semiconductor Devices. I Critical Review S. TESZNER Centre National #Etudes des Telecommunications, France AND
J. L. TESZNER Direction des Recherches et Moyens d'Essais, Paris, France
Introduction .................................................................................... 291 Two-Terminal Devices ................... 293 A. Basic Considerations .................................... 293 B. Transferred Electron C. Transferred Electron .................. 301 References for Section I 11. Junction Diodes .........................................................
References for Section II,A ........................................ B. Punch-Through BARITT Diodes ......................................................
352
............... 364 C. Varactor Diodes ................................................................ ............................ 311 References for Section II,C
Introduction The field of microwave power semiconductor devices has undergone far reaching and swift expansion recently. Both higher frequencies and power levels have been reached. Moreover, new development prospects have arisen, and it is not unreasonable to envisage a long term possibility of almost total replacement of vacuum tube devices by semiconductor devices, even in the microwave power field. Mainly two types of structural configuration have been developed: two terminal and three terminal, the latter may be extrapolated sooner or later to 29 1
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S. TESZNER AND J. L. TESZNER
a greater number of terminals. Three types of application are concerned-(a) oscillators as power sources, (b) amplifiers, and (c) switching devices. The two-terminal devices are mostly used to make oscillators. However, with certain provisos, they may also be used for amplifiers or switching devices, although the circuits involved may increase in complexity. On the other hand, the three-terminal devices, presently operational, may be used either for oscillators or amplifiers or for switching devices, without any increased circuit complication. These structures, however, are inherently more complex than the two-terminal ones. The first category includes: (a) Bulk diodes, utilizing the Gunn effect, presently named transferred electron devices (TED); and (b) Diodes containing one or more rectifying junction (np, metal-semiconductor, metalinsulator-semiconductor junctions): (1) Avalanche diodes with one rectifying junction-the so-called IMPATT-mode and TRAPATT-mode diodes; (2) Punch-through diodes with two back-to-back rectifying junctions-the so-called BARITT diodes, the development of which in the power field is presently at the laboratory stage; and (3) Varactor diodes with one rectifying junction, where use is made of the junction capacitance variation with the applied voltage to provide a nonlinear element. The second category includes: (a) Bipolar transistors ; and (b) Fieldeffect transistors. Each of these groups of devices comprises several types of structure. We will obviously only be considering those devices already operational, at least at a laboratory stage, in the microwave power field. However, it is important to note that there is no prejudice against other devices currently undergoing preliminary developmental work. For all the devices considered we shall try to explain the basic principles, to set forth briefly the theory (emphasizing in particular the physical mechanism of their operation, while keeping the analytical developments to the necessary minimum), to outline generally the manufacturing processes, and finally to specify the present state of the art concerning characteristics and performance levels thus far reached. Each section will end with a brief discussion of advantages and disadvantages, as well as of the apparent limitations, of the devices under consideration, from a point of view of principle and from a technological aspect. Moreover, the concluding section of this review will be devoted to an outline of present trends in the evolution of power semiconductor devices in the microwave region as well as our personal views on future developments in this field. This review has been split into two parts: Part I concerns the two-terminal devices and is presented in this volume; Part I1 includes the three-terminal devices and the conclusion and is planned for a subsequent volume.
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
293
Two-Terminal Devices I. BULKDIODES A. Basic Considerations Transferred electron devices (TED) have been extensively studied since the discovery of microwave oscillations in GaAs and InP by J. B. Gunn (I) in 1963. Twelve years after this discovery, it is worthwhile looking at what has been done in this area, what are now the components under development, what their possibilities may be in the semiconductor devices market, and, what incentive still remains for further research. Our perspectives will, however, be restricted to the general aims of this review, i.e., power semiconductor devices. A brief summary of the basic physics of the phenomenon will be given first. The behavior of TED as oscillators (TEO) or amplifiers (TEA) results from the existence of a negative differential resistance (NDR). This NDR is due to the very structure of 111-V compounds such as GaAs or InP. Looking at the structure of the conduction band of GaAs with its two valleys, Ridley and Watkins (2) and Hilsum (3) predicted the “Gunn” effect before experimental evidence was obtained. Figure I.la describes the typical form of a two-valley conduction band, the lower valley having a light effective mass (0.07m0for GaAs), the upper one a high effective mass ( 1.2mo for GaAs). As long as the electrons in the conduction band have not gained an energy greater than the intervally gap (A = 0.36 eV for GaAs) they remain in the lower valley. As the applied voltage is increased, they acquire sufficient energy to scatter to the upper valley with a lower mobility. The instantaneous velocity of these hot ” electrons is given by “
with n , ( E ) + n2(E)= no as doping concentration. The V ( E )characteristic may thus be plotted and it is obvious that above threshold level, where E > A, a differential negative resistance should be observed. Numerous computations and experiments have enabled an accurate determination of the shape of this curve. The validity of these computations, however, has an upper limit in the millimeter range since the variations of the electrical field should be low with respect to the intervalley scattering time, estimated at 10- l 2 sec (4). As for the experiments, different
294
S. TESZNER AND J. L. TESZNER
Energy
- - -’- -.?L , ev strong coupling
Conduction ‘aland
I
i Band gap t4OeV I
j
Band gap1.33eV
!
FIG.1.1. Band structure of G a A s and InP.
methods have been used (5-7). For GaAs, resulting curves show good agreement between theoretical and experimental results ;such agreement has not been observed, so far, for InP. In 1970, Hilsum and Rees (7) proposed a more sophisticated conduction band model which should apply to InP. Improvement of the field velocity characteristics (ie., higher peak-to-valley ratio) was to be obtained by the fact that transitions occurred between three levels, the lowest energy level having a light mass, the intermediate level a heavy one, and light coupling with the other two levels. Figure I.lb gives the proposed scheme for a three-level conduction band structure for InP. The three levels r,L, and X are such that the energy separation between any set is greater than 4kT so as to minimize thermal transfer, and the rX intervalley gap is small enough to avoid an avalanching process. With such a structure a field threshold of 4 kV cm-’ was predicted and a peak-to-valley current ratio of 3 : 1. As it appears from Fig. 1.2, experimental results show a
295
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
-\
0
5
',-',
I
I
I
10
15
20
I
E
*
25kVcm-1
FIG.1.2. uE characteristics for InP u-Electron drift velocity; E-Electric field. Experimental and theoretical results. Experimental: E ,-space-charge measures (Boers, 75); E,-microwave 35 GHz (Glover, 76); E,-microwave 9 GHz (Nielsen, 77); E,-domain probe (Prew, 78). Theoretical: T,-2 level theory (Fawcett and Herbert, 79); T,-3 level theory (Rees and Hilsum, 80).
great discrepancy and discussion of these results has appeared in the literature (8). Nevertheless, an experimental ratio of 4 : 1 is obtained whatever the theoretical model. Such a ratio should lead to higher efficiency and the experimental results will be discussed in the following sections. Finally, a brief mention should be given to Ga,In, - ,Sb. Thus far, few results have been published; however, in France a group is actively working on the subject (9) and work is also beginning in Japan (10). The main features of this material should be a very high mobility at room temperature 40,00050,000 cm2 V- sec- ; a predicted high negative differential mobility, 12,000 cm2 V- sec- ; a low threshold field 600-700 V cmSince results have been extensively published on GaAs, this material will mainly be discussed and will serve as a reference for the theoretical discussion. As has been mentioned, when the applied field is such that the threshold field is attained in such a material, a negative differential mobility is observed within the sample. If there is no local perturbation, this structure may continue indefinitely. However, if a high field perturbation exists locally, resulting, for instance, from a doping homogeneity, the electrons in this region are slowed down with respect to electrons outside this region. Consecutively, a local increase of the electron concentration is obtained while a depletion zone at the front end of the perturbation is observed. From Poisson's equation, it may be shown that the electrical field is increased locally. The growth of the electrical field within the perturbed zone, or "domain," is stopped when it reaches maturity. This notion will be explained later.
' '
'
'
-
'.
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S. TESZNER AND J. L. TESZNER
The details of domain growth are outside the scope of the present review. Readers are referred to several exhaustive papers, for instance, the theoretical works of Kurowaka ( 11) and Butcher and Fawcett (12, 13). Within a small signal approximation, the experimental growth time constant is given by
It should be noted that for a local low field perturbation, the value of du/dE would have been positive and the growth evanescent. Considering that the diffusion constant is negligible, the rear end of the domain is much narrower than the front end and it may be shown that, if V,, is excess voltage within the domain, the time derivative of V,, is
Ed being the maximum field within the domain, Eo the field outside the domain, and @ I ), the corresponding electron velocity. This equation has been derived by Butcher, Fawcett, and Hilsum (12), as well as by Kurokawa (11). It shows that the steady state is obtained for d t x / d t = 0 (equal-area value) (Fig. 1.3). The advantage of Kurokawa's method is to show how this steady state is reached. A detailed study of the transient state is given by Heinle (14, Tarnay and Szekely (15) and, more recently, by Slope )I(
I
1
L i m i t operatins paint
traiector
\drumrdDparrtlnsetralectory
FIG.1.3. Velocity-field characteristics. E,, is the steady state field outside the domain; V,, is the corresponding electron velocity; E , is the threshold field for Gunn effect; V, is the corresponding velocity; Edslis the steady state peak field; V,, is the corresponding velocity; V,,, E,,, V,,, Ed, refer to the transient state; y, and y2 correspond to electron mobility, respectively, in the central valley and in the upper valleys. [From Teszner and Boccon-Gibod (16). reprinted with permission].
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
297
Teszner and Boccon-Gibod (16).This transit domain is an important feature of the Gunn effect since it explains that any increase of this time may lead to the fact that the domain never matures. Thus, instead of oscillations, stabilization is realized. The first aspect, transferred electron oscillators, will now be examined. The possibility of stabilization leading to amplifiers will be discussed thereafter.
B. Transferred Electron Oscillators 1. Oscillation Modes
The simplest case, the first observed by Gunn, involves the transit-time domain mode. When the growth time of the domain is less than the transit time of the domain across the sample (supposing the drift velocity of the domain to be equal to the drift velocity of electrons outside the domain), the electric field within the sample will become unstable and traveling domains , obtains the uniwill be observed. Comparing the period v / L with T ~ one dimensional criterion for domain formation : n L 2 1.6 x 10’’ cm-’.
The domain nucleates at the cathode, grows exponentially while moving toward the anode, and disappears at the anode. At that time the condition for creating a new domain is satisfied and so on. During the propagation of the first domain, the field outside the domain drops below the threshold value and again reaches E,, when the first domain disappears. Current oscillations, therefore, are obtained, the period being determined by the transit time of the domain and the peak-to-valley current ratio corresponding to the ohmic current without and with the presence of a domain. The transit-time mode which is detected by oscillations at the load resistance is of no further interest since the related efficiency is negligible. These oscillations are still studied, however, in theory, but are hardly ever used in a practical oscillator circuit. If the resistive load is replaced by a resonant circuit with a high Q factor, the efficiency may be increased considerably. Moreover, resonant operation may entirely modify the oscillation process as has been described by Gunn (1 7). Extinction of domain may now result, not from its arrival at the anode, but from the variation of the applied voltage. In these circumstances the period of oscillation will now be determined by the resonant behavior of the external circuit. Two types of resonant operation may occur related to the mode of domain extinction: the under-voltage resonant mode when it results from a decrease of the voltage under l$, and the over-voltage resonant mode when it results from the fact that the applied
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voltage is too high for steady domain propagation. The two modes have been experimentally observed by Gunn, the latter one being, however, still difficult to explain. The preceding modes are such that the oscillation period is large compared to domain growth time. When the frequency of the resonant circuit becomes comparable to the reciprocal of zg, the device voltage does not stay significantly long above threshold to allow domain formation ; this is called the hybrid mode (18). Multiple domains may coexist in the sample, separated spatially by the transit time per period of oscillation. In contrast with the LSA* mode, which will be discussed subsequently, the traveling domain may form but is cancelled during the passive portion of the rf period. This mode, where there exists a definite space-charge accumulation, is actually a continuum of modes between the transit-time mode and LSA. It has the specific advantages of frequency tunability over a broad band and of wide operation with respect to the rf circuit and load. It is now widely used for TEO. We will now give a brief survey of the LSA mode discovered by Copeland in 1966 (19), the aim being to make a status report after the initial results. By contrast with the previously described modes, the space-charge region is not allowed to grow by means of a superimposition of a sinusoidal waveform on the dc bias. This bias is such that the field within the sample is above threshold. However, the rf voltage swing is such that during a portion of the period, the actual field is below the threshold field value. Under these conditions, the space-charge accumulation region, which is allowed to grow during the first part of the cycle, is subsequently quenched. The growth factor initially given may now be used to determine the necessary condition for the space-charge region to be quenched. The relative growth (or decrease) of the perturbation in the time interval t , - t2 is given by
Between 0 and t , (time for which the actual field becomes less than Eth)the growth factor is G,. During the rest of the period, t,-T, the decrease factor is G,. The product G , G, should be less than one. Moreover,
should be less than
* Limited space-charge accumulation.
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
299
which leads to the condition found by Copeland: 2x
sec cmp3 < n o J < 2 x loi5sec ~ m - ~ .
Under these conditions no space charge is created and the I V curve adequately fits the V ( E ) curve (20). What are the advantages of the LSA mode? 0 The LSA mode should have the most efficient dc to rf power conversion for single-frequency operation. Results will be given subsequently. 0 The frequency of operation is independent of the length of the sample and is determined only by the circuit. 0 An LSA diode can be made much thicker than the transit-time mode, which means that there is no thickness limitation. 0 High-frequency generation may also be obtained up to the V band.? The results, however, are far from the theory and this may be explained by thermal degradation especially after CW operation. The power output and efficiencies of CW LSA diodes are much lower than that of pulsed LSA oscillators. In such conditions highly homogeneous material is necessary and this condition appears to be strongly restrictive for industrial development.
2. EfJiciency Considerations The output power being the real part of the complex product VI*, optimum output is obtained when current and voltage are perfectly out of phase. The simplest waveform which may be used is a square wave where the minimum voltage is equal to the threshold voltage and maximum voltage being thus over y,, avoiding any dissipation resulting from the positive resistance dissipation under v h . However, this does not give the highest value at the fundamental frequency since, for a square wave, current and voltage have odd harmonics which give way to power dissipation. More precisely, the output is
and the power output at the fundamental frequency is poutc
=
Pout(8/aZ)?
t In the LSA mode, hot electrons should relax before a new accumulation layer has had the opportunity to develop. Such a condition should limit the LSA mode in GaAs to 20 GHz. A detailed analysis ( 2 1 ) of wavelength modes of TED has recently been published showing that three regions of operation may exist in the frequency range 10-40 GHz.
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S. TESZNER AND J. L. TESZNER
since the input power isequal to$(Il
+ I,)(
Vl
+ V,), the resulting efficiency is
Taking 11/1, z 2, V, z 3 kV cm- l , V, -+ oc) for GaAs, an efficiency of 26% for the fundamental may be obtained (22). For InP, the corresponding efficiency should theoretically exceed 40%. Experimental results are, however, up to now far below the theoretical levels for both materials. For GaAs, the best results are still of the same order as those obtained by Kino and Kuru (22), i.e., 18%; since they used diodes, the length of which exceeded 200 pm, the maximum value of Vl was severely restricted by avalanching and the theoretical value was not realistic. It should be pointed out that for optimum conversion to the fundamental, the square-wave voltage form is still not the most efficient. Kino and Kuru have shown that the voltage waveform must contain either the fundamental and even harmonics or, at worst, an odd harmonic component, 90"out of phase with the harmonic components of the current waveform. In this case, there will be no real power supplied to the harmonics. These conditions are satisfied by a half sinusoidd voltage waveform. Readers are referred to the original paper by Kino and Kuru (22). The conversion efficiency to the fundamental is now equivalent to the total conversion efficiency for square-wave operation. Similar work has been carried out on the LSA mode by Copeland (23) and Camp (24). Copeland showed that the efficiency and the nlfrange of an LSA oscillator could be extended by using the appropriate nonsinusoidal voltage waveform, with a circuit having an appropriate resistive impedance at just a harmonic frequency and a low impedance of all others. It appears that the greatest improvement is obtained by introduction of the second harmonic up to 12 kV cm- dc bias. For higher bias fields, a third harmonic insertion should be more valuable. Copeland predicted at that time that with the second harmonic present, a 30% efficiency at 11 kV cm-' bias was possible. The approach of Camp is different, since, instead of comparing oscillations with different voltage waveforms, he compared oscillations with different loads. I n his opinion, this method of optimization gives a more meaningful approach to improve efficiency. Using an I V characteristic identical to the v-E curve (which is adequate for the LSA mode) he obtains theoretically and experimentally,a conversion efficiency limit slightly lower than 20% for a ratio peak-to-valley current equal to 2. This result is already well below Copelands predicted values! More recent work on the subject has been carried out by Johnston and Riddle (25). They show that the efficiency may be doubled using reactive terminations for second and third harmonics. As a result, the research on optimum efficiency appears to lead to more and more sophisticated circuits which, in turn, may be questionable from an industrial basis.
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
30 1
3. Temperature Considerations
Because the experimental results are still far below the theoretical predictions, the rather poor efficiency of Gunn diodes leads to the following fact: Most of the dc input power is dissipated in the device as heat. Since it is necessary to avoid any local heating within the active material and because the high and uneven temperature of the device during operation limits its performance, the technological problem of dissipating heat still has to be solved adequately. If this condition is satisfied, higher efficiencies may be obtained since the temperature effect results mainly in a decrease of the mobility, i.e., a decrease in the peak-to-valley current ratio. It is noteworthy to emphasize that, up to now, commercially available TEOs have typical CW efficiencies of 2-3% with occasional results exceeding 10%. Since 1968, different types of solutions have been proposed. Most early Gunn diodes were realized from a n’nn sandwich system on bulk material. However, heat evacuation from the bulk appeared difficult. Coplanar structure was thought to be the solution. For instance, Norton er al. (26) and T. B. Ramachandran (27) proposed in 1968 two different coplanar structures. The first is classical, with an epitaxial active layer grown on a highly resistive GaAs substrate. For a 1 GHz device, the transit length is about 100 pm, and in these conditions “the hottest part of the thin epitaxial layer is little more than 50 pm away from heat sinks over two equal paths.” Thermal resistance as low as 5” to 10°C per dissipated watt has been estimated. Peak pulse powers as high as 125 W with efficienciesaround 19% at 1 GHz have been observed. The other structure mentioned before is original since it is a coplanar one on a heteroepitaxial structure, GaAs on silicon. The heat is carried down to the silicon (which has a thermal conductivity three times higher than that of GaAs), cooling the active GaAs portion of the device. Under such conditions, 3.5 W of continuous output power at 1.5 GHz have been obtained with, however, a poor efficiency (less than 1%). Today, what is the situation? One must admit that coplanar structures for high-power devices are no longer seriously considered: Heteroepitaxy of GaAs on silicon is still an unsolved problem in terms of the quality of the epitaxial layer and the usual homoepitaxial coplanar structures lead to highly inhomogeneous field distribution within the sample with high-field regions under the electrodes. For high-power operation such distribution is highly objectionable and proposed solutions to avoid this problem remain still in the laboratory domain. Thus, for power devices, sandwich structures are used with a heat sink. Recently a new fabrication process has been developed in which heat sinks are plated into the devices before the wafer is separated into chips, the chip and heat sink being then bonded directly into packages. A paper recently published by Narayan and Paczkowski (28), has strengthened interest in
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S. TESZNER AND J. L. TESZNER
this integral heat sink (IHS) technology. Moreover, the introduction of a dual heat sink structure by electroforming the thick top contact has a drastic effect on T,,, which is nearly halved for a mesa diameter of 100 pm. The simplified process schedule for integral heat sink TEOs is given in Fig. 1.4. Detailed numerical results will be given in the next paragraph, which summarizes the experimental results on Gunn and LSA diodes.
Etch thin N+-l’ubtnb.
L contact muk
Sintor top contact, dice
I
PKk.0.
FIG. 1.4. Simplified process schedule for integral heat-sink TEOs. [From Narayan and Pankowski (28),reprinted with permission].
4. Electrical Performances
Before giving any results, it must be once more emphasized that an evident gap exists between laboratory results and commercially available diodes. This remark applies to Gunn diodes. For LSA diodes, however, only laboratory results are actually available. No discrepancy therefore may be observed. a. Gunn diodes. i. GaAs diodes. Early Gunn diodes delivered CW output power of a few milliwatts with an efficiency of less than 1%. From that time, great improvement has been achieved on the material, substrate, and epitaxial layers, as well as in device technology. TEO’s now deliver a few hundred milliwatts of CW output power with an average efficiency of a few percent. Therefore, their use is no longer restricted to local oscillators and may be extended to the transmitters and pumps of parametric amplifiers. Moreover, at the level of local oscillators, the high power in multichannel receivers will allow several mixers, in systems, to be driven simultaneously. By combination of power from several diodes, the power level has been extended to a medium power range (1-2 W) and Gunn diodes are therefore
303
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
finding new applications as communication transmitters or as radars. For the individual diode, as defined recently by Ramachandran (29), state-ofthe-art CW output power is in excess of 0.5 W, at frequencies within the C band and output powers greater than 0.35 W in the X band. Concomitantly with the increase of output power, a major trend in the research work of the past few years has been the development of high-frequency TEO from X band up to V band. The following laboratory results (28-30, 72), given in Table 1.1, do not pretend to be exhaustive; they are given to support the general analysis presented above. TABLE 1 . 1 OSCILLATOR GUNNGAASDIODES PERFORMANCES’
Frequency range Oscillator output power CW Po,, (mW) Oscillator power efficiency q%
C Bandb 5 GHz (29)
8-12 GHz‘ (28, 30, 72)
25-54 GHz (30, 72)
780
w loo
4.5
146.9
w 1 5 0 6.1-4
a Commercially available diodes appear to have CW characteristics not too far removed from results, given by laboratories, in the X and Ku bands (efficiencies are not always mentioned). For higher frequencies a huge gap remains (above 40 GHz no diodes are commercially available up to now). It should be noted that Gunn diodes may find applications as pulse emitters (73) and are already commercialized for such a purpose. For a 1% duty cycle, peak output power of a few tens of watts in the X and Ku bands are obtained. At lower frequencies, the main problem is the heat removal from the center of the sample. This explains that relatively low efficiency is obtained. Under 4 GHz, CW operation is difficult to obtain. For this range of frequency the discrepancy of results, when the product output powerefficiency is considered, is due to the fact that the higher the frequency at which the diode works, the more difficult it is to realize an adequate material.
ii. InP diodes. I n P TEO diodes have been extensively studied in Great Britain since the original paper of Hilsum and Rees in 1970 (7). Four years later the results appear to be effective but the opportunity of industrial development remains questionable. Most of the initial results have been given for pulse conditions since the threshold field in InP is much higher than in GaAs, as has been already pointed out (10 kV/cm instead of 3 kV/cm). However, recently highefficiency CW TEO diodes have been realized by the Plessey Co. (private communication). Such apparently “anomalous ” results are attributed to
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S. TESZNER AND J. L. TESZNER
contact effects, which may be explained by the existence of a barrier potential at the cathode contact being 0.5 eV rather than the usual 3 gap value as in GaAs. Such contacts are obtained by Ag + Sn or Ga.* However, Plessey is now working on ohmic contacts-waiting from the Royal Radar Establishment (RRE) the proofs that " anomalous " contacts are reliable for relatively high power levels.? The state-of-the-art results at Plessey and RRE are given in Table 1.2 following. b. LSA diodes. LSA has been one of the most promising solid-state sources for peak powers exceeding 100 W in the X band and higher frequency ranges. Jeppsson and Jeppesen first reported peak powers of 6 kW at 2 GHz and 2 kW in the X band (31). Christensson et al. (32) published the first results on high peak power LSA operation from epitaxial GaAs. Results have also been obtained in the Q band (33) and the V band (34) whatever the real mode of operation. Comparison of results on dc and pulse operation showed that a twofold increase in efficiency was usually observed, passing from dc to pulse bias with a few percent duty factor, while three- or fourfold increases in power output would be obtained. What appears, therefore, is the strong dependance of the device behavior on the quality of the active layer and on the temperature of this layer. High-quality 150 pm thick n-GaAs layers should be grown on an n+ substrate with less than 5 % variation on electron density. Moreover, the device operation relies critically on heatsinking for dc bias and frequency stability versus temperature. Up till now, the poor temperature reliability and the difficulty in material are such that no commercially available diodes have been issued. The results given in Table 1.3 are laboratory results, the extension to production diodes being still questionable. A brief mention should be made at this stage of the noise figure of Gunn oscillators, since it is an important parameter for choosing a Gunn diode as an osci,llatorby comparison with IMPATT diodes as we will see in the next section. Two types of noise are dominant-the modulation noise and the Johnson noise.
* The explanation given by RRE is the following. Since the electrons have to acquire a high energy to scatter from the lower valley to the upper one, the necessary finite time induces a "dead region" within the sample, the effect being a decrease in efficiency. Since the threshold field in InP is higher than in GaAs, the width of this region should be greater (2 pm, instead of 1 pm, roughly speaking). Through control of the cathode barrier, it is possible to inject hot electrons. However, to get high efficiencies, it is necessary for electrons to transit from a low-field situation to a high-field situation. Under these conditions one must have at the cathode the possibility to inject hot electrons with a low-field situation. This is obtained by the Schottky barrier with an n diffusion region at the cathode. t Last results published at the 4th European Microwave Conference by Brookbanks and White (74) show that anomalous diodes do not seem to be adequate for high-power operation.
2
TABLE 1.2
$E
OSCILLATOR GUNNINP DIODESPERFORMANCES Frequency (GHz)
11
11.5
12.4
14.5
Oscillator output power Po,, (W) Mode of operation Efficiency q% Contact Origin
8.3 Pulsed 14.3 Anomalous
3.6-7.9 Pulsed 17.6 to 18.4 Anomalous
3.2 Pulsed 16.2 Anomalous
2.6 Pulsed 15.6 Anomalous
RRE
RRE
RRE
RRE
22
28.5
0.2
0.25
cw
cw
10 Anomalous Plessey
3.5 Ohmic Plessey
2.3 Ohmic Plessey
cw
41
w
m
0.1
uC
9
B
8
2 iP2 M
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S. TESZNER AND J. L. TESZNER
TABLE 1.3 OSCILLATOR LSA
DIODES
PERFORMANCES
~
Frequency (GHz)
Oscillator output power Peak power (W) Average power (mW) Efficiency q% Reference
9.4
400 8.5 (32)
9
High duty cycle
Low duty cycle
200 700 13
500-600 16.5
(35)
36.2
64
200 3.5 (33)
62 2.2 (34)
The first one comes from the bias voltage which induces modulation of both oscillation frequency (FM noise) and oscillation amplitude (AM noise). FM noise results partly from local density fluctuations in the bulk and at the surface; thus it depends greatly on the quality of the material and on the contacts. Expressions of FM noise and AM noise are, respectively, given as the mean frequency deviation AL,, and as an amplitude noise to carrier DSB measured both in a 1 kHz double sideband (DSB) from ratio (N/C)AM, the carrier. The Johnson noise is an intrinsic noise which results from the high electron temperature in the high-field region and from the scattering of electrons from the central valley to the satellite ones. For state-of-the-art Gunn diodes the modulation noise is still predominant. Extensive results have been given by Ohtomo (36). For instance, at the X band for an output power of 275 mW, Af,,,,,(Hz/@) is equal to 1.53, (N/C)AM,DSB(dB/Hz)is equal to - 170.5 dB and the effective noisetemperature ratio for white noise N/kT, = 29 dB. The corresponding value for IMPATT diodes will be discussed in the following section.
5. General Discussion and Conclusion on TEO We will now discuss the present advantages and disadvantages of TEO with respect to some other microwave semiconductor device for power applications. Initially Gunn diodes were limited in their applications to receiver systems where they were used as local oscillators. At that time, advantage was taken of their ability to work at relatively high frequencies with low output power. The present state of the art is quite different-Gunn diodes giving relatively large output power may be used as transmitters or pumps for parametric amplifiers and, on the other hand, as local oscillators. As for relatively high power applications, TEOs, as opposed to
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
307
IMPATT diodes (cf. Section II,A), still give a lower output power with a lower efficiency, especially after the recent results obtained on IMPATT GaAs diodes (cf., Section II,A,3). But the noise figure of Gunn diodes is still much lower. A detailed comparison has been made by Ohtomo (36). Thus Si IMPATT diodes appear to have FM noise much greater than Gunn diodes at the same level of output power (8.2 H z / e z compared to 1.5). GaAs IMPATT diodes have FM noise comparable to Gunn diodes but exhibit AM noise 20 dB higher than Gunn diodes (Si IMPATT diodes having AM noise even higher). As for the ratio ( N / k T , ) , Ohtomo found that its value for Gunn diodes, Si IMPATT diodes, and GaAs IMPATT diodes was, respectively, 23-29 dB, 41-51 dB, and 38-44 dB, which shows a definite advantage of Gunn diodes for the noise figure. Now as mentioned already, the origin of the FM part of the noise figure for Gunn diodes is to be found in the nucleation of domains, in the quality of the contacts, and in the quality of the material. Therefore progress shall be made by improving the quality of the active layer and by improving the quality of contacts with n+ layers. Circuit techniques may also be used as, for instance, injection locking. Thus Gunn diodes have undeniable advantage for applications where medium output power is required with relatively low noise figure. As for local oscillators versus BARITT diodes (cf., Section II,B), a recent comparison has been made by Weissglass (37). The advantage of TEO’s relies mainly on the wide frequency range of operation and a higher efficiency.Gunn diodes were operated with lower bias voltages than classical BARITT diodes ; however, recent results on complementary structures for BARITT diodes (cf., Section II,B,3) give the same range of voltage as for TEOs. Moreover, Gunn diode oscillators have a poorer noise figure-at the same level of output power, 23-29 dB versus 10-12 dB for BARITT diodes. Finally, BARITT diodes should be more reliable since they use a simpler material (silicon) with comparable structures. In conclusion, at the present time, BARITT diodes appear to be quite competitive with TEOs for local oscillator purposes. C . Transferred Electron AmpliJers
Two years after Gunn’s publication on GaAs oscillators, Thim et al. achieved amplification in bulk GaAs (39). After over ten years of work on the subject, it now appears possible to summarize what has been done and what are the definite possibilities of TEA. First of all, a theoretical approach will be given to show the different methods for diode stabilization. This first step for amplification appears to be necessary, exceptions to this rule being extremely rare and without any practical applications (40). Of the different
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methods which will be discussed, some already appear to belong to the past as, for instance, subcritical diodes; some are now being actively developed as, for instance, overcritically doped diodes with injection limitation at the cathode; and, some structures which are still at the perspective level, the traveling-wave transistor and other solid-state amplifiers derived from the Gunn effect-the future of their development still being questionable. Specific attention will be given to the present results of TEA's and their possibilities as power devices. In conclusion, a comparison will be given with other solid-state amplifiers-IMPATT and field effect transistors (FET) especially-in order to define what place TEA's may obtain in the microwave power devices field. 1 . Methods of Stabilization
a. Field distribution in a stabilized diode. In a unidimensional structure, the field profile depends strongly on the contacts and on the doping profile. For uniform doping, the 1V characteristics would follow the uE curve. However, this theoretical approach is different from reality. For an n'n type contact, the electric field at the cathode may be considered to be equal to zero. Since the average field distribution is Eo within the bulk of the sample, there is a definite region localized near the cathode where the field gradient is steep. Thus, from Poisson's equation, an accumulation of electrons occurs near the cathode. These excess electrons are such that the current density remains constant through the diode. As long as Eo does not exceed the threshold value E,,, the same situation is observed. Since J is constant, u increases with E, and n decreases. Then, from Poisson's equation d E / d x decreases. When the threshold is crossed, u decreases, n increases, and d E / d x increases with n (Fig. 1.5). From these considerations, the static 1 I/ curve may be obtained. From Fig. 1.5, it appears that when the field has increased, the point where E = E t h , which corresponds to the constant value of u = uth, is such that local electronic density increases. Therefore the current density J increases and the static conductivity above threshold appears to be positive. This is an important result to be borne in mind: In a stabilized diode, a differential negative conductivity exists simultaneously with a static positive conductivity. In such a diode biased above threshold, when an ac field is applied, the frequency being greater than the transit frequency, the space-charge fluctuation induced by the ac field does not have the time to stabilize. Thus it increases from cathode toward 'anode, while propagating (Fig. 1.6). The Gunn diode behaves under these conditions as an amplifier with a gain
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
Cathode
Anode
309
x
FIG. 1.5. Electrical field (a) and donors distribution (b) in a subcritically doped Gunn diode. From Levinshtein and Shur (56a), reprinted with permission.
I
I&!
hodr
I Anode
Anode
f
X
FIG.1.6. Space-charge growth in subcritically doped diode. From Levinshtein and Shur (56a), reprinted with permission.
3 10
S. TESZNER AND J. L. TESZNER
increasing exponentially with the length of the sample. Before looking at the actual results of such an amplifier, the different methods of stabilization will be described. b. Stabilization of a Gunn diode. i. Subcritically doped diode. It has already been shown that the growth time of a domain within the bulk of the sample is
A growing domain does not reach maturity and therefore is inhibited if the time growth is greater than the transit frequency T - of the perturbation, i.e.
with T = L/Vo, the period of oscillation. This inhibition condition may be written as follows: &
4
I
I
pndr
> L/V,
or 1012 cm-2
=
EVO ~
1
&dr
14
> n,L.
Diodes satisfying this condition are the so-called subcritically doped diodes. They are short enough with a low doping level and, in these conditions, act as amplifiers. Experimental evidence of amplification with subcritically doped diodes has been given in two different ways. (a) Using the diode as a reflection amplifier, the input and output are at that time disconnected by a circulator and the diode terminates the coaxial line. The global circuit used in these conditions is a broadband circuit and has been extensively studied and described by Thim and co-workers (39-41 ). Moreover, Thim has used this circuit for series-connected diodes operating globally as a reflection-type amplifier (42). (b) Using the diode as a traveling-wave amplifier or two-port amplifier, this structure has been realized by Robson et al. (43). In this configuration the most acute problem is the disconnection between dc and ac which is obtained by a capacitive coupling of the ac input. Typically, bulk samples 1 mm long with a very low doping level of 10l2 cm-3 have been used. In such diodes the signal should grow exponentially with distance whatever the variation of donor density. A similar structure, though more sophisticated, has been used by Kanbe et al. (44), which was supposed to combine the desired advantages of conventional and two-terminal amplifiers.
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MICROWAVE POWER SEMICONDUCTOR DEVICES. I
ii. Stabilization by an external load. In this case stabilization is obtained by waveguide cavity circuits, the starting point of a cavity Gunn amplifier being naturally a cavity Gunn oscillator. The typical circuit is described by Sweet et al. (45), Fig. 1.7. In this circuit, the iris coupling transformer is an adequate means to obtain G , > 1 -Gd 1, i.e., the conductance load seen from the Gunn diode exceeding the negative conductance of the diode. The oscillations are, thus, inhibited and the circuit may be used as a reflection amplifier circuit with a circulator. In theory this result is obtained easily, but experimentally the solution is not easy to find since suppressing oscillations at the transit-time frequency is not sufficient, higher harmonic oscillations having to be prevented simultaneously. This solution now appears to be widely developed and present results will be discussed in a following paragraph.
Gd
‘Circulator iris coupling transfc
w Source
FIG. 1.7, Single resonant equivalent circuit of a cavity Gunn amplifier. From Sweet, Collinet, and Wallace (45), reprinted with permission.
iii. LSA amplijier. This method has been proposed in 1970 by Hashizume and Kataoka (46).Since no experimental results have been published up to now, the proposed solution will be only mentioned. A large microwave field is superimposed onto a direct bias field. No growing domain may occur provided the microwave field is large enough to swing through the positive conductivity region for a certain fraction of the period and that the frequency of oscillations is “sufficiently high.” As in the LSA mode, the IV curve therefore follows the uE characteristics. Low noise due to space-charge inhibition and intrinsic high power handling capacity were expected. iu. Supercritically doped diodes. In 1967 Thim and Lehner ( 4 7 ) obtained amplification in diodes which actually did not satisfy the nL criterion. They explained such a result by modification of the epitaxial doping profile, reducing the effective length of the active layer. Two years later, Walsh er al. (48) did stabilize diodes with an nL product as high as 4 x 10” cm- with linear amplification in the C band. Perlman developed this structure and obtained
’
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S. TESZNER AND J. L. TESZNER
in 1971 definite high-power results in CW operation (49). At that time, however, “ the specific physical mechanism associated with the achievement of a nonoscillatory negative resistance mode operation with supercritically doped GaAs devices appear not fully understood.” Three years later, the mechanism for microwave amplification in supercritically doped n-GaAs diodes appear definitely to be determined by the electrodes of TEA. Diffusion has been extensively invoked (50). In Thim’s theory, the accumulation layer should adjust more quickly then it moves into the anode. In this case the upstream field remains below threshold, preventing a new domain to be nucleated. The accumulation layer at the anode should not disappear. It is assumed, therefore, that it keeps moving toward the anode, maintaining a high-field region near this electrode. The mathematical condition may be roughly written as E
41 - A n
3
ED
.;Jim
a few Debye lengths
It results from this condition that for n-GaAs, the doping level should be greater than 5 x 1014cm-3. This phenomenon at the anode, however, does not appear now to be the determining factor for controlling stability in Gunn supercritically doped diodes. In 1973, Spitalnik et al. (51) obtained good agreement between theory and experimental results with an explanation based on a contact phenomenon. With a moderate cathode doping notch, they obtained, by increasing the applied voltage, stable, unstable, and, lastly, stable regions of operation, with gain in both regions of stability. The determining importance of the contacts had already been foreseen theoretically by Boer and Voos (52) and Conwell (53) and experimentally by Shaw, Solomon, and Grubin (54). The purpose of the present review is not to give theoretical results, especially on the field direction method. Readers are referred to the papers mentioned above. Experimentally, the high field at the cathode (51) is obtained by an appropriate doping notch. This realization is technologically difficult and presupposes elaborate skill in epitaxial technology ! The high-field region becomes unstable when a critical current is reached, provided a critical domain width is exceeded. If sufficient bias is applied, the high-field region fills the whole sample and a stable solution is obtained with maximum field at the anode. However, the practical results were obtained at high voltages for devices which were good oscillators at lower voltages. Numerical results will be given in a following paragraph. v. Stabilization by the nd criterion. In 1968, Kino and Robson (55) published a short theoretical paper in which they showed that diode operation
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
3 13
could be modified drastically as soon as the thickness of the sample became the order of the wavelength of the space-charge wave associated with drifting carriers. As the threshold field is reached the transverse mobility remains equal to u/E, while the longitudinal one becomes equal to its differential value du/dE and thus becomes negative. By taking into account surface charges with the quasi-static approximation and a small-signal theory, they derived a condition for domain inhibition in their samples nd I 0.8 x 10" cm-2. This means that even if the nL criterion is satisfied for domain formation, the device will not oscillate if its thickness d is small enough. For instance, for a doping level around l O I 4 cm-3, d should be less than 1 pm. Technological problems at that level become determinant. They may be avoided by dielectrical loading or two-stream interaction. Levinshtein and Shur in their review paper on the Gunn effect (560) have shown, in a simple manner, that domain inhibition can be obtained by dielectric loading as well as by reduction of the diode thickness. More specifically, a space charge Q in a dielectric loaded sample induces a longitudinal field smaller than in the unidimensional case. In an unidimensional sample, the field is equal to Q/2&d,d being the thickness of the sample. In a bidimensional case, following Green's theorem it is possible to write Q
= 2d&E,
+ 2AEdEy,
being the permittivity of the external medium, A a characteristic dimension of the sample which may be the space-charge wavelength or the domain dimension. If Ex 2: E , , the value of Ex may be derived:
&d
then the longitudinal component of the electrical field is decreased by a factor equal to
The electrical result is a short cut of the field lines by the external medium, the field within the sample being decreased, inducing domain inhibition. The space-charge growth being decreased by the same factor, the stabilization criterion may be written as
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S. TESZNER AND J. L. TESZNER
If Ed2/&dis much greater than 1, it becomes
thus domain inhibition may be obtained as well by dielectric loading as by thickness reduction. Experimental evidence of this fact has been given from 1968. At that time Vlaardingerbroeck et al. (56b) and Kataoka e f al. (57) showed that domain inhibition could be obtained by dielectric loading. Kataoka was able to inhibit domain formation with a BaTiO, sheet (relative dielectric permittivity 12000).Thedccurrent saturation in the ZV characteristic occurred at the threshold level. In 1969, Kuru and Tajima (58) gave clear evidence of the role of the parameters mentioned above: load and thickness. They studied samples with thicknesses in the range 20-200 pm and showed that oscillations were inhibited if the diodes were thin enough. The critical thickness depends on the diode resistivity and on the permittivity. When the thickness is greater than the critical value, the threshold voltage increases with decreasing d. Moreover, the growth time of the domain increases by an order of magnitude4.2 nsec -,2 nsec. A potential distribution study reveals that, in this case, there is a high-field region at the anode. This inhomogeneous distribution has been explained by Hoffmann (59) by considering the two structures shown in Fig. 1.8. When the dielectric is not metalized (Fig. 1.8a) near cathode and anode, the electrical field has an important transverse component. Under such conditions, an accumulation layer is created near the cathode with a depletion layer near the anode which disappears by metalization of the dielectric (Fig. 1.8~).A potential probing gives evidence of this phenomenon (Figs. I.8b and d). vi. Two-stream stabilization. This method was suggested in 1970 by Gueret (60) and Teszner (61).In this configuration, the two-stream interaction between an active GaAs diode and a passive one can give way to a decrease of the space-charge factor and thereby inhibit domain formation. If the surface charges in the passive slab are not drifted, the criterion for domain formation becomes nd > 0.8 x 10"(0,/0)~ cm-',
-
where w , is equal to u,/E, (a, being conductivity and E , dielectric permittivity relative to the passive medium) and w is the pulsation of the applied rf signal. When the surface charges in the external slab are drifted, it appears that, for a relative velocity close to 1, the domain formation is drastically inhibited and an important gain in space-charge amplification may be obtained.
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
metal
metal
mltmde
anode
315
(d anode
96. 72.
I
region
24.
distance horncathode (bm)
FIG.1.8. Field and potential distribution in a Gunn diode epitaxial layer with dielectric loading. (a) and (b)-without dielectric metalization; (c) and (d)-with dielectric metalization. From K. R. Hoffmann (59), reprinted with permission.
Some 5 years after the first publication relative to an nd structure, few experimental results on amplification have been published; see, however, Dean (64-67) on a thin layer two-port amplifier structure. Results will be discussed in the next paragraph. Before this discussion a brief mention will be given on other possible methods to obtain amplification without stabilization. One has already been mentioned (40), i.e., the traveling domain amplifier described by Thim in 1967. The other is the self-pumped parametric amplifier (62,63).In the first case, Thim takes advantage of the fact that when the applied voltage is increased during the transit of the diode, the excess voltage within the domain increases and the current decreases. Thus the sample has a negative resistance, the effect of which adds to the differential negative resistance of the diode. In the second case, advantage is taken of the fact that the domain behaves as a capacitance, the value of which is modulated when the domain
3 16
S. TESZNER AND J. L. TESZNER
is formed or when it disappears. A self-pumped capacitance is thus created and this is the basic effect of the parametric amplifier. 2. Electrical Performances Having shown the methods of stabilization for Gunn oscillators, we will now present the results of Gunn devices as amplifiers. However, as has already been said, only the structures that have given definite results at relatively high power levels will be mentioned, i.e., devices stabilized by external loading and supercritically doped diodes.* As for the other types of amplifiers, subcritical ones have never been able to give more than a few milliwatts of output power. The nl criterion is far too restrictive to allow high-power devices. As for the nd criterion, the structures studied belong more to the future than to the present; up to now the experimental results have been unsatisfactory [Dean et al. (64-67)]-the output power is only a few milliwatts, there is a poor output to input isolation, and the transducers are somewhat ineffective due to high reflection. This type of device should be, however, the most powerful. It has inherently a broad bandwidth. On the other hand, according to Dean, “the space-charge wave can be made to grow with extremely high gain, but the ability to obtain high spacecharge wave gain is not nearly so important as the ability to move the signal over a large distance with some gain.” In our opinion, to obtain such high gain implies necessarily a successful control of the finest epitaxial technology, 1 pm homogeneous since it requires the growth of thin epitaxial layers over a lo00 prn length and more. Moreover, the control of the uniformity of the electrical field within the device by using a third control does not appear to be up to now completely realized. As for the two-stream amplifier that has been mentioned, it should be possible to realize now since heteroepitaxial technique is presently under development. A comment should be made at this time on the noise figure of TEA amplifiers and the possibility of using InP instead of GaAs. Detailed study has been done by Robson who has shown that a close solution between the noise figure and the diffusion coefficient exists. Since for E/E,,, equal to 1, the diffusion coefficient ratio for GaAs versus InP is equal to 2, a 3 dB difference in the noise figure should be observed. This has been actually observed by Baskaran and Robson in 1972 (70), the noise figure for an InP reflection amplifier being 7.5 dB with 23 dB gain. However, the 1 dB m output power was only 0.5 mW which is far from the scope of microwave power amplifiers. Moreover, the latest results on CW InP amplifiers (71) give a noise figure of 10.7 dB at 14 GHz, very near to GaAs diode performances, however, with a saturated output power at 3 dB gain of only 8 rnW, still quite out of the power-device field.
-
See Table 1.4.
TABLE 1.4 PERFORMANCES OF SUPERCRITICAL AND STABILIZED REFLECTION-TYPE GaAs GUNNDIODEAMPLIFIERS m
Supercritical Stage Number Amplifier Frequency (GHz) Output power (mW) Power gain (dB) 3 dB bandwidth (GHz) Noise figure (N/kT,) (dB) Reference
$
Reflection-Type
1
1
1
1
4
2
2
4.5-8
8- 12 250
7-11 100 (at 1 dB m)
7 245 (at 1 dB m) 9.5
10.3-11.8 loo0
13.3 500
30
10
0.1 13 (45)
0.3
35 170 (100 at 1 dB m) 13 0.18
loo0 (250 at 1 dB m) 8
6
3.5 15
4
(49)
(49)
4 10.5 (68)
(68)
16.2
(45)
E
ij
2U c
9 8
(69)
s
&I
P
I
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S. TESZNER AND J. L. TESZNER
3. General Considerations and Conclusion on TEA’s
For communication systems, TEA’s have to compete with IMPATT diodes, FET’s, and bipolar transistors. Two frequency ranges may be considered, up to the Ku band and beyond. Below 18 GHz recent progress on bipolar transistors and FET’s have provided competitive solutions for power amplification-for a time through varactor multipliers (Section II,C) for high-frequency range. The present results of TEA’s at these frequencies are not good enough to show a definite advantage to their use. However the broad bandwidth capacity is an advantage for TEA. At higher frequencies, TEA’s have to compete with IMPATT devices. The main disadvantage of TEA versus IMPATT diodes is their lower power capability. In return, their great advantage is naturally their lower noise figure, 16 dB at 35 GHz in comparison with 26 dB for GaAs IMPATT and > 30 dB for Si IMPATT diodes (Section II,A,3). The development of TEA’s has up to now borne no comparison with IMPATT diodes or transistors. The market for TEA’s is still not well defined and this is the reason that most components manufacturers have not developed the TEA. However, after an unsuccessful launching in 1972, TEA’s have reappeared on the market in 1974. What success the device may have is still difficult to estimate especially in the first range of frequency. More precisely, TEA are reflection-type amplifiers and the corresponding electronic circuit is more delicate than for a FET. Therefore, the main advantage of TEA, the wide bandwidth, could just be illusory.* In the last section of this review (to be published in Part 11), our personal view on the future trends of TEA will be given and definite applications which may have industrial use will be emphasized.
REFERENCES FOR BULK
I. 2. 3. 4. 5. 6.
7. 8. 9.
SECTION
I
DIODES
J. B. Gunn, Solid-State Commun. 1, 88 (1963). B. K. Ridley and T. B. Watkins, Proc. Phys. Soc. (London) 78, 293 (1961). C. Hilsum, Proc. I R E 50, 185 (1962). B. W. Hakki, J . Appl. Phys. 38,808 (1967). J . B. Gunn and B. T. Eliott, Phys. Lett. 22, 366 (1966). J . G. Roch and G. S. Kino, Appl. Phys. Lett. 10, 40 (1967). C. Hilsum and H. D. Rees, Electron. Lett. 6, 277 (1970). P. M. Boers, Electron. Lett. 9, 134 (1973). A. Joullie, P. Esquirol. and G . Bougnot, Mater. Res. Bull. 9, 241 (1974).
-
* Commercially available TEA in 1974 have a narrow bandwidth 500 MHz which should be compared to the laboratory results, i.e., 4 GHz bandwidth in the X band.
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.
53. 54. 55.
3 19
A. Hojo and 1. Kuru, Electron. Lett., 10, 61 (1974). K. Kurokawa, Bell Sysr. Tech. J . 46,2235 (1967). P. N. Butcher, W. Fawcett, and C. Hilsum, Brit. J . Appl. Phys. 17, 841 (1966). P. N. Butcher, W. Fawcett, and N. R. Ogg, Brit. J . Appl. Phys. 18, 755 (1967). N . Heinle, Solid-State Electron. 11, 583 (1968). V. Szekely and K. Tarnay, Electron. Lett. 4, 492 (1968). J . L. Teszner and D. Boccon-Gibod, J . Appl. Phys. 44, 2765 (1973). J. B. Gunn, 1BM J. Res. Develop., 300 (1966). H. C. Huang and L. A. MacKenzie, Proc. IEEE (Lett.)56, 1232 (1968). J. A. Copeland, Proc. IEEE (Lett.) 54, 1479 (1966). W. 0. Camp, IEEE Trans. Electron Devices 18, 1175 (1971). D. Jones and H. D. Rees, Electron. Lett. 9, 105 (1973). G. S. Kino and 1. Kuru, IEEE Trans. Electron. Deuices 16, 735 (1969). J. A. Copeland, Proc. IEEE (Lett.) 57, 1666 (1969). W. 0. Camp, IEEE Trans. Electron Devices 18, 1175 (1971). R. H. Johnston and E. R. Riddle, Proc. IEEE (Lett.) 60, 1449 (1972). L. E. Norton, R. E. Enstrom, and I. J. Hegyi, Proc. IEEE (Lett.)56, 543 (1968). Electronics, 41, 52 Feb. 5 (1968). S. Yegna Narayan and John P. Paczkowski, RCA Rev. 33, 752 (1972). T. B. Ramachandran, Proc. IEEE (Lett.) 56, 336 (1968). T. G. Ruttan, IEEE Trans. Microwaue Theory Tech. 22, 142 (1974). B. Jeppsson and P. Jeppesen, Proc. IEEE (Lett.), 54, 1479 (1969). S. Christensson, D. W. Noodward, and L. F. Eastman, IEEE Trans. Electron Devices 17, 732 (1970). M. Shyam, Electron. Lett. 6, 315 (1970). J. J. Barrera, IEEE Trans. Electron Devices 18, 866 (1971). L. F. Eastman, IEEE Int. Conf. Electron Devices, Philadelphia, 1973. M. Ohtomo, IEEE Trans. Microwave Theory Tech. 20,425 (1972). P. Weissglas, Proc. European Microwave Con$ Brussels, Belgium, 1973, p. A.2.1. Ad Hoc Committee, Materials and Processes for Electron Devices, Nat. Acad. Sci., Washington, 1972. H. W. Thim, M. R. Barber, B. W. Hakki, S. Kright, and M. Venohora, Appl. Phys.Latt.,7, 167 (1965). H. W. Thim, IEEE Trans. Electron Devices 14, 517 (1967). H. W. Thim and M. R. Barber, IEEE Trans. Electron Devices 13, 110 (1966). H. W. Thim, Proc. IEEE (Lett.) 56, 1245 (1968). P. N. Robson, G . S. Kino, and B. Fay, IEEE Trans. Electron Devices 14, 612 (1967). H. Kanbe. K. Kumabe, and R. Nii, Proc. Conf. Solid-State Devices, 2nd, Tokyo, 1970, p. 114. A. A. Sweet, J. C. Collinet, and R. N. Wallace, IEEE J. Solid-State Circuits 8, 20 (1973). N. Hashizume and S. Kataoka, Electron. Left., 6, 34 (1970). H. W. Thim and H. H. Lehner, Proc. IEEE (Lett.) 55, 718 (1967). T. E. Walsh, B. S. Perlman, and R. E. Enstrom, IEEE J . Solid-State Circuits 4,374 (1969). B. S. Perlman, R C A Rev. 32,3 (1971). H. W. Thim, Proc. IEEE (Lett.) 59, 1285 (1971). R. Spitalnik, M. P. Shaw, A. Rabier, and J. Magarschack, Appl. Phys. Lett. 22, 162 (1973). K. W. Boer and P. Voss, Phys. Status Solidi, 30,291 (1968). E. M. Conwell, IEEE Trans. Electron Devices 17, 262 (1970). M. P. Shaw, P. R. Solomon, and H. L. Groben, IBM J. Res. Develop. 13,587 (1969). G. S. Kin0 a n d P. N. Robson, Proc. IEEE (Left.),56, 2056 (1968).
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S. TESZNER AND J. L. TESZNER
56a. M. E. Levinshtein and M'. S . Shur, Sov. Phys.-Semicond. 5, 1561 (1972).
566. M. T. Vlaardingerbroeck, G. A. Acket, K. R. Hoffmann, and P. M. Boers, Phys. Lett. 28A, 97 (1968). 57. S. Kataoka, H. Tateno, and M. Kawashima, Electron. Lett. 5, 48 (1969). 58. I. Kuru and Y.Tajima, Proc. IEEE (Lett.), 57, 1215 (1969). 59. K. R. Hoffmann, Electron. Lett. 5, 227 (1969). 60. P. Gueret, Electron. Lett. 20, 1970 (1970). 61. J. L. Teszner, Solid-state Electron. 13, 1471 (1970). 62. H. J. Kuno, Electron. Lett. 5, 232 (1969). 63. C. S. Aitchison, C. D. Corbey, and B. H. Newton, Electron. Lett. 5, 36 (1969). 64. R. H. Dean, A. B. Dreeben, J. F. Kuminski, and H. Triano, Electron. Lett. 6, 777 (1970). 65. R. H. Dean, IEEE Trans. Electron Devices 19, 1148 (1972). 66. R. H. Dean and R. J. Matarese, Proc. IEEE 60, 1486 (1972). 67. R. H. Dean and B. B. Robinson, IEEE Trans. Electron. Devices 21, 61 (1974). 68. A. Rabier and R. Spitalnik, Proc. European Microwave Con$ Brussels, 1973, p. A62. 69. F. E. Rosztoczy and R. E. Goldwasser, WESCON Con$, 1973, 23 p . E-I. 70. S . Baskaran and P. N. Robson, Electron. Lett. 8, 137 (1972). 71. R. M. Corbett, I. Griffith, and J. J. Purcell, Electron. Lett. 10,307 (1974). 72. F. E. Rosztoczy and J. Kinoshita, J . Electrochem. SOC. 121,439 (1974). 73. R. Stevens and F. A. Myers, Proc. European Microwave Con$ Montreux, Switzerland, September, 1974, p. 257. 74. D. M. Brookbanks and P. M. White, Proc. European Microwave Con$ Montreux, Switzerland, 1974, p. 227. 75. P. M. Boers, Electron. Lett. 7 , 625 (1971). 76. G. H. Glover, Appl. Phys. Lett. 20, 224 (1972). 77. L. D. Nielsen, Solid State Commun. 10, 169 (1972). 78. B. A. Prew, Electron. Lett. 8, 592 (1972). 79. W. Fawcett and D. C. W. Herbert, Electron. Lett. 9, 308 (1973). 80. H. D. Rees and C. Hilsum, Electron. Lett. 7 , 437 (1971).
-
11. JUNCTION DIODES
A. Avalanche Diodes: I M P A T T and T R A P A T T 1. Basic Considerations
Avalanche junction diodes appear as a direct consequence of the extension to semiconductor materials ( 5 , 6 ) of spark triggering theory in gases by electron avalanche formation (1-4). There is an undoubted link between the first avalanche transit-time diodes, proposed by Read (8) and the vacuum diode, by Llewellyn and Bowen (7) 20 years previous. In both cases, moreover, the aim was the same, to produce ultrahigh-frequencyoscillations by making use of a negative differential resistance in the diode structure. The theory of avalanche diodes has already been studied in many articles following that by Read [Misawa (9), and more particularly, Scharfetter and Gummel (10), Blue ( I I ) , and Delagebeaudef (12) on large-signal analysis].
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
321
Because of the complication of the system of equations under large-signal operation with realistic semiconductor parameters, the numerical approach, rather than the analytical approach, is generally used for the large-signal problem examination. We will restrict ourselves, on this point, to a recapitulation of the principle and a short examination of the physical mechanism underlying the large-signal phenomena present, in the various diode structures, from the most simple to the most complex. Any avalanche breakdown of a diode generates through the thermal effect a reverse volt-ampere characteristic showing a negative differential reistance. Nevertheless its triggering effect cannot in fact be used in the case of semiconductor junction diodes unless the heat dissipation is limited to avoid the diode destruction through the mechanism of so-called second breakdown. This is, in fact, a preliminary condition for using the avalanche breakdown as triggering to excite and maintain oscillations in a resonant circuit. However, in order that this possibility be conveniently used two fundamental conditions must be fulfilled: (a) The triggering must be reproducible at a frequency of the same order as the resonant diode circuit frequency, and (b) to enhance the negative dynamic conductance, the transit time of the carriers generated in the avalanche through the diode circuit must bear a suitable ratio to the rf period. Moreover, where large-signal operation is concerned, it would be advisable that the microwave power efficiency and also the output powerimpedance-square frequency product be as high as possible. a. IMPATT mode. Let us mention briefly the operational process of an IMPATT* diode. The avalanche breakdown condition is given by either formulas (1) or (2), expressing the multiplication rate through the impact electron (1) or hole (2) ionization, depending on whether the avalanche process is initiated by electrons or by holes; this multiplication ratio tending toward infinity:
(- ./. exp ( ./.
WA
a,(E) exp ap(E)
1
(a, - a p ) dx’ d x = 1,
X
X
-
(up - a,) dx’
0
In these formulas, u, and upare, respectively, electron and hole ionization rates. W, is the width of the avalanching depletion layer (cf., Fig. 11.1) representing the most simple diode structure, and for the moment the most common, where the avalanche process is initiated by electrons. a, and u p , giving the number of ionizing collisions per centimeter length * Impact Ionization Avalanche Transit Time.
322
S. TESZNER AND J. L. TESZNER
made by a single charge carrier, are approximately exponential functions of the field E: a@) = a. exp
[ - (EO/E)'"],
(3)
where a o , E o , and rn are constants. Note that for materials presently used, i.e., Ge, Si, and GaAs, a,, z a, for Ge and GaAs, whereas for Si the ratio a,, /ap varies from 10 to 2, for E varying from 3 x lo5 V/cm to 6 x lo5 V/cm. In accordance with formulas (1) or (2), for Ge and GaAs the width W, is much narrower than for Si. Nevertheless, for the EB (breakdown field) characterizing each of these materials, the absolute value of a,, is but approximately twice as high for Si as for Ge or As. This tends to attenuate, to some extent, the difference between the corresponding W, values. As we know, the formation of the avalanche is initiated as soon as a field is high enough to impart to the electrons sufficient velocity to set up a self-sustaining ionization process through successive generating electronhole pairs ionizing collisions. The avalanche formation time is reduced through a cumulative process arising from the electric field increasing with the electron multiplication, which in turn makes the coefficient a,, (or possibly, a p ) increase and this accelerates the avalanche formation. Therefore the ionization process becomes self-sustaining with a relatively narrow avalanche zone. A localized plasma with highdensity charge carriers is produced. We may note for memory the basic carrier formulas for the structure seen in Fig. II.la, considered as a one-dimensional structure : (a) Continuity equutionsdn dt
ldl, = 9, q dx
(4)
dp + -1 dP l =.g, dt q d x
-
where I,, (electron current) = qnu,,,, , the total current being I = I,
I, (hole current) = qpu,, ,
(6)
+ I,;
g (electron-hole pairs creation rate per unit volume) = anunsIn
+ ap~pslp,
(7)
u,,,, upst being electron and hole scattering limited velocity. Note that the generation of carrier pairs and their recombination by structural faults in the material are neglected.
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
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(b) Poisson’s equation-
Let us now consider six categories of structure for the avalanche diode, which can be represented schematically in Fig. II.la-f.
nt
FIG. 11.1. Block diagram of the six IMPATT diode structure categories (a, b, c, d)-single drift zone; (e, + d o u b l e drift zone.
The first, most simple (Fig. II.la) has already been mentioned above. The second (Fig. II.lb) is the original Read diode (8), characterized by the subdivision of the median region of the diode into two parts with very different doping levels. The third (Fig. 11.1~)of recent design* (14-16), is an improved form of the Read diode, which consists of an increase in the doping concentration of the median region, while keeping their graduation to some degree; this improvement is aimed at increasing power efficiency. Likewise for the fourth structure shown (Fig. II.ld) which is also recent, and characterized by alternating doping levels: low-high-low (14). All four types of structures are said to have a single drift zone, as distinguished from those with a double drift zone, represented by Fig. II.le (17, 18) and with an increased degree of complication, in Fig. II.lf. Both structural types correspond in double drift zone configuration, to the types shown in Fig. II.la,c.
* In fact, a similar structure had been studied previously, but simply with a view to investigating oscillatory phenomena below the transit-time cutoff (13).
324
S. TESZNER AND J. L. TESZNER
Structural complications with respect to Fig. II.la are all designed to reduce the W,/W, ratio, where WDis the drift zone and W, + WD= W (total width of the median diode region). All tend to increase the power efficiency as we shall see later on. Figures II.Za,b show the distribution function of the electric field in the median region of the six types of structure for reverse bias voltage V,, slightly lower than V, , the breakdown voltage, the junction being assumed abrupt and the W zone punched through.
E
EB-
P:
n:
e
I ,
W
FIG.11.2. Static electric field distribution of the six IMPATT diode structure categories presented in Figs. II.la-f. Note: By way of simplification it is assumed that there are only two different widths W for the total depletion layer: one for the single drift-zone case (a, b, c, d), the other for the double drift-zone case (e, f), where it is further supposed that n = p ; a, = a P ; USl"
=
v.1p.
A slight increase in the voltage at the terminals, from either an applied pulse or an overvoltage associated with switching on the bias circuit, triggers the avalanche process. Figures II.3a-f show modifications in the field distribution due to the formation of the avalanche in the six structural variants. We note that the ratio p = W,/W, varies appreciably according to the type of structure. The avalanche zone W, constitutes the rf power source which is shaped and transmitted into the external circuit through the drift zone W, . In order to obtain as high as possible a power efficiency, the V,,
X
FIG.11.3. Dynamic characteristic (the avalanche being formed) of the electric field distribution for the six IMPATT diode structure categories presented in Fig. 11.1.
326
S. TESZNER AND J. L. TESZNER
contribution across the avalanche zone must be reduced as much as possible. This is done by increasing the doping concentration in the avalanche zone, not only its relative value with respect to concentration in the drift zone, but also in absolute value. From this point of view, the structure of Fig. 11.1~is a marked improvement, as long as the thickness of the over-doped layer W,, corresponds as closely as possible to the avalanche zone W,. In effect, if W,, < W,, the avalanche zone will stretch well beyond and will increase the ratio p consequently. If, on the contrary, W,, > W,, the contribution of the voltage V,, across the relatively slightly doped drift zone will be reduced, and this may prevent a punch through of W, before breakdown. Now, punch through is necessary so that the charge carrier speed in the drift zone be equal to the maximum velocity V,, throughout the whole of this zone; by doing this, the parasitic resistance of the structure is restricted. The structure in Fig. II.1d (14) gives another solution to the problem of increasing efficiency. In this case, the field is kept more or less constant at a high level in the W, zone, and at the low level in the W, zone. There are two results ; in the W, zone, the field-increase mechanism during the avalanche formation period being no longer limited by the space charge, this increase is more rapid,* and continuous, until self-sustaining ionization is reached ; the time for the avalanche to set up and the thickness of the W, zone are thus reduced. Moreover, the punch through of the W, zone is much more easily reached. However, two points need to be made. The space charge in the avalanche zone being minimized, the field will be able to reach particularly high values and the ionized carrier density can be very high. By a mechanism close to that of the TRAPATT mode which we shall examine later, the plasma velocity in the avalanche zone could then reach some higher value than us, in the drift zone. Consequently at least a part of this plasma could propagate through the diode. This disturbance of the IMPATT mode will in turn result in an increase in power efficiency. Moreover, there is not a priori assurance that the low level of the field characterizing the W, zone will be sufficient for the limit carrier velocity us! to be reached there. So that this condition may be complied with, one has to determine structural parameters. The structure with two drift zones W,, and W,, and one avalanche zone centrally located (Figs. II.le,f) provides a radically different solution to the same problem of increasing efficiency. The ratio p = W,/(W,, + W,,) is reduced even further here. In effect, with respect to the single drift zone structure, with other conditions being the same, the increase in ac output power is close to loO%, whereas that of the power delivered by the dc source
* The analogy with the spark-triggering mechanism (4) is all the more striking.
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
327
is notably less. Lastly, the arrangement in Fig. II.lf combines the advantages of the double drift zone with those of Fig. 11.1~.The ratio p is then minimized. Let us now consider the transfer of charge in the drift zone W,; we will restrict our comments on this subject to giving the data related to an optimal transit-time choice. In the most simple case, that we shall take as an example, we will only be looking at electron transport, since the holes produced in the avalanche are almost immediately collected by the p z + layer. By supposing that the field E throughout W, is at least equal to the field E, (approximately lo4 V/cm), for which the speed I/nsL will be reached, the formula
is obviously applicable, where tD is the transit time through the drift zone. To obtain the overall time that determines the dephasing between voltage and circuit current, the time formation of the avalanche t Amust be added to t,. On the other hand, it will be possible to leave out the extremely low time for the voltage rise from V,, to V,. The choice of tD for a given operational frequency is predetermined essentially by maximizing the value for the negative conductance of the diode. This conductance is given by the fundamental term in the Fourier series, expressed in the following simplified formula ;
where a is a constant at a given voltage. Then the following optimal condition holds for
tD:
1
TD = - - T A .
2f
Having stated this, the relationship between the current and the voltage at rf in a steady state will be illustrated in Fig. 11.4. This requires a few words of explanation. During the positive half-wave (not represented here) in the rf voltage, the avalanche will be set up and then the charge carriers set up will
FIG.11.4. Ac voltage and external current induced in IMPATT diode electrodes.
328
S. TESZNER AND J. L. TESZNER
be moved through the drift zone. The current in the external circuit only appears after a time (zA + zD) = 1/2ffrom the avalanche starting time. The flow of this current will continue during the negative half-wave for the voltage. During the following positive half-wave, the avalanche will be reformed, followed by retransmission of the charge carriers through the drift zone. In principle, during this half-wave, the current should be nil. In fact, there is a small residual current, corresponding to evacuation of residual charge superimposed on the displacement current. This series of phenomena is reproduced periodically. The choice of zD for a given tA,obviously determines the operational frequency. However, there can be no question of reducing tD to zero, since the existence of a minimum drift zone is necessary to avoid burnout. It is important to consider here the frequency power product Pfand, on the other hand, the product PXf2 (where X is the reactance of the diode) that is often put forward as the quality criterion for microwave power oscillating and amplifying devices (19). The power that can be delivered for a given frequency depends essentially on the diode cross section, the current density, and the power efficiency. The maximum useful area for a diode chip is limited by the skin effect, and also, by the inherent risk of current concentration into filaments of locally very high intensity (where this is a danger of burnout). The procedure therefore is rather to couple in parallel, wherever possible, several chips with reduced dimensions. Beforehand, however, all other possibilities of increasing the current density and bias current will be explored up to the thermal limitation of the diode, and for up to ultimate nonthermal space-charge limited current density (20). Increasing the operational frequency raises yet another more difficult problem if one refers to the relation expressed as PXf’ = const for a given device. In effect, X = 1/2nfC, where C = E, A / W, A being the useful area of the diode and E, the semiconductor permittivity. Now the optimal value of W is a function of the frequency W = Bf- where B is a constant. Hence,
’,
If it is required that X be unchanged, then A must be decreased in proportion tof’ when the frequency has to be raised. Theoretically the power output would be reduced by a proportionate amount. In order to maintain the latter at the same level, the current density must also be increased, in the same way, in proportion tof’. In fact, the prevailing operational conditions are less stringent, and it is rarely necessary to maintain the given diode reactance. Moreover, when the
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diode structure remains unchanged, the variation in its output power is closer to a n f - law than to a n f - law, for a wide range of frequencies (20). It is nontheless true that one should tend, insofar as is possible, to increase the PX product, which characterizes a diode for a given frequency. Furthermore, the double drift-zone structure has its own specific advantages. It is to be noted that both P and X values for a double drift-zone diode are near enough double those for the corresponding single drift-zone device. The PX product would, therefore, be at least tripled in value. This advantage proves particularly attractive in millimetric wavelength operation. The foregoing remarks serve to confirm the basic need to go for maximum power efficiency. From our previous discussion on this topic, we saw that this efficiency is proportional to VD/(VA + VD), where VA and VD are dc voltages developed, respectively, across the avalanche zone and across the drift zone. In this light q becomes
’
where Pa,, V,, , I,, are the ac output power, ac external voltage, and current; P,, , Vd, ,Idc are the dc power, voltage, and current delivered by the dc power source. For ideal diode structure and operating IMPATT-mode conditions at the optimum frequency, with a sinusoidal variation of the drift voltage for a square wave of particle current, we have
Hence, for V,, = f V Dand VAof negligible value compared with V,, we have, qlim ‘V
1 - ‘V
72
31%.
It is only recently that this apparent limit has been approached ( 1 4 2 1 ~ 77, 80). In fact, it must be stressed that it is only valid for the conditions set out above. There are therefore cases where the “limit” can be exceeded,* especially if one departs somewhat from the stringent IMPATT mode conditions, and if Vd, drops for part of the V,, cycle. Indeed, these are the basics for high q values reached in the operating mode, as distinct from the fundamental IMPATT mode and called now TRAPATT mode (for Trapped Plasma Avalanche Triggered Transit). At the Cornell Conference, June 1973, C. K. Kim, using a two-step field profile structure, put forward a figure of 35% for q . This performance value has been confirmed lately by Goldwasser and Rosztony (77).
3 30
S. TESZNER AND J. L. TESZNER
At the time when “ high-level”efficiency values were put forward (22) for the first time (around 40% to begin with), the IMPATT-mode values scarcely exceeded q = 8%. At that time, although the operational frequency for the new mode was only about one third that of the IMPA’IT mode, the TRAPATT mode proved most attractive. However, there has been some marked progress of late in IMPATT technique and the TRAPATT mode has become less attractive. Notwithstanding this foregoing proviso, we give below the outline description for the TRAPATT mode. b. TRAPATT mode. Although the phenomena of highefficiency power generation with avalanche diodes in a TRAPATT mode have been discovered recently, there have been several experimental, as well as theoretical, studies. Note should be made of the experimental work by Johnston and Scharfetter (23), and of the theoretical studies by Bartelink and Scharfetter (24), Clorfeine, Ikola, and Napoli (25), De Loach and Scharfetter (26),and Scharfetter (27). The TRAPATT mode requires essentially that the drift velocity u, of the avalanche shock front be much higher than the scattering limited carrier velocity us(. For this, the avalanche must be set up periodically by an overvoltage, amplitude, and rate of rise which should be sufficiently high, and generated from a high enough power source to set up a current of density J binding down the wave velocity u, to greatly exceed usl. This density should be maintained for a time at least equal to that required for the avalanche plasma to fill the drift zone. The current, of density J set up by an overvoltage, complies with the following relationship:
AE At
J=Es-,
where AE/At is the increase of the electric field for a time interval At. Moreover, variation of the electric field as a function of distance into the depletion region is given by Poisson’s equation
-A_E - q n , Ax
E,
’
where n d is the doping concentration in that region. From this it follows that
A -x-- I ) , = - . J At qnd In order that u, be much higher than usl, the requirement is that
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Now, q = 1.6 x C and if we take, for example, nd = 10’’ cm-3 and usl = lo7 cm/sec (silicon), we obtain J % 1.6 x lo3 A cm-’.
’
A value of J = 8 x lo3A cm- is easily attainable, then u, = 5 x lo7cm/sec will be reached. If this is the avalanche shock-front velocity, the electron and hole plasmas will completely fill up the depletion region and bring about a sudden drop in the voltage across the terminals (Fig. IISa, period 1) as well
AvtLL perio 1 period
o period
period
period
period 3
u
FIG. 11.5. Diode voltage (a) and current (b) vs. time during TRAPATT operation AV-pulse overvoltage triggering the operation.
as of the field E in the avalanche and drift zones (Fig. II.6a) and a concomitant increase in the current up to the saturation value (Fig. IISb, period 1). The value of the field E is then much lower than that of E, and thus the plasma will be trapped, since extraction of the carriers will take place at relatively low velocity, that is, u, = p E < us,, p being the electron mobility. However, since the voltage and the field E are recovering in parallel with extraction of the plasma, the velocity u, will increase simultaneously and with that the rate of rise of recovery voltage and field will be consequently increased (Fig. IISa, period 2 and Fig. II.6b, curves 1, 2, and 3). During this period which lasts much longer than that for the voltage recovering in the IMPATT mode, the current (Fig. IISb, period 2) will stay close to its maximum value but suddenly drops toward the end of the period, to a very low value, corresponding to a small displacement current. Finally, in period 3 in Fig. 11.5, the current remains low and the voltage stays at a high level, slightly below the breakdown value V’ . At the end of
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S. TESZNER AND J. L. TESZNER
FIG. 11.6. Electric field vs. position in the TRAPATT'operation. (a) during carrier plasma trapping; (b) during recovery period: (1) at the start of residual carriers extraction; (2) during this extraction; (3) at the end of this extraction.
this period, the situation returns exactly to that found at the start of period 1, only to be set in motion again by a new overvoltage and current J pulse. The periodic regeneration of this overvoltage and of the J current can be assured by an external generator of appropriate design (for example, a capacitor next to the diode). It can, however, also be set up by a diode operating in IMPATT mode at the beginning of each TRAPATT period and at a frequency corresponding to the harmonic third or fifth of the TRAPATT oscillation (23). This will depend on the external circuit used, components for which have been studied on several occasions (28,29)and are noticeably more complex than for the IMPATT mode. The oscillation one has to generate will be rich in harmonics and, at the limit, will produce a square wave by compounding the fundamental and the third and fifth harmonic (this explains the squarewave form shown in Fig. 11.5). On the other hand, the power efficiency could then reach a very high value (theoretically about 80%), mainly because of the decreased power P d , , since to a high current Id, corresponds an almost cancelled voltage V,, and, reciprocally, to a high voltage vd, corresponds an almost cancelled current I d , . However, the mechanism as explained above, is itself a drawback, as far
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as the operating frequency is reduced and through the plasma trapping process, the noise level is increased. A comparison of the advantages and disadvantages of both modes will be brought on in later sections.
2. Device Fabrication a. I M P A T T mode. IMPATT diodes are fabricated using epitaxial growth and diffusion or ion implantation, or all three techniques successively. Figures II.7a-c show cross sections of the three categories, taken as representative examples. Figure II.7a corresponds to Fig. 11.l a (single drift
n n
(C)
FIG. 11.7. Some IMPATT diode shapes: (a) single drift-zone; (b) double drift-zone (as alternative single epitaxial + ion implantation and shallow diffusion); (c) improved Read diode Schottky barrier.
+
zone) with an n2+ layer (substrate) of low resistivity (from 0.001 to 0.003 R cm) on which an n layer is epitaxially deposited (doping concentration - 5 x 1015 cm-3 at f = 6 GHz up to - 2 x l O ’ ’ ~ m - ~ at f = 100 GHz*). A p2+ layer is then diffused into this layer of low rcsistivity value. Ohmic contacts are then formed on the top and bottom surfaces. For microwave applications, the thicknesses of the n2+ and p2+ layers will be minimized in order to diminish, as far as possible, any parasitic series resistance and nonuniform current flows through skin effect. Figure II.7b corresponds to Fig. II.le (double drift zone), which also has an n2 layer, on which an n layer is epitaxially deposited (doping concentration range as above). Two variants are possible-Either there is a second +
* This increase in the doping concentration n, with frequency, is motivated, as was explained on p. 328, by the need to increase the current density by decreasing the diode cross section as frequency is increased; moreover, it is also motivated by the reduction of the width W,in an almost inverse ratio to frequency change, the aim being then to increase the breakdown field E , (from 3 x 10’ V/cm to 6 x 10’ V/cm) and the punch-through field E,, (proportionally to nl/*).
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epitaxial deposition (31-33) of a p layer with doping concentration level +W,) similar to that of the n layer, the n layer thickness being en z (W,, while the p layer has a thickness ep z (W,, fW,); or, the second variant (30, 34a, 34b), the n layer has double thickness, corresponding to en 2 (WDN + W,, + W,) plus a thin p 2 + layer. In this case, the p layer could be formed by ion implantation with appropriate concentration and energy levels, followed by shallow diffusion to realize the p 2 + layer. One should note that the use of the double drift-zone structure has received widespread acceptance since first implemented. Figure 11.7~corresponds, in like manner, to Fig. 1 1 . 1 (two-step ~ doping), in which a rectifying metal-semiconductor contact (a Schottky barrier) replaces an n-p junction. An n layer (doping concentration 5 x 1015up to 1017 cmW3)is deposited on a substrate heavily doped: A thin n+ layer loi7up to 5 x loi7 cm-3) is diffused into this (doping concentration layer ; a rectifying metal-semiconductor contact is formed on that layer. This contact presents some advantages: The junction is abrupt, the parasitic resistance is reduced, and the heat-generated evacuation is facilitated. Moreover, the minority carrier storage in the p 2 + layer during the avalanche process is eliminated; otherwise, the out diffusion in the drift zone of those carriers would give rise to an increased reverse saturation current and thus to a drop in power efficiency (35). From a technological point of view, although it is not easy to make a suitable Schottky-barrier contact, there is the compensation of doing away with the second epitaxial deposit. Thus, this technique would appear to be gaining ground (21, 32, 3 6 3 9 , 76, 78, 80). However, despite the better thermal resistance of the Schottky-barrier diodes, their power handling capability seems presently to be somewhat lower than that of the np junction diodes, through a lower working temperature limit. In effect, the GaAs Schottky diodes are unable to withstand barrier temperature in excess of 200°C (instead 300°C for the pn junction) through the increased risks of the burnout (74) and the movement of the Schottky-barrier contact into the semiconductor (75) degrading the highefficiency IMPATT diodes' performances. Obviously, all the other structural categories, whose functions have been explained in the foregoing paragraph, can be implemented using these technologies. Moreover, instead of n-type IMPATT diodes, the complementary p-type ( p + p n + or p + p M ) diodes can also be fabricated. A recent experimental work (79) has brought in a proof that the electrical performances of those diodes in the millimeter wave range are comparable to the best values obtained with the n-type diode of the similar structure model. All such devices can now be fabricated using three materials-silicon, gallium arsenide, and germanium. However, for power devices the practical choice is limited to the first two. Silicon is more commonly utilized, but
+
+
-
-
-
-
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recently the use of gallium arsenide is spreading in the low microwave region-up to 30 GHz (21a-21c, 36-39, 74, 76, 77, 80). From a technical point of view, the use of GaAs ensures an avalanche zone that is appreciably more restricted in the frequency range under consideration. On the other hand, this advantage is somewhat lost as frequency rises, and is totally lost around 100 GHz (40). This is the result of the increase of field E , with increased operating frequency, which, as was mentioned previously, for silicon lead not only to an increase in the ionization rates, a, and a p , but also to an improvement in the results through a noticeable reduction in the a, /ap ratio. In fact, the ionization coefficients of Si and GaAs, and consequently the thicknesses of corresponding W,, practically become equivalent around 100 GHz. However, because the u,,~for GaAs is slightly lower than for Si (8 x lo6 cm/sec compared with lo7 cm/sec) the optimal length for W, will be slightly higher, and thus the ratio W, /W, will be slightly lower. Moreover, electron mobility at low field values for GaAs is notably higher than for Si, and the losses due to parasitic resistance in the outer n z + layer will be much lower for GaAs. This comparison proves an overall advantage for GaAs compared with Si. However, the thermal conductibility for GaAs, is only one third that of Si, which is a definite handicap for power device applications. The thickness of the epitaxial layer which corresponds to the width W, will obviously be a function of frequency, of the material employed and of the category of the structure concerned. It will vary within very large 4 8 pm at 6 GHz up to 0.4 and 0.7 pm at 100 GHz. limits-between The part of the avalanche zone in the width W will depend essentially on the category of the structure and will lie between 5 and 40% at 6 GHz; on the other hand, it will increase with frequency to reach 25-75% at 100 GHz. b. TRAPATT mode. The diode structure is qualitatively similar to that of the IMPATT diodes; however, there are some notable differences concerning the width Wand the median layer doping concentration (41). Also, the importance of the bonding to ensure adequate dissipation of the heat generated (42) should be pointed out. Figure II.8a,b shows two crosssectional views of the TRAPATT structure. The structure in Fig. II.8a is the simplest, and is qualitatively identical to that in Fig. II.7a.* It has a single median layer, generally of n-type material as shown here, the width being W 6 pm at 1 GHz up to 0.6 pm at 10 GHz and the doping concentration nd 6 x 10'4-10'5 at 1 GHz up to 6 x 1015-1016 cm-3 at 10 GHz.
-
-
-
-
--
* However, it will be noted that, likewise, the p-type TRAPATT diodes have been developed.
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S. TESZNER AND J. L. TESZNER
heal sink
(0
1
/
heat sink
(b)
FIG.11.8. (a) Conventional TRAPATT diode shape; (b) IMPATT-TRAPATT back-toback diodes shape.
We note that the width W is much lower, for a given frequency, than in the IMPATT diode structure. There are two reasons for this: First, the evacuation of carriers is carried out here at a speed u, 4 unsl;second, because it is important to maintain within certain limits the product nd W determining the space charge in the median zone. These limit values are to be deduced from the following considerations. The space charge should be sufficiently low (a) To avoid too rapid a decrease of the field E with distance, which would require, to compensate for this decrease, a very high voltage overshoot to start the TRAPATT operation, and which would reduce the zone WA too much where the avalanche is formed and, consequently, the quantity of ionized carriers generated; (b) to increase power efficiency, the rf power being a decreasing function of nd W ( 4 1 ) ; (c) to avoid the avalanche re-striking before the carrier plasma has been removed from the median region (41). On the other hand, this space charge should not be too low, since it is important to avoid the zone WA spreading over the entire width W, as this would undermine the very principle of the TRAPATT mode. The doping concentrations of the median layer are relatively moderate for the same reasons and, also, because it is generally convenient to work here with as low a power density as possible. In effect, when one considers the relatively high power levels that such diodes are designed to handle, it is important to minimize the dissipation per unit area. This is, of course, indispensable only for CW operation ; however, in the short-pulse case, the current density quoted below can be practically attained without sophisticated heat sinking. Moreover, for devices operating in a CW mode, special attention must be given to the bonding material of the heat sink, in copper or in diamond; the latter will guarantee almost double level power dissipation (42). The structure shown in Fig. II.8b is of more recent design (43, 44) and has a median zone formed of two layers as back-to-back diodes. There is
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certainly an analogy with Fig. II.7b, but this is only apparent. In effect, these two diodes operate in different modes. One of them with a p layer is thus optimized for efficient IMPATT operation and the second with the n layer is optimized for TRAPATT highefficiency oscillation. Recently, such an IMPATTPRAPATT device, fabricated by implantation of boron ions into an epitaxial n layer for operation in the X band, has been presented ( 4 4 , the widths Wpand W, and the doping concentration p and n of these two layers being as follows: Wpz 0.5 pm, p z 8 x 10l6cm- 3, W, z 0.6 pm, n = 8 x 1015~ m - One ~ . side of such a structure is thus designed for IMPATT oscillation at a sufficiently high frequency and a sufficiently high amplitude to be appropriate for initiating the TRAPATT mode of the other side of the structure. Hence this device contains internal triggering for the TRAPATT-mode operation. Since, for a given frequency of the TRAPATT operation, the total thickness of the median layers is much higher here than for a diode of the type shown in Fig. II.8a, a much larger cross section can be allowed without particularly interfering with the diode-circuit impedance-matching problem. It is thus possible to obtain much higher power levels, particularly in a CW operation. Regarding the material chosen, it appears that only silicon has been used, to date, for all TRAPATT structures. Without prejudicing future possibilities and looking at this from a purely technical point of view, one must note that the reasons which favor the use of gallium arsenide, mentioned in the previous paragraph, lose a lot of weight, whereas the disadvantages for its use remain valid. 3. Electrical Characteristics and PeNormances
By the term “electrical characteristics,” we understand breakdown voltage, operating voltage, and bias current density for the different operating frequency ranges. As with fabrication parameters, these characteristics vary greatly, not only as a function of the operating frequency, but also according to the structural category chosen and the implementation parameters, notably the choice of semiconductor material. Likewise, concerning the performance values, we understand rf power output and efficiency at a given frequency in an oscillator mode, together with a possible amplifier function. In the latter case, the power gain, the bandwidth, as well as the nonlinearity parameters will also be specified. Thus, we will generally give not a specific value, for the characteristics and the performance values, but quite a wide range of values. Obviously, the optimal performances will correspond to experimental devices. Also, the problem of noise behavior will be examined and data given on noise figures,
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S. TESZNER AND J. L. TESZNER
both in the oscillator and amplifier functions. Lastly, mention will be made of the temperature effect and the reliability problem will be touched up on, notably concerning the burnout risks. a. ZMPATT diodes. Table 11.1 sets out the electrical characteristics which take the latest publications available into consideration-namely ( 1 4, 21a-21c, 30-32,34a,b, 37-39,45-47, 74, 76-80). TABLE 11.1 ELECTRICAL STATIC
CHARACTERISTICS OF SILICON AND DIODES (AT
ROOM
GALLIUM ARSENIDEIMPATT
TEMPERATURE^
Frequency range (GHz)
Breakdown voltage V, (V) Si diodes GaAs diodes Operating voltage V, (V) Si diodes GaAs diodes Bias current density J , (A cm-’) Si diodes GaAs diodes
8-12
30-60
90-110
135-75 110-35
30-15 25-12
20-8
150-85 125-45
35- 16
22-9
W1500 300-1200
30-14 8000-20000 5000-15000
30000-60000
The upper operating ranges for V, and V, diodes correspond to double-drift-zone structures. The lower ranges to single-drift-zonestructures.Inversely, for the J , ranges.
Concerning performance figures, both operational modes will be considered in turn-first the oscillator mode and then the amplifier mode. We set out the oscillator performance figures in summary form in Table 11.2. The operation of IMPATT diodes as nonlinear power amplifiers puts to best effect the negative resistance characteristic. This problem has already been given theoretical treatment and experimentation in several cases, notably (48-52) and these have recently led to applications of these diodes as amplifier devices. These have been of the reflection type using an external circulator to separate input and output (reflected) signals (Fig. 11.9). Two operational modes are then possible-stable or injection locked. In the first case, without input power the diode is stable-that is, when the resistance of the load is greater than the magnitude of the negative resistance of the packaged diode at resonance frequency. In the second case, however, a free running oscillation takes place unless an input signal of adequate power in the proper frequency band is applied; nevertheless, when the diode is
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TABLE 11.2 OSCILLATOR IMPATT DIODESPERFORMANCES‘ Frequency range (GHz)
Oscillator output power Po,, (W) Si diodes Single drift zone Double drift zone GaAs diodes Single drift zone Double drift zone Oscillator power efficiency q% Si diodes Single drift zone Double drift zone GaAs diodes Single drift zone Double drift zone
8- 12
3&60
90-110
4.7-2.7b 6.gb
0.5-0.3 1-0.5
0.24-0.16 0.38-0.2
8-4” 2.3‘
0.75-0.2
11-8‘ 12
-
35-20 16‘
--
11-8‘ 1410‘
9-7 --12-10
16-11
a The performances specified above concern experimental devices, except concerning the singledrift-zone silicon X-band diodes. As a general rule, for a given structure model, the output power and efficiency decrease with increasing frequency. These values are taken in CW operation. In pulsed-wave operation, a power level of 10 W has been obtained. These values are also taken in CW operation. In pulsed-wave operation Po,, = 4.1 W with q = 21 % has been obtained. It will be noted that some GaAs double drift-zone IMPATT diodes have been recently made for an operating frequency of 20 GHz. The maximum output power of 1.2 W at 21 GHz, with an efficiency of 15.6%has been obtained (76). dThe power values obtained are in CW operation, but with heat-sink cooling. The maximum value for power efficiency has been obtained with lowhigh-low structure samples (Zlc, 77). ‘These values of power efficiency are already two years older than the other values specified.
-
suitably driven, this oscillation is locked and the device will deliver a coherent single-frequency amplified power output. The output powers can thus be obtained with an acceptable amplification gain ( 2 5 dB) and would appear almost equal to the oscillator output powers, at those same frequencies (Fig. II.lO).* This is an interesting However, to date, the output powers in the amplifiers mode are at a considerably lower level than the maximum output levels obtained by oscillators elsewhere.
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S. TESZNER AND J. L. TESZNER Input Matching
Circulator IMPATT Diode output
FIG.11.9. Schematic structure of the reflection-type IMPATT diode amplifier.
result which justifies the complex circuit structure. In return, the system does have three drawbacks-a relatively high amplification distortion, a relatively poor gain-bandwidth product, and a relatively high noise level (for the power devices, the latter is especially troublesome in oscillator applications). We will examine these briefly. The amplified signal distortion is a result of the inherent nonlinearity associated with large-signal device operations. The susceptance varying with rf amplitude and the signal level being dependent on the diode admittance, strong parametric effects are observed in the high-power devices. On the other hand, the introduction of two, or more, nonharmonically related signals into the input of such an amplifier, results in the generation of intermodulation signals in the amplifier output (50, 55). The power dissipated in those spurious signals is supplied mostly by the fundamental signals; the available output power of the amplifier is then decreased and the power gain reduced. This phenomenon, therefore, presents a double drawback. Amplifier Power Gain
5-
0
Amplifier pOut I
05
1
I
1’5
Oscillator Po?
FIG.11.10. Typical curve of the gain per stage vs. normalized power for IMPATT diode reflection-type amplifier (X band). Continuous stroke available for the stable as well as for the injection-locked amplifier; hatching-strokes for the stable amplifier only.
MICROWAVE POWER SEMICONDUCTOR
DEVICES.
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Regarding the voltage gain-bandwidth product in the locked-injection amplifier, it is basically determined by the Q factor (reactance/resistance slope) of the oscillator. As the intrinsic diode Q value is rather high (Q 15 in the X band and Q 22 in the Ku band) the gain-bandwidth product is relatively low. This product is slightly higher for the stable amplifier, but it still remains a low absolute value. Figure 11.11 shows, as an example, the
-
-
'h I
20
10
-
-
\
FIG. 11.1 1. Voltage gain-3 dB bandwidth product vs. power gain per stage for reflectiontype X band IMPATT diode power stable amplifier.
graph for this product typical for reflection-type X-band power-stable amplifiers. The graph has been drawn within the gain limits that are useful for power amplifiers: Higher gain values correspond to small signal operations, while lower gain values correspond to the power output operations zone, where the gain value drops rapidly when load conductance is increased. The operations noise problem, both in the amplifier mode and in the oscillator mode, merits special attention because avalanche diodes are very noisy devices. For this reason, a lot of studies, both theoretical and experimental, have been undertaken. We will cite a restricted list, including some of the more recent publications-(16, 36, 52-60, 81). The noise referred to is a result of the physical mechanism of the diode operation, i.e., of the impact ionizing collisions inside the avalanche zone during avalanche formation. Since the number of collisions is directly related to the ion multiplication coefficient, noise increases with the amplitude, which results from the statistical nature of the generation rates of the ionized carriers. One sees, therefore, that the noise value will increase with the bias current density, at least beyond a certain value of the latter. Under these conditions, the noise value will increase with the doping concentration
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L. TESZNER
of the avalanche zone; this is, indeed, the case for Ge and GaAs, the case of Si being a little special from this point of view. We shall come back to this later. Moreover, two minor sources of noise should be mentioned-First, the up conversion of low-frequency noise, and second, thermal noise. The first source is due to the inherent nonlinearity of the diode characteristics and the up conversion of both avalanche and thermal low-frequency noise generated within the diode. This phenomenon will downgrade noticeably the noise performance of the large-signal IMPATT amplifier. The second minor source is rf thermal noise in the diode, but this noise can generally be considered negligible with respect to that produced by the avalanche mode. Noise in the oscillators normally comes through amplitude modulation and frequency modulation (AM and FM). AM noise is characterized by the signal-to-noise power ratio measured in a window of known bandwidth. The AM noise figure is relatively fair. Thus, for oscilllators operating in the frequency range 6 1 2 GHz, the measured values of this power ratio (double sideband, in the 100 Hz window) are *120-140dB according to the frequency from the carrier; the noise figure is slightly lower ( - 3 dB) for GaAs than it is for Si. In the Ka band (26.5-40 GHz), the noise figure seems to be quite similar, 115-135 dB. On the other hand, the FM noise figure is high. This noise is characterized by the rms frequency deviation Ahms, defined by the FM noise measurement
-
-
wheref, is the carrier frequency, B is the window bandwidth used to measure Afrrms,kTo is the thermal noise energy at a standard ambient temperature, To = 290"K, and Q,, is the oscillator external quality factor. Thus, the best M,M reported values in the 6-12 GHz frequency range are 36 dB for Si (59) and 30 dB for GaAs (36). The noise figures seem to increase slowly with the diode transit-time frequency: At 30 GHz, for instance, MFM is approximately equal to 36 dB for GaAs (56). The situation is somewhat similar for reflection-type IMPATT amplifiers. Thus, the noise figure in the X band is, for the best diode samples available, 27 dB for Si and 17 dB for GaAs (51). For higher frequency diodes, the best results obtained so far are 26 dB for GaAs at 30 GHz (56) and 36 dB for Si at 53 GHz (60). This noise figure appears to be highly dependent on the output power. Thus, it ranges from 35-55 dB for Si diodes in the X band, when one raises the rf-dc voltage ratio from 10 to 30% (53, 55).
-
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It seems, moreover, that the difference between these noise figures for Si and GaAs diodes tends to diminish and even disappear as the power output value rises. Figure II.l2a, deduced from the experimental results (36), is a graphic illustration of this phenomenon. The number of experimental results available concerning this subject is certainly too restricted to draw any formal conclusions for the present. Nevertheless, one can bring together these experimental data with the recent forecasts concerning the effect of the doping concentration increase on the noise figure, which will be discussed below.
Pout I
1
2w
FIG. 11.12. (a) M,, noise figure vs. power output for 6 GHz Si and GaAs IMPATT diodes-after experimental results (36). (b) M,, noise figure vs. power efficiency for Si and GaAs IMPATT diodes high-low structures for 6 GHz-after computed results (16, 58).
The measurements mentioned have all been taken on homogeneous median-band diodes (Fig. ILla), with relatively low power efficiency values. However, far more sophisticated diode structures, shown in Fig. 11.1~ (called high-low) or Fig. 11. Id (called low-high-low), have already undergone study concerning noise figure, and this has provided some complementary information which merits brief discussion.
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S. TESZNER AND J. L. TESZNER
The structure in Fig. 11.1~is essentially characterized by a notably increased doping concentration in the avalanche zone. This allows a considerably improved power efficiency to be obtained, but as was already pointed out, this is countered by the noise figure increasing when the diode is implemented in GaAs (16). In effect, if the doping concentration of this zone is increased tenfold, one can forecast an increase of the power efficiency from around 18 to 28% for a 6 GHz structure, but the noise figure will rise by approximately 8 dB. On the other hand, for a Si diode with the same initial doping concentration, variation of the noise figure with increase of the doping concentration level would be far less noticeable. Thus, the advantage that one would apparently gain by using GaAs instead of Si, as regards the noise figure, would appear to be considerably limited. Figure 11.12b, which derives from recently published analysis (16, 58), illustrates this prediction. The supposed phenomenon can easily be explained, since it is similar to that described previously of balancing around the 100 GHz frequency level of the avalanche-zone width in both GaAs and Si. In effect, the advantages for GaAs at low doping concentration level from a noise figure point of view, result essentially from the consideration that the ionization coefficient a, / a p , characteristic of this material, is of the order 1, whereas for Si, this ratio goes up to 10. However, when the doping concentration level is increased in Si, this same ratio diminishes and finally tends toward 1; for this reason, the increase in the noise figure inherent in the increased concentration is compensated to a certain degree, and even beyond that degree, by the contrary effect resulting from a decrease in the ratio a,/ap. On the contrary, for the GaAs case, where this ratio remains practically unchanged E 1, an increase in the doping concentration figure brings about a considerably degraded noise figure. The case of the low-high-low structure in Fig. II.ld is different. In effect, we note here that the doping concentration of the avalanche zone is relatively low and is of the same degree of magnitude as that characterizing the homogeneous median zone of Fig. II.la. At this point, the advantage of GaAs with respect to Si, from a noise point of view, as noted on the simple structures used in the diode, should normally remain valid for sophisticated structures as in Fig. II.ld. It now remains for us to consider briefly the temperature effect and reliability problems. The temperature of the n-p junction of the diode has an influence, at once, on the electrical performances and on the reliability factor. The temperature influence on electrical performances has been studied experimentally (61-63). It has been proved that increase in temperature produces a certain decrease in the transit-time frequency, because uSl decreases by 15% for Si and by 25% for GaAs when one goes from 25" to 200°C. However, the considerable increase of power efficiency and power
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
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output with temperature that one noted in the first investigations seems to be apparent only in diodes, where the drift zone is not totally punched through at room temperature (63), and this nowadays tends to be considered as a technological flaw in the structure. It is possible, in effect, to correct this fault by raising the temperature, since the voltage then applied to the drift zone at the terminals is increased through the breakdown voltage V, rising with temperature. At this point, the drift zone could be punched through and the parasitic resistance of the unswept portion of the drift zone would be eliminated, which would explain the notable improvement measured. In this case, moreover, one can gain in power efficiency with external heating of the heat sink (63). O n the other hand, if the drift zone is punched through at room temperature, increase in output power is solely due to an increase in the voltage V,, and possibly also in bias current. Moreover, it would not seem that the increase of power efficiency must then necessarily accompany that of output power values. To optimize the result, it would then be preferable, in contradistinction to the foregoing considerations, to operate with the heat sink cooled (Zlc), so as to be able to work with the highest bias current density without exceeding the limiting internal diode temperature. It is, moreover, obvious that to put to best use the power and frequency possibilities of a diode, it is of advantage to evacuate generated heat as much as possible. It is, in effect, important not to exceed an average temperature level recognized as the present-day limit case, beyond which the burnout risks increase rapidly (200°C for pn junction and somewhat lower for MS junction), within the diode and notably, at the junction. This phenomenon is particularly relevant when the carrier current is not uniformly spread through the cross section of the diode and includes filaments of locally very high current intensity. Because of the physical mechanism of the diode operation in the setting up of the avalanche, it is, moreover, difficult to design or conceive uniformity for current distribution. The uniformity disparity will therefore be all the more likely to favor high current filament formation if the operational temperature is high, a fortiori, when the diode is overheated. These flaws can be made worse by a cumulative process whereby the current concentration in filament form brings about overheating which, in turn, tends to increase the current density in the filaments. This cumulative phenomenon will be all the more striking when dc negative conductance appears, because in those local regions the breakdown voltage is decreasing when the current density is increasing. In the peculiar case of the Schottky-barrier diode, an overheating could be, moreover, injurious through a rapid metalsemiconductor contact degradation. Thus, any flaw in the structural uniformity and hence overheating and in particular localized overheating, will prove serious drawbacks to the reliability factor.
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S. TESZNER AND J. L. TESZNER
b. TRAPATT diodes. Experimental data concerning electrical characteristics and performances of TRAPATT diodes are much more restricted and sparse than those concerned in the IMPATT diodes. We will limit ourselves here to mentioning some of those likely to characterize the situation in this domain (44, 72). Table 11.3 summarizes the main data elements relating to TRAPATT diode devices operating as oscillators. TABLE 11.3 SOMEELECTRICAL CHARACTERISTICS AND PERFORMANCES OF SILICONTRAPATT DIODES (ATROOMTEMPERATURE) Frequency range (GHz)
Breakdown voltage V, (Vyl Conventional structure IMPATT-TRAPATT structure Bias Current density J, (A cm-’) Conventional structure IMPATT-TRAPATT structure Oscillator output power Po,, (w)b Conventional structure IMPATT-TRAPATT structure Oscillator power efficiency v% Conventional structure IMPATT-TRAPATT structure
1-2
3-6
8-1 1
140-90
66-45
45-30 35
5000
- 1 m
- 2 m 15000
3W120 12w
100-30
16-7
-
-
-
15-8 60-25
-
2 33 t 29
55-25
-
Operating voltage Vo for conventional structure-V, 1.5 V , , for IMPATT-TRAPATT structure-V, 1.2 Vs All power values mentioned are the peak values relating to the pulsedwave operation. This performance has been attained at 1.1 GHz with a series-stacked fivediode unit; = 25% (64).
It is to be noted that contrary to IMPATT diode structures, we have given the output power in pulse wave operation and duty cycles which are low for the highest level powers cited. Thus, for example, for P,, = 1200 W the duty cycle is only a few thousandths at most. It is only under these operational conditions that TRAPATT diodes have power outputs that justify and bring out the advantages of such devices, since they exceed the possibilities of the IMPATT structures by a considerable margin. Moreover, we have intentionally left aside any consideration of operational cases with a high frequency-power product, but where the power
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
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efficiencyis correspondingly diminished with respect to the values now possible with newer structures of IMPATT diodes-judging by the results of recent experiments (21c, 77). Power may be obtained from a TRAPATT diode, either at the fundamental operational frequency, or possibly at the second or third harmonic level (68). Although this latter-mentioned approach raises the operational frequency of the device, the power efficiency is decreased below those values obtained at this time with IMPATT diodes (cf., Table 11.2). There would not, therefore, appear to be any apparent advantage in this approach. Table 11.3 deals only with silicon devices, this being almost the only material used in the fabrication of TRAPATT diodes. For technical and cost reasons, this material has appeared the most practical. But, although only oscillator characteristics have been specified, there have been TRAPATT reflection-type amplifiers, in particular in pulsed-wave operation or in class C, with a relatively broad bandwidth. Peak output power values of 60-80 W have been noted at 3.3 GHz, with a power gain of from 5 to 6.5 dB and a 3 dB bandwidth of 12-14%, the voltage gainbandwidth product attaining 2 GHz and the power efficiency 18% (69). These are certainly noteworthy results, yet such figures can be attained also with IMPATT diodes with one of several recently presented structures (1416, 21a-21c) and, in so doing, use relatively less complex circuits to obtain comparable results. The noise figure for TRAPATT diodes is apparently worse than for IMPATT diodes. This feature may be considered normal, when the relatively longer duration of the carrier plasma is taken into consideration. The best results, for instance, for f = 4 GHz with Po,, = 1 W are N F A M = - 145dB and NFFM z 60dB, defined by M value (cf. 20). Now, for IMPATT structures, NF,, is exactly in the same order of magnitude, but in return NF,, is some 20-30 dB lower. Lastly, a few comments may be made concerning temperature effect and its consequences. Power efficiency apparently decreases quite rapidly with temperature, and this can be brought into evidence by increasing the pulsewidth, for a given pulse frequency repetition rate, and noting that beyond a certain threshold there is a rapid drop in efficiency (66).This is probably the consequence of mismatching due to heating, and enhanced by a cumulative process. In effect, the increase of the duty cycle beyond a certain critical value for a given power level and a given diode structure brings about heating which in turn considerably increases V , , hence, as a direct result, the ratio V, /V, is no longer optimum and there is a drop in power efficiency, an increase in dissipated power, and further heating enhancement. Such a phenomenon is difficult to control, even if the V, value is adjusted each time the pulsewidth value is changed. For these reasons, TRAPATT diodes are gen-
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S. TESZNER AND J. L. TESZNER
erally used in pulsed wave operation with high peak power values but, nevertheless, relatively low average power values. It must also be noted that when the operational frequency is increased, the power dissipated increases per unit of cross section of the diode structure, since the current density .lo and the breakdown voltage V, increase and the power efficiency decreases. Despite a reduction of the drift-zone width W, which favors evacuation of the heat generated, the average output power has to be still further reduced, in order that excess temperatures be avoided. Generally speaking, to obviate serious burnout risks, it appears advisable to keep the maximum TRAPATT structural temperature within the limit value noted for IMPATT structures, i.e., 200°C. Indeed, it is the nonhomogeneous distribution of instantaneous power densities that lead to greater risks (compared with IMPATT diodes) of overheating by current concentration in filaments, as previously described. 4. General Discussion and Conclusions
Let us recapitulate the advantages and inconveniences of IMPATT and TRAPATT diodes. Concerning IMPATT diodes, we note that their advantages, apparently, are: (a) very wide range of operational frequencies in the microwave region, extending to date up to around 110 GHz; (b) appreciable output power levels ( N 5 W in CW at the bottom end of the range, 0.3 W at the top end) with the possibility of increasing these power levels by using several diodes in parallel configuration, this being favored by the fact that the IMPATT device is a majority carrier component, or in series configuration, where the advantage is a higher output impedance which helps output matching; (c) an equally appreciable power efficiency ( 30%at the bottom end of the range, 10% at the top end); (d) relatively simple structure, which facilitates fabrication. These advantages hold both in the amplifier and in the oscillator modes of application. However, concerning the disadvantages, a distinction has to be drawn regarding these two operating modes; so we will first discuss the general case and then the particular case of the amplifier. The general disadvantages are: (a) high noise figure (FM), at least 30 dB at the bottom of the frequency range, increasing with operating frequency, output power, and the structure doping concentration ; (b) certain risk of burnout occurring. The disadvantages specific to the amplifier application are: (a) small gain-bandwidth product, the bandwidth reaching 10% at the
-
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
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bottom of the frequency scale, but this relative value falls off very quickly with increasing frequency ; (b) high level distortion due to the characteristics inherent nonlinearity and low intermodulation ratio; (c) relative complexity of circuits compared with circuits utilizing threeterminal devices. Certainly, it is possible to eliminate, at least partly, some of these specific disadvantages, but the circuit then becomes appreciably more complex, for example, if one adds more tuning elements to the circuit to widen the bandpass, or distributes the diodes in a nonresonant propagating circuit formed by the iteration of a large number of networks with each section containing one diode (51, 73). However, the improvement of one deficient characteristic will bring about deterioration of another-notably power gain, power output, or power efficiency. Where TRAPATT diodes are concerned, their advantages are: (a) high power output levels, in excess of loo0 W (with five diodes in series configuration) at the bottom end of the frequency range (1 GHz) and still capable of 15 W (per diode) at the top end of the range ( - 10 GHz), but still in pulsed-wave operation; (b) a high power efficiency figure at the bottom end of the frequency range ( - 60% at 1 GHz) but falling off when either output power or frequency are raised (q I 30% at LO GHz). In this rather narrow frequency range where their applications could be proposed, their disadvantages are similar to those of the IMPATT diode already mentioned, but are, in fact, more prominent-notably, the noise level is much higher and the risk ofburnout is much greater, and when in the amplifier application, they give a bandpass relatively just as narrow but at a lower frequency comparatively and with still greater distortion. Lastly, their fabrication is more delicate and their utilization requires a more sophisticated circuit. This brief recapitulation enables us to draw the following conclusions, valid for microwave power semiconductor devices. As oscillators, the comparison of advantages and inconveniences of IMPATT devices comes out positive. Thus application can be envisaged throughout the frequency ranges under consideration, up to their limit values as set out above. For millimeter wave applications, IMPATT diodes are, for the moment, the preferred devices for power solid-state sources. However, for lower frequency range and moderate output power (some hundred milliwatts), Gunn diodes are competitive with IMPATT diodes because of their much better noise figure. As amplifiers, the value judgment has to be tempered with caution, since the advantages and disadvantages are on a par. IMPATT diode application
-
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S. TESZNER AND J. L. TESZNER
can, therefore, be envisaged throughout the frequency range and power values under consideration where three terminal devices do not as yet exist. Other solutions have of course to be taken into account, notably with three terminal devices driving varactor multipliers. Moreover, Gunn diodes are also competitive in the same frequency and power range as for the oscillators, namely through their lower noise figure and larger gain-bandwidth product. It is nevertheless possible that IMPATT diodes be preferably used for amplifier applications as for oscillator applications, in the millimeter waveband. Concerning TRAPATT diodes, however, the intrinsic technical attraction of the structure can be doubted to some degree. Certainly, if one compares their characteristics with those of IMPATT diodes, the attractiveness is justifiable at the bottom end of the microwave frequency region (from 1 to 6 GHz). However, when one extends the comparison to other devices that will be considered in this review, the judgment must be given nuance. Thus, the conclusions drawn only remain positive in a very narrow frequency band and it would be possible that even this restricted preferential range of application would disappear in a near future. We shall touch on this again in the last section of the review (to be published in Part 11).
REFERENCESFOR SECTION II,A AVALANCHE DIODES 1. J. S. Thownsend, “Electricity in Gases.” Oxford Univ. Press, London and New York, 1914. 2. W. Rogowski, Arch. Elektrotech. (Berlin) 25, 55 (1931); 26, 643 (1932). 3. L. B. Loeb and J. Meek, J . Appl. Phys. 11, 438, 459 (1940). 4. S. Teszner, Bull. Soc. Pr. Elec. 6, 61 (1946). 5. K. G. McKay, Phys. Rev. 94, 877 (1954). 6. S. L. Miller, Phys. Rev. 99, 1234 (1955). 7. F. B. Llewellyn and A. E. Bowen, Bell Syst. Tech. J . 18, 280 (1939). 8. W. T. Read, Bell Syst. Tech. J . 37, 401 (1958). 9. T. Misawa, I E E E Trans. Electron Devices 13, 137, 143 (1966). 10. D. L. Scharfetter and H. K. Gummel, I E E E Trans. Electron Devices 16, 64 (196 ). 2 1. J. L. Blue, Bell Syst. Tech. J . 48, 383 (1969). 12. D. Delagebeaudef, Rev. Tech. Thornson-CSF 1, 309 (1969). 13. B. Hoefflinger, IEEE Trans. Electron Devices 13, 151 (1966). 14. G . Salmer, 3. Pribetich, A. Farrayre, and B. Kramer, J . Appl. Phys. 44, 314 (1973). I S . H. C. Huang, I E E E Trans. Electron Devices 20, 482 (1973). 16. S. Su and S. Sze, I E E E Trans. Electron Devices 20, 541 (1973). 17. D. L. Scharfetter, W. J. Evans, and R. L. Johnston, Proc. IEEE (Leti.) 58, 1131 (1970). 18. W. J. Evans, IEEE Trans. Electron Devices 19, 746 (1972). 19. E. D. Johnson, R C A Reo. 26, 163 (1965). 20. D. L. Scharfetter, I E E E Trans. Electron Devices 18, 536 (1971).
MICROWAVE POWER SEMICONDUCTOR DEVICES. 1
35 1
2 l a . W. R. Wisseman, D. W. Shaw, R. L. Adams. and T. E. Hasty, Solid-State Res. ConJ Denver, Colorado. 1973, Abstr. 29. 21b. J. C. Irvin, L. C. Luther, and D. J. Coleman, Jr., Solid-State Res. ConJ Denver, Colorado, 1973. Abstr. 30. 21c. C. K. Kim, W. G . Mattei. and R. Steele, Cornell Elec. Eng. High Frequency Generation Amp/$ ConJ, 1973, p. 43. 22. H. J. Prager, K. K. N. Chang, and S. Weisbrod. Proc. IEEE (Lett.) 55, 586 (1967). 23. R. L. Johnston and D. L. Scharfetter, IEEE Trans. Electron Devices 16, 905 (1969). 24. D. J. Bartelink and D. L. Scharfetter, Appl. Phys. Lett. 14. 320 (1969). 25. A. S. Clorfeine, R. J. Ikola, and L. S. Napoli, RCA Rev. 30, 397 (1969). 26. B. C. De Loach, Jr. and D. L. Scharfetter. IEEE Trans. Electron Devices 17, 9 (1970). 27. D. L. Scharfetter, Bell Syst. Tech. J . 49, 799 (1970). 28. W. J. Evans, IEEE Trans. Microwave Theory Tech. 17, 1060 (1969). 29. B. C. De Loach, Jr., IEEE J . Solid-State Circuits 4, 376 (1969). 30. T. E. Seidel, R. E. Davis, and D. E. Iglesias, Proc. I E E E 59, 1222 (1971). 31. R. S. Ying, Proc. IEEE (Lett.) 60,1104 (1972). 32. M. Omori, F. Rosztoczy, and R. Hayashi, Proc. IEEE (Lett.) 61, 255 (1973). 33. J. V. Bouvet, Proc. European Microwave ConJ, Brussels, Belgium, 1973. p. A.8.1. 34a. W. C. Niehaus, T. E. Seidel. and D. E. Iglesias, IEEE Trans. Electron Devices 20, 765 (1973). 34h. T. E. Seidel, W. C. Niehaus, and D. E. Iglesias, Solid-Stare Res. ConJ Denver, Colorado. 1973, Abstr. 31. 35. T. Misawa, Solid-Stare Electron. 13, 1363, 1369 (1970). 36. J. C. Irvin, D. J. Coleman, Jr., W. A. Johnson, I. Tatsuguchi, D. R. Decker, and C. N. Dunn, Proc. IEEE 59, 1212 (1971). 37. H. C. Huang, P. A. Levine, A. B. Gobat, and J. B. Klatskin, Proc. IEEE ( L e f t . )60, 464 (1972). 38. M. Migitaka, M. Nakamura, K. Saito, and K. Sekine, Proc. IEEE (Lett.)60,1448 (1972). 39. M. Migitaka, M. Nakamura, K. Saito, and K. Sekine, Proc. European Microwave Con$, Brussels, Belgium, 1973, p. A.8.5. 40. J. Grierson and S. O’Hara, Solid-State Electron. 16, 719 (1973). 41. A. S. Clorfeine, IEEE Trans. Electron Devices 18, 550 (1971). 42. J. M. Assour, J. Murr. Jr., and D. Tarangioli, RCA Re;. 31, 499 (1970). 43. W. J. Evans, T. E. Seidel, and D. L. Scharfetter, Proc. IEEE 58, 1294 (1970). 44. T. T. Fong, R. S. Ying, and D. H. Lee, Proc. I E E E 61, 1044 (1973). 45. L. P. Marinaccio, Proc. IEEE (Lett.) 59, 94 (1971). 46. W. E. Schroeder and G . I. Haddad, Proc. IEEE 61, 153 (1973). 47. F. J. Hilsden and C. M. Butler, Proc. European Microwave ConJ, Brussels, Belgium, 1973, p. A.8.2. 48. M. E. Hines, IEEE Trans. Electron Deoices 17, 1 (1970). 49. Y. Takayama. I E E E Trans. Microwave Theory Tech. 20, 266 (1972). 50. R. J. Trew, N. A. Masnari, and G . 1. Haddad, IEEE Trans. Microwave Theory Tech. 20, 805 (1972). 5 1 . H. C. Bowers and W. H. Lockyear. Int. Solid-State Circuits. Con$ Philadelphia, 1973. 52. M. E. Hines, I E E E Trans. Electron Devices 13, 158 (1966). 53. A. M. Cowley, Z. A. Fazarine, R. D. Hall, S. A. Hamilton, Chu-Sun Yen, R. A. Zettler, IEEE J . Solid-State Circuits 5, 338 (1970). 54. H. Johnson and B. B. Robinson, Proc. IEEE (Lett.)59, 1272 (1971). 55. R. G . Giblin, E. F. Scherer, and R . L. Wierich, IEEE Trans. Electron Devices 20, 404 (1973).
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56. K. P. Weller, I E E E Trans. Electron Devices 20, 517 (1973). 57. J. S. Goedbloed, I E E E Trans. Electron Devices 20. 752 (1973). 58. S. Su and S. M. Sze, I E E E Trans. Electron Devices (Lett.)20, 755 (1973). 59. W. Harth and G. Ulrich, Electron. Lett. 5, 7 (1969). 60. T . P. Lee, R. D. Stanley, and T. Misawa, I E E E Trans. Electron Devices 15, 741 (1968). 61. E. Allamando, E. Constant, and M. Lefebvre, Onde Elec. 48, 496 (1968). 62. W. E. Schroeder and G. 1. Haddad, Proc. I E E E (Lett.) 59, 1242 (1971). 63. W. J. Chudobiak, R. McKillican, and V. Makios, Proc. I E E E (Lett.)60, 340 (1972). 64. S. G. Liu and J. J. Risko, R C A Rev. 31, 3 (1970). 65. G. Gibbons and M. 1. Grace, Proc. I E E E (Lett.) 58. 512 (1970). 66. R. S. Ying and N. B. Kramer, Proc. I E E E (Lett.), 58, 1285 (1970). 67. M. 1. Grace, H. Kroger. and J. Telio, Proc. I E E E (Lett.) 60, 1444 (1972). 68. S. G. Liu, Proc. I E E E 59, 1216 (1971). 69. A. Rosen, J. F. Reynolds, S. G. Liu, and G. E. Theriault, R C A Rev. 33, 729 (1972). 70. W. J. Evans, Proc. I E E E (Lett.)60, 125 (1972). 71. P. K . Chaturverdi and W. S. Khokle, I E E E Trans. Electron Devices 20, 353 (1973). 72. R. J. Chaffin, Proc. I E E E (Lett.)59, 1270 (1971). 73. S. F. Paik, I E E E Trans. Microwave Theory Tech. 20, 202 (1972). 74. K. P. Weller, A. B. Dreeben, H. L. Davis, and W. M. Anderson, I E E E Trans. Electron Devices 21, 25 (1974). 75. D. J. Coleman, Jr., W. R. Wisseman, and D. W. Shaw, Appl. Phys. Lett. 24, 355 (1974). 76. M. Migitaka, A. Doi. K. Saito, and K. Sekine, Proc. I E E E (Lett.)62, 141 (1974). 77. R. E. Goldwasser and F. Rosztoczy, Appl. Phys. Lett. 25, 92 (1974). 78. T . Watanabe, H.Kodera, and M. Migitaka, Electron. Lett. 10, 7 (1974). 79. G. A. Swartz, Y. S. Chiang, C. P. Wen, and A. Gonzalez, I E E E Trans. Electron Devices 21, 165 (1974). 80. W. R. Wisseman, D. W.Shaw, R. L. Adams, and T. E. Hasty, I E E E Trans. EIectron Devices 21, 317 (1974). 81. F. Diamand, Electron. Lett.. 9, 405 (1973).
B. Punch-Through B A R I T T Diodes
1. Basic Considerations The basic principle of BARITT diodes (Barrier Injection Transit Time) is space-charge limited emission in semiconductors and negative conductance arising from transit time of injected carriers. There is a certain analogy with the principles underlying IMPATT diodes; like the latter, the idea was inspired from vacuum tube physics. Indeed, Shockley ( 1 , 2) proposed such a device in 1953-1954-the basic structure of two p + n diodes, connected back-to-back (Fig. II.13a). This idea was developed in 1964 ,by Yoshimura (3). Recently it was extended ( 4 ) to metal-semiconductor-metal structures (MSM) (Fig. 11.13b),and even more recently, several studies on small-signal characteristics of such diodes were published almost simultaneously, as far as the p + n p + structures (5-7), or MSM (8, 9). We may further point out a theoretical study on both types of
MICROWAVE POWER SEMICONDUCTOR
DEVICES. I
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structure (10). However, to our knowledge, no theoretical study has yet been carried out concerning large-signalcharacteristics. What is certain, however, is that the equations in this case become nonlinear, and only numerical solutions are possible. However, it is true that the BARITT diode does not seem as well qualified as the IMPATT and TRAPATT diodes where highlevel power outputs are required. The physical mechanism controlling the function of the BARITTdiode is apparently very simple. Let us, for example, study the case of the structure shown in the block diagram of Fig. I1.13a. When no bias voltage is applied, the electric field profile of this structure will be determined exclusively by the built-in field region of the p + n and n p + junctions; this is shown in Fig. I1.14a. When a bias voltage is applied, the conditions change radically; one of the junctions then becomes forward biased and the other reverse biased. The graph in Fig. II.14b shows the case where this bias potential is
FIG.11.13. Block diagram of the two BARITT diode common structure categories.
FIG.11.14. Static electric field distribution qualitatively available for both of the BARITT diode common structures presented in Fig. II.13a.b; (a) VB = 0; (b) Vs = VpT; (c) VB 5 VpT.
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S. TESZNER AND J. L. TESZNER
sufficiently high to deplete all majority carriers (electrons in that case) from the median region. The structure now becomes punched through; further bias potential increase will diminish the potential barrier of the forwardbiased junction and will finally cancel it, producing a reverse electric field step (see Fig. 11.14~).The injection of the minority carriers (holes) in the median region increases rapidly during the decrease of the potential barrier, but much slower during the formation of the reverse electric field. Modulation of this injection by varying the bias potential increase (obviously below the avalanche breakdown voltage) will give rise to a corresponding variation of the current induced in the external circuit. However, in order that this phenomenon be used to set up oscillations or to amplify rf signals, this structure would need to provide a negative conductance, preferably as large as possible. The largest negative conductance value occurs when the phase shift between the output external current and the input voltage is cp = n and, as in the avalanche diodes, this phase shift can be obtained by the carrier transit time through the median region (Fig. II.13a). However, in contrast to the avalanche diodes, the current injected at the source plane is in phase with the voltage; hence the expression giving the conductance of the diode as a function of the transit time becomes in this case
where tlNis the carrier injection time from the source plane to the depletion region; T~ is the transit time of the depletion region; and b is a constant for a given voltage. Now, the maximum negative conductance value is obtained with
Having set out these considerations, it would seem necessary to refer to some formulas to have a better understanding of the mechanism of the BARITT diode. We will assume that the structure in Fig. II.13a has abrupt junctions and is potential biased beyond the punch through at the intermediate situation between Fig. II.l4b,c. The equations then giving the hole injected current density from the forward-biased junction will be
J,
= qpv -
KT aP -
ax’
where p is the hole density, u its velocity, p its mobility, and q the electron charge. The first term represents the carrier current and the second term the
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
355
diffusion current in that low-field region of the device; u will be expressed there as
u
(3)
= pE,
E being the electric field. Poisson's equation giving the electric field divergence will here be
where E, is the permittivity of the semiconductor and n, is the doping concentration of the median region. The total current J , is given by J, =J,
+
E,
aE at .
Now assuming the potential barrier to be canceled, the scattering limited velocity us, can be reached over the whole length of the median region. The current density J , will then only be space-charge limited and proportional to 0.
Before the situation is reached, and throughout the working zone where the potential barrier remains, it will obviously be highly desirable that the operational conditions characterized by ap/dx N 0, and u = us[ be reached over most of the length of the median region. In other words, the aim is to minimize the width of the low-velocity carrier region (see Fig. 11.15) with a
"t
t I
* - ' s v L-l !FBJ : I
iLF
'
-HF--~
4
FIG.11.15. Velocity carrier variation at the punch through in the median region of a BARITT diode common structure; FBJ-forward-biased junction; LF-low-field region; HF-high-field region.
view to increasing power efficiency and operating frequency of the diode. In order to do this, one must increase n,; however, so as not to increase the punch through of the voltage too much, this n, increase must be limited to the low-velocity carrier region. If one modifies the structure in Fig. II.13a accordingly, one obtains the structure shown in Fig. II.16a. There is an analogy here with the sophisticated structure of the Read IMPATT diode described in Section II,A,la.
356
FIG.
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11.16. Block diagram of the two BARITT diode sophisticated structure categories.
Some BARITT diodes of that type have already been produced (11, 12); Fig. II.17a-c give field profiles of such structures for similar operation steps as those of the common structure in Figs. II.B.2a-c. The same reasoning may be applied to the MSM structure, with, however, some slight rearrangements. On one hand, from a theoretical point of view, one has to take into account as the barrier potential, the metalsemiconductor interface barrier height in addition to the built-in junction potential; moreover, the thermionic emission from the forward-biased junction, must be considered a priori to play an important role in the particles' injection. From a practical point of view also, an MS junction is really
tE
FIG. 11.17. Static electric field distribution qualitatively available for both of the BARITT diode sophisticated structure presented in Fig. II.I6a,b; (a) V, = 0; (b) V, = VpT; (c) V, % VpT.
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abrupt, whereas it is much more difficult to obtain such a junction in p 2 + n structures. The differences from the theoretical aspects are certain, but they do not bring about notable changes in practice. The built-in potential, in effect, is lower in that case than in a p2+njunction and the interface barrier is reduced by the semiconductor surface states effect, and the total barrier height is quite comparable to that ofa p2+njunction. On the other hand, the injection mechanism is similar in both cases, under high-injection conditions, i.e., for power devices. From a practical point of view, the advantage noted should be retained, although it is possible today to make fairly abrupt junctions using lowtemperature epitaxial techniques. Whatever the case, i.e., advantage or not, it will only concern the forward-biased junction. Having stated that, one variant of the p2+ntnp2+ structure in Fig. II.l6a, will be the structure Mn+np2+in Fig. II.16b. The case with an n-type material median region was only taken as an example. From an analytical point of view, this choice has the advantage of concerning positive injected carriers, and this does away with numerous negative signs in the equation. This was, moreover, the reason given by Shockley in his choice of structural type; his example was copied later. However, the n2+pn2+seems to be of equal or even greater validity. In effect, there is a mobility of injected carriers (electrons) which is markedly higher in the low-field region and, consequently, the scattering limited velocity is reached for a lower electric field value. In principle, then, the relative part of the drift zone in the median region would be larger, and the operating voltage could be lower (26). It was by referring to these electron transport phenomena, among other properties, that an n2+pnz+ gallium arsenide structure was recently proposed (11). I t is obvious that in the case of this material, the advantage noted above would be all the stronger because of an electron mobility in low fields which is some four times higher than for silicon. Unfortunately, there was practically no experimental confirmation of the promising theoretical forecast until now. 2. Device Fabrication
To our knowledge, the fabrication of microwave power BARITT diodes seems to be for the moment limited to laboratory production (4,7,12,13,15, 16,22)and the fabrication process is rather similar to that used for IMPATT diodes, the difference being that here there are two rectifying junctions instead of only one.
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Thus to produce the structure shown in Fig. II.13a, an epitaxial layer of n-type silicon (doping concentration approximately 6 x 10'4-1015 cm- ') is deposited on a silicon p z + substrate (with a resistivity in the order of some lop3R cm), in which epitaxial layer boron is diffused to form a p 2 + layer (with resistivity comparable to that in the substrate). In order to obtain the four-layer structure shown in Fig. II.l6a, there are two methods possible: epitaxial deposition of two layers successively, with their doping concentration gradation [n (3-6) x 1014cm-', n+ (3-6) x 1015cm-'1; or by an epitaxial deposit of only one layer, the thickness of which is equal to that of both epitaxial layers in the previous case, and the doping concentration equal to that of the first layer in the previous case, this deposit being followed by phosphor diffusion with a view to obtaining a doping concentration equal to that of the second layer in the previous case. In both cases the process will finish with diffusion of boron, so as to form the outer p 2 + layer. Moreover, to make the structure in accordance with Fig. II.13b, first, an n-type silicon substrate is prepared (doping concentration similar to that in the epitaxial layer of the structure shown in Fig. 11.13a), in which the thickness has been reduced by polishing and etching until the appropriate dimension for the given operating frequency is obtained (see further comment below). Then both sides of the wafer are cleaned and a platinum film is deposited by sputtering to form a platinum-silicon layer around 10oO di thick on each side. Afterwards, a very thin layer of chromium is deposited on both sides in turn, followed by a 3-5000 di unit layer of gold. As for the structure shown in Fig. II.l6b, the process is similar to that in Fig. II.l6a, except that the diffusion of the p z + layer is replaced by a metalsemiconductor junction formation following the process set out above. We note at this point that certain fabrication techniques (13, 16) leave out the prior deposit of a PtSi layer and limit themselves to deposits of Cr and Au following the chemical cleaning of the wafer. The width of the median region, n in Fig. II.l3a,b and nn+ in Fig. II.l6a,b, depends strictly on the operating frequency for a given structure. This varies in accordance with the inverse ratio of the frequency from 12 pm for 4 GHz in the case of the n median region (Fig. II.l3a,b); the optimum operating frequency is relatively higher for a given width in the case of the nn+ median region, since the part of the high-field region where the carrier scattering limited velocity is reached is then increased. We should add that the structures described above do have basic variants, one in terms of structural complementarity-npn or M p M instead of pnp or Mnm; the other in terms of the material used-notably GaAs instead of Si. In the foregoing paragraph we have pointed out the interest that would arise in an experimental approach to these two possibilities which remain unexplored, to all intents and purposes.
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3. Electrical Characteristics and Performance The experimental results relating to the BARITT diode are, as yet, not numerous, especially concerning large-signal operation. Nevertheless, they do enable us to give a state-of-the art report on the matter, and to give a first appreciation both for positive and negative aspects of the structure, without being in a position to clearly set out the characteristics and performance tables as in the previous sections. By electrical characteristics here, we understand punch-through voltage VpT,operating voltage Vo,bias-current density J o , and the frequency range. We include as part of the performance, rf power output Po,, and efficiency q in a given frequency range as an oscillator and possibly as an amplifier. In parallel, a few brief comments are made about temperature effect. Finally, we will examine the noise-behavior parameter. Our basis for this rapid examination will be papers published in this area, and notably concerning electrical characteristics as well as oscillator power output and efficiency (9-18) and concerning noise behavior (6,9,10, 12, 19-24). We will distinguish here only two types of structure-three-layer and four-layer structures, without differentiating them according to the mode used to fabricate the rectifying junction. Indeed, it is not said that there is a fundamental difference between the characteristics and performances of structures with Mn or p 2 + n junctions, respectively, as long as the injecting junction remains practically abrupt and that, moreover, the parameters of the median region are similar. We will take the frequency range to be just the 4 8 GHz band, where all the results in large-signal operation are known. In this frequency band, from bottom to top, VPT decreases approximately from 100 to 50 V, V, being 15-20% higher. Concerning J,,, its optimum value (i.e., giving maximum power output) will lie between 100 and 150 A cm-2. We immediately note that J o is much lower here than in the case of the IMPATT diodes. It is, in effect, essentially limited by the very mechanism underlying BARITT diode operation so that to reach values of J,, of the same order of magnitude as the J o of the avalanche diodes, V, would have to rise to a level exceeding the avalanche breakdown voltage and this of course could not be allowed here. Moreover, J o is still limited by temperature effects. This effect has several components, some of which act to limit J o , while others intervene in the opposite direction. Thus, raising the temperature will raise the breakdown voltage V, and will increase the operations margin between V, and VpT. However, this expansion will be somewhat attenuated because of the parallel increase of VpT, due to a slight increase in carrier concentration. O n the
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other hand, mobility is decreased, and this brings about an extension of the zone where u < us,, whence a decrease in u. In return, usI is reduced and this compensates somewhat for the extension. Moreover, with temperature rising, the injecting current density is increased exponentially, and this gives rise to a risk of runaway, even though we are dealing here with a unipolar device. To be precise then, the result of the temperature effect seems unfavorable, and increasing the temperature would appear to bring about a drop in output power and efficiency. This may be explained by certain experimental observations attributed to self-heating (13), which we shall discuss later. This effect appears above all to upset the optimal operating frequency and also the operating voltage. However, it seems difficult and especially premature to translate the temperature effect into figures in a manner of a worked example, since too many factors intervene in conflict with each other and, moreover, certain particular structures, for example, those which give rise to ohmic loss, may lead to a false interpretation of the experimental data. Having noted this, we may conceive that the swing of oscillatory voltage and current must be relatively small, limiting both the rf power output and power efficiency. Figure 11.18 shows qualitatively what we have been just J
FIG.11.18. Qualitative illustration of the bias current J , and output power Po,, with the voltage V,, evolution.
discussing. The current density J o which increases rapidly as soon as V,, exceeds VPT,is essentially limited by the variation in Po,, which goes through a maximum for reasons we explained above; saturation of Po,, followed by a drop is a consequence of the drop in negative conductance G.We should point out here that the shape of the graph J o ( V,J and Po,, ( V,,) is valid to all intents and purposes for all variants in BARITT diode structures.
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1.6 1.4 1.2
40
/+'02 /
0.8 0.4-
/ b/
o
I tro 0
I
I
I
1
-
1
FIG.11.19. Examples of the power efficiency and power output Po,, evolution as function of bias current density J , for one three-layer TLS and one four-layer FLS BARITT diode cm2; operating frequency: 5.4 GHz for TLS and 6.6 GHz for FLSstructure: area 3 x after experimental results (13).
On the other hand, in Figs. 11.19 and 11.20, which were drawn up from recent experimental data for large-signal operation (viz., Reference 1 3 ) and which illustrate quantitatively two important aspects of the problem, the case of the three-layer and the four-layer structure are differentiated. We note in Fig. 11.19 that: (a) The structure Poutmax is a saturation value which is to be found in the .Io zone where dq/dJo is already markedly negative. (b) Between these two structures, the four-layer provides both higher Po,, and higher q simultaneously. Concerning Fig. 11.20, this confirms an earlier remark made for other power semiconductor devices, that beyond a certain device area, Po,, density falls as does q ; then, instead of increasing the area, it is preferable to put
'i I
0
6.
,-----* I
5
I
10
A
i5 x i Z m 2
FIG.11.20. Power efficiency q as a function of the BARITT diode area A for similar three-layer and four-layer structures as in Fig. 11.19-after experimental results (13).
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some chips in series or in parallel. It is interesting to add that the thermal as a direct function drift (f-f,,,)/fof such an oscillator is 10-4-5 x of output power. Concerning operation in the reflection-type amplifier mode, Po,, and q at a minimum acceptable power gain level (- 5 dB), must normally be of the same order of magnitude as in the free-running oscillator types. However, experimental results in large-signal operation are still too limited in this case, and it is not possible to specify here the characteristics such as gainbandwidth product, intermodulation, and signal distortion for the amplifier case. Having set this out, we must now go on to note that the values of Po,, specified in Figs. 11.19 and 11.20, especially the products, are very far below the performances cited for avalanche diodes. However, and this is its essential advantage, the BARITT diode has a considerably better noise-behavior figure. There are two noise sources in the BARITT diode: One source is called the injection noise of the injected carriers; the other, called diffusion noise, is the random velocity fluctuations of the carriers in the low- and high-field regions. Both sources are of relatively minor importance compared to that arising from the avalanche formation in the avalanche diodes. The noise figure can be expressed as the noise voltage normalized to the thermal noise voltage spectrum, i.e., the optimum noise measure will be given by (25):
where li21 is the mean-square value for the noise current per unit bandwidth and I G [ is the negative diode conductance. Now the shot noise will be smoothed through the space charge in the low-field region, thus decreasing when J o increases. On the other hand, diffusion noise increases with Joythus becoming prominent at higher J o values. Consequently, the total noise figure increases more and more rapidly as the current density increases. In order to illustrate the foregoing by a numerical example, Fig. 11.21 shows two curves giving the BARITT diode small-signal amplifier optimumnoise measure M as function of J o . These noise figures have been measured (23) for two types of BARITT, one with three-layer, the other with four-layer structure. The lowest noise figures measured (10) at J o 1 to 10 A cm-* in a free-running oscillator with BARITT diode (three-layerstructure) are somewhat inferior to 5 dB.
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,:[ 0
10
20
30
40
50
60 A cm-2
FIG.11.21. Optimum noise measure M as function of the current density J , for a three layer TLS and a four-layer BARITT diode structure, largest power delivered being, respectively, 44 and I15 rnw with the same device area of 7.5 x cm’-after experimental results (23).
4. Discussion and Conclusion
Summing up, the advantages of the BARITT diode seem to be that (a)Its noise figure is quite low; and (b) its fabrication is quite simple. The noise figure in the power applications is about l(r15 dB lower than that of avalanche diodes, but only in the range of 6 dB below that of TED. The fabrication is quite simple because it is similar to that of the avalanche diodes. In return, its disadvantages seem to be: low power efficiency, low RF output power, and low operating frequency. The power efficiency presently obtained is around 2%, which is very low compared to the 20-30% obtained with avalanche diodes. A lot of progress has still to be made in that area as we have pointed out. However, we have also shown that the BARITT diode has limitations embodied in its very operational principles-in the limitations of J o and V,, . According to some theoretical calculations ( l o ) , power efficiency of the order of 10% could be reached. But besides the fact that these calculations are very simplified, they abstract the parasitic parameters of the structure and leave out the self-heating effect which accentuates the limitations of J o and thus one must use appropriate heat-sink structures. Thus unless there are structural modifications themselves in terms of principle, it would seem more reasonable to set an upper limit at around 5 % only for power applications, and this for some time to come. The maximum output power presently obtained is a little higher than 100 mW at around 7 GHz and this may be compared with around 5 W at the same frequency for IMPATT diodes. This limitation of Po is a consequence of both the low values of q and J o . Increasing one other will, of
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course, increase P o , but when we take into account all the foregoing comments, one cannot really expect to exceed much more than Po = 500 mW for a single chip. The operating frequency for the power applications is presently limited to around 8 GHz. A notable increase in this frequency will bring about an equally notable drop in q, since the drift zone will be diminished in the median region. This increase in frequency will be disturbed by the relatively high value of the output capacity, this being another direct consequence of the low current density. Certainly one may expect to find an extension of the frequency range over the X band, but it seems to be very difficult to go much beyond. The possibilities offered by BARITT diodes would seem to be quite restricted in this area compared with IMPATT diodes, for which the operating frequency is presently reaching 100 GHz, with output power and efficiency values higher than those of C-band BARITT diodes. In fact, the performance of BARITT diodes is better compared with TED, in which power efficiency is higher, and where the noise level is also, however, slightly higher for TEA than for the BARI'IT diode structure. Considering the frequency-power performance, the TED (Section I,B,4) is now approaching 1 W per single chip in the C band and 500 mW in the X band. Moreover, the operating frequency reaches the V Band (54 GHz) with Po > 100 mW. However, we should point out an advantage of the BARITT diode compared with the TED-its quite simpler fabrication using a less sophisticated material. When there is a choice between two devices, the choice is based on factors beyond the technique involved. The choice will obviously be a function of the evolution of one or another type of device. We shall touch on this again in the last section of the review (Part 11).
REFERENCESFOR SECTIONII,B PUNCH-THROUGH BARITT DIODES 1 . W. Shockley and R. C. Prim, Phys. Rev. 90, 753 (1953). 2. W. Shockley, Bell Syst. Tech. J . 33, 799 (1954). 3. H. Yosimura, IEEE Trans. Electron Devices 11, 414 (1964). 4. S. M. Sze, D. J. Coleman, Jr., and A. Loya, Solid-State Electron. 14, 1209 (1971). 5. G. T. Wright, Solid-State Electron. 16, 903 (1973). 6. A. Sjolund, Solid-State Electron. 16, 559 (1973). 7 . G. T. Wright and N. B. Sultan, Solid-State Electron. 16, 535 (1973). 8. K. P. Weller, R C A Rev. 32, 372 (1971). 9. D. J. Coleman, Jr., J . Appl. Phys. 43, 1812 (1972). 10. J. L. Chu and S. M. Sze, Solid-State Electron. 16, 85 (1973). 11. C. G. Englefield, P. N. Robson, and J. Sitch, Proc. European Microwave Con$ Brussels, Belgium, 1973, p. A10.4. 12. 5. G. Liu and J. J. Risco, RCA Rev. 32, 636 (1971).
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13. C. P. Snapp and P. Weissglas, IEEE Trans. Electron Devices 19, 1109 (1972). 14. N. B. Sultan and G. T. Wright, Electron. Lett. 8, 24 (1972). 15. D. J. Coleman, Jr. and S. M. Sze, Bell Syst. Tech. J. (Brief) 50, 1695 (1971). 16. C. P. Snapp and P. Weissglas, Electron. Lett. 7 , 743 (1971). 17. N. B. Sultan and G . T. Wright, IEEE Trans. Electron Devices (Short Pap.) 19, 773 (1972). 18. G. T. Wright and N. B. Sultan, Int. Solid-State Circuits Con$ Philadelphia, 1973, THAM 7.3 p. 78. 19. C. A. Lee and G. C. Dalman, Electron. Lett. 7 , 565 (1971). 20. H. A. Haus, H. Statz, and R. A. Pucel, Electron. Lett. 7 , 667 (1971). 21. J. Helmcke, H. Herbst, M. Cloassen, and W. Harth, Electron. Lett. 8, 158 (1972). 22. H. Statz, R. A. Pucel, and H. A. Haus, Proc. IEEE (Lett.) 60,644 (1972). 23. G. Bjorkmann and C. P. Snapp, Electron. Lett. 8, 501 (1972). 24. A. Sjolund and F. Sellberg, Proc. European Microwave Con$ Brussels, Belgium, 1973, p. A10.3. 25. B. C. De Loach, IRE Trans. Electron Devices 9, 366 (1962). 26. D. Delagebeaudeuf, Electron. Lett. 10, 166 (1974).
C. Varactor Diodes 1. Basic Considerations
Voltage variable capacitors (varactors) are doubtless the best known microwave power semiconductor diodes at the present time. Certainly, given their importance in semiconductor electronics, they could not be left out of this overall critical review. Nevertheless, this section will restrict itself to updating our knowledge of this domain and to present trends in varactor development. A varactor can be made up of a pn junction (Fig. II.22a), or a rectifying MS junction (Fig. II.22b), or even a MIS junction (Fig. 11.22~).Based on (a)
FIG.11.22. Block diagram of the three varactor diode structure categories: M-metal layer; I-insulator layer.
these constructions, the theory of the varactor diode operation derives from either basic p n junction theory (1-4, that of the rectifying metalsemiconductor junction (I, 5-10), or that of the metal-semiconductor junction through an insulating layer ( I I , I 2 ) . The first and second cases are well known and will not be described herein; the last case was considered in an
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extensive study on this type of varactor, in a recent volume of these Advances (13),and subsequently the reader will find references to this work. In all three cases, the basic principle is that of the junction capacitance as a function of applied voltage used to provide a nonlinear circuit element.* However, depending on the structure and the technology utilized, the capacitance-voltage relationship can be very different. It will likewise be affected by spreading capacitance and resistances within the structure concerned. We shall deal briefly with the effects of these various factors. For a p n plane junction, one side of which is presupposed heavily doped (the p side, for instance, Fig. II.23), the basic expressions, respectively, for
FIG.11.23. Three impurity distributions for p-n varactor diodes: m = 1, linear graded junction; m = 0, abrupt junction; m = -3, hyperabrupt junction. W-width of the n layer.
depletion layer width W and differential capacitance per unit area C at the reverse bias are given as
and
In these equations, Q, is the charge per unit area, Ki is the built-in potential, V the applied voltage, and B the background doping concentration on the n side, the generalized doping distribution being N = Bx". The power factor rn defines the doping profile, and by the same token the junction characteristic (cf., Fig. 11.23). Thus, for rn = 1, the junction is
* Among the pioneers to be cited are Giacoletto and OConnel (14). McMahon and Staube (IS), and Moll ( 1 1 ) and Pfann and Garnett (16) for MIS structures.
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linearly graded; for m = 0, the n side with uniform doping corresponds to an abrupt junction; for m < 0, the junction is called “hyperabrupt,” the case m = -2 represented in Fig. 11.23 being of particular interest, as will be explained below. Numerous studies have been done on the variation of the differential capacitance C of a p n junction as a function of I/-junctions whatever their profile (27-29), or more particularly abrupt junctions (20, 22), or yet again hyperabrupt junctions (22, 23). Figure II.24a-c shows the plotted graphs C
J
8
I
6
4
b
2
Volts
C ‘Cjo
-8 J
-6
-4
2
<
v
Volts
FIG. 11.24. Differential capacitance C as function of applied reverse bias-voltage V for pn varactor diode with: (a) m = -4; (b) m = 0 ; (c) m = 1; three doping profiles shown in Fig. 11.23. Note: Cj, (a) > Cj, (b) > Cj, (c). It is also noteworthy that the capacitance variation ratio, and this is especially the case for m = -3, is greatly reduced in fact through the resistance R, (see Fig. 11.26) effect.
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(Vbi + V ) at the reverse bias, for the three doping profiles contained in Fig. 11.23. For rn = -2, C varies as the bias voltage VBi = ( & + V )according to a parabolic law, and the resonant frequency of the varactor is linearly dependent on the junction bias voltage. It follows that the distortion may be kept to a minimum; moreover, the slope dC/dVBi,since it is directly proportional to VBi, can reach relatively high values. In the case of an MS junction, only the abrupt and hyperabrupt junctions are to be considered. Now, the C( VBi)graphs are quite similar to those corresponding to the pn junction. The built-in potential Vbi is obviously somewhat different, but this will have only a slight effect on VBi. Thus the abrupt and hyperabrupt graphs in Fig. II.24a,b will remain valid in this case. In contradistinction, the case of an MIS diode is fundamentally different. As this case has been extensively dealt with in a previous volume in this series (13), we shall restrict our notes to a reminder of the conclusions relating to very high-frequency signals, which is the only case which concerns us here. When there is no applied voltage V , the diode capacitance is made up of that of the insulating layer Ci, in series with the capacitance C, of the surface barrier in the semiconductor. Now, the forward bias V' applying tends to reduce the surface barrier potential, and consequently the corresponding capacitance is increased. Then, the overall capacitance C ,max approaches Ci . On the other hand, a reverse bias V, tends to set up a depletion layer inside the semiconductor, the width of which will increase as I V, I increases; consequently the space-charge capacitance tends to diminish. Thus, the overall capacitance can be reduced to a small fraction of Ci . Figure 11.25 illustrates this evolution of Ct as a function of VBr. Following this reminder, we shall now define the figures of merit for varactors, using the varactor equivalent circuit shown in Fig. 11.26. In the microwave region, the wafer equivalent circuit is reduced to a capacitance
FIG.11.25. Differentialcapacitance C, as a function of bias voltage V', for the MIS varactor diode.
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FIG. 11.26. Simplified varactor equivalent circuit at microwave frequencies: R , , Cj-series resistance and differential capacitance of the wafer; C,-case capacitance; L,-lead inductance.
Cj, in series with a resistance R, . In fact, the parallel equivalent resistance R , , of the generation-recombination current, the diffusion current, and the surface current may not be taken into consideration since it greatly exceeds that of the junction current. The quality factor Q, being a ratio of energy stored to energy dissipated, is expressed as 1
Q=a
(3)
The cut-off frequency, at which Q = 1 for a varactor diode at a given bias voltage VB,, is then
The cut-off frequency f, between bias extremes, frequently called the dynamic cut-off frequency, is given as
It will be seen, thus, that to ensure the highest possible values for Q and f & ,then one must try to first decrease R , and then to decrease Cj V,,. Also, to
raise the dynamic cut-off frequency f,, the Cj ,,JCj ratio must be increased as far as possible. In order to decrease R,, one should use a semiconductor material in which the carrier mobility is as high as possible (therefore, necessarily n type and preferably, GaAs rather than Si). Likewise, one should try to reduce the thickness of the semiconductor layer and/or increase the cross-sectional area of the diode. However, in order to decrease Cjmin,one must have a sufficiently wide depletion layer and thus the semiconductor layer thickness should be increased, with the proviso that the corresponding punch-through voltage remains below the breakdown voltage. It would, moreover, be advisable to reduce the diode cross section as far as possible. Thus one should be seeking a trade off between these two contradictory requirements for R , and Cj respectively.
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Otherwise, one could try specifically to raise Cjmax in order to increase the Cj ratio. At first sight, the MIS structure would doubtless seem to be more appropriate. In effect, by superimposing a thin insulating film upon a lightly doped semiconductor layer it would seem that the result sought would be more easily reached. However, the development, relatively recent [although the idea itself goes back some way now (25, 26)], of socalled charge-storage step recovery, but generally termed snap, snapback, or snapoff diodes, has considerably modified the problem. This development calls for some brief comment. When a pn varactor diode is forward biased through the driving voltage, the minority carriers are injected from one layer into another and stored within each layer for a time 7 s 6 TL(minority-carrier lifetime),the condition being that their diffusion will not be taken into account. Then, it would be convenient to have 30 TL>-, An
whereJ;, is the input frequency. Now, in the reverse bias portion of the waveform of the driving voltage the storage charge will be withdrawn from the junction into the external circuit (cf. Fig. 11.27). The current then decreases rapidly during a very short
FIG.11.27. Current i vs. time t in a charge-storage step-recovery (snap) varactor diode: 7"harge-storage time; T,-charge-withdrawal time; T,,-depletion-layer charging time; &,depletion-layer discharging time; T,-transition time called snap-time.
transition time and after the depletion layer charging and discharging periods, the forward biasing with charge-storage process resumes (Fig. 11.27). This operation results in a substantial increase of Cj mBx. It requires a step impurity profile and a narrow pn junction. Further, with a view to reducing the charging and discharging periods of the depletion layer, its space charge must be reduced. However, if one makes the depletion layer very narrow
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with this intention in mind, the breakdown voltage will be low and the power handling capability very limited. Thus a much better solution seems to be substantially reducing the layer doping concentration; one then realizes a p'n-n diode structure. Figure 11.28 shows a schematic presentation of
FIG.11.28. Differential capacitance Cj as function of applied bias-voltage V for snap-diode varactor.
the Cj variation for the forward- and reverse-bias voltages in that case; the very sharp Cj variation at V = 0 and the very little Cj change at reverse bias are quite remarkable. Thus the transition time is likewise reduced. Both conversion efficiency and the output frequency stability are improved (cf., Section 11,C,3). Finally, concerning the frequency limitation, it is noteworthy that f, is not the most restrictive factor. The other limitation arises from the carriers' transit time for depletion or for refilling the space-charge region. Then we shall have e where e is the width of that region and uSIis the carrier scattering limited velocity. Now, obviously
At this point, a further advantage of the structure and of the operational mode inherent in snap diodes becomes apparent. In effect, because of the variation of capacitance which almost exclusively occurs at forward bias, the depletion layer at reverse bias may be greatly reduced. Moreover, the p n junction is very narrow. Thus, all the requirements to ensure as high a value ofJim as possible would appear to be fulfilled.
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It should be stressed that the charge-storage and the step-recovery mechanism is only valid and the techniques which derive from them are only applicable to p n diodes and for a single material, silicon, in presently available technology. There is indeed no charge storage in a MS junction. Moreover, it is only in silicon, at least for the moment, that the minoritysec) in order carrier lifetime is sufficiently long (from lo-' sec to some that condition (6) be satisfied throughout the microwave region. 2. Device Fabrication This review is limited to the varactor diodes designed for harmonic generation with a view to frequency multiplication requirements in the power microwave region. The technology most frequently used nowadays is based on the epitaxial growth of an n layer on an nz+ substrate. Bearing this in mind, one sees three types of structure in Fig. II.29a-c, structures which were mentioned in the previous paragraph: a-the p n junction; b-the MS junction; c-the MIS junction. ohmic
fi
roc tifylng contact
__-__
n or n-
hmic contact
(b)
(a I
n
FIG.11.29. Three varactor diode shapes: (a) pn junction diode; (b) MS junction diode; (c)
MIS diode.
Figure II.29a is a planar structure with a hyperabrupt junction which was seen to be particularly attractive. One can, however, note that a mesa structure can also be used (especially when a relatively high breakdown voltage is sought). Likewise, other doping profiles, notably abrupt, are permissible. The structure shown has an n z + low resistivity (some 0.001 R cm) substrate on which three epitaxial layers are grown, in the case of the hyperabrupt junction. The first layer, n- or n has a doping concentration of 10i4-5x 1015cm-3 and a thickness that may range from
-
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
-
373
1-10 pm as a reverse function of the doping concentration. The second layer is very thin, a fraction of a micrometer, and relatively overdoped, ND = -2 x 10'6-10'7 ~ m - Last, ~ . the third layer is a p 2 + layer with a doping concentration value ranging from 5 x lo'* to 5 x 1019 cm-3. It is possible to substitute for the second and third layer epitaxial depositions, a double diffusion process with first an n+ diffusion followed by p 2 + shallow diffusion. Moreover, at least one of these diffusion processes could be replaced by ion implantation (27, 28). It should also be noted that the fabrication process for an abrupt junction will only differ from that described above in that the n+ layer will not be produced. In Fig. II.29b, the p 2 + layer has been replaced by a rectifying metalsemiconductor contact. This fact already implies that the mesa structure has been substituted for the planar structure. The MIS structure, as indicated by Fig. II.29c, is vastly different (13). Onto an n epitaxial layer with a doping concentration of 10l6 cm-3 and thickness 1-4 pm, deposited on an n2+ substrate, a very thin SiO, film (of 200 d;) is formed, on which an amorphus Si,N, or Al,03 layer (of loo0 d;) is deposited. One of the diode electrodes is applied on that layer, the other one obviously being at the substrate base. It is a mesa structure, but a planar structure can also be realized (13). Concerning the choice of basic material for the varactor, in practice only silicon and gallium arsenide are used at present. The use of GaAs has the advantage of a considerable drop in R, because of the relatively good carrier mobility, hence a higher conversion power efficiency, as we shall see later. However this advantage cannot be put to best use in charge-storage steprecovery diodes because, for the moment at least, of the too-low value of the minority carrier lifetime in GaAs, scarcely 10sec (29). The inequality (6) then shows that for this undoubtedly attractive mode of operation, the use of GaAs only should be considered for applications far beyond 30 GHz. For this reason, the scope for GaAs in multiplier varactor diode technology is rather limited.
-
--
-
-
N
3. Electrical Characteristics and Performances
By electrical characteristics we understand the cut-off frequency f, and the operating frequency f, range, the breakdown voltage V,, the overall Cjmax/Cjmin ratio and, in the case of snap .varactors, their snap-time. Concerning the performance values in a multiplier frequency mode, we mean the rf power output and conversion efficiency at given input and output frequencies. Problems of noise and of reliability will then also be briefly discussed. Regarding electrical characteristics, the present limits for the cut-off
374
S. TESZNER AND J. L. TESZNER
frequency of the commercially available microwave power pn and MS varactor diodes are the following: 100 GHz ~ f I ,300 GHz. At the laboratory stage, the value off, = 800 GHz for a GaAs diode has been reached (30) and it seems to be possible to go even higher. On the other handJim z 500 GHz. However, the operating frequency f, limit actually reached is substantially lower. For the commercially available diodes, f, 50 GHz; at the laboratory stagef, 130 GHz. This frequency is otherwise limited through packaging parasitic elements (essentially inductance and capacitance), as well as through multiplier circuit problems. The breakdown voltage range is quite wide: 15 V I V , I 300 V (particularly for the hyperabrupt junctions one has 15 V IV, I 100 V). The higher the frequency, the lower the V, because of the n-layer thickness reduction and NDincrease. Then, to avoid or to limit a parallel increase of Cjmin, the area of that layer must be reduced at the same time (and thus the power output is likewise reduced, as discussed below). Concerning the Cjmax/Cjmin ratio, there are two different cases to discuss-(a) the common pn or the MS junctions and, (b) the charge-storage snap junction. In the first case, the varactor diode operates almost only under reverse bias, the capacitance variation as a function of the applied voltage being progressive. Then (Cjmax/Cjmin) will reach a value generally between 4 and 7. On the other hand, in the second case the capacitance changes very little at reverse bias. It is when conducting in the forward direction that the capacitance rapidly and greatly increases through the charge storage. As explained above (Section II,C,l, Figs. 11.27 and II.28), it conducts for a short time at reverse bias until this charge is swept out and, immediately afterward, conduction ceases. At the same time the capacitance drops practically to Cjmin.The time of switching between conducting and nonconducting states is called the snap time T,. Depending on the output frequencyf,, T, must be < l/f, (and preferably substantially lower). Regarding the electrical performances, we note the following. (a) The multiplier order range-n = 2-8. In particular, the snap diodes are most useful for high-order (narrow band) multipliers, through their high degree of nonlinearity, but they can be likewise advantageously used for low-order multiplication. (b) The output power goes beyond 25 W at the bottom level of the microwave region (1-3 GHz). It falls off to 0.5 W at the 20-30 GHz level; 100 mW at 50 GHz and 50 mW at 60 GHz (28). it is presently of However, the output power may be greatly increased without alteration of the operating frequency by the series interconnection, i.e., stacking of varactor diodes (31). Theoretically this power increase is proportional to a’, where a is the number of varactors stacked. In fact, the gain is a little lower through a slight increase of the conversion loss.
-
-
-
- -
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
375
(c) The varactor diode conversion efficiency depends on several parameters. The operational frequency fo (and in particular thefo/h ratio), the Po,, level, and last, the diode structure itself-its type and its basic material. If moreover, we are looking, as usual, at the overall multiplier efficiency and if we take the losses in the multiplier circuit into consideration, the efficiency drops considerably. In effect, those losses can be appreciable, especially if one aims at realizing a relatively high order of multiplication ( n > 4) and if (32) one sets about frequency mixing by using idling circuits (discussion on such techniques lying outside the scope of this review). In that case, moreover, overall efficiency will also depend notably on the multiplier order. In this way, the range of values for conversion efficiency is relatively quite wide. For laboratory results, it is noteworthy that: (a) As a function of frequency efficiency-q E 80% for GaAs diodes and q z 70% for common p n or MS Si diodes in the low microwave range (foul 10 GHz) and low Po,, 50-100 mW; q = 54% for snap diodes at 60 GHz (28) for Po,, of the same order of the magnitude z 25 mW. (b) Depending on Po,,-q decreases but relatively slowly as Po,, increases, e.g., at low frequency range q decreases to 50% for Po,, z 20 W. (c) Depending on the diode structure, snap diodes generally provide higher efficiencies than the common pn and MS diodes. This is especially remarkable for the overall efficiency of the high-order multipliers. We shall cite, as an example, some commercially available minimum values of the overall efficiency for snap diodes in the low range of the microwave region :
-
-
-
2 GHz, n = 5; = 2 GHz, n = 6;
foul = foul foul foul
6 GHz, n = 3; = 13 GHz, n = 4; =
Po,, = 8 W Po,, = 4 W, Po,, = 1.5 W, Po,, = 0.3 W,
qmin= 27%; qmin= 27%;
qmin= 30%; qmin= 15%.
The comparison of the data leads to the following comments. The multiplier order value has a certain repercussion on the overall efficiency, but it does not appear to be striking in the figures given; on the other hand, the overall efficiency drops quite quickly when the output frequency is increased and that despite the output-power lowering. However, the recent laboratory result with a snap diode at foul = 60 GHz (28) enables a prediction that, at some future date, we shall be seeing the above-mentioned drop in overall efficiency attenuated with frequency increase. The overall efficiency of course includes all of the multiplier circuit, but not necessarily the input and output pass-band filters. These filters are generally required to limit the FM noise level. We must now turn our attention
376
S. TESZNER AND J. L. TESZNER
to this important varactor operation parameter. The noise problem and especially FM noise in multiplier frequency varactors is important for two reasons. (a) Frequency multiplication by a factor n leads to a multiplication by n2 of the input noise level. It is thus very necessary to provide an input filter and this lowers the overall efficiency. In the example given (28), the efficiency value 54% drops to 49% because of the input filter. Now, for high-order multiplier varactors, a high Q input filter is necessary. (b) However, one must take the noise generated in the varactor diode into consideration. The shot noise level which arises in all rectifying diodes is not high, but the FM noise level resulting from a lack of stability in output frequency could be very troubling. In this area, snap diodes still seem advantageous. Regarding varactor diode reliability, it is generally satisfactory if the inequality VBl < VB holds true at all operating conditions. The operating mode of MIS varactors in the microwave region is quite similar, and hence their potential electrical characteristics and performances should be comparable to that of snap diodes. Since this topic has been dealt with recently and quite fully in Reference (13), we shall not take up the subject here.
4. Discussion and Conclusion As a frequency multiplier, the varactor diode is a complementary device designed to improve the frequency performance of a main device, such as a transistor or a negative-resistance diode. The application of such varactors has been, therefore, mainly developed as an extension of the transistor, or sometimes diode, amplifier or oscillator. Frequency multipliers were largely used in the past in the microwave region, in the frequency range from 1 to 30 GHz. However, things have changed now that the range of powertransistor operating frequencies has been extended to 5 GHz and even beyond, as we shall see in Part 11, and moreover the operating frequencies of IMPA'IT diodes and TED have gone well beyond 30 GHz as we saw in Sections I and I1,A. Presently, power varactor frequency multipliers are seen as possible complements only for transistor power amplifiers or oscillators. From this angle, the advantages and disadvantages of multiplier varactors can be summarized as the following. Their advantages appear to be: 0 high operating frequency limit (well beyond 100 GHz) 0 relatively high power efficiency (- 50% at 60 GHz) 0 high frequency multiplier order, especially with charge-storage steprecovery (snap) diodes
MICROWAVE POWER SEMICONDUCTOR DEVICES. I
377
relatively low specific noise level quite simple fabrication. Their disadvantages appear to be: 0 complementary device applications only 0 quite complex frequency multiplier circuit necessary input filter, preferably with high Q, because input noise is multiplied by the square of the multiplier order 0 quite low output power beyond 30 GHz. However, the recent development of snap varactors leads us to think that these disadvantages could be attenuated. Thus, the multiplier order has been raised as far as 8 without an overcomplicated circuit, while one is still availed of an acceptable overall efficiency level. The output power is certainly still low for frequencies beyond 50 GHz, but there is every reason to assume that this could be gradually increased. Finally, because f, and Jim are raised, the operating frequency could be effectively extended beyond 100 GHz. This line of improvement of multiplier varactors still holds promise. Moreover, it is also possible that MIS varactor development may provide some quite interesting results. In fact, this progress noted in performance figures is most necessary since transistor performance values continue to climb and multiplier varactors should follow up these improvements. We shall take up this subject again in Part I1 of this review. 0 0
REFERENCESFOR SECTION II,C VARACTOR DIODES 1. B. Davydow, Sou. Phys.-Tech. Phys. 5, 87 (1938). 2. W. Shockley, Bell Syst. Tech. J. 28, 489 (1949). 3. F. S. Goucher, G. L. Pearson, M. Sparks, G. K. Teal, and W. Shockley, Phys. Reu.81,637 (195 1). 4. C. T. Sah, R. N. Noyce, and W. Shockley, Proc. IRE, 45, 1228 (1957). 5. W. Schottky, Naturwissenschajien 26, 843 (1938). 6. W. Schottky and E. Spenke, Wiss. Veroeff: Siemens- Werken 18,225 (1939). 7. N. F. Mott, Proc. Roy. Soc. London 171, 27 (1939). 8. J. Bardeen, Phys. Rev. 71, 717 (1947). 9. S. Teszner, Rev. Gen. Elec. 63, 319 (1954). 10. A. M. Cowley and S. M. Sze, J . Appl. Phys. 36, 3212 (1965). 11. J. L. Moll, IRE Wescon Conu. Rec., pt. 3, 32 (1959). 12. R. Lindner, Bell Syst. Tech. J . 41, 803 (1962). 13. H. G. Unger and W. Harth, Adu. Electron. Electron Phys. 34, 281 (1973). 14. L. J. Giacoletto and J. OConnel, R C A Reu. 21, 68 (1956). 15. M. E. McMahon and G. F. Straube, IRE Wescon Conu. Rec., pt. 3, 72 (1958). 16. W. G . Pfann and C. G. B. Garnett, Proc. IRE 47, 2011 (1959). 17. L. J. Giacoletto, Proc. IRE, 45, 207 (1957). 18. H. Lawrence and R. M. Warner, Jr, Bell Syst. Tech. J . 39, 389 (1960).
378 19. 20. 21. 22. 23.
24. 25. 26. 27. 28. 29. 30. 31. 32.
S. TESZNER AND J. L. TESZNER
T. P. Lee,IEEE Trans. Electron Devices 13, 881 (1966). Y. F. Chang, J . Appl. Phys. 31, 2337 (1966). H. K. Gummel and D. L. Scharfetter, J . Appl. Phys. 38, 2148 (1967). T. Sukegawa, K. Fujikawa, and J. Nishizawa, Solid-State Electron. 6, 1 (1963). T. Sukegawa, T . Sakurai, and J. Nishizawa, IEEE Trans. Electron Devices 13,988 (1966) (Corr). D. Kahng, Solid State Electron. 6, 281 (1963). J. L. Moll, S. Krakauer, and R. Shen, Proc. IRE 50, 45 (1962). S. Krakauer, Proc. IRE 50, 1665 (1962). R. A. Moline and G. F. Foxhall, IEEE Trans. Electron Devices 19, 267 (1972). H. L. Stover and H. M. Leedy, National Telecommun. Con/: Atlanta, 1973, p. 23 A-1. I. W . Pence, Jr., and P. I. Greiling, Proc. IEEE (Lett.)62, 1030 (1974). C. B. Swan, Int. Solid State Circuits Con/: Dig. Tech. Pap. 8, 106 (1965). J. C. Irvin and C. B. Swan, IEEE Trans. Electron Devices 13, 471 (1966). I. Kaufman and D. Douthett, Proc. IRE 48, 790 (1960).
AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed. A Acket, G. A., 315(56b), 320 Adams, R. L., 329(80), 334(80), 335 (21a, 80), 338(21a, 80), 347(21a), 351, 352 Ahmed, H., 75(36), 118 Aitchison, C. S., 316(63), 320 Allamando, E., 334(61), 352 Allan, D. W., 229, 234, 244, 245 Alsop, L. E., 220, 229, 244 Amboss, K., 75(31), 80(31), 118 Amelinckx, S., 60(61), 71 Anderson, J. M., 133, 179 Anderson, W. H. J., 73(13), 74(13), 114(13), I18 Anderson, W. M., 334(74), 335(74), 338(74), 352 Angelo, E. J., 277(23), 288 Art, A., 60(61), 71 Assour, J. M., 335(42), 336(42), 351 Audoin, C., 221, 224 Auerbach, D., 192,244 Awender, H., 235, 244
B
Bachynski, M. P.,126, 133, 181 Baier, W., 136, 178 Bajgar, V., 218, 223, 240, 245, 251 Barber, M. R., 307(39), 310(39, 41), 319 Barchukov, A. I., 197, 199, 222, 224, 233, 245 Bardeen, J., 254(1), 265(1), 288, 365(8), 377 Bardo, W. S . , 188, 191, 226, 230, 231, 242, 245, 249 Bardsley, J. N., 153, 176, 178, 180 Barnes, F. S., 195, 224, 225, 245
Barnes, J. A., 234, 237, 238, 239, 245, 249 Barnes, R . S . , 66(66), 67(66), 72 Barrera, J. J., 304(34), 306(34), 319 Bartelink, D. J . , 330, 351 Bashkin, A. S., 203, 245 Basinski, Z . S . , 18, 71 Baskaran, S., 317, 320 Basov, N. G., 184, 191, 193, 195, 203, 219, 223, 225, 226, 227, 229, 231, 235, 239, 245 Bates, D. R., 148, 151, 153, 175, 178 Beale, J. R. A., 278(26), 288 Beaty, E. C . , 146, 147, 178 Beaufoy, R., 259, 271,288 Beck, A. R., 73(23), 118 Beck, A. H. W., 75(36), 118 Becker, G., 191, 195, 196,223, 225, 228, 232, 245 Bederson, B., 123, 178 Beehler, R., 234, 238, 249 Beers, Y . ,209, 245 Bekefi, G., 139, I78 Belenov, E. M., 228, 245 Bell, K. L., 175, 178 Ben-Reuven, A., 192, 202, 243, 244, 245, 25 I Berlande, J., 157, 178 Bershteyn, I. L., 237, 245 Bethe, H. A., 4(1), 5 , 24, 32, 70 Bhattacharya, A. K., 140, 141, 144, 146, 1 78 Biondi, M. A., 128, 134, 137, 143, 144, 146, 153, 154, 155, 156, 157, 178, 179, 180, 181 Bjorkmann, G., 359(23), 362(23), 363(23), 365 Bland, G . F., 238,246 Blaquiere, A., 229, 247 Blaser, J. P., 238, 245 Blodgett, K., 90, 118
379
380
AUTHOR INDEX
Blue, J. L., 321,350 Bluyssen, H., 190, 205, 207, 214, 245, 251 Boccon-Gibod, D., 296, 297, 319 Boer, K. W., 312, 319 Boers, P. M . , 295(8), 315(56b), 318, 320 Boersch, H., 4, 8(22, 23), 70, 71, 73(1), 74(1), 114, 117 Bohm, D., 45, 71 Bollmann, W., 57, 71 Bonanomi, J., 195, 211, 223, 228, 234, 238, 245, 246 Booker, G. R., 4(4), 70 Borisenko, M. I . , 235, 245 Borrmann, G., 47, 71 Bostanjoglo, O., 8(22), 71 Bougnot, G., 295(9), 318 Boulmer, J., 123, 130, 135, 151, 158, 159, 178, 179, 181 Bouvet, J. V., 334(33), 351 Bowen, A. E., 320, 350 Bowers, H. C., 338(51), 342(51), 349(51), 351 Bowler, D. L., 287(43), 289 Brattain, W. H., 254(1), 265(1), 288 Brewer, G . R., 80(39), 96(39), 118 Bromberg, E.A., 192, 244 Brookbanks, D. M . , 304, 320 Brown, C. M., 162, I79 Brown, S. C., 130, 139, 140, 180 Browne, J. C., 146, 147, I78 Bryzzhev, L. D., 238, 246 Bullough, R., 66, 72 Burton, R., 135, 139, 159, 179 Butcher, P. N., 296, 319 Butler, C. M., 338(47), 351
C Cahill, P., 202, 205, 251 Camp, W. O., 299(20), 300, 319 Carlsten, J. L., 173, 180 Carnahan, C. W., 239, 246 Castleton, K. H., 206, 218, 246, 248 Cedarholm, J . P., 238, 239, 246 Cederberg, J. W., 203, 205, 246 Chaffin, R. J., 346(72), 352 Chandra, S., 205,246
Chang, K. K. N., 330(22), 351 Chang, Y. F., 367(20), 378 Chanin, L. M., 143, 144, 146, 147, 178, 179,180 Chaturverdi, P. K., 352 Chawla, B. R., 282(33), 288 Chen, C. L., 157, 179 Cheremiskin, I. V., 219, 224, 250, 251 Cheret, M . , 157, 178 Cherrington, B. E., 173, 180 Chiang, Y. S., 334(79), 338(79), 352 Chibisov, M. I., 175, 180 Chikhakhev, B. M., 235,239,245 Chikin, A. I., 229, 246 Christensson, S.,304, 306(32), 319 Chu, J. L., 353,(10) 359(10), 362(10), 363(10), 364 Chudobiak, W. J., 280(28), 288, 344(63), 345(63), 352 Chu-Sun Yen, 341(53), 342(53), 351 Clarebrough, L. M., 64(63), 72 Clark, G. L., 133, 179 Cloassen, M., 359(21), 365 Clorfeine, A. S., 330, 335(41), 336(41), 351 Cockayne, D. J . H., 23(33), 24(34), 67, 68 (67, 68), 71, 72 Colella, R., 69(70), 72 Coleman, D. J., Jr., 334(36, 7 3 , 335(21b, 36), 338(21b), 341(36), 342(36), 343(36), 347(21b), 351, 352, 352(4, 9), 357(4, 15), 359(9, 15), 364, 365 Collier, R. J., 220, 221, 246 Collin, J. E., 73(14), 114(14), 118 Collinet, J. C., 311, 313(45), 319 Collins, C. B., 135, 139, 157, 159, 170, 173, 179 Connor, T. R., 154, I79 Constant, E., 344(61), 352 Conwell, E. M., 312, 319 Copeland, J. A., 298, 299, 300, 319 Corbett, R. M . , 317(71), 320 Corbey, C. D., 316(63), 320 Cowley, A. M., 341(53), 342(53), 351, 365(10), 377 Cowley, J. M., 4(6), 70 Crewe, A. V., 73(25), 118 Crompton, R.W., 123, 179 Crosswhite, H. M . , 160, 162, 169, 179 Cunningham, A. J., 155, 156, 179, 180 Cupp, R. E.,199, 236,250
38 1
AUTHOR INDEX
D Daams, H., 228, 247 Dalgarno, A., 146, 147, 153, 178 Dalman, G . C., 359(19), 365 Darwin, C. G . , 5, 24, 30, 70 Davis, H. L., 334(74), 335(74), 338(74), 352 Davis, P. J., 86(44), I12(44), 118 Davis, R. E., 334(30), 338(30), 351 Davisson, C., 4(2), 5(2), 70 Davy, P., 158, I78 Davydow, B., 365(1), 377 Dean, R. H., 316, 317, 320 Decker, D. R.,334(36), 335(36), 341(36), 342(36), 343(36), 351 de Graaff, H. C., 280(27), 281(29), 288 Delagebeaudef, D., 321, 350, 357(26), 365 De Loach, B. C., Jr., 330, 332(29), 351, 362(25), 365 Deloche, R., 157, 178 Delpech, J.-F., 123, 133, 135, 140, 151, 156, 158, 159, 178, 179, 181 De Lucia, F. C., 190, 192, 199, 201, 202, 203, 205, 206, 207, 210, 212, 213, 217, 222, 223, 246 De Prins, J., 195, 223, 228, 234, 238, 245, 246 Devos, F., 123, 135, 156, 179, 181 De Vos, J. C., 110(54), 119 DeVries, C. P., 145, 179 De Zafra, R. L., 203, 246 Diamand, F., 341(81), 352 Dicke, R. H . , 197,246 Dietrich, W., 73(6), 114(6), 117 Dimock, D., 173, 179 Director, S. W., 287(43), 289 Dixon, J. R., 171,179 Doi, A., 334(76), 335(76), 338(76), 339(76), 352 Donaghay, R. H., 5(10, I I), 70 Dougal, A . A,, 132, 179 Dousek, O., 223, 245 Douthett, D., 375(32), 378 Dreeben, A. B., 316(64), 317(64), 320, 334(74), 335(74), 338(74), 352 Dreicer, H., 185, 250 Drum, C . M.,19, 58, 71 Dryagin, Y . L., 237, 245 Dubonosov, S. P., 235, 245 Dudenkova, A . V., 235, 246
Dunn, C. N., 334(36), 335(36), 341(36), 342(36), 343(36), 351 Dyke, T. R., 187, 246 Dymanus, A., 187, 190, 200,201, 203, 205, 206, 207, 208, 213, 214, 215, 217, 218, 245, 246, 247, 250, 251
E Early, J. M.,257(6), 288 Eastman, L. F., 304(32), 306(32, 3 3 , 319 Ebers, J. J., 258, 270, 271, 274, 286, 288 Elford, M. T., 123, 179 Eliott, B. T., 294(5), 318 Ellenbroek, A. W., 203,215, 250 Ellis, E., 171, 175, 177, 179 Engl, W. L., 283(40), 289 Englefield, C. G . , 356(11), 357(11), 359(11), 364 English, T. C., 186, 246 Enstrom, R. E., 301(26), 311(48), 319 Epszstein, B., 73(7), 114(7), 117 Esquirol, P., 295(9), 318 Evans, W. J., 323(17, 18), 332(28), 336(43), 350, 351, 352 Evenson, K. M.,203, 218, 251
F Fack, H . , 73(4), 114(4), 117 Falconer, W. E., 187, 246 Farrayre, A., 323(14), 326(14), 329(14), 338(14), 347(14), 350 Fawcett, W., 295, 296,319, 320 Fay, B., 310(43), 319 Fazarine, 2.A., 341(53), 342(53), 351 Fedorenko, G. M., 235,245 Fehsenfeld, F. C., 157, 179 Ferguson, E. E., 157, 179 Ferrell, R. A., 46, 71 Fertik, N. S . , 229, 247 Fey, L., 234, 238, 249 Fisher, M., 73(17), 114(17), 118 Fitzsimmons, W. A., 171, 174, 176, 179 Fong, T. T., 336(44), 337(44), 346(44), 351 Forwood, C. T., 64(63), 72 Fossum, J. G . , 287(44), 289
3 82
AUTHOR INDEX
Fowler, H. A., 46(46), 71 Foxhall, G. F., 373(27), 378 Franzen, W., 73(19), 74(19, 32, 33), 80(33), 90(33), 97(33), 102(32), 118 Freeman, A. J., 45, 71 Frommhold, L., 128, 134, 137, 154, 155, I 79 Frost, L. S., 123, 179 Fujikawa, K., 367(22), 378 Fukuhara, A., 8(24), 71 C
Gaigerov, B. A., 236, 251 Gaines, L., 206, 246 Gallagher, J. J., 199, 236, 250 Gambling, W. A., 221,246 Garnett, C. G. B., 366, 377 Garrison, B. J., 175, 179 Gamey, R. M., 202,205, 206, 246 Gaukler, K. H., 73(15), 114(15), 118 Gauthier, J.-C., 123, 130, 133, 135, 140, 156, 158, 178, 179, 181 Gerardo, J. B., 127, 129, 130, 141, 142, 156, 157, 174, 179, 180 Gerber, R.A., 130, 141, 142, 145, 146, 175, 179 Germer, L. H., 4(2), 5(2), 70 Gevers, R., 60, 71 Giacoletto, L. J., 258(8), 268(8), 288, 366, 367(17), 377 Gibbons, G., 352 Gibbons, J. F., 275(20), 288 Giblin, R. G., 340(55), 342(55), 351 Gilardini, A., 123, 179 Ginter, D. S., 162, 179 Ginter, M. L., 162, 179 Ginzburg, V. L., 133, 179 Giordmaine, J. A., 220, 229, 244 Glauber, R.,7(19), 71 Glover, G. H., 295, 320 Gobat, A. B., 334(37), 335(37), 338(37), 351 Goedbloed, J. S., 341(57), 352 Goldenberg, H. M., 221, 246 Goldstein, L., 130, 132, 133, 157, 173, 179, 180
Goldwasser, R. E., 313(69), 320, 329, 335 (77), 338(77), 339(77), 347(77), 352
Gonfalone, A., 157, 178 Gonzalez, A., 334(79), 338(79), 352 Gordon, J. P., 183, 184, 190, 193, 199, 202, 208,219, 220, 223, 227,229,246 Gordy, W., 190, 192, 201,203, 205, 207, 210, 211,213, 217, 222,223,246 Goringe, M. J., 8(25), 71 Goucher, F. S., 365(3), 377 Grace, M . I., 352 Grant, F. A., 171, 179 Grasyuk, A. Z., 230, 246 Gray, E. P., 129, 154, 179 Greiling, P. I., 373(29), 378 Griem, H. R., 163, 168, 179 Grierson, J., 335(40), 351 Griffith, I., 317(71), 320 Grigor’yants, V. V., 188, 236, 238, 246, 247,251 Grivet, P., 221, 229, 244, 247 Groben, H. L., 312, 319 Grossheim, T. R., 158, 179 Groves, G. W., 66, 72 Gryzinski, M., 149, 151, 179 Gubkin, A. N., 193, 247 Gueret, P., 315, 320 Gummel, H. K., 277(21), 281(31), 282(32), 288, 321(10), 350, 367(21), 378 Gunn, J. B., 293, 294(5), 297, 298, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 316, 317, 318, 319 Gupta, R.,73(19), 74(19), 118 Gurevitch, A. V., 133, 179 Gusinow, M. A., 127, 129, 141, 145, 146, 156, 175, 179, 180 Guttwein, G. K., 235, 247
H
Haas, G. A., 91(48), 118 Hackam, R.,146, 179 Haddad, G. I., 338(46, 50), 340(50), 344(62), 351, 352 Hansch, T. W., 172, 181 Hagebeuk, H. J. L., 136, 180 Hahn, L. A., 278(25), 288 Hakki, B. W.,293(4), 307(39), 310(39), 318,319
383
AUTHOR INDEX
Hall, C. R., 47, 71 Hall, R. D., 341(53), 342(53), 351 Hamilton, D. J., 263(14), 271(14), 283(37), 288,289 Hamilton, S . A., 341(53), 342(53), 351 Hanszen, K.-J., 73(16), 118 Harada, K., 234, 250 Hardin, J., 221, 236, 241, 247, 249, 250 Harrington, R. F., 136, 137, 179 Harris, H. H., 158, 179 Harth, W., 341(59), 342(59), 352, 359(21), 365, 366(13), 368(13), 373(13), 376(13), 377 Hartl, W. A. M.,73(12), 114(12), 118 Hartwig, D., 73(8, 9), 114(9), I18 Hashi, T., 223, 251 Hashimoto, H., 49, 51, 54, 55, 56, 59, 60, 71 Hashizume, N., 311, 319 Hasker, J., 74(26, 27, 28, 29), 118 Hasty, T. E., 329(80), 334(80), 335(21a, 80), 338(21a, 80), 347(21a), 351, 352 Haus, H. A., 357(22), 359(20,22), 365 Havens, B. L., 238, 246 Hayashi, R., 334(32), 338(32), 351 Head, A. K.,64(63), 65, 71, 72 Heald, M. A., 132, 179 Hegyi, I. J., 301(26), 319 Heidenreich, R. D., 15(27), 24, 71 Heim, L. E., 239, 245 Heinle, N., 296, 319 Hellwig, H., 228, 234, 235, 238, 239, 247 Helmcke, J., 359(21), 365 Helmer, J. C . , 186, 190, 192, 219, 220, 224, 225, 227, 242, 247 Herbert, D. C . W., 295, 320 Herbst, H., 359(21), 365 Herrmann, J., 195, 211, 223, 228, 245, 246 Heuvel, J. E. M., 201, 206, 213, 247 Hicks, H. S . , 135, 139, 157, 159, 179 Higa, W. H., 226, 231, 247 Hilsden, F. J., 338(47), 351 Hilsum, C., 293, 294, 295, 296, 303, 318, 319,320 Hines, M. E., 338(48, 52), 341(52), 351 Hinnov, E., 158, 159, 173, 179, 180 Hirsch, P . B., 2, 4(4, 9, 5(16), 15, 16, 17, 18(29), 19(28), 20, 21, 22, 23(16), 29, 40, 47, 51, 60,64, 66(16), 70, 71
Hirschberg, J. G., 158, 159, 180 Hobson, R. M., 155, 156, 179, 180 Hoefflinger, B., 323(13), 350 Hoerni, J. A., 7(20, 21), 71 Hoffmann, K. R., 315(56b), 316, 320 Hojo, A., 295(10), 319 Holstein, T., 143, 154, 174, 178, 180 Holuj, F., 228, 247 Hopfer, G., 235, 247 Hornbeck, J. A., 144, 178, 180 Howie, A., 2(16), 5(16), 15(16), 16(29), 17, 18(14, 15, 29), 20, 21(29), 22(29), 23(16, 29), 24, 28, 40(16), 47, 49(59), 54(59), 55(59), 56(59), 57, 59(59), 60(16, 59), 62, 64 (29), 66(16, 66), 67(66), 70, 71, 72 Huang, H. C., 298(18), 319,323(15), 334(37), 335(37), 338(37), 347(15), 350, 351 Humble, P., 64(63), 71 Humphreys, C. J., 4(5), 47, 51, 70, 71 Hurt, W. B., 157, 170, 175, 179, 180 Hutson, A. R., 73(2), 117
I Ibers, J. A., 7(20, 21), 71 Iglesias, D. E., 334(30, 34a, 34b), 338(30, 34a, 34b), 351 Ikola, R. J., 330, 351 Ingraham, J. C., 130, 139, 180 Irvin, J. C . , 334(36), 335(21b, 36), 338(21b), 341(36), 342(36), 343(36), 347(21b), 351, 374(31), 378 Isaacson, M., 73(25), 118 Ito, T., 223, 247 Ivanov, N. E., 235,245
J Jacobus, F. B., 186, 190, 192, 242, 247 Javan, A., 199,203, 205, 251 Javan, J., 199,203,251 Jenkins, M. L., 68(67, 69), 72 Jeppesen, P., 304, 319 Jeppsson, B., 304,319 Jepsen, D. W., 69, 72 Jeschke, G., 8(23), 71 Johnson, A. W., 157, 174, 180 Johnson, B. W., 173, 179
384
AUTHOR INDEX
Johnson, D., 73(25), 118 Johnson, E. D., 328(19), 350 Johnson, H., 341(54), 351 Johnson, L. C., 173, 179 Johnson, S., 234, 235, 247 Johnson, W. A., 334(36), 335(36), 341(36), 342(36), 343(36), 351 Johnston, R. H., 300, 319 Johnston, R. L., 323(17), 330, 332(23), 350, 351 Johnston, T. W., 126, 133, 181 Jones, D., 299(21), 319 Joullie, A., 295(9), 318 Joy, D. C., 5(12), 70
K Kahng, D., 378 Kainuma, Y.,47, 71 Kakati, D., 192, 247 Kalra, S. N., 228, 247 Kanbe, H., 310,319 Kartashoff, P., 195, 223, 228, 234, 238, 246 Kataoka, S., 311, 315, 319, 320 Kaufman, I., 375(32), 378 Kawashima, M., 315(57), 320 Kazachok, V. S., 228, 240,247 Keck, J. C., 149, 150, 159, 180 Kelly, A., 66, 72 Kerr, D. E., 129, 154, 179 Khokle, W. S., 352 Kholodova, G. K., 193,247 Kieffer, L. J., 123, 178 Kim, C. K., 329(21c), 335(21c), 338(21c), 339(21c), 345(21c), 347(21c), 351 Kinderdijk, H. M. J., 136, 180 Kingston, A. E., 151, 175, 178 Kino, G. S., 294(6), 300, 310(43), 312, 318, 319 Kinoshita, J., 303(72), 320 Kioustelidis, J. K., 283(40), 289 Kirk, C. T., Jr., 278(24), 288 Klaassen, F. M., 281(30), 288 Klatskin, J. B., 334(37), 335(37), 338(37), 351 Kleiman, A. S., 229, 247 Klemperer, W., 187, 246 Kleppner, D., 221,246
Klyshko, D. H., 236, 238, 251 Kobayashi, M., 195, 234,238, 250 Kobayashi, S., 204, 218, 250 Kodera, H., 334(78), 338(78), 352 Koehler, D., 263(15), 271(15), 288 Kohno, N., 238,250 Kondo, K., 202,218,247 Konstantinov, A. I., 238,247 Koshurinov, E. I., 193, 227,247 Krakauer, S., 370(25, 26), 378 Kramer, B., 323(14), 326(14), 329(14), 338(14), 347(14), 350 Kramer, N. B., 347(66), 352 Krause, W. H. U., 226,232,247 Kright, S., 307(39), 310(39), 319 Krisher, L. C., 195, 199,202, 203,204, 205, 207, 209,212, 213, 216,251 Kromer, H., 257(5), 288 Kroger, H., 352 Krokhin, 0. N., 239, 245 Kroll, N. M., 239, 247 Krupnov, A. F., 188, 191, 193, 194, 197, 198,201,202, 204, 213,214, 216, 219, 220, 222, 223, 227, 228, 232, 234, 235, 240,247,248,250 Ku, R.T., 173, 180 Kuckes, A. F., 158, 180 Kukolich, S. G., 187, 192, 196, 201, 202, 203, 204, 205, 206, 207,211,213, 215, 218, 240,243, 244, 245,246, 248,249, 250,251 Kumabe, K., 310(44), 319 Kuminski, J. F., 316(64), 317(64), 320 Kunkel, W. B., 159, 180 Kuno, H. J., 316(62), 320 Kurokawa, K., 296,319 Kurtz, C. V., 194, 205, 249 Kuru, I., 295(10), 300, 315, 319 Kuyatt, C. E.,73(11), 75(35), 103(35), 114(11), I18
L Laink, D. C., 184, 188, 191, 192, 193, 197, 198, 199, 200,217, 222,224, 225, 226, 229, 230, 231, 232, 233, 234, 241, 242, 243, 245,247, 248, 249,250 Lamb, W. E., 239,247
385
AUTHOR INDEX
Lane, W. F., 171, 174, 176, I79 Langenberg, D. N., 122, 181 Langmuir, I., 90, I18 Lauer, R., 73(16), I18 Lawrence, H., 367(18), 377 Lecar, H., 199, 203, 204, 205, 251 Lee, C. A., 359(19), 365 Lee, D. H., 336(44), 337(44), 346(44), 351 Lee, T. P., 341(60), 342(60), 352, 367(19), 378 Leedy, H. M., 373(28), 374(28), 375(28), 376(28), 378 Lefebvre, M., 344(61), 352 Lefrkre, P. R., 224, 231, 232, 233, 241, 242, 249 Lehner, H. H., 311,319 Leiby, C. C., 157, I79 Leikin, A. Y., 229, 238, 246, 247, 249 Lenz, F., 45(44), 71 Le Poole, J. B., 4, 70 Leventhal, J. J., 158, I79 Levi, R.,91(47), 99(47, 501, 106(47), I18 Levine, P. A., 334(37), 335(37), 338(37), 35I Levinshtein, M. E., 309, 314, 320 Lichtenstein, M., 199, 236, 250 Lindholm, F. A., 263(14), 271(14), 283 (38, 39), 287(43), 288, 289 Lindner, R., 365(12), 377 Lindsay, P. A., 73(20), 80, I18 Linvill, J. G., 275, 288 Liu, B., 162, I80 Liu, S. G., 346(64), 347(68, 69), 352, 356 (12), 357(12), 359(12), 364 Llewellyn, F. B., 320, 350 Lochte-Holtgreven, W., 168, I80 Lockyear, W. H., 338(51), 342(51), 349(51), 351 Loeb, L. B., 320(3), 350 Loeffler, K. H., 114(58), I19 Logachev, V. A., 224, 227,228, 232, 248, 249 Logan, J., 273(17, 18), 274(17, 18), 277(23), 280(17), 286(17), 287(17, 18), 288 Loubser, J. H. N., 199, 203, 204, 205, 207, 251 Loya, A., 352(4), 357(4), 364 Luther, L. C., 335(21b), 338(21b), 347(21b), 35I LYM, D. K., 283(37), 289
M McDaniel, E. W., 123, 128, 143, I80 MacGillavry, C. H., 24, 71 McIlrath, T. J., 173, I80 McKay, K. G., 320(5), 350 MacKenzie, L. A., 298(18), 319 McKillican, R., 344(63), 345(63), 352 McMahon, M. E., 366,377 McRae, E. G., 5(17), 71 McWhirter, R. W. P., 151, I78 Madson, J. M., 144, 147, 180 Magarschack, J., 312(51), 319 Magnee, A., 73(14), 114(14), I18 Mahan, B. H., 144, I80 Maher, D. M., 66, 72 Makios, V., 344(63), 345(63), 352 Maloney, C. E., 73(23), I18 Mansbach, P., 149, 150, 159, I80 Manus, C., 157, 178 Marcus, P. M., 69, 72 Marcuse, D., 191, 195, 199, 205, 223, 249 Marinaccio, L. P., 338(45), 351 Maroof, A. K. H., 188,232,234,249 MartiSovitS, V., 141, I80 Marton, L., 46, 71 Masnari, N. A., 338(50), 340(50), 351 Massey, H. S. W., 7(18), 71 Matarese, R. J., 316(66), 317(66), 320 Mattei, W. G., 329(21c), 335(21c), 338(21c), 339(21c), 345(21c), 347(21c), 35I Mazey, D. J., 66, 67(66), 72 Meckwood, J. C., 283(35), 288 Mednikov, 0. I., 191, 249 Meek, J., 320(3), 350 Mehr, F. J., 128, 134, 155, 156, 179, I80 Menoud, C., 234,246 Metherell, A. J. F.,51, 71 Meyer, C. S.,283,286,289 Middlebrook, R. D., 259(11), 270(11), 288 Mielczarek, S. R., 75(35), 103(35), I18 Migitaka, M., 334(38, 39, 76, 78), 335(38, 39, 76), 338(38, 39, 76, 78), 339(76), 351, 352 Miller, S.L., 320(6), 350 Miller, W. H., 175, I79 Misawa, T., 320, 334(35), 341(60), 342(60), 350, 351,352 Misell, D. L., 8, 71
386
AUTHOR INDEX
Mitchell, A. C. G., 167, 168, 172, I80 Mittelstadt, V. R., 154, 156, 180 Mockler, R. C., 234, 238, 249 Moiseiwitsch, B. L., 126, 180 Molikre, K., 47, 71 Moline, R. A., 373(27), 378 Moll, J. L., 258, 270, 271(16), 274, 286, 288, 365(11), 366, 370(25), 377, 378 Msller, C., 239, 249 Mollier, J. C., 221, 241, 249 Molnar, J. P., 144, 175, 176, 178, 180 Moore, C. E., 160, 180 Morgan, A. H., 238, 249 Morozov, V. N., 202, 224, 227,231, 232, 245,249 Morton, A. J., 64(63), 72 Mosharrafa, M. A., 140, 180 Motley, R. W., 158, 180 Mott, N. F., 7(18), 71, 365(7), 377 Muller, E. W . , 73(24), 118 Mukhamedgalieva, A. F., 221, 228, 240, 249 Muller, M. W., 220, 247 Mulliken, R. S . , 157, 162, 180 Murin, I. D., 236, 249 Murphy, B. T., 277(21, 22), 288 Murr, J., Jr., 335(42), 336(42), 351 Myers, F. A,, 303(73), 320
Noodward, D. W., 304(32), 306(32), 319 Norton, L. E., 301, 319 Nottingham, W. B., 80(40),96(40), 106(40), 114(57), 118, 119 Novak, M. M., 193,247 Noyce, R. N., 365(4), 377 0
Oates, D. E., 192, 202, 243, 244, 248, 251 Oberoi, R. S . , 176, 180 O’Connel, J., 366, 377 Ogg, N. R., 296(13), 319 O’Hara, S., 335(40), 351 Ohtorno, M., 306, 307, 319 Okaya, A., 199, 203, 251 Olivier, M., 236, 247 O’Malley, T. F., 153, 155, 156, 180 Ornori, M., 334(32), 338(32), 351 Oraevskii, A. N., 192, 193, 202, 221, 225, 226, 227, 228, 229, 230, 231, 234, 239, 245,246,249 Oskarn, H. J., 129, 130, 140, 144, 145, 146, 147, 153, 154, 156, 160,179,180, 181 Ovcharov, M. Y.,216, 223, 227, 228, 234, 235, 240, 248
P N Nakamura, M., 334(38, 39), 335(38, 39), 338(38, 39), 351 Napoli, L. S . , 330, 351 Narayan, S. Y . ,301, 302, 303(28), 319 Narud, J. A., 263(14), 271(14), 286, 288 Naumov, A. I., 188, 235, 248,250 Nelson, A. C., 187, 203, 204, 206, 218, 248,249 Nesbet, R. K., 176, 180 Newton, B. H., 316(63), 320 Nicholson, R. B., 2(16), 5(16), 15(16), 23 (16),’40(16),60(16), 66(16), 71 Niehaus, W. C., 334(34a, 34b), 338(34a, 34b), 351 Nielsen, L. D., 295, 320 Nii, R., 310(44), 319 Nikitin, V. V., 227, 228, 234, 236, 249 Nishizawa, J., 367(22, 23), 378
Paczkowski, J. P., 301, 302, 303(28), 319 Paik, S . F., 349(73), 352 Palkina, L. A., 175, 180 Parker, J. H., 102(52), 119 Parker, W. H., 122, 181 Parygin, V. H., 191, 249 Pashley, D. W., 2(16), 5(16), 15(16), 23(16), 40(16), 60(16), 66(16), 71 Patterson, P. L., 146, 162, 178, 180 Pearson, G . L., 365(3), 377 Pence, I. W., Jr., 373(29), 378 Pendry, J. B., 36, 71 Perlman, B. S . , 311(48), 312(48), 313(49), 319 Perrin, R. C., 66, 72 Persky, G., 283(34), 288 Pestovskii, U. I., 229, 247 Pfann, W. G . , 366, 377 Phelps, A. V., 123, 140, 171, 175, 176, 177, 179, 180
387
AUTHOR INDEX
Philbrick, J., 155, 156, 180 Pierce, J. R., 80(41), 96(41), 118 Pines, D., 45, 71 Pitaevskii, L. P., 150, 180 Plotkin, H. H., 235, 247 Polonsky, l., 86(44), 112(44), 118 Poon, H. C . , 282(32), 283(35, 36), 288 Porter, J. H., 73(19), 74(19, 32, 33), 80, 90, 97(33), 102(32), 118 Poukey, J. W., 127, 129, 180 Prager, H. J., 330(22), 351 Prew, B. A., 295, 320 Pribetich, J., 323(14), 326(14), 329(14), 338(14), 347(14), 350 Prim, R. C., 352(1), 364 Pritchard, R. L., 257(7), 288 Prokhorov, A. M., 184, 197, 199,219, 222, 224, 233, 245, 249 Pucel, R. A., 357(22), 359(20, 22), 365 Purcell, E. M., 103, 119 Purcell, J. J., 317(71), 320
R Rabier, A., 312(51), 313(68), 319, 320 Radford, H. E., 194, 205, 249 Raith, H., 8(22, 23), 71 Rakova, G. K., 195,250 Ramachandran, T. B., 301, 303(29), 319 Ramsey, N. F., 79(37), 96(37), 118, 201, 214, 215, 221,243, 244, 246, 250 Ravaut, R., 236,247 Ray, I. L. F., 23(33), 67, 68(33, 67, 68), 71, 72 Read, W. T., 320, 323, 350 Rees, H. D., 294, 295, 299(21), 303, 318, 319,320 Reynders, J. M. H., 203, 215, 250 Reynolds, J. F., 347(69), 352 Riddle, E. R., 300, 319 Ridley, B. K., 293, 318 Risco, J. J., 346(64), 356(12), 357(12), 359(12), 352, 364 Rittner, E. S., 99(50), 118 Robben, F., 158, 159, 168, 169, 171, 180, 181
Robertson, A. G., 123, 179, 180 Robertson, W. W., 157, 177, 179, 181
Robinson, B. B., 316(67), 317(66), 320, 341(54), 351 Robinson, E. J., 176, 181 Robinson, N. O., 236, 237, 250 Robson, P. N., 310, 312, 317, 319, 320, 356(11), 357(11), 359(11), 364 Roch, J. G . , 294(6), 318 Rogers, W. A., 157, 181 Rogowski, W., 320(2), 350 Rosen, A., 347(69), 352 Rossel, J., 195, 228, 246 Rosztoczy, F. E., 303(72), 313(69), 329, 334(32), 335(77), 338(32, 77), 339(77), 347(77), 320, 351, 352 Ruben, D. J., 196, 201, 203, 204, 206, 211, 2 15, 248, 249, 250 Rutgers, G . A. W., 110(54), 119 Ruttan, T. G . , 303(30), 319
S
Saburi, Y.,195, 234, 238, 250 Sah, C. T., 365(4), 377 Saito, K., 334(38, 39, 76), 335(38, 39, 76), 338(38, 39, 76), 339(76), 351, 352 Sakurai, T., 367(23), 378 Salazar, H., 234, 238, 249 Salmer, G., 323(14), 326(14), 329(14), 338(14), 347(14), 350 Sauter, G. F., 146, 179 Savranskii, V. V., 197, 199, 222, 224, 233, 245 Sawa, V. A,, 224,227, 232, 249 Schaeffer, H. F., 175, 179 Scharfetter, D. L., 320, 321(10), 323(17), 328(20), 329(20), 330, 332(23), 336(43), 347(20), 350, 351, 367(21), 378 Scherer, E. F., 340(55), 342(55), 351 Schermann, J. P., 221, 244 Schiske, P., 114(56), 119 Schmeltekopf, A. L., 157, 179 Schomaker, V., 7(19), 71 Schottky, W., 365(5, 6), 372(6), 373(6), 377 Schroeder, W. E., 338(46), 344(62), 351, 352 Schulson, E. M., 5(10, l l ) , 70 Schulz, G . J., 75(34), 103(34), 105(34), 112(34), 118 Seaton, M. J., 150, 181
388
AUTHOR INDEX
Seidel, T. E., 334(30, 34a, 34b), 336(43), 338(30, 34a, 34b), 351 Sekine, K.,334(38, 39, 76), 335(38, 39, 76), 338(38, 39, 76), 339(76), 351, 352 Sellberg, F., 359(24), 365 Shakhov, V. O., 230,250 Shaw, A. M. B., 4(4), 70 Shaw, D. W., 329(80), 334(75, 80), 335 (21a, 80), 338(21a, 80), 347(21a), 351, 352 Shaw, M. J., 173, 179 Shaw, M. P., 312(51), 319 Shcheglov, V. A., 191, 250 Shchuko, 0. B., 223,232,240, 247 Shelton, H., 73(22), 75, 106(22), 118 Shen, R., 370(25), 378 Sher, N., 220, 250 Sheronov, A. P., 216,223, 227, 228,234, 235, 240, 248 Shigenari, T., 203,204,210,213, 218,250, 251 Shimizu, T., 190,194, 201, 203, 204, 211, 213, 217,218,250, 251 Shimoda, K., 190, 191, 194, 195, 196, 197, 202,203,204, 209, 211, 213, 218, 220, 222,223,224, 225, 227, 229, 238, 240, 241,247,250,251 Shkarofsky, I. P., 126, 133, 181 Shockley, W., 254(2, 4), 259, 288, 352, 357, 364, 365(2, 3,4), 377 Shurnyatskii, P. S., 227, 250 Shur, M. S., 309, 314,320 Shyam, M., 304(33), 306(33), 319 Sibiryakov, V. L., 237, 245 Silvey, W., 236, 237, 250 Simpson, J. A., 46(46), 71, 73(11), 75(35), 103(35), 114(11), 118 Sinegubko, L. A., 194, 201, 202, 213, 214, 240, 248 Sircar, P., 241, 250 Sitch, J., 356(11), 357(11), 359(11), 364 Sjolund, A., 352(6), 359(24), 364, 365 Skvortsov, V. A., 188, 191, 193, 194, 197, 198, 201, 202, 204, 213, 214, 216, 219, 220,222,223, 227, 228,234, 235, 240, 247, 248, 250 Slatter, J. A. G., 278(26), 288 Slotboom, J. W., 285(42), 289 Smart, G. D. S., 191, 197, 198, 199, 222, 249
Smirnov, B. M., 175, 180 Smith, A. L. S., 226, 229,230,232,249, 250 Smith, G. F., 73(3), 117 Smith, S. J., 126, 180 Snapp, C. P., 357(13, 16), 358(13, 16), 359 (13, 16, 23), 360(13), 361(13), 362(23), 363(23), 365 Sol, C., 123, 130, 135, 178, 181 Solomon, P. R., 312, 319 Sopel'nikov, M. D., 238, 246 Sparkes, J. J., 259, 271, 288 Sparks, M., 254(4), 288, 365(3), 377 Speidel, R., 73(15), 114(15), 118 Spencer, J. P., 4(5), 70 Spenke, E., 365(6), 372(6), 373(6), 377 Spitalnik, R., 312, 313(68), 319, 320 Srivastava, R. C., 225, 249 Stanley, R. D., 341(60), 342(60), 352 Statz, H., 357(22), 359(20, 22), 365 Steele, R., 329(21c), 335(21c), 338(21c), 339(21c), 345(21c), 347(21c), 351 Stevefelt, J., 151, 158, 159, 169, 171, 179, 181
Stevens, R., 303(73), 320 Stitch, M. L., 236, 237, 250 Stover, H. L., 373(28), 374(28), 375(28), 376(28), 378 Strakhovskii, G. M., 193, 196, 202, 219, 221, 224, 225, 226, 227, 228, 229, 232, 234, 235, 239, 240, 242,245,249, 250 Strandberg, M. W. P., 185, 250 Straube, G. F., 366, 377 Strauch, R. G., 199, 236, 250 Sturkey, L., 24, 71 Sturrock, P. A., 186, 190, 192, 242, 247 Su,S., 323(16), 341(16, 58), 343(16, 58), 344(16, 58), 347(16), 350, 352 Suchkin, G. L., 195, 250 Sudvilovskii, V. Y.,234,250 Sukegawa, T., 367(22, 23), 378 Sultan, N. B., 352(7), 357(7), 359(14, 17, 18), 364, 365 Sussman, K. W., 277(23), 288 Suzuki, S., 223, 251 Sverchkov, E. I., 236, 238, 251 Sverdlov, Y.L., 236, 238,251 Svidinskii, K. K., 191, 195, 223, 245 Swan, C. B., 374(30, 31), 378 Swanson, N., 46(46), 71
389
AUTHOR INDEX
Swartz, G. A., 334(79), 338(79), 352 Sweet, A. A , , 31 I , 313(45), 319 Sweeting, R. C., 192, 193, 243, 249 Sze, S., 323(16), 341(16, 58), 343(16, 58), 344(16, 58), 347(16), 350, 352 Sze, S. M., 352(4), 353(10), 357(4, 15), 359(10, 15), 362(10), 363(10), 364, 365, 365(10), 377 Szekely, V., 296, 319
T Tajima, Y., 315, 320 Takagi, S., 18, 71 Takahashi, I., 223, 251 Takahasi, H., 220, 250 Takami, M., 190, 203, 217, 218,251 Takayarna, Y., 338(49), 351 Takurna, H., 194, 203, 204, 211, 213,218, 250, 251 Talbot, L., 159, 180 Tansley, D. W., 5(12), 70 Tarangioli, D., 335(42), 336(42), 351 Tarnay, K., 296, 319 Tatarenkov, V. M., 193, 196, 202, 225, 226, 227, 228, 229, 234, 242, 245, 249, 250 Tateno, H., 315(57), 320 Tatsuguchi, I., 334(36), 335(36), 341(36), 342(36), 343(36), 351 Taylor, B. N., 122, 181 Taylor, E. R. G., 235, 251 Teal, G. K., 254(4), 288, 365(3), 377 Telio, J., 352 ter Meulen, J. J., 205, 208, 251 Teszner, J. L., 296, 291, 315, 319, 320 Teszner, S., 320(4), 326(4), 350, 365(9), 377 Teter, M. P., 177, 181 te Winkel, J., 284(41), 289 Thaddeus, P., 187, 190, 193, 195, 199, 202, 203, 204, 205, 207, 209, 212, 213, 216, 217, 222, 223, 2S1 Theriault, G. E., 347(69), 352 Thim, H. W., 307, 310, 311, 312, 316, 319 Thomas, D. E., 271(16), 288 Thomassen, K. I., 136, 181 Thomson, G. P., 4(3), 70 Thownsend, J. S., 320(1), 350
Tolnas, E. L., 229, 2.51 Tomasevich, G. R., 187, 190, 193, 196, 204, 205, 206, 217, 222, 223, 246, 251 Tournarie, M., 69, 72 Townes, C. H., 183, 184, 195, 196,209, 219, 220, 221, 222, 223, 224, 221, 229, 238, 239,244, 246, 250, 251 Trew, R. J., 338(50), 340(50), 351 Triano, H., 316(64), 317(64), 320 Trkal, V., 218, 223, 240, 245, 251 Troitskii, V. S., 229, 251 Tucker, K. D., 187, 190, 193, 196, 204, 205, 217, 222, 223,251 Tiixen, O., 144, f81 Twiddy, N. D., 171, 175, 177, 179
U Uebersfeld, J., 221, 241, 247, 249 Ulrner, K., 73(8, 9, lo), 114(9, lo), 118 Ulrich, G., 341(59), 342(59), 352 Ul'yanov, A. A., 216, 223, 221, 228, 234, 235, 240, 248 Unger, H. G., 366(13), 368(13), 373(13), 376(13), 377
V van Essen, C. G., 5(10, l l ) , 70 van Roosbroeck, W., 254(3), 288 Vasneva, G. A., 236, 238, 251 Vauge, C., 162, 181 Vavra, A., 240,251 Vavra, V., 218, 223, 245 Veatch, G. E., 144, 181 Veith, W., 73(5), 114(55), 117, I19 Venohora, M., 307(39), 310(39), 319 Verdeyen, J. T., 173, 180 Verhoeven, J., 190, 205, 207, 214, 218, 245, 25I Vessot, R. F. C., 239, 251 Vitols, A. P., 145, 146, 160, 181 Vlaardingerbroeck, M. T., 315, 320 Vlasov, V. P., 235, 245 Vonbun, F. O., 196,251 von Laue, M., 10, 11, 13, 24, 27, 30, 31, 37, 39, 41, 51, 71 Voss, P., 312,319
390
AUTHOR INDEX
W Wainwright, A. E., 234, 245 Wallace, R. N., 311, 313(45), 319 Wallenstein, R., 172, 181 Walsh, T.E., 311,319 Walters, G. K., 171, 174, 176, 179 Wang, J. H. S., 192, 202, 203, 206, 243,
Wilmshurst, T. H., 220, 221, 246 Wisseman, W. R., 329(80), 334(75, 80), 335(21a, 80), 338(21a, 80), 347(21a), 351, 352
Wofsy, S. C., 201,202,248 Woolf, R. J., 5(12), 70 Wright, G. T., 352(5, 7), 357(7), 359 (14, 17, 18), 364, 365
244,248,251
Wang, T. C., 195, 196,209,222,223,224, 227,229, 241, 250, 251 Warner, R. M., Jr., 367(18), 377 Warner, R. W., 102(52), 119 Watanabe, T.,334(78), 338(78), 352 Watkins, T. B., 293, 318 Weber, C., 80(38), 89(38), 96(38), 118 Weisbrod, S., 330(22), 351 Weissglas, P., 307, 319, 357(13, 16), 358 (13, 16), 359(13, 16), 360(13), 361(13), 365
Weller, K. P., 334(74), 335(74), 338(74), 341(56), 342(56), 352, 352(8), 364 Wells, W. E., 135, 139, 157, 159, 179 Wen, C. P., 334(79), 338(79), 352 Wenstrup, R. S., 73(19), 74(19, 33), 80(33), 90(33), 97(33), 118 Wharton, C. B., 132, 179 Wharton, L., 192, 244 Whelan, M. J., 2(16), 4(4), 5(16), 12, 13, 15(16), 16(29), 17, 18(14, 15, 29), 19(28), 20(29), 21(29), 22, 23(16, 29, 33), 24, 28, 40(16), 45(42), 47, 49(59), 51, 54(59), 55(59), 56691, 57, 58, 59(59), 60(16, 59), 62, 64(29), 66(16), 67, 68(33), 70, 71 White, L. D., 188, 219, 220, 246, 251 White, P. M., 304, 320 Whitten, J. L., 162, 181 Wierich, R. L.,340(55), 342(55), 351 Williams, J. R., 206, 248
Y Yamamoto, M., 223, 247, 251 Yamanashi, B. S., 206, 248 Yarnano, M., 223, 251 Yasuda, Y., 234, 250 Yelkin, G. A., 236, 251 Ying, R. S., 334(31), 336(44), 337(44), 338(31), 346(44), 347(66), 351, 352 Yoshioka, H., 47, 71 Yosimura, H., 352, 364 Young, R. D., 73(21, 24), 118
Z
Zawels, J., 259(13), 288 Zeiger, H. J., 183, 184, 219, 223, 227, 229, 246
Zemansky, M. W., 167, 168, 172, 180 Zettler, R. A., 341(53), 342(53), 351 Zhabotinskii, M. E., 188, 236, 238, 246, 247, 251
Zimmermann, B., 73(10, 18), 75(30), 80(18), 114(10), 115, 118, 119 Zorn, J. C., 186, 246 Zuev, V. S.,191, 195,203,219,223, 245, 251
SUBJECT INDEX A
base width modulation in, 261 charge-control concept in, 253-288 Early effect in, 261 Linvill lumped model and, 275-277 Bloch wave in electron diffraction, 34-36, 38, 48-49 perfect crystal and, 55 Block wave vector, 48 Bohr’s radius afterglow and, 160 in electron diffraction, 44 value for, 122 Born approximation for fast electrons, 6 in inelastic scattering, 44 in wave-mechanical theory, 33 Borrmann transmission, electrical diffraction and, 47 Bragg condition, in electron diflraction, 24, 30-31 Bragg law, in kinematical scattering, 10 Bragg reflection, 27, 67 in aluminum foil, 53 in transmission electron microscopy, 58-59 Bragg scattering, in electron diffraction, 24-25 Burgers vector, of screw dislocation, 20, 23, 61-67
ABM, see Atomic beam maser Absorption spectroscopy, rare-gas stationary afterglow and, 170-172 Afterglow, see Rare-gas stationary afterglow; Stationary afterglow Afterglow spectroscopy, 162-1 68 Doppler broadening in, 163 Aluminum crystal, electron diffraction pattern in, 3 Argon, metastable atoms of, 177 Atomic beam maser, 221 see also Molecular beam masers Atomic ion recombination, in stationary afterglows, 158-159 Avalanche diodes, 320-350 .see also IMPATT diodes; TRAPATT diodes electrical characteristics and performances of, 337-348 structure of, 323
B
BARITT (barrier injection transit time) diodes, 292, 307 advantages and disadvantages of, 363-364 electrical characteristics and performance of, 359-362 fabrication of, 357-358 noise sources in, 362 physical mechanism in, 353-354 punch-through, 352-364 Beam-maser spectroscopy, 199-219 see also Molecular beam maser spectroscopy Bend extinction contours, 14-15 Bethe theory, in electron diffraction, 3 2 4 2 Bipolar transistors, 292 see also Charge-control concept
C
Cathode in electron gun, 99-102 hemispherical structure around, 97-98 spherical, 95-99 thermionic, see Thermionic cathode Cathode temperature see also Thermionic cathode potential and, 92-93 Richardson’s law and, 91
391
392
SUBJECT INDEX
Charge-control concept in bipolar transistor, 253-288 in computer-aided analysis of transistors and circuits, 284-287 Ebers-Moll model derived from, 269-271 Gummel's charge-control relation in, 28 1-283 lightly doped collector and, 278-281 miscellaneous applications of, 277-284 origin of, 258-263 physical background of, 263-266 self-analog or separation technique in, 277-278 for time-dependent response, 283-284 Charge-control models general remarks on, 274 large-signal, 271-273 Charge-control representation, hybrid-pi equivalent circuit as small-signal version of, 266-269 Closed-cavity resonators, in molecular beam masers, 197-199 Collisional-radiative recombination, of electrons and ions, 148-149 Collision frequency measurements, stationary afterglow and, 133-139 Complex lattice potential, in inelastic scattering, 47-48 Coulomb collisions, 126-128 electron energy distribution and, 127-128 rates in, 126-127 Crystal($ extinction distances in, 14 imperfect, 16-23,40 inelastic scattering of, 47-52 kinematical scattering by, 9-16 perfect, in electron diffraction, 9-16, 25 Crystalline materials, electron microscope and, 1-70 Crystal-vacuum interface, wave vectors at, 34
D Debye-Waller factor, 33 Diffraction, electron, see Electron diffraction Diffusion, ambipolar, in rare-gas stationary afterglows, 129-130
Diffusion equations, in rare-gas stationary afterglows, 128-129 Diode lasers, solid state, 173 Diodes, doped, 310-3 12 see also Germanium arsenide diodes; Gunn diodes; IMPATT diodes; TRAPATT diodes; Varactor diodes Dislocation lines, in transmission electron miscroscopy, 61-68 Dispersing equations, in electron diffraction, 68-69 Doped diodes, 310-312 Doppler broadening, in afterglow spectroscopy, 163-164
E
Early effect, in bipolar transistors, 261 Ebers-Moll equivalent circuits, 274 Ebers-Moll large-signal equivalent circuit, 258 Ebers-Moll model, from charge-control concept, 269-271,286 Effusers, multichannel, in molecular beam masers, 186-1 87 Elastic electron neutral collisions, rates in, 124-125 Elastic scattering by metastable rare-gas atoms, 176 by single atoms, 5-9 Electrode, annular, 96 see also Cathode; Thermionic cathode Electron(s) emission of by hot cathode, 73-1 17 focusing and scattering of, 107 potential barrier passage by, 93-94 thermionic, 73-1 17 wavelengths for, 6 Electron continuity equation, 178 Electron diffraction, 1-70 for aluminum crystal, 3 amplitude-phase diagram for, 19-20 application to electron-microscope image contrast of crystalline materials, 52-68 Bloch waves in, 34-38,48 Bragg case in, 24, 30-31 Bragg scattering and, 24-25 Bragg solution in, 30-3 1
393
SUBJECT INDEX
dispersing equations in, 68-70 dispersion surface in, 27 dynamical equilibrium in, 24 dynamical theory of, 24-42 dynamical theory including adsorption in, 50-51 elastic scattering and, 5-9 image formation and, 2 inelastic scattering and, 43-52 kinematic theory and, 5-24 Laue case solution in, 27-29, 37-39 Laue solution with adsorption in, 51-52 in perfect crystal, 9-16, 25-26 phonon scattering in, 4 6 4 7 planar defects in, 57-61 plasmon excitation in, 45-46 screw dislocation in, 20-22, 63-67 stacking fault in, 18-19 theories of, 5-52 wave-mechanical (Bethe) theory in, 3 2 4 2 wave-optical (Darwin) theory of, 24-31 Electron density measurements data interpretation in, 136-139 experimental systems in, 134-1 35 nonresonant systems in, 135 stationary afterglows and, 133-1 39 Electron energy analysis, 103-109 Electron energy balance, and rare-gas stationary afterglows, 130-1 32 Electron energy distribution Coulomb collisions and, 127-128 results of study in, 109-113 surface barrier and, 75-80 Electron energy spectra, 93-95 Electron gun construction of, 99-102 converging and diverging beams in, 114-115 spherically symmetrical, 95-99 Electron-ion recombination collisional-radiative, 148-1 52 dissociative, 152-1 53 experimental results in, 154-160 radiative processes in, 150 in stationary gas afterglows, 147-160 Electron microscope bend extinction coutours in, 14-15 contrast production in, 3 crystalline materials and, 1-70 electron diffraction theory and, 1-70
image formation in, 2 kinematical scattering and, 9-16 scanning, see Scanning electron microscope stacking fault and, 18-19 transmission, see Transmission electron microscope wedge fringes in, 15 Electron monochromator, 74, 103 focusing and scattering of electrons in, 107 Electron-neutral elastic collisions, 123-124 Electron-neutral inelastic collisions, 126 Electron spectrometer, Purcell type, 103 see also Spectrometer; Spectroscopy Electron temperature measurements, stationary afterglows and, 139-140 Emission spectroscopy, and rare-gas stationary afterglows, 168-170 Ewald sphere, 27 Excited states populations, 160-168 and energy level of rare-gas atoms and molecules, 160-162
F
Fermi-Dirac distributions, in thermionic emission, 80 Fermi-Thomas model, 45 Field-effect transistors, 292, 308
G Gallium arsenide heteroepitaxy of, on silicon, 301 microwave oscillations in, 293-295 Gallium arsenide diodes, 302-304, 375 doped, 310-312 Gallium arsenide IMPATT diodes, 338 Gauss-Hermite quadrature function, 112 Gummel’s charge-control relation, 281-283 Gunn diodes, 301-302, 309 noise figure for, 306-307, 350 stabilization of, 310 Gunn diode amplifiers, performance of, 317
394
SUBJECT INDEX
Gunn effect, 313 in microwave power semiconductor devices, 293 Gunn oscillators, stabilization for, 316
H Helium electronic recombination of, 156-157 metastable atoms and, 177 nonmetastable atoms and, 173 Helium afterglows, room-temperature, 157 Helium resonances, recording of, 108 Hemispherical structure, for cathode, 97-98 Hot cathode, energy spectrum of electrons emitted by, 73-1 17 see also Thermionic cathode Hybrid-pi equivalent circuit, as smallsignal version of charge-control representation, 266-269
Ion diagnostics experimental results in, 144-147 by mass spectroscopy, 140-142 Ionic mobilities experimental values of, 146 in rare-gas stationary afterglows, 142-147 Ionic population ionic mobilities and, 142-144 in rare-gas stationary afterglows, 140-160
J
Junction diodes, 320-350, 352-377 see also BARITT diodes; IMPATT diodes; TRAPATT diodes
K Kikuchi lines, 3, 47 Kinematical scattering, 9-16 by imperfect crystal, 16-22 limitations of theory in, 23
I IMPATT (impact ionization avalanche transit time) diodes, 292, 304, 307-308, 320-350
advantages and disadvantages of, 348-349
electrical characteristics and performances of, 337-348 fabrication of, 333 noise figure for, 347 IMPATT/TRAPATT diodes or devices, 337 Imperfect crystal(s) equations for, 3 1 kinematical scattering by, 16-22 in wave-mechanical theory, 40 Indium phosphide diodes, 303,316 Inelastic electron-neutral collisions, 126 Inelastic scattering absorption due to, 43-52 by crystals, 47-52 singleelectron excitation and, 4 3 4 7 Integral heat-sink technology, 302 International crystallographic tables, 7 Ion conversion, in rare-gas stationary afterglows, 144
L Langevin formula, in ionic mobilities, 143 Laser, tunable, in active spectroscopy, 172-173
Laue case, in electron diffraction, 27-29,37 Laue solution. in electron diffraction, 51-52
LEED (low-energy electron diffraction), 5 Light, absorption and emission of by plasma, 164-168 Linvill lumped model, 275-277 Low-energy electron diffraction, 5 LSA amplifier, 311 LSA diodes, 302-306 LSA node, 298-299
M Maser, molecular beam, see Molecular beam masers Mass spectrometry, ionic diagnostics by, 140-142
395
SUBJECT INDEX
Maxwell-Boltzmann distribution, in thermionic emission, 73-74 MBM, see Molecular beam masers Metal-semiconductor-metal structures, 352 Metastable atoms, reactions involving, 177 Metastable conversion, atomic-tomolecular, 175-176 Metastable diffusion coefficients, 175 Metastable population, in rare-gas stationary afterglows, 173-175 Microwave diagnostic techniques electron density and collision frequency measurements in, 133-139 in rare-gas stationary afterglows, 132-140 Microwave power semiconductor devices, 291-377 two-terminal devices and, 293-31 8 Microwave propagation, in plasma, 132-140 Mobility, ionic, 142- 143 Molecular beam maser(s), 183-244 amplifiers for, 219-221 beam-maser spectroscopy and, 199-219 closed resonators in, 194-196 component parts of, 185 defocusing in, 189 focusing in, 189-193 as frequency standard, 237-239 frequency translation with, 236-237 as laboratory standard of frequency, 233-234 linewidth in, 2W208 millimeter wave, I99 molecular beam sources in, 186-188 multichannel effusers for, 186-187 open resonators in, 196199 in population studies, 241-242 as portable frequency standard, 235-236 principles and techniques of, 184-199 quadruple electrode configuration in, 190 in quantum electronics, 241 relaxation effects and, 243-244 resonant systems in, 194-199 sensitivity in, 208-209 as spectrometer, 199-219 spectroscopic applications of, 24&241 as spectrum analyzer, 239 state separators or focusers in, 189-193
Molecular beam maser amplifier gain, bandwidth, and noise in, 219-220 Molecular beam maser oscillators, 221-241 amplitude and frequency characteristics of, 224-229 amplitude modulation,effects in, 231-233 conditions for oscillation in, 221-224 dynamic properties of, 229-233 frequency characteristics of, 227-229 systems and applications in, 233-241 transient properties of, 230-231 Molecular beam maser spectrometer, 185-1 87 high-resolution, 210 narrow spectral lines in, 201 Ramsey separated cavity scheme in, 215 systems in, 209-216 Molecular beam maser spectroscopy depolarization modulation in, 213-214 gases studied by, 202-207 high-resolution, 201 molecules investigated in, 216-218 Molecular beam maser Stark spectroscopy, 194 Molecular beam method, basic problem of, 185 Molecular ion recombination, in stationary afterglows, 154-1 56 Molecules, rare-gas, 160-162 Monochromator, 74, 103-104, 107 Morse formula, in inelastic scattering, 44
N Narud-Meyer model, in charge-control concept, 283 Negative differential resistance, 293 Neon, metastable atoms of, 177 npn transistor, 257, 263-264, 281
0 Overrelaxation constant, 89
P Pendellosung fringes, 56
396
SUBJECT INDEX
Perfect crystal contrast effects in, 52-57 defined, 52 thickness extinction contours in, 56-57 Phonon scattering, in electron diffraction, 46-47 Pittsburgh, University of, 134 Planar defects, images of in electron microscope, 57-61 Plasma absorption and emission of light by, 164-168 microwave propagation in, 132-140 Plasma absorptivity, in electron temperature measurements, 139 Plasma dielectric constant, in rare-gas stationary afterglows, 132-1 33 Plasmon excitation, 45-46 pn junction, 255-256, 345 pnp structures, 352 Poisson’s equation for BARITT diode, 355 thermionic cathode and, 87-90 Population studies, molecular beam maser and, 241-242 Punch-through diodes, see BARITT diodes Purcell electron monochromator, 103
Q Quantum electronics, molecular beam maser in, 241
R Radial kinetic energy, thermionic cathode and, 80-81 Ramsauer effect, 123 Rare-gas atomic spectroscopic data, 163 Rare-gas atoms elastic scattering by, 176 energy level for, 160-162 Rare-gas molecular ions, recombination rates for, 156 Rare-gas molecules, energy levels of, 161-162 Rare-gas stationary afterglows, 121-178 see also Stationary afterglows
Rare gases, afterglow spectroscopy for, 162-168 Relaxation effects, molecular beam maser and, 243-244 Resonators, in molecular beam masers, 196-199 Richardson’s constant, 78, 182 Richardson’s law cathode temperature and, 91 in thermionic emission, 78
S
Scanning electron microscope, 1-2 Scattering inelastic, see Inelastic scattering kinematical, 9-1 6 Schrodinger equation, in wave-mechanical diffraction theory, 32 Screw dislocation Burgers vector and, 63-67 electron diffraction and, 20-22 Self-analog or separation technique, in charge-control concept, 277-278 SEM,see Scanning electron microscope Semiconductor, see Microwave power semiconductor devices; pn junctions Snap diodes, 373-374 Source modulation, in molecular beam maser spectrometer, 212 Space charge self-consistent solution for, 90-93 underrelaxation and, 90 Space-charge barrier, electron energy spectrum of, 93-95 Space-charge density, for spherical cathode 83-86 Spherical cathode desing of, 95-99 physical situation outside, 80-83 space-charge density for, 83-86 Spectrometer, MBM, see Molecular beam maser spectrometer Spectroscopy see also Stationary afterglows afterglow, 162- 168 beam maser, 199-219 Stacking fault, 60-61 in electron microscopy, 57-59
SUBJECT INDEX
Stark modulation, in molecular beam maser spectrometer, 213 Stark spectroscopy, 194 Stationary afterglows see also Rare-gas atoms; Rare-gas molecules active spectroscopy and, 172-173 atomic-to-molecular metastable conversion and, 175-176 Coulomb collisions and, 126-128 elastic electron-neutral collisions and, 123-126 electron energy balance and, 130-1 32 electron-ion recombination in, 147-160 electron temperature measurements in, 139-140 excited states populations and, 160-168 general description of, 123-132 helium and, 157 ionic populations in, 140-160 metastable population in, 173-176 microwave diagnostics techniques in, 132-140 molecular ion recombination and, 154-1 56 and particle diffusion to walls, 128-130 spectroscopic diagnostic techniques and, 168-173 State separators, in molecular beam masers, 189-193 Surface barrier, in electron energy distribution, 75
T TED, see Transferred electron devices TEM, see Transmission electron microscope Thermionic cathode energy spectrum of electrons emitted by, 73-1 17 hemispherical, 74 physical situation outside, 80-83 Poisson’s equation and, 87-90 radial kinetic energy and, 80-81 and self-consistent solution for space charge and potential, 90-93 Thermionic electrons, energy spectrum for, 73-1 17
397
Thermionic emission Richard’s law in, 78 temperatures in, 78 Thickness extinction contours, in electron microscopy, 56-57 Time-dependent response, charge-control concept in, 283-284 Transferred electron amplifiers, 307-3 18 electrical performances of, 316 market for, 318 stabilization in, 308-316 Transferred electron devices, 292-297 Transferred electron oscillators, 293, 297-307 efficiency considerations in, 299-300 electrical performances of, 302-306 oscillation modes in, 297-298 temperature considerations in, 301 Transistor(s) see also Gunn diodes; IMPATT diodes; TRAPATT diodes bipolar, see Bipolar transistors charge-control concept in analysis of, 284-287 early years of, 254-258 field effect, 292, 308 npn, 251, 263-264,281 pn and pnp structures, 255-256, 345, 352 Transistor circuits, charge-control concept in analysis of, 286-287 Transmission electron microscope, 1-2 see alsd Electron diffraction; Electron microscope Bragg reflecting position in, 58-59, 67 crystalline materials in, 52-68 dislocation lines in, 61-68 kinematical theory and, 23 stacking-fault images in, 57-59 weak-beam images in, 67-68 TRAPATT (trapped plasma avalanche triggered transit) diodes, 292, 320-325, 329-331, 335 advantages and disadvantages of, 349-350 electrical characteristics and performances of, 337-348 noise figure for, 347 structure of, 335-336
398
SUBJECT INDEX
U
Underrelaxation, space charge and, 90
fabrication of, 372-373 snap type, 373 Voltage variable capacitors, see Varactor diodes
V Varactor (voltage variable capacitor) diodes, 365-377 advantages and disadvantages of, 376-377 differential capacitance in, 371 electrical characteristics and performances of, 373-376
W
Wave-mechanical theory, in electron diffraction, 3 2 4 2 Wave-optical theory, of electron diffraction, 2 4 31 Wedge fringes, 15
A
5
6 6 c 7 D
8
E 9 F O G
1
H 2 1 3 J
4
