ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS
VOLUME 51
CONTRIBUTORS TO THISVOLUME
R. Stephen Berry Robert W . Brodersen Marvin L. Cohen A . P. Gnadinger H . L. Grubin L. Ronchi A. M . Scheggi M . P. Shaw P. R. Solomon Richard M. White
Advances in
Electronics and Electron Physics EDITEDB Y L. MARTON A N D C. MARTON Stnirhsotii(rti Itistitiitioti Wmhingtotr , D . C .
EDITORIAL BOARD E. R. Piore T. E. Allibone M. Ponte H . B. G. Casimir A. Rose W. G. Dow L. P. Smith A. 0. C. Nier F. K. Willenbrock
VOLUME 51
1980
ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers
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COPYRlGHT @ 1980, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
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ISBN 0-12-01465 1-7 PRINTED IN THE UNITED STATES OF AMERICA
80 81 82 83
9 8 7 6 5 4 3 2 1
CONTENTS CONTRIBUTORS TO VOLUME 51
. . . . . . . . . . . . . .
FOREWORD . . . . . . . . . . . . . . . . . . . . .
vii ix
Electrons at Interfaces MARVINL . COHEN
I. I1. I11. IV . V.
Introduction . . . . . . . . . . . . . . Semiconductor Surfaces and Theoretical Techniques Semiconductor-Metal Interfaces . . . . . . . Semiconductor-Semiconductor Interfaces . . . . Summary and General Discussion . . . . . . . References . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
1
2 13 32 58 60
Beam Waveguides and Guided Propagation L . RONCHIA N D A . M . SCHEGGI
I . Introduction . . . . . . . . . . I1 . Theoretical Background . . . . . . I I1. Some Typical Longitudinal Structures . IV . Metallic Waveguides . . . . . . .
V. VI . VII . VIII . 'IX . X.
. . . .
. . . .
. . . .
Dielectric Rods and Fibers . . . . . . . . Two-Dimensional Waveguides with Metallic Walls Two-Dimensional Dielectric Structures . . . . Wave Guiding by Transverse Structures . . . Guiding Media . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . Appendix I . Ray Tracing Method . . . . . . Appendix I1 . The WKB Approximation Applied to Propagation in a Slab . . . . . References . . . . . . . . . . . . .
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64 66 73 73 80 88 91 94 110 127 128
Modal
. . . . . . . . . . . .
130 133
.
Elementary Attachment and Detachment Processes I R . STEPHENBERRY
1.Goals. . . . . . . . . . . . . . . I1. Classification of Processes . . . . . . . I11. Orders of Magnitude: General Considerations . IV . Specific Processes . . . . . . . . . . References . . . . . . . . . . . . V
. . . . . . . . . . . . . . . . . . . .
. . . .
. . . . . . . . . .
137 138 143 145 177
CONTENTS
vi
Electronic Watches and Clocks A . P . GNADINGER 1. Introduction . . . . . . . . . . . . . . . . . . . 183 I1 . Some History of Timekeeping . . . . . . . . . . . . . 184 I11 . Electrical Clocks . . . . . . . . . . . . . . . . . 186 IV . The Electronic Watch . . . . . . . . . . . . . . . . 186 V . Conclusion . . . . . . . . . . . . . . . . . . . 258 References . . . . . . . . . . . . . . . . . . . 259
Charge Transfer and Surface Acoustic-Wave Signal-Processing Techniques ROBERTW . BRODERSEN A N D RICHARDM . W H I T E I . Introduction . . . . . . . . . . . . . . . . . . . 265 I1 . Charge-Coupled Device Principles . . . . . . . . . . . . . 266 Ill . CCD Performance Limitations . . . . . . . . . . . . . 270 IV . -Surface Acoustic-Wave Principles . . . . . . . . . . . . 279 V . Transversal Filtering . . . . . . . . . . . . . . . . 286 VI . CCD Transversal Filters . . . . . . . . . . . . . . . 287 VII . SAW Transversal Filters . . . . . . . . . . . . . . . 295 VIII . SAW Oscillators, Resonators. and High-Q Filters . . . . . . 301 IX . Conclusions . . . . . . . . . . . . . . . . . . . 306 References . . . . . . . . . . . . . . . . . . 306
Gunn- Hilsum Effect Electronics M . P . SHAW. H . L . G R U B I NA. N D P . R . SOLOMON I . Negative Differential Mobility (NDM) in Semiconductors . . . . I1 . The NDM Element's Environment: Circuits and Boundaries . . . 111. The Behavior of an NDM Element in a Circuit . . . . . . . IV . NDM Devices . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . AUTHOR INDEX .
. . . . . . . . . . . . . . . . . . .
SUBJECTINDEX . . . . . . . . . . . . . . . . . . . .
310 329 340 367 427
435 448
CONTRIBUTORS TO VOLUME 51 Numbers in parentheses indicate the pages on which the authors’ contributions begin.
R. STEPHEN BERRY,Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (137) ROBERTW. BRODERSEN, Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720 (265) MARVINL. COHEN,Department of Physics, University of California, and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, California 94720, and Department of Physics and Astronomy, University of Hawaii, Honolulu, Hawaii 96822 (1)
A. P. GNADINGER, INMOS Corporation, 2860 South Circle Drive, Colorado Springs, Colorado 80906 ( 1 83) H. L. GRUBIN,United Technologies Research Center, East Hartford, Connecticut 06 108 (309) L. RONCHI,Istituto di Ricerca sulle Onde Elettromagnetiche C.N.R., Florence, Italy (63) A. M. SCHEGGI, Istituto di Ricerca sulle Onde Elettromagnetiche C.N.R., Florence, Italy (63) M. P. SHAW,Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202 (309)
P. R. SOLOMON, Advanced Fuel Research, Inc., East Hartford, Connecticut 06108 (309) RICHARD M. WHITE,Department of Electrical Engineering a n d Computer Sciences, University of California, Berkeley, California 94720 (265)
vii
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FOREWORD This volume spans a range of topics from the purely industry oriented, such as the lovely paper by A . P. Gnadinger on electronic watches and clocks, to the purely research oriented, such as the paper on electrons at interfaces, clearly expounded by Marvin L. Cohen. In between, in ascending order to the more technological, are the splendidly presented papers by R. Stephen Berry; Robert W. Brodersen and Richard M. White: M. P. Shaw, H. L. Grubin, and P. R. Solomon; and L. Ronchi and A. M. Scheggi. We trust our readers will find some interest in each of the topics and that the volume will serve to provide a broad background in these active fields. We thank each of our authors. As is our custom, we include here a list of articles to appear in future volumes. Criticul Review's: A Review of Application of Superconductivity Sonar Electron-Beam-Controlled Lasers Amorphous Semiconductors Design Automation of Digital Systems. I and 11
Spin Effects in Electron- Atom Collision Processes Review of Hydromagnetic Shocks and Waves Seeing with Sound Large Molecules in Space Recent Advances and Basic Studies of Photoemitters Josephson Effect Electronics Present Stage of High Voltage Electron Microscopy Noise Fluctuations in Semiconductor Laser and LED Light Sources X-Ray Laser Research The Impact of Integrated Electronics in Medicine Ionic Photodetachment and Photodissociation Electron Storage Rings Radiation Damage in Semiconductors Solid-state Imaging Devices Cyclotron Resonance Devices Heavy Doping Effects in Silicon Spectroscopy of Electrons from High Energy Atomic Collisions Solid Surfaces Analysis Surface Analysis Using Charged Particle Beams
ix
W. B. Fowler F. N. Spiess C . A. Cason H. Scher and G. H s t e r W. G. Magnuson and Robert J. Smith H. Kleinpoppen A . Jaumotte & Hirsch A. F. Brown M. and G. Winnewisser H. Timan M. Nisenoff B. Jouffrey
H. Melchior Ch. Cason and M. Scully J . D. Meindl T. M. Miller D. Trines N . D. Wilsey and J. W. Corbett E. H. Snow R. S. Symous and H . R. Jory R. Van Overstraeten D. Berenyi M. H. Higatsberger F. P. Viehbock and F Riidenauer
X
FOREWORD
Sputtering Photovoltaic Effect Electron Irradiation Effect in MOS Systems Light Valve Technology High Power Lasers Visualization of Single Heavy Atoms with the Electron Microscope Spin Polarized Low Energy Electron Scattering Defect Centers in Ill-V Semiconductors Atomic Frequency Standards Reliability Microwave Imaging of Subsurface Features Novel MW Techniques for Industrial Measurements Electron Scattering and Nuclear Structure Electrical Structure of the Middle Atmosphere Microwave Superconducting Electronics Biomedical Engineering Using Microwaves. I1 Computer Microscopy Collisional Detachment of Negative Ions International Landing Systems for Aircraft Impact of Ion Implantation on Very Large Scale Integration Ultrasensitive Detection Physics and Techniques of Magnetic Bubble Devices Radioastronomy in Millimeter Wavelengths Energy Losses in Electron Microscopy Long Life High Current Density Cathodes Interactions of Measurement Principles Low Energy Atomic Beam Spectroscopy History of Photoelectricity Fiber Optic Communications Photoiodes for Optical Communication Electron Microscopy of Thin Films
G. H. Wehner R. H. Bube J. N. Churchill, F. E. Holmstrom, and T. W. Collins J. Grinberg V. N . Smiley
J. S. Wall D. T. Pierce and R. J. Celotta J. Schneider and V. Kaufmann C. Audoin H. Wilde A . P. Anderson W. Schilz and B. Schiek G . A. Peterson L. C. Hale R. Adde M. Gautherie and A. Priou E. M. Glasser R. L. Champion H. W. Redlien and R. J. Kelly H. Ryssel K. H. Purser M. H. Kryder E. J. Blum B. Jouffrey R. T. Longo W. G. Wolber E. M. Hod and E. Semerad W. E. Spicer G. Siege1 J. Miiller M. P. Shaw
Supplementary Volumes:
Image Transmission Systems Applied Charged Particle Optics Microwave Field Effect Transistors
W. K. Pratt A. Septier J. Frey
Volume 53:
Particle Beam Fusion The Free Electron Laser: A High Power Submillimeter Radiation Source The Biological Effects of Microwaves and Related Questions Ion Optical Properites of Quadrupole Mass Filters
A. J. Toepfer T. C. Marshall, S. P. Schlesinger, and D. B. McDermott
H. Frohlich P. H. Dawson
FOREWORD Spread Spectrum Communication Systems Electron Interference Volume 54: Magnetic Reconnection Experiments Electron Physics in Device Fabrication. I1 Solar Physics Aspects of Resonant Multiphoton Processes
Fundamentals and Applications of Auger Electron Spectroscopy
Xi
P. W. Baier and M. Pandit M. C. Li P. J. Baum and A. Bratenahl P. R. Thornton L. E. Cram A. T. Georges and P. Lambropoulos P. H. Holloway
As in the past, we have enjoyed the friendly cooperation and advice of many friends and colleagues. Our heartfelt thanks go to them, since without their help it would have been almost impossible to issue a volume such as the present one.
L. MARTON C. MARTON
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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL. 51
Electrons at Interfaces MARVIN L. COHEN Department of Physics, University of California and Materials and Molecular Research Division Lawrence Berkeley Laboratory Berkeley , California and Department of Physics and Astronomy University of Hawaii Honolulu, Hawaii
I . Introduction . . . .. .. .. .. .. .
........................
11. Semiconductor Surfaces and A. Introduction . . . . . . . . . . . . . . . .
B. Self-consistent Pseudopotentia ................... C. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Discussion . . . . . ............................ 111. Semiconductor-Metal Interfaces . . . . . . . . . . . . . . . . . ................ A . Introduction .............................................. B. SCPMfor Se -Metal Interfaces C. Results .................................... ................... .............................. D. Further Results and Discussion A. Introduction
....
. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .
B. SCPM for Semiconductor-Semiconductor Interfaces ....................
. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . ........................ ............................
D. Further Results and Discussion V . Summary and General Discussion . . . References. . . . . . . . . . . . . . . . .
1 2 2 5 8 12 13 13 14 16
23 32 32 31 40 58 58
60
I . INTRODUCTION The field of interface study is a very mature one. The work by Braun (1874) on contacts between metal wires and crystals is over 100 years old. Research on surfaces and interfaces was particularly active in the 1930s and 1940s, and pioneers like Tamm (1932), Schottky (1939), Davydov (1939), Mott (1939), Shockley (1939), and Bardeen (1947) established the
foundations on which much further work rested. A large amount of this I
Copyright @ 1980 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-014651-7
2
MARVIN L. COHEN
work was on semiconductor-metal contacts. More recently, experimental work on clean surfaces became more reliable than in the past and considerable understanding of the electronic structure of clean surfaces and interfaces resulted. Theoretical models became more realistic and relations between clean surface properties and interface properties were found. This past decade has been a very active period for research and development in this field in physics, chemistry, and engineering. Microscopic models are now available that have significant predictive power. The problems are still not all solved. In fact, many problems and interesting areas remain to be formulated. However, the recent progress is encouraging and an attempt is made here to describe part of it. The focus of this review is on the electronic structure of interfaces in which one of the constituents is a semiconductor. Emphasis is placed on studies of semiconductor-metal (Schottky barriers) and semiconductorsemiconductor (heterojunctions) interfaces. Some discussion of the clean-surface properties of semiconductors is given as background since many concepts developed for the “semiconductor-vacuum interface” are needed to describe the semiconductor-solid cases. In most cases, theory is emphasized. The systems are considered to have ideal properties. Interfaces are assumed to be abrupt and free of contamination and defects. The pseudopotential approach for the electronic structure calculations is stressed; experimental results are described only as they apply to specific points. Hence, the discussion does not represent a good review of experimental research or a broad survey of theory. Section I1 briefly describes the semiconductor-vacuum case, emphasizing the aspects that bear on Schottky barrier and heterojunction studies and that give much of the theoretical background for the methods used. Sections I11 and IV are mostly on Schottky barriers and heterojunctions, respectively. Finally, Section V contains conclusions and some general comments.
11. SEMICONDUCTOR SURFACES AND THEORETICAL
TECHNIQUES A . Introduction
The main task for a theory of surfaces is to describe the behavior of electrons near the surface in as realistic a manner as possible so that comparisons with experiment can be made. Model calculations are useful and tests have been used to explore the basic theory, but there is a wealth of experimental detail on a variety of materials and some of this research has
ELECTRONS AT INTERFACES
3
revealed curious and important effects that need theoretical exploration. Hence, many theorists have attempted to do detailed electronic calculations relying on measurements to give models for the atomic positions at the surface. Although some progress has been made using total energy calculations to also compute the positions of surface atoms, this work is in its infancy and is not reliable enough to use as a starting point for most theoretical calculations. Consequently, the majority of calculations of surface electronic structure begin with a specific model for the structure of the surface, and most models are based on LEED (low-energy electron diffraction) and other measurements. We begin with an ideal model for the surface structure and use the Si(11 I ) surface as a prototype. In the ideal case, no reconstruction of the surface is assumed and from the structural point of view, this model regards the Si(111) surface as the end of a perfect silicon crystal (Fig. I). In the representative ball and stick model of Fig. 1, the balls are silicon cores, while the sticks represent bonds. It was necessary to cut bonds to form the surface and in this model, halves of sticks or bonds are pictured as pointing out of the (11 1) surface. The "dangling bonds" pictured this way do not readjust to the surface perturbation. However, we do not expect, in reality, to be able to describe the surface as the end of a perfect crystal. Electrons can react to the surface perturbation and adjust to it. Charge can flow and the atomic positions can relax back or even change or reconstruct. In addition, we know from the
2
3 3
DANGLING BOND
FIG.1. Perspective view of the silicon crystal structure projected on a ( I 10) plane. The ( 1 1 1 ) direction is vertical, and the ( 1 11) surface is obtained by cutting the vertical bonds in a
horizontal plane.
4
MARVIN L. COHEN
work of Tamm (1932) and Shockley (1939) that the surface can bind new states-surface states. These states are localized at the surface and decay both into the bulk crystal and into the vacuum outside the surface. Therefore, electronic calculations are expected to determine the readjustment of the electronic charge, locate the surface states in energy, describe their characteristics, and explain or predict (or both) properties of the surface. One major problem that must be faced in calculating the electronic energy spectrum of a surface is the destruction of translational invariance because of the formation of the surface. Most bulk electron energy band calculations assume translational symmetry and exploit it by working in Fourier space using reciprocal lattice vectors. Another major ingredient in working with surface calculations is self-consistency . The surface perturbs the electron density. The density readjusts and this changes the potential that electrons near the surface feel. Hence, the potential near the surface can be different than the bulk potential and the effect through the charge density of the electrons changes the potential itself. This requires the use of self-consistent approaches where the charge density is calculated from a potential; then the changes are fed back into the potential in a repetitive scheme until self-consistency is achieved. One way to avoid the problems connected with the loss of translational symmetry is to deal with a finite slab or cluster. The slab geometry and the empirical tight-binding method (e.g., Pandey and Phillips, 1974) has been very successful. Tight-binding parameters are fit to reproduce the bulk band structure, and then the electronic structure of the surface is computed. Although this method is not inherently self-consistent, adjustments of the parameters near the surface are possible. There are other ways to simulate the surface using Green's functions and Bethe lattice techniques (e.g., Falicov and Yndurain, 1975) along with the tight-binding model. Self-consistency and the availability of electron density profiles of the total charge density and of individual surface states are not generally featured in tight-binding or localized orbital calculations, but these are two of the strongest aspects of pseudopotential calculations. In one form of the pseudopotential approach (Appelbaum and Hamann, 1976), the wavefunctions near the surface are matched to the bulk to account for the surface perturbation. The electronic structure is calculated self-consistently, and total charge densities, wavefunctions for surface states, and the energy spectrum for bulk and surface states are computed. Another pseudopotential approach uses the concept of a "supercell" to simulate the surface (Schliiter et a / . , 1975) or other localized configuration (Cohen et a / . , 1975). The central idea is to mathematically construct a large cell with the localized configuration of interest placed in the cell. The cell itself is partially made of vacuum or space with no atoms in it.
ELECTRONS AT INTERFACES
5
This is done to separate the configuration of interest, e.g., a molecule, from neighboring cells. This supercell is then repeated infinitely, allowing the use of Fourier space techniques; this overcomes the problems connected with loss of translational symmetry. In the case of a molecule, the space assures separation of the molecules, and even though the system is solved via a band structure approach, the overlap of molecular states between neighboring cells can be minimized. For a surface, the supercell is a slab with two surfaces. Part of the atomic configuration shown in Fig. 1 is put into the supercell geometry and the energy spectrum is calculated using the self-consistent pseudopotential method (SCPM). The slab geometry assumes the standard bulk periodicity parallel to the surface, and the perpendicular geometry is achieved through the use of the supercell. We concentrate on this technique in exploring the interface problem.
B . Selfconsistent Pseudopotential Method (SCPM) The pseudopotential method is based on the Phillips cancellation theorem (Phillips and Kleinman, 1959), which demonstrates that the strong atomic potential felt by a valence electron is partially canceled by a repulsive potential, which results from the condition that the valence electron states must be orthogonal to the core electron states of the atom. The resulting pseudopotential is weak and plane waves can form a convenient basis for the wavefunction. The pseudopotential can be obtained through calculations based on atomic wavefunctions or fitted to experiment (the empirical pseudopotential method, EPM), or through the use of atomic spectra. Several variations of the above and much fundamental theory have been explored (for a review, see Cohen and Heine, 1970). The SCPM used here is an outgrowth of this work, which has wide applicability. The concepts associated with the SCPM are given in Section I. The associated techniques are not complex and these are outlined here. Further details are given in the literature (e.g., Schluter et al., 1975; Cohen et al., 1975). In the pseudopotential approximation, the one-electron Hamiltonian can be written in the form H
=
P 2 / 2 m + V,,
+
V,
+
V,
(1)
The pseudopotential V,, is taken to be a superposition of ionic pseudopotentials V,,, , representing an ionic core dike Si4+),
6
MARVIN L. COHEN
where the R, and r1 represent the lattice and basis vectors in the primitive cell (Kittel, 1976). The ionic pseudopotential is screened by the valence electrons through a Hartree potential VH and a local exchange potential Vx . Both potentials can be obtained from the valence charge density. For the Hartree potential, Poisson’s equation yields,
v2v H ( f )
=
-41re2p(r)
(3)
and the Vx can be approximated by the Slater form,
where the parameter (Y is assumed variable. For many calculations it is chosen to be 0.794, but this is not a general result. The exchange potential approximation can be altered to include approximations to correlation. Several functionals depending only on the density that give reasonably good local potential approximations for electron exchange and correlation have been developed. We use Eq. (4) and ignore correlation effects. The basis set is a sum of plane waves
where n, k, and G are the band index, the wavevector of the state under consideration, and the reciprocal lattice vector, respectively. The expansion coefficients & k are evaluated, by solving Eq. (1) using the standard secular equation approach (Cohen and Heine, 1970),
The matrix elements Hc,c, require a reciprocal lattice space expansion for the potentials,
where, using Eq. (3),
(8)
VH(G) = 4 ~ e ~ p ( G ) / l G 1 ~ V,(G) =
3 21r
- - (3lr2 )113
~
z21
[ p(r)]1’3e-ffi*rdr
(9)
ELECTRONS AT INTERFACES
7
where p(G) is the Fourier form factor of the charge density p(r). The ionic pseudopotential in reciprocal space is fitted to a four-parameter model of the form Vjon(q) =
(a,/q*)[cos(uzq) + a31 exp(a4q4)
(10)
The pseudopotential form factors ViO,(G) can then be evaluated from inspection of this fit. The self-consistency procedure is illustrated in the block diagram of Fig. 2. The starting potential is an empirical potential obtained from studying bulk properties of the solid of interest. A structure factor (Kittel, 1976) S(G) is used to "put" the atoms in a slab geometry. The matrix equation, Eq. (6), is solved and the wavefunctions are used to construct the charge density p(r). From the charge density the potentials V,(r) and Vx(r) are calculated using Eqs. (3), (4),(7), (8), and (9). The pseudopotential Vps(r)[Eq. (2)] is then added to the screening potential V, + V, and this total potential is used to begin the calculation again. The procedure is repeated until self-consistency is achieved, i.e., the input and output screening potentials agree with each other. This usually requires around six or seven cycles for reasonable accuracy.
STEPS IN ACHIEVING SELF-CONSISTENCY
Solve
HJIZEJI
HARTREE POTENTIAL EXCHANGE t CORRELATION SLATER p"
e g,
t Model parometers Structure, V,,
FIG.2. Self-consistentloop in calculating the electronic structure of surfaces and interfaces.
8
MARVIN L. COHEN
C . Results The theoretical approach discussed in Section II,B yields energy eigenvalues and wavefunctions corresponding to the geometry examined. These can be used to construct: (1) band structures for the surface states to study the dependence of the energy on wavevector, (2) the total charge density (for all the states), which can be displayed using contour maps, (3) the charge density of individual states, specifically surface states, and (4) the local density of states (LDOS).
The above list is far from being exhaustive, but it represents the major outputs of a theoretical calculation, which can be used for comparisons with experimental data. In the discussion that follows, ideal Si(ll1) is used as a prototype for a homopolar semiconductor surface and GaAs( 110) represents the heteropolar semiconductors. The details of the band structures are not emphasized, but the total charge density, LDOS, and charge density of individual states are examined. Although the ideal Si(11I ) surface shown in Fig. 1 is stable only above 80O0C, it does provide a good model for theoretical understanding of the properties of the ideal silicon surface, reconstructed silicon surfaces, and even other semiconductor surfaces. The total charge density is displayed in Fig. 3. The atomic cores are positioned in a similar geometry to the ball and stick model of Fig. 1. The charge density is plotted in a (110) plane that intersects the (111) surface at a right angle. The area displayed begins in the vacuum above the surface and extends four and one-half atomic layers into the bulk. Comparison of the charge density of Fig. 1 with charge density calculations for bulk silicon (Chelikowsky and Cohen, 1974) shows that after allowing for differences in calculational approaches, the charge densities are very similar after proceeding only a few layers into the crystal. Near the surface there is a sharp decay of electron density. One prominent feature not seen in Fig. 3 but shown in Fig. 1 is the presence of cut “dangling bonds” sticking out of the surface. The selfconsistent calculations demonstrate that the charge readjusts and “heals” the cut bond. Charge is redistributed, resulting in a smooth surface. The topology of the region near the surface is otherwise quite similar to the ideal structure with clearly displayed channels from the surface into the bulk (which may be the paths for foreign impurities entering the crystal). Surface states are not discernible in the total charge density plots since
ELECTRONS AT INTERFACES
9
Si (111) SURFACE TOTAL VALENCE CHARGE
FIG.3. Total valence charge distribution for an unrelaxed Si(l11) surface. Charge is plotted as contours in a ( I 10) plane intersecting the ( I I I ) surface at right angles. Plotting area starts in the vacuum and extends about 4.5 atomic layers into the crystal. Atomic positions and bond directions are indicated by dots and heavy lines, respectively. Shaded circles represent atomic cores. Contours are normalized to electrons per silicon bulk unit cell volume.
their weight is small compared to the large number of decaying bulk states. The energy location of surface states can be revealed by examining the LDOS. The LDOS is displayed as a series of curves that gives the density of states as a function of position or layer starting in the bulk and proceeding to the surface. The LDOS given in Fig. 4 was calculated for Si(11 1) with a small relaxation of the outermost atoms. This relaxation of about one-third of an angstrom is believed to be a reasonable approximation to the actual situation for this surface. Beginning with layer 1, which is six layers into the surface, the density of states (DOS) is quite similar to bulk silicon (Chelikowsky and Cohen, 1974). Starting near the valence band maximum at 0 eV there is a peak about 5 eV wide containing mostly p-like electrons. The next peak down to about - 8.5 eV represents a mixture of s and p electrons, while the lowest energy peak is s-like in character. The gap above 0 eV is the fundamental gap of silicon and no states are shown in this gap. However, the situation changes dramatically as one proceeds toward the surface. A peak grows in the gap region and it becomes the most conspicuous feature of the spectrum at the surface. This peak represents the dangling bond surface state. The reason for the name
MARVIN L. COHEN
10
-14 -12-10
-e -6 - 4 -2 o Energy (eV)
2
FIG.4. Local density of states (LDOS) for six atomic layers parallel to the Si(1I I ) surface. The most prominent surface states are indicated.
chosen will become clearer later. Other features in the surface DOS appear that were not present in the bulk DOS. These surface features represent surface states or resonances that are localized near the surface and decay both into the bulk and into the vacuum. We focus on the most prominent surface state, i.e., the dangling-bond surface state in the gap. The wavefunction for this state can be isolated from the bulk by considering states only in the gap region near the surface. This allows the construction of a charge density plot for this state. A contour plot of the electron density appears in Fig. 5 , which clearly illustrates the surface localization of this state. The state decays both into the vacuum and into the bulk crystal. It is concentrated near the outermost silicon core and juts out into the vacuum, hence the name dangling-bond surface state. This state was observed and studied experimentally by Eastman and Grobman (1972) and by Wagner and Spicer (1972). Other surface states exist that are associated with other regions near the surface, e.g., transverse-back-bond states localized in the bonds between the
ELECTRONS AT INTERFACES
I1
Si 0111 surface
DANGIING EON SURFACE STATE
FIG. 5 . Charge density contour plot of the dangling-bond state.
outermost and second layers of silicon cores and longitudinal-back-bond states localized in the bonds between the second and third layers of cores. The dangling bond surface state is the one most affected by overlayers and is important in studying interface behavior. Calculations have also been done for geometries that are not ideal like the one shown in Fig. 1. One example is the relaxed geometry mentioned before and another is the (2 x I ) reconstructed geometry. Si( 1 1 1) is known to reconstruct and LEED data have been interpreted in terms of a buckled surface. This leads to a primitive surface cell with two atoms in it instead of one as in the ideal case. One of the atoms is pushed out, while the other is pulled back. The dangling bond peak in the LDOS of Fig. 4 is split into two peaks by this reconstruction. This splitting and the resulting gap in the peak explains why Si(1 1 1 ) is a surface with semiconductor properties rather than metallic properties. The atom that is pushed out from the surface is associated with the occupied dangling-bond surface state while the atom that is pulled back is associated with a dangling-bond surface state that is empty. The ideal geometry, where all surface atoms are equivalent, gives a peak and a band that is half-filled with electrons and hence a metallic surface. Other properties of this (2 x 1) reconstruction have been computed and comparisons with measured data support the picture described above. The details of the topology of the reconstruction and the exact shifts of the cores are not completely known. Other re-
12
MARVIN L. COHEN
constructions like the (7 x 7) reconstruction have not been studied as extensively. Surfaces other than the (1 11) surface are also of interest. For silicon, the (100) surface has received considerable attention, while for zincblende crystals the (11l), ( l a ) , and (1 10) surfaces have been studied. The (1 11) zincblende surface is interesting since it can terminate in an anion or cation (e.g., As or Ga for GaAs). The (110) surface is nonpolar and has both anion and cation components. For GaAs, the (110) surface is by far the one studied most extensively and we review the results briefly to illustrate the major features of this prototype zincblende surface. The question of the existence and energy location of empty cationderived surface states has been a widely publicized debate. Filled anion-derived surface states are found in both experimental and theoretical studies, but the situation regarding the empty surface states has only recently become clear. Initial measurements by Eastman and Freeouf (1974) and by Gregory et al. (1974) indicated that intrinsic surface states exist near midgap in GaAs. However, currently it is believed that these experiments should be reinterpreted and that intrinsic surface states do not exist in the gap. The theoretical situation also evolved considerably during this period. Both tight-binding and SCPM results gave empty surface states in the gap. However, these calculations assumed an ideal geometry. Analysis of LEED data reveals that although the GaAs (1 10) surface does not reconstruct, there is relaxation at the surface. The arsenic atoms move out while the gallium atoms move in, but the symmetry is still (1 x 1) or ideal. Using the latest LEED data for the structure of the surface, SCPM calculations have been done (Chelikowsky and Cohen, 1979) for the relaxed geometry. These reveal that the empty surface states move out of the gap region to the bottom of the conduction band. Hence, experiment and theory are currently in agreement regarding the (110) surface of GaAs. Occupied anion-derived surface states exist near the top of the valence band, while empty cation-derived surface states are found near the bottom of the conduction band. Studies of 11-VI zincblende surfaces are not as prevalent as those of group IV or 111-V materials. A SCPM calculation has been done for the ideal (1 10) surface of ZnSe (Chelikowsky and Cohen, 1976a). The results are similar to the ideal GaAs (1 10).
D. Discussion The results from the above description of clean surfaces will be useful in the following sections on semiconductor-metal and semiconductor-
ELECTRONS AT INTERFACES
13
semiconductor interfaces. In particular, the theoretical methods described for the interface calculations are similar to those used for the clean surface, and the results for the clean surface yield information on the properties of surface states that are important to the discussion of interfaces. 111. SEMICONDUCTOR-METAL INTERFACES
A . fntroduction
Semiconductor-metal (s-m) interfaces are of great interest in the technology of devices because of their rectifying properties. Although real interfaces used in devices are not ideally abrupt or clean, all the theoretical discussion presented here will assume ideal geometries and clean interfaces. Recent advances in ultrahigh vacuum techniques have allowed experimental studies that approach these theoretical models. However, even with the best current equipment, the samples studied are not ideal and therefore it should be emphasized, as a caveat, that the theoretical models are assumed to be ideal. The Schottky barrier height 4Bof a s-m interface will be defined as the energy separation between the Fermi level EF and the minimum of the semiconductor conduction band. For covalent semiconductors like silicon or germanium, 4Bwas found to be roughly independent of the metal contact. Bardeen (1947) attributed this behavior to the presence of surface states in the semiconductor. He suggested that extra charge coming from the metal contact could be accommodated by the semiconductor surface states. These states would in effect “pin” the Fermi level. Thus the Bardeen pinning model could account for the insensitivity of 4Bto the metal work function. The Bardeen model required a high density of surface states. Some experimental work did show a small dependence on metal work function, and in an attempt to explain this behavior, Cowley and Sze (1966) proposed a model assuming a lower density of surface (or interface) states uniformly distributed in the semiconductor energy gap. Heine (1965) questioned the existence of surface states on a semiconductor covered with a metallic overlayer. Heine’s analysis suggested that interface states of another type could be responsible for pinning the Ferrni level. These states would propagate on the metallic side and be bulklike, but would decay into the semiconductor with a decay length of order 2 10 A. Other pinning theories were proposed, starting from completely dif-
14
MARVIN L. COHEN
ferent points of view (Inkson, 1974; Phillips, 1974; Harrison, 1976). Inkson proposed that the pinning of EF arose from a narrowing of the semiconductor gap at the interface. Phillips suggested that chemical binding forces between the semiconductor and the metal were responsible for the pinning. Harrison’s proposal involved dangling-bond hybrids that were shifted in the valence and conduction bands, keeping the Fermi level in the band gap. These approaches do not involve interface states and do not bear directly on the Bardeen suggestion, but rather they represent alternative models. The calculation that did bear directly on the Bardeen and Heine work was a SCPM calculation for a Si-AI Schottky barrier (Louie and Cohen, 1976). This calculation showed that the intrinsic surface states on silicon were quenched by the metal and new metal-induced-gap states (MIGS) were responsible for Fermi level pinning. The Louie-Cohen states or MIGS are hybrid states made up of the tails of the penetrating metal wavefunctions and decaying surface-like states in the semiconductor. These states were similar in form to the states suggested by Heine, but the decay lengths were only on the order of one semiconductor bond length. (Also, Heine’s work was done before dangling-bond surface states had been explored.) On the semiconductor side, this short decay length is not unlike the lengths characteristic of clean surface states. Hence, the LouieCohen model has characteristics of both the Bardeen and Heine models. The SCPM calculation of Louie and Cohen used a smeared out positive background plus a free electron gas to simulate the metal, i.e., jellium. Other SCPM calculations have studied overlayers of aluminum on silicon using pseudopotentials for the aluminum side (Appelbaum and Hamann, 1974; Chelikowsky, 1977b; Zhang and Schliiter, 1978). These calculations described the detailed nature of the bonding between the silicon and the aluminum and addressed the question of the geometries of the atoms at the interface. The Louie-Cohen model was extended to 111-V and 11-V semiconductors to explore the effects of ionicity on Schottky barrier behavior (Louie et al., 1976, 1977). This latter question has been the focus of many theoretical and experimental studies. Specifically, the dependence of the Schottky barrier height on ionicity is still an open question at this time. The theoretical calculations and the relation of the results to recent theories of Schottky barrier behavior are described in the next sections.
B. SCPM for Semiconductor -Metal Interfaces The first fully self-consistent calculation for an s-m interface (Louie and Cohen, 1976) used the SCPM described in Section I1,B. The interface
ELECTRONS AT INTERFACES
15
considered was Si(11 1) in contact with aluminum. The silicon component was modeled using silicon pseudopotentials in a manner similar to the clean surface case described in Section 11. For aluminum, a simplified model was used. Because the main feature of the metallic contact, in a s-m interface with a covalent semiconductor, is the free-electron nature of the metal, aluminum was represented by a jellium model. The jellium consisted of a positive smeared-out background of charge and a free electron gas having the same density as aluminum. Following the procedure discussed in Section II,B, we model the interface with a unit cell consisting of a slab of silicon with its (1 11) surfaces exposed to the aluminum (jellium). This unit cell is then repeated periodically. The cell consists of 12 layers of silicon plus an equivalent component of aluminum (jellium). The geometry of the cell is shown schematically in Fig. 6. As shown in Fig. 6, the usual pictures of band bending are drawn in figures representing regions on the order of thousands of angstroms, whereas the calculation described here represents a region less than 10 A. Hence, the region studied is very close to the interface. The Si-Si bond distance is 2.35 A, and the edge of the jellium is taken to be at a distance equal to one-half of the Si-Si bond distance away from
METAL
FIG.6. Schematic model of a rn-s interface and the AI-Si supercell used in the calculation discussed in the text.
16
MARVIN L. COHEN
the atoms on the Si(1 11) surface. This is approximately the distance of an AI-Si bpnd. The self-consistent cycle (Fig. 2) is started with an empirical potent& for silicon and an aluminum potential, which gives a uniform charge confined to the aluminum slab. Since the Hartree part of the electronic contribution (Eq. 3) cancels the positive jellium background, the starting potential for aluminum contains only an exchange term of the form given by Eq. (4). Hence, the s-m interface calculations proceed in much the same fashion as the clean semiconductor surface calculations. The electrons from the semiconductor and from the jellium side are allowed to readjust to the interface because of the self-consistent scheme. In addition to the Si- Al calculation described above, calculations of aluminum modeled by jellium in contact with zincblende materials were also done. These include GaAs(1 lo), ZnSe( 1lo), and ZnS( 110) (Louie et al., 1976, 1977). Applications to other covalent semiconductors include germanium and diamond (Ihm el al., 1978a, c). In all cases, band structures, LDOS functions, and charge density plots for both the total electronic distribution and for individual states are computed. These are then used to analyze the behavior of the interface. C . Results
1. Si-A1 Interface
In Fig. 3, the total charge density for the clean Si( 111) surface is given. The analogous plot for Si-AI reveals that the charge density in a (110) plane is similar to the silicon vacuum results in that the charge density away from the interface resembles the bulk charge density and it is the interface region that is altered. On the aluminum side the charge density is
2.0
1
'
'
'
'
'
'
'
'
'
"
' . '
"
A h / Interface
PlOliL(4 I
1.0 7
0. . . . . . .
I
; I
I
I
I
I
I
I
I I
.I.
A
I
ELECTRONS AT INTERFACES
17
constant over much of the region, demonstrating the metallic character of the jellium. This is shown more explicitly in Fig. 7 where &(z), which is the total charge density averaged parallel to the interface, is displayed; the z coordinate is perpendicular to the interface. The peaks in pbt(z)on the silicon side arise from the contributions of the semiconductor bond charges and the constant value near unity on the aluminum side illustrates the free-electron nature of aluminum as discussed above. The doubledashed line represents the jellium edge. There is some charge transfer from the aluminum to the silicon side of the interface. Using the divisions of the cell given in Figs. 6 and 7, we find on the aluminum side that regions I and I1 each contain 7.9% of the total charge in the cell, while regions V and VI each have 8.8%. Using these regions as standards for aluminum and silicon, we examine regions I11 and IV and find that I11 has 7.6% of the charge while IV has 9.1%. The charge transfer of 0.3% appears to be going to the dangling-bond sites and the empty channels (see Fig. 3). The charge in the dangling bond is suggestive of a metallic-covalent-like bond between the silicon and the jellium. This is reminiscent of the model proposed by Phillips (1974). As in the case for the free silicon surface, the surface or, in this case, the interface states do not show up in the total charge density plots. Again the LDOS (e.g., Fig. 4) is used to determine the energy locations of these states. In the Si-AI case we make use of the division into six regions to search for interface states. The histogram LDOS is given in Fig. 8 for each region corresponding to Figs. 6 and 7. To facilitate comparisons, the density of states of silicon is superimposed on the LDOS of regions is superimposed on IV-VI, while a free-electron DOS [i.e., N ( E ) the LDOS of regions 1-111. The Fermi level is given by the dashed line. Examination of region I reveals a bulk aluminum-like DOS, while region VI looks just like bulk silicon. The major change takes place between regions 111 and IV, i.e., the interface region. A prominent state SK appears around - 8 . 5 eV. The charge density of this state is given in Fig. 9; its k space location is at the K point of the surface Brillouin zone. In real space it is localized near the interface mostly in region IV. This state is an interface state and is analogous to the surface states found before. However, because this state exists in the filled-valence-band region of the DOS, it does not contribute significantly to the properties of the Schottky barrier. We expect that the semiconductor gap region would be the principal energy range for states determining the Schottky barrier height. The state SK does, however, show that interface states can exist and is an excellent example of an interface state that decays on both sides of the interface.
- a]
18
MARVIN L. COHEN
FIG.8. Local density of states in arbitrary units. The regions are as shown in Figs. 6 and I .
FIG.9. Charge density contours for the interface state in AI-Si.
ELECTRONS AT INTERFACES
19
The most striking feature of region IV when comparing it with the surface LDOS of Fig. 4 is the absence of the dangling-bond surfice state peak that was present in the gap for the silicon vacuum case. The overlayers of aluminum appear to have suppressed this peak and the gap region is filled with new states. These states are the metal-induced-gap states (MIGS), which have a charge density that is metallic-like on the aluminum side (i.e., constant charge density) and becomes dangling-bondlike near the silicon surface, and then decays rapidly to zero in the silicon slab. The charge density for these states in the thermal gap (0- 1.2 eV) is given in Fig. 10 along with p ( z ) , which is the charge density averaged parallel to the interface. The state resembles a hybrid of a dangling-bond free-surface state matched to the continuum of metallic states. It is these states that determine much of the Schottky barrier behavior since they influence the position of the Fermi level. The calculated barrier height at the interface is the energy separation of the Fermi level and the bottom of the conduction band. The value given by this calculation is 0.6 2 0.1 eV, which is in excellent agreement with the recent experimental result of 0.61 eV (Thanailakis, 1975). Considering the fact that no experimental information other than atomic data (to establish the pseudopotentials) is used in this calculation, the agreement between theory and experiment is very impressive. 2. Ge-A1 Interface A calculation similar to the Si-AI calculation was done for Ge-A1 (Ihm et u l . , 1978a, c) and comparison was made with calculations for the free Ge(l11) surface (Ihmet al.. 1978b; Chelikowsky, 1977a). MIGS dominate the band gap region and determine the Schottky barrier height. The results are similar to the AI-Si case. Once again the peak found in the
1.0 -
MIGS
0.
-
INTERFACE REGION
FIG. 10. Charge density contours for "gap" states with energy between 0 and 1.2 eV. The charge density is averaged parallel to the interface and plotted along the direction perpendicular to the interface.
20
MARVIN L. COHEN
free-surface case (in the gap region) is suppressed by the aluminum overlayer (Fig. 11). JELLIUMIGe (111) SURFACE DENSITY OF STATES
26
25
t
EP'
Energy (eV)
FIG. 11. Surface density of states Dsof the jellium-Ge(ll1) interface in the gap. Also shown is the density of states of the clean Ge(ll1) surface states in the gap. It is clear from the fisure that the surface state peak of the clean germanium is drastically reduced by the metal contact. The new metal-induced gap states give a more or less uniform 4.@* and @ are the Fermi levels of the metal-germanium and the clean germanium surface, respectively.
3. Zincblende Semiconductor-Metal Interfaces Although the Ge-AI interface yields no qualitatively new features compared to the Si-AI case, the zincblende semiconductors do. As described earlier, a great deal of research has been done on the dependence of the Schottky barrier properties on ionicity. Since both silicon and germanium have zero ionicity, these materials give the features of the covalent end of the ionicity spectrum. GaAs, being a III-V semiconductor, is partially ionic and serves as a prototype for III-V compounds. The II-VIs ZnSe and ZnS are more ionic, with ZnS being even more ionic than ZnSe (Phillips, 1973). This series of materials is used to explore trends in Schottky barrier behavior.
21
ELECTRONS AT INTERFACES
First, we briefly describe the major results of the SCPM calculations for the zincblende s-m interfaces, and then the question of the role of ionicity is discussed. In the silicon and germanium cases, the surface used for the interface culculation was a (1 11) face. For the zincblendes, the ( 1 10) face is chosen because of its nonpolar nature. This causes some problems in comparing the zincblendes and group IV Schottky barriers, but these are not major. In all cases, the results are qualitatively similar in that the intrinsic surface states that existed in the fundamental gap of these materials for the clean surface are removed by the presence of the metal, and the metal-semiconductor hybrid states occur in this energy range. Localized interface states similar to the SK state found in silicon are found in the low-energy regions of the valence bands, but since these do not have much effect on interface properties such as the barrier height, they are not discussed here. LDOS functions similar to Fig. 8 reveal that the interface region is changed most, and the LDOS curves for the zincblendes (Louie P t al., 1976, 1977) are qualitatively the same as the silicon or germanium s-m interfaces. The decay lengths of the metal-semiconductor hybrid states or MIGS change depending on the semiconductor. Figure 12, which is analogous to and contains part of Fig. 10, displays the charge distribution of the penetrating tails of the MIGS into the semiconductor. The function p ( z ) is the charge density for the states in the thermal gap averaged parallel to the
0
t
5
10
15
Z (Atomic units)
INTERFACE
FIG.12. Charge distributions of the penetrating tails of the MIGS in the semiconductor thermal gap. i j ( z ) is the total charge density for these states averaged parallel to the interface with z = 0 at the edge of the jellium core.
22
MARVIN L. COHEN
interface. The point z = 0 corresponds to the jellium edge and p(z)/p(O)is given for Schottky barriers for aluminum with silicon, GaAs, ZnSe, and ZnS (Fig. 12). The overall behavior of p(z),lp(O) is similar for silicon and GaAs. The differences in the short-range oscillations of the function arise from charge density associated with the two different geometries studied, i.e., the (111) for silicon vs. the (110) surface for GaAs. The penetration depth 6 defined by p(S)/p(O)= e-l is equal to approximately 3.0 and 2.8 A for silicon and GaAs, respectively. As the semiconductor ionicity increases, S decreases. The values for ZnSe and ZnS are approximately 1.9 and 0.9 A, respectively. The Schottky barrier heights are evaluated using the energy splitting between the calculated E , and the conduction band minimum. The theoretical values for GaAs-Al, ZnSe-Al, and ZnS-A1 are 0.8 -+ 0.2, 0.2 & 0.2, and 0.5 5 0.2. The measured values for the GaAs and ZnS cases are both 0.8 (Sze, 1969); no measured value for the ZnSe case appears to be available. Another quantity that will be important to our discussion of ionicity dependent behavior is the surface density of states D , ( E ) . For energies in the semiconductor thermal gap, we define r
rm
qg--qa
where N ( E , r) is the local density of states, A is the surface area, and the integral over z is to be evaluated from the interface, z = 0, to deep into the bulk of the semiconductor. Hence, - eD,(E) gives the density of localized surface charge per unit energy on the semiconductor surface. In Fig. 13, the D,(E) functions for aluminum interfaces with silicon, GaAs, and ZnS are shown for comparison. The D,(E) for Al-ZnSe, which is not
>,
28
.-z 4 6
;6
-
a"2
0
0
04 08
0
0 4 08
12 0
10 2 0 30 40
Energy (eV)
FIG. 13.
Surface density of states, as defined in the text.
23
ELECTRONS AT INTERFACES
shown, is essentially the same as AI-ZnS but -30% higher. The main features common to D,(E) for the systems studied are that the dangling-bond surface state peak that appears in this region for the clean surface is absent, and the magnitude of D,(E) in the gap region decreases with increasing ionicity. The D , ( E ) function also becomes flat and relatively featureless for the more ionic cases. These considerations together with the decrease in 6 with ionicity are used to study the experimental situation described below. D . Further Results and Discussion
Studies of Schottky barrier heights have yielded an empirical linear theory M m , s)
=
S(s)X(m) +
40b)
(12)
where rn and s refer to metal and semiconductor, X(m) is the PaulingGordy electronegativity (Pauling, 1960), and S(s) and 40(s) are constants depending on the semiconductor used in the interface. As an illustration of this linear theory for the four semiconductors silicon, GaAs, ZnSe, and ZnS, Fig. 14 displays the measured barrier heights for various metals (Sze, 1969; Thanailakis, 1975). The slopes of the lines in Fig. 14 give the interface index S in Eq. (12). As shown in Fig. 14, the covalent semiconductor silicon has a very small slope, i.e., S - 0. I , and is hence independ-
o ZnS A ZnSe GaAs
su1.0
cu
0
0
1.o
2.0
3
3
xnl
FIG. 14. Experimental values of the bamer heights for four semiconductors in contact with various metals. X,,,is the electronegativity of the metal in the Pauling-Gordy scale. Data were taken from Thanailakis (1975) for silicon and Sze (1%9) for GaAs, ZnSe, and ZnS .
24
MARVIN L. COHEN
ent of the metal electronegativity. This independence suggests that the Fermi level is pinned in the band gap and the metal characteristics do not affect the position. The ZnSe and ZnS cases yield a larger value for S, suggesting more sensitivity to the metal contact. Kurtin et al. (1969) suggested that S is a function of the electronegativity difference between anions and cations in the semiconductor, AX = XA - X B .Since AX is a measure of the ionicity of the semiconductor, it was expected that S should also be a function of ionicity. The results of the data analysis of Kurtin et al. (1969) appear in Fig. 15. This curve has been the subject of many studies, but the sharp transition at AX 0.7 and the saturation at large AX are features that have not been completely explained. The question of the reliability of the data base and the analysis giving these features has been raised recently by Schluter (1978); we return to this question later. What do the SCPM calculations predict about the interface index S? Physically, the barrier height 4Bis determined by the requirement that in equilibrium, the Fermi levels of the metal and semiconductor are lined up. This is achieved by creating an electric dipole at the interface. Since in the SCPM results, it is the MIGS in the semiconductor gap that pin E F , the penetration depth S and the density of surface states 0,are both essential to a theory of & since they describe the properties of the MIGS. One approach (similar to that used by Cowley and Sze, 1966) for exploring the barrier height dependence on 0,and S was developed by Louie et al. (1977). In this model, use is made of the empirical relation between the metal work function +, and the electronegativity X(m),i.e.,
-
1.0 0.0
S 0.6
1
I I
*ZnSe
0.4
AX FIG. 15. Index of interface behavior S (Kurtiner al., 1969)
ELECTRONS AT INTERFACES
25
+ B where the choice A = 2.27 and B = 0.34 (Pauling, 1960) is made. Assuming an intrinsic or slightly n-type semiconductor of electron affinity, X ( s ) in contact with a metal, the dipole potential established at the junction is
(bm = AX(m)
A = X(S)
+ (bB
-
AX(m) - B
(13)
The change in A for a small change in the X(m) of the metallic contact is given by
d A = d(bB
-A
dX(m)
(14)
Using electrostatic arguments, d A can be related to the change in the charge distribution at the interface db = -4.rre2Ds tieffd(bB
(15)
where seffis the effective distance between the negative charge transferred to the semiconductor (because of the change in + B ) and the positive charge left behind in the metal. The effective distance is the actual distance divided by the dielectric constant for the s and m regions, i.e., Setf = trn/Ern
+
ts/es
(16)
The semiconductor distance t, can be replaced by the calculated 6 and tm/em can be approximated by the typical screening length in a metal (-0.5 A). Equations (12), (141, (15), and (16) yield S=
2.3
1
+ 4ae2Ds(t,/em + 6/cs)
The dielectric screening for short-distance fluctuations of the order of a few angstroms can be estimated using the calculations of Walter and Cohen (1970) to be es 2. Hence, S can be calculated directly from microscopic quantities, i.e., D, and 6. The results (Louie et al., 1977) give agreement between theory and experiment, which is quite good considering that the model for S is an approximate one. Hence, the SCPM and the resulting s-m hybrid states (or MIGS) appear to give a consistent picture of Schottky barrier behavior. The quenching of surface states by just a few layers of metal and the introduction of metal-induced states was basically confirmed by the experiments of Rowe et al. (1975) and later by measurements of Brillson (1978). More detailed theoretical studies using aluminum pseudopotentials to model the metallic side of the Schottky barrier were done for Al-GaAs (Chelikowsky et al., 1976) and for AI-Si (Chelikowsky, 1977b; Zhang and Schliiter, 1978). These calculations also conclude that the intrinsic surface states were replaced by metal-induced interface states. Because of the
-
MARVIN L. COHEN
26
atomic nature of the model for aluminum, other interface states not seen in the jellium calculation were also discovered. Many of these states were also found in the experimental measurements, but these are not important for determining Schottky barrier properties. Returning to the interface index S, the SCPM calculations gave values in general agreement with the measurements, indicating that S does increase with ionicity. Since only a few cases were explored, i.e., silicon, germanium, GaAs, ZnSe, and ZnS, there were not sufficient points to theoretically establish the main features of Fig. 15. For example, it is unclear whether S is a function of ionicity or band gap based on the above calculations, since the band gap increases in the series of semiconductors considered as does the ionicity. For most materials, large gaps are associated with large ionicity ; therefore, correlations with ionicity and band gap are hard to determine spearately. Diamond is unusual in this respect since it has a large gap, but zero ionicity. The behavior of the diamond-metal interface was therefore considered to be a critical probe of Schottky barrier theory. Specifically in the interface index vs. ionicity plot (Fig. 15), if ionicity is in fact the relevent property, the diamond Schottky barrier should have S 0. However, this would imply a high density of interface states and significant penetration of the MIGS into the semiconducting (or insulating) side [see Eq. (17)]. A large penetration is not expected because of the large band gap. On the other hand, if penetration were small and the density of surface states were also small, this would imply that diamond should be in the Schottky limit and hence in the high ionicity region of Fig. 15. Unfortunately, the experimental situation for diamond Schottky barriers is not completely settled. Some early measurements were reanalyzed (Mead and McGill, 1976), giving a barrier height ranging from 1.9 to 2.2 eV and an S value near zero. However, the authors point out that they do not consider the S measurement to be conclusive. New data on this interface would be very helpful. The theoretical situation is quite interesting (and lively). Estimates can vary considerably if based on the arguments described above. Hence, a SCPM calculation was done to attempt to fix +B and S (Ihm et al., 1978a, c). The results for 0,had the same qualitative features as previous studies, i.e., the large clean-surface-state peak in the gap region was suppressed by the metal contact and a lower almost constant density of states appeared throughout the entire gap region (Fig. 16). MIGS were also found and were similar in form to those found in the cases previously studied. The calculation was not expected to be as accurate as the other covalent semiconductor calculations (i.e., silicon and germanium), because the stronger potentials and small lattice constant result in poorer
-
ELECTRONS AT INTERFACES
27
SURFACE DENSITY OF STATES 38
Energy (eV)
FIG.16. Surface density of states 0,of the jellium-diamond(] 11) interface in the gap. Also shown is the density of states of the clean diamond(] 1 1 ) surface states in the gap. Again, the clean diamond surface state peak is drastically reduced by the metal contact. The new metal-induced gap states give almost uniform Ds.EFd and E l are defined as in Fig. 1 I .
convergence for the diamond case. Nevertheless, the calculated barrier height of 2.2 eV is in satisfactory agreement with the measured values. The penetration of the metal-induced gap states into the diamond side 1.37 A. This value is about half of the correof the s-m interface is 6 sponding decay lengths for A-Si (-3 A) and AI-Ge (-2.7 A). The decay length of the dangling-bond surface state into the bulk for the clean diamond surface is 1.0 A, whereas the corresponding decay length for silicon or germanium is -2.5 A. Hence when a s-m contact is formed, 6 increases by a small but nonnegligible amount in all cases studied. The average & ( E ) value for diamond (Fig. 16) is 2.3 x IOl4 states/eV-cm'. Using Eq. (17), this yields a value for the interface index S = 0.38 2 0.1. The relatively large error bounds reflect the uncertainties in 6 , D s, and E, . A value of S 0.4 for A X = 0 is far from the curve drawn in Kurtin et al. (1969) given in Fig. 15. Hence the calculation of Ihm et al. (1978a, c) presents a challenge to the experimentalists to determine S for diamond and also questions the validity of an analysis such as the one represented by Fig. 15. If the calculation is confirmed by experiment, this would support the conclusion that the interface index is not primarily determined by ionicity, and the sharp transition between covalent and ionic materials of the type seen in Fig. 16 is not an appropriate description. These results would also conflict with attempts to parametrize the barrier behavior using the chemical reactivity (Brillson, 1978).
-
-
-
28
MARVIN L. COHEN
It is useful to attempt to get a physical picture to estimate 8 and 0, using the results of the SCPM. Ihm et al. (1978a, c) have suggested that the results can be interpreted in terms of the energy gap E , and the lattice constant a,. The s-m SCPM calculations suggest that the charge transfer per unit area to the semiconductor arising from the MIGS in the thermal gap is proportional to the number of semiconductor surface atoms per unit area. Each surface atom receives a fixed amount of charge, and it is estimated that 0.6-0.7 electronic states are available on the semiconductor side in the thermal gap for each surface atom. If 0,is assumed uniform, then since 0,is the number of states per unit energy interval and a constant number of states is available for the entire gap region, 0, is inversely proportional to the thermal gap. Hence 0,of diamond would be 3 the 0,of silicon. (The use of the thermal gap is only approximate here.) Since diamond has a smaller lattice constant than silicon, it has a larger number of surface atoms per unit area; hence 0,of diamond increases by a factor that is the ratio of the square of the lattice constants -2.3. The net 0,for diamond should be about one-half that for silicon, and this is consistent with the SCPM results (i.e., 2.3 x l O I 4 vs. 4.5 x IOl4 states/eV cm2 for silicon). The other parameter of interest is 6, which can be estimated using a one-dimensional WKB approximation, i.e., 6/26 = (mEg)1/2,whereE, is the Phillips average gap (Phillips, 1973). A more appropriate gap would be the average gap of the two-dimensional projected band structure, but this is more difficult to obtain. Using the above analysis for D,(E) and 6, it is possible to estimate S. It is important that D,(E) be approximately constant in the thermal gap region. This is roughly satisfied by the SCPM calculations and by other theoretical work (Louis et al., 1976) It is interesting to use the arguments presented to examine a case where SCPM calculations are not available. S i c is such a case and it is useful to consider since experiments suggest (Mead and McGill, 1976) that S(SiC) = S(Si) even though the ionicity and gap of S i c is larger than that of silicon. Using silicon as a reference material, we find
-
-
For 8, the ratio of the square roots of the Phillips gaps gives 8(SiC) - 0.728(Si). This yields S(SiC) 0.2, which is half the corresponding value for diamond and closer to silicon (calculated, S 0.13; measured, S 0.1). We now return again to the interface index and the empirical observations of Kurtin ef al. (1969). Recently, Schliiter (1978) has reexamined the experimental data used by Kurtin et al. (1969) and does not find the abrupt
-
-
-
29
ELECTRONS AT INTERFACES
transition between covalent and ionic semiconductors (Fig. 15). He also suggests that the saturation of Fig. 15 at S = 1 is spurious and a saturation at S = 2.0-3.0 is more appropriate. A least-squares fit of the data was made by Schliiter (Fig. 17) as a function of electronegativity and total semiconductor polarizability . The revised values for S alter the topology of the curve of Kurtin er ul. (Fig. 15) and neither the saturation at S = 1 nor the sharp transition is present. The same changes would occur if the Schliiter values for S were used in Brillson's (1978) analysis based on heats of formation, as Brillson's observation of a covalent-ionic transition and a saturation at S = 1 are also dependent on the Kurtin et ul. estimates of S. In Schliiters' analysis he explores the relation between the metal electronegativity and work function, i.e., the A and B parameters of Eq. (13). Since S is dominated by A in the Schottky limit, the detailed relationship between &, and X(m) is quite relevant. In particular, the choice of measured or internal metal work functions affects this value. Schliiter points out that A = 2.0-3.0 yields reasonable results for most theoretical work (e.g., SCPM calculations used A = 2.3); he uses A = 2.86, which is the value obtained by Nethercot (1974). Values for S in this range led Schliiter to expect a Schottky limit value of S = 2.0-3.0, i.e., S = A. The largest S in Fig. 17 is 1.52 2 0.3; hence Schluter concludes that saturation has not yet been observed. An argument can be made for a lower maximum value of S, S, . By examining Eq. (17) and noting that there are physical limits on the minimum values of 0,and 6, S, does not reach 2.3 or whatever the value of A is in Eq. (13). From extrapolations of the SCPM calculations, minimum values for 0,and the effective penetration length are about 5 x l O I 3 states/eV-cm2 and 1 A. Further justification for these estimates can be obtained by using the approximate relations 0, (Eminaz)-l and S (Eg)-1/2 to scale the diamond Schottky barrier results. The largest rea20 eV, and a, 4 or 5 A. These sonable values are Emin l Ry, E, 1.25 and S, (a, = 5.5 A) 1.5. choices give S,,, ( a , = 4.5 A) These results depend critically on the minimum 0,. If the Nethercot (1974) values of A are used, the above estimates would be increased by -25%. Ionicity effects are not explicitly taken into account here and these could also increase S, . The above analysis would then suggest has already been reached experimentally. that S, Mele and Joannopoulos (1978) have contributed a theoretical model calculation to determine the behavior of the interface index S and to construct a theory of s-m interfaces. Their work rests on the concept of metal-induced gap states, but these states appear to differ somewhat from those of Louie and Cohen (1976). The MIGS used are derived from clean semiconductor surface states that have been broadened by interaction
-
-
-
-
-
-
-
30
MARVIN L. COHEN 2 0
1.5
n 10-
0.5
-
1
1
10
05
1
1
1,s
0.0
A t.5
AX 2.0
-
l.5-
1.0-
0.5 -
o
"
'
"
~
5"
' a.
'
-1
~
"10 '
'
'
IS
FIG. 17. Least-squares-fitted experimental S parameters plotted vs. electronegativity difference AX, and vs. total semiconductor polarizability c0 - I (Schluter, 1978).
31
ELECTRONS AT INTERFACES
with the metallic contact. The broadening parameter r is calculated using a “golden rule” approach and shown to be of the order of 1 eV. For small-gap materials, it is difficult to distinguish these states from the Louie-Cohen states since the latter fill the gap region and would resemble the broadened clean surface dangling bond states. Differences occur for large-gap materials if r is not too big, as the Mele-Joannopoulos model would give a resonantly broadened peak, while the Louie-Cohen approach leads to featureless LDOS functions like that found for diamond in Fig. 16. Using the above approach and a new ionicity parameter, Mele and Joannopoulos are able to construct an S vs. ionicity curve for the tetrahedral compounds that is in satisfactory agreement with experiment (Fig. 18). The cadmium compounds do not lie close to the curve and this is attributed to the possible occurrence of interface bonds or atomic distortions at the interface. ZnS is also a “bad actor,” and theoretical calculations using the SCPM and other analyses (e.g., Tejedoret al., 1977) fail to find d large difference in S between ZnS and ZnSe. Perhaps an experimental reexamination of ZnS is warranted.
-10
-0
-6
-4
-2 0 2 X IONlClTY ( eV )
4
6
8
FIG. 18. Dependence of the interface index S on the Mele-Joannopoulos ionicity parameter (Mele and Joannopoulos, 1978). The theoretical values are given by the curve. The filled circles are from Kurtin et a/. (1969); the open circles are the revised values given by Schluter (1978).
32
MARVIN L. COHEN
The Mele-Joannopoulos calculation does give a sharp distinction between covalent and ionic materials; this is evident by the change in the dependence of S on their ionicity parameter for positive and negative ionicities (Fig. 18). The zero ionicity level in the Mele-Joannopoulos definition corresponds to the transition between materials in which the s-p mixing causes the semiconductor gap and those more ionic materials where a gap between the p-anion energy level and the s-cation energy level exists even with no s-p mixing (the s-p mixingjust increases the gap). Diamond and S i c are again interesting to consider because they lie at negative ionicity in Fig. 18, but have large gaps. The S value for S i c lies close to the value estimated using results for the SCPM calculations for diamond. The diamond S value itself is almost one-half of the SCPM result; however, it. is not clear at present whether the above differences are outside the uncertainties inherent in the calculations. For example, a large r value in the Mele-Joannopoulos calculation could bring the S parameters into agreement. Hence, significant progress has recently been made in the microscopic understanding of s-m interfaces. The introduction of metalsemiconductor hybrid states or MIGS with short decay lengths (Louie and Cohen, 1975), experimental confirmation that clean-surface states are replaced by metal-induced states (Rowe et al., 1975), and applications of these theoretical ideas and more detailed experiments, have led to a clearer picture of the electronic structure of a s-m interface. Although considerable attention has been focused on the ionicity dependence of Schottky barrier properties and the S parameter, the resultant picture is not completely clear. The efforts of Louie et al. (1977), Schluter (19781, Brillson (1978), and Mele and Joannopoulos (1978) have refined the questions, but one is still not left with a theoretical curve that goes through all the experimental data. Part of the problem is the lack of reliable data. Hopefully, new and more accurate determinations of S will emerge to refine our theoretical picture. Another point of view would be to abandon the pursuit of S and ask new questions or suggest new measurements to test the current models.
IV. SEMICONDUCTOR-SEMICONDUCTOR INTERFACES A . Introduction There has been a great increase in fundamental research on semiconductor-semiconductor (s -s) interfaces in just the past three
ELECTRONS AT INTERFACES
33
years, and the availability of molecular-beam epitaxy (MBE) apparatus in many laboratories promises increased research participation in this general area. The semiconductors forming the s-s interface or heterojunction can have very different electronic properties; hence the various choices of semiconductors can give junctions with a variety of characteristics. Here theoretical work can be very useful if the properties of the interface can be predicted before fabrication. This also implies that a close interplay between theory and experiment is valuable both to decide on the important directions for research and for collaboration on ways to obtain microscopic information about s-s interfaces. Such interaction has in fact occurred, and a great deal of progress has resulted. A review of work up to 1973 appears in Sharma and Purohit (1974), and the book by Milnes and Feucht (1972) has become a standard reference in this field. The earlier work is therefore not reviewed here, and this review focuses on current research. We begin by pointing out the prominent problems and describing some of the progress made with regard to their solutions. The discussion is restricted to diamond and zincblende semiconductors. Lattice matching is important to the formation of good heterojunctions and this suggests the following groups (Frensley, 1976): lattice 5.4 A, Si, Gap, ZnS; a, 5.65 A, Ge, GaAs, AIAs, ZnSe; constant a, u, 6.1 A, GaSb, AlSb, InAs, ZnTe, CdSe; a, 6.4 A, InSb, CdTe; and others like InP, CdS. The degree of lattice matching is related to the number of misfit dislocations. Analysis of lattice-mismatched interfaces using transmission electron microscopy (TEM) (Petroff, 1977) in itself is an interesting aspect of s-s interface research. New techniques are available for analyzing interface roughness and interface diffusion on an atomic scale, and the use of MBE and TEM can lead to fabrication and study of extremely thin films. Abrupt interfaces can be formed and their detailed structures studied. Unfortunately, from the theorist’s point of view, these studies leave one with the impression that many of the observed s-s properties may be significantly influenced by characteristics of the interface not normally modeled in a theoretical calculation, e.g., misfit dislocations. In fact, most theoretical studies ignore these features of heterojunctions and assume ideal, abrupt interfaces with perfect lattice matching. This should be a reasonable approximation for systems with small lattice mismatch. We describe such systems and most of the theoretical calculations assume ideal, abrupt geometries. Some of the same questions raised in the previous section on Schottky barriers can be asked here. In particular, the question of the existence of interface states and their possible role in determining the properties of the s-s interface. In the Schottky barrier study, we explored two types of states, which decayed on both sides of the interface into the metal and
-
-
-
-
34
MARVIN L. COHEN
into the semiconductor. The other type were the s-m hybrid states or MIGS, which decayed only on the semiconductor side. For the s-s case, we focus on interface states, which are primarily of the former kind, i.e., they decay on both sides of the interface. New states with electron density “running” parallel to the interface also occur. By far the features of the s-s interface that have received the most attention are the discontinuities in the band edges. Specifically, discontinuities in both the valence band maximum and conduction band minimum must result if two semiconductors having different band gaps are brought into contact. After contact, the Fermi levels of the two semiconductors line up as shown in Fig. 19, producing the discontinuities in the conduction band (cb) AEc , and in the valence band (vb) AEv. Figure 19 is schematic and is based on a Ge-GaAs model of an interface. Ordinary band bending takes place over hundreds or thousands of hgstroms, but as in the case of the s-m interface, we are interested in the behavior close to the interface, i.e., - I0 A. Knowledge of AEc and AEv is important to the design of heterojunc-
SEMICONDUCTOR / SEMICONDUCTOR
+-
1000 A
__I
FIG.19. Schematic drawing of a s-s interface (heterojunction)and the region near the interface for a Ge-GaAs interface.
ELECTRONS AT INTEHFACES
35
tions. The discontinuities satisfy the obvious relation
AE,
+-
AE, = AEg
(19)
i.e., their sum is just equal to the difference in the semiconductor gaps. So, the problem remains of how to solve for one of the discontinuities, i.e., AE, or AE,. One of the simplest and most used approaches for evaluating AE, was suggested by Anderson (1962) and it has been critically discussed at length in the literature (e.g., Kroemer, 1975; Shay et al., 1976: Frensley and Kroemer, 1976, 1977). This method equates the conduction band discontinuity to the difference in the electron affinities of the semiconductors ( 1 and 2),
AE,
XI -
X2
(20)
For example, in Fig. 19, AE, = X, - XGaAs.Equation (20) has motivated new estimates of electron affinities and new measurements of AE, to compare with the electron affinity rule (EAR); e.g., Shay et a / . (1976) have estimated electron affinities for a series of chaliopyrite 11-IV-V semiconductors and have also shown that the InP-CdS interface obeys the EAR. The EAR has been questioned by several authors, notably Kroemer (1973, who points out the following: the electron affinity is a characteristic of the clean-semiconductor interface rather than a s-s interface. The resulting AE, is a small number obtained from the difference between two values much larger in magnitude, and hence uncertainties give the EAR little predictive power. Electron affinities are measured on natural cleavage planes, which can be different for different semiconductors [e.g., (1 1 1 ) for germanium and ( I 10) for GaAs]. The electron affinity is sensitive to differences in the surfaces used. It should also be noted (and added to Kroemer's' list) that Grant et al. (1978) have shown, using x-ray photoelectron spectroscopy, that AE, is also a function of orientation. The orientation variation of the band-gap discontinuities between the ( 1 1 l ) , ( 1 lo), and (100) faces of GaAs for a Ge-GaAs heterojunction is significant fraction (-4) of the total band-gap discontinuity. In addition, some measurements of AE, disagree with the EAR. For example, a recent measurement on Ge-GaAs by Perfetti et d.(1978) using angle-resolved ultraviolet photoemission techniques measures AE, = 0.50 eV, in good agreement with the SCPM predictions (Pickett ef al., 1977, 1978), but in conflict with the EAR. The experimental work and the questions related to the EAR have inspired several theoretical calculations in this area. In addition to trying to provide some estimates for the magnitudes of the band edge discontinuities and interface dipoles that cause them, some theoretical calculations
36
MARVIN L. COHEN
have explored the interface states and their role in interface behavior. We list afew of the calculations as examples of the various approaches. Dobrzynski et a / . (1976) and Tejedor and Flores (1978) using the matching of surface Green's functions are able to extract some general conditions related to the existence of interface states and make some predictions about band edge discontinuities. Louis (1977) uses a tight-binding approach and the Bethe lattice approximation to study the Ge-GaAs (111) interface. Herman and Kasowski (1978) consider the (1 10)Ge-GaAs interface using a linear combination of muffin tin orbitals. Harrison (1977) employs an LCAO theory to develop a scheme for estimating band edge discontinuities. To verify and support his LCAO or tight-binding approach, Harrison then uses a pseudopotential model to compute valence band maxima. His scheme has predictive power and he makes a series of estimates of the band edge discontinuities that compare reasonably well with experiment. Frensley and Kroemer (1976, 1977) use a pseudopotential approach and consider the problem of matching potentials across an interface in great detail. In particular, these authors choose interstitial points, i.e., the points farthest away from the surrounding atoms, and compare electrostatic potentials at these points on both sides of the interface. The band structures of the semiconductor components are calculated selfconsistently, but the entire interface structure is not computed selfconsistently. The Frensley-Kroemer calculation does not provide information on interface states, whereas the fully self-consistent interface calculations do. The first fully self-consistent calculation for a realistic interface was done by Baraff et u / . (1977) for the (100) Ge-GaAs interface. The (100) unreconstructed geometry necessarily leads to a metallic interface with a partially occupied interface bond. Experimentally, all Ge-GaAs interfaces are found to be semiconducting. To eliminate the high density of localized interface states in the band gap, relaxations or reconstructions are necessary; however, the calculation of Baraff ef al. was only for an ideal polar (100) surface. This calculation did allow analysis of the interface dipole, the band edge discontinuities for this metallic interface, and a study of the behavior of the_ bonds at the interface. The first SCPM calculations for a nonpolar interface were done by Pickett et a / . (1977) for Ge-GaAs (110). The calculation revealed a variety of interface states, estimated band edge discontinuities that were later verified by experiment (Perfetti et a / . , 1978), and gave a detailed analysis of the various bonds at the interface. These calculations were subsequently extended to a study of the AIAs-GaAs interface (Pickett et a/., 1978), Ge-ZnSe (Pickett and Cohen, 1978a), and GaAs-ZnSe (Ihm and Cohen, 1979a). The SCPM approach was also used to calculate the effects
ELECTRONS AT INTERFACES
37
of relaxation at the Ge-GaAs( 110) interface (Pickett and Cohen, 1978b) and the systematics of the various SCPM results for the interfaces were analyzed €or trends and to explore the underlying theory (Pickett and Cohen, 197%). These results and the calculational methods involved are described in the next section. A number of calculations have been done on repeated heterojunctions or semiconductor superlattices. In principle, the SCPM calculations using supercells are in fact superlattices, and this aspect is used to give some information on superlattices. Most calculations designed specifically for superlattices have concentrated on structures composed of GaAs and AlAs (e.g., Caruthers and Lin-Chung, 1978; Herman and Kasowski, 1978; Schulman and McGiU, 1977). Much of this work has been motivated by the extremely interesting systems fabricated using MBE techniques (Esaki and Tsu, 1970; Chang et a/., 1973; Dingle et al., 1974; Gossard et al., 1976). Although interest in these systems is focused on the superlattice behavior, studies also yield information about interface structure. For example, investigations (Sai-Halasz et a / ., 1977) of complex heterojunction superlattices of selected alloys have given information about systems where the cb of one semiconductor is below the vb maximum of the second semiconductor forming the heterostructure. An example of such a pair of semiconductors is InAs-GaSb. A recent SCPM calculation (Ihm et a/., 1979) for this system explores the electronic structure near the interface. Despite the inherent importance of superlattices, these structures and their properties are outside the scope of this review. We emphasize, however, that there is cross fertilization between these subareas of semiconductor research, and some references to superlattices are made.
B. SCPM for Semiconductor -Semiconductor Interfaces We again concentrate on one type of theoretical approach. The s - s interface is modeled in a similar fashion to the semiconductor-vacuum and semiconductor-metal cases discussed previously. The calculation is then done self-consistently using the scheme outlined in Fig. 2. Pseudopotentials are used for both semiconductors of the heterojunction since it is necessary to have an accurate description of the electronic structure of both components. The potentials are designed to give good descriptions of the individual atomic and some ionic configurations and the bulk serniconductor. Once the pseudopotentials are chosen, no further adjustments are made to describe interface features. Hence, the calculation proceeds much like the semiconductor-jellium calculation described before, but there are no jellium approximations. The systems considered are Ge-
38
MARVIN L. COHEN
GaAs, AIAs-GaAs, Ge-ZnSe, and GaAs-ZnSe. Because the lattice mismatch in all of these systems is very small (0.1-0.2%), atomic disorder (misfit dislocations) should be minimal and the ideal geometry should be a reasonable approximation to a real system. The Ge-GaAs(l10) interface is regarded as a prototype, and both the calculational procedures and results for this interface are described in more detail than the other systems considered. The supercell for the Ge-GaAs is taken to have 18 layers (nine of each material) in the z direction. As each (110) plane contains two atoms per cell, the supercell contains 36 atoms and two interfaces. Because the cell is reasonably large, the two interfaces do not interfere significantly. Tests with five layers of germanium and five layers of GaAs reveal that the total potential and charge density are essentially the same as for the 18 layer supercell, but it is difficult to make reliable estimates of band-edge discontinuities with the 10 layer cell. Baraff et al. (1977) have suggested that two or three atomic layers on each side of the interfaces are sufficient to give realistic potentials and charge densities. These authors use a three-layer film of germanium terminated on a (100) gallium plane of GaAs; they also use hydrogen potentials to artificially cap the dangling germanium bonds. The purpose of the hydrogen is to remove the partially occupied surface states that would otherwise be expected at the Ge( 100) surface. Baraff et al. claim that the spatial density disturbance caused by the hydrogen does not extend into the interface. The supercell calculations using the SCPM assumed the following numbers of layers: Ge-GaAs, (9, 9); AlAs-GaAs, (9, 9); Ge-ZnSe, ( 5 , 5 ) ; and GaAs-ZnSe, (7, 7). These calculations and further tests indicate that ( 5 , 5 ) is essentially the smallest number of layers that can be used in a calculation of this kind consistent with reliable results. Hence, the Ge -ZnSe calculation is the least accurate of those discussed here, and some problems arise in interpreting results for this system. Smaller cells require less computer time and are sometimes necessary for cases involving strong potentials, which in themselves require added computer power. A schematic drawing of the Ge-GaAs unit cell appears in Fig. 20. Because the real space cell is so large in the z direction, the corresponding Brillouin zone (BZ) has almost negligible width and is effectively two dimensional. The BZ is also shown schematically in Fig. 20 and the labels of the center (r),edges (X andz’), and corner (M) are given. These will be used later in describing the band structure for the interface states. The atomic positions near the Ge-GaAs(l10) interface are given in Fig. 21. As shown explicitly in this figure, each atomic layer contains equal numbers of cations and anions and hence is nonpolar. For nonpolar interfaces, all bonds remain saturated, at least on the average; this suggests that the ideal
UNIT CELL
BRlLLOUlN ZONE
FIG.20. Schematic drawing of the Ge-GaAs(l10) unit cell or supercell used for calculating the electronic structure. The Brillouin zone associated with this cell is shown, and the labeling of the points of the two-dimensional Brillouin zone is given.
Ge-GaAs (110) Interface OGa
@Ge
.As
7 First
Interface
FIG.21. Atomic positions near the Ge-GaAs(l10) interface. Bonds are denoted by heavy solid lines, except bonds across the interface are shown as heavy dashed lines. The chains ABAB and CDCD are the two independent bonding chains perpendicular to the interface, containing the Ge-Ga and Ge-As bonds, respectively. The x, y , and z directions used in setting up the unit cell are shown at bottom.
40
MARVIN L. COHEN
geometry will yield a semiconducting interface in agreement with experiment (this is in fact the case). This is the primary reason for the (110) choice for most SCPM calculations. Polar interfaces [e.g., (100) or ( 1 1 I)] that are unreconstructed should be metallic, as noted by Baraff et al. (1977). It is assumed that atomic rearrangement ultimately gives rise to a semiconducting interface for polar configurations. Once the unit cell and pseudopotentials are chosen, the selfconsistency loop of Fig. 2 can be started. Because of the nonpolar geometry, the interface is semiconducting at each iteration and hence the charge density can be computed more simply using a special k-point scheme (Chadi and Cohen, 1973). This charge density is used to form the Hartree and exchange potentials described earlier. Self-consistency is normally achieved in five to ten iterations. The starting empirical potentials for the two semiconductors differ considerably near the interface. Since the empirical potential represents the bulk potential of each semiconductor, when the s-s interface is formed, the potential changes at the interface over a characteristic length that is of the order of a bond length. Through the self-consistent procedure, the potential is allowed to readjust, the charge density also adjusts, and the final self-consistent potential changes rather slowly in crossing the interface. A typical length here is about four atomic planes (in Ge-GaAs). The above features are shown in Fig. 22 for the Ge -GaAs and AlAs-GaAs empirical and self-consistent potentials.
C. Results The results of the calculations for the various interfaces are given separately. Ge-GaAs is retained as a prototype and more details are given for this case. 1. G e - G a A s ( l l 0 ) The total self-consistent valence electron charge density for the GeGaAs(ll0) interface is given in Fig. 23. This is the figure for a s-s interface analogous to Figs. 3 and 7 for the semiconductor-vacuum and s-m interfaces. The charge density is shown in two planes to illustrate the two types of bonds at the interface, i.e., Ge-Ga and Ge-As. The charge density returns to its bulk values (Chelikowsky and Cohen, 1976a) at about two atomic layers away from the interface. This illustrates the approximate size of the perturbation of the interface. The bonds near the interface differ from the bulk values for the Ge-Ge or Ga-As bonds. The values are about 10% lower for the Ge-Ga bond and about 8% larger for the Ge-As bond (Fig. 23). On the basis of elec-
ELECTRONS AT INTERFACES
-
-1
Empirical
I
41
’
J
Fic. 22. Empirical and self-consistent potentials, averaged parallel to the interface, for the ( I 10) interfaces of (a) Ge-GaAs and (b) AlAs-GaAs. The large arrow denotes the geometric interface, while the smaller arrows show the positions of atomic planes. One-half of the unit cell in the z direction is pictured.
tron counting using a simple chemical bond picture, gallium, germanium, and arsenic donate 3, 4, and 5 electrons to a bond, respectively. For a tetrahedral coordination this yields 0.75, 1.0, and 1.25 electrons/bond for the three elements. Hence, the Ge-Ga bond is expected to have 1.75 electrons, while the Ge-As bond would have 2.25 electrons. Detailed calculations of the charge in the bonds shown in Fig. 23 yield 1.77 and 2.23 electrons for the Ge-Ga and Ge-As bonds, respectively. The Ge-Ge and Ga-As bonds parallel to and bordering the interface contain two electrons. We now turn to the question of the existence of interface states. The conclusion is that these stutes do exist, and this is one of the most dramatic discoveries of this calculation. A convenient way to search for interface states and to demonstrate their properties is to display them on a projected band structure (PBS). The PBS for Ge-GaAs(l10) appears in Fig. 24. This figure represents a projection of the bulk band structure for germanium and for GaAs on the ( 1 10) surface BZ shown in Fig. 20. True interface states can exist in gaps (white regions) in the PBS where no bulk
42
MARVIN L. COHEN
(110) Ge-GaAs Total Charge Density
FIG.23. Contour plot of the total self-consistent valence charge density of Ge-GaAs, pictured in the planes perpendicular to the interface containing the ABAB (a) and CDCD (b) chains of Fig. 21. Only one-third of the unit cell, centered at an interface, is pictured. The average charge density is normalized to unity; successive contours are separated by 0.2 units. Note that the Ge-Ga and Ge-As bonds across the interface are unlike both the Ge-Ge or Ga-As bonds. The maximum charge density in the bonding regions is quoted to the nearest 0.05 units.
states are present. Interface states that occur in the shaded regions can resonate with bulk states and become interface resonances. The PBS of Fig. 24 has several distinct gap regions. The fundamental gap ( - 1 to 2 eV) is derived from the bulk semiconductor gap. The minimum gap is at r. Another region of interest for interface state study is the “stomach” gap ( - 2 to - 6 eV). Other regions include the “lower gap” (-7 to - 10 eV) and the region below the valence bands (< - 11 eV). There are other gaps in the conduction band region, but these are not used here. In Fig. 24, the interface states are represented by heavy solid lines and for those states that have considerable decay lengths into the bulk regions by heavy dashed lines. The interface states appear to be mostly associated with regions along the edges of the BZ (Fig. 20), i.e., along +?I? + x’. There are basically six types of states, which are described below. The charge densities for the interface states are displayed in Figs. 25 and 26, and it is evident from these figures that the states are localized in the interface region. The S , and S2 states are s-like about arsenic and gallium, respectively, as can be seen in Fig. 25. S, is pulled below the
x
ELECTRONS AT INTERFACES
43
FIG.24. Interface states of ( 1 10) Ge-GaAs relative to the projected band structures of bulk germanium and GaAs from self-consistent calculations. The dispersion of the interface states is denoted by heavy solid lines; heavy dashed lines indicate interface states that have a long decay length into the bulk. Symmetry points (in reduced units) are r = (0, 0). 3 = (4, 0). = (4,t), = (0, 1).The interface states S , , S,, B,, B,, PI,P,, as well as the stomach gap ( - 2 to - 6 eV) and the lower gap (- 7 to - 10 eV) are described in the text.
bottom of the arsenic s-like bulk bands, while S, is pushed up into the stomach gap (Fig. 24). The B, and B2 states are p-like bonding states localized at the interface on the Ge-As and Ge-Ga bonds, respectively. One view of the origin of the energy locations of the interface states relative to the bulk is to note that the arsenic potential is more attractive than the germanium potential, which in turn is stronger than the gallium potential, and this should influence the energy positions of the interface states. Since the B, and S, states involve the arsenic site, their energies should be lowered, while the gallium-derived B, and S, states should be raised relative to similar bulk states. This is demonstrated in the PBS of Fig. 24. Another class of interface states is shown in Fig. 26. These states originate from Ge-Ge and Ga-As bonds parallel to and adjacent to the interface. Their location in the PBS (Fig. 24) is almost completely confined to
44
MARVIN L. COHEN
-B2
Ge-Ga Interface State 0--.
n
1
[(b)
B, Ge-As Interface State
I
)(d)
S i As s-like lnterfoce State
1
FIG.25. Contour plots, perpendicular to the interface, of the charge densities of the interface states s,, s,, B , , B,. Each averaged charge density is normalized to unity: successive contours are separated by 2.0 units. Straight lines denote bond directions. The interface states derived from the gallium (respectively, arsenic) are plotted in the plane of the ABAB (respectively, CDCD) chain in Fig. 21. In each case the charge density in the plane that is not shown is <5% of that in the plane shown.
the region near X‘ in the BZ. The charge density plot in Fig. 26 is displayed in a plane parallel to the interface; very little charge related to these states lies in the interface itself. The origin of these “parallel” states is not completely clear physically if one tries to use the potential model to visualize the binding of the states as we did for B,, B2,S,, and S2. The LDOS for the Ge-GaAs(110) interface appears in Fig. 27. This figure for the s-s interface is analogous to Fig. 4 for the clean surface and Fig. 8 for the Schottky barrier. The region labelled “excess” LDOS is defined as the increase (if positive) in the LDOS above the LDOS contributions from both bulk semiconductors. The LDOS of both second-layer GaAs (see Fig. 21 for the geometry) and second-layer germanium resemble their bulk DOS with little excess states except that the resonances R , and R , begin in second-layer GaAs. The first layers of germanium and
ELECTRONS AT INTERFACES
45
FIG.26. Contour plots. in planes parallel to and adjacent to the interface, of the parallel bonding interface states P2(a, b) and P, (c. d). The average charge density of each state is normalized to unity; successive contours are separated by 2.0 units. Less than 10%of the charge of each state lies in the other atomic layers combined. The states are described more fully in the text.
GaAs, i.e., the layers adjacent to the interface contain some contributions from the interface states, and the interface layer itself displays the structure arising from the interface states. The positions of the peaks in the excess contributions to the LDOS are easily associated with interface states in the PBS of Fig. 24. All of the interface states found lie outside the minimum semiconductor gap. The B2 and P2 states lie in the fundamental gap but below the band edge, i.e., the valance band maximum. Hence these states are not expected to influence transport properties. The absence of interface states in the minimum gap is consistent with experiment. Hopefully, techniques such as angular resolved photoemission can be used to detect the interface states predicted by the SCPM calculation. Returning to the question raised at the outset regarding band edge discontinuities, the SCPM calculations give estimates for AE, (and hence AEv using the measured band gap difference AEg = 0.75 eV). This information is obtained from the average potential difference WGe) - V(GaAs) = 0.25 eV (Fig. 22) plus a contribution from the electrostatic dipole across the interface, which is 5 0 . 1 eV. The dipole contribution d is evaluated by integrating the average charge density parallel
MARVIN L. COHEN
46
Ge-GaAs (110)Interface
4 ,
3-
I
I
I
I
I
I
I
Second Ge l a y e r
(a) ''Excess''
-
4
3
2 I
, I
0 41
I
I
I
I
I
1
I
I
Interface Layer
1
-I
2 1
0 41
I
I
I
I
I
I
I
I
First GaA s Layer
I
1
2 1
0 4
3
2 1
0 -12
-I0
-8
-6 -4 -2 ENERGY (eV)
0
2
FIG. 27. Local density of states for five layers surrounding the Ge-GaAs( 110) interface. The "excess" denotes the density of states localized at the layer as described in the text. The designations interface layer, first Ge layer, etc., are as pictured in Fig. 21. The positions of the interface states (Sl, s t , B1,B,) and resonances (Rl, R a )from which localized states arise are pictured. The local density of states in layers farther from the interface is essentially bulklike.
ELECTRONS AT INTERFACES
47
to the interface, d
= 4we2
( z - y ) p ( z ) dz
(21)
where Z is the position of the interface. (As an aside, if the charge in each layer were evaluated by integrating the charge density, each layer would be found to contain eight electrons to an accuracy of -0.3%.) Using the values for AEg, d, and the differences in the average potentials, by aligning the vb maxima, we get AE, = 0.35 eV and AE, = 0.40 eV, with an estimated error of -0.1 to -0.2 eV. Capacitance-voltage and current-voltage measurements give a range of values for AE, (Milnes and Feucht, 1972). A “most probable value” estimate was made by Pickett et al. (1978) of AE, = 0.2 2 0.15 eV. A recent measurement using angular resolved photoemission techniques gives a value of AEc = 0.50 eV (Perfetti et d.,1978). The effect of relaxation of the bonds at the interface on the band edge discontinuities was considered by Pickett and Cohen (1978b). A simple but overly large relaxation of the bonds at the interface was assumed and a supercell was constructed having this geometry. The relaxation consisted of a 20% increase in the distance between the germanium and GaAs planes in Fig. 21. Experimental estimates for a suitable relaxation are not available. The main aspect studied was the averaged self-consistent potentials shown in Fig. 28 for both the ideal and relaxed geometries. The large peak at the interface for the relaxed case arises from the weakening of the attractive ionic potential because of the relaxation. The feature of
FIG.28. The self-consistent potentials, averaged.paralle1 to the interface, for the relaxed and the ideal geometry. The small arrows denote the positions of atomic planes parallel to the interface. In each case one-half of the unit is pictured.
48
MARVIN L. COHEN
the averaged potential that influences the band edge discontinuities is the difference in the average potentials, A T = T(Ge) - V(GaAs) as discussed previously. The related geometry result is AVrelax= 0.13 + 0.1 eV, compared with AV,ldeal= 0.27 eV. The small difference between this latter value and the value given before and the large uncertainty in AVideal arise from the use of a 10 layer supercell in the Pickett and Cohen (1978b) calculation. The band-edge discontinuity shifts arising from relaxation are of the order of the change in AT, which is 0.14 ? 0.10 eV. It can be argued that the direction of this shift is correct since it is approaching the EAR value that would hold at large separation. The experimental value (Gobeli and Allen, 1965) for the difference of the electron affinities is 0.06 eV. The results for the ideal and relaxed geometries are AE, = 0.4 eV and 0.26 ? 0.10 eV, respectively. However, the relaxed geometry represents a gross overestimate of the relaxations expected to occur. The 20% relaxation was chosen to emphasize the effect to ascertain its magnitude; a reasonable estimate would only be a small fraction of this value. Hence, an estimate of the AE, change for relaxations in the expected range is quite small, i.e., a small fraction of 0.14 ? 0.10, and the conclusion of Pickett and Cohen (1978b) is that the effect is unimportant. This implies that the value of AE, for the ideal geometry is a firm result of the calculation, and that changes in AE, arising from relaxation cannot be used effectively to determine the value of the relaxation. Further discussion of Ge-GaAs( 110) is given as part of the discussion of results for other interfaces and in Section V. 2. AlAs-GaAs(ll0)
The matching of the lattice and ionicity for GaAs and AlAs results in a relatively small perturbation when an ideal interface of these materials is formed. Studies of the total charge density (Pickett et a!., 1978) show that there is virtually no disruption of the charge density at the interface of the kind found for Ge-GaAs(110) (Fig. 23). In addition, new types of bonds like those found for Ge-GaAs are not formed in the AlAs-GaAs system; the bonds are either Ga-As or AI-As bonds. In addition, the calculated charge transfer from the Ga-As to the AI-As bond is negligible (S0.002 electrons). The valence band structure of these two compounds is very similar, leading to an excellent matching of the bulk electronic states in forming the PBS (Fig. 29). The gaps line up well, and it is apparent from Fig. 29 that no interface states are found in any of the gaps. That is also evident in the LDOS plots in Fig. 30. The excess contributions are very small. Inte-
49
ELECTRONS AT INTERFACES AIAs-GaAs Proiected Band Structure
@ GaAs 2
0 -2 I
> Q I
>- -4
z w
I-4
t
I
I
I
ii
-X
-
-
-t
-10
-f2
M
X'
-
r
5; FIG.29. (1 10) projected band structures of bulk AlAs and GaAs from self-consistent calculations. No interface states are found to exist in either the fundamental gap (0 to 2 eV), the stomach gap (- 6 to - 2 eV), or the antisymmetric gap ( - 10 to - 7 eV), or below the valence bands (< I I eV).
gration of the charge in each layer yields eight electrons per layer and evaluation of the electrostatic dipole [Eq. (21)] gives a negligible value for this interface. The difference in the measured minimum gaps for these compounds is AEg = 0.65 eV; however, AlAs is an indirect material whereas GaAs is direct. The measured band-edge discontinuities were evaluated for the Al, Gal-, As -GaAs (x = 0.2) in which the minimum gap is direct. Hence, the direct gap at r is used to study the band-edge discontinuities and its = 1.45 eV. Recent value is linearly extrapolated to x + 1 giving AEiireC1 experiments (Dingle et al., 1974, 1975; Tsu et a / . , 1975) give AE, = (0.15 -e 0.03) AEgdirect.The value obtained for the AlAsGaAs(l10) calculation is AEv = 0.25 eV or AE, = 0.17 AEgdirect,which is in excellent agreement with the experimental results.
3. Ge-ZnSe(lI0) The geometry for the Ge-ZnSe(l10) interface is basically the same as for the Ge-GaAs(l10) interface shown in Fig. 21. The lattice mismatch is about twice as large in the Ge-ZnSe case, -0.2%, but this difference is
50
MARVIN L. COHEN
FIG.30. Local density of states for three layers surrounding the AIAs-GaAs( 110) interface. The localized states as described by the excess local density of states is extremely small for this interface. The density of states at layers farther from the interface than those shown is essentially bulklike.
again ignored. The results for Ge-ZnSe are also less accurate because of the reduced number of layers considered, ( 5 , 5). The difference in the average potentials yields an electrostic dipole of 0.25 2 0.1 eV and a valence band discontinuity of AEv = 2.0 ? 0.3 eV. No experimental values are available for AE, ; however, the result is close to the EAR value of 1.90 eV (Milnes and Feucht, 1972) and a value of 1.84 eV obtained by Frensley and Kroemer (1977). Harrison’s (1977) model gives a smaller value of 1.46 eV. An experimental measurement of this discontinuity would help to provide a test for the various theories. As is the case for Ge-GaAs(l10) no interface states are found in the minimum gap of Ge-ZnSe(ll0). Other interface states are found, but the results for the ZnSe case are not a simple extension of the GaAs interface. The PBS for Ge-ZnSe(l10) is given in Fig. 31 ; the matching of the bulk bands is clearly worse than in the two previous cases considered. A prominent interface state associated with an s-like selenium band splits off at - 16 eV. This state is similar to the S1arsenic-derived state in the Ge-GaAs interface. The selenium state is well separated from the bottom bulk bands, and for this reason, should be identifiable in an angular re-
51
ELECTRONS AT INTERFACES
-
Ge-ZnSe lnterfoce States
G e m Z n S e h S (110) Propcted Bond Structure 0 -2
-2 -r
-A
-6
P -a
LLI C
-10 -12 -lA
-16
-
r
X
M
x'
-
r
-c
k
FIG. 31. Spectrum of interface states in Ge-ZnSe(I10) in relation to the projected band structures of germanium and ZnSe.
solved photoemission measurements of this interface. A p state associated with the Ge-Zn bond, analogous to the Ge-GaAs B, state, appears at -0.8 eV in the fundamental gap. The charge densities for the Ge-Zn and selenium s-like state are given in Fig. 32. Before discussing the above interface states and other resonances found, it should be emphasized that the ( 5 , 5 ) geometry does limit the accuracy of the analysis. For example, the bulk bands given in the PBS of Fig. 31 are uncertain to -0.3 eV. This can blur the distinction between the identification of an interface state as a resonance or as a true interface state. The P resonance shown in Fig. 31 represents a in-Se bonding state parallel to the interface; its charge density is given in Fig. 32. Since the P resonance lies close to the band edge and considering the localization of its charge, this state would probably be a true interface state for a single isolated interface. The LDOS in Fig. 33 illustrates the energy and layer locations for the interface states. The s-like selenium state and the Zn-Se P state are located on the ZnSe side of the interface, while the Ge-Zn interface bond state is located in the interface layer. There are also resonant states on the
52
MARVIN L. COHEN
(ZnSe parallel bond)
FIG.32. Contour plots of the charge density of the interface states shown in Fig. 31. (a) and (b) are plotted in planes perpendicular to the interface, while (c) is plotted in the ZnSe atomic layer adjacent to the interface. At least 90% of the charge of each of these states is confined to the plane that is shown, and the charge of each state is normalized to unity in the unit cell.
germanium side, which decay rapidly on the ZnSe side in a fashion similar to the MIGS in the s-m case. In analogy with the Ge-GaAs(l10) interface a zinc s-like state and a Ge-Se bonding state are expected for the Ge-ZnSe(ll0) case; however, these states are not found. It is likely that these anticipated states are reduced to weak resonances because of the poor overlap of the “stomach” gaps in Ge-ZnSe. As in the Ge-GaAs system, the Ge-ZnSe interface contains bonds that have no bulk counterparts. In the bulk, zinc, germanium, and selenium contribute 0.50, 1.00, and 1.50 electrons to each of their four bonds in the tetrahedral structure. Without charge transfer, Ge-Zn and Ge-Se bonds would have 1.50 and 2.50 electrons. Integration of the calculated charge density gives 1.54 and 2.46 electrons; hence 0.04 electrons are transferred from the Ge-Se bond to the Ge-Zn bond. 4. GaAs -ZnSe(IIO) In the previous studies, interface states were found for the GeGaAs(l10) and the Ge-ZnSe(ll0) interfaces, while no interface states were found for the AlAs-GaAs(l10) system. Since GaAs and AlAs have similar ionicities and the same crystal structure, the perturbation felt by the electronic structure because of the formation of an interface is rather small. In both the Ge -GaAs and Ge-ZnSe systems the ionicities and the crystal symmetries change in going from one member of the heterojunction to the other. Hence, it is not clear on the basis of the above calcula-
53
ELECTRONS AT INTERFACES Ge-ZnSe (1101 Interface
-16
-14
-12
-10
-8
G e Atomic Plane #
2
G e Atomic Plane #
1
-6
-4
-2
0
2
Energy ( e V )
FIG.33. Local density of states near and at the Ge-ZnSe( 110) interface. The "excess" denotes localized states. The notation and interpretation is given in the text.
tions whether interface states arise from changes in ionicity or changes in crystal symmetry or both. Since the junctions are assumed ideal and the lattice mismatch is small, symmetry and ionicity changes appear to be the only likely candidates for causing interface states. (The lining up of the bulk structures in the PBS is also important.) The GaAs-ZnSe system is therefore useful to study to separate the above effects because the ionicity change is large compared to the AlAs-GaAs case, and the symmetry does not change. An additional benefit of studying this interface is that comparisons of the three interfaces Ge-GaAs, Ge-ZnSe, and GaAs-ZnSe can illustrate trends in interface behavior (Ihm and Cohen, 1979a).
MARVIN L. COHEN
54
The geometry of the GaAs-ZnSe( 110) interface is given in Fig. 34, where the atomic positions and the Zn-As and Ga-Se bonds across the interface are shown. The supercell used is a (7, 7) layer structure containing 28 atoms, Charge density studies confirm that the major changes in the charge configuration occur in or very near the interface as is the case for the other interfaces studied (e.g., Fig. 23). Integration of the charge yields 2.23 and 1.78 electrons for the Ga-Se and Zn-As bonds, compared to the simple chemical valence estimates of 2.25 and 1.75 electrons, respectively. The interface state spectrum is shown in the PBS of Fig. 35. Unlike the Ge-GaAs and Ge-ZnSe cases, no interface states are found in the fundamental gap between the top valence and bottom conduction bands. This result, which may not be valid if the assumption of an ideal abrupt interface is not made, can be understood in terms of bad-gap line-up and ionicity change across the interface. The valence band discontinuities AE, for Ge-GaAs, Ge-ZnSe, and GaAs-ZnSe are -0.4, -2.1, and -2.1 eV, respectively. Hence, the GaAs-ZnSe heterojunction has approximately the same band-gap line-up as Ge-ZnSe for the upper valence bands as is shown in Figs. 31 and 35. However, the ionicity change for GaAs-ZnSe is (110) &As-ZnSe OGo
.As
interface
@Zn
@Se
FIG.34. Atomic positions near the GaAs-ZnSe( 110) interface. Bonds are denoted by heavy solid lines except bonds across the interface, which are shown as heavy dashed lines. The chains ABAB and CDCD are the two independent bonding chains perpendicular to the interface, containing the Ga-Se and As-Zn bonds, respectively. z is the direction perpendicular to the interface.
55
ELECTRONS AT INTERFACES
-
GaAs-ZnSe interface states
(110) Proiected band structure ZnSe-
G a A s m 2
0 -2
-4
-%
-6
z -8
e C UJ
-10 -12 -14 -16
- 18
r
P
-
-k M
Ti'
I1
FIG.35. Interface states of GaAs-ZnSe( 110) relative to the projected band structures of bulk GaAs and ZnSe from self-consistent calculations. The dispersion of the interface states is denoted by heavy solid lines; heavy dashed lines indicate interface resonances.
approximately the same as Ge-GaAs and hence about half of the GeZnSe value. For Ge-GaAs, the disruption in the bonds at the interface is sufficient to push interface states into the gap. However, AE, is only 0.4 for this case and therefore the energy shift required to split states off into the gap is small. It is larger for Ge-ZnSe, but in this case the ionicity change is larger (and unlike GaAs-ZnSe there is also a symmetry change); hence the disruption in Ge-ZnSe appears to be strong enough to cause interface states to lie in the fundamental gap. For GaAs-ZnSe, the effects of the potential changes at the interface are not sufficient (within the accuracy of the calculation for the band line-ups) to push the Zn-As and a bond p-like state up into the fundamental gap. This state at Ga-As parallel bond state at x'remain below the vb maximum as resonances and no interface states are found in the fundamental gap between the top valence and bottom conduction band. While no interface states are found in the fundamental gap region, true interface states are found in other regions of the PBS (Fig. 35). An s-like
x
56
MARVIN L. COHEN
selenium state splits off the bottom of the valence band; this state is similar to the s-like selenium state found in the Ge-ZnSe calculation (Fig. 32). A Ga-Se p-like state exists in the “ionic” gap of both zincblende semiconductors. This gap, which is shown in the PBS of Fig. 35, is a feature peculiar to zincblende-zincblende interfaces and also exists for AlAsGaAs. For the germanium heterojunctions, the germanium states fill this region, leaving only a partial gap referred to as the “lower gap” (Figs. 24 and 31). Since this gap in the GaAs-ZnSe PBS is empty except for the Ga-Se p-like interface state, the interface state should be resolvable above the background using angular resolved photoemission techniques. The same should be true of the low-lying selenium s-like state, which splits off from the lowest valence band. Interface states are also found in the stomach gap near the X andX‘ points of the BZ (Fig. 35). If the band line-up is not precisely as shown, these states could move into the continuum of bulk states and become resonances. The band discontinuities calculated for GaAs-ZnSe are in disagreement with the EAR. The electron affinity values (Milnes and Feucht, 1972) give AE, = 1.29 eV, while the SCPM calculation reports a value of AEv = 2.1 eV. D . Further Results and Discussion It is clear on the basis of the GaAs-ZnSe(l10) calculation that even if the smaller number of layers used in this calculation leads to some inaccuracies in band line-up, etc., the question of the existence of interface states for this system has been answered. Interface states do exist and even though the crystal symmetry of the two components is the same (zincblende), the ionicity or differences in the potentials are sufficient to cause enough of a disruption at the interface to bind localized interface states in this region. The existence of specific interface states is dependent on band line-up and the details of the calculation. What about the relation between symmetry changes across the interface and the existence of interface states? In the examples considered, when the symmetry of the two components of the heterojunction was different so was the ionicity. What is needed to answer the above question is an interface with no ionicity change, but a symmetry change. An example of such a system would be zincblende ZnS or ZnSe in contact with wurtzite ZnS or ZnSe. Another potential system for analysis would be zincblende GaAs and wurtzite ZnSe. A comparison with the zincblendezincblende case discussed above could give useful information; InP-CdSe is also a possibility. Any heterojunction composed of the same compound or element on both sides but with each in a different crystal
ELECTRONS AT INTERFACES
57
structure would be ideal to study if the calculations of the electronic properties are feasible. It is, of course, possible that either a symmetry or an ionicity change will yield interface states. If this is the case then an interesting study would be to determine whether specific types of states are induced by the two different changes. It is known that atomic coordination changes can cause significant density of states changes and even cause the formation of localized states. The introduction of five- and seven-membered “rings” into an otherwise six-membered diamond-type lattice can upset the normal phase relationships and introduce new states (Joannopoulos and Cohen, 1976). A change of symmetry in crossing an interface can introduce an “effective disorder,” independent of the potential change, and hence the symmetry change might induce the formation of localized states in the gaps of the bulk structure. In the results discussed, band edge discontinuities calculated with the SCPM were found to be in good agreement with measured values for those cases where they are available, and the interface dipole found for all the calculations was quite small, -0.1 eV. The GaAs-ZnSe results are likely to be the least reliable because of the limitation in supercell size and convergence uncertainties in the energy level determinations. The band structure line-up could be shifted, which would lead to major changes. However, the Ge-GaAs and AIAs-GaAs calculations represent a much better test since convergence and cell size are in the desirable range. An interesting case for further study of band edge discontinuities would be heterojunctions composed of InAs and GaSb (Sai-Halaszer al., 1977; Ihm et a l . , 1979). These systems are expected to have the conduction band minimum of one semiconductor below the valence band maximum of the other. Superlattice solutions of this system are also interesting as the size of the repeat distance can influence its semiconducting or metallic behavior. One interesting feature of the calculations discussed is that the calculated charge in the bonds across the interface compared well with a simple valence model for the elements involved. Using the valence values Zn(2), Ga(3), Ge(4), A(5), Se(6), and Al(3) and assuming tetrahedral coordination, i.e., dividing the valences by 4, it is possible to estimate the number of electrons each element contributes to an interface bond. The SCPM calculated values were consistent with these estimates, illustrating that atomic characteristics can be used to predict bond properties and that there is no need at present to compute bulk properties to understand the gross features of the bond charges. Since the interface bonds generally have no bulk counterparts, there is not a great deal of information about their characteristics. The charge transfer between these bonds suggests
58
MARVIN L. COHEN
some small unbalanced forces, which can cause relaxation of some of the bonds at the interface. There is at present little experimental information on the interfacial morphology; hopefully future experimental and theoretical studies will help determine the detailed characteristics of the interface bonds. This should, in turn, allow even more realistic interface calculations of the electronic properties. V. SUMMARY A N D GENERAL DISCUSSION This chapter has been mainly a review of theoretical progress, and therefore a great deal of important and ingenious experimental measurements have been ignored. Only data bearing on a particular issue relevant to the theoretical results have been described. Even the discussion of the theory has not been global in any sense. Self-consistent pseudopotential results have been stressed, and other methods involving tight-binding or LCAO approximations, Bethe-lattice approaches, Green’s functions, and various other methods have only been mentioned if discussed at all. Despite the paucity of experimental results given, the reader should still have a sense of the large amount of recent activity in the study of interfaces and the very effective interplay between theory and experiment. The emergence of reliable experimental techniques to produce reproducible data has in a sense been matched by the development of theoretical approaches that can describe the properties of real physical systems. Some specific general comments and suggestions for future research are in order for this summary section. The description of clean surfaces given here was far from being an exhaustive survey. The purpose of the discussion on clean surfaces was to present a background for the description of the interface research that followed, and hence in no way reflected the vast amount of effort being applied in this area. The interdependence of the study of clean surfaces and solid-solid interfaces is large, and it is important for researchers in these areas to be aware of developments in both areas. The current calculations on clean surfaces have focused on the properties of surface states. These states have been observed experimentally, and their properties have been studied for a number of surfaces. This is in contrast to the s-s interface, where the predicted interface states described in detail here have not yet been observed. The situation for s-m interfaces is somewhere between the above two cases. Observations of Fermi level pinning, photoemission studies, etc., all indicate the existence of metal “associated” interface states. The question of whether these states are identical to the Louie-Cohen metal-induced gap states is not completely answered yet, but results thus far indicate consistency between the experimental and theoretical pictures.
ELECTRONS AT INTERFACES
59
In most of the descriptions, ideal surfaces and interfaces have been assumed. For Si(111). the (2 x 1) reconstruction of the clean surface greatly affects the physical properties of the surface. For example, the ideal surface is metallic, whereas the reconstructed surface is semiconducting. Relaxation of the surface atoms in GaAs(ll0) causes empty surface states, which are located in the gap region for the ideal case, to be removed and relocated in the conduction band region in agreement with experiment. Hence, the geometry of-the surface can be critical to a correct description of the electronic properties. At present, LEED studies are necessary for geometry determination. In the future, hopefully calculations will determine geometry through a minimization of the total energy. Such calculations would then be truly self-consistent in that the geometry would be used to compute the electronic properties and then the electronic properties in turn would determine the geometry. Some progress in total energy calculations has recently been made using tight-binding (Chadi, 1978) and self-consistent pseudopotential (Ihm and Cohen, 1979b) approaches. These calculations are not yet at the stage where predictions of surface structure are possible for a wide class of materials, but they do represent a first step in calculating surface total energies and geometries. For s-m and s-s interfaces, all the geometries considered were taken to be ideal. Therefore, we have not yet even explored the effect of a (2 x 1) reconstruction on a Si-A1 Schottky barrier. Calculations of this type remain to be done. A relaxation of the Ge-GaAs( 110) interface distance was explored and the dependence of some Schottky barrier properties on the Si-AI distance was evaluated, but again these are only first steps. For s-m and s-s, as in the case of clean surfaces, it would be very enlightening to know how the electronic properties depend on the geometry. For the s-s polar interface, elementary considerations show that this interface should be metallic if relaxation or reconstruction is not considered. Thus, geometry is fundamental to the properties of these systems since they are observed to be semiconductors. Chernisorption and defects are areas that were only briefly touched on here, but these areas are related to the properties of real systems. Questions about the precise way metals are absorbed on semiconductors or how s-s interfaces are first formed and what impurities do to the properties of interfaces are all important to a realistic description of a physical interface. Experimental data in this area are becoming more precise and abundant. There are several obvious areas that need new answers and research. More data and analysis of the interface parameter S are necessary before predictive theories and rules about Schottky barrier behavior can be developed. It seems likely that this very active area will evolve consider-
60
MARVIN L. COHEN
ably, and an S theory may result or be shown to be irrelevant in the light of new data. Superlattice results and their relation to the isolated heterojunction will hopefully occupy experimental and theoretical researchers so that refined analysis of band discontinuities, interface states, and interface bonds can be done. Thus, the major call is for more detailed and precise data and more realistic theory that do not rely on experimental input. This is a standard wish in science, but as this review may have shown, this is precisely what has been occurring in this active area of solid-solid interface research. ACKNOWLEDGMENTS I would like to thank Professor W. Pong and the members of the Department of Physics at the University of Hawaii for their hospitality; much of this review was written there. The contributions to the research and comments on this manuscript made by my graduate student and postdoctoral collaborators at Berkeley are gratefully acknowledged. I would also like to acknowledge the support of the National Science Foundation (Grant DMR7822465), the Division of Materials Sciences, Office of Basic Energy Sciences, U.S.Department of Energy (Grant W-7405-ENG-48), and the Guggenheim Foundation.
REFERENCES Anderson, R. L. (1962). Solid-State Electron. 5 , 341. Appelbaum, J. A., and Hamann, D. R. (1974). f r o c . Int. Conf. Phys. Semicond., 12th. 1974 p. 675. Appelbaum, J. A., and Hamann, D. R. (1976). Rev. Mod. Phys. 48, 3. Baraff, G. A., Appelbaum, J. A., and Hamann, D. R. (1977). Phys. Rev. Lett. 38, 237. Bardeen, J. (1947). f h y s . Rev. 71, 717. Braun, F. (1874). Ann. Phys. (Leipzig) [2] 153, 556. Brillson, L. J. (1978). f h y s . Rev. Lett. 40, 260. Caruthers, E.,and Lin-Chung, P. J. (1978). Phys. Rev. E 17, 2705. Chadi, D. J. (1978). f h y s . Rev. Lett. 41, 1062. Chadi, D. J., and Cohen, M. L . (1973). f h y s . Rev. B 7 , 5847. Chang, L. L., Esaki, L., Howard, W. E., Ludeke, R., and Schul, G. (1973). J. Vac. Sci. Techno/. 10, 655. Chelikowsky. J. R. (1977a). f h y s . Rev. E 15, 3236. Chelikowsky, J. R. (1977b). f h y s . Rev. E 16, 3618. Chelikowsky, J. R., and Cohen, M. L. (1974). Phys. Rev. E 10, 5095. Chelikowsky, J. R., and Cohen, M. L. (1976a). Phys. Rev. E 13,826 Chelikowsky, J. R., and Cohen, M. L. (1976b). Phys. Rev. E 14, 556. Chelikowsky, J. R., and Cohen, M. L. (1979). Solid State Commun. 29,267; Erratum, Solid Stute Commtm. 30, 819 (1979). Chelikowsky, J. R., Louie, S. G., and Cohen, M. L. (1976). So/id Stute Commtm. 20.641. Cohen, M. L., and Heine, V. (1970). Solid Stute f h y s . 24, 37. Cohen, M. L., Schliiter, M., Chelikowsky, J. R., and Louie, S. G. (1975). Phys. Rev. B 12, 5575. Cowley, A. M., and Sze, S . M. (1966). J . Appl. Phys. 36, 3212.
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Davydov. B. (1939). J. Phys. (Moscow-)1, 167. Dingle. R., Wiegmann, W., and Henry, C. H. (1974). Phys. Reif. L e f t . 33, 827. Dingle, R., Gossard, A. C., and Wiegmann, W. (1975). Phys. Re\,. Len. 34, 1327. Dobrzynski, L., Cunningham, S. L., and Weinberg, W. H. (1976). Surf. Sci. 61, 550. Eastman, D. E., and Freeouf, J. L. (1974). Phys. ReV. Lett. 33, 1601. Eastman, D. E., and Grobman, W. D. (1972). Phys. Rev. L e f t . 28, 1378. Esaki, L.,and Tsu, R. (1970). IBM J . Res. &is. 14, 61. Falicov, L. M., and Yndurain. F. (1975). J. Phys. C 8, 147. Frensley, W. R. (1976). Ph.D. Thesis, University of Colorado, Boulder (unpublished). Frensley, W. R.. and Kroemer, H. (1976). J . Virc. Sci. Technol. 14, 810. Frensley, W.R., and Kroemer, H. (1977). Phys. Rev. 16, 2642. Gobeli, G. W.. and Allen, F. G. (1965). Phys. Rev. 137, A245. Gossard, A. C., Petroff, P. M., Wiegmann, W.,Dingle, R., and Savage, A. (1976).Appl. Phys. Letl. 29, 323. Grant, R. W., Waldrop, J. R., and Kraut, E. A. (1978). Phys. Rev. Lett. 40,656. Gregory, P. E., Spicer, W. E., and Harrison, W. (1974). A p p l . Phys. L e u . 25, 51 1. Harrison. W. A. (1976). Phys. Rev. L e f t . 37, 312. Harrison, W. A. (1977). J. Vac. Sci. Technol. 14, 1016. Heine, V. (1965). Phys. Rev. 138, A1689. Herman, F., and Kasowski, R. V., (1978). Phys. Re\,. B17, 672. Ihm, J., and Cohen, M. L. (1979a). Phys. Rev. BU), 729. Ihm, J., and Cohen, M. L. (1979b). Solid Srute Commun. 29, 711. Ihm, J., Louie, S. G., and Cohen, M. L. (1978a). Phys. Rev. Letf. 40, 1208. Ihm. J., Louie, S. G., and Cohen, M. L. (1978b). Phys. Rev. B 17, 769. Ihm, J., Louie, S. G., and Cohen, M. L. (1978~).Phys. Rev. B 18,4172. Ihm, J., Lam, P. K., and Cohen, M. L . (1979). Phys. Rev. B20, 4120. Inkson, J. C. (1974). J. Vac. Sci. Technol. 11, 943. Joannopoulos, J. D., and Cohen, M. L. (1976). Solid State Phys. 31, 71. Kittel, C. (1976). “Introduction to Solid State Physics.” Wiley, New York. Kroemer, H. (1975). Crir. Rev. Solid State Sci. 5, 555. Kurtin, S.,McGill, T. C., and Mead, C. A. (1969). Phys. Rev. Lett. 22, 1433. Louie. S. G . , and Cohen, M. L. (1975). Phys. Rev. Lefr. 35, 866. Louie, S. G., and Cohen, M. L. (1976). Phys. Rev. B 13, 2461. Louie, S.G . , Chelikowsky, J. R., and Cohen, M. L . (1976). J. Vac. Sci. Technol. 13, 790. Louie, S. G., Chelikowsky, J. R., and Cohen, M. L . (1977). Phys. Rev. 15, 2154. Louis, E . (1977). Solid State Commun. 24, 849. .Louis, E., Yndurain, F., and Flores, F. (1976). Phys. Rev. B 13, 4408. Mead, C. A., and McGill, T. C. (1976). Phys. Lett. A 58, 249. Mele, E. J., and Joannopoulos, J. D. (1978). Phys. Rev. B 17, 1528. Milnes, A. G., and Feucht, D. L. (1972). ”Heterojunctions and Metal-Semiconductor Junctions.” Academic Press, New York. Mott, N. F. (1939). Proc. R . Soc. London. Ser. A 171, 27. Nethercot, A. H. (1974). Phys. Rev. Lett. 33, 1088. Pandey, K. C., and Phillips, J. C. (1974). Phys. Rev. Leu. 32, 1433. Pauling, L. (1960). “The Nature of the Chemical Bond,” 3rd ed. Cornell Univ. Press, Ithaca, New York. Perfetti, p., Denley, D., Mills, K. A., and Shirley, D. A. (1978). Appl. Phys. L e u . 33, 667. Petroff, P. M. (1977). J. Vac. Sci. Technnl. 14, 973. Phillips, J. C. (1973). “Bonds and Bands in Semiconductors.” Academic Press, New York. Phillips, J. C. (1974). J . Vac. Sci. Technol. 11, 947. Phillips, J. C., and Kleinman, L. (1959). Phys. Rev. 116, 287.
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Pickett, W. E., and Cohen, M. L. (1978a). Phys. Rev. B 18, 939. Pickett, W. E., and Cohen, M. L. (1978b). Solid State Commun. 25, 225. Pickett, W. E., and Cohen, M. L. (1978~). J . Vac. Sci. Techno/. 15, 1437. Pickett, W. E., Louie, S. G . . and Cohen, M. L. (1977). Phys. Rev. Lett. 39, 109. Pickett, W. E., Louie, S. G., and Cohen, M. L. (1978). Phys. Rev. B 15,815. Rowe, J. E., Christman, S. B., and Margaritondo, G. (1975). Phys. Rev. Left. 35, 1471. Sai-Halasz, G. A., Tsu, R., and Esaki, L. (1977). Appl. Phys. Lett. 30, 651. Schockley, W. (1939). Phys. Rev. 56, 317. Schottky, W. (1939). Z. Phys. 113, 367. Schliiter, M. (1978). Phys. Rev. B 17, 5044. Schliiter, M.. Chelikowsky, J. R., Louie, S. G., and Cohen, M. L. (1975).Phys. Rev. B 12, 4200. Schulman, J. N., and McGill, T. C. (1977). Phys. Rev. Lett. 39, 1680. Sharma, B. L., and Purohit, R. K. (1974). “Semiconductor Heterojunctions.” Pergamon, Oxford. Shay, J. L., Wagner, S., and Phillips, J. C. (1976). Appl. Phys. Letr. 28, 31. Sze, S. M. (1969). “Physics of Semiconductor Devices.” Wiley (Interscience), New York. Tamm, I. (1932). Phys. Z. Sowetunion 1, 733. Tejedor, C., and Flores, F. (1978). J. Phys. C 11, L19. Tejedor, C., Flores, F., and Louis, E. (1977). J . Phys. C 10, 2163. Thanailakis, A. (1975). J . Phys. C 8, 655. Tsu, R., Chang, L. L., Sai-Halasz, G. A., and Esaki, L. (1975). Phys. Rev. Lett. 34, 1509. Wagner, L. F., and Spicer, W. E. (1972). Phys. Rev. Left. 28, 1381. Walter, J. P., and Cohen, M. L. (1970). Phys. Rev. B 2, 1821. Zhang, H. I., and Schliiter, M.(1978). Phys. Rev. B 15, 1923.
ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS. VOL. 51
Beam Waveguides and Guided Propagation L. RONCHI
AND
A. M. SCHEGGI
Istituto di Ricerca sulk Onde Elettromagnetiche C . N . R . Florence. Italy
I. Introduction .......................................................... B. The Source
64
.....
111. Some Typical Longitudinal Structures ..... IV. Metallic Waveguides ..................... V. Dielectric Rods and Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Homogeneous Medium, Circular Cross Section ... B. Homogeneous Medium, Noncircular Cross Section .................... C. Inhomogeneous Medium . . VI. Two-Dimensional Waveguides s ......................... VII. Two-Dimensional Dielectric St A. Planar Structure with Homogeneous Medium ......................... ............... B. Planar Structures with Graded-Index VIII. Wave Guiding by Transverse Structures A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Reiterative Wave Beams ........................................ C. The Reiterative Beams as Eigenfunctions of an Integral Equation D. The Dielectric-Frame Beam Waveguide .............................. E. Propagation in an Open-Beam Waveguide . F. Propagation in an Open-Bea
A. Introduction
.......................................................
80
88 91
97 104
., ,.
B. Theory of Modal Propagation in a Two-Dimensional Graded-Index Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Propagation in a Parabolic Two-Dimensional Medium . . . D. Guided Propagation in a Nonparabolic (Two-Dimensional) Medium . . . . . . E. Guided Beams in a Three-Dimensional Graded-Index Medium . . . . . . ee-Dimensional Medium .......................... ......................................... X. Conclusion d ..................... Appendix 11. The WKB Approximation Applied to Modal ................................................. in a Slab . . . ................................................. References . 63
108 110
I10 112 113 119 122 123 127 128
I30 133
Copyright @ 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0- 12-01465 1-7
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I. INTRODUCTION From the point of view of propagation, electromagnetic waves can be divided into two classes: guided and free-space waves. The important point of distinction is that a guided wave is substantially confined within or along the guiding structure despite any change of direction, whereas a free space wave spreads out in many directions and its course cannot be changed except by introduction of special devices (such as mirrors) in the path of the beam. It is helpful to recognize that electromagnetic waves can be guided mainly by two distinct processes: reflection between highly conducting-reflecting walls or total reflection at the interface between two dielectrics. For instance, a coaxial cable and a waveguide are two particular embodiments of the first class, whereas a dielectric rod guide is a member of the second class. Another more particular guiding process is that of directing a beam by an iterative procedure using a series of irises or alternatively lenses (“phase transformers”) spaced along the required route. The criteria of the design of guiding systems are the same whether one deals with radio or optical frequencies. In principle all the methods of transmission could be used throughout the whole electromagnetic spectrum and the choice of the method is almost entirely determined by the availability of suitable materials or techniques. In general the difficulties of transmission increase with frequency : mechanical tolerances on the guiding structure get smaller, reflection and absorption losses increase, while stability of the source of oscillations becomes more difficult. Communication systems are based both on free-space radiation and on guided propagation. For the first type, free-space microwave links are used and satellite communications working in the same part of the spectrum have been developed with a tendency at present to higher-frequency regions (> 10 GHz). The invention of the laser in 1960 offered new, very interesting possibilities due to the immense information capacities of optical frequency radiation. However, light transmission in the open atmosphere was recognized as unreliable due to the variable weather conditions and atmospheric turbulence. A controlled atmosphere was proposed using pipes with periodic refocusing of the beam, but practical considerations precluded system utilization (See Section VIII). To obviate the complications of free space propagation and at the same time to satisfy the rapidly growing demand for telecommunication circuits capable of operating over wider-frequency bands, guided propagation appeared of relevant importance. Moreover, screened channels can be duplicated as traffic grows and can be of great help in resolving future problems of interference between long-distance links. Although
BEAM WAVEGUIDES AND GUIDED PROPAGATION
65
coaxial cables for long-distance transmission have been widely and successfully used, the waveguides may offer an actractive alternative due to the wider bandwidths allowed. The concept of directing a traveling wave along a chosen path by introducing boundary surfaces is due to Lord Rayleigh (1897). However, the practical development and detailed understanding of microwaves trapped inside closed pipes were achieved only in the early 1930s by Schelkunoff (1934) and Southworth (1936). During the second world war years, much effort was put in the use of microwaves for radar, and after the war the great knowledge acquired in the field was turned toward civil applications. Long-distance telecommunications by means of the so-called low-loss circular TEIo waveguide fist suggested by Barlow (1947) was one of them. The basic problems were to solve the difficulty of preserving the desired mode in a highly overmoded waveguide suitable for practical applications and to find means of utilizing effectively the wide spectrum available, providing for the necessary terminal equipments, regenerators, etc. A great effort was made in various countries for more than 20 years to establish a practical low-loss waveguide system for long-distance communications. In 1960 a new class of overmoded guiding structures, the “beam wave-guides,’’ was proposed (Goubau and Schwering, 1961; Christian and Goubau, 1%1), where the guidance is performed by reconstruction of the beam at periodic intervals using a series of lenses. These waveguides may be suitable for large capacity and long-distance communication systems either at millimeter or at optical frequencies. However, the structure is extremely sensitive to slight movements of the supports and to turbulence of the medium between adjacent elements, thus requiring for a good performance the introduction of servomechanisms on the elements for automatic correction (Goubau, 1968). Another type of optical waveguides that in the near future will most likely replace and expand the existing telecommunication systems is represented by optical fibers. Guidance is achieved by total reflection at two dielectric interfaces, or by continuous refraction in a transverse gradually varying medium. Optical fibers had been investigated for image transport application since 1950 (Kapany, 1967). The first communication expert who proposed glass fibers for long-distance communications was Kao (Kao and Davies, 1968). At that time, typical fiber losses were above 1000 db/km. but Kao suggested that purer materials should permit much lower losses. The realization of 20 db/km fibers by Corning in 1970 opened a wide horizon for optical communications by providing for a stable, flexible, low-loss transmission medium. Thereafter, progress in science and technology of fiber transmission developed along a broad front, ranging
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L. RONCHI AND A. M. SCHEGGI
from system components (fibers, cables, sources, detectors) to prototype and in situ system demonstrations (Miller et al., 1973; Giallorenzi, 1978). As is well known, fibers find interesting applications in many other fields, such as short and medium-distance systems for visual and data transmission (Amemiya et al., 1978; Carrat and Tache, 1978; Checcacci, 1978; Giertz et al., 1978; Ishio et al., 1978; Rawson et al., 1978) or scientific uses as passive sensors of strain, pressure, and temperature (Culshaw et al., 1978) or other exotic applications as suggested by Ostrowsky at the 1978 meeting in Cargese (Corsica). From the preceding remarks, it appears evident how huge is the subject of the electromagnetic wave-guided propagation and hence how treating the general problem is unavoidable, making a certain selection of subjects. This may reflect some preference of the authors due to personal experience or to the prevailing interest in some subjects with respect to others. The purpose of this chapter is to give a sufficiently wide picture on guided propagation that can provide also an idea of the direction in which the present research work is going. After a schematization of the general problem, methods of analytical approach are outlined which can then be applied for the solution of particular problems encountered when different guiding structures are considered. Continuous longitudinal (planar and cylindrical) structures are examined, suitable for different frequency ranges as well as transverse periodical structures. Indefinite guiding media, which can be of interest for a first approach and understanding of the behavior of practical structures like optical fibers, dye laser, etc., are also examined. 11. THEORETICAL BACKGROUND The general problem of guided-wave propagation in a longitudinal structure consists in finding the solution of the inhomogeneous Maxwell equations, with the source properly described, which satisfies the boundary conditions imposed by the presence and by the features of the structure. Then one has to find the “sources” generating the fields, which suffer the least spatial spreading and remain concentrated inside and/or in the vicinity of the structure as much as possible up to its end. Due to the finite dimensions of the structure, the above problem is generally a very complicated diffraction problem. However, a procedure may be followed by which the various difficulties are approached in a sequence, passing from the easiest to the hardest ones, rather than as a whole. In this procedure, the approximations may be introduced as late as possible and their effects may be evaluated.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
67
A . The Modes of the Infinitely Extended Structure The first step consists in taking into consideration the unlimited structure, a portion of which constitutes the real structure under study, and then, after the choice of the most convenient system of (curvilinear) coordinates, in solving the homogeneous (source-free) Maxwell equations in that system of coordinates. The general solution of Maxwell’s equations turns out to depend on a number of parameters, which are then to be chosen so as to satisfy the boundary conditions, including generally the radiation condition. From the point of view of the radiation condition, two different classes of problems may be individuated. If one has in mind problems with the source inside the structure, the radiation condition is expressed by imposing that the outer field propagates by going away from the structure, or, if the phase propagates along the structure, that the field attenuates outwards. In these cases, the solutions of the homogeneous Maxwell’s equations satisfying the boundary conditions are termed modes of propagation of the structure and constitute, as will be seen later, a discrete spectrum. If on the contrary, the problem has an external source, the external field turns out to be constituted by a wave approaching the structure and by a wave going away from it (reflected wave), so that there is an additional parameter in the problem, namely, the reflection coefficient. Consequently, the solutions satisfying the boundary conditions constitute a continuous spectrum. This first step of the procedure may be exactly treated when a system of orthogonal curvilinear coordinates 6 , q , 5 can be found such that the structure corresponds to a coordinate surface, and the Maxwell equations admit separate-variable solutions [it may be shown that this occurs in 11 systems of coordinates (Stratton, 1941)l. In all these cases, the boundary conditions may be imposed in the easiest way. A trivial example is a cylindrical rod, whose lateral surface, in a system of cylindrical coordinates p , 4, z, coaxial to the rod (Fig. 1) has p = const. A little less trivial is the case of a rectangular structure (Fig. 2), where one can choose the Cartesian reference system with the x and y axes normal to the structure wall. In this case, the wall can be divided into four sections such that either x or y remains constant over each of them. However, each section is not a coordinate surface, but only a strip of coordinate surface. Now, if the wall is or can be considered perfectly conducting, the above remark is irrelevant, since the space inside the structure is insulated from that outside. Hence, in evaluating the field inside, one can imagine the conducting wall to extend all over the corresponding coordinate planes. If, on the contrary, the wall cannot be considered as perfectly conducting, an exact solution cannot be found, and the problem can be solved only approximately.
68
L. RONCHI AND A . M. SCHEGGI
FIG. 1 . Cylindrical coordinate system.
Whenever it is possible, the vectorial treatment is reduced to a scalar treatment, either as an approximation (mainly at optical frequencies) or by introducing suitable potential vectors, such as the Hertz vector or the Fitzgerald vector, which are specified by their component in a single coordinate direction. Such a component satisfies a differential equation, derivable from Maxwell’s equations, which may be generally denoted as a wave equation. In the sequel we refer to monochromatic solutions of such a wave equation, satisfying the appropriate boundary conditions, and we deal with scalar functions of the type (1) u ( t , 7,5) = X(t)Y(?-/)Z(C) A consequence is that its distribution across a transverse surface normal to the longitudinal coordinate 5 changes only by a proportionality factor, depending on 5. The function Z(5) is a complex function, which plays a very important role in the problems we are dealing with. In most practical cases, the coordinate 5 can be so chosen to put Z(5) in the form
Z(5) = exp(iky0 (21 and the ratio between the field distributions across two successive surfaces 5 = 5’ and 5 = { ‘ I , with 5‘ < i”, can be written as
FIG.2. Rectangular coordinate system.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
69
where k = w(ro,)l/z denotes the free-space wavenumber. Accordingly, Re y accounts for the phase velocity in the 5 direction, whereas Im y describes the attenuation (or amplification) of the mode per wavelength. According to the sign of Re y , the fields are divided into progressive (Re y > 0) and regressive (Re y < 0) fields with respect to 5 [the time dependence exp(- i o t ) is understood]. The variable-separation procedure of solving the wave equation introduces in the analytical treatment three separation constants, one for each coordinate [the parameter y introduced in Eq. (3) is one of them], related to one another by a single equation (representing the condition for u ( f , 7, C) to satisfy the wave equation). In the cases of no field impinging onto the structure from outside, the imposition of the boundary conditions renders them a discrete (doubly (numerable) set. Accordingly, a mode and the separation constant associated with it are labeled by a pair of indexes n, m: unm((7
77,
5; w ) = Xn(6; wIYm(77;
0)
ex~[iky,~(w)51
(4)
The function Xn(()Y,,,(q)is termed the (n,m)th eigenfunction and ynmthe (n,m)th eigenvalue. They generally depend on w , and therefore on k , as indicated in Eq. (4). As a solution of the homogeneous wave equation, a mode is determined apart from a constant factor, which allows one to introduce normalization relations such as
where the asterisk denotes complex conjugates. When there is a field impinging onto the structure from outside, y assumes a continuous spectrum of values, and Eq(4) can be rewritten as
~ ( 6q,, 5 ; w ) = X ( g ; y.; w ) Y ( q ; y ; 0 ) exp(iky5)
(6)
The normalization condition takes the form
B . The Source The source is generally treated as boundary conditions across a surface 5 = const = t o ,where the impressed field d((, q , 5 0 , f) is assumed known. In this case, the source is also termed aperture.
L. RONCHI AND A. M. SCHEGGI
70
The fist step of the procedure consists in expanding ui in monchromatic Fourier components: u'(4, 77, to, t ) =
1
+m
2lr
A ( w ) u * ( t 77, , C0;
dw
(8)
--m
Then, if the source is internal to the structure, one has to write each component as a superposition of mode distributions X , ( t ; w ) Y m ( 9 ;w): A ( u ) u ' ( ~77, , (0;
OJ)
= A(w)
2n m
Cnm({o;
w)Xn((; w)J'm(?;
0)
(9)
By virtue of Eq. (9), the propagation of the field of the source, for example in the positive { direction, it easily treated by multiplying each term of the series of the propagation factor
z({) = exp[ikynmb){1
(Re y n m > 0)
(10)
In conclusion, the source establishes the complex amplitudes ; w) of the excited modes at the initial cross section { = {o. At any other section { > to,the various modes turn out to be present with a complex amplitude
A (W)C,,({~
Anm(w;
50, 5 )
= A(w)cnm([o; w)
exp[ikynm(o)({- {o)]
(11)
with Re ynm> 0, where A ( w ) is defmed by Eq. (8) and ~ ~ ~ W) ( by6 Eq. ~ ; (9). Accordingly, the field distribution v " ( f , 77, [, t ) at { is given by
When the source is external, the series are to be substituted for by integrals. In the sequel, we mainly refer to the modal propagation.
C . Dispersion and Attenuation
It appears from Eqs. (8) and (1 1) that if the initially launched field u' is not a single-mode monochromatic field, it deforms during the propagation because of the fact that the parameter y is a function of both w and the mode numbers n and m. For the sake of simplicity, two ideal limit cases can be considered: signals that are transmitted as a band of frequencies in
BEAM WAVEGUIDES AND GUIDED PROPAGATION
71
a single mode [say the ( n o ,rno)th mode], and images that are transmitted as a group of modes at a single (angular) frequency (o0). Consider briefly the signal transmission. Since yno,mo is a function of W , the signal after a length L is deformed with respect to the input signal. By working at frequencies sufficiently higher than the cutoff frequency w C , the distortion of the signal is mainly due to the dispersion of the structure, ( o )on 0 , rather than to the attenunamely, to the dependence of Re ynoBmo ation. Being mainly responsible for the distortion of the signals, the dispersion limits the band allowed by the structure. However, in principle, the dispersion can be compensated for by a suitable processing of the signal at the receiver (Lucky ec a/., (1968). For the dispersion changes (deterministically) the phase relationships of the various Fourier components of the signal, with respect to those of the input signal, but it does not introduce loss of information. This type of problem is treated in terms of the so-called group delay 7 g , defined as 79
odL ,
=-
c
dk
In the case of image transmission, the main role is played by the attenuation, namely, by Im ynm(wo), which is a function of n and m.It turns out that at sufficiently high values of n and m , say for n > TI and rn > 3, Im ynm(uo)is so large that the associated mode is practically nonexistent in the signal at the receiver even after moderate paths L. This means that the structure is able to transmit only the modes with n < Ti; m < iii, which constitutes a loss of information. Due to the "noise," the process aimed at extracting from the signal at the receiver information about the modes with n > Ti-and m > Z contained in the source (superresolution) is destined to require a very big effort with poor results. Dispersion is due both to the medium where the waves propagate, if its dielectric constant is a function of the frequency, and to the geometry of the structure. In fact, the boundary conditions are expressed by an equation containing o (or k). Generally speaking, such an equation contains the transverse dimensions of the structure measured in free-space wavelengths A. It turns out that the dispersion due to the structure is more important with respect to that due to the dielectric, the smaller are the dimensions of the structure. The attenuation of a mode is partly due to the losses by the Joule effect in the medium (if it is not perfectly dielectric) and in the walls of the structure (if they are not perfectly conducting), and partly to the radiation losses, which are constituted by radiation leaving the structure through the boundaries.
72
L. RONCHI AND A. M. SCHEGGI
D. Effects of the Finite Length of the Structure Consider now a real case where the waveguide is constituted either by a portion of unlimited structure, or by a number of portions of unlimited structures, possibly different from one another. Let us denote by toand tout the values oft; at the two ends of a section of waveguide, and assume the (n,m)th mode to be excited in it and to propagate, say, to the right. Such a (progressive) mode propagates in the structure with the modalitites summarized in Section B until it arrives at t; = tout,where it is partly reflected and partly radiated in free space or into the successive section of the guide. The reflection may be treated (Ragan, 1946; Wainstein, 1966) by introducing the “transformation coefficients” Rn,m;nt,,t, representing the ratio of the complex amplitude of the regressive n’,m’-th mode to the complex amplitude of the (n,m)th progressive mode, both evaluated at 6 = (out. The reflection coefficient R,, of a mode into itself is a particular case of transformation coefficient, R,, = Rn,,;,,, . Analogously, the transmission to the successive portion of guide, if any, can be treated in terms of a of which represents the ratio of transmission matrix, each term Tn,m;nt,ml the complex amplitude of the (n’,m’)th progressive mode for t; > toutto the complex amplitude of the (n,m)th progressive mode, evaluated at
5 = (out.
The calculation Of Rn,,;n,,m! and Tn,m;nt,mt is not very easy and depends on the features of the passage from one portion of waveguide to the other. The reflected field, propagating into the negative t; direction, arrives at 6 = t;,, , where it is partly radiated in free space or in the nearby portion of the waveguide, and partly transformed in a progressive field. At the open ends, currents may be originated that circulate along the external surface of the walls, and generate nonguided fields. However, they are generally minimized and their effects neglected. In these back and forth passages, the field may encounter and be scattered by the source, if it is materially present within the section of waveguide under consideration. In most practical cases, with the remarkable exception of the open optical resonators (Wainstein, 1963), the complications arising from the reflections at the discontinuities of the structure are overcomed by locally altering the structure so as to reduce as far as possible the modulus of the transformation and reflection coefficients. The purpose is that as much as possible of the energy and/or information supplied by the source at one end, and reaching the other end, be utilized by the receiver. This purpose is reached in the microwave technique by suitable impedance adaptors between two different sections and at the open ends (Ragan, 1946).
BEAM WAVEGUIDES AND GUIDED PROPAGATION
73
111. SOMETYPICALLONGITUDINAL STRUCTURES
The longitudinal structures of most common use may be divided into two groups: cylindrical and planar. Cylindrical structures include metallic pipes and dielectric rods. Metallic pipes are of very large use in the microwave technique, where they are mostly designed to operate in a single mode, the lowest order or fundamental one, for the coherent transmission of energy and signals. However, multimode cylindrical waveguides are also used and, in the field of single-mode waveguides, the advantages of operating with a particular high-order mode have been indicated (Barlow, 1947; Collin, 1960). Dielectric rods are used in the antenna field, for example, to realize directive feeders (Fradin, 1961), but their main application is in the field of optical communications, where optical fibers are used for both energy and signal and image transmission. Planar structures too find applications both in microwave and in optical band. At microwave frequencies, they have mostly metallic boundaries, and, if the boundary encloses a dielectric medium, it is generally homogeneous. Examples are the geodesic lenses (Toraldo di Francia, 1955a) and the microstrip lines (Collin, 1960; Guston, 1972). At optical frequencies, the applications of planar structures cover the field of integrated optics. Generally, only one boundary, if any, is metallic, and the dielectric may be homogeneous or present a gradient of the refractive index in the plane transverse to the direction of propagation. In the planar structures, the dependence on one transverse coordinate (that parallel to the boundaries, say 77) is of minor importance, and the wave propagation constitutes an essentially two-dimensional problem. Accordingly, the eigenvalues and the associated eigenfunctions are specified by a single mode number, say n .
IV. METALLIC WAVEGUIDES Metallic waveguides have been known for a long time. The theory of wave propagation in them is very well and extensively described in a large number of textbooks, especially regarding cylindrical waveguides with either circular or rectangular cross section (Stratton, 1941; Toraldo di Francia, 1953; Borgnis and Papas, 1958: Schelkunoff, 1963; Ram0 et uf., 1965; Jackson, 1975). Generally speaking, the electromagnetic field E,H is completely determined by its longitudinal components E,, H , . When there are no losses, neither in the homogeneous medium or in the walls, the various modes
74
L. RONCHI AND A. M. SCHEGGI
can be divided in TE (E, = 0) and TM (H,= 0) modes. Moreover, the modes are orthogonal to one another, which facilitates the treatment of energy propagation. When there are losses in the medium or in the walls, the modes are not exactly orthogonal to one another. However, if the losses are sufficiently low, namely, if the conductivity T, of the walls is high and the conductivity rdof the dielectric is low, the losses can be evaluated separately for each mode by a perturbative method. This consists in evaluating dispersion and losses by using the expressions of the fields valid in the ideal lossless case. Let us first determine the fields and propagation constant kv of a lossless structure. The value of 7 can be determined as follows. Let us write, with self-explanatory symbols,
and consider separately TE and TM modes. For TE modes (e, = 0) h, has to be determined by solving the wave equation
V2h, + P(1 - 7*)h, = 0
(15)
and by looking for the values of 7 that satisfy the boundary conditions. Then eT and hT are given by eT
=
hT
k(l =
iz - 7')
k(1
-
7
i,A grad h,
)
grad h,
where Z = ( P / C ) ~denotes '~ the impedance of the dielectric inside the medium. Analogously, for TM modes ( h , = 0), e , is a solution of the wave equation Vze,
+ P(1 - r2)e, = 0
(17) and the values of 7 are to be found by imposing the boundary conditions. Then + and hT are given by
.-
eT
=
' grad e, k(1 - 7 1 i
hT =
kZ(1 - T 2 )
(18)
i,A grad e,
BEAM WAVEGUIDES AND GUIDED PROPAGATION
75
Since both e , and h, are determined apart from an arbitrary constant factor, one can impose normalization conditions, which are generally chosen in the form
il
e T *e+ d Z =
I
//
or
I;
x
eT e? d Z = 1721
(19)
z
for TE and TM modes, respectively, and correspondingly,
L
L
where 2 indicates the cross section of the waveguide. We finally recall that a critical wavenumber k, is usually introduced as
7') (21) such that if k, < k , 7 is real, while if k, > k, 7 is purely imaginary. kZ, = k2(1
-
Let us now consider a lossy structure with the same configuration and size as the lossless one previously examined. It turns out that, for one and the same mode, the effects of the losses on Re y are generally of the second order and are usually neglected. Accordingly, the dispersion is almost the same as in the case of no losses. Re y = Re 7
(22)
As for the attenuation, the power dissipated in the dielectric and in the walls as well as that stored in the structure are evaluated by assuming for the field of each mode the expressions valid in an ideally lossless structure, namely, Eqs. (14)-(18). Thus, by evaluating the flux of the Poynting vector of the no-loss field through the walls, one arrives at the following expression of the power W, dissipated per unit length of waveguide:
W, = -2l R , f H * H*dl
(23)
where the integration is to be extended to the contour of a cross section of the waveguide, and R, denotes the intrinsic resistance of the wall: R, =
(~op/r,)l/~
(24)
Analogously, the power w e dissipated in the dielectric per unit length of waveguide can be written as w d
=$
rd
I// V
E. E* dV
(25)
L. RONCHI AND A. M. SCHEGGI
76
where V denotes the volume of a unit-length section of waveguide. Finally, the energy U stored in the same unit-length volume V is given by
u = u, + urn
(26)
with
V
V
When W, , w d , and U are evaluated, one can evaluate the quality factor Q defined as
Then, the attenuation of the mode turns out to be (Y
where
0, =
=
l o 2 Qve
k Im y = --
‘yc represents the energy velocity. Hence
Im y = 1/2Q7 Equation (29) holds when 3 is real. When 7 is purely imaginary, 7 = the attenuation is simply given by
(29)
il71,
Im Y = 171 and the losses play a role of minor importance. The above expressions can be written in explicit form when considering some specific structures, namely, rectangular and circular crosssection waveguides. For a rectangular waveguide with metallic walls the boundary conditions at x = 2 a and y = A b (Fig. 3) are
FIG.3. A rectangular waveguide.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
tu
ah,/ax = 0
at x
ahJay = 0
at y = + b
=
77
(30)
for TE modes, and e, = O
at x = * u ,
ae,/ay
=
0
at .r
= +-a
ae,/ax
=
0
at y
=
y = t b
(31)
+b
for TM modes. [For a better approximation, one could use the so-called Leontovich boundary conditions (Leontovich, 19441.1The eigenfunctions are
and therefore the eigenvalues for a lossless waveguide are given for both TE and TM modes by
with n and m integers. It can be noted that one cannot choose n = m = 0 , and therefore = Too = 1. In this case, one would have e, = 0 for TM modes and h, = 0 for TE modes, namely, the mode could be TEM, but one would have also eT = 0 and hT = 0, as appears by a passage to the limit in Eqs. (14) and (18). Moreover, TM modes do not exist with either n = 0 or m = 0, while TE modes may exist. As is well known, guided TEM modes may exist in structures whose walls are composed by two or more conductors, such as coaxial cables and bifilar lines (Toraldo di Francia, 1953; Collin, 1960; Jackson, 1975). As to the lossy waveguides, in a first approximation one has, for k > k,,
rnm
L. RONCHI AND A. M. SCHEGGI
78
and, by neglecting the attenuation in the medium, namely, by assuming rd= 0,
for TE and TM modes, respectively. Eigenfunctions and eigenvalues are well known also when the waveguide is constituted by a conducting cylinder with circular cross section (Fig. 4). The eigenfunctions are of the form
h,
= AJm(wr)e’fmm
e,
=
(TE)
A’Jm(wr)e+*mm
(35)
(TM)
where J m denotes the Bessel function of order m, A and A ’ are two constants to be chosen as to satisfy Eq. (19), and w = k(l -
y2) 1’ 2
The boundary conditions are written ah,/ar = 0,
at r = a
(TE)
ae,/a+ = h e , = 0,
at r = a
(TM)
(37)
where a denotes the radius of the waveguide. Accordingly, the eigenvalue,; of the lossless waveguide are given by -
Ynm =
[1 - (pnm/ka)21”2,
kc =
knm =
pnm/a
(38)
where pnmdenotes the nth root of J , in the case of TM modes, and the nth root of JA in the case of TE modes. For the lossy waveguides, assuming
FIG.4. A cylindrical waveguide.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
79
t
FREQUENCY
FIG.5. Attenuation vs. frequency in a cylindrical waveguide with metallic wall, for three low-order modes (the first index is the radial one).
r,
= 0, one finds with the help of Eqs. (23)-(29) and ( 3 9 ,
The behavior of the eigenvalues as a function of n and m when the size of the waveguide cross section, measured in A, is of the order of unity is well known, and justifies the choice of a A/2, b = A/2, for single-mode rectangular waveguides, or a = (plI/7r)A/2 [with pll = 1.84 (Abramowitz and Stegun, 1WS)]for single-mode TE,, circular waveguides. When a %- A, a large number of modes are allowed to propagate inside the waveguide, which implies distortion of the signal both due to the dispersion and to the attenuation. It is worth noting that the modes TEnoturn out to have attenuation decreasing with increasing frequency (Fig. 5 ) and radius (Fig. 6). The least attenuation pertains clearly to the TEIo mode (Collin, 1960; Van Bladel, 1964). In practical cases, the behavior of a multimode waveguide is disturbed and complicated by the fact that, even if one excites by a suitable source, 2 :
RADIUS ( a )
FIG.6. Attenuation vs. radius in a cylindrical waveguide with metallic wall, for three low-order modes (the first index is the radial one).
L. RONCHI AND A. M. SCHEGGI
80
the only lowest-attenuation mode, that mode excites all other modes at any alteration and imperfection of the structure, in particular of the walls. On the other hand, deviations of the walls from the ideal smooth shape considered above may be introduced to obtain particular effects. An example of this principle is constituted by the so-called corrugated waveguides or by the periodic slow-wave structures (Johnson, 1965; Ram0 et al., 1965;Bryant, 1969;Clarricoats, 1963;Clarricoats and Slinn, 1965).
V. DIELECTRIC RODS AND FIBERS The treatment of wave propagation in dielectric rods and (optical) fibers may be divided into two groups, according as the dielectric may be considered homogeneous or of the graded-index type. In turn, each group may be divided into two subgroups, according as the cross section is circular or of some other shape. A . Homogeneous Medium, Circular Cross Section
Let us denote by E the dielectric constant of the rod (core), by ee that of the medium (cladding) around the core, by a the radius of the cross section of the core. The main difference with the metallic pipes of circular cross section is that, in the case of rods, the modes of propagation cannot be separated in TE and TM modes, but are hybrid modes, the only exception being the modes whose longitudinal components do not depend on 4. As in Section IV, Eq. (33,we can introduce two functions h, and e, given by h, = AJ,(wr)e", e , = BJm(wr)efM (40)
for r
5
a, and
h,
=
A'Hgf(wer)efmm, e, = BfHg)(wer)e**
(41)
for r 2 a, where the z dependence exp(ikyvz) is omitted, u denoting the refractive index of the core, and
U,
indicating the refractive index of the cladding. In terms of h, and e , , the transverse components of the field are given
BEAM WAVEGUIDES AND GUIDED PROPAGATION
81
by the well-known expressions (see, for example, Marcuse, 1972)
Accordingly, for r
5
a , by substitution of Eq. (41), one has
and for r > a similar expressions, with w replaced by w e , E replaced by te,and J,(wr) by H g ) ( w e r ) ,while v y does not change. The boundary conditions require 4 , . em,h, , and h, to be continuous at r = a , which yields four (complex) homogeneous equations for the five parameters A , B, A ’ , B’, and y . Hence one cannot generally choose either A = 0 or B = 0, but in the case rn = 0, for which A,A’ and B,B’ appear in separate equations. Consequently, only TE,, and TM,, modes exist, all other modes being hybrid. From the continuity conditions at r = a one finds for y the well-known equation
where J , stands for J , ( w a ) and H$ for H g ) ( w e a ) . In the case rn = 0, Eq. (45) splits into the two equations
-we J O ’ - Hi”’ w Jo Hi1)’
When
IW,CIJ
for TE modes
is sufficiently large, so that H&l)(wer) --. ( 2 / 7 r w ~ ) ~x’ ’
L. RONCHI AND A. M. SCHEGGI
82
exp[i(w,r
- ~r/4)],Eq. (46)
can be written as
6 we Ji --Qe
~
i
(TM)
w JO
(47)
Each of these equations is a transcendental equation for y . A way to solve them is to prescribe y and to consider ve as unknown: in this way they can be treated as two algebraic equations for ve (Consortini et al., 1976). Such forms are particularly useful for a numerical or graphical determination of the eigenvalues yno. As an example, Fig. 7 shows a graphical solution of Eq. (47) for TE modes when w and y are real while we is purely imaginary: the function w a J o ( w a ) / J ~ ( w ais) plotted vs. wa (solid line). Dotted lines represent w,a/i plotted vs. wa, for a few values of the “normalized frequency” u (Gloge, 1971) defined by 2,
= ka(v2
-
uy2
in terms of which one can write wea/i = (u2
-
10
..O 3 \ 0
.
3 x
0
FIG.7. Graphical determination of the solutions w,a of the second Eq. (47).
(48)
BEAM WAVEGUIDES AND GUIDED PROPAGATION
83
The abscissas w,a of the intersection points represent the values of w a for which the second Eq. (47) is satisfied, and are related to the eigenvalues Yno by Eqs. (421, namely,
wt YEo = 1 - k2u2 In the general case, it appears that Eq. (45) may be satisfied by real values of y (when Y, c, Y , , ee are real), if we may assume purely imaginary values, which requires Ye
(49)
As is well known, condition (49) is the condition under which waves inside the dielectric Y may suffer total reflection at the interface with a dielectric Y e . When we is purely imaginary, the external field is constituted by evanescent waves. The corresponding field inside the rod propagates without attenuation, and therefore is a ''guided" mode in the most stringent sense of the word. In any other case, when y has a (positive) imaginary part, the external field will be constituted by a leaky wave (Re we > 0, Im we < 0). This follows, for example, from Eq. (42), which can be written as
w,' = k2v,2 - k2u2y2
(50)
which shows that Im we has sign opposite to that of Im y , since, as required by the radiation condition, Re we > 0. In conclusion, the attenuation of the field is zero for n up to a value n(m)for which yn,* = u e / u , and then increases rapidly due to the radiation losses. The number N = ZmE(m) of no-attenuation modes turns out to be of the order of (Gloge, 1971)
N =+v2
(51)
where u is given by Eq. (48). Equation (51) holds better the larger is N, and takes into account all possible polarizations of the field. As to the group delay, defined by Eq. (13), its evaluation requires knowing the solutions of Eq. (45) in a sufficiently close form. An approximate expression, apparently valid for large Iweal and [wal is (Drougard and Potter, 1%7) Wnma
2
+ 3m + 4)
~ ( f i
Hence k v y = [k2v2 - (n
1
+ -2 rn +
i)2 21 =2
1/2
L. RONCHI AND A. M . SCHEGGI
84
which yields C
A better approximation is given by Snyder (1969). In the case of weak guidance, namely, A = (Y - ue)/ue 4 1, Gloge (1971) gives for T~ the following expression:
The propagation in dielectric rods is often treated in the ray-tracing approximation (see Appendix I), which is particularly useful in multimode operations. Due to its simplicity, that method allows one to reach very significant, even if approximate, results in a straightforward way. As an example, a characteristic of some field is to have an internal (cylindrical) caustic surface. From the point of view of wave theory, this means that there are fields that have a nonoscillatory behavior inside an axial region of radius r, (
FIG.8. Normal projection of a multireflected ray and of the internal caustic in a cylindrical waveguide.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
85
It may be worth recalling that the parameters of a mode, which in the wave treatment are deduced from the boundary conditions, in the raytracing method are deduced by imposing some “resonance” conditions. Such resonance conditions, which find their justification just in the wave theory, express the uniqueness of each wavefunction and can be written as (Maurer and Felsen, 1967) k fc
Vx dl
=
27m
7T +m ’ + i 2 In pj 2
(531
j
where C represents a closed curve, V x is a vector of modulus Y along the ray direction, m ’denotes the number of times a ray touches a caustic, and i In p r is the phase shift introduced by thejth reflection at the boundary.
B. Homogeneous Medium, Noncircular Cross Section The increasing interest in optical fibers with circular cross section has also attracted attention to the fibers with noncircular, in particular elliptical, cross section. Slight ellipticities may be created in the process of fiber drawing; strong ellipticities may be useful when the fibers are to be fed by injection lasers or similar sources. A ray-tracing treatment of the propagation in fibers of this type, with the determination of the eigenvalues and group delays, has been worked out by Checcacci et al., (1979). The wave treatment has been developed by Yeh (1962). However, it is very complicated since the field components inside and outside the rod cannot be simply expressed by separate-variable functions, but by infinite-product terms of Mathieu and modified Mathieu functions. A consequence is that finding the eigenvalues and the eigenfunctions requires finding the roots of an infinite determinant. In Yeh’s paper, the problem is solved numerically, and the modal wavelengths inside the rod are determined for several values of the parameters involved, for the dominant even and the dominant odd modes, where dominant means with the lowest cutoff frequency. The dominant mode turns out to be the hybrid mode with n = m = 1, as in the case of metallic pipes with circular cross section (Sec. IV). Note that, unlike what happens in the circular rods, all modes of elliptical dielectric rods are hybrid. An approximate treatment, valid when the outer refractive index is only slightly different from the inner one, has been worked out by Snyder and Young (1978).
86
L. RONCHI AND
A.
M. SCHEGGI
C . lnhomogeneous Medium
When the medium constituting the longitudinal structure is not homogeneous, but of the graded-index type, the guiding properties are partly due to the medium and partly to the boundaries. As far as mode guidance is concerned, graded-index media may be divided into normal and inverted media (Marcuse, 1972). In a normal medium, the real part of u decreases away from the axis. In an inverted medium, the opposite occurs, and the real part of u increases away from the axis. In the case of normal media, the main guiding properties are due to total internal reflection. In other words, modes may exist that are bounded by an (external) caustic surface: if the radius of the caustic is smaller than the radius of the fiber, the boundary of the fiber is expected not to affect the mode propagation too much. However, since the radius of the caustic depends on the order of the mode, mainly on the radial number n, and increases when n increases, one concludes that, in normal media, the role of the boundary is important for high-order modes and less important for low-order modes. In the case of inverted media, the boundary is expected to play the main role also on the lowest-order mode. However, the role of the boundary may be reduced, if the imaginary part of the refractive index u has a suitable trend, namely, if the losses increase (or, in the case of an amplifying medium, if the gain decreases) with increasing distance from the axis (Marcuse, 1972). When many modes exist that are not markedly affected by the presence of the boundary, the dispersion properties of the waveguide turn out to be quite different from those of the homogeneous dielectric rods because they depend on the parameters specifying the medium. Consider, for example, a graded-index fiber, whose refractive index u is given by u2 = u% - u2r2
(54)
with vo and u2 real constants, whose relevant properties have been studied by several authors leading to the invention of the so-called Selfoc fibers (Miller, 1%5; Kawakami and Nishizawa, 1968; Uchida et al., 1969; Gloge and Marcatili, 1973). For an insight into such properties let us apply the ray equation (see Appendix I), which in view of Eq. (54) takes the form
Hence r and 4 are periodic functions of z , with period p given by
BEAM WAVEGUIDES AND GUIDED PROPAGATION
87
Since p does not depend on the initial conditions of the ray, it turns out that, if there is a point source S on the axis at say z = 0, all rays emerging from S are focused at the point of the axis z = p (and, obviously, at all points of the axis with z = Np, N being an integer). It can be noted however, that Eq. ( 5 5 ) is approximate in two respects. First, it is the paraxial ray equation; second, in the right-hand side the denominator vt should be replaced by vi - v z r p . A more precise wave treatment (see Section IX,F)indicates that the eigenvalue of the (n,m)th mode is given by
[if the z dependence of the mode is assumed of the form exp(ikvoynmz)], which gives approximately 1
kvoynm= kvo - - ~ . j ’ ~ ( 2+nm
+
1)
(56)
yo
The group velocity deduced from (56) is therefore Tnm =
L
C VO
(57)
and is clearly independent of the mode numbers n,m. However this result is valid with some approximation, but optimized index distributions can be found that give rise to group velocity equalization with better approximations (Gloge and Marcatili, 1973; Olshansky and Keck, 1976; Arnaud, 1976, 1977; Arnaud and Fleming, 1976; Keck, 1977; Marcatili, 1977). Finally, it is to be noted that not all rays emerging from the point source reach their own caustic and suffer total internal reflection. Some rays have the caustic radius larger than the radius of the fiber and therefore are reflected at the boundary. This corresponds to the fact that high-order modes interact with the boundary, which implies for them a dispersion of the order of that of the homogeneous rods. For a wave treatment of the wave propagation in an inhomogeneous medium with transverse gradient of the refractive index, one has to start noting that the wave equations for E and H inside the rod have the form
VZE + k V E
+ V(E
V In
4)
=0
V2H + k Z v 2 H - iw VEAE = 0
(58)
The terms V(E V In C) and VaAE play an important role, since they control the polarization properties of the modal fields. However, they complicate the analytical treatment so much that they are generally neglected. Even so, the solutions of the scalar equation for the longitudinal compo-
88
L. RONCHI AND A. M. SCHEGGI
nents of E and H satisfying the boundary conditions are not easily found. Several approximations have been suggested. Snyder and Young (1978) have elaborated a method valid when the maximum value of the refractive index of the core is close to that of the cladding. This method is applicable to both circular and noncircular cross sections of the fiber. Another method, whose accuracy is generally limited by numerical accuracy, consists in replacing the continuous medium with a stratified medium (Clarricoats and Chan, 1970) formed by cylindrical shells sufficiently thin to assume the refractive index to be practically constant inside each of them. In this case the problem is to look for the optimal number of shells (Arnaud and Mammel, 1976; Bianciardi and Rizzoli, 1977). Clearly, this method is particularly useful when the refractive index is not continuous up to the boundary but presents a number of discontinuities, such as when the fiber is composed by a core and a possibly stratified cladding. In many practical applications, however, the trend is not to introduce a discontinuity of the refractive index between the gradedindex core and the cladding. Other methods may be thought of, such as the use of series expansions. The method most commonly used is based on the WKB approximation (see Appendix 11) (Kurtz and Streifer, 1%9a,b; Gloge and Marcatili, 1973). However, a method that looks more efficient, especially in the cases where the WKB approximation fails (Hashimoto, 1976), is the socalled regularized WKBJ method (Maslow, 1965; Ikuno, 1978).
VI. TWO-DIMENSIONAL WAVEGUIDES WITH METALLIC WALLS Planar waveguides with metallic walls have been known for a long time and are described in most textbooks on the theory of electromagnetic wave propagation (Toraldo di Francia, 1953). Their main use is not to transmit energy, signals, or images, but as image-forming systems, namely as “lenses.” In this respect, two large classes can be differentiated. In the first class, propagation occurs with the electric field parallel to the walls (TM waves), so that the phase velocity is different from that in free space and depends upon the spacing of the walls. Thus originate the well-known microwave lenses (Fig. 9). Another class uses TEM propagation (the electric field is normal to the walls) and the phase velocity is not critically dependent on the spacing of the walls: the optical properties depend upon the configuration of the walls, or better, on the configuration of the surface Z midway between the walls. For this reason, such lenses have been called configuration lenses (Brown, 1953). The other name, geodesic lenses
BEAM WAVEGUIDES AND GUIDED PROPAGATION
89
t' Fic. 9. Metallic-plate microwave lens.
(Righini et al., 1972), derives from the fact that, since t..e two walls are generally kept at a distance as small as possible from one another, the propagation is very close to a two-dimensional propagation along 2,and by virtue of the Fermat principle, the phase paths coincide with the geodesics of 8. In general, the geodesic lenses have a revolution symmetry around an axis. The profile of the meridional curve can be so designed as to obtain optical systems free from aberrations, at least for a class of couples of conjugated points. The simplest example is a sphere, illuminated by a point source S (Fig. 10) placed on it. The radiation is constrained to travel along the sphere, and the rays are the largest circles through S. Accordingly, all rays cross together at the point I diametrically opposite to S. Clearly, Z is the image of S given by the spherical surface, and is perfect for any position of S on the sphere. The analogy of such a lens with Maxwell's fish-eye (Maxwell, 1854) is well known. Another interesting lens is the so-called Rinehart-Luneberg lens Rinehart, 19481, which transforms a beam diverging from a point source S into a beam of parallel rays propagating alone a plane. It is composed of a central curved portion (Fig. 11) and a disk, and the source S is to be located at a point of the parallel 1 common to the central portion and to the disk. The output beam acquires spherical aberration if S is moved from the parallel 1. (A generalization of these lenses has been found by Toraldo di Francia, 1955a.)
FIG. 10. Maxwell's fish-eye.
90
L. RONCHI AND A. M. SCHEGGI
FIG. 11. Rinehart-Luneberg lens.
The same behavior as a collimator free from spherical aberration is presented by Toraldo’s lens (Fig. 12), which has the advantage, with respect to the lens of Fig. 11, that the passage from the central curved portion to the plane occurs through a toroidal junction, which increases the luminosity of the system (Toraldo di Francia, 1957; Scheggi and Toraldo di Francia, 1960). Simpler configurations present poorer optical properties. For example, a revolution surface formed by two coaxial cones joined along a parallel has the same optical properties as a thin lens (“conflection” lens; Toraldo di Francia, 1955b). It examines the possibility of obtaining systems corrected for the third-order spherical aberration by suitably joining two or more thin lenses (Toraldo di Francia, 1955b; Ronchi and Scheggi, 1956). It can be noted that there is a particular conflection lens, formed by a disk and a cone of 60” aperture, which has no aberrations when the source is either on the edge of the disk or at infinity on the cone (DeVore and Iams, 1948). A toroidal junction between two conical surfaces behaves as a thick lens (Ronchi, 1955), and a “doublet” made up with two toroidal junctions may have a luminosity better than a “conflection doublet.” In the above order of ideas, it easily appears that the systems in Figs. 11 and 12 can be considered as analogous to optical doublets formed by a thick aspherical lens (the central portion) and either a conflection lens (Fig. 11) or a toroidal junction (Fig. 12). The aspherical lens exactly compensates for the spherical aberration of the second element. Lenses like that in Fig. 12 may be used as antennas for the rapid scanning of the horizon, since rotating the source along its parallel lets the emerging beam rotate, but does not introduce aberrations in the emerging
FIG. 12. Toraldo’s geodesic lens
BEAM WAVEGUIDES AND GUIDED PROPAGATION
91
beam. A stack of lenses (Scheggi and Toraldo di Francia, 1360) may be used to regain the third dimension.
VII. TWO-DIMENSIONAL DIELECTRIC STRUCTURES A. Planar Structure with Homogeneous Medium Planar dielectric structures, mainly thin films are being largely used in the field of integrated optics, for optical signal generation and processing. In this case, the dielectric is mostly homogeneous, and the guiding properties are based on the total reflection at the walls. In practice, these guiding structures are not symmetric, in the sense that they are composed by a thin film, of refractive index v, deposited on a substrate of refractive index ua < U , while the upper medium is air (Fig. 13). Accordingly, the modes are neither even nor odd functions of the transverse coordinate x, but rather are to be expressed as the superposition of two plane waves, A, exp[ikv(w + yz)] and A 2 exp[iku(- ax + yz)], satisfying the boundary conditions at x = + d / 2 . The boundary conditions at x = d / 2 and x = - d / 2 yield
4 eikvad
=
A2
+
CY1
VCY - ( ~ 1 ’
4 e{ b a d = va - vaa2 Ag
U(Y
respectively, where a1 = (1 -
FIG. 13. Rays in a dielectric film.
+
vaag
(59)
92
L. RONCHI A N D A. M. SCHEGGI
represent the direction cosine with respect to the x axis of the plane waves in the air and in the substrate, respectively, The mode is guided when both a1and azare purely (positive) imaginary, namely, when Y
'
(61)
%/JJ
In this case the incidence angle e' at the upper and lower boundaries is larger than both the critical angles 6, = arcsin 1/u, & = arcsin UJU. If Eq. (61) is satisfied, the right-hand sides of Eqs. (59) have unit modulus, and the resonance condition, expressing the conditions under which system (59) has solutions, can be written as 2kvad
-
41 - c $ ~ = 2 n ~
(62)
where
41 = 2 arctan(Iall/va) = 2 arctan[v2 -
rPz = 2 arctan(u,la21/ua)
=
vZa2- 1)1/2/ua]
2 arctan[(u2 - v2a2-
(63) V ~ > ~ / ~ / V ~ ]
In practical applications, in order to take as much advantage as possible of the features of two-dimensional propagation, especially with regard to luminosity, propagation has to occur by modes that do not radiate energy in the outer media. It follows that one of the most critical points is how to efficiently excite the modes in the structure. Since the field in the outer media corresponding to a mode inside the film is constituted by an evanescent wave, the best way of exciting a mode seems to be the use of a total-reflection prism, as indicated in Fig. 14. In the same way, namely, by frustrating the total reflection, one can extract the energy from the film, after its utilization. The main feature of propagation inside the film is that it may be controlled from outside. For example, since the phase velocity depends upon the (optical) thickness of the film, it can be locally varied by varying the thickness: accordingly, a region of the film may behave as a lens if a suitably shaped coating is deposited over that region.
FIG. 14. Modal excitation and power extraction in a dielectric film.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
93
\
FIG. 15. Bending in a thin film.
Another feature of the total-reflection guiding is that the film may be bent without introducing substantial radiation losses: to this purpose, it is sufficient that the curvature of a bend be sufficiently small so that the incident angles Of,8*, . . . (Fig. 15) remain larger than the critical angles O, and @. Such a feature opens the possibility, among other things, of realizing and using geodesic lenses (Righini et d.,1972, 1973) (see Section VI), which in addition to all good optical properties they have, do not require variable thicknesses, and therefore can be realized with the simplest uniform-coating techniques.
B . Planar Structures with Graded-Index Medium Another field where planar dielectric structures find application is the laser field (see, for example, the dye lasers). In this case, the refractive index of the dielectric is not real and generally, due to the features of the pump light absorption, has a transverse gradient, with real part decreasing and gain increasing toward the boundaries (Devlin et al., 1962; Pratesi and Ronchi, 1976). Moreover, the outer index of refraction v, may happen to be appreciably larger than the inner index U, so that total reflection cannot occur at the discontinuities’ dye cladding but only, if any, internally to the dye. As a consequence, many of the approximations useful in other physical problems cannot be applied. The main problem for a theoretical study of the modal propagation in a graded-index slab is to solve the wave equation. It can be noted that in the case of a graded-index medium, where the refractive index depends upon a single transverse coordinate, say x, Maxwell’s equations admit TE and TM solutions, with either E or H parallel to the y axis. Assuming a z dependence of the type exp(ikyz), namely,
E(x, z)
=
A(x)e*””i,
(64)
for TE modes, A ( x )turns out to satisfy a scalar wave equation of the usual type (for TM modes, see Section IX,B). This wave equation may be exactly solved only in the case of a square-law medium, where the square
94
L. RONCHI AND A. M. SCHEGGI
refractive index v2 is given by v2(x) = v;
-
vzx2
(65)
In this case, the wave equation reduces to the Weber equation (Abramowitz and Stegun, 1965), whose solutions are the parabolic cylinder functions (Kirchhoff, 1972). Then the transcendental equation that expresses the continuity of E,, H , , and H , at the boundary can be solved with the help of an electronic computer. This approach is valid for both real and complex media. In a more general case, one has to use approximate methods, such as the ray-tracing method (Appendix I) or the WKB approximation (Appendix 11). The ray-tracing method is generally applied, at least in the form presently found in the literature, when the medium is real. A trial to apply it to complex media has been done by Pratesi and Ronchi (1976), who traced the rays as if the medium were real and then evaluated the gain by integrating the amplification along the rays. As to the WKB method, it can be noted that its application is generally laborous and it too requires the use of an electronic computer, except for the modes bounded by a caustic internal to the slab and far enough from the boundaries. In this case, one may neglect the presence of the boundaries; in other words, one may assume the modes and the eigenvalues to be only slightly different with respect to those of an unbounded medium. Propagation in an unbounded medium is treated with some detail in Sec. IX.Here we only note that this last approximation is able to provide a good idea of the propagation within the slab except, obviously, for what concerns modal gain and losses.
VIII. WAVEGUIDINGBY TRANSVERSE STRUCTURES A . Introduction
In the years 1960-1961 a new type of waveguide was proposed by Goubau and Schwering (1961) to transmit electromagnetic energy over long distances with low attenuation. This new waveguide was essentially constituted by a periodic sequence of optical lenses, illuminated by a proper aperture. A similar idea had been proposed by Pierce for the guide of electron beams (Pierce, 1961). Since the lenses were placed orthogonally to the mean direction of propagation of the radiation and no lateral (longitudinal) wall had to contribute to the wave guidance, such structures have been denoted as "open" beam waveguides.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
95
LLIJ-L-! FIG.16. A dielectric-lens beam waveguide.
In the same years Fox and Li (1960, 1961) and Boyd, Gordon, and Kogelnik (Boyd and Gordon, 1961; Boyd and Kogelnik, 1962) published the basic theory of open resonators. It immediately appeared that the two fields-open-beam waveguides and open resonators -though addressed to completely different purposes, had many points in common, first of all, the mathematical theory. Accordingly, the development of one field contributed to the progress of the other one. The first proposed open-beam waveguide consisted of a periodic sequence of optical converging lenses (Fig. 16), and the theoretical expectations were checked (Christian and Goubau, 1961) by experiments on a structure formed by two spherical mirrors, each having the curvature center on the other one (Fig. 18): this pair of spherical mirrors constitutes a so-called confocal resonator (Boyd and Gordon, 1961). On the other hand, a Fabry -Perot resonator, formed by a pair of plane -parallel mirrors facing one another, turned out to have, by definition, the same modes as a structure formed by a sequence of irises (Fig. 18) in equispaced absorbing screens (Fox and Li, 1961). Analogously, the rimmed open resonators (Checcacci and Scheggi, 1969; Checcacci et a / . , 1971) yielded as a logical consequence the so-called dielectric-frame beam waveguides (Checcacci er a / . , 1972) sketched in Fig. 19. From the technical point of view, on the contrary, the two fields presented quite different problems. In the case of open-beam waveguides
FIG.17. A confocal open resonator.
L. RONCMI AND A. M. SCHEGGI
96
FIG.18. An iris beam waveguide.
formed by lenses, the main problem has been to build low-loss largeaperture lenses. This has brought about the design and realization of gas lenses (Berreman, 1964a,b), where the optical properties were obtained by means of a transverse thermal gradient of the refractive index of the gas. Thus began the studies on propagation in graded-index media, which became so important in the field of guided propagation even as guiding media (Section IX and also Sections V,C and VIILB). A severe limit to the practical applications of open-beam waveguides is imposed, however, by the turbulence and convection motions of the medium between the optical elements of the sequence (and inside the optical elements themselves, if they are gas lenses). Thus, the main applications they lind are over short distances, in the version using mirrors instead of lenses (Fig. 20) (Degenford et al., 1964). In spite of these limitations, their importance is great, both for ideas and for all the theoretical studies they originated. In particular, they contributed to the birth of Gaussian beams as approximate solutions of the free-space wave equation and to the delinition of their main properties.
L
-
L
L
L
-
J
FIG.19. A dielectric-frame beam waveguide.
BEAM WAVEGUIDES A N D GUIDED PROPAGATION
97
FIG.20. A Degenford beam waveguide.
B. The Reiterative Wave Beams The main idea underlying the design of an open-beam waveguide is that there exist reiterative beams, i.e., beams whose real amplitude distribution is repeated at a distance, say L, from the aperture. The existence of such beams has been quite simply proved by Goubau and Schwering (1971) in the case of cylindrical symmetry and by Schwering (1961) in the case of rectangular symmetry. Let us consider cylindrical symmetry around the z axis, and introduce cylindrical coordinates r, 4, z (unit vectora i,, b, iz).The general solution of the free-space homogeneous Maxwell equations can be written as (Stratton, 1941)
m4
E, = ET(r, z) cos m4,
H, =
T
YoE:(r, z) sin
Em = E$(r, z) sin m4,
H,,, =
4
YoE$(r, z) cos m 4
z ) cos m6,
H, =
7
YoEz(r,z) sin mQ,
E, = E:(r,
(66)
with Yo = ( ~ ~ and / b ) ~ / ~
E: ( r , z)
=
2i
lom
f+(w)J,(wr) exp(ikyz)w2 dw
x exp(ikyz)w d w
x exp(ikyz)w dw
where w is related to y by
(67)
98
L. RONCHI AND A. M. SCHEGGI
The beams of Eqs. (66)and (67) are hybrid beams. However, one can obtain a pure TM or a pure TE beam by summing two hybrid beams with f+ = Tf-,respectively. It appears from the first Eq. (62)that w2f+(w)dw and w2f-(w)dw represent the amplitude of the elementary cylindrical waves forming either beam in Eqs. (66) and therefore the amplitude spectra. Iff+(w) are real and different from zero only in an interval of values such that y too is real, Eqs. (67) clearly say that the field at any value z = L / 2 is the complex conjugate of the field at z = - L / 2 and therefore the real amplitude transverse distribution is the same. Accordingly, the beams whose spatial frequency spectrum is real and limited to a finite angle are reiterative beams. The main feature of such reiterative beams is that the field distribution at the plane z = L / 2 differs from the field distribution at the plane z = - L / 2 (“aperture”) only by the phase distribution. Accordingly, the aperture can in principle be reconstituted at z = L / 2 by a suitable phase transformer put across the beam at z = L / 2 . I€ so, the aperture amplitude distribution repeats at z = 3 L / 2 , and the aperture can be reconstituted at z = 5 L / 2 by a phase transformer equal to the preceding one put at z = 3 L / 2 . Iterating the process, namely, by passing the beam along a sequence of phase transformers uniformly spaced by L, one can guide the beam in the z direction, by avoiding the energy spreading that occurs in free propagation (Fig. 21). There are many questions to be answered about the physical realizability of phase transformers. It is to be noted that the transformer has to reconstitute the phase distribution of, say, the transverse electric field E,i, + E m b ,which, as is well known, uniquely determines the total electromagnetic field. In this concern, Goubau and Schwering noted that, if the spectrum is limited to a very small angle, so that w 2 + kZ, Eqs. (67) yield E$(r, z ) =
FIG.21.
? E: ( r , z)
A reiterative beam in a periodical sequence of lenses.
99
BEAM WAVEGUIDES AND GUIDED PROPAGATION
Accordingly, any phase transformer suitable for E , is suitable also for E m , and may reconstruct the complete field aperture. Another question regards the complexity of the phase transformation to be operated. In this regard, it has been shown (Goubau and Schwering, 1961) that there exist beams that simply require square-law phase transformers. To this end, it is sufficient to note that, when w -e k so that one can write y = 1
1 wz 2 k2
in the phase factors of Eqs. (67), and w = k in the amplitude factors, Eqs. (67) can be evaluated if one chooses
where w o is a constant and L,, denotes the generalized Laguerre polynomial of orders n and m (Magnus and Oberhettinger, 1955). Introducing Eq. (70) into Eqs. (67) and integrating, one obtains (Erdelyi, 1954), apart from an integration constant factor,
X
exp
(- pr2 + ik')2 R
exp(i&,)
(71)
where Wo is an integration constant, related to w o by Wo = d / w o ,
and
&,
=
-(2n
+ m + 1 * 1) arctan-kW; LL
(73)
Moreover,
E$
=
E $ ( r , z) sin m&
(74)
Formulas (71)-(74) describe the well-known Gaussian beam of orders n , m , with cylindrical symmetry around the z axis. The reiterative properties of such beams follow from the parity of the function W ( z )with respect to Z.
The phase transformers are only required to introduce at z
=
L/2 a
I00
L. RONCHI AND A. M. SCHEGGI
phase distribution T(r) given by
where qois a constant and Ro = R ( L / 2 ) = (L/2)[1 + (kW2,/L)2].In practice, they can be realized by optical lenses of focal length fgiven by
The above theory proves the possibility of guiding electromagnetic radiation from a suitable aperture over a long distance, by avoiding free-propagation spreading, since the lenses periodically refocus the beam (Fig. 21). In practice, however, such a spreading cannot be completely avoided. It could be avoided if beams could Se generated with a spectrum limited to a finite angle, but this property of the spectrum implies an infinitely extended aperture, and consequently, infinitely extended phase transformers. In practice, one can think to realize only a “truncated” aperture and “truncated” phase transformers. The effects of the truncations are diffractional effects, which are the more important the higher the field is at the edges of the aperture and of the lenses. Now the Gaussian beams of Eq. (71) have an important feature, namely, they are bounded by a caustic surface, as follows from the fact that the real amplitude decreases exponentially when r + w. Let us denote by r, the radius of the pupil of the aperture and of the phase transformers and by rc,nm(z)the radius of the caustic of the (n,rn)th modal beam. It turns out that r,,,, ( W / 4 ) ( 2 n + m ? l)lm, due to the polynomial factor rm+iL,,,*1(2r2/W2).The diffractional effects turn out to be important and to substantially alter the propagation features for the beams with n , m such that
-
rc,nm(L/2) 2 Conversely, the beams for which
ra
(77)
rc,nm@/2) Q ra
(78)
are altered by diffraction but not to a substantial extent. To go deeper in this problem, one should apply the treatment described in Section VIII,E. Due to the above recalled diffraction effects, not all the radiation emerging from the aperture or from a lens is collected and refocused by the subsequent lens, but there is some spill-over radiation. Consequently, the beam propagating in thejth “cell,” namely, from the (j- 1)th to the
BEAM WAVEGUIDES AND GUIDED PROPAGATION
101
jth lens, is attenuated with respect to that propagating in the ( j - 1)th cell. The spillover radiation, however, is not the only cause of the attenuation per cell of an open-beam waveguide. Other important causes are the power reflection that generally occurs at the incidence of the beam onto a lens, which can be minimized by the use of gas lenses (Berreman, 1964a,b); irregularities such as displacements or misalignments (Hirano and Fukatzu, 1964; Steier, 1966; Gloge, 1966) due, for example, to movements of the ground, which may be corrected by servomechanisms (Marcatili, 1966; Goubau, 1968); inhomogeneities of the medium in the cells, such as stratification or turbulence (Goubau and Christian, 1964), and so on. All these factors are to be taken into account in the evaluation of the performances of a periodic sequence of lenses as open-beam waveguide, in comparison with other types of waveguides. C . The Reiterative Beams as Eigenfunctions of an Integral Equation
The theoretical approach of Goubau and Schwering to open-beam waveguides consists, as described in Section VIII,B, in looking for the sequence of optical elements able to reconstitute cell by cell a prescribed reiterative beam. The development of the theory of open resonators, as proposed by Fox and Li (1960, 1961), has shown that any periodic sequence of “optical elements” (in the most general sense of the words) sustains beams that repeat cell by cell, and therefore may be used as open-beam waveguide. Consider an open resonator formed by two equal mirrors facing each other and separated by a distance L (Fig. 44) and look for the modes it can sustain. This problem is generally treated in the scalar approximation. By definition, a mode of the resonator is a field distribution u(r, 4) across a mirror [namely, with P ( r , 4) belonging to the mirror surface], which gives rise to a field distribution u(r, 4) across the other mirror, which differs from u(r, 4) only by a constant factor r: ~ ( r4) ,
=
W r , 4)
(79)
The factor r, denoted “eigenvalue” of that mode, is generally complex and describes with its square modulus the “power attenuation per transit,” and with its argument the phase variation due to the propagation from one to the other mirror (Fox and Li, 1961). It is usual to introduce the phase shift 6, related to r by
5 = arg r
-
kL
(80)
102
L. RONCHI AND A. M . SCHEGGI
The field between the two mirrors is generally composed by a beam propagating in the positive z direction, from Z to 2’and by a beam propagating in the opposite direction from 2’to 2. When the field distributions across the mirrors satisfy Eq. (79), the beams are “modal” beams. Consider now a periodic sequence of optical elements, of periodicity L, each operating in transmission as each mirror of Fig. 22 operates in reflection. If in some way we generate at the input surface of the first element a modal distribution of the field, it gives rise to a modal beam that propagates in the structure by repeating itself cell by cell apart from the constant factor r. In order to find the modal distributions that can be sustained by an open resonator or by the “equivalent” sequence of optical elements, one has to find the field distributions that satisfy Eq. (79). To this end, it is to be noted that if u(r, 4) denotes the field distribution at the points of the pupil in the input plane X of an optical element (Fig. 23) the field distribution u(r, 4) at the points of the input plane Z’ of the subsequent element can be written as
where a denotes the area of the pupil and K(r, 4; r‘, 4 ’ ) ~dr’ ’ d4’ describes propagation through the optical element just after Z and the propagation from it to Z’. Accordingly, the modal condition Eq. (79) can be written as
when the point P(r, 4) belongs to the pupil of the optical element. Equation (82) is an integral equation of Fredholm type (Fredholm, 1903), which
-
L
L
FIG.22. An open resonator.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
L
L
I03
J
FIG.23. A “cell” of a beam waveguide formed by diaphragmed optical elements.
may be solved by an iterative numerical procedure (Fox and Li, 1960), once the expression of the kernel K ( r , 4; r‘, 4’) is known. As to the kernel K, one generally uses the Huygens -FresneI principle and some approximate forms, appliable when the propagation does not involve large angles. However, various approximate expressions have been shown to yield results that may be appreciably different, depending on the parameters of the structure (Consortini and Pasqualetti, 1972, 1978). Due to the fact that the integration field a has finite size, a property of the eigenfunctions of Eq. (82) (Morse and Feshbach, 1953) is that they form an infinite discrete doubly numerable set of functions, which will be denoted as unm(r,4). The set is complete, since the kernel is compact. Generally speaking, the 4 dependence is of the type cos m 4 , sin m4, or exp( & im4). The eigenvalues, which are labeled by the same indices as the associated eigenfunctions, have zero as accumulation point external to the ensemble. For each value of m ,the eigenfunctions are numbered in such a way that larger n correspcr,J to smaller. Thus the eigenfunction with minimum attenuation per cell (fundamental mode) is denoted by urnand the corresponding eigenvalue by Too. It can be noted in Eq. (82) that since the integration region is the pupil a of the optical elements (the spill-over radiation can be physically eliminated by using optical elements inserted in infinitely extended absorbing screens, as sketched in Fig. 23), the eigenfunctions of Eq. (82) are characteristic of a “truncated” optical element. (The main approximations of the above approach are the scalar approximation, the expression for K, and neglecting the radiation reflected at the transverse discontinuities of
104
L. RONCHI AND
A.
M. SCHEGGI
the structure). Accordingly, with the above treatment, one can evaluate the effects of the “truncation” of the aperture and of the lenses in Goubau and Schwering’s open-beam waveguide of Section VIII,B. It turns out that, if the argument of K is approximated to the second order in the transverse coordinates, the eigenfunctions of Eq. (82) are the so-called hyperspheroidal functions (Heurtley, 1965; Heurtley and Steifer, 1965; Wainstein, 1965), which, when condition (78) is satisfied, are well approximated by the Laguerre -Gaussian polynomials of Eq. (71). For such beams, the associated eigenvalues r turn out to be very close to 1 in modulus. For the beams for which condition (77) is satisfied, the associated eigenvalues turn out to be vanishingly small. Such highorder beams practically are not guided by the structure, because their energy is practically completely intercepted by the absorbing screens. There follows an easy way of realizing a single-mode propagation: to this end, one has to choose the size of the pupils of the optical elements in such a way that only a mode exists with eigenvalue close to 1 in modulus (Siegman, 1965). Analogous (and preceding) studies carried out on open resonators formed by cylindrical mirrors with circular cross section (Boyd and Gordon, 1961; Boyd and Kogelnik, 1962), equivalent to open-beam waveguides formed by cylindrical lenses (curved only in the x, z plane), showed that the eigenfunctions of the associated integral equation are the spheroidal functions (Flammer, 1957; Slepian, 1964, 1965; Slepian and Pollak, 1961; Slepian and Sonnenblick, 19651, which, when the diffraction effects due to the finiteness of mirror size are neglected, are well approximated by the well-known Hermite-Gaussian beams. D . The Dielectric-Frame Beam Waveguide
Several types of open resonators and equivalent open-beam waveguides have been studied by means of Eq. (82), starting with the Fabry -Perot resonator and the equivalent iris open-beam waveguide (Fig. 18). An interesting result is that for a prescribed value of L and of the transverse size of the guiding elements, the Goubau and Schwering waveguide is the structure where the modal beam of a given order has the minimum spill-over losses. This is probably the merit of the Gaussian profile of the spectrum [see Eq. (70)] and of the consequent presence of the caustic surface. However, there is another beam waveguide that has interesting properties from the point of view of the minimum hardware. This waveguide, the dielectric-frame beam waveguide (Fig. 19) has been suggested by a study of the resonators formed by plane mirrors with a stepped rim (Checcacci et al., 1971) (Fig. 24).
BEAM WAVEGUIDES AND GUIDED PROPAGATION
I05
FIG.24. (a) The rimmed open resonator. (b) A cell of the equivalent beam waveguide.
By numerically solving the integral equation associated with such a resonator, it was found that the eigenvalues were critically dependent upon the thickness 6 of the rims, but only slightly dependent upon the width I of the rims, if I was sufficiently, but not excessively, large. The of the eigenvalues turned out to be a quasi-periodic function modulus lrnrnl of 6, with period of the order of h/2, and a maximum per period (Fig. 25). The equivalent open-beam waveguide is represented in Fig. 24b: the plane mirrors are simply to be replaced by pupils, the rims by dielectric frames (with refractive index V ) of thickness 6' = 26/v - 1) and width I , the outer regions of mirror planes by absorbing screens. However, the fact that, in practice, the eigenfunctions of the open resonator turned out for one and the same 6 to be independent of I, for I larger than a few wavelengths, suggested that the outer regions do not contribute to the features
I06
L. RONCHI AND A. M. SCHEGGI
(SIX)
FIG.25. Power losses vs thickness of the frame in a dielectric-frame beam waveguide, for the two lawest-order modes.
of the resonator, and therefore of the open-beam waveguide. Accordingly, the “optical” elements of the equivalent waveguide turned out to be simply constituted by the dielectric frames. The main favorable characteristics of this structure are the low weight and the practically nonexistent reflection losses. Experiments done with a model in the K, band (Checcacci and Scheggi, 1971) confirmed all theoretical expectations. Further studies allowed replacing of the dielectric frames by a portion of them (Checcacci et al., 1973) with low-efficiency loss, and showed the possibility of guiding the radiation along curved paths in the simplest way (Checcacci el al., 1974). In the above way one arrives at the conclusion that a dielectric frame is an “optical element.” Its optical properties may be understood by noting that (Consortini et al., 1975a,b) the presence of a dielectric obstacle bends a wavefront impinging on it (Fig. 26) in such a way that a caustic may be created, extending a certain distance L’ behind the obstacle. The existence of the caustic is strictly related to the value of the phase shift k(u - 1)6’ introduced by the frame, which determines the curvature of the wavefront (Figs. 26a,b). When the caustic exists and the distance L between two successive frames is smaller than L‘, the beam propagating in the structure is bounded by the caustic and suffers very low radiation losses.
E . Propagation in an Open-Beam Waveguide In the preceding sections it has been shown that a periodic sequence of optical elements sustains beams that in propagation from cell to cell are altered only by a constant factor.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
I07
(bl
(0)
FIG.26. Formation of a caustic behind a dielectric obstacle.
In analogy with the conventional waveguides described in Section I, two problems arise: one in finding a way to create the modal field distribution across the first element of the sequence: the other one in studying what happens when the fist element is not properly illuminated by a modal distribution. The first question has been bnswered by Goubau (19681, who suggested illuminating an open-beam waveguide through the equivalent open resonator. The other question can be answered with the help of Eq. (81). Let us refer, as in Section VII1,C to cylindrical coordinates r, 4, z. If u(r’, 4f) denotes the field distribution created by a “source” across the input plane of the first element, the field distribution across the input plane of the second element is given by Eq. (81). Then, to obtain the field across the input plane of the third element one has simply to substitute u for u in the right-hand side of Eq. (81), and so on. In general, if U , - ~ ( T ’ , 4’) denotes the field across the input pupil of the (j- 1)th element, one can write uj(r, 4) =
/I
uj-dr’,
4f)K(r’,4’; r , 4)r’ dr’ d4’
(83)
This equation can be written for any sequence, even not periodic, of optical elements. In the case of nonperiodicity, K and/or a turn out to depend uponj. The application of the above method requires the use of electronic computers of good capacity. As first noted by Fox and Li (1961),whichever is the initially launched field distribution, the result of a sufficiently high number of iterations is always a distribution that satisfies Eq. (82), namely, a modal distribution. More precisely, it is the modal distribution with highest modulus of the eigenvalue present in the initial distribution.
108
L. RONCHI AND A. M. SCHEGGI
This is easily proved as follows. Since the eigenfunctions of Eq. (82) form a complete set for the functions defined in the region u the initially launched field u ( r ’ , 4’) can be written as
Since, after propagation in a cell, each mode turns out to be multiplied by the eigenvalue r,,, the field at the entrance of thejth cell will be given by
Increasing j diminishes the contribution to u j ( r ’ , 4’) of any mode u,, with respect to that of the fundamental mode, if a,, # 0. If a,, = 0, the leading term in the sum on the right-hand side of Eq. (85) turns out to be that corresponding to the highest-modulus eigenvalue in the initial field. F . Propagation in an Open-Beam Waveguide without Absorbing Screens around the Optical Elements
While Eq. (83) appears completely appropriate to describe the back and forth propagation of a beam between two equal mirrors, it can be applied in principle to a periodic sequence of optical elements only if they are inserted in absorbing screens. In this case, only the field across the pupil u of one optical element contributes to the field at the points of the plane of the successive element. A method to treat the propagation in a sequence of optical elements without external absorbers has also been worked out (Consortini et al., 1975a,b), by assuming the optical elements to be infinitely thin. The lines of the approach are as follows: Assume thejth optical element to occupy only a region 3 (possibly constituted by separate subregions Bl, ?&, . . . ) of the plane z = j L (Fig. 27) with j integer, and denote by u j ( x , y, z) the field impinging onto such a plane from say the left. The field distribution across the plane z = j L - 0 is u j ( x , y , j L ) , while the field distribution across the plane z = j L + 0 is
where T(x, y ) is a generally complex function describing the optical transparency, or better, the optical properties of the element of the sequence. The field u j + l(x, y , z) impinging onto the successive element can be
BEAM WAVEGUIDES AND GUIDED PROPAGATION
109
FIG.27. Sketch of a undiaphragmed element forming a general dielectric beam waveguide.
obviously written as
wherez describes the free-space propagation from the point (x‘, y ’ , j L ) to the point (x, y z ) with z 2 j L . The first integration, however, is not easily or accurately carried out by means of an electronic computer of limited capacity, since the integration field extends up to infinity. This practical difficulty can be overcomed by noting that such an integration can be clearly written as an integration to the whole plane 2,which is simply given by u j ( x , y , z), minus an integration to the finite-extent region 9. Accordingly, Eq. (86) can be rewritten as uj +
l(X9
y , z) =
U j k
x
y , z)
+
JJ 9
Uj(X’,
y ’ , j L ) [ T ( x ’y, ’ ) - 11
K(x, y ; x’, y ’ ) dx’ dy’
(87)
for z 2 j L , which may be easily introduced in a computer. In particular, the method has been applied (Consortini et al., 1975a,b) to treat the propagation in a dielectric-frame beam waveguide. For simplicity, only the bidimensional case has been considered, where each
L. RONCHI AND
110
A.
M. SCHEGGI
frame is replaced by two dielectric rods, in a plane z = j L , parallel to the y axis and symmetrically placed with respect to the z axis (Fig. 28). By neglecting the reflection at the rods, the function T ( x , y ) is given by
T(x, y )
=
T ( x ) = exp[ik(v - 1) 8’1
(88)
for 3a 5 1x1 5 ) a + 1, if 8’ denotes the actual thickness of the rods, as in Fig. 24b u its refractive index, a the internal width of the frame, and 1 the width of the frame. The results of the numerical calculations confirmed the validity of the results obtained by means of Eq. (83), namely by assuming the optical elements to be inserted in an absorbing screen.
IX. GUIDINGMEDIA A . Introduction
The guiding properties of the longitudinal structures recalled in Sections I-VII depend mainly upon the sharp discontinuity constituted by the boundaries. In the transverse structures of Section VIII, the guiding properties are strictly related to the possibility of creating beams laterally bounded by a caustic surface (“narrow” beams) and to that of avoiding the natural spread of the beams by periodically refocusing them. However, as has been well known since the beginning of the present
x FIG.28.
A cell of the dielectric-frame beam waveguide.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
111
century, there also exist “guiding media,” which avoid the spread of a beam without the help,of any boundary. The role of the ionosphere in the guidance of radio waves around the earth has been known since the Marconi experiments in 1902 (Heaviside, 1902; Kennelly, 1902), while the possibility of the existence of “ducts” within the ionosphere and troposphere due to its inhomogeneities or stratification was demonstrated in 1919 (see, e.g., Gerson, 1962). Recently, with the advent of integrated and fiber optics, artificial graded-index media became of great practical interest. Clearly, in practice, the graded-index media are limited to a finite region of space, in the form of a planar slab, or a cylindrical rod or a more complicated configuration. However, knowledge of the features of wave propagation in an infinitely extended medium, conceived as the analytical prolongation of the medium or the slab or of the rod, gives an immediate, even if approximate insight into the properties of propagation in the slab or in the rod. A great deal of attention was paid to square-law (or parabolic) media, where the square refractive index is a quadratic function of the transverse coordinates. Such media, if “normal” (see Section V,C), have been proved (Marcuse, 1972) to sustain “modal” Gaussian beams, which propagate without changes of the field transverse distribution, in other words, without diffractional spreading. Such beams turn out to be symmetric with respect to the axis or planes of symmetry of the medium (on-axis beams) (Kogelnik, 1965) and to have characteristics strictly related to the parameters of the medium. In addition, such media may guide on-axis beams with parameters different from those of the modal beams, and also off-axis beams. These beams change during propagation, but if the medium is normal they oscillate around the symmetry axis or planes by remaining of the narrow-beam type, namely, bounded by a caustic surface (Tien et al., 1965; Casperson, 1973, 1976). Clearly, the effects due to the actual presence of the boundaries of the graded-index medium can be treated as a perturbation and possibly neglected, only if the external caustic of the beam is internal to the boundaries and sufficiently far from them so that any “tunnel” effect is negligible. Stationary and “oscillating” beams are solutions of the scalar wave equation even in “aberrating” media, where the square refractive index is represented by a polynomial expression of degree larger than 2 in the transverse coordinates. However, they are approximate and not exact solutions as in the case of a parabolic medium. An initially Gaussian beam, while propagating in a nonparabolic medium, acquires an amount of aberrations that depend, in particular, on the aberrations of the medium, defined as the departure of v 2 from the
112
L. RONCHI AND A. M. SCHEGGI
square law and/or from the axial symmetry. Accordingly, the measurement of the aberrations acquired by the beam may serve to determine the aberrations of the medium, and hence may possibly constitute a method for the diagnostic of the medium itself. In the sequel we treat mainly parabolic media and some aberrating media. For other graded-index media, such as those whose profile minimizes the dispersion characteristics, one can refer to the appropriate bibliography (see Section VII1,E). B . Theory of Modal Propagation in a Two-Dimensional Graded-Index Medium Let us assume the space to be filled with a medium whose square refractive index v2 depends upon a single transverse coordinate, say x , =
v2
v2(x)
(89)
The field propagating in it can be divided into TE and TM fields, with either E or H parallel to the y direction, namely,
E = Eyj H = Hyj
(TE) (TM)
The wave equation for the TE modes has the usual form A2E,
+ k2vz(x)Ey= 0
(TE)
(91)
For TM modes, it is expedient to introduce the quantity
-
H, = H , / v ( x )
(92)
which may be easily proved, with the help of Maxwell’s equations, to satisfy the equation
A 2 g u + k 2 N 2 ( x ) z ,= 0
(TM)
(931
I
(94 1
where (Janta and Ctyroky, 1978)
[
N 2 ( x )= vz 1
- 2v12 + v”v k2v4
Accordingly, without loss of generality, one can deal with the only wave equation vzu
+ k2W(x)u = 0
(95)
with N ( x ) = V ( X ) for TE modes and N ( x ) as given in Eq. (94) when TM modes are involved.
113
BEAM WAVEGUIDES AND GUIDED PROPAGATION
In the general case, solutions of Eq. (95) are looked for in the form of quasi-plane waves, namely, waves of the type ux, z) = W x , z) exp(ikyz)
(96)
with Y(x,z) so slowly varying with z that a 2 q / a z 2 can be neglected in comparison with k aY/az (transverse diffusion approximation). For each solutions, the wave equation takes the parabolic form
?ax2 ?!
+ 2iky-aaz9 + k(NZ - y 2 ) 9 = 0
(97)
However, in the particular case of a parabolic medium, where
N2 = Nt
-
N2 X2
(98)
exact solutions of Eq. (95) exist, with Y = U(x), independent of z, which therefore do not deform during propagation and will be denoted as modal beams. Their derivation will be recalled in the next section. C. Propagation in a Parabolic Two-Dimensional Medium
1. Modal Propagation It is well known (Marcuse, 1972)that Eq. (95) with N2(x)given by Eq. (98) admits solutions of the type (96) with Jr independent of z. By introducing Eq. (96) into the wave equation (95)one obtains for $(x) the equation
It may be easily verified that Eq. (99) is satisfied by
where H, denotes the Hermite polynomial of order n, provided that
y2 = N:
-
K (2n
+
1)NiI2
(102)
In order to verify Eqs. (100)-(102), one has simply to recall that the Hermite polynomial of order n satisfies the differential equation
HC - 2XHA
+ 2nH,
=
0
(103)
where the primes indicate derivatives with respect to the argument X.
114
L. RONCHI AND A. M. SCHEGGI
In Eqs. (101) and (102), Nil2 indicates the root with positive real part. The other root of Nz corresponds in Eq. (100) to a field amplitude that exponentially diverges for 1x1 + m, and therefore cannot be associated with a field in indefinite space. Note also that, as follows from Eq. (102), the two fields with sign opposite to Nil2 have a different phase velocity in the z direction. By writing Nil2
=a
- ib
(104)
and substituting into Eqs. (101) and (100) the quantity ( h / ~ a )turns ” ~ out to represent the beam width, and l / b the (axial) curvature of the wavefronts. It can be noted that the transverse field distribution across a plane z = const of the beam Eq. (100) is a different from that of the usual Gaussian beams propagating in free space, known from the theory of open resonators (Kogelnik and Li, 1966). If we write a usual free-space Gaussian beam in the form
the quantities u , q, and P depend on z in the well-known way (Kogelnik and Li, 1966), but for any z,
In other words, the “scale” of the Hermite polynomial coincides with the beam width w , and therefore is real. The consequence is that the transverse distribution of lu(x, z)I2 has the well-known trend reported in Fig. 29
+I
FIG.29. Intensity transverse distribution of the Hermite-Gaussian beam of order n = 10.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
I15
X -
IWI of intensity transverse distribution of a generalized HermiteExamples FIG. 30. Gaussian beam.
in a particular case, i.e., it is constituted by a sequence of n + 1 maxima, separated by zeros, the values of the maxima increasing with increasing 1x1. On the contrary, a beam of type Eq. (100) has a scale of the Hermite polynomial that may be complex (Siegman, 1973) and therefore the intensity transverse distribution may be qualitatively quite different as appears, for example, from Fig. 30 (Pratesi and Ronchi, 1977).
2. “Oscillating” Guided Beams in a Parabolic Two-Dimensional Medium The beams described in Section IX,C,1 are modal beams in the sense that the z dependence is simply represented by the factor exp(ikyz). It has been shown, however (Kogelnik, 1965), that a parabolic medium may also sustain beams bounded by a caustic, with parameters dependent upon z . Such beams may be divided into two classes, namely, on-axis beams, which remain symmetric with respect to the z axis, and off-axis beams (Casperson, 1973, 1976). Such beams describe the propagation in a half-space originating at the plane I: normal to the z axis (Fig. 31), when a free-space Gaussian beam enters it through Z. The general theory may be described as follows. Let us start with Eqs. (96) and (97) and look for solutions of Eq. (97) of the form $(x, z) =
H,
[q
(x -
[
5 ) ] exp ik
(5
+ Sx) + iP]
(107)
with u = u(z), 4 = q(z), 5 = t(z), S = S(z), and P = P ( z ) . (The introduction of the phase term P allows us to put y = N o without loss of generality). Equation (107) describes a beam whose axis oscillates around the z axis, and whose beam-width and curvature of the wavefronts change al-
L. RONCHI AND A. M. SCHEGGI
I16
FREE-SPACE
c FIG.3 1. A free-space Gaussian beam impinging onto a graded-index medium through a plane interface.
most periodically with increasing z (see, for example, Fig. 32). Even the structure of the transverse amplitude distribution changes almost periodically with z. By introducing Eq. (102) into Eq. (97), with N 2given by Eq. (98) and taking into account Eq. (103), we obtain the following set of differential
t
FIG.32. Example of the variation with z of a parameter (half-beamwidth)of an oscillating Gaussian beam in a graded-index medium.
BEAM WAVEGUIDES AND GUIDED PROPAGATION
117
equations:
which may be solved in sequence. The general solution is given by (Kogelnik, 1965)
c tan #4z) 1
9(z) =
(10%
with C +(z) =-z
C = N;", and
c$o
NO
+ 40
( 1 10)
an arbitrary, generally complex, integration constant. Then S(z) =
where 90 = 9(z
=
so
0) = (1/C) tan
$o
and So = S(Z = 0 ) ,
and finally,
1
n+1/2
iP = iPo - In (1 + c9 c2q2)1/2
ik s,Z9; 1 + In un + -2 c 1 c29;;f? (114)
+
The physical meaning of the various parameters 9 0 , uo , t o So, , and Po can be easily understood by matching at z = 0 (Fig. 31) a free-space Gaussian beam, impinging at a general height h with a general incidence angle 0 onto the slab from say the left, with the beam Eq. (107). The only require-
118
L. RONCHI AND A. M. SCHEGGI
ment for e is to be so small that the variations of the parameters of the impinging beam over its section in the plane z = 0 are negligible. If h = 0 and e = 0, one has on-axis beams within the parabolic medium. Otherwise, one has to deal with off-axis beams. It can be noted that
whichever is the initial value q o .In Eq. (1 15), the upper (lower) sign holds when Im C / N o is positive (negative). Since a beam is bounded by a caustic surface if
[see Eq. (106)] it turns out that the beams in the parabolic medium are bounded if Im C / N o < 0, and that q ( z ) tends to - i / C , namely to the value of q for the modal beams [see Eq. (lol)]. As to u , it turns out from Eq. ( 1 12) that it generally tends to infinity (since 1 + C2qz+ 0 when z + m), which means that whichever is the (even) order of the beam impinging onto I:the beam excited on-axis in the parabolic medium generally tends to the fundamental one, since
only if
does u 2 ( z ) not tend to infinity, when z + m, but to 2/kC = 2/kN4‘2, namely to the value of uz for the modal beams, as given by Eq. (101). For free-space Gaussian beams, however, u2 is real and related to q by Eq. (106). Accordingly, the only way of exciting a modal beam of order n # 0 in a half-space filled with a parabolic medium appears to be to illuminate it with an on-axis Gaussian beam of order n, with beam waist at the plane, and with spot size given by
Clearly, thjs is only possible if N2 is real and positive, namely, if the graded-index is real and “normal” (see Section VII1,A).
BEAM WAVEGUIDES A N D GUIDED PROPAGATION
I19
D . Guided Propcigarion in u Nonpurabolic (Two-Dimensional) Medium 1. Modal Propugation
When the square-refractive index of the medium is not represented by a square law of the transverse coordinate x , exact solutions of the wave equation cannot in general be found. In a more general case, one can expand N z ( x )in series of powers of x, truncate the series at a given term (of degree larger than 2), and look for (approximate) solutions of the wave equation. Let us assume N 2 ( x )to be approximated by
Nz (x) = Ng- N2x2-N4x4N6xs-
*
=
Pl (x)
( 1 18)
with I even and look for modal beams of the type u ( x , z) = + ( x ) exp(ikyz)
( 1 19)
The wave equation yields
d2JI + k2[P!(X)- y 2 ] + = 0 dx2 and solutions of it can be found of the form (Pratesi and Ronchi, 1978)
where K, is an (even or odd) polynomial of order n and 0, a polynomial of orderj, whose coefficients are the unknowns of our problem. In the sequel, for simplicity, we assume both K, and n, to be even polynomials of X.
Equation (121) represents guided beams, bounded by a caustic surface, if @(x) has an imaginary part such that
lim Im R,(x)
=
CQ
(122)
for 1x1 + CQ. Introduction of Eq. (121) into Eq. (120) yields a polynomial equation in x, namely
2 K;' + 2 WZ
d C n;K; W
+ ik0;'K, + k2(Prnj2- yZ)K, = 0
(123)
Such an equation, however, gives rise, when the coefficient of each power of x is put equal to zero, to a number of equations larger than the number of unknowns (which are the 4 2 coefficients of K , , the j/2 coefficients of
L. RONCHI AND A. M. SCHEGGI
I20
Q,, and y). Accordingly, it cannot be exactly solved, but only up to the (n + 0th power of x included. For certain purposes, such as the diagnostic of graded-index media, it is sufficient to assume j = 1. Consider, for example, the case 1 = 4 . Equation (123) indicates thatj has to be taken larger than 2, which means that a nonparabolic medium sustains aberrating Gaussian beams. If one is interested in the determination of the coefficient N4 of N2(x), which is expected to introduce some amount of third-order spherical aberration, one can limit himself to consider the fourth-degree term of Q, and therefore can neglect the terms with j > 4 . In other words, one can choose j = 4 . This case has been treated in detail, with the following results. If we write x2
Q4(x)
=-
2q
+4Q3 x4
it turns out that
and
On the other hand, K , turn out to satisfy the differential equation
K;
-
2Xp(X)Kk
+ T(X)K,= 0
(127)
with X = d ? x / w , and 1 = -ik wz
2q
p(x) = 1
’
N4 +--2k N:12
{
T ( X ) = 2n 1
x2
“
+--2k N:/2 N4 ‘z (n - 1 )
+XI)
Clearly, the polynomial K , can be considered as a perturbed Hermite polynomial of order n , and tends to it when N4+ 0.
2 . General Aberrating Gaussian Beams in a Nonparabolic Medium When solutions of Eq. (97) are looked for in the form of Eq. (96) with JI depending upon both x and z, the problem may be treated in an analogous even if more laborious way. By limiting ourselves to the on-axis case and
BEAM WAVEGUIDES AND GUIDED PROPAGATION
121
to the fundamental mode n = 0, it can be easily verified, in analogy to the cylindrical case (Ronchi, 1978b), that I/&,
[
(3+ 4-3
z ) = exp ik -
-
with q = 9 ( z ) and Q = Q ( z ) , satisfies Eq. (97), provided that
where C#J = +(z) and C are given by Eqs. (1 lo), F ( z ) is given by F(z) =
J,'sin4 4 d+
and terms of the relative order l/k have been neglected. The second Eq. (130) describes how the fourth-order coefficient of the complex phase f14 of $ evolves during the propagation within the graded-index medium, while, clearly, ao/sin4 c $ ~ measures the amount of aberration at the input plane of the impinging beam. In other words, if the half-space of Fig. 32 is illuminated by a Gaussian beam free from aberrations, one has to choose a0 = 0, and the second Eq. (130) indicates that the amount of spherical aberration of the beam in the graded-index medium is proportional to N 4 . Strictly speaking, one should not say that the beam Eq. (129) inside the graded-index medium is affected by spherical aberration, but rather that if it emerges from the graded-index medium after a certain path it gives rise to a free-space Gaussian beam affected by spherical aberration (Ronchi, 1978~).As a matter of fact, such emerging beam can be written as
where 9 = q ( z ) = qo + z ,
1/4Q3 = c 0 / q 4 ( z )
(133)
Since, for z + a,1/4@ diverges as z 4 , one concludes that the real part of the term x4/4Q3 in the complex phase of ue represents third-order spherical aberration (Toraldo di Francia, 1958). Clearly, the coefficient of the fourth-order term of the phase of the emerging beam is in direct relation with that of the beam inside the graded-index medium, and therefore with
122
L. RONCHI AND A. M. SCHEGGI
N 4 . Accordingly, the measurements of N4 may be reduced to that of the spherical aberration of a beam. This problem has been treated in detail for the three-dimensional case by Ronchi (1978b).
E . Guided Beams in a Three-Dimensional Graded-Index Medium The analysis of Sections IX,B -IX,Dcan be extended to media where the square refractive index N 2 is a function of both transverse coordinates x and y: (134)
N 2 = N 2 ( x ,y )
Two cases may be distinguished, according as the dependence on x and y occurs through r2 = x2 + y 2 (stigmatic case) or through x2 and y2 separately (astigmatic case). In both cases, it is easily proved that the medium may sustain bounded beams, both of the modal type, which propagate without deformations of the transverse field distribution, and oscillating beams, whose parameters depend upon z , but in a quasi-periodic way. From Maxwell's equations, assuming p = po = const, E = E(r), one obtains
V2E + k2N2(r)E = -grad
(I:- grad
V2H + k2N2(r)H= io grad e h E
Q
1
E
(135) (1 36)
where N 2 ( r )= 4 r ) / e o
(137)
Equations (135) and (136) reduce to the standard form of the vectorial wave equation when E varies sufficiently slow with r, so that one can neglect the right-hand sides of them. In other words, when the gradient of the refractive index has two Cartesian transverse components, the wave equation in the usual form holds only approximately (see also Kurtz and Streifer, 1969a,b). In analogy with the two-dimensional problem, the wave equation
V'U
+ k2N2u= 0
(138)
can be solved exactly only when N 2 is represented by a square-law of the transverse coordinates, and modal solutions are looked for. In any other case, one introduces the transverse diffusion approximation (see Section IX,B) and replaces Eq. (138) by the differential equation of parabolic type x2
+ ??!ay2 b + 2ikNo*az + k2(N2- N$)$= 0
I23
BEAM WAVEGUIDES AND GUIDED PROPAGATION
where No = N ( 0 ) and @ is the “amplitude” of u, i.e., is related to u by u(x, y , z) = +(x, y , z)eikNoz
(140)
and is assumed to vary only slowly with z .
F. Propagation in a Three-Dimensional Medium 1. Modal Beams in a Parabolic Medium
Let us consider a medium specified by =
NB
-
N2r2
In analogy with the two-dimensional case, it is easily verified that the scalar wave equation is satisfied by u(r, z ) =
LE ( 2 6 ) exp (ikyz
(+)m
+ ik-r2 + imd) 2q
(142)
where [Ln(p)Im denotes the generalized Laguerre polynomial, obeying the differential equation
By introducing Eq. (142) into the wave equation, one h d s
_1 -- _ _ik v2
2q
k --N;12, 2
2 y 2 = N i - k (2n
+ m + 1)N112
(144)
When N 2 is given by Eq. (141), the wave equation also admits solutions separated in x and y , namely of the form ikyz
u(x, y , z ) = H ,
+ ik- x 2
+
2q
”)
(145)
with H , and H,,, denoting the Hermite polynomials of order n and m, respectively. For V and q , the first Eq. (144) still holds, while y is given by y2
=
NB - 2 Nll2 ( n + m + l )
(146)
As in the two-dimensional case, the above beams are of the bounded type if Nil2 denotes the square root of N z with positive real part. Equation (145) is easily extended to treat the case of “elliptical” media, where
W = NB - Nz1X2 - Nzzy2
(147)
L. RONCHI AND A. M. SCHEGGI
124
In this case, solutions of the wave equation of the form
exist, provided that l/VT = -ik/2ql = (k/2)N?J2
(Re N i f > 0)
l/V%= - i k / 2 q 2 = (k/2)N:L2
(Re N?J > 0)
(149)
and
Clearly, Eq. (148) represents a beam affected by an astigmatism that depends only on the ellipticity of the medium. Measurements of the astigmatism of the beam may be carried out to study how much N2,is different from N,, in a practical case. In practice, however, some problem may arise in exciting in the medium one of these modal beams, say the fundamental one. If we have a half-space originating at a plane normal to the z axis filled with a parabolic medium (Fig. 31) and illuminate it through Z with a free-space Gaussian beam, this beam may excite the modal beam only if it is astigmatic with the proper astigmatism at 2.If one illuminates the medium by a free-space stigmatic Gaussian beam, the beam inside the medium turns out not to be a modal beam, but a beam of the oscillating type, which is described next. A consequence is that the astigmatism of the field inside the medium and that of the beam emerging from it through a plane Z' parallel to Z are functions not only of N,, and N,,, but also of the distance from C to Z'. An interferometric method for deducing the parameters of the medium from measurements of the astigmatism of the emerging beam has been described (Ronchi, 1978a).
2 . Oscillating Gaussian Beams in a Generally Elliptical Square-Law Medium For simplicity, we limit our analysis to the fundamental on-axis beam. In the usual way, it is easily verified that Eq. (139), with N Z given by Eq. (14) is satisfied by a function of the type (Casperson, 1973)
BEAM WAVEGUIDES AND GUIDED PROPAGATION
provided that q1 = q l ( z ) , q2 = q 2 ( z ) ,q 3 ( z ) , and p lowing system of differential equations:
=
12s
P ( z ) satisfy the fol-
A general solution of the system (152) has not yet been described in the literature. A particular solution of interest in the diagnostic of a graded-index medium, corresponding to a stigmatic free-space Gaussian beam impinging on-axis onto the plane C of Fig. 31 is (Ronchi, 1978d) 1lq3 = 0
(154)
which yields q1 =
1
tan dil(z),
1 q2 = -tan +2(z) N:k2
(155)
with
This solution describes an astigmatic beam whose principal planes do not rotate during the propagation in the graded-index medium, even if its astigmatism turns out to vary with z. Another solution that describes astigmatic beams whose principal planes rotate with z (Arnaud and Kogelnik, 1969; Arnaud, 1969) is (Ronchi, 1978d).
L. RONCHI AND A. M. SCHEGGI
i 26
3 . Guided Beams in a Nonparabolic Medium
Consider finally the case of a medium whose square refractive index
Nz is well described by a fourth-order polynomial of the cylindrical coordinate r : N 2 = N: - N 2r2 - N4r2
(159)
By limiting ourselves to look for the fundamental beam, and introducing the usual approximations (see Section IX,B), we write the solutions of Eq. (139) in the form (Choudari and Felsen, 1974; Felsen, 1976)
with S = S(x, y, z ) and A, = A,(x, y , z ) independent of k . By introducing Eq. (160) into Eq. (139) and equating to zero the coefficients of the various powers of ik, one finds for S the eikonal equation
(g)'+ (g)'+
2 N0 @ az
+ N2r2 + N4r4 = 0
(161)
Equation (161) admits approximate solutions of the form
s = - r2 2q
r4
+4e"
(162)
if q = q ( z ) and Q = Q ( z ) are solutions of the following differential equations:
Hence q is given, as in the two-dimensional case, by 1 4 ( z ) = - tan
C
4(z)
with C = Nil2 and $ 4 2 ) = ( C / N a ) z + &, and integration constant, whereas l/Q3 is given by
(164)
+,, indicating an arbitrary
-=o--1 a N 4 F(z) (165) 4Q3
sin4 4
2C sin4 $,
with F ( z ) given by Eq. (131). As already discussed in the two-dimensional case, if the beam (165)
BEAM WAVEGUIDES A N D GUIDED PROPAGATION
127
passes through a plane interface to free space, it gives rise to a free-space Gaussian beam affected by spherical aberration. Measurements of the spherical aberration of the free-space beam may be used to determine the coefficient N4 in the expression (159) of NZ.To this end, the usual interferometric methods (Ronchi, 1964; Toraldo di Francia, 1958) should be generalized to take into account the Gaussian envelope of the beams (Ronchi, 1978b). When z + m, q ( z ) --* i / C [if, as we assume, Im(C/v,) < 01, and 1/4@ + iN4/8C, which requires Im(N,/C) > 0 for guided beams. It may be easily verified that a beam with JI given by Eqs. (160) and (162) and with i 4=-c,
- 1-
4Q3
i N4
-sc
represents a modal beam of the medium specified by Eq. (159) even if only approximately. With respect to the results found in the two-dimensional case [see Eqs. (125)], Eqs. (166) have been derived by neglecting terms of the order of I l k in comparison with unity (asymptotic approximation).
X. CONCLUSION The subject of this chapter has been a review on electromagnetic wave-guided propagation, aimed at introducing the related physical problems and at giving the mathematical algorithms necessary to the solution of such problems in a variety of cases of interest. Two main types of guided propagation have been considered: in the first the guidance is performed by means of a structure, whereas in the second propagation occurs in an infinite medium having particular guiding properties. As for the first type, metallic waveguides and dielectric rods as well as open transverse structures, the so-called beam waveguides, have been considered. While in the metallic waveguides (which are assumed to be filled with a homogeneous medium) the guidance properties depend only on the geometry of the guide, in the dielectric waveguides the guidance depends both on the geometry and on the medium properties constituting the dielectric rod. Accordingly, it seemed to us of particular interest to examine also propagation in unbounded media having transverse refractive index distribution of the same type as that presented by dielectric waveguides (in particular, optical fibers) because knowledge of the propagation characteristics in the unbounded medium can give interesting even if approximate information on propagation in the dielectric waveguides. At the same time it has been shown that one can utilize such
128
L. RONCHI AND A. M. SCHEGGI
propagation properties for developing methods for the medium diagnostic. As for the beam waveguides, apart from the particular principle of operation on which such guiding structures are based, they are of interest also in view of the thematic they originated. In the course of the chapter basic concepts have been presented, for which the reader can refer to the literature cited, and also the state of the art of current research has been reported with a rather wide list of contribution references. However, we are well aware that the literature presented is not complete. This does not depend on a willful neglect of some papers but rather on an unavoidable omission due to the breadth of the subject and of the related literature. In conclusion, we have attempted to provide a review that can be useful, for different reasons, both to people who are entering to the field and to specialists.
APPENDIX1. RAY TRACING METHOD This section is concerned with a description of the ray method, which generally speaking constitutes a valid tool for studying complicated propagation and diffraction problems of high-frequency waves in terms of local plane waves and their associated trajectories. The method up to now has been applied only to scalar fields and consequently has not taken into account, for instance, polarization and related effects. However, it turns out to be very efficient and may offer a visualization of the physical problems. Starting with a description of the method in general, we then consider its application to the particular case of propagation in dielectric waveguides. Let us consider the scalar-wave equation relative to a field in a region exterior to the sources and in a regime where the properties of the (lossless, isotropic) medium change slowly over the distance of the order of the local wavelength, V2u(r) + k2NZ(r)u(r)= 0
(167)
where r denotes the generic point. The high-frequency asymptotic solution (k + m) can be written in the form (Felsen and Marcuvitz, 1973)
BEAM WAVEGUIDES A N D GUIDED PROPAGATION
I29
where uj(r) and X(r)are assumed to be independent of the wavenumber k. By substitution of Eq. (168) into Eq. (167) one gets
This equation can be satisfied by equating to zero independently the coefficient of each power of k. Accordingly, one obtains
-
(V vx (V * Vy
(VXY = N2
+ 2vx ’ V)uo = 0 + 2Vx V)uj = - V2uj-1,
( 170)
j 2 1
The first equation is the “eikonal equation” of geometrical optics (Born and Wolf, 1959) while the second and third equations represent the transport equations for the amplitude coefficients in Eq. (168). The lowestorder solution of Eq. (167) in the high-frequency limit is u = u,,e*& ,which dominates over the remaining terms in Eq. (168) if JVNJIN is small compared with the total wavelength, that is, if IVNl/kNZ4 1 . From the eiconal equations one has V x = Ns, where s = dr/ds denotes the unit vector in the direction of VX. Then, after some manipulation (Born and Wolf, 1959), one gets the ray equation
$(Ng)
=
VN
For what concerns the amplitude uo(r)of the field, one can easily write the second Eq. (170) in the form of the energy conservation statement: V
(lu0lZNs) =0 = V S
(172)
where S denotes the time-averaged energy flux density. From Eq. (172) one obtains that the intensity at two points r2 and rl along a ray is related by the ratio of the infinitesimal areas A 2 , A l intersected by a narrow ray tube on two wavefronts through rl , r2, precisely (Fig. 33)
A2
At
FIG.33.
A narrow ray tube in a graded-index medium.
130
L. RONCHI AND A. M. SCHEGGI
It is evident from this formula that the condition A(r) # 0 must be satisfied, which means that when the ray tube converges on a line or a point (caustic or focus), that is, for A(r) + 0, a more detailed analysis is necessary. When considering propagation in a graded-index dielectric waveguide, the ray method is particularly suitable for evaluating the dispersion characteristics of the different modes. In fact, once the ray paths have been evaluated through Eq. (171) one can impose the resonance condition (see Section V,A) k
f
+ r n 7r’ +~ iC In p r
c Vx *dl= 2 1 ~ m
(174)
I
where C is any closed curve within the set of ray families (for more details, see Maurer and Felsen, 1967). The third term on the right-hand side takes into account the reflections at the boundaries of the waveguide, whereas the second term accounts for the ?r/2 phase jump experienced by the ray after touching a caustic, if any. In practice it may be convenient to consider the ray projection on the transverse cross section of the waveguide and write a relation of the type of Eq. (174) for each transverse coordinate. In particular, for a waveguide with diameter very large with respect to A, and with a transverse graded index, the rays are trapped by a continuous refraction and tangent to an external caustic without practically reaching the boundaries except near cutoff. Accordingly, a further approximation can be made by eliminating the term on the right-hand side of Eq. (174) corresponding to reflections at the boundaries because its contribution can be, with respect to the other two terms, neglegible even for rays with high inclination with respect to the waveguide axis (Scheggi et al., 1975). This is equivalent to considering an unbounded guiding medium. APPENDIX11. THE WKB APPROXIMATION APPLIEDTO MODALPROPAGATION IN A SLAB Let us start with the wave equation in the form
V 2 u + k2N2u= 0 and consider the cases when N depends upon a single transverse coordinate, say x . Modal solutions of Eq. (167) are of the form (168) with
x
=
$44 + yz
(175)
BEAM WAVEGUIDES AND GUIDED PROPAGATION
131
From the first Eq. (170) one obtains Q, = k
i,” ( N 2
- y2)112dx
(176)
By integration of the second Eq. (170) one fhds in the WKB (or optical) approximation, valid for k + 03, the solutions of Eq. (169) in the form
where
w
= W(X) =
[ N 2 ( x )- Y * ] ” ~
(Re w > 0)
(178)
In a slab waveguide the field will be represented by =1 (Cefk@ +
De-lk@),-rkYZ =
W1/2
JI(x)eikyS
(17%
with C and D constants. Since the WKB approximation ceases to be applicable in the vicinity of the values of x , if any, for which ~ ( x =) 0 (“turning” points), which specify the position of the caustics, the problem arises of “joining” the solution on one side of a caustic with the solution on the other side. Note that, sufficiently far from the caustics, on both sides of each caustic, the solutions have the same form Eq. (179), but with different coefficients C and D. The above problem is generally treated for real media and real y, but the procedure may easily be extended to complex N and/or y. The results are as follows. Let us denote by x, a generally complex root of the equation ~ ( x =) 0, and use Eq. (179) for say x < Re xc , whereas for x > Re x , the field will be assumed to be expressed by 1 +(x) = p (C’elk@ D’ e - f w )
+
If we put
it turns out that C’ and D‘ are related to C and D by C’ = iC + De-zfks,
D’
=
f(Ce2‘kS + iD)
(182)
If the expression (180) of + ( x ) has to hold up to x = m (since there are not other caustics at any x > Re xc , and the medium is not limited to a
L. RONCHI AND A. M. SCHEGGI
132
slab), the radiation condition indicates that
D’= 0 since the term D’exp( - ik(p)represents an “incoming” wave. In the applications to the wave propagation in a planar slab of graded-index medium, the above expressions are to be used to impose the boundary conditions at say x = _+d,which allow one to determine the ratio C / D and the eigenvalue y . Note that Eq. (183), which constitutes a remarkable simplification [for some expressions of N 2 ( x ) it allows the approximate determination of y without the use of a computer] is often used even for propagation in a slab, when the most external caustics are inside the slab, sufficiently far from the boundaries ()Re x , ) e d ) . The degree of approximation obtainable with the WKB method has been discussed by several authors (Gedeon, 1974; Janta and Ctyroky, 1978). If the complex-amplitude factor w - ” ~ appearing in Eqs. (179) and (180) is not taken into account in writing the boundary conditions, namely, the continuity of the field and of its normal derivative at x = d, between a slab of graded-index medium and a homogeneous medium (of refractive index ve),one obtains tan k((pd + (po) = where
(pd =
- i- w1d [v:
- y2]’”
(184)
(p(x = d ) , wd = w ( x = d ) , and (po is defhed by
according as Eq. (178) or Eq. (180) is used. The WKB formulas generally given in the literature can be derived from these relations. In the above formulas, all quantities are in general complex except x and z. Note that for d Re x, , the preceding treatment no longer yields a good approximation. As is well known, in place of Eqs. (179) and (180) one has to write
-
+(x) = a A i ( - X )
+ bBi(-X)
(186) where Ai and Bi denote the Airy functio.ns (Abramowitz and Stegun, 1965),
x=
k2l3f13(xc
- x)
2 (larg X I< 3 r)
(187)
BEAM WAVEGUIDES AND GUIDED PROPAGATION
I33
with -d
f =dx N 2 ( x c ) and a =
b = -kl‘sh f ll6
+ D exp[-
{C exp[i(k$
- m/4)]
{C exp[i(k6
+ ~ / 4 ) ]+ D exp[-
J
ik$ i<@
- rr/4)}
+ m/4)}
( 189)
As a final remark, it is to be noted that the above formulas may be applied only if the caustics are sufficiently far from one another. When two caustics approach one another, the analysis is more complicated (Felsen and Marcuvitz, 1973).
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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS, VOL. 51
Elementary Attachment and Detachment Processes. I R. STEPHEN BERRY Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois
............
................ ............................................... 111. Orders of Magnitude: General Considerations . .................... IV. Specific Processes. ..................................... A. Radiationless Transitions .................... B. Autoionization or Preionization and Autodetachment . . . . . . . . . . . . C. Predissociation .................................... I . Goals
11. Classification of Pro
D. Inverse Autoionization, Dielectronic Recombination, Attachment ....................................... E. Inverse Predissociation ............................................... F. Radiative Recombination and Attachment ..................... G . Associative Ionization, Chemiioni Penning Ionization . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .........
137 I38 143 145 145 146 151 155 156 158
I59 177
I. GOALS This is the first part of a two-part review of the mechanisms whereby electrons attach and detach themselves in atomic and molecular processes. We especially consider processes in which the particle energies are all relatively low, up to no more than a few tens of volts, so that the processes considered here are the kind that occur in glow discharges, arcs, conventional plasmas, and shock-heated gases. The subject includes the mechanisms of ionization and recombination associated with formation and destruction of positive ions, with the attachment and detachment processes that form or destroy negative ions, and with ion pair formation. The literature on these subjects is so vast that it would be quite impossible to present an exhaustive survey of what has been written. Instead, the following discussion presents the current conI37 Copyrighi 0 1980 by Academic Press, Inc All rights of reproduction in any form reserved ISBN 0-12-014651-7
138
R. STEPHEN BERRY
ception of the subject, with emphasis on the general unifying ideas that help one to intuit from known systems what is likely to occur with new substances or conditions. Selected examples illustrate the specific processes. The magnitudes of the rates of elementary processes are discussed, hopefully with enough interpretation to allow the reader to infer whether known systems can be used as suitable analogs or starting points for thinking about an unstudied system. We have not attempted to review the extensive phenomenological and macroscopic interpretations of attachment and detachment processes. For example, the recombination of electrons with positive ions is, under many conditions, the result of several elementary radiative and collisional reactions occurring in concert, with electrons acting as the third body in three-body capture reactions and as the principal carrier of excitation energy in inelastic collisions. The action of the entire set of these reactions is called collisional-radiative recombination (Bates and Kingston, 1961 ; D’Angelo, 1961; McWhirter, 1961; Bates et al., 1962). We shall refer to such processes only when the aggregated representation bears some direct connection to the specific microscopic processes. Other reviews may be of assistance to the reader. Recombination generally has been surveyed by Bates ( 1975); dissociative recombination was reviewed by Bardsley and Biondi (1970); Moseley er al. (1975) discussed ion-ion recombination. Berry ( 1974b) has reviewed the elementary mechanisms of ionization at low energies. Chemiionization has been reviewed recently by Peeters et al. (1969), Berry (1970), and Fontijn (1971), and Penning ionization by Niehaus (1973), Berry (1974a), Hotop (1974), and Manus (1976). Because the text is not intended to be exhaustive, the author apologizes in advance to those researchers whose works have gone unmentioned in the text and are omitted from the bibliography. 11. CLASSIFICATION OF PROCESSES
The processes of interest to us here can be described for the most part in terms of the traditional quantum-mechanical models for atomic and molecular collision processes. Well before or after a reactive encounter, the system can be described by the specification of a quantum state, which in turn consists of specifications of the electronic state and the state of the nuclear system, meaning translation and especially rotation and vibration. This separation is often only valid well before or well after the reactive event. Any theoretical description of the transition from one state to another requires that there is an intermediate interval in which the atomic system is in a superposition of its asymptotic states. However, the
ATTACHMENT AND DETACHMENT PROCESSES. I
139
asymptotic states are what we normally observe; moreover they provide a natural vocabulary for the classification of attachment and detachment processes. This language refers to the character of the nuclear (vibrational-rotational) states, the electronic states, and the chemical composition of the initial and final species, in terms of which things are free and which are stuck together. Table I is a catalog of these processes, with some of the names commonly assigned to them. We shall return to these processes, to give important examples, to describe how they are studied experimentally and theoretically, and to present typical cross sections and rates. A second way to classify the processes is according to the duration of a compound intermediate state. When the duration of such a state is short, comparable with the duration of elastic atom-atom or atomelectron encounters-whichever is appropriate for the case in pointthen the process is called direct. When an intermediate compound state can be identified and have a lifetime comparable to many Bohr periods (for ionization or detachment) or for many vibrational periods (for heavy-particle processes), then the process is said to be indirect. All but three groups of processes in Table I are more or less indirect; the direct processes are collisional ionization and detachment, photoionization and detachment, and photodissociation. Penning ionization and detachment may occur at long enough distances to be considered direct, in many instances. As we shall see, the indirect processes play a very important role, often the dominant role, in attachment and detachment processes at thermal energies. Indirect processes can be thought of as those in which the exit channel of interests is energetically allowed, but is not readily available because the requisite energy is not all in the appropriate degree of freedom. The system can only go off to its asymptotic region in the desired exit channel when enough energy has collected in one (or a very few) degrees of freedom; if more than one degree of freedom is involved, even the relative phases of the different modes may have to lie within particular ranges. The indirect processes can be usefully characterized not only by their final channels, but by the kinds of degree of freedom in which energy is stored in their intermediate states, so as to retard passage to the final states. Thus there are processes known in which energy is stored in electronic excitation, vibrational excitation, rotational excitation, and even in hyperfine excitation. One can think of the indirect process as involving the leakage of energy from such an internal degree of freedom to an ionizing or dissociating mode. When the energy is stored in electronic excitation, as in the doubly excited states of the helium atom (Madden and Codling, 1965; Sinanoglu and Herrick, 1973; Wulfman, 1973; Herrick and
TABLE I CLASSIFICATION OP ELEMENTARY ATOMIC/MOLECULAR Initial state
Rocess number
6 0
Nuclear
PROCESSES
Final state
Electronic
Nuclear
Electronic
Special conditions
Name and process
1
bound
bound, excited
bound, excited
bound
2
bound
bound, excited
bound
free
free
bound
predissociation ABorAB*+A+ B
bound, excited
inverse autoionization or preionization; dielectronic recombination AB+ + e + AB*; (radiationless, nondissociative) attachment AB + e + ABinverse predissociation A + B + AB* or AB
3
4
or bound, excited bound
or
radiationless transition; internal conversion (intersystem crossing if forbidden with respect to change of electron spin) AB + AB* autoionization, preionization AB or AB* + AB+ + e; autodetachment AB-+ AB + e
bound bound, excited
bound, excited
bound
bound
free
bound
or
bound, excited or
bound bound, excited
5
free
bound
bound
bound, excited
bound
6
free
bound
bound
bound
photon released
radiative recombination A + B + AB + hv, AB+ + e + AB radiative attachment AB + e + AB- + hv
+ hv,
7
free
bound
bound
free
associative ionization, chemiionization, Hornbeck-Molnar process A or A* + B + AB + e; associative detachment
8
free, excited
bound
bound
free
Penning ionization A* Penning detachment
9
free
bound, excited
bound
10
bound
free
free, rearranged free
11
free
bound
free
free
12
free
bound
free
bound, ionic
13
free
bound, ionic
free
bound
14
bound
bound, excited
free
bound, ionic
15
bound
bound
bound
free
photon absorbed
16
bound
bound
free
bound
photon absorbed
17
bound
bound
free
bound
A+B-+AB+e
+ B + A + B+ + e;
+
A* + B - + A + B e rearrangement ionization, Kuprianov process A* + BC + AB+ + C + e dissociative recombination AB+ + e + A + B; dissociative attachment AB +
bound
e+A+B-
coliisional ionization
A + B + A + B+ + e; collisional detachment A + B--+ A + B + e ion-pair formation A
+B
-+
A+
+ B-
ion-ion neutralization (occasionally recombination) A+ + B - + A + B (or AB) (dissociative) ion-pair formation AB* + A+
+ B-
photoionization AB + hv + AB+ + e; photodetachment AB- + hv -+ AB + e photodissociation AB + hv + A + B or AB
+ hv+
A+ + B-
dissociative ionization AB* -+ A + B+ + e
I42
R. STEPHEN BERRY
Sinanoglu, 1975, and references therein), the negative-ion resonance states of He- (Schulz, 1963; Massey, 1976), or the rapidly autoionizing states of N, (Duzy and Berry, 1976), the mechanism of passage to the decaying state is the transfer of energy via electron correlation from one electron to another. When the energy is stored in vibration, as in most of the autoionizing states of H2(Dehmer and Chupka, 1976, 1977; Berry and Nielsen, 1970) or in some of the autoionizing states of N, (Duzy and Berry, 1976) the mechanism is one of energy transfer from vibrational excitation to electronic energy, promoting the active electron from an excited bound level such as a hydrogen-atom-like Rydberg orbital into another. Rotationally induced autoionization for Rydberg levels of H, fall above the first ionization limit but below the energy of HZ in its first excited vibrational state (Herzberg and Jungen, 1972). Here, rotational excitation passes into electronic excitation. Both vibrationally and rotationally induced processes involve the breakdown of the Born -0ppenheimer approximation; the active electron does not follow the moving nuclear frame precisely and is therefore not in a stationary state. Its state, rather, is a mixture usually dominated by a single bound component that would be stationary if the nuclei were standing still and that is responsible for the long life of the intermediate, together with free components that correspond to the final channel. That channel may be free with respect to an electron (ionization) or a heavy particle (dissociation), or it may even be bound but have its energy dispersed among so many degrees of freedom that the system has negligible probability of finding its way back to the intermediate state (radiationless transitions; Freed, 1976). We shall not dwell in detail here on the mathematical representations of the direct and various indirect processes. However, it is useful to introduce a little of the language and viewpoint of the theoretical approaches to clarify some of the later discussion. In a direct process, the system is described initially by a Hamiltonian X I , and a state VIthat, for an initially free system, is factorable into at least two functions @fi and @& at times well before the encounter of A and B, whatever they may be. The final state qFis not necessarily factorable into a single @,“c and @fj; the final state produced by the interaction-represented by a term X’ in the total Hamiltonian X-may be a superposition of many channels of the separated particles C and D, or even a superposition of channels corresponding to different products C + D , C’ + D’, etc. The probability of finding a particular final (observable) stationary state of the products, denoted by C(1) D(*)with wavefunction @p)@g) onto the actual finalstate wavefunction is given by
+
**,
P(c‘”,D‘h’) = I( VF\@V)@g))12
(1)
A'ITACHMENT AND DETACHMENT PROCESSES. I
I43
There may be constants required to normalize this probability, depending on the choice of scaling of the wavefunctions, especially with regard to the number or density of final states. The corresponding flux of a particular particle to the detector is given by this probability, multiplied by the appropriate velocity of the particle in the laboratory system, which must generally be derived by transformation from a velocity or momentum in a coordinate system in particle-fixed coordinates. The functions @y) and @,6k) can usually be presumed to be known, at least in approximate form and frequently, for atoms, diatomic and simple triatomic systems, in quite accurate form. The mathematical problem is one of finding qFfrom a knowledge of @i, ah, XI,and X'.Most of the theoretical work on attachment and detachment processes in gases has been directed toward ways to obtain qF;only a small fraction has dealt with the phenomenological problems or the coupling of the primary ionization, detachment, and attachment processes with other relaxation processes in complex systems. 111. ORDERSOF MAGNITUDE: GENERAL CONSIDERATIONS
In general, two factors govern the rates of most of the indirect processes in our list: the square of the quantum amplitude given explicitly in Eq. (1) and the density of final states pF-the number of states per unit of energy-which is contained implicitly in (1). The final state q F c a nbe represented as the state that evolves from a known initial state WI by the action of a succession of physical processes, which we can denote by a mathematical operator X, so (qF(@y)@jf)) can be written as (qIlXI@y)@,6h)). If we simplify X to be a single event, such as the one-step transfer of a quantum from an excited electron to a normal vibrational mode, or a single transition induced by a collision, then we can replace the sometimes-complicatedoperator X with the effective potential operator that couples the two degrees of freedom between which energy flows. ) , a time-dependent operator that may reflect Instead of ( q I I X l @ c @ Dwith the system's passage through a long sequence of states, we can often D ) , Y- is a rather simple mathematical operator, write ( ~ , l V l @ c @ where , in the case of such as the electron-electron Coulomb repulsion e 2 / r i j or, the radiationless vibronic transitions, the nuclear kinetic energy operator 2,(-h2/2pJ)ViJ. (The sum is taken over all motions except translations and p, is an effective mass.) If the density of final states pFis kept explicit, the rate of the process is given by the "golden rule" (Wentzel, 1926, 1927):
R. STEPHEN BERRY
144
Rate of I
+C
+D
27T h
= - I(Y,~Srl@,CgD,)lzp,
In most of the processes in Table I, both the matrix element and pF govern the behavior of the process. The more internal degrees of freedom, the larger is pF. For radiationless transitions in polyatomics, for example, pF is usually dominated by the vibrational states, in the sense that the more normal modes in the system, the more ways a given amount of energy can be distributed among those modes. The matrix element for electronic-vibration coupling, the kinetic energy operator for nuclear motion in the normal vibrational modes, can be transformed so that its normally dominant part has the form of a vibrational transition dipole between the initial and final states of the mode changing its quantum state. Unlike an ordinary optical transition of the vibrational mode, this transition has an additional factor in its probability amplitude, the nuclear momentum operator acting between the initial and final electronic states. This factor can be expressed as the transition amplitude of the “force” based on the gradient of the effective potential for the electrons arising from a displacement in the nuclei; or, in other words, on the probability amplitudes for breakdown of the Born-Oppenheimer approximation that assumes fixed nuclei to determine the electronic states. If this electronic factor were constant over the amplitude of the active normal mode, the selection rules for vibronic radiationless transitions would be the same as those for infrared spectroscopy. Even when the electronic factor is not constant, the general pattern imposed on the radiationless transitions, as well as other related processes discussed below, is similar to that for infrared transitions. The situation.is one of several in this subject in which it is useful to recognize a systematic hierarchy of transition probabilities. Any rule defining such a hierarchy has come to be known as apropensity rule (Berry, 1966), to distinguish it from a rule governing the yes-no character of a single type of transition. Some very general relations can be considered propensity rules. For example, for indirect processes, the upper limit of the rate for the process is fixed by the rate of motion of the internal degree of freedom that supplies or absorbs the energy for the system to reach its final state, e.g., to set an electron free or to trap an electron into a bound state. The degrees of freedom that can participate in enhancing lifetimes during indirect processes are electronic, vibrational, rotational, and nuclear-magnetic, in order of decreasing rates and decreasing characteristic energies. The upper limits of the rates of energy exchange with these degrees of freedom are their characteristic rates of motion: the Bohr frequency for electronic
145
ATTACHMENT AND DETACHMENT PROCESSES. I
coupling, the vibrational and rotational frequencies for those modes, and the Larmor frequency in the appropriate internal field, for hyperfine or nuclear-magnetic coupling. The propensity rule for vibronic coupling, both in radiationless transitions and in processes that generate or remove free electrons or heavy particles, is this: the probability of transitions involving transfer of one vibrational quantum is much larger (e.g., one to two orders of magnitude) than transitions involving transfer of two vibrational quanta, which in turn are much more probable than processes transferring three vibrational quanta, and so forth. This rule was originally believed to apply well for changes of only a few quanta but only erratically for larger changes (Berry and Nielsen, 1970). However, the one extensive experimental test of the rule gives clear evidence that in at least one system, the vibrationally autoionizing states of H,, the rule applies to all states for which autoionization can be observed (Dehmer and Chupka, 1976, 1977).
IV. SPECIFIC PROCESSES The 17 processes in Table 1 include virtually all the significant simple mechanisms by which electrons attach or detach themselves in gaseous collision processes; complex mechanisms such as collisional-radiativerecombination are excluded. Their roles are in no way equally important. To understand their relative importance, it is worth considering their rates, under typical conditions, and the parameters that govern these rates. We take them in order. A . Radiationless Trunsitions
Radiationless transitions are responsible for almost all the conversion of electronic excitation into vibrational, rotational, and translational excitation in gases, liquids, and solids composed of polyatomic molecules. These transitions provide what is usually the fastest path for turning light into heat. Most radiationless processes that do not involve ionization (analogous ionizing processes fall into categories 2, 7, 9, and sometimes 1 1 ) usually involve coupling of electronic and vibrational motion. Hence the time scale for such vibronic coupling has an upper limit set by vibrations, the slower of these motions. Other modes of motion such as molecular rotation or nuclear spin have frequencies lower than those of vibrations, so that the fastest radiationless transitions occur in the range loi3 sec-I, i.e., in a few vibrational periods. These high rates occur when a
-
I46
R. STEPHEN BERRY
single vibrational quantum has enough energy to match the interval between two electronic energy levels. Radiationless conversion of electronic energy to vibrational energy or to energy in modes of motion with still lower frequencies can be very slow. Even the rate of energy transfer to vibrations can be as low as lo9 or lo7 sec-l. However, in the present context, we are concerned with those radiationless processes in our first category because they sometimes compete with processes of ionization; consequently we are interested in rapid, rather than slow radiationless processes. A general review of this subject was given by Freed (1976).
B. Autoionization or Preionization and Autodetachment Autoionization or preionization and autodetachment are processes in which a bound system absorbs energy and goes into an excited state that remains bound for a period long enough that it can be characterized by bound-state quantum numbers at least for the excited electron, and frequently for vibration and rotation, if it is a molecule that is excited. The distinction between the radiationless processes of category 1 and these is only that an electron is set free in the final state in category 2. In either case, energy must flow between an electronic excitation and another degree of freedom. Autoionization, like many of the processes in Table I, was recognized in the 1920s. Its explanation was even the specific motivation for Wentzel’s development of the expression for the rate of any bound-to-free process from first-order time-dependent perturbation theory, the expression called the “golden rule” since Fermi so named it in his lecture notes on quantum mechanics:
like Eq. (21, where pF is again the density of final states. Autoionization may involve the coupling of one electron’s motion with another-socalled correlation-driven autoionization-or the same couplings just cited for the radiationless processes of category 1. Because the ionizing or detaching electron may pick up its energy from another electron, whose characteristic period is of order sec, the fastest autoionizing processes may occur in almost as short a time. Some examples will illustrate the range of rates of autoionization processes: The widths of spectral lines of autoionizing states in doubly excited helium span the range of autoionization rates for processes in which the free and bound levels are coupled by electron correlation, i.e., in which the energy needed to ionize one electron is temporarily stored and
ATTACHMENT AND DETACHMENT PROCESSES. I
147
shared as the excitation energy of the second electron. The widths are as small as lo-' eV, for the few states nearly forbidden to autoionize, but most are in the range 10-3-0.2 eV, corresponding to lifetimes of 3 x 10-13 sec for the range of most sec for the slowest-decaying states but 1.6 x of the doubly excited states of helium. Rates of vibrationally induced autoionization are highest in the H2 molecule, when it is excited with radiation whose wavelength is shorter than about 82.4 nm, but still corresponds to excitation of less than about 3-4 eV. The widest lines whose width is clearly due to autoionization are about 28 cm-' full width at half-maximum, (FWHM), corresponding to upper-state lifetimes of about 9.5 x 10-14 sec. These, however, only constitute a fraction, probably less than one-tenth of the lines due to autoionizing states. Moreover the potentially competitive decay process of spontaneous fluorescence is much slower, of order sec at fastest, in a molecule. There are some four orders of magnitude within which autoionizing processes may lie and be slow compared with their maximum rate but still very fast on the time scale of the alternative decay process. A portion of the photoionization spectrum of cold H2 is shown in Fig. 1. One can see how dense the autoionizing peaks are in the absorption spectrum of this molecule, and one must keep in mind that the transitions that appear in the photoionization spectrum are only the optically allowed ones. Many others that occur via electron impact are electric-dipole forbidden and can be excited in electric discharges, for example, and thereby contribute to the mechanisms of electron production in molecular gases. Hence one can expect niany molecular ionization cross sections to be dominated, at least in the first few electron volts above threshold, by the peaks associated with autoionization. For one system, superexcited N2, the rates of electron- and vibration-induced autoionization may be compared. Duzy and Berry (1976) evaluated rates of autoionization of the lower states of N: that can decay to the "2,+ ground state or to the first excited A2n, state, and compared their theoretical analysis with the ultraviolet spectroscopic studies of P. K. Carroll (private communication, 1977) and of Dehmer and Chupka (1976, 1977). The rate of the vibronic coupling processes are computed to fall primarily in the range ~09-10*1sec-' if one vibrational quantum is exchanged, and to drop roughly an order of magnitude for each additional vibrational quantum that must pass from vibrational to electronic excitation. These rates are illustrated in Fig. 2 for the npu and n p r Rydberg series. Autoionization of N2by electron correlation is, in some instances, considerably faster, as Fig. 3 shows. Rydberg states built on the A2n, excited N: core can autoionize to N,+(X2Xg+)+ e at rates well above 10" sec-', possibly as high as 3 x 10l2sec-I. The series origi-
-
148
R. STEPHEN BERRY
7;
l cw
np2
1
BOO
"
"
~
'
"
'
801
"
"
I
802
803
L2
l,~.! 804
,
,
. ( ,a ) ._ 805
(d) FIG. 1. A portion of the photoionization spectrum of para-Ht, at 78 K , with a wavelength resolution of 0.016 A. Note the differences in widths, for example, between the narrow npO, u = 1 lines and the broad npO, u = 4 and u = 5 lines, especially at the onset of the series. Taken with permission from Dehmer and Chupka (1976). PHOTON
WAVELENGTH
I49
ATTACHMENT AND DETACHMENT PROCESSES. I
4PV.8
, l J L
70
77 I
8PW
i
75
v
-
np2 3
776
772
7;3
774
28
24 22 21 20 19
777
778
w
-T;;T;xT--
17
16
775
15
779
I0
782 PHOTON
WAVELENGTH
L
784 . .
WE
780 np2 -V=2 - npO V. 2
5PO v.4
-
14
(b)
+ 785
(8)
FIG.1 (Continurd).
nally called O,(N') and O,(v') by Carroll and co-workers (Carroll and Yoshino, 1967, 1972; Carroll and Collins, 1969), which are [N$A211,]ncr,, *nu, and 3nu,respectively, are indeed composed of strong but diffuse lines, while lines in the series built on the ionic ground state exhibit no significant broadening at the resolution used thus far. It is attractive to examine lines associated with autoionizing or predissociating states in terms of the Beutler-Fano line shape
150
R. STEPHEN BERRY 12
m
16
15
A.
745
746
747
748
12
13
:!:
9
npO
np2 v.5 npO v.5
h
749
10
I1
A! 14
750
= " : I
v.5
6 ~ " v.6
I
A
A
750
751
752
753
.-. r
754
14
*
E0
17
12
16
15
14
13 l2
z -_ A >
,
y 755 w
.
.
.
.
I
.
756
.
.
.
1
757
'
'
'
'
I
758
759
0
765
755
n
np2 v.4 wO v.4
760
np2
766
767 PHOTON
768 WAVELENGTH
769
770
18)
FIG.1 (Conrinued).
-
(q + E)2/(1 + E2), where q is essentially the bound-free coupling strength and E the energy away from resonance, both in units of the linewidth r/2. The Beutler-Fano line (Fano, 1961) is generally asymmetric; the sign of the asymmetry parameter q indicates the sign of the coupling. One may, in favorable cases, be able to use the line-shape and especially the sign of the asymmetry of a line to distinguish which decay mechanism is principally responsible for carrying away the excitation energy of a state. Unfortunately, in the case of the Rydberg states of N2, the two
Z
151
ATTACHMENT AND DETACHMENT PROCESSES. I
t
-
""0
0.01
-
'
0.02
0.03
E (a.u.1
E(0.u.)
FIG. 2. Rates of vibronically induced autoionization of Nf leading to N : ( e ~ i ) + e. (a) npo Rydberg series; (b) npa Rydberg series. The energy on the abscissa is taken relative to the ionization threshold. Taken with permission from Duzy and Berry (1976).
decay mechanisms give rise to Beutler-Fano shapes that are probably too similar to be distinguished experimentally. Nonetheless the tool may well be a useful one with other systems. C . Predissociation
Predissociation, while strictly not an attachment or detachment process that involves changing the number of charged particles, needs to be considered here because predissociation sometimes competes with autoionization as a mechanism to carry off excitation energy. The isoergic transition from a bound electronically excited state to the vibrational (dissociation) continuum of another electronic state may be as slow as radiative decay or almost as fast as autoionization. In some situations, predissociation is faster than autoionization, and the excited molecules dissociate rather than ionize. While several studies have dealt with the theoretical analysis of predissociation, and the subject has been reviewed for diatomics (Child, 1975), only a few attacks have been made on the competition of predissociation and autoionization. Berry and Nielsen (1970), using a full quantum-mechanical treatment but in the framework of a
152
R. STEPHEN BERRY
c
a
IO'O
Io9
L-
1
,
, 0.01
,
I
0.02
,
I
0.03
- E (u.u.1 FIG. 3. Rates of electron-correlation-induced autoionization of N: leading to N$(AzrI,,) + e . Numbers in parentheses are the principal quantum numbers of the autoionizing states. The zero of energy is the ionization threshold. Taken with permission from Duzy and Berry (1976).
time-dependent or golden rule perturbation formulation, studied the rates of these two processes for vibrationally induced bound -free coupling for Nakamura (1974, 1975a) developed a the bonding Rydberg states of Hz. formulation more easily applied to arbitrary coupling mechanisms and to more complicated diatomic molecules, first by using the Landau-Zener approximation and the adiabatic approximation, especially for the predissociation channel. Then, by using a multichannel optical model he was able to drop the Landau-Zener approximation and retain the BornOppenheimer approximation for predissociation, but in its semiclassical form (Nakamura, 1975b). The essential physics of the competition is this: a molecule, especially a diatomic molecule, can predissociate with a very high probability in a narrow energy band very close to the energy at which the predissociation becomes energetically possible. In that narrow energy band, which may be comparable to the vibrational spacing of the initial bound state, predissociation may be as fast as the molecular vibrations allow, occurring in only a few vibrational periods. If autoionization is not extremely fast in that energy region, then predissociation will be the dominant decay mechanism there. However, the rate of predissociation falls rather rapidly as
ATTACHMENT AND DETACHMENT PROCESSES. I
153
the molecule’s energy is increased above threshold and autoionization will generally become dominant as the excitation energy gets larger. The situation is not quite that simple, as the energy goes still higher. Bandrauk and Child (1970) showed that predissociation rates may exhibit oscillations with excitation energy, over ranges studied in ordinary optical spectroscopy. No attempts have yet been made to take explicit account of this phenomenon by making an analysis of the relative rates of autoionization and predissociation (the autoionization/predissociation branching ratio) as a function of the energy above the two thresholds. A starting point from which this could be done is contained in Nakamura’s formulation (Nakamura, 1975b). The contrast between autoionization and predissociation comes about largely because the former involves a transfer of energy that sets a previously excited electron free, while the latter involves the transfer of electronic energy to translational energy of nuclei, which are initially in a well-defined rotational-vibrational state. The predissociation process therefore requires the transfer of momentum from the electrons to heavy particles, a notably inefficient process unless the transition occurs when the requisite momentum change is very small. For this momentum change to be small, the nuclei must have nearly the same relative velocity in initial and final state, which in turn means that the two states must have very close and nearly identical effective potential surfaces, or the relative velocity of the nuclei must be close to zero in both the initial and the final state. This is the more common situation, but it occurs only when the two surfaces cross or nearly cross, and the kinetic energy of the nuclei is low enough that they pass through the region of the potential surface crossing at an energy that requires them to move very slowly in the transition region. If the nuclei move too fast, or if the potential curves are too widely separated, the probability of transition is very low, because (in classical terms) the required momentum change is too large. Figure 4 shows how autoionization and predissociation compete as decay mechanisms for H, excited enough to fall into either exit channel. Examples of predissociation by rotational -electronic coupling and by nuclear magnetic -electronic coupling are now known. The water molecule excited in the wavelength region 1070- 1240 A has rotational lines in its spectrum that vary strongly with the rotational state, and the inference is that the excited water molecule may predissociate to H + OH, but only from certain of the levels. The coupling here is due to breakdown of the adiabatic approximation, that electronic motion is so fast that it is not coupled to the nuclear motion. In those excited states, the Coriolis coupling, represented mathematically by action of the nuclear rotational
R. STEPHEN BERRY
154
Autoionization and Predissociation i n H,:puStates
4
6
8
10
I2
14
16
V
FIG.4. Examples of predissociation rates (open circles) and autoionization rates (solid circles) for npu Rydberg states of HI. The horizontal axis is labeled by vibrational quantum number u of the initial state. Taken with permission from Berry and Nielsen (1970).
kinetic energy on the electronic and rotational wavefunction, causes energy to flow between the rotational and electronic degrees of freedom, giving the system enough energy to cross onto a dissociating potential curve. This process was invoked by Oka (1973) as a possible mechanism for the production of inverted rotational populations of OH in the interstellar medium. Predissociation of I, by hyperfine interaction has been demonstrated by Broyer et al. (1976). The hyperfine predissociation rate is discernible as a difference in the J = 0 limit of the decay rates of the ortho and para The predissociation rate is of the same order as the states of 12(B3110+,,). radiative decay rate of the states of this system, lo8 sec-'.
-
ATTACHMENT AND DETACHMENT PROCESSES. 1
I55
D. Inverse Autoionization, Dielectronic Recombination, and Radiationless Attachment
Inverse autoionization, dielectronic recombination, and radiationless attachment are names for the inverses of the processes we have just discussed. No clear distinction can be made between inverse autoionization and dielectronic recombination: an electron strikes an atom or molecule, transfers part of its energy into some degree of freedom other than its own motion, and sticks, at least for a time. The process was suggested by Sayers, and reported by Massey and Bates (1943), described by Bates (1962b), by Bates and Dalgarno (Bates, 1962a), and by Burgess (1964a,b, 1965) for atoms. It has been proposed as a mechanism significant for the neutralization of interstellar CH+ (Solomon and Klemperer, 1972), for which a rate of about 5.7 x 10-TO.'cm3sec-' was suggested as a reasonable magnitude. The rate coefficient is proportional to T-3'2exp(- E / k T ) T - ' , where E is the energy of the quasistable bound state of the compound system, relative to the energy of free ion and electron (with no kinetic energy, naturally) and T is the lifetime of that compound state toward radiation and all other processes that could stabilize the compound system. Bates and Dalgarno (Bates, 1962a)estimate the upper limits of the rate coefficient of this process to be of order 10-lo cm3/sec at 250 K, lo-" at 1000 K, at 4000 K, and -2 x 10-13 at 16,000 K. They point out that the mechanism depends on there being a quasi-stable bound state within about kT of the energy of free ion plus electron, in order for these rates to obtain; if there is such a level, then dielectronic recombination may be a more important mechanism for removing charge than the direct radiative recomand bination mechanism, which ranges between about 5 x 3 x 10-13 cm3/sec over the same range of temperatures. This mechanism was examined by R. S. Berry, J . C. Mackie, and R. L. Taylor (unpublished, 1965) as a way that electrons might recombine with oxygen atoms, when the electron temperature is over 1 eV or more. The process they proposed was O+(2sx2p3,'S)
+e
-D
O*(2sx2p3,ID; 3p, SD or 3F)
0*(2s*2p, 3D;3p3D or 3F)-D O*(2s22p3,' S ; 3p. sD)
(4a) (4b)
The latter step is strongly allowed from the 3F state with a core in an excited term to the 3D with a normal core. The process apparently has not yet been found in decaying hot plasmas, but the 1.6 eV of excitation energy required to carry O+ + e to the neutral state on the right hand side of
I56
R. STEPHEN BERRY
(4a) should be available under many discharge conditions now studied, and the corresponding emission lines of oxygen at 794.8, 795.1, and 795.2 nm should exhibit rather high intensities if this process does indeed occur. Dielectronic recombination has been invoked as the predominant mechanism of recombination of electrons with HJ (McGowan et al., 1976; Auerbach et al., 1977) and with HJ (Caudano et al., 1975; Auerbach et al., 1977). In both cases, McGowan et a/. obtain the same envelopes to the curves of the recombination cross section as do Peart and Dolder (1974a,b). However McGowan et al. report an elaborate series of resonances for the e + H i recombination process, particularly when they look only at the states of HJ with vibrational quantum number v = 0, 1, and 2. They interpret these resonances as excitations of Rydberg states, probably singlet Rydberg states with configurations l s m p o and lsonpr. (Other Rydberg states may also be involved, of course.) It is quite plausible to interpret the resonances as being due to the inverse of the vibronically coupled autoionization studied by Berry and Nielsen (1970; see also Nielsen and Berry, 1968), and observed in high-resolution photoionization spectroscopy, i.e., in the autoionization direction, by Chupka er af. (1975; Dehmer and Chupka; 1976). However, the experiments will require still better resolution or the selection of single initial vibrational states of Hi to establish whether this interpretation is actually right. The cross sections vary as Ez&, and are approximately cm2 at relative collision energies of about 2-3 eV (McGowan et al., 1979). E . Inverse Predissociation Inverse predissociation is, like predissociation, only secondarily relevant to the present discussion. As a precursor to associative ionization or Penning ionization, it may play a very important role in stabilizing a vibrationally bound state for a time long enough to permit ionization to occur. This aspect is discussed in Section G. Rates of inverse predissociation can be expressed as collision frequencies multiplied by transition probabilities (transition rates x mean collision lifetimes). The transition rates to be used here are related to those for predissociation; if the colliding species could be prescribed to be in a specific quantum state with respect to energy and angular momentum, the inverse predissociation rate would be exactly equal to the rate of the inverse predissociation connecting the same pair of free and bound quantum states. However, colliding systems cannot be assigned to precise quantum states of angular momentum, and the best one can do is to estimate what portion of the collisions occur with that angular momentum
ATTACHMENT AND DETACHMENT PROCESSES. I
IS7
value J,, associated with the channel that exhibits inverse predissociation. Suppose the colliding pair of particles has energy E, relative velocity u , and reduced mass p. Furthermore, suppose that the collisions that are used to define the collision frequency occur with impact parameters 6 less than or equal to some maximum b,,,, which we can assume is approximately equal to the distance of closest approach R,,, , for a classical trajectory with that impact parameter. Within the classical approximation, we define the “reactive” impact parameter b, by Jo
(5)
= pvbo/fi
and the increment 660 by
(6)
1 = PV 66o/h
and so the fraction of collisions with angular momentum quantum number Jo is the area of the ring with radius b, and width 660, divided by the area of the circle with radius 6,ax or Rmax:
6n( Jo)/n = 2 ~ 6 660/(~R&,,) , = 2h2Jo/(p2u2R&,,)
(7)
The lifetime of the collision 7 ~ 0 1 1is approximately 2R,,,/v and the number of collisions per second experienced by a B particle in an A-B collision is
nB = ITR;,, u[A]
(8)
where [A] is the concentration, in particles/cm3, with which the B might collide, and the total number of A-B collisions is n,[B] = n,[A] (unless A and B are identical, in which case a factor of 1/2 must be included to avoid double counting). If P(E, J,) is the probability, per unit of energy, per second, that a bound excited compound AB” particle will predissociate into free A + B, then the order of magnitude of the rate of inverse predissociation is
zz 2
10-2sJoP(E, J,)[A][B]
(9) which is still a rather small number even if P ( E , J o ) 1OI2 or 1013because the number of colliding pairs having relative energy in the bandwidth that just reaches the predissociating level is only a narrow slice of the entire Maxwell-Boltzmann distribution. By including the Maxwellian distribution of velocities, Julienne and Krauss (1973) derived the expression for the total rate, including both the X
-
158
R. STEPHEN BERRY
sticky collision that generates the transient excited species and the subsequent emission of radiation: Rate of production of stabilized AB = (COnSt)(~kT)-3'2(T~a~ - Tt,,,)7A2d exp(-E,/kT)
(10)
where 7rad and T ~are, ~ , respectively, the radiative lifetime and lifetime due to all decay paths, of the predissociating level n at energy En above the zero of energy at the dissociation limit. This treatment includes both rotational nonadiabatic coupling and spin -orbit coupling. The vibrational nonadiabatic coupling mechanism was not included, but can play an important role in some systems. Julienne and Krause estimate inverse predissociation rates for the formation of OH with v = 1 that are in the range 5 x cm3sec-' to temperatures as low as about 50 K, and then fall as T falls. Their calculated rate for the recombination of C + H to form CH by inverse predissociation is much faster, of order lo-" at 100 K; CN and CO are expected to have similar rates. From their measured rates of predissociation of CH, Brzozowski et a]. (1976) estimate that the most likely rate of inverse predissociation and radiative stabilization of the A2A state of CH, at 100 K (the temperature they take for their interstellar region is considerably lower than that esti[C][H] cm3sec-*, but mated by Julienne and Krause) is about 2 x if the lowest predissociating the coefficient could be as low as 5 x level of the A state were the thirteenth instead of the twelfth excited rotational state. In other words, the rate of inverse predissociation is very sensitive to the details of the potential curve and to which level is the lowest into which the colliding pair may go. This is especially true when the recombining particles are at a low temperature.
F. Radiative Recombination and Attachment Radiative recombination and attachment was long assumed to be a very improbable process because the transit time at thermal energies of a colliding pair of heavy particles through their region of interaction, 10-12sec,is so much shorter than the fastest radiative lifetimes for visible or ultraviolet radiation, 10-8-10-esec, and the transit time for passage of an electron past an ion is 40-50 times shorter. The two-body process, which is the inverse of photoionization or photodetachment, has been studied since the earliest development of the quantum theory of radiation, beginning in 1923. Bates (1975) has reviewed this history, and cites most of the recent studies relevant to atoms. He also gives the rate coefficients for e + H+ + H ( n , I) as a function of n and I, for electron temperatures of lo4 and 2 x lo4 K, computed from the earlier Burgess
-
-
ATTACHMENT AND DETACHMENT PROCESSES. I
159
(1958, 1964a) functions. The rates have a maximum of about 10-13 cm3/sec for recombination to H(ls) and fall with n and with f for any given n , but for n 2 3, have a maximum at an intermediate f value. For example, the rate coefficient at 104K to give H(6, 0) is 12.2 x cm3/sec, a maximum at 42.9 x 10-l8 to give H(6, 2) and 5.85 x 10-l8 to give H(6, 5 ) , among the rates to give n = 6 atoms. Because the process is essentially a radiative transition to a Rydberg state, the rates for e + H+can probably be used as good guides for any other substance, for those levels well outside the atomic core. Radiative recombination is more important for atoms than for molecules because of the alternative paths open, in molecular systems, to carry the excess energy of the recombination. In fact, dissociative recombination (see below) is far more frequently observed with molecules than radiative recombinat ion. The theory of radiative recombination has been reviewed by Seaton (1968) in the context of atomic line emission from gaseous nebulae, where the low density makes the radiative two-body mechanism the most important. Seaton gives the rate coefficient a(,to produce the atomic level i, in terms of the degeneracies g , and g+ of the ith atomic level and the ion precursor, the electron temperature T e , the light frequency v, the threshold frequency vifor photoionization of the ith level, and the photoionization cross section
The reduced mass p is almost that of the electron, of course.
G. Associative Ionization, Chemiionization, and Associative Detachment; Penning Ionization Associative ionization, chemiionization, and associative detachment are among the most important attachment and detachment processes in ordinary discharges, arcs, and sparks. These processes, together with the very closely related Penning ionization process and the inverse processes of dissociative recombination and attachment, have been studied and reviewed extensively during the past decade (Rundel and Stebbings, 1972; Niehaus, 1973; Berry, 1974b; Hotop, 1974; Shaw, 1974; Manus, 1976), and are now understood as well as any attachment/detachment process with the exception of the radiation-induced processes. Penning ionization
160
R. STEPHEN BERRY
is so closely related that it is convenient to combine it with associative ionization for our discussion here. Several distinctions are worth making at this point. First, the term “chemiionization,” which has been given many meanings in the literature, is used here in a very restricted way, to denote only those ionization processes that can occur spontaneously under conventional ambient temperatures (including those of discharges and, marginally, arcs), i.e., as chemical reactions that occur without external sources of excitation. This definition allows for ionization through collisionsf species in their ground states but excludes most excited-state processes, particularly those involving very energetic metastables such as the He(23S) and He(2lS) states. It is stricter than the definition used, for example, by Fontijn (1974). Moreover, this definition also excludes ionizing collisions of ground-state species at very high relative velocity. With this definition, examples of chemiionization are rather uncommon, albeit important, but their counterparts in dissociative attachment are rather frequently seen. Second, we distinguish associative and Penning ionization simply on the basis that the final state of the former lies in the vibrational bound state manifold, while the latter is in the vibrational continuum. Third, we can distinguish processes involving potential curves that, at least in some approximation, cross each other, from processes involving potential curves that never cross. This distinction is more important for selecting how the rate of the process should be computed than for fixing the physics of the system. Finally, among the associative and Penning processes, one can distinguish processes in which the initial excitation is stored in a metastable species from those in which a species in an optically allowed excited state provides the energy required for ionization. Chemiionization is particularly important in conditions in which small neutral radicals are present, because their recombination is an effective source for the energy of ionization in flames (Peeters er ul., 1969; Fontijn, 1971) and the interstellar medium (Oppenheimer and Dalgarno, 1977). The classic example of chemiionization is the reaction CH + 0 + HCO+ + e, which is the primary source of ions and electrons in most hydrocarbon flames. Evidence for this process is summarized by Fontijn (1971, 1974); it is only marginally exothermic, about 0.2 k 0.2 eV, according to Fontijn (1974). The rate coefficient has been inferred as about 3 x (Peeters er ul., 1969) or possibly as low as 8 x (Miller, 1968), but has not been directly measured for this process. A semiempirical computation was made by MacGregor and Berry (1973)in which a wide variety ofparameters of the potential surface were tested, and a sort of sensitivity analysis of the rate coefficient to these parameters was carried out. The authors favor a rate
161
ATTACHMENT AND DETACHMENT PROCESSES. I
coefficient higher than the experimental estimates, about 2-3 x lo-" cm3sec-' for temperatures between 250and 1000 K. At lower temperatures, the computed rate is very sensitive to the shape of the potential, especially to the presence of any barriers. It is possible that the rate falls as low as lo-', at 3 K. This rate is presumably important for any models of the formation of interstellar HCO+, formerly the mysterious X-ogen (Buhl and Snyder, 1970; Klemperer, 1970; Hollis et al., 1975; Woods et al., 1975), which is presumably produced by the chemiionization reaction of CH and 0 (Dalgarno et al., 1973). Other chemiionization reactions yielding free electrons are such associative reactions as Ti + 0 + TiO+ + e, Zr + 0 + ZrO+ + e, U + 0 + UO+ + e, U + O2 + UO: + e, Gd + 0 + GdO+ + e, Th + 0 + Tho+ + e, Th + O2 + Tho: + e, and U + NO + UNO+ + e, and rearrangement chemiionization such as Th + N 2 0 + Tho+ + N2 + e and U + SF6 + UF: + e+ + SF,. The history of these reactions was reviewed by Fontijin (1974). Cross sections for several of the metal-oxygen chemiionizations were estimated from beam collisions by Fite and his coworkers (1974; Lo and Fite, 1974). However, these must be used with some care because they are all measured with respect to the ratio of the electron impact ionization cross sections of the metal in question and uranium. For Ti + 0, uioniz = 4.53 -+ 1.51 x lo+; for Zr + 0, the cross Oxygen atoms and oxygen section is much larger, 1.29 2 0.38 x molecules have very different behavior: for Th + 0, (T = 1.03 0.32 x but Th + O2 has a cross section of 1.49 2 0.28 x 10-17 (Lo and Fite, 1974); similarly, U 0 gives UO+ with a cross section of 1.62 -+ 0.41 X lo-',, but U + O2gives U 0 2 with a hundredfold smaller cross section, 1.68 -+ 0.27 x 10-17. The sharp differences in these cross sections suggest that there may be two distinct mechanisms by which such chemiionization proceeds, of which only one is open to Zr + 0, Th + 0, and U + 0. It has been suggested (Oppenheimer and Dalgarno, 1977) that chemiionization reactions of metal atoms with oxygen could play an important role in generating free electrons in interstellar clouds. Another class of reactions often called chemiionization are those associated with production of ion pairs. These are discussed later, as process 12, Moutinho et al. (1971), Liner al. (1974), Reck et u l . (1977), and Dispert and Lacmann (1977) all use this terminology. Associative ionization involving one excited species was probably the first ionization process to be identified in terms of a microscopic mechanism. Mohler et ul. (1926) and then Mohler and Boeckner (1930) found ionization in cesium vapor illuminated with light whose frequency lay slightly below the ionization potential of the cesium atom. James Franck correctly suggested that the ionization process is Cs* Cs -+
*
+
+
I62
R. STEPHEN BERRY
Cs: + e. Twice later the process again came to light: Arnot and M'Ewen (1938, 1939) observed He: in a mass spectrometer with electron energies below the ionization potential of helium, and attributed the result to He* + He + He: + e. Still later, Hornbeck and Molnar (1951), whose names are sometimes attached to the associative ionization process, observed He:, Ne:, and Art at appearance potentials below those of He+, Ne+, and Ar+, and, as cause, cited the mechanism invoked by Arnot and M'Ewen and by Franck. Herman and Cermak (1963a) detected the associative process for heteronuclear systems and distinguished it from the Penning process by mass analysis of the product ions. The associative ionization process is known to occur with excited hydrogen atoms (Comes et al., 1968; Chupka er al., 1968; Comes and Wenning, 1969), with rare gas atoms, usually in their metastable excited states (Hornbeck and Molnar, 1951; Kaul and Taubert, 1962; Munson er al., 1%3; Teter ef al., 1966; Herman and Cermak, 1966), with metal atoms in their ground states (Fite P r a/., 1974; Patterson and Siegel, 1976; Siegel and Fite, 1976), with metal atoms in optically allowed excited states (Lee and Mahan, 1965; Fontijn et al., 1975), with metastable excited states of molecules (Herman and Cermak, 1963a; Cermak, 1965), and even with two metastable rare gas atoms colliding with one another (Neynaber and Magnuson, 1975b). High Rydberg states can also be involved as the precursor system's way of storing the requisite energy (Dalidchik and Sayasov, 1965; Hotop and Niehaus, 1968; Matsuzawa, 1971). Even molecular vibration may provide the energy required for ionization, in the case of N2 with alkalis (Schmidt er al., 1974), where the cross sections may be between 30 and 150 Biz. Nakamura (1971) has distinguished associative ionization (AI) processes of A* + B as (1) associative ionization of the first kind, in which the excitation energy E(A*) is greater than the ionization potential of B, ZP(B),and (2) associative ionization of the second kind, the Mohler-Franck or Hornbeck-Molnar process, in which E(A*) < ZP(B). Associative ionization of the second kind can be subdivided usefully into type 2a in which the Born-Oppenheimer potential curves of A* + B and A + B+ do not cross, and type 2b in which the curves do cross. Associative ionization of the first kind might exhibit examples of curve crossing, but these seem to be rare and not very important at present.
Examples of A1 of the first kind are He(2%) or He(2lS) with hydrogen or argon to give HeH+ or HeAr+; A1 of type 2a is exemplified by H(n = 3) + H(ls) to give H i , and A1 of type 2b occurs with N + O* to give NO+ or, in the case of associative detachment, with H- + H to give H,.
ATTACHMENT AND DETACHMENT PROCESSES. I
I63
The process requires that sufficient energy flow from nuclear motion into electronic excitation to leave the nuclei in a bound state. Thus, associative ionization requires that the outgoing electron, in its work function or ionization potential plus its kinetic energy, carry off more than some minimum amount of energy that initially was nuclear kinetic energy. In Penning ionization, the process is the same except that the initially free nuclei are left free. Hence if both Penning and associative ionization can occur with the same final electronic state, the Penning process is (qualitatively) associated with the slow electrons and the associative ionization with the fast emerging electrons. Theoretical analyses of associative and Penning ionization are manifold. They are generally carried out so that the theory relates the associative ionization process to its inverse, dissociative recombination, or to Penning ionization. The subject prior to 1962 was summarized in a volume of modest size (Bates, 1962a). Since then, this problem has been treated by many authors indeed (Smirnov and Firsov, 1965; Katsuura, 1965; Nielsen and Dahler, 1965; Sheldon, 1966; Berry, 1966; Bardsley, 1967; Watanabe, 1967; Watanabe and Katsuura, 1967; Russek er al., 1968; Bell et al., 1968; Mori, 1%9; Nakamura, 1969; Bell, 1970; Janev et al., 1970; Fujii et al., 1970; Miller, 1970; Miller and Schaefer, 1970; Micha er al., 1971; Cohen and Lane, 1971; Blaney and Berry, 1971; Nielsen and Berry, 1971; Nakamura, 1971; Millerer a / . , 1972; Olson, 1972a,b; Niehaus, 1973; Cohen and Lane, 1973; Micha and Nakamura, 1975; Preston and Cohen, 1976; Hickman and Morgner, 1976; Bellum and Micha, 1977a,b; Kohmoto and Watanabe, 1977; Cohen and Lane, 1977; Hickman et al., 1977a,b). For heteronuclear systems, the physical picture from these analyses is one in which the excited particle and its collision partner may undergo an Augerlike process while they are together in a transient compound state. The active electron of the excited particle is ejected as the other colliding particle deposits a valence electron into the hole in the normally occupied shell of the “hot” species. The alternative model would involve deexcitation of the hot partner and ionization of the ground-state species, requiring that an electron be extracted from a normally occupied level and put into the continuum. These two alternatives, often called the exchange and direct mechanisms, respectively, are illustrated in Fig. 5 . The importance of the exchange mechanism was pointed out by Hotop, Niehaus, and Schmeltekof (Hotop and Niehaus, 1969c; Hotop et al., 1969) and others, who noted that the similarity of the behavior of the He(2lS) and He(2%) was too great for a simple direct dipole mechanism (Sheldon, 1966; Smirnov and Firsov, 1965; Watanabe and Katsuura, 1967) to account for the process at a microscopic level. A mechanism had to be invoked to explain the ease with which the spin selection rule is violated: the calculations of Katsuura (1969, Watanabe and Katsuura (19671, Mat-
I64
R. STEPHEN BERRY
----a FIG.5 . Schematic representation of the direct and exchange mechanisms of Penning and associative ionization: (a) direct, with A*, the initial metastable, going directly to A in its ground state while B loses its initially unexcited electron; (b) exchange, in which an excited electron of A* is set free while an electron from B moves into a low-lying level on A. Taken with permission from Berry (1974a).
suzawa and Katsuura (1970), Sheldon (1966) and, most recently, Kohmot0 and Watanabe (1977) are all based on a dipole-dipole radiative transfer of energy as would occur if the process were direct. The cross sections for ionization, Penning plus associative, calculated this way for metastable rare gases colliding at a velocity of lo5 cm sec-' with alkali atoms are 11 -22 A2 (Sheldon, 1966). Helium in its 2IP resonance state has a computed cross section for ionizing argon at 300 K of 21.9 Biz at a velocity of approximately 1.4 x lo5 cm/sec, 7.61 A2 at a velocity of 1.4 X los and 3.64 A2at a velocity of 1.4 x lo7cm/sec. The cross sections for total ionization by He(2IS) on argon is approximately 10 A2 at a velocity of 1.5 x 105cm/sec and for He(2%) on argon about 3 A at the same velocity (Pesnelle et al., 1975). Kaul et al. (1963) had reported a cross section of 6 Aa for He* + He + He: + e, where the metastable helium, produced by electron bombardment, was at about 500 K. In other words, the processes that would be optically forbidden for well-separated atomic particles have cross sections at least as large as the processes that are optically allowed, so one cannot expect a direct radiative transfer model to give a generally accurate representation of the process. By preparing polarized metastable triplet helium atoms, and then determining the polarization of the released electrons, Keliher et al. (1975a,b) showed that the exchange mechanism is at least a significant contributor to the total associative and Penning ionization process, and may be the overwhelming contributor when dipole transitions are forbidden. In essence, they found that the outgoing electron carries the spin polarization initially given to the helium metastable. Most treatments of associative and Penning ionization of the first kind that did not limit themselves to optically allowed energy donors have, in
ATTACHMENT AND DETACHMENT PROCESSES. I
165
varying degrees of sophistication, derived from the notion of a distance-dependent cumulative probability of ionization P ( R ) ,and a classical velocity v ( R ) :
P(R) = 1
-
exp{ - 2
I
T(R)[hv(R)]-'d R )
(12)
where T ( R )is 2.rrJV(R)I2, the lifetime of the species toward autoionization if it could be frozen at R. Beyond a classical-phenomenological estimation of P ( R ) are the semiclassical models of Miller (1970), Fujii et al. (19701, and with the stationary phase approximation, the model of Miller et a / . (1972) and the complex potential formulations of Nakamura (1971), both of which have been used with considerable success in comparing theoretical expectations with experiments involving helium metastables (Pesnelle et al., 1975; Magnuson and Neynaber, 1974). Nonclassical quantum-mechanical calculations of specific systems were rather sparse at the time this writer reviewed the state of the theory of Penning and associative ionization (Berry, 1974a,b). The work of Bell (1970) on He(23S) + H and of Nielsen and Berry (1971) on H* (several states) + H were of this type. The latter unfortunately treates only the states of H, with attractive potentials, and so omits many of the states most important for the Penning process [however see Zhdanov and Chibisov (1978) for repulsive states]. A semiclassical approach that would be appropriate to treat the interaction of bound with repulsive states of H* H was actually applied to the more complex problem of He* + He by Cohen (1976), whose approach was based on the qualitative picture presented by Mulliken (1964). In essence, a homonuclear diatomic molecule ion always has two kinds of states, those in which the odd electron is in an antibonding level and pairs occupy the bonding levels, and those in which the odd electron is in a bonding orbital and a pair resides in an antibonding level. The former or A-core states naturally tend to exhibit strong bonding, and the latter, or B-core states tend to have repulsive potential curves. Excited electronic states of the neutral homonuclear diatomic can be built on A or B cores by adding a last electron to a molecule-ion in either kind of state. The process of associative ionization occurs in this model when an excited atom collides with another atom along a B-core repulsive trajectory with enough energy to reach an A-core state of the ion, and the last electron falls off into the continuum as the core makes its transition from B to A type. The coupling of the A- and B-type channels in Cohen's model is driven by electron correlation, which leads to much larger cross sections-when it is available-than vibration-electronic coupling, such as was invoked by Nielsen and Berry. The complexify of the problem
+
I66
R. STEPHEN BERRY
arises because the B-core excited state may cross many other states, so that the problem becomes one of many channels rather than only a few. This problem has been treated briefly in general terms by Demkov and Komarov (1966), but Cohen applied the calculations to an important specific example. He found that the thermally averaged cross sections for associative ionization due to He(ls2) + He(ls31) or (ls41) are of order 0.1-3 x 10-15 cm2, in reasonable agreement with measurements of Wellenstein and Robertson (1972), partial agreement with the earlier measurements of Teter et al. (1966), but definitely in disagreement with those of DeCorpo and Lampe (1969), whose cross section for He(1‘s) + He(3”) + He: + e is 50 times larger than theoretical value. The theoretical interpretation of the rate of probability of ionization hinges on the coupling between the bound and free states-the r of Eq. (12), or the IViffzof the Wentzel golden rule-both its mechanistic origin and its dependence on internuclear distance. By 1974, although its evaluation was difficult and probabilities obtained by that time by different procedures were not always in harmony, it seemed likely that the ionization probability would become an accessible quantity. Indeed, this hope is almost satisfied in a quantitative way for very simple systems, and a formal procedure has been developed (Bellum and Micha, 1977a)to interpret the Penning and associative processes through molecular orbital correlation diagrams. For collisions of simple metastable atoms with simple atoms or molecules, semiempirical methods based on parametrized potentials, such as the formulation by Nakamura (1971) or the critical radius model of Bell et al. (1968) have been supplanted by methods based on potential surfaces from a b initio calculations, and on coupling matrix elements computed from wavefunctions. The wavefunctions themselves are generally derived by methods that force bound character onto discrete states associated with the initial channel, e.g., He* ( P S ) + Hz, and free character onto the final channel, in this case He(1’S) + HZ++ e, HeH+ + H + e, or HeH: + e if a reactive process is involved. The primitive example of Penning and associative ionization is the collision of metastable helium (either z3S or 2lS) with hydrogen atoms, and considerable effort has been expended on this system both experimentally (Shaw et al,, 1971; Hotopet al., 1971; Howardet al., 1973; Magnuson and NeynaHer, 1974; Forter al., 1976a,b)and, even more, theoretically (Ferguson, 1962; Fujii et al., 1970; Bell, 1970; Miller and Schaefer, 1970; Nakamura, 1971; Cohen and Lane, 1971, 1973; Miller et al., 1972). However in the present context, two more complex examples can more usefully illustrate the status of the subject. One is He (2% and 2 9 ) + Ar, long a subject
ATTACHMENT AND DETACHMENT PROCESSES. I
167
of both experimental and theoretical studies; the other is He(2% and 2's) + H2,well-studied experimentally but only recently the object of theoretical investigations. The total cross section for ionization, including Penning and associative processes, was the first quantity to be studied for the system of metastable He + Ar. Among the techniques used to measure this cross section have been optical absorption of metastables (Benton et al., 1962), collection of ions from a flowing afterglow (Schmeltekopfand Fehsenfeld, 19701, a combination of both these techniques (Bolden et af., 1970), and cross-beam scattering (Sholette and Muschlitz, 1962;, Rothe and Neynaber, 1965; Dunning and Smith, 1970; Howard et al., 1972; Neynaberet al., 1972; Neynaber and Magnuson, 1975a,b, 1976).The early work indicated that the cross sections for singlet and triplet helium are similar, but the more recent results, beginning with the electron energy analyses of Hotop et al. (1969) and the ion measurements of Schmeltekopf and Fehsenfeld (1970), indicate that the shorter-lived singlet reacts with a cross section two to three times that of the triplet. This finding has been confirmed by measurements of the differential scattering cross section (Chen et al., 1974; Wanget al., 1976)from which the total ionization cross sections for singlet and triplet helium on argon at 65 meV, are, respectively, 30 and 15 A2,both +20%. The temperature dependence or velocity dependence of the ionization cross section has been an unsettled question. Experiments by Stebbings' group at Rice (Cook et af., 1974; Riola et a / . , 1974) and by Schmeltekopf and Fehsenfeld (1970) gave quite different ratios of the ionization cross sections for singlet and triplet metastable helium or argon. This discrepancy was apparently resolved when Lindinger et al. (1974) found that the rate of loss of He(23S)increases rapidly with temperature; the differences between the kinetic energies of the metastables in the SchmeltekopfFehsenfeld experiment and the Rice group seemed to explain the differences in the relative cross sections. Then, more recently, the velocity dependence of the cross section for ionization of argon by He(29) has been measured by three groups (Illenberger and Niehaus, 1975; Pesnelle et ul., 1975; Wang et al., 1976). Wang et ul. carried out their analysis by determining differential cross sections at two energies, fitting the parameters of a complex potential to the data, and inferring the total cross section at other energies from their potential. The other two groups made measurements of total ionization cross sections at several relative velocities. All three agree that the cross section rises rapidly with relative energy for low velocities; it doubles, approximately, when the velocity increases from 1500 to about 2200 m/sec. Moreover there is general agreement
I68
R. STEPHEN BERRY
E (kcal/mole) FIG.6. The ratio of the cross section for associative ionization uAlto the total ionization cross section utOt = oA,+ uPennlng, for metastable helium atoms colliding with argon. Taken with permission from Chen er a / . (1974).
regarding the ratio of associative and Penning cross sections; the form of this distribution, as derived by Chen et al., is shown in Fig. 6. This will be discussed below. At higher energies, the determinations of total cross sections are not in unanimous agreement; those of Pesnelle et al. show a decided fall-off at high velocities (3000 m/sec) for He(2IS) that is much less marked in the result of Wang et al. or Illenberger and Niehaus. The question apparently can only be settled by further measurements at relative velocities of 3000 m/sec or higher (relative energies above about 160 meV, or temperatures above about 1800 K). The method used by Chen et al. (1974) deserves some comment, partly because of its power and partly because it brings us to the next level of analysis of Penning and associative ionization, the determination of the relative cross sections for the two processes. We saw one example of such results for helium metastables with argon in Fig. 6; the form is apparently rather general, as judged by the second example, Fig. 7, Ne* + Ar, as measured by Neynaber and Magnuson (1975a). The method of analyzing the differential cross section is an application of the ingenious approach introduced by Greene et al. (1966) and Rosenfeld and Ross (1966). The differential elastic scattering cross section is determined experimentally for a wide range of angles. The cross section for
169
ATTACHMENT AND DETACHMENT PROCESSES. I
t-
V
w
0.1 :D . D
v)
v)
8
0
0 U V
w
$
0.01 r
X
a
ASSOCIATIVE IONIZATION PENNING IONIZATION TOTAL I ON I Z AT1ON
D
1
w U
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'"""1
0.I '
'
I
' ' 1 1 ' ' 1
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' " " " 1
' " l " ' l
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1000
' i L 1 e J
INTERACTION ENERGY, W (eV) (a)
r lrllll, (u
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PRESENT EXPERIMENT * MOSELEY, PETERSON, LORENTS, AND HOLLSTEIN TANG, MARCUS, AND MUSCHLITZ - OLSON (THEORY) 0
.
(b) FIG. 7. Behavior of the cross sections for Penning and associative ionization for Ne* + Ar: (a) relative cross sections Q A ~ + , QNeAr+, and QT for Ne* + Ar collisions; (b) total absolute cross sections for production of both Ar+ and NeAr+ from Ne* + Ar. The discrepancy between the results of Tang, Marcus, and Muschlitz and of Neynaber and Magnuson and Moseley et a / . has not been resolved. Taken with permission from Neynaber and Magnuson (1975a).
170
R. STEPHEN BERRY
small scattering angles, in practice 10-40" with thermal beams, is used to determine the elastic contribution to the differential cross section at larger angles. The actual large-angle elastic scattering is depleted, relative to what this hypothetical pure elastic scattering would be. The difference is attributed to reactive scattering, in this case to Penning and associative ionization. By making a classical association of scattering angle with relative velocity and impact parameter, one can determine from the "missing elastic contribution" the range of interparticle distances in which the onset of reaction occurs, and the probability of reaction as a function of the distance of closest approach of the particles. The inferred shapes of these ionization probabilities, reported by Chen et al. in terms of opacities, as functions of R, , the classical turning point or distance of closest approach, are given in Fig. 8. In the He* on Ar system, the opacities go from their maxima to zero rather sharply on a narrow range of R, ; this seems to be the more usual but by no means the only way the opacities behave. The real and imaginary parts of the complex potential for He(2lS) + Ar (Brutschy et al., 1976) are not simple: the real part has a minimum at 10 bohr, a local maximum at about 7.5 bohr, and a narrow, shallow minimum near 6 bohr; the imaginary part has a minimum at 8 bohr. Determining the relative amounts of associative and Penning ionization, which is to say the fractions of products in which the relative nuclear motion is bound as a vibration or free as a translation, can be done by
Rc FIG.8. The probability of ionization as a function of the distance of closest approach R, for collisions of argon atoms with He(2lS) (solid lines) and with He(ZSS)(dashed lines). Taken with permission from Chen er al. (1974).
ATTACHMENT AND DETACHMENT PROCESSES. I
171
direct measurement of the product ion masses (Sholette and Muschlitz, 1962; Herman and Cermak, 1963a,b, 1968; Hotop and Niehaus, 1968; Hotop et al., 1969; Kramer et al., 1972). A second approach depends on the Franck-Condon character of the associative and Penning processes: so long as the ionization process involves negligible change in nuclear relative positions or momenta, the process can be represented as a vertical transition from one potential curve to another in a conventional diagram, such as that shown in Fig. 9. The Franck-Condon condition requires that the nuclear kinetic energy in the final state be the same as that in the initial state, so that at each internuclear distance, the nuclear kinetic energy in the final state is represented by a point whose distance above the A + B+ + e curve equals the distance of the corresponding initial line (1 or 2 in the figure) above the A* + B potential at that point. If the kinetic energy of the final state lies below the dissociation energy of A + B+ + e, the product from that collision at that distance corresponds to associative ionization; if the final nuclear kinetic energy exceeds the dissociation enenergy of A + B+ + e, the products are dissociated, i.e., are the Penning
I
R
FIG.9. Schematic representation of Penning and associative ionization as vertical (Franck-Condon) processes. A* and B approaching with higher energy react at their classical turning point to produce A + B4 + e above the dissociation limit of AB+ (Penning ionization);A* and B approaching with lower energy react to produce AB+ + e (associative ionization). Transitions at 1 and 2 indicate these two cases.
172
R. STEPHEN BERRY
products. The difference between the total transition energy Ebt and the nuclear kinetic energy is the electron energy E, . Hence by selecting the initial energy and measuring the distribution of E,, and knowing the potential curves moderately well, one can determine the distribution of products between associative and Penning ionization. Figure 10 shows and aPennlng for He(2IS, the energy dependence of the cross sections aAr 23S) + Ar. This analysis was proposed by Herman and Cermak (1966) and later expanded (Cermak and Herman, 1968; Hotop and Niehaus, 1969a) not only to determine relative amounts of associative and Penning ionization, but to determine the shapes of unknown potential curves (Hotop and Niehaus, 1970a). The method was first used in the laboratory by Cermak (1966b), then by Fuchs and Niehaus (1968), and combined with mass spectrometry to find the relative Penning and associative ionization cross sections (Hotop et al., 1969). The method has been exploited widely as an alternative to photoelectron spectroscopy (Cermak, 1967a, 1968; Cermak and Ozenne, 1971; Brionetal., 1972; Yee and Brion (1975); Yee et al., 1976; Cermak et al., 1976; Cermak and Yencha, 1976, 1977; Brion and Crowley , 1977). "Penning ionization electron spectroscopy" has been a useful supplement to the more widely used photoelectron energy analysis, partly because it provides a means to excite states difficult to produce by optical (electric dipole) excitation. The validity of the assumption of the Franck-Condon principle has been called into question by the ionization of HCl and HBr by metastable helium (Richardson et u l . , 1971; Richardson and Setser, 1973). Nonverticality was inferred for these systems because the emission spectra from
E (kcol/rnole
I
FIG.10. Energy dependence of the cross sections for associative ionization aA,and for
Penning ionization uPennlng, from collisions of He(2'S) + Ar (solid curves) and He(2") (dashed curves). Taken with permission from Chen et al. (1974).
+ Ar
ATTACHMENT AND DETACHMENT PROCESSES. I
I73
HCI+ and HBr+ produced in photoionization and Penning ionization differed, whereas the spectra observed previously for N: (Robertson, 1966; Schmeltekopf er ul., 1968) and 0; (Robertson, 1966) were essentially independent of whether ionization occurred optically or by collision with a metastable. A similar phenomenon has been found with He(2%) and NO (Coxon et d.,1975). There is an equally or more plausible alternative explanation, which leaves the supposition of Franck -Condon ionizing transitions intact: that the difference in population distribution among the states of HCI+ and HBr+ produced by photo- and Penning ionization is due not to the ionizing process itself but to interactions between the HCl+ or HBr+ and the outgoing helium. That is, the deviations seen by Richardson et a / . are probably due to final-state interactions rather than to the intermediate process itself. This mechanism (Hotop, 1974) has been shown recently to be the most probable case, from the agreement of the electron energy distributions from photoionization and Penning ionization of the hydrogen halides (Hotop et ul., 1975; Cermak, 1976a). Hence the model of associative and Penning ionization based on vertical FranckCondon transitions remains a generally acceptable one. That nuclear products may leave with a range of final kinetic energies is shown neatly in the He(2lS, 23S) + Ar system. The electron energy spectrum shows clearly the doublet peaks associated with the formation of Ar+(2P312) and Ar+(2P112), both by singlet and triplet helium atoms. The doublets from the two initial states of He* are separated by 800 meV; the peaks of each doublet are separated by 200 meV. The individual peaks from the singlet are narrower than those from the triplet (Fuchs and Niehaus, 1968; Cermak and Ozenne, 1971; Brion et al., 1972). The singlet-peak widths, reported as 35 meV by Hotop and Niehaus (1969a), are comparable to those from photoionization, which is 43 meV in the work of Brion et al., but those from the triplet have widths of about 25 meV greater. The broadening in the triplet case is asymmetric, occurring largely on the high-electron-energy side, and is due to the associative ionization that presumably can occur along the potential for He(23S) + Ar but effectively not along the curve for He(2IS) + Ar. Brion et al. argue this by analogy with the case of helium metastable with hydrogen atoms, which had been calculated by Miller and Schaefer (1970). Penning ionization cross sections for helium in its first optically allowed excited state He(3IP) on all the rare gases have been measured by Kubota et al. (1975). For an argon target, the cross section is 55.6 Biz, nearly twice the cross section of the He(2lS) on the same target. The measured cross section is about 25% larger than that calculated by Nakamura (1969), which is in turn about a hundredfold larger than the photoionization cross section. A similar process, associative ionization from the opti-
174
R. STEPHEN BERRY
cally allowed excited states of potassium, has been observed recently by Klucharev et al. (1977). The systematic study of associative and Penning ionization of H, by metastable helium began with the complementary experiments of Kupnanov (1965, 1966) detecting ion masses, Cermak (1968) measuring the kinetic energy of the electrons (and comparing that energy with the energy of electrons from photoionization), and Penton and Muschlitz (1968)determining the kinetic energy of the ions. The electron energy distribution appeared broad under the 0.2 eV resolution available at that time to Cermak, and it was not possible to distinguish electrons corresponding to different final vibrational states of the H t . The H,+ ions observed by Penton and Muschlitz had translational energy well above that of the initial thermal H,. Together the results suggested that the final state He + H,+ + e has a somewhat more repulsive He-H, potential than the initial He*-H, potential, at least in the region of internuclear distance where the Franck-Condon transition occurs. In the next years, Hotop and Niehaus (1969b) removed the He(2'S) metastable and improved the resolution of the Penning electron energy measurements to 12-20 meV, enough to permit the clear observation of steps corresponding to individual vibrational states of the H t ions. They also made the first measurements (Hotop and Niehaus, 1968) of the branching ratio for the competing processes He(2?S)
+ Hz
--f
-+ --f
He + H i + e HeH+ + H + e HeH: + e
The kinetic energy measurements confirmed that the interaction between He and H t is strongly repulsive. These were pursued further (Hotop and Niehaus, 197Ob), particularly with comparisons from photoelectron spectroscopy, to show that the outgoing Penning electrons pick up about 70 meV from the translational energy of relative motion of the He* and H,. This excess energy is given to the electrons independent of the final vibrational state of the H,+. The effect is about the same for He(2%) as it is for He(2IS) with Hi, HD, and D, (Yee et al., 1975a,b) but this cannot be generalized: with N, ,the singlet and triplet of helium exchange rather different amounts of energy between translation and electron motion. This independence of the H t is very strongly suggestive that the outgoing electron leaves from the helium atom and has very little interaction with the hydrogen molecule. This in turn is what one expects if the process occurs by the exchange mechanism, rather than the direct mechanism. Measurements of the absolute cross section for ionization of H2 by He metastables, particularly by He(2%), were carried out by Lindinger et al.
ATTACHMENT AND DETACHMENT PROCESSES. I
I75
(1974), Cooker al. (1974), Riolaer at. (19741, and West ef al. (1975). Two groups were involved, one using crossed beams and the other flowing afterglows like the results for He* + Ar, the two measurements seemed for some time to show significant differences, but were reconciled when it was found that the cross sections for ionization of several species are strongly energy and temperature dependent. Lindinger ef al. found that the rate coefficient for He(23S) + H, at 300 K is 2.9 x lo-" cm3/sec. The branching ratios were measured with considerable precision by Hotop et al. (1969) and by West et al. (1975). The overwhelming ionic product, -88%, is H i , followed by HeH+ (8.1%from Hotop er al., 10% from West et ul.), with much less HeHJ (1.5-1.6%) and H+ (2% from Hotop et al., 0 from West ef al.). The first theoretical study of the associative + Penning process of a molecule in three dimensions (Preston and Cohen, 1976) is also the only one that attempts to dissect the process to give these branching ratios. The semiclassical model used by Preston and Cohen, involving a stochastic leakage of the compound HeHz resonance into any of several continua, gives a remarkably good representation of the branching, with predictions of 9O.5-91.4% HJ, 8.6-9.5% HeH+, and < 0.7% of HeH: or H+. Moreover, the computed rate coefficient for total ionization at 300 K is 2.8 x lo-" cm3/sec. Two other rather elaborate theoretical calculations have been made of the total ionization cross section, by Hickman et al. (1977b) and by Cohen and Lane (1977). The two calculations are rather similar, but differ in their numerical values of the cross section. Cohen and Lane's values are very close to those of Lindinger et al.; Hickman et al., also using a complex potential but with somewhat smaller real and imaginary parts, computed rate coefficients for temperatures between 300 and 900 K that are only about half the values measured by Lindinger et ul. The Hickman calculation uses the formalism of Arthurs and Dalgarno (1960) for atommolecule collisions. Cohen and Lane explored the interactions by studying each spherical harmonic component of the effective complex potential of interaction between He* and H,. Cohen and Lane's calculations indicate that only the spherically symmetric part of the potential and not the higher spherical harmonics is required to account for the process. Moreover, only close collisions, corresponding to scattering angles greater than 40", lead to Penning ionization, according to these results. A few other recent developments connected with Penning and associative ionization deserve mention here. One, very important for the critical comparison of theory with experiment but of somewhat less importance in most applications in electron physics, is the study of the angular distribution of electrons released by Penning and associative ionization. Here, experiments preceded theory; the Freiburg group (Hotop and Niehaus,
I76
R. STEPHEN BERRY
1971; Ebding and Niehaus, 1974) carried out these challenging experiments, comparing, for example, the angular distributions from He(23S) on Ar, Kr, Xe, Hg, Nz, and CO with those from He(2IS) on the same targets. The triplet metastable gives rise to more anisotropic distributions, with the greatest intensity in the direction of approach of the helium with all targets except mercury, which gives rather isotropic distributions. The formal theory of Miller (1970) is capable of yielding angular distributions, and was used successfully by Ebding and Niehaus as a phenomenological representation to rationalize the observed distributions. Micha and Nakamura (1975), like Ebding and Niehaus, used a classical description of the collision trajectories to develop another phenomenological picture differing from Ebding and Niehaus principally in the spherical harmonic terms that are kept. Morgner (1978) extended the analysis to a quantum-mechanical representation of the relative motion of metastable and target; presumably this method will be applied to specific examples soon. Applications of Penning ionization include its use as an alternative to photoionization for electron spectroscopy, which has been discussed previously; as a means to bracket thermochemical quantities, such as the heat of formation of the CN radical (Setser and Stedman, 1968); its application as the principal mechanism for production of ions in one of the most widely used source chambers for ion beams (Anonymous, 1976; Heinecke et al., 1976); for the detection of long-lived states of molecules that can play the same role as metastable rare-gas atoms (Kuprianov, 1965; Cermak, 1966a); recently, as a source of polarized electrons (McCusker et al., 1972; Keliher et al., 1975b); and finally, for the production of coherent excitation and the measurement of excited-ion lifetimes from the modulation of emission created by that coherence (Schearer and Riseberg, 1971; Parks and Schearer, 1972). Schearer’s method was used by Hamel and co-workers (1974; Hamel and Barrat, 1976) to measure lifetimes of 4d05s2 states of Cd+ and 4f states of Mg+. The Penning process may prove useful for the detection of core ionization, if the interpretation by Johnson et al. (1978) of the He(23S)-alkali atom process proves correct. The rate coefficients for the Penning ionization of sodium, potassium, rubidium, and cesium are 5.1 2 1.0, 7.7 ? 1.5, 12.0 ? 2.4, and 4.5 k 0.9 x 10-lo cm3/sec, respectively, at 350 K. If the cross sections are relatively independent of energy in the thermal range, the corresponding cross sections are 33 2 6,55 2 10,93 2 18, and 34 & 7 x cm2.The suggestion is made by Johnson et al. that the large cross sections for potassium and rubidium are due to core ionization processes available to these two species with the 19.8 eV of the helium triplet:
ATTACHMENT AND DETACHMENT PROCESSES. I He*
+ Hz+ He + H $ t + e +
HeH+ + H
I77
+e
The latter process seems to follow a spectator stripping model, insofar as the HeH+ ions tend to be found in the forward direction, that is, along the direction in which the He* atoms are moving.
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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS, VOL. 51
Electronic Watches and Clocks A. P. GNADINGER Faselec Corporation Zurich, Swirzerland*
I. Introduction ............................................................
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......................... ......................... IV. The Electronic Watch ........ ............................ A. Time Base .....................................................
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G . Future Trends . . . . . . . . . . . . ................................. V. Conclusion ........................................... References . . . .
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11. Some History of Timekeeping 111. Electrical Clocks ............
I. INTRODUCTION Only little more than a decade has passed since the first commercial electronic watch appeared on the market (I). Those early models were manufactured in small quantities and were of no significance compared to the large numbers of mechanical watches then produced annually. Even in the early 1970s, electronic watches were merely considered a technical toy and not taken seriously by traditional watch makers. However, in the last few years, a stormy development in the field of electronic watches has taken place and at present (1979) a substantial fraction of the annual production of clocks and watches is electronic. The reasons for this unusual success of the electronic watch are basically threefold: (1) Using an electronic time base it is easy and therefore inexpensive to achieve accuracies in timekeeping that are far better than what can be realized with mechanical means. *
Present address:
INMOS Corporation, Colorado Springs, Colorado 80906. I83
Copyright 0 1980 by Academic R e r s . Inc All rights of reproduction in any form reserved ISBN 0-12-014651-7
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(2) Electronic watches are inherently reliable. They have essentially no or very few moving parts that are subject to wear. The integrated circuit, the heart of an electronic watch, has also profited from the vast development effort in applications such as aerospace and computers where high reliability is essential. (3) Electronic watches are versatile. It is easy to include additional functions such as automatic calendar, alarm functions, and stop watch functions, which could only be realized with great difficulties in mechanical watches.
All these advantages are fully exploited in today’s electronic clocks and watches with ever-decreasing costs. Electronic watches follow essentially the trend of semiconductor components, which have always shown a strong decrease in manufacturing costs with maturing of manufacturing technology (2). The industrial realization, however, of electronic watches-particularly filly electronic digital watches-was only feasible with the advent of large-scale integration (LSI) in silicon integrated circuits. A watch circuit requires from several hundred to several thousand components. The restricted volume available in a watch case prohibits the use of discrete elements or small-scale integrated circuits. The impact of the electronic concept on the traditional watch industry that has developed the mechanical watch to perfection over a timespan that has to be measured in centuries was very profound. Even so, this revolution is far from being completed; watch and clock technology has matured substantially, so that this is an appropriate point in time to review the present status of the electronic watch and assess the trend for future developments. N o attempt is made to cover all aspects of electronic watches and clocks in a well-balanced manner. Emphasis is clearly put on the heart of every electronic watch-the integrated electronic circuit. 11. SOMEHISTORYOF TIMEKEEPING Clocks and watches have remained relatively unchanged for many centuries. They have seen, of course, gradual improvements both in style and size and in accuracy and reliability. Until the mid-1960s they were purely mechanical devices. The first mechanical clocks made their appearance in Europe during the eleventh century. They were tower and cathedral clocks of substantial size and weight. Their accuracy was quite poor and they had to be reset daily by sundial or other astronomical observation. The energy for driving these clocks was usually provided by the potential energy of falling
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weights. This motion was transmitted to the indicator directly over a gear train. Since there was no means of accurately metering the descent of the falling weight, the indicated time fluctuated widely. A big step in the evolution of mechanical clocks was made with the application of the pendulum, combined with an escapement. This invention is usually credited to Christian Huygens of Holland (1658). Now the descent of the falling weight is used to keep the pendulum in motion, which in turn determines very accurately the motion of the gear train. In other words, the pendulum serves as a time base. The accuracy of such a mechanical clock is further improved by proper design and clever choice of materials to compensate for temperature variation and influence of humidity. The zenith of the development of the mechanical pendulum clock was probably reached in the early twentieth century. The observatory clock built around 1920 by Johann Meindl of Vienna is an outstanding example of the degree of perfection reached in mechanical pendulum clocks: this clock, now at the Antique and Modern Clock Co. of Los Angeles, California, still has an accuracy of one second per month! The evolution of the mechanical wristwatch started when Peter Henlein introduced his egg-shaped version in 151 1. Instead of gravitational energy of falling weights, a spring coil is now used to store potential energy and serves as an energy source to drive the watch. Again early wristwatches did not use a time base to meter the unwinding of the coil and their accuracy was, therefore, quite poor. As a parallel development to the pendulum as a time base for clocks, a torsion pendulum in the form of a balance wheel was introduced in wristwatches. The balance wheel is linked to an escapement that both meters the unwinding of the coil and provides impulses that drive the indicator over an appropriate gear train. Again, the introduction of the balance wheel as a time base proved to be a major improvement in the performance of the mechanical wristwatch. Since then, mechanical clocks and watches have undergone very little change. They have become smaller and more reliable. Their accuracy has improved steadily and manufacturing costs have decreased with the start of industrial production. A modern mechanical watch still consists of the same four basic parts as it did a few centuries ago, as shown in Fig. 1. These parts are an energy source, a time base, counters, and an indicator. The energy source is usually potential energy stored either in falling weights or in a helical coil spring. In later years the self-winding feature has been added to the coil spring. The motion of the bearers arm rewinds the coil continuously and keeps the watch running as long as it is not put to rest for more than approximately 48 hours. The time base is either a pendulum or a balance wheel combined with an escapement mechanism. The counter consists of
A. P. GNADINGER
186 TIME EASE
’
COUNTER
(GEAR TRAIN)
’
INDICATOR
FACE AND HANDS
FIG. 1. Block diagram of a mechanical watch.
a mechanical gear train that divides the speed of rotation of the escapement wheel down to the appropriate speed to drive the indicator. The indicator-at last-is in most cases the familiar numbered face with moving hands.
111. ELECTRICAL CLOCKS Electrical clocks do not really satisfy the definition of a clock, since one vary essential component is missing-the time base. As a time base, the frequency of an AC power line is used. This frequency is usually controlled by the power-generating companies to such a degree that a clock deviates only within about + 2 sec from the exact time. Electric clocks consist of a synchronous electric motor, gear train, dial, and hands. The motor runs at a constant speed that is a function of the power line frequency, not its voltage. A second class of electrical clocks and watches uses an electrically driven spring balance movement as a time base. They satisfy the definition of a watch and can be represented by a block diagram similar to the one shown in Fig. 1. The power source is now an electrical battery or power supply. Power is transmitted to the spring balance movement serving as a time base via electric contacts operated by the balance wheel itself. These clocks are employed very often in automobiles. Although electrical clocks and watches are used widely and are still very popular today, they are not treated further here.
IV. THE ELECTRONIC WATCH The block diagram of an electronic watch or clock is given in Fig. 2. Every electronic watch possesses an energy source, a time base, a display and an electronic circuit that connects and controls these parts in a suitable manner.
187
ELECTRONIC WATCHES AND CLOCKS TIME BASE
>
ELECTRONIC CIRCUIT
>
DISPLAY
SOURCE (BATTERY)
FIG.2. Block diagram of an electronic watch or clock.
The different types of electronic clocks and watches, also termed generations, distinguish themselves through the type of time base used, the
different types of displays, and the complexity of the electronic circuit. So far, four generations of electronic watches have been defined; their main characteristics are summarized in Fig. 3. The first watch generation-introduced in approximately 1964 (/)-uses a conventional mechanical display, that is, hands for hours, minutes, and seconds driven in conventional manner by a gear train. The time base is a synchronized balance wheel. Power is transmitted to the balance wheel electromagnetically. The difference between an electronic watch and electrical watches described in Section I11 can be made clear at this point. In both cases power is transmitted to the balance wheel electromagnetically. However, in an electrical watch the coil is switched via electric contacts operated by the balance wheel itself, whereas in the electronic version these switches are replaced by an electronic switch, usually a transistor. The energy is provided by an electrical battery. The fre-
Time Base
Complexity of the circuit
Typical frequency f
Af I f
lhcorelicai inaccuracy
Watch generation
01 Ihe walch
Tuning fork
Ouarlz Resonator
l-IOTrans1slors
300Hz
up lo aboul 500 Transistors
32kHz
2000 - 4000
4MHz
1
>.lo5
-6
1-3.10
10 mlnlyear
2
30-90scrlycar
3
I0
Transtslors
FIG.3. The four generations of electronic watches and clocks.
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quency of the balance wheel is usually 4 cycles per second with an accuracy Af/f = This frequency variation corresponds to a theoretical accuracy of the watch of 50 min/year. The electronic circuit needed is quite simple, very often only one discrete transistor. For the second watch generation the balance wheel is replaced by a tuning fork as time base (4). The tuning fork oscillates at a frequency of approximately 300 Hz, which is much higher than the balance wheel frequency. The oscillation is transmitted mechanically by means of pawls. Again the energy is provided by a battery and switched with a simple electronic circuit. The stability of the tuning fork movement is better, typically 4 f / f = 2 x lov5,which corresponds to a theoretical accuracy of the watch of 10 min/year. Before the advent of the quartz-crystal-controlled movements this watch generation was quite a success. Now it is largely replaced by watches of the third and fourth generation, mainly because the mechanical coupling of oscillator and divider gear train is quite expensive. The third watch generation still employs an analog display -a face and moving hands-but the rest is electronic. The time base is now a quartz-controlled oscillator with a frequency in the kilohertz to megahertz range. The electronic circuit is much more complex-it contains several hundred components -since the oscillations have to be divided electronically. The second (or minute) hand is driven by a vibrating or stepping motor, which interfaces directly with the electronic circuit. The stability of a quartz-controlled oscillator is much better than that of the time base of the two previous generations, typically Af/f = 1 x which corresponds to a theoretical accuracy of the watch of 30-90 sec/year. With the fourth generation, the last mechanical part-the display -is now replaced by electronic solid-state display. The degree of complexity of the electronic circuit is again considerably higher; it contains about 3000-4000 active components. The typical frequency and the accuracy of a digital watch are identical to those of analog watches of the third generation. The main advantages of electronic over mechanical watches as already mentioned in the introduction become again evident from the representation in Fig. 3: (1) The excellent stability of a quartz-controlled oscillator is the reason for the inherently high accuracy of an electronic watch. (2) The integrated circuit follows the laws of batch processing, as will be explained in Section IV,C, and is therefore inexpensive to produce. (3) Electronic watches are very versatile, since it is not difficult to add some one hundred devices more to a circuit already containing several thousand components and implement in such a way additional functions
ELECTRONIC WATCHES AND CLOCKS
that could only be realized-if ical means.
at all-with
I89
great difficulty with mechan-
The main functional blocks of electronic watches and clocks are covered in detail in the following sections. Emphasis is placed on watches and clocks of the third generation-analog watches-and of the fourth generation-digital watches. A . Time Base The most commonly used timing element in electronic watches and clocks is the quartz crystal. As mentioned in the previous section, time bases employing tuning forks and electronically driven spring balance movements were used in the past for electronic watches of the first and second generation, but are of little importance today. They are not treated any further. As already indicated in Fig. 3, the commonly employed frequencies for a quartz-crystal-controlled oscillator for analog and digital watches of the third and fourth generation are in the range from 32 kHz to 4 MHz. In fact, the frequency of 32 kHz is now generally accepted as a standard, at least for wristwatches. 32 kHz seems to be a good compromise between power consumption, stability of the oscillator, and cost and size of the quartz crystal. The current consumption of an integrated circuit including the quartz oscillator circuit and the subsequent divider chain increases with oscillation frequency, at Ieast for the now commonly employed complementary integrated technologies such as CMOS, as is shown in more detail in Section IV,C. It is therefore desirable to use a rather low frequency in order to minimize power consumption and maximize the battery life. On the other hand, the quartz crystal size and also the manufacturing and packaging costs of the quartz crystals fall and crystal performance increases as frequency increases. it is also more and more difficult to achieve adequate stability of the oscillator and utilize fully the inherent stability of the quartz crystal resonator, if the frequency is decreased. 32 kHz seems now to be a good compromise between these conflicting requirements, at least for wristwatches. For clocks, on the other hand, a frequency of 4 MHz is almost universally accepted as standard. Power consumption requirements of clock circuits are not as stringent as those of wristwatch circuits. Clocks can be driven by much larger batteries so that the lower cost and the lower temperature sensitivity of a 4 MHz quartz is more important than minimum power consumption. There is, however, a tendency to also lower the frequency of clock oscillators to 32 kHz. The costs of 32 kHz quartz crystals have decreased and the technical performance has increased dramatically
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A.
P. GNADINGER
during the last few years, because of large volume production of these crystals for wristwatch applications. With only a marginal prize and performance difference between 4 MHz and 32 kHz crystals the lower current consumption of the 32 kHz circuits would therefore also be favorable for clock circuits. The early electronic wristwatches utilizing a quartz-crystal-controlled oscillator were using 8192 Hz as oscillating frequency (56). At that time it was difficult to handle higher frequencies because the current consumption would have been too high since the circuits were realized in a noncomplementary standard bipolar technology and the oscillator designs were not well suited for higher frequencies (3). Although 32 kHz is now generally accepted as a standard for wristwatches, in special cases higher frequencies such as 524 kHz (7), 786 kHz (8),2.4 MHz (9), and even 4.2 MHz (10) are employed. For watches where the main emphasis is placed on accuracy of timekeeping, frequencies higher than 32 kHz are desirable because the better performance of the high frequency quartzes can then be utilized. All these designs have to use much more elaborate circuit techniques in order to keep the current consumption within reasonable bounds. The upper frequency limit is obviously given by the circuit. An overall current drain of 10 pA at 1.4 V and 4.2 MHz has been obtained with a commercially available SOS circuit (10). Similar results were obtained in production with bulk silicon-gate technology, but much more complicated circuit techniques were necessary [6 p A at 1.4 V and 2.4 MHz ( I ] ) ] . A wristwatch introduced on the market in 1974 uses a very stable AT quartz resonator at 2.4 MHz with a guaranteed accuracy of the watch of 1 sec per month (9). This very high timekeeping accuracy, however, can only be achieved at fairly high costs and rather high current consumption. With a quartz crystal as timing element almost any type of oscillator circuit could in principle be used. However, as mentioned before, there are some important restrictions. The current consumption should be as low as possible. Second, the circuit has to be capable of integration. Therefore, circuits using inductances are not suited. Also, due to the volume restrictions in a wristwatch the oscillator should require as few external components as possible. In addition to the quartz crystal, at least one more external component, a trimmer capacitor for the frequency adjustment, is, however, almost always needed. 1. Oscillator Circuits
The most popular quartz oscillator circuit used in electronic wristwatches today is the Pierce oscillator shown in Fig. 4 (3). Figure 4a
ELECTRONIC WATCHES AND CLOCKS
191
p or n c,hannel transistors
+V
Yl1
FIG. 4. (a) Three-point oscillator circuit. (b) Current source implementations. (c) Pierce oscillator circuit.
presents the basic three-point equivalent circuit. The current source can be implemented in several different ways. The choice depends mainly on the technology used to implement the main electronic circuit since it is desirable to integrate the current source together with the oscillator circuit on the same chip. Some possibilities are indicated in Fig. 4b. The most common is the CMOS inverter, which offers a transconductance g, equal to the sum of the transconductances of both transistors. The feedback admittance Y2is basically given by the quartz crystal. As mentioned before, it is undesirable to use inductances in an electronic watch circuit. Therefore, the admittances Yland Y, are best realized by capacitances. For os-
A. P. GNADINGER
192
cillations to take place, Y2 must then be inductive. The equivalent circuit of a quartz crystal can best be represented as shown in Fig. 5a. The reactance X 2of this circuit as a function of frequency is shown in Fig. 5b. The intersection of this curve with the abscissa, where X , ( w ) = 0, are the series and parallel resonant radian frequencies w, and up,respectively. us and wp are given by (12) w, = I/(LC)1’2
w, = w,(l
+ C/C3)1’2
(1)
w,(l
+ C/2C3)
(2)
where L is the inductance of the quartz crystal, C the dynamic, and C 3the shunt capacitance, respectively. If now Yl and Y, in Fig. 4a are susceptances of equal sign (capacitances), Y, (w) must be of opposite sign for oscillations to take place and the oscillation frequency wo has to lie between ws and w,, as indicated in Fig. 5b. In Fig. 4c a practical realization of the Pierce oscillator circuit is represented. The current source is made up of an inverter in CMOS technology. A feedback resistor R, is needed in order to bias the inverter into the linear mode of the transition characteristics so that it can act as a current source. This is shown also in Fig. 6, where the transfer characteristic of a
a)
FIG. 5. (a) Equivalent circuit of a quartz crystal. (b) Reactance of the quartz equivalent circuit as function of frequency.
ELECTRONIC WATCHES AND CLOCKS
193
Vout
'VOO
I
'+l
k I I
i
I /
I
1
I I
I
I
voo
-Vin
I
FIG.6. Transfer characteristic and drain current of a CMOS inverter.
CMOS inverter is plotted. At the biasing point Q the current I, in the lower half of the drawing reaches a maximum. This peak current I, is essentially the quiescent current flowing if the quartz crystal is removed and the oscillator is not oscillating. It is to first order linearly dependent on the current consumption of the oscillator in operation as illustrated in Fig. 7, where some experimental data (13)of oscillator current I vs. quiescent inverter current I, are plotted. The slope of this curve depends in a rather complicated way on threshold voltages and gain factors of the oscillator circuit as well as the electrical parameters of the quartz. Quite often-to simplify testing-watch oscillator circuits are tested without quartz resonator and a correlation such as the one shown in Fig. 7 is used to extrapolate to the actual current consumption of the circuit. The biasing resistor R , in Fig. 4c does not influence the current consumption of the circuit as long as its value exceeds a few hundred kiloohms. Its effect on the frequency of the oscillator is also quite small. This resistor is conveniently integrated into the monolithic chip. Because of the high resistance value required, a standard diffused resistor would use up too much silicon area. In nearly all cases it is therefore realized by an active component, preferably a transmission gate as shown in Fig. 8. This complementary pair of transistors is connected in parallel and acts as a
194
A. P. GNADINGER
3
0
I
I
I
I
I
I
1
2
3
4
5
6
-
IQ IPAl
FIG.7. Quiescent current la and corresponding oscillator current I for Pierce oscillator.
high ohmic resistor. The nonlinear characteristic of this element does not pose a disadvantage in this application. Of course, the biasing resistor R , can also be a discrete component bonded externally to the chip. Provision must be made to adjust the oscillator frequency in order to compensate for quartz manufacturing tolerance and aging effects. In the Pierce oscillator configuration of Fig. 4c, the capacitances C , and C, can be combined with C,of the quartz equivalent circuit (Fig. Sa) as a capacitance of value C, in parallel with the quartz. The oscillation radian frequency wo is then given by wo =
w,(l
+ C/2CP)
(3)
where C, is the equivalent value of C1, C,, and C , combined as a capacitance in parallel with the crystal: 9
T9 P FIG.8. Transmission gate.
ELECTRONIC WATCHES AND CLOCKS
c, =
c 3
I95
+ C , C , / ( C , + C2)
(4)
and Eq. (3) reads (wo -
4 / w s
=
c/2cp= c/r2c3 + 2C,C2/(C, + C2)I
(5)
The oscillator frequency can be adjusted by either making the input or output capacitance C , or C2 variable as evident from Eq. ( 5 ) . However, the output capacitance C2 determines the power dissipation and is best integrated into the chip to ensure that it remains at a constant low value. Therefore, the input capacitance C,is usually made variable. The change in oo,Amo, with a variation of C1for maximum and minimum values of C, , C, and Cp is given by
AOO = 0
C 6 7
(l/Cpmin - l/Cpmax)
(6)
This trimming range doo must be wide enough to encompass drifts and manufacturing tolerances but not so wide that the crystal is made to operate outside its stable range. A trimming range of about 70 ppm is easily obtained with a standard 32 kHz quartz with a load capacitance variation of about 6- 14 pF. An actual curve of frequency vs. load capacitance Cpis shown in Fig. 9 (12). Equation ( 5 ) also shows that the frequency of the Pierce circuit is dependent not only on the quartz and its mechanical stability but also on the stability of the relative circuit components C , and C2. Assuming for simplicity that C, = C2,from Eq. ( 5 ) it follows that the expected frequency variation due to circuit capacitance changes is Ao/w
2:
-(C/Ci)(ACi/Ci)
(7)
300
E, n
200
I
3 . 3
a 100
1 I
15 l o a d Capacitance. Cp ( p F ) FIG.
20
9. The frequency variation vs. load capacitance for the type MTQ21 crystal.
A. P. GNADINGER
I96
C may be of the order of 0.003 pF and Cz= 5-40 pF. A frequency stability of the order of therefore calls for a capacitor stability of the order of 1%, which can be realized without difficulties in today’s technology. As mentioned in Section IV, the aim should be to design an oscillator with minimum supply current to reach a maximum battery life. The supply current of a standard Pierce oscillator (Fig. 4c), however, is a rather strong function of the supply voltage. Figure 10 shows the typically observed variation of current drain I and output voltage V , of Fig. 4c with supply voltage V ( 1 4 ) .If V is increased, the current I is small and no oscillations take place until a critical value V , is reached. After passing V , the current Z increases sharply and oscillations start with a peak to peak amplitude V , . There is a small built-in hysteresis to the I-V characteristics. By reducing the supply voltage V oscillations can be maintained down to a voltage that is lower than V,, which is mainly determined by the technology used to implement the current source. For a CMOS inverter V , is essentially given by the sum of Lile p and n threshold voltages. V, must be kept below the minimum supply voltage. To allow for manufacturing tolerances a margin of a few tenths of a volt must be provided to ensure the building up of oscillations under worst-case conditions. As evident from Fig. 10 this implies an appreciable increase of nominal current due to the steepness of the I - V characteristics. This rather strong sensitivity of current drain on supply voltage variations is one of the disadvantages of the simple Pierce oscillator.
“2
FIG.10. Typical variations of current drain [and output amplitude V s with supply voltage V, observed with a standard oscillator of Fig. 4c. After Vittoz (M), with permission.
ELECTRONIC WATCHES AND CLOCKS
I97
A second condition has to be satisfied in order to start and sustain oscillations: The transconductance g, of the current source in Fig. 4a must exceed a certain critical value g,, , which is given by (14)
where Q is the quality factor of the quartz crystal, given by
Q
(9) Inserting Eq. (9) into (8), the critical transconductance can be expressed as a function of the relative frequency pulling range Aw/o: = I/(osRC)
g m c = 0 s C/QCAW/WO)~
(10)
Equation ( 10) shows that the critical transconductance g,, increases with increasing frequency and decreasing quality factor Q . The relative frequency pulling range A o / o o cannot be changed at will. It is usually limited to 100-200 ppm in order to avoid frequency instabilities due to the variations of capacitances C1, C 2 , and Ct. The drain current necessary to achieve the critical transconductance g,, can be decreased by increasing the amplification factor of the transistors. In CMOS technology this is accomplished by increasing the channel width. There is, however, a minimum value of g, reached if the transistor operates in weak inversion (15). This minimum critical value of g, can be considered as a measure for the minimum current necessary to sustain oscillations. For 32 kHz this minimum current is of the order of 20 nA for standard frequency-pulling ranges. For g , > g,, the osciilations grow up until they are limited by the nonlinearities in the circuit. The main nonlinearity is the sharp increase in the output conductance of the inverter transistors in the peaks of the oscillations if the peak voltage is close to the supply voltage. This variation of the output conductance cannot be well controlled. Therefore, the frequency of the oscillation of the simple Piece oscillator is a rather sensitive function of supply voltage, which is an undesirable feature of a quartzcontrolled oscillator that should be capable of using the high inherent accuracy of the quartz crystal as a timing element. Various additions to the simple Pierce oscillator circuit are possible in order to improve the frequency sensitivity. The simplest solution is adding a series resistor R2 (100-300 kfi) to the output of the inverter (Fig. 4c). A better solution is the addition of two series transistors that are always conducting and operating in the linear mode to the inverter as shown in Fig. 11. This solution is also better suited for integration in actual circuit technologies such as CMOS. The series transistors add a resistance to the inverter circuit that is only weakly dependent on supply voltage and therefore decreases the overall sensitivity of the frequency on the supply voltage.
I98
A. P. GNADINGER
{ P
&P :;
*Out
T"
FIG.11. Inverter oscillator for 32 kHz.
Other improvements of the circuit of Fig. 4c have been proposed (16, 17) with the goal of decreasing the nominal value of the current I and decreasing the sensitivity of this current on supply voltage variations. An alternative realization of the Pierce oscillator of Fig. 4a consists of using a single active transistor driven by a constant-current source as shown in Fig. 12. The current I can be chosen such that the amplitude V , is sufficient to drive the next stage-an inverter, for example. The advantage of this circuit is that the current Z is not influenced by the nonlinearities of the oscillator circuit but is essentially determined by the constant current source. The best results for minimum current consumption of an oscillator cir-
FIG.12. Simple constant-current oscillator circuit.
ELECTRONIC WATCHES AND CLOCKS
I99
cuit are obtained by means of an amplitude feedback scheme that automatically adjusts the current I to the designed value (18).These oscillators are much more complex than the Pierce oscillator. Quite often they consist of up to 40 active elements. A further disadvantage is the fact that the start up of the oscillator is quite slow. It may take up to several seconds-depending on temperature and power supply voltage-to build up the oscillations with the correct amplitude. In large-scale manufacture this may be a prohibitively long time in testing the circuit and may add considerably to the cost of the circuit. Nevertheless, for high-frequency wristwatch oscillators, it is the only reasonable way to keep the current level at an acceptable value. Using this concept, oscillator currents of 1.5 p A at 2.4 MHz have already been achieved in a commercial wristwatch (9) in 1974. An interesting possibility is combining the amplitude feedback scheme with an oscillator circuit where the active transistors operate in weak inversion (15). As mentioned earlier, weak inversion operation can ease the oscillator start up problem by ensuring that g, > g , in Eq. (8) for minimum drain current. The circuit diagram of such an oscillator is shown in Fig. 13. The quartz Q, the transistor T I ,and the two capacitances C3and C4constitute a simple constant-current oscillator as described in Fig. 12 biased by the resistor R, and the current source T z . The DC gain of the closed loop made up of transistors T, , T 3 , T4, and Tz is greater than one. The currents in both branches therefore increase until they are limited by the output characteristics of Tz and T 3 . The drain current ID,of T , is high so that oscillations start to build up. As the amplitude [I,at the gate of T , increases, the average gate voltage V,, of T , must decrease to keep the average current drain of this transistor constant in spite of the nonlinear transfer characteristics. The AC component of the oscillation at VG1is filtered out by the low-pass R z C z . Therefore, the gate voltage VG3of transistor T3 is
1I
-
_
44-7-
"n
FIG. 13. Low-current quartz oscillator using an amplitude feedback scheme. After Vittoz and Fellrath ( 1 9 , reprinted with permission.
200
A.
P. GNADINGER
equal to Vcl . Hence, the drain current of T3decreases as U1increases. As U1 reaches a critical value UlC, the drain current of T4 overcomes that of T3 and the drain voltage of T3jumps to a value close to the supply voltage Vcc . This transition effects a sudden drop of ID,down to a value just necessary to keep the amplitude of oscillation U1 close to the critical value UIC ' For a sinusoidal oscillation signal U, sin wt and for transistors T , and T3 operating in weak inversion with V,, = V,, = 0 (sources connected to a common p well) and v D 3 S UTthe critical value UlCis given by (15) where Z, is the zero order modified Bessel function (Z9),UT = k T / q , and pI-p4 are the gain factors of the transistors T,-T4, respectively. These gain factors are basically given by the geometrical shape factors (effective width and length of the channel) of the transistors. n is the slope factor of the gate transfer characteristics of a transistor operating in weak inversion (20) and is only dependent on technological parameters. The drain current I D , of the transistor T , operating in weak inversion is related to the transconductance g,, by (20) is somewhat larger than the critical transconductance g, necessary for oscillations to be sustained given in Eq. (8). The advantage of using weak inversion operation can now clearly be seen. The ratioDI/&,! represents a figure of merit for a watch oscillator circuit indicating minimum power consumption. This value now reaches a maximum in the weak inversion regime. The directly coupled amplifier stage T,-T,, in Fig. 13 is needed to reach the necessary logic swing for driving the following divider chain. The noncritical resistors R, and R , can be implemented in various ways depending on the technology used. A differential resistance of 1-10 GR at zero voltage is quite adequate, so that, for example, polycrystalline diodes (2Z) or high ohmic implanted-polycrystalline resistors (22) can be used if the oscillator circuit is realized in silicon gate technology (23). Fig. 14 shows the experimental results obtained with the circuit of Fig. 13. It is seen that the amplitude of oscillation U , at the gate and the current I are both fairly independent of the supply voltage Vcc in the interesting voltage range of 1-3 V. With typical technological parameters, the current drain is of the order of 30 nA. The total current consumption including the current for the output amplifier T,-T, can easily be kept below 100 nA at 32 kHz, employing a standard quartz resonator. So far, as an example, it was assumed that the oscillator circuits g,,
20 I
ELECTRONIC WATCHES AND CLOCKS I
I
u, [mvl! i
I
J
1000
I (without quartz)
I I
I
to-’.
(calculated)
00
0
1.0
2:o
FIG. 14. Amplitude of oscillation and total current as a function of supply voltage for the quartz oscillator of Fig. 13. After Vittoz and Fellrath ( I 3 , reprinted with permission.
described in this section were realized in CMOS silicon gate technology. However, the basic three-point oscillator circuit of Fig. 4 can also be implemented using other MOS or bipolar technologies, as described in Section IV,C. For standard bipolar circuits an alternative solution to the Pierce oscillator would be the zero-phase shift circuits as described by Forrer (3).These circuits, which were quite popular in the first electronic wrist watches (5, 6), consume considerably more power and are therefore of no practical use today. For high-frequency applications other possibilities apart from the popular Pierce oscillator have been tried. An interesting example makes use of the Clapp oscillator (24). Again, this circuit cannot easily be implemented in standard IC technology and has therefore no practical use. 2. Digital Tuning Several attempts have been made to eliminate the variable trimmer capacitor that is needed for the frequency adjustments of the oscillator. This trimmer capacitor quite often poses problems because its stability
202
A. P. GNADINGER
might not be high enough to match the inherently good stability of the quartz resonator. It also adds to the manufacturing costs of the watch module. The most promising suggestion in this direction seems to be the digital tuning technique illustrated in Fig. 15 (14,25). The quartz crystal is now used only as a very stable frequency reference whose absolute value has to match the frequency desired only approximately, which means that manufacturing tolerances of the quartz and also of the capacitances C1 and Cz(Fig. 4) can be relaxed. The subsequent divider chain is now made adjustable in such a way that the frequency division ratio matches the quartz frequency. The adjustment is retained in an N-bit alterable memory defining 2N tuning steps. The increment per step is the reverse value of the division ratio, so that a 500,000 ratio is necessary to achieve an accu(0.09 sec/day). There are many advantages to this scheme. racy of Apart from the trimmer capacitor that can be avoided, a better optimization of the oscillator is possible, especially at high frequencies. The tuning range can be widened without degradation of stability and an unadjusted, less-expensive quartz can be used. The adjustable divider can be realized as a preset counter by inhibiting part of the pulses supplied by the oscillator or by adding pulses. The alterable memory should be integrated with the watch circuit in order to take full advantage of the potential simplification of the watch module that can be achieved by avoiding the trimmer capacitor. The number of bits is relatively small and does not consume much area on the silicon chip. Therefore, it can easily be implemented as a static random access memory circuit using standard flip-flops as memory cells. If it is desired to keep the tuning information stored when the battery has to be changed, nonvolatile memories such as FAMOS structures (26) or MNOS (27) techniques have to be applied. These techniques are, however, not fully compatible with standard CMOS processing or will at least add considerably to the manufacturing costs of the integrated circuit. It is also possible to use a second battery for the memory part of the circuit
FIG. IS. Principle of digital tuning. After Vittoz (14), reprinted with permission.
ELECTRONIC WATCHES AND CLOCKS
203
(28).This battery will have a very long lifetime since it has to supply only the leakage current of a few nanoamperes. It adds, however, considerably to the cost of the watch module. Probably the most satisfactory solution is to accept the volatility of the tuning information. This is not a great disadvantage since the time between battery changes approaches 2 to 3 years nowadays. The concept of Fig. 15 can now be extended to an automatic digital tuning scheme as shown in Fig. 16. An acquisition circuit is added to the system that is integrated together with the main watch circuit. This circuit compares the output period of the divider with an external reference and computes the time difference between these two periods. It then transfers this correction to the memory, which in turn adjusts the divider accordingly. The function of the acquisition circuit can even be partly combined with the divider so that the memory circuits are simplified as well. This solution has two major advantages: first, the external equipment necessary to tune the watch is a simple reference generator; second, the information goes unidirectionally into the watch, so that nongalvanic coupling is possible. The concept described has been implemented in a commercial watch module (28). This analog watch with automatic tuning uses a quartz with a nominal frequency at 532 kHz. 14 bits of memory are needed to provide a tuning range of 3% with an accuracy of & The circuit is realized in silicon-gate CMOS technology. The total current drain is less than 4 p,A due to the use of dynamic dividers and the use of amplitude feedback in the oscillator. The tuning of the watch is extremely simple. The watch is placed on top of the reference source and a single button is pushed. This initiates the transmission of a short reference pulse. This pulse is picked
OSCILLATOR DIVIDER
DISPLAY
Q MEMORY
FIG.
16. Automatic digital tuning. After Vittoz (14), reprinted with permission.
204
A. P. GNADINGER
up by a small coil that is included inside the watch and fed to the acquisition circuit. The inductive coupling even makes it unnecessary to open the watch case. 3. New Time Buse Systems The quartz-stabilized oscillator has found such a wide application in any kind of electronic watch or clock that thoughts and suggestions concerning different time bases have not found much interest so far. The concept of a fully integrated time base avoiding quartz as a resonator seems at first quite intriguing because it would eliminate a bulky, rather expensive, and shock-sensitive mechanical component. However, there is such a wide gap between the stability that can be obtained with the best possible RC oscillator on one hand and with the simplest quartz oscillator on the other hand, that a major advantage of the electronic watchaccuracy-would have to be sacrificed. Furthermore, using such an RC oscillator would run contrary to the general trend mentioned before: decreasing the energy consumption to the lowest possible value in order to extend the battery life, because an RC oscillator would definitely consume more energy than a quartz-stabilized one. Nevertheless, at least one manufacturer (29) has expressed confidence in the feasibility of an oscillator of this kind, but with quite a moderate goal for stability. In addition, an accurately trimmed discrete resistor is needed that would cancel nearly all advantages in cost and volume reduction. Other suggestions such as using a piezoelectric substance that would be deposited onto the silicon wafer during wafer fabrication and would serve as a frequency-controlling element have not proven feasible-at least not until now. Alternatives still using a quartz resonator but assembling it together with the integrated circuit into a common package are, of course, quite attractive in terms of cost and volume reduction. Solutions of this kind are under development at various semiconductor manufacturers and will probably be introduced into commercial watches within the next few years. Time base systems based on radioactive sources have been proposed in the literature (30),but they have not yet been realized. Many problems still have to be solved and with the high development cost and the low prices of quartz resonators, the advantages of such a time base are not obvious at all. 4. Quartz Crystals
The quartz crystal resonators that are now used nearly exclusively in electronic watches and clocks owe their frequency-stabilizing properties
205
ELECTRONIC WATCHES AND CLOCKS
to the piezoelectric effect discovered in 1880 by P. and J. Curie. The theory of piezoelectricity is covered in any elementary physics textbook; it is not treated here. In 1921, W. Cady succeeded in utilizing quartz crystals for the frequency stabilization of a tube oscillator and in the early 1930s, W. A. Marrison and A. Scheibe built the first quartz clocks. The application of quartz crystal resonators in electronic wristwatches had to await the advent of integrated circuits and was only realized in the late 1960s (5, 6). Starting material for any quartz crystal resonator is either natural quartz, mostly found in Brazil, or synthetically grown quartz. The synthetic quartz is particularly suited for industrial applications because of the equal size of the crystals, the absence of cracks and other defects, and in general a much better fabricational yield. The large quartz crystals are cut into small units, where special emphasis has to be placed on accurately controlling the crystalline orientation. This is usually accomplished by X-ray diffraction techniques. There are 12 known crystal cuts differing in the orientation of the crystalline axes. The resulting quartz crystal resonators vary widely in their size and shape and their mechanical and electrical properties. In order to stimulate oscillations, these crystals are covered with electrodes in a suitable manner. Depending on the cut of the crystal and the size and shape of the electrodes, four different modes of oscillations can be induced: flexion, displacement, planar shear, and thickness shear, as illustrated in Fig. 17 (31).These modes do not exist in a pure form. They are, moreover, coupled to each other, the coupling constants being strongly dependent on the geometric dimensions and the ( c ) Planar shear
( a 1 Longitudinal flexure L
_ _ - - - - -_ _ _ _ _
z
t $--------L
Y
4 *
-I +I
A
Z
( b ) Longitudinal displocement
-EX
(d)
Y-
Thickness sheor
-
FIG.17. The various forms of oscillation of quartz crystals. After Glaser (311, reprinted with permission.
206
A.
P. GNADINGER
orientation of the crystal (32). Depending on the crystal cut and the oscillation mode chosen, a frequency range starting at I kHz (xy flexure) up to about 200 MHz (employing the ninth harmonic of an AT-cut crystal in a thickness shear mode) can be covered. The equivalent electrical circuit of a quartz crystal resonator has already been shown in Fig. 5a. The inductance L in Fig. 5a may vary from 300,000 H (xy flexure at 1 kHz) to 5 x H at 200 MHz. For the same frequency range, the dynamical capacitance C varies from 0.05 fF to 0.2 pF, the static or shunt capacitance C3from 1 to 50 pF, and the series resistance R from a few hundred ohms to about 1 M a . This is to illustrate the wide range of the electrical parameters that can be covered. The mechanical properties of a quartz resonator can also vary drastically, depending on the crystal cut and the oscillation mode. The most important mechanical properties are shock resistance, size, temperature dependance, and aging behavior. For 32 kHz wristwatch applications, the crystal type used in the early developments was the xy-flexure mode crystal (X 5” cut) shown in Fig. 18a. This type of crystal, however, is very difficult to mount since any mechanical supports have to be located exactly at the nodes of the mechanical oscillations. The high positioning accuracy required renders this type of quartz shock sensitive and expensive. A big improvement was made with the development of the tuning-fork crystal. The base of the tuning fork provides a rigid body for easy and shockproof mounting. The early versions of tuning-fork crystals, shown in Fig. 18b, were cut out of a flat piece of quartz and covered with electrodes on both sides in a suitable configuration in order to stimulate oscillations in the flexure mode, A much improved version of the tuning-fork crystal, shown in Fig. 18c, has been pioneered by Statek Corp. in recent years (33).These crystals are fabricated many units at the same time, employing an essentially planar technology: First the electrode films are deposited by vacuum evaporation onto one side of a flat quartz plate, then the structure is defined by photolithography, and etched using standard wet chemical or plasma etching methods. Finally, the individual units are separated from each other by etching through the quartz plate with an anisotropic etchant. The direction of the fastest etch rate has to be perpendicular to the quartz plate surface in order to prevent underetching of the masked areas. This requirement necessitated the use of a 2 cut instead of the previously employed NT cut and, as a consequence, a new piezoelectric concept had to be found. With this batch processing scheme, more than 100 units can be produced simultaneously on one quartz plate, resulting in low manufacturing costs. An additional advantage of tuningfork crystal resonators is the fact that their frequency can easily be ad-
+
ELECTRONIC WATCHES AND CLOCKS (0)
207
(b)
Oscillator excitation X Y mode
-I
-A
01
+-
-+
a m +- t
with ‘ flexure
FIG.18. The various types of quartz crystals used in electronic clocks and watches. (a) Quartz crystal oscillating in the flexure mode ( X + Y)-cut. (b) Sawed tuning fork crystal (X + 5”)-cut. (c) Etched tuning fork crystal (x + 5“)-cut.(d) Quartz-crystal oscillating in the thickness shear mode AT-cut. After Glaser (31).reprinted with permission.
justed by laser trimming. Using a semiautomatic adjustment scheme, the mass of the tines and therefore the resonance frequency can be altered by removing a controlled amount of electrode mass provided for this purpose at the end of the tines by laser evaporation. In principle, this can even be done through a clear window after the crystal has been put into a hermetic package. It has also been suggested (34) to use this feature on a finished module, rendering the trimmer capacitor for frequency adjustment obsolete. However, the unavoidable aging of a quartz crystal would probably still make an easy frequency adjustment of a finished watch desirable. For clock applications, where minimum power consumption is not as stringent as for wristwatches, 4 MHz quartz crystals have succeeded as standard as mentioned before. These crystals employ an AT cut and oscillate in the thickness shear mode. They are rather small and quite shock insensitive. The main problem in achieving high accuracy in electronic watches or clocks is the dependence of the quartz crystal frequency on temperature.
208
A. P. GNADINGER
+
This dependency is a quadratic parabola for the (X So)cut and a cubic parabola for the AT cut, as shown in Fig. 19. The turning point or the inflection point, respectively, can be put close to the operating temperature. A proper choice of the cutting angle ensures a rather flat characteristic around these points, as evident from Fig. 19. It can clearly be seen that the AT-cut crystal shows a much better performance over a wide temperature range. This is one of the reasons for choosing this type of crystal in electronic clocks. Especially in automotive applications where an extremely wide temperature range is specified, the tuning-fork NT-cut -20
-40
hflf
=
0
20
40
60
98OoC
k(9--80)2
10-5 6
35O 10'
Af/f
35O 12'
4
35O 14' 35O 16' 350 18' 35O- 20' 35O 22' 35O 24' 35O 26'
2
0 -2 -4
-6 -60
-40 -20 0
20 40 60
80 100'
9O
FIG. 19. Frequency variation as a function of temperature with cutting angle as parameter. (A) CT (X + 5")-, BT-, DT-cut. (b) AT-cut. After Glaser (31), reprinted with permission.
ELECTRONIC WATCHES AND CLOCKS
209
crystal would not qualify because of its inadequate temperature range. For wristwatches, however, the practical temperature range is much narrower due to the temperature-stabilizing effect of the human body. The frequency deviation due to the parabolic characteristic of Fig. 19a is therefore small enough and can in most cases be tolerated. The lower power consumption due to the lower frequency of the NT-cut crystal is, for wristwatches, in most cases more important than the better temperature characteristics of the AT-cut crystal. It seems feasible to compensate for temperature variations by special compensating networks possibly integrated on the main electronic circuit, rendering in such a way the watch or clock more accurate even with nonoptimum quartzes and over a wide temperature range. However, such solutions have not been successful so far, mainly due to the increased complexity of the integrated circuit. Such schemes could have more success with digital tuning as described in Section IV,A,2, but temperature sensors and possibly a microprocessor solution will be necessary. In any case, these inventions would only be used for expensive watches where high precision is wanted. The frequency of a given quartz crystal will also change slightly with time. This aging effect is caused by many factors and is dependent on crystal cut, frequency, manufacturing technology, and mounting and encapsulation methods. All aging factors can be reduced to the following: mass transfer from or to the quartz surface, mass transport along the surface, and structural changes in the electrode films and the mounting structure. These factors have to be minimized, which is usually done by appropriate heat-treatments. First, the crystals are cleaned carefully after cutting and mounting and a first heat-treatment is performed. After cooling down, the crystals are possibly tuned to the proper frequency and hermetically packaged by cold-welding under vacuum. The sealed quartz crystals are now exposed a second time to a high temperature. This way any tension in the electrode films or the mounting structure can be decreased. A typical aging characteristic of a 32 kHz xy' flexure quartz is shown in Fig. 20. Further factors that can influence the frequency of a quartz oscillator are mechanical shocks and vibrations that act upon the quartz crystal. As mentioned before, xy flexure mode crystals as shown in Fig. 18a are the most shock-sensitive units because the wire mounting structure forms an integral part of the resonating system. The shortest possible wire length is h / 4 , i.e., a quarter of the wavelength corresponding to the oscillating frequency. Shorter wire lengths would reduce the Q value to unacceptably low values. This quarter-wavelength restriction actually determines the lowest resonating frequency of the supporting structure. Fortunately, at
210
A.
P. GNADINGER
5 -
I
0
3
6
1
I
9
I 12
Time (months)
' t
I
0
I
1
I
I
1
2
I
3
I
1
4
1
I
5
Time (years)
FIG.20. Aging of quartz watch crystals 32 kHz,XY' flexure mode.
32 kHz this value is high enough (>500 Hz) to be in the nondestructive region. Frequency changes due to mechanical vibrations are usually below 5 ppm for xy flexure mode quartzes. Larger deviations are encountered if a quartz resonator is subjected to a single mechanical shock of, for example, 3000 g , where g is the acceleration factor due to gravity. Here, the maximum frequency deviations can be as large as 20 ppm, which
ELECTRONIC WATCHES AND CLOCKS
21 1
amounts to a time deviation of a 32 kHz watch of 1.73 sec/day. Moreover, it takes approximately 7 days for the quartz to recover (35) due to a memory effect that takes place in the strongly loaded and slightly deformed supporting wire. All these effects are greatly reduced for the tuning-fork crystals (33)because of the rigid base available for mounting purposes. Shock sensitivity also decreases with increasing crystal frequency because the mounting structure no longer influences the resonating system. B. Electronic Watch and Clock Circuits As evident from Fig. 2, the integrated electronic circuit connects and controls the other components of the watch in a suitable manner. Its main functions are (1) to divide the high-frequency output signal of the time base to the low value necessary to drive the display, (2) to feed the energy provided by the battery in a suitable form to the time base, (3) to provide the proper signals for displaying minutes, hours, date, etc., by using various counters, (4) to decode the signals to drive the solid-state display in case of digital watches, and ( 5 ) to drive the stepping motor in case of an analog display.
In additian, there are various auxiliary functions incorporated in the electronic circuit. All these functions are realized by standard digital techniques. The oscillator circuits described in the previous section are actually also part of the integrated circuit of the watch. They are analog circuits. In the early electronic watches they were usually integrated in a separate circuit. In recent years, the manufacturing technologies have been optimized to such a degree that it became feasible to integrate the oscillator circuit on the same chip with the main circuit, so that practically all watch and clock modules nowadays contain only one integrated circuit. Referring again to Fig. 2, the time base and the watch circuit are therefore partly merged. The various functional blocks of the electronic circuit are now described in detail. 1. Frequency Dividers and Counters
One of the main functions of the electronic watch circuit is to reduce the frequency of the output signal of the oscillator in the range from 32
A. P. GNADINGER
212
kHz up to several MHz to a value of the order of a few cycles per second or less, depending on the type of display used. This is accomplished by cascading a number of divider cells where each cell divides the frequency of the signal by a given factor. This factor can be any number from 2 to about 9. The values most often used are 2, 3, and 5 . Keeping in mind that it is highly desirable to keep the number of active components as small as possible in order to minimize power consumption and chip area, employing dynamic logic circuit techniques would be advantageous. However, because of the rather low-frequency operation of most of the divider stages in the chain, dynamic logic can only be employed for the first few stages. The lower-frequency limit is approximately 1-5 kHz, depending on the technology used. It is, however, still worthwhile employing dynamic dividers even for a very small part of the divider chain, since the first few stages will consume most of the power, as is shown later. For high-frequency applications (2-4 MHz) it is even mandatory in order to keep the current consumption within reasonable limits (510 PA). A few examples of divider stages as they are used in actual watch circuits are now given. First, the static version is described and the dynamic version deduced therefrom. A very popular divide-by-2 cell is given in Fig. 21 (36). This combination of inverters and single or two-level gates corresponds to a total number of 22 transistors and realizes the following set of logic equations:
=E,
A
B
=
(I
+ D)A,
D=B,
C = IE
+ AB
(13)
E=C i
I
A A
A
I -
‘-0
I
-
I
I
HE
FIG.21. Basic circuit of scale-of-two CMOS stage. After Vittoz et al. (36), reprinted with permission.
ELECTRONIC WATCHES A N D CLOCKS
213
where 1 is the input signal and each of the variables A , B, C, D, and E can be used as output signal, since they all switch at a frequency half that of the input. This structure has been arrived at by means of a special algorithm based on the standard Huffman (37) method. The complete switching cycle of each variable is given in Table I. In the first state each equation of the set is satisfied. It is thus a stable state, which is preserved as long as I does not change. If Igoes from 0 to 1 (line 2), the equation for B is no longer satisfied. B, therefore, tends to change state, so that the second state is unstable. This leads to another unstable state where the equation for D is no longer satisfied. Transition of D leads to a second stable state (line 4), which will be maintained as long as I is 1. If 1 switches back to 0, the cycle goes on through two more unstable states and then reaches the original state where the cycle had started. It can be seen clearly that this structure is free of any logical hazards: during any of the unstable states only one equation of the set is not satisfied. Hence, a single variable tends to change state and there exist no rating conditions between the variables. This is an important consideration in all divider circuits. In order to ensure proper operation of a divider cell in a real circuit with variation of the technological parameters, racing conditions have to be strictly avoided. A further advantage of the divider cell of Fig. 21 is the use of only the uncomplemented form of the input variable I. This avoids the use of an additional inverter to drive the first stage, which would enhance the power consumption. Another advantage is the possibility of cancelling the dynamic power consumption of the first stage by combining the gate capacitance of the first stage with the functional capacitance Cz of the Pierce oscillator (Fig. 4c). The circuit of Fig. 21 can be simplified by combining some transistors, yielding the 18-transistor circuit of Fig. 22. One transistor has been eliminated by changing the static gate giving C by a dynamic gate (38), where the functions C+ of the block of p-type transistors and C - of the n-type transistors
C + = % i- 2,
C-
=
A B f IE
(14)
have been introduced. It can be shown that node C is floating only during a transient state and the static behavior of the cell is therefore not affected. The dynamic version of the circuit of Fig. 22 can be arrived at by omitting all feedback loops that are required to maintain the stable states. The internal capacitances associated with each node of the circuits will then maintain the state of the divider. This dynamic version is drawn in thick lines in Fig. 22. It is very simple and requires only nine transistors.
214
A. P. GNADINGER TABLE I
SWITCHINGCYCLEOF VARIABLES Type of state
I
A
B
C
D
E
Stable
0
1 I
I I
0 0
0 0
1 1
0
I I
I
O
O
O
I
1
I
0
0
1
I
O
I
O
O
I
I
0
0
1
0
1
I
I
0
0
1
0
1
I
0
1
I
0
1
I
0
1
0
0
1
1
0
1
0
1
1
I
0
1
0
1
1
0
0
0
0
1
1
0
0
0
1
I
1
0
0
0
1
I
0
0
0
0
1
I
0
0
1
Unstable
Path of the switching current"
0
12, 16 0 19
0
5, 9
22 0
2, 4
13 20 0
1
10, 12
21 ~
~
~
~
0 ~~~~
~~~
The last column indicates the transistors through which the current charging or discharging the switching node is flowing. a
Divider stages that divide by a ratio other than 2 are described in the literature (14, 36, 38, 39). An example of a dynamic divide-by-3 cell is shown in Fig. 23 (38, 39). This circuit can be extended to any odd dividing ratio.
ELECTRONIC WATCHES AND CLOCKS
215
E I
FIG.22. Five-gate race free divide-by-2cell. The simplest dynamic divider is drawn in thick lines. After Vittoz (14). reprinted with permission.
The circuits described so far are realized in a complementary technology (CMOS). Some technologies such as 12L provide only single-level gates, that is, NOR or NAND gates, depending on the definition of the logic levels. The most common divide-by-2 cell in this technology is a standard six-gate structure. A simpler circuit using four NOR gates with a single input I is shown in Fig. 24. This circuit is not completely race free. Forbidden transitions have to be suppressed by slowing down the 0 to 1 transitions of variables A and D . This structure has been realized in an actual circuit in Z2L technology with a very small cell size of less than 8000 pm2 (40).
2 . Decoder and Driving Circuits The decoder and driving circuits are the interface between the integrated circuit and the display. There are quite different requirements put on these circuits, depending on the type of display used, the most obvious difference existing between those used in analog clocks and watches and those required to drive a solid-state display. In particular, the maximum current they have to be able to supply can be orders of magnitude apart. Peak current levels for analog watch drivers are in the range of 1 mA, whereas the peak current for LEDs (light emitting diode displays) can be as high as 70 mA. In the case of LCDs (liquid crystal displays), the maximum currents are much lower, mostly around 1 pA, but the complexity of the circuit is far greater since every segment of the display basically needs its own driver stage. The three basic types of decoding and driving circuits used for analog displays, LEDs and LCDs, are now treated in somewhat more detail.
216
A. P. GNADINGER
10-
Fig. 23. Dynamic divide-by-3 cell; it can be extended to any odd dividing ratio. After Vittoz (14). reprinted with permission.
a . Analog Watch Drivers. For analog watches and clocks, no special decoding circuitry is needed since the output signal of the divider chain-usually a short current pulse of 1 Hz-is directly fed to the driver stage, which in turn interfaces with a stepping motor that advances the fastest moving hand of the watch, usually the second hand, in discrete steps. A miniaturized stepping motor is by far the most common device to drive the display of an analog watch. Various designs of those micromotors have been described in the literature ( # 1 , 4 2 ) . They consist basically of a wound coil that is mounted between two permanent magnets. The coil is energized by a short current pulse provided by the drive circuit and moves thus for a predetermined angle. Means are provided so that
A=
B*c
-
B = A*C*D
C= m D
L
D=
FIG.24. Controlled-race divide-by-2 cell using four NOR gates: forbidden transitions (in dashed lines) are avoided by slowing down the 0 to 1 transitions of variables A and D . After Vittoz ( / 4 ) , reprinted with permission.
ELECTRONIC WATCHES A N D CLOCKS
217
this movement is unidirectional. To stop the movement after each pulse, the motor is short-circuited and the resulting magnetic field provides a retarding moment to the coil. Most motors use bipolar pulses; that means that pulses of alternate polarity flow through the motor. However, at least one supplier (42) has specialized in stepping motor designs that employ unipolar current pulses. Figure 25 shows a standard circuit used to drive a bipolar stepping motor M of an analog watch. At each step, the input of one of the two inverters is alternately switched on and current pulses of alternate polarity flow through the motor M. Between pulses, the motor is kept in shortcircuit through the two conducting n-channel transistors. The pulse amplitude is very much dependent on the motor design and is of course kept as low as possible. A minimum peak current of 0.5 mA is typical at the time of this writing. The period of the pulses is usually 1 sec for analog watches with a second hand (in rare cases more than one pulse per second is required, but this is disadvantageous because the current consumption is higher). For watches with only minute and hour hands, the pulse period is longer, 4 , 5 , or 10 sec being typical. It is quite obvious that such a watch has a considerable advantage since the power consumption is reduced in proportion to the pulse period for all parameters (pulse amplitude and width) being equal. For a given peak current, the pulse width is determined by the minimum moment required to drive the second (or minute) hand and the associated gear train. For bipolar motors in wristwatches a pulse width of the
VL (a)
FIG.25. (a) Analog-watch driving circuit. (b) Pulse train of analog-watch driver
218
A. P. GNADINGER
order of 3-30 msec is required. In electronic clocks the driving circuit and the stepping motor are capacitively coupled so that the pulse width is essentially determined by the value of the coupling capacitor and is not fixed by the integrated circuit. The internal resistance of a stepping motor has decreased steadily in the course of time and is at present approximately 1000-2000 fk for bipolar stepping motors and 500- 1000 fk for unipolar motors. In order to be able to supply the maximum required current into such a load, the driver circuit has to have an extremely low internal resistance. The maximum voltage drop in the conducting transistors must be kept below 50- 100 mV to ensure reliable operation from a low-voltage power supply. The channel resistance of an MOS transistor is to first order given by the relation r = L/[pWCo(V - V d l
(15)
where L and W are the length and width of the channel, p the mobility, and Co the oxide capacitance per unit area. V and VT are the supply voltage and the threshold voltage, respectively. For a minimum value of V of 1.1 V and a maximum threshold voltage of 0.7 V (worst-case condition), Co = 530 pF/mm2, p = 200 cm2/V sec, and an effective channel length L of 3.5 pm, the channel width W must be at least 16.5 mm to keep the voltage drop below 50 mV at 1 mA. This extremely wide transistor is realized by a meander-shaped interdigitated structure. The resistances of the source and drain-diffused fingers have to be kept as low as possible, which is usually accomplished by connecting them to a low resistance metal line at as many points as possible. In any case, the driver inverters have to be carefully optimized for minimum size. Even then the total size of the driving circuit very often exceeds 1 mm2 and often covers more than 25% of the total chip area. This can also clearly be seen in Fig. 36, where an example of an actual analog watch circuit is shown. b. LED Decoding and Driving Circuits. The digital watch circuits have to supply decoded signals able to drive a minimum of 34 digits, corresponding to 23 segments plus some special signs such as flashing colon, the AM/PM flag, or the name of the day. LED displays have a big advantage compared to LCDs: they can be multiplexed quite easily. This reduces the complexity of the decoder considerably. A standard multiplexing scheme is shown in Fig. 26 for a display with six digits (14). A single four bits/seven segment decoder is employed sequentially for each digit. The multiplexing switches can best be implemented by means of transmission gates (Fig. 8). A strobing frequency of 64 Hz is used most often. The decoder itself is realized with standard techniques. A compact ratioed CMOS technique has proven
ELECTRONIC WATCHES AND CLOCKS
219
COUNTER
k
FIG.26. Standard counter-multiplexer-decoder circuit as used in watches with LED display. After Vittoz (14), reprinted with permission.
well suited where transistors of one type are used as passive loads switched on simultaneously with the display. The decoded signals are then fed to the segment drivers as shown in Fig. 26. As mentioned earlier, LED drivers have to provide extremely high currents: currents of the order of 10 mA have to be switched by the seven segment drivers and up to 70 mA by the four to eight digit drivers. Therefore, most existing CMOS circuits need external bipolar transistors for the digit drivers or for both digit and segment drivers. There has been one attempt to incorporate all driver circuits onto a single CMOS chip (43),but this solution does not seem to be economical. Too much silicon area is consumed by the CMOS transistors, which have to be laid out so that they can provide the required high currents. Multichip solutions seem therefore to be a better choice; even so, packaging the chips creates additional costs. A manufacturing technology that is ideally suited for LED drivers is the Z2Ltechnology that is covered in more detail in Section IV,C. Being a bipolar technology, it can provide high-current transistors on a rather small area. Yet the multistage amplifiers necessary to scale up the currents from the 10 nA logic level to the 70 mA drive level still need an area of roughly 1-2 mm2 (40). Figure 27 shows a schematic of such an 12L digit driver (40). A low signal at the base of transistor Ql enables QI1to sink the digit drive current of 70 mA. Transistors Ql-Q, are minimum size P L transistors, con-
220
A. P. GNADINGER
stantly drawing 5-10 nA per transistor. With the signal at the base of
Q, going high, the isolated transistors Qs-QI1 are turning off. The digit driver in the off state then draws 50 nA. Transistors Ql-Q3 serve the purpose that any gate signal less than two gate delays long will not turn the stage on. The same configuration is implemented with the segment drivers, preventing the turning on of incorrect segments due to timing has to be errors. Under worst-case conditions, the output transistor designed with a gain factor p > 23 at 70 rnA, which can easily be done on a fairly small chip area. c. LCD Decoders and Drivers. Contrary to LED displays, LCDs cannot be multiplexed easily, as is shown in Section IV,D. The signal for each segment must therefore be decoded separately, unless each output is buffered by a latch circuit that is updated periodically. Each segment of a nonmultiplexed LCD requires a separate driving circuit. Up to 70 external connections between the integrated circuit and the LCD are thus necessary for a complex digital watch. Digital wristwatches are usually operated with a minimum supply voltage of 1.3 V. The LCD, however, requires a voltage of at least 2.5 V, as shown in Section IV,D. It is therefore necessary to use two 1.5 V (1.3 V) batteries in series, or a voltage doubler incorporated into the main circuit. An example of a voltage doubler that is now preferred in wristwatch circuits is described in the next section. It is also mandatory to drive LCDs with alternating voltages. Direct voltage drive would lead to an electrolytic decomposition of the electrodes that would drastically reduce the life expectance of an LCD. If the
ell
ELECTRONIC WATCHES AND CLOCKS
22 I
electrodes of the display were covered with an insulating lacquer to prevent this decomposition, the voltage across the liquid crystal would strongly decrease due to the voltage drop across the double layer formed near the electrodes. An example of a driving and decoding circuit (44) used to drive an LCD in a parallel (nonmultiplexed) fashion is shown in Fig. 28 ( 1 1 ) . A rectangular voltage with 0 and 5 3 V levels is applied to the common back-electrode. In position 2 of the switch, the same voltage is applied to the middle segment of the front electrode, which is considered as an example. In the overlapping region of the two electrodes no voltage drop exists across the liquid crystal and the segment is not excited. If the switch is put in position 1, voltages of opposite polarities are applied to back and front electrode and the segment under consideration is exercised. The segment effectively sees a rectangular alternating voltage of + 3 and -3 V. Standard BCD to segment decoders together with an exclusive OR gate are necessary to drive one digit in the above-mentioned way. The minimum driving frequency of an LCD is given by the sensitivity of the human eye to variations in light intensity. Nonmultiplexed displays LCO P A R A L L E L ORlVE +3v
nr
A V = V,
-3v
v1
- Vz
'iVN
FIG.28. Driving (top) and decoding (bottom) circuit for LCD display in parallel (nonmultiplexed) fashion.
222
A. P. GNADINGER
that are driven by symmetrical rectangular voltage pulses are practically subjected to the same uninterrupted voltage since the electrooptical effect is independent of polarity. The light intensity varies only slightly during a short fraction of time when the capacitances associated with the electrodes are charged and discharged. The human eye can barely see these variations, even at frequencies of around 30 Hz.A convenient frequency that can easily be deduced from the divider chain of the circuit is 32 Hz. This is now usually employed in most circuits. Higher frequencies are not desirable because the current consumption of the display would increase considerably because of the fairly large capacitance of the electrodes. For multiplexed LCDs, the rules are different because a clocked signal is now applied to the display. Figure 29 shows an example of an LCD with N equal digits. In order to allow for XY selection, equal segments are connected together (rows of the matrix). Each digit possesses a separate back electrode (columns of the matrix). The back electrodes are biased in a cyclic fashion independent of the digits to be displayed. For the columns selection, the clock cycle is therefore 1 : N. If the voltage applied to the back electrode is opposite to the voltage at the row electrodes, the desired digit is displayed. The big advantage of multiplexing the LCD is a considerable reduction of the number of external connections. For a multiplexed display this number is given by N + 7, instead of 7N + 1 for a nonmultiplexed version. However, multiplexing a display requires bipolar cyclic pulses with controlled amplitudes because of possible interference with neighboring elements. This leads to a considerable increase in the complexity of the driving and decoding circuitry of LCD displays. Further disLCD Multiplexed Drive
I
I
I I I I
I I I
-1 Segment G +V -v
Digit 1
I
+2q-L-jj v -2
Digit 2
Digit N
FIG.29. The principle of driving a multiplexed display.
ELECTRONIC WATCHES A N D CLOCKS
223
advantages of multiplexed LCDs more related to the physics of liquid crystals are described in Section IV,D.
3. Auxiliary Circuits Apart from the main functions described so far, each watch and clock circuit contains a variety of auxiliary parts such as logic level shifters, voltage multipliers for LCDs, special circuitry for fast automatic testing, and battery voltage indicators. For multifunctional, digital wristwatches, the additional logic circuitry needed can constitute a major part of the circuit and occupy a considerable chip area. Logic level shifters are required as an interface between the lowvoltage high-frequency part and the high-voltage low-frequency part of a circuit for LED displays. Figure 30 shows an example of such a circuit in CMOS technology (14, 45). Transistors T, and T2 are driven by the low-level signal X and are designed so that they can sink the current provided by the transistors T3 and T4,which are themselves driven by the high-level signal X’.Good pulse shaping and short transit time are given by the inherent regeneration. For LCDs the high voltage is usually supplied by an electronic voltage multiplier allowing the circuit to be operated by a single battery. Modern displays require not more than 2.5-3 V and a voltage doubler can be used. Figure 31 shows an efficient voltage doubler using active switching (46). A first capacitance C, is alternatively connected across the lowvoltage and the low- and high-positive-voltage levels. In the first state it is charged; in the second state it shares its charge with that on capacitance Cz.If C1 and C2 are equal, the voltage on the high positive terminal is
X ’ W 1
C
x 4 L --iL -..
--
Tl
T2
-
224
A. P. GNADINGER
H
FIG.31. Voltage doubler circuit for digital wristwatch with LCD display. Courtesy of Faselec Corp., Zurich, Switzerland.
effectively double the voltage across the low positive terminal and ground. Use is made of the level shifter circuit shown in Fig. 30, which is used to drive the two p-channel switches connected to the high positive terminal. The circuit has to be operated with a certain minimum frequency. A signal of 512 Hz, which can readily be deduced from the divider chain, serves as a clock signal H. Reasonably high values of the capacitances cannot, of course, be integrated into the same chip. They have to be realized by miniaturized discrete components bonded externally to the chip. An important auxiliary circuit that can occupy a considerable amount of chip area is the circuitry needed to set the watch. Most of the timesetting systems use a scheme to accelerate the particular time function to be set or to be corrected. For digital watches at least two push buttons are provided on the watch case that have to be pressed singly or simultaneously in order to enter the so-called set mode, to select the appropriate function (hours, minutes, data, etc.), and to change this function. For analog watches a simpler scheme with a mechanical crown can be employed to set the hours and minute hands as is customarily done with ordinary mechanical watches. To synchronize the second hand to a reference time
225
ELECTRONIC WATCHES AND CLOCKS
signal, a push button may be provided. The circuitry corresponding to these time-setting features is normally implemented in a standard manner employing random-logic gates. A particular example is described in the next section. The push buttons of an electronic wristwatch or of a clock cannot be manufactured with reasonable costs to ensure that no spurious signals enter the circuit. Therefore, antibounce interface circuits must be provided on those input pads of the circuit that are connected to the push buttons. The push buttons are in most cases simple single-pole singlethrow switches without spring-loaded fast-switching action. Upon closing of this switch, in most cases a spurious pulse train is generated that could lead to a malfunctioning of the watch and that has to be suppressed. A circuit employing shift registers could be employed to filter out those unwanted signals. A simpler solution that is well suited for this particular application is shown in Fig. 32. The two transmission gates are clocked with a signal CP of approximately 10 Hz. While closing the push button contact, a logic 1 is generated at the input. This state is stored in the first latch, the master flip-flop, independent of the point of time and the duration of the signal. The information is transferred to the second latch, the slave flip-flop, when the clock pulse is going to 0. To set the circuit back to the initial condition, a 0 has to appear at the input during the negative transition of the clock pulse. In the actual situation a 1 at the input is never considered a disturbance and is therefore transmitted asynchronously. A 0 is only transmitted during the negative transition of the clock signal. A transmission of a disturbance is thereby not totally impossible but highly unlikely. CP Push
to main 4circuit
button
CP tranwnissim Qata
FIG.32. Antibounce circuit.
226
A . P. GNADINGER
Testing of an integrated watch circuit can be a time-consuming and expensive operation as mentioned earlier. In order to speed up the testing procedure, special circuitry has to be provided. This is especially true for analog watch circuits, where the output of the driver transistor is in most cases a 1 Hz signal. In order to determine whether a circuit is properly functioning, a measuring time of at least 1 sec is necessary-the time needed so that at least one pulse leaves the output stage. This time is much too long for economic testing. The method ordinarily chosen to shorten testing time is to bridge N stages of the divider chain and increase by this means the output frequency by a factor 2 N .The upper limit of this scheme is reached if transit time effects inherent in the technology used are influencing the pulse shapes and the timing of the circuit to such an extent that distinguishing good and bad circuits cannot be done with a reasonable degree of confidence. To preserve the total number of external connections, this special test pin is quite often combined with some other function of the watch. The selection of the test mode is then accomplished by a special combination of test voltages applied to the bonding pads. An example of such a circuit for accelerated testing is given in the next section. 4. Examples of Actual Clock and Watch Circuits
In Fig. 33, the block diagram of an actual clock circuit, the MB 9B of Faselec Corp., Switzerland, is shown as an example.
I
d p
t-I
Vp or VN
3
""T
M
Alarm out
. I IV.
M
I
I -
I
-
'"lY
I
-I
FIG.33. Block diagram of 4 MHz clock circuit MB 9B. Courtesy of Faselec Corp., Zurich, Switzerland.
ELECTRONIC WATCHES AND CLOCKS
227
Connected to terminals 1 and 2 are the positive and negative terminal of the power supply V, and V, , respectively. The circuit is operated from a single Leclanche-type battery with 1.5 V nominal voltage. External to the circuit and connected to pins 7 and 8 is the quartz crystal with a frequency of 4.19 MHz, which serves as a time base. As mentioned before, this frequency has become a standard for clock operations. The two capacitances Ci, and C,,, are also external to the circuit and connected to pins 7 and 8 in a Pierce configuration. C,,,is a variable trimmer capacitor for accurate frequency adjustment. Integrated on the circuit and connected to pins 7 and 8 as well is the actual oscillator circuit-a simple inverter oscillator with an integrated feedback resistor. The output signal of the oscillator is then divided down to 1 Hz by means of a 23-stage divider circuit as described in Section IV,B,l. The output signal at the end of the divider chain is fed to a bridge-type output driver for a stepping motor requiring alternating drive pulses. This motor is capacitively coupled to the integrated circuit and connected to pins 3 and 5 . At pin 6 an alarm signal of typically 512 Hz is available that can be connected to an alarm transducer. The alarm output contains a very low impedance nchannel driver transistor and a protection diode to short-circuit current spikes from an inductive load so that the alarm transducer can be driven in most cases directly without an external driving transistor. At terminal 4 a push button is connected that is used for accurate time setting and for rapid testing. Left open, it gives the normal running condition. Connected to V,, it interrupts the motor output. This is used for accurate time setting (the first output pulse appears after release of the switch) and to reduce power consumption of the clock during storage. If terminal 4 is connected to a voltage about 1 V higher than the positive supply voltage V , , the test logic is activated, which short-circuits essentially four stages of the divider chain and in that way speeds up the motor and alarm outputs by a factor of 32. Figure 34 shows a photomicrograph of the Faselec circuit MB 9B. The chip measures 1.8 x 2.76 mm and contains approximately 400 active elements. About # of the chip area is occupied by the large output drivers for the motor and the alarm that can clearly be seen at the bottom and the right hand side of the chip. The current consumption of this circuit is typically 30 @, determined largely by the oscillator and the first few divider stages. For clock application, this current consumption is low enough to ensure a battery life of more than a year. For wristwatch applications, the current consumption has to be reduced by more than an order of magnitude to ensure adequate battery life. Figure 35 shows a block diagram of a typical integrated circuit for an analog wristwatch-the MB 4B manufactured by Faselec Corp. The circuit
228
A.
P. GNADINGER
FIG.34. Photomicrograph of the 4 MHz clock circuit MB 9B. Courtesy of Faselec Corp., Zurich, Switzerland.
architecture is similar to that of Fig. 33 but the circuit is now optimized for minimum power consumption. The oscillating frequency of the quartzcrystal-controlled oscillator is 32 kHz. The oscillator is amplitude regulated. This complicates the circuit design considerably but ensures minimum power consumption of the oscillator. One capacitance of the Pierce-type oscillator is integrated on the chip itself; the second one-a variable trimmer capacitor-is connected externally. The current consumption of this circuit is typically 0.6 @, ensuring a battery life of at least two years. A photomicrograph of the circuit MB 4B is shown in Fig. 36. The chip measures 1.7 x 2.0 mm and contains about 400 active elements. As an example of a digital watch circuit, a block diagram of the Faselec circuit MJ l l is shown in Fig. 37. This circuit is a single-chip silicon gate CMOS watch circuit designed to drive a six-digit in-line LCD. It incorporates six functions: hours, minutes, seconds, date, month, and weekdays. The oscillator circuit is basically identical to the concept shown in Fig. 11 with one variable capacitance integrated on the chip itself. The oscillation frequency is 32 kHz. The output signal of the oscillator is fed to a divider chain made up of static cells as described in Section IV,B,1 and di-
ELECTRONIC WATCHES AND CLOCKS
229
FIG.35. Block diagram of 32 kHz watch circuit MB 4B. Courtesy of Faselec C o p . Zurich, Switzerland.
FIG.36. Photomicrograph of the 32 kHz watch circuit MB 4B. Courtesy of Faselec Corp., Zurich, Switzerland.
0sc.h
O~C.OU1
I ATST
DISPLAY MODES
/
\
/Min/Sec/Weekday
--Vp
/M inlDateNVeekday
12 hour or 24 hour display operation
FIG.37. Block diagram of six digit/six function digital watch circuit MJ 1 1 . Courtesy of Faselec Corp., Zurich, Switzerland.
ELECTRONIC WATCHES AND CLOCKS
23 1
vided down t o a 1 Hz signal in a fashion similar to analog watch circuits. The 1 H z signal, however, is now fed to various counters, where the minutes, hours, weekday, date, and month information is generated. The variable length of the months is thereby properly accounted for so that the watch has to be reset only once every four years, during a leap year. The outputs of these various counters are connected to the decoders and driving stages as described in Section IV,B,2, which in turn drive the various segments of the LCD. A bonding option (shown on the right-hand side of Fig. 37) allows the selection of a 12- or 24-hour display mode. The circuit operates with a supply voltage of nominal 1.5 V generated from a battery connected between pins Vp and VL. The LCD requires a higher operating voltage of typically 3 V as described before. This voltage is generated by a voltage doubler (Fig. 31), which needs two external capacitors C2 and C3. The 512 H z input signal of the voltage doubler is deduced from the divider chain. To provide for readability of the watch in the dark, a backlight is provided that is connected to the 1.5 V battery through switch B. The backlight is simply a miniaturized incandescent light bulb mounted at the side of the display. Illumination and operation controls are accomplished by two singlepole single-throw switches A and B. The operation diagram is shown in Fig. 38. Switch A determines what will be displayed, whereas B is used for setting and illumination. If A and B are pressed at the same time, the setting modes are called up. Both inputs have an internal pulldown, which allows these pins to float during normal operation. The watch will start (after power up o r test reset) by displaying hours/minutes/seconds and weekday with the colon continuously on as shown in Fig. 38. Pressing and releasing switch A will change the display to hours/minutes/date and weekday with the colon flashing at a 1 Hz rate. Pressing and releasing switch A again will change the display back to the original information. By pressing switches A and B simultaneously, the circuit enters the set mode. First, minutes and seconds are displayed. If B is pressed, the seconds counter will be reset to zero, the minutes counter will advance by one or will remain unchanged, depending on whether the seconds count was greater or less than 30 sec at the time B was pressed. The seconds counter will immediately resume operation after reset. This feature greatly simplifies accurate setting of the watch for the user, since the timing error will rarely be more than 30 seconds. Pressing and releasing A again will call up the next set mode. The display shows minutes and the colon flashing. Pressing B will advance the minutes at a 2 Hz rate. At the same time, the seconds counter is reset to zero and the watch stops timekeeping operation. This is indicated by the colon being continuously on.
232
A. B Watch starts atter itOD1
P. GNADINGER
4H H . M M S A'I
&r )' ~
I
Display modes
A.B I A and B pressed ar the =me time1
Error mun be less then 2 30 seconds.)
Setmodes
< +
B I S 1weekday1
nB lSet date1
-.("?"I-
B (Set month1
FIG.38. Operation diagram of the six digit/six function digital watch circuit MJ 1 1 . T = Timer reset to display mode 10-20 seconds after the release of switch A or B (only if the watch was not stopped by setting the minutes). Courtesy of Faselec Corp., Zurich, Switzerland.
Pressing and releasing A again will call up the third set mode. The display shows hours and an A or P respectively at the position of unit minutes to indicate AM or PM. The colon is flashing or on, depending on whether the watch was stopped or not in the minutes set mode. B will again advance hours at a 2 Hz rate. In the same manner the weekday, the date, and the month information can be set. In the date-set mode the date counter counts independently of the month information to 3 1. In all set modes, timekeeping is unchanged as long as B is not used. Using B,each counter is set separately and no carry signals from and to other counters are accepted or generated. In all set modes, a timer reset signal is generated 10-20 sec after releasing A or B (unless the watch was stopped in the minutes-set mode), which will automatically return the watch into the display mode hours/minutes/seconds. Referring back to Fig. 37, a test input (TST) is furnished to facilitate high-speed testing of the circuit by short circuiting part of the divider chain in a similar fashion as described for analog watch circuits. A test
ELECTRONIC WATCHES AND CLOCKS
233
FIG. 39. Photomicrograph of the digital watch circuit MJ 11. Courtesy of Faselec Corp., Zurich, Switzerland.
reset input (TR) resets the circuit in a defined state: January 1, 1:OO (AM), 00 sec for the 12-hour operation and 0:OO ( A M ) for the 24-hour operation. Figure 39 shows a photomicrograph of this circuit. The chip measures 3.6 x 3.9 mm and contains roughly 3000 active elements. Because a nonmultiplexed display is used, the total number of bonding pads is fairly large, 58 in this example. Compared to an analog watch circuit as shown in Figs. 34 and 36, the complexity of this circuit is considerably larger. This is mainly due to the increased demand on control circuitry, additional counters, and the considerably more complex decoding and driving circuitry. The total current consumption of the MJ 11 circuit is typically less than 1.5 p4, which guarantees a battery life of at least two years provided the backlight is not operated too often. C . Manufacturing Technologies
The manufacturing technologies chosen to realize the integrated circuits described in the previous chapters have to satisfy certain requirements that are unique for watch and clock circuits. As mentioned earlier, the overriding and most important requirement put on the technology is
234
A.
P. GNADINGER
minimum power consumption, since it determines to a large degree the life expectancy of the battery (at least for analog watches and digital watches with LCDs where the display requires only little power). The second requirement put on a watch circuit technology is low-voltage operation. Watch batteries have typically 1.35 or 1.5 V nominal voltages, so that watch circuits should still operate satisfactorily with minimum voltages of about 1.1 V. Third, a watch circuit technology should be capable of large-scale integration. The most complex circuits such as circuits for digital watches with stop watch and alarm functions contain up to 10,000 individual active components with the corresponding interconnections. Since the major part of such a circuit consists of random logic, the interconnections require a considerable amount of chip area. These circuits can be as large as 20 mm2 or more. Considering these rather severe requirements, only a few of the integrated-circuit technologies developed during the last 10- 15 years qualify for watch circuit applications. In Table I1 some of the technologies are listed that have been developed for these purposes. The first technology in this list is a special bipolar technology optimized for low current consumption (47). This is accomplished by replacing the collector load resistors by current sources implemented as lateral p-n-p transistors. This technology is capable of medium-scale integration (up to a few hundred active components); it can be used for 32 kHz oscillator circuits and it is most suited for watches of the first, second, and third generation (analog display). Its power consumption is rather high, as is typical for any bipolar technology at these low frequencies. With a battery voltage of 1.3 V, the current drawn is approximately 20 nA/kHz for one divider stage. For a typical watch circuit (47) this corresponds to about 10 pA total current consumption. A more recent bipolar technology that is well suited for digital watch circuits intended to drive LED displays is the integrated injection logic ( P L ) technology invented in 1972 (48-50). As already mentioned in Section IV,B, 12Loffers the big advantage that the segment and digit drivers can be integrated on the main integrated circuit. The current consumption, however, is still rather high, typically 15 nA/KHz per divider stage. Recently, an 12L watch circuit with a considerably reduced current consumption has been described (40) where special techniques such as current "starving" have been used to reduce power consumption. With the display off, this circuit draws only an average of 7 pA and will operate correctly to below 5 p A over the temperature range. However, this performance has to be bought by rather tight processing requirements and dense design rules. The more modern technologies that are used for nearly all of the
TABLE I1
INTEGRATED CIRCUITTECHNOLOGIES Degree of integration"
Current consumption for one static fiequency divider stage
Preferred display
Oscillator frequency
Watch generation
Technology
Subgroup
SSI ~400 gates MSI s1OOO gates
=20 nA/kHz
Mechanical
5 3 2 kHz
( I -3)
Bipolar
Standard (micropower) IfL
-25 nA/kHz
LED
5 3 2 kHz
(41
Metal-gate Silicon-gate Silicon on sapphire (SOS)
LSI c5OOO gates MSI 5 IOOO gates
-0.1 nA/kHz -0.4 nA/kHz 4 . 2 nA/kHz
Mechanical LCD
s l MHz 5 4 MHz >4 MHZ
(3.4) (3,4)
CMOS
~~~~
~
~
LSI, Large scale integration; MSI,Medium scale integration; SSI, Small scale integration.
236
A.
P. GNADINGER
present watch and clock circuits are the complementary MOS technologies (CMOS). Of the many subgroups of CMOS technologies, three variations are listed in Table I1 that are most widely used: metal-gate, silicon-gate, and silicon on sapphire (SOS) technology. Before going into a detailed description, some common features of all three variations are described. All three CMOS technologies are well suited for medium- or largescale integration and are therefore applicable for analog and digital watches. They require considerably less power than any of the bipolar technologies. The extraordinary potential of CMOS technology for micropower applications was pointed out as early as 1963 (51). The fundamental difference from bipolar circuits lies in the fact that the current consumption is largely dynamic, being proportional to the number of switching operations, that is, proportional to the operating frequency f. The basic element of any CMOS logic gate is the inverter shown in Fig. 40. Its power consumption consists of three parts: If the inverter rests in one of its logic states, basically no connecting path exists between the positive and negative pole of the power supply and only a very small leakage current ILis flowing through the blocked transistors. If the inverter changes its logic states, the two transistors are both conducting during a short time and a current can flow. This current IT is also dependent on the supply voltage VDDand the shape of the input and output signal. The third contribution to the total current is the current through the output impedance I c . Since MOS circuits have nearly exclusively capacitive loads, it is easy to see that this current I c is proportional to the number of switching operations, that is, the frequency f a n d also to the supply voltage V D D to which the capacitance Cis being charged. This third VDD
P
i FIG.40. CMOS inverter.
ELECTRONIC WATCHES AND CLOCKS
237
component is now by far the dominating contribution provided that the switching time is small compared to the repetition period l/ft so that the power consumption of the inverter in Fig. 40 can be approximated by
P = fCVBI, independently of the transistor device parameters. It has been shown (52) that this remains a good approximation even if the input signal has a finite rise time. The power relation (16) can be extended to a frequency divider stage as described in Section IV,B,l. Assuming a binary divider for simplicity, we obtain a power consumption of wherefis the input frequency and C an appropriately weighted sum of the circuit capacitances. Equation (17) shows that the power consumption of an infinitely long divider chain will at most be twice that of the first stage. This is an important result, indicating that the power consumption of a watch circuit will largely be determined by the first stages of the divider chain (apart from the oscillator). The rest of the circuit that can occupy the major part of an integrated watch circuit in the case of a digital watch will contribute only a small part to the total current consumption since it operates at a very low frequency. This would not be the case in bipolar technologies, where all the gates will contribute about equally to the current consumption. A comparison of the bipolar and CMOS technologies listed in Table I1 is again given in Fig. 41, where the speed power product is plotted as a function of frequency. For the bipolar technologies, the power consumption is nearly constant over a wide range of frequencies. Only at frequencies above a few MHz can the influence of a capacitive current be seen. For the CMOS technologies, the above-mentioned proportional increase of power with frequency can clearly be seen. IzL seems to be more favorable at higher frequencies than CMOS but as mentioned before we have to keep in mind that most of the watch circuit operates at low to very low frequency, where any of the CMOS technologies is far superior. The requirement of low operating voltages (1.1 - 1.5 V) has made the industrial realization of watch and clock circuits in CMOS technology very difficult and delayed the introduction of these technologies considerably. The threshold voltages of the p- as well as the n-channel transistors must obviously be well below the supply voltage. The problems that had to be overcome were therefore the reproducible fabrication of weakly doped ( 1 8 x 1015atoms/cm2) wells in a silicon substrate of opposite conductivity type, the realization of clean gate oxides, the control of the
A.
238
I
10
P. GNADINGER
100
1000
FREQUENCY (kHz1
FIG. 41. Speed-power product as a function of frequency for the five technologies listed in Table 11. '
silicon-oxide interfaces, as well as the stability of the devices. All this had to be realized in a production environment. The first laboratory models of CMOS watch circuits were already made in the early 1960s by the pioneering work of the Centre Electronique Horloget- (6). However, the industrial realization only started some ten years later. It is fair to say that the main obstacle, the realization of controlled weakly doped wells, had to await the introduction of ion implantation techniques (53)into production as replacement for the diffusion technology. This was accomplished in the early 1970s. The three CMOS technologies listed in Table I1 -especially the silicon-gate technology-are now described in somewhat more detail. 1. Silicon-Gate Technology
As mentioned before, the most successful and best suited technology for watch and clock circuit applications is the silicon-gate CMOS technology. Silicon-gate technology was first described by Sarace ct NI. (23) and Faggin and Klein ( 5 4 , 5 5 )for single-channel MOS and combined with the complementary principle in 1971 to form silicon-gate CMOS (56). The first silicon-gate CMOS technology, however, was not suited for watch circuit applications since the threshold voltages were in the range of 1.0-1.5 V, requiring a power supply of more than 3 V.
ELECTRONIC WATCHES AND CLOCKS
239
A silicon-gate CMOS technology suited for watch circuit applications by controlling the threshold voltages to values of approximately 0.5 V was first described in 1972 by Vittoz ef al. (36). In order to describe the silicon-gate CMOS process in detail, a CMOS inverter (Fig. 40)-the basic building block of any watch circuit-is used as an example. Figure 42 (56a) shows a crosssection and Fig. 43 a scanning electron microscope (SEM) photograph of such an inverter. The starting material is a single crystal silicon wafer with (100) orientation and a uniform n-type background doping of about 2 x l O I 5 atoms/cm2. The right-hand part of Fig. 42 and the upper part of Fig. 43c show the p-channel transistor situated in the n-type substrate. It contains the p+-doped source and drain regions forming p+n diodes with a junction depth of typically 1 pm. The gate electrode is made out of polycrystalline silicon, approximately 0.5 pm thick and also doped p+. It is separated from the single-crystal substrate by the gate oxide, which is formed by a thin layer (-700 A) of thermal SiOz. This oxide is an integral part of the active p-channel transistor. By applying a negative voltage of sufficient magnitude (larger than the threshold voltage), the silicon surface between source and drain is inverted and transistor action can take place. The left-hand part of Fig. 42 and the lower part of Fig. 43c show the n-channel transistor situated in a well weakly doped p-type, typically 5-6 pm deep. The source and drain areas are now heavily doped n-type, forming n+pjunctions approximately 1 pm deep. The polycrystalline gate electrode is now n+ doped-the same polarity as source and drain. It is again separated from the channel area by the thin gate oxide. By applying a positive voltage of sufficient magnitude (larger than the threshold voltage), the silicon surface between source and drain is inverted and the
nGHA"EL-TRANSISTOR p-CHANNELTRANSISTOR SOURCE DRAIN
a b c
d
DRAIN
c f
SOURCE
B
FIG.42. Cross section through a CMOS-inverter in silicon-gate technology. a, n substrate; b, p well; c , metal interconnection: d, polysilicon gates; e , field oxide; f, intermediate oxide; g. gate oxide; \ \ \, n-doped; / / /, p-doped silicon. After Liischerera/. (56a) reprinted with permission.
240
A. P. GNADINGER
FIG.43. CMOS inverter in silicon-gate technology. (a) Electrical symbol. (b) Electrical circuit diagram. (c) SEM photomicrograph, 6 0 0 ~After . Luscher er a / . (56a), reprinted with permission.
n-channel transistor turns on. The drains of the p- and n-channel transistors are connected with a short metal line. It is necessary to look briefly at the basic equations governing the operation of the MOS transistor in the view of low-voltage operation. The fundamental parameter of the MOS transistor VTn is dependent on the processing parameters as follows (57):
VTn =
4MS
+
2+Fp
-
+
Qss
QB
C O
where 4MS is the polysilicon -semiconductor work function difference, +m the Fermi potential for holes, Q B the bulk charge density, and Qss the charge density at the SiOz-Si interface. C o is the capacitance of the gate oxide per unit area, given by co =
€,/to
(19)
where e0 is the dielectric constant of the gate oxide and c, the oxide thickness. Both #Fp and QBare dependent on the surface concentration of the p well: $FP
= ( k T / q ) In(NA/ni)
(20)
QB
= -[2Es4NA(2&p)]1’2
(21)
ELECTRONIC WATCHES AND CLOCKS
24 1
where N A is the effective surface concentration of the acceptors in the p well, es the dielectric constant of silicon, and q the electronic charge. For the p-channel transistor, the above equations remain essentially the same with the exception of substituting N Aby N D ,the effective donor concentration at the surface of the n substrate, and appropriate changes in signs. To achieve low threshold voltages, it is advantageous to use as thin a gate oxide as possible. Typical values are around 600 A. Qss-the fixed charge at the Si-SiOz interface-has to be kept as low as possible. With (100) orientation of the substrate, Q s s is normally 5 x 1O1O cmb2 with a variation of about _+2 x 1 O l o cm-2. Adjustment of QB is then made to obtain the desired threshold voltage of typically 0.5 5 0.1 V for both types of transistors. It is advantageous to have a p-doped silicon gate for the p-channel and an n-doped silicon gate for the n-channel device. The doping levels required under these circumstances are N D = 1 x 10*scm-2 for the p-channel and N A 2 x 10l8cm-2 for the n-channel device. Early CMOS technologies had severe problems with control of both Qss and QB. Q s s came under control with improvements in processing techniques, clean oxide growth, and control of annealing. The control over the low surface concentration required in the p wells caused severe problems that could only be overcome with the introduction of ion implantation techniques (53). If the polysilicon gates are to be uniformly doped (e.g., n+ type), a special threshold adjust implantation through the gate oxide is necessary in order to compensate €or the 1 V work function difference between n+ poly and n substrate. Such an additional implantation may, however, be justified due to the increase in packing density, since the metal bridge short-circuiting the n+p+ polysilicon diodes can be deleted. These metal bridges can clearly be seen in Fig. 43c. With the use of fairly low-doped material, parasitic threshold voltages in the field oxide regions may become a problem. However, since modern LCDs require only voltages up to 3 V (see Section IV,D), parasitic threshold voltages of 5 V are, in most cases, sufficient. Channel stopper or guard ring diffusions as they are usually applied in standard CMOS technologies for nonwatch applications are therefore not necessary. The major processing steps necessary to fabricate a CMOS silicongate watch circuit are now briefly described. First, the silicon wafer is oxidized and the information regarding the p wells transferred from a photomask to the wafer by standard photolithographic techniques (58). This first window is etched into the oxide and the whole wafer implanted with boron ions, the remaining oxide acting as a mask and preventing implantation outside the p-well regions. The wafer is 1 15OOC) driving the boron to then subjected to a high-temperature step 2 :
(2
242
A.
P. GNADINGER
the required depth. After that, the field oxide is formed in a second thermal oxidation step and the active regions of the transistors are etched out of this field oxide requiring a second photolithographic step. The photomask containing the pattern information has to be aligned to the p well very accurately. The gate oxide is then grown and the whole wafer covered with a layer of undoped polycrystalline silicon. The most common method for producing this layer is by a low-pressure chemical vapor deposition (CVD) reaction in a hot-wall furnace tube (59) with SiH4 as a source. The next photolithographic step, again requiring exact alignment of the appropriate photomask, defines the polysilicon runners that serve as gates as well as the first level of interconnections. The gate, source, and drain regions are now doped n+ or p+, separately for the n-channel and p-channel devices, requiring in general one or two additional masking steps. The doping sources can be gaseous, doped oxides, or ion implantation. As a next step, the intermediate oxide is deposited onto the wafer-again employing a chemical vapor deposition-and contact hoies etched through this layer where contacts to source, drain, or gates are intended. The whole wafer is then covered with a metal layer, preferably aluminum. This can be accomplished by evaporation, sputtering, or any other convenient means (60). The next masking step defines the metal interconnections. As a protection against mechanical and chemical attack, the wafer is covered with a layer of silicondioxide or silicon nitride in a low-temperature process and via holes etched to the bonding pads. The wafers are finally subjected to an annealing treatment in hydrogen, which serves to alloy the metal-silicon contacts and reduces the Si-Si02 interface states as well as the radiation damage that may have been introduced during metalization. The finished wafers are then tested, separated from each other, and mounted on suitable substrates. The packaging technology for watch circuits are covered in more detail in a subsequent section.
2 . Silicon-on-Sapphire (SOS) Technology An integrated-circuit technology using monocrystalline silicon films on a sapphire substrate was proposed about ten years ago (61). For some circuit applications, this technique has found industrial applications, especially where high speed and low power are important ( 6 2 , 6 3 ) . At first sight, SOS technology seems very attractive for watch circuit applications as well, particularly if it is combined with complementary MOS technology. SOS would offer a strongly reduced power consumption for a given
ELECTRONIC WATCHES AND CLOCKS
243
frequency or allow higher frequency operation of a watch circuit at given current drain level. This is due to the fact that source and drain regions have very little diffused capacitance associated with them. The silicon islands are grown on the sapphire substrate-an insulator-where no depletion regions can form that would contribute to the capacitance. A second advantage of SOS technology is the absence of parasitic transistor effects, since the silicon islands are separated from each other by an insulator. The performance advantages of combining CMOS transistors with SOS technology were first discussed by BoIeky (64).The original process technology used to fabricate CMOS/SOS circuits involved only one epitaxial silicon layer and contained p-channel deep depletion devices (65)as well as aluminum gates. The resulting threshold voltages were extremely low, making the technology well suited for watch circuit applications. However, a fundamental problem associated with this SOS technology, the inability to control the leakage currents of the p-deep-depletion transistors, made this process impractical. An improvement had been made in this respect by the introduction of the double epitaxial aluminum gate process (64), and the double epitaxial, self-aligned polysilicon gate process (66).These processes, however, are not suited for watch circuit applications because they lead to higher threshold voltages. A controllable process with low enough threshold voltages for watch circuit applications can be accomplished by incorporating n layers and p+-doped polysilicon, as demonstrated by Ipri and Sarace (10).It is supenor to the original technology with deep-depletion p-MOS transistors because only a small fraction of the n-silicon films is electrically conducting and the leakage problem is thereby reduced. Figure 44 shows a cross section through such a CMOS/SOS inverter as described by Ipri and Sarace (10). It combines the simplicity of the deep-depletion approach and the shift in n- and p-channel threshold voltages resulting from an all p+silicon-gate structure. The deep-depletion n-channel device is made from n-epitaxial silicon, source-drain areas are n+ and the gate is p+ polysilicon. The p-channel device is a standard enhancement-type PMOS structure. The processing sequence for the structure shown in Fig. 44 can have several forms. In this particular example, the polysilicon layer is uniformly doped p+ from a boron glass source. After patterning of the polysilicon gates and the interconnections, source and drain areas of the p- and n-channel devices are simultaneously doped from phosphorous (n+) and boron (p+)doped oxides. Care has to be taken that the n+ concentration is low enough to not overcompensate the p+ polysilicon on the n-channel device. By proper choice of the thickness and carrier concentration in the
A. P. GNADINGER
244
Silicon Epitaxy
Masking Oxide
I = j (?-<,
Island Definition
n-
Channel
n-
Oxidation
P+ doped oxide
Poly-Silicon
Gate Definition
p <doped / Poly-SI
x
doped Poly-Si
k-,
n i d o p e d oxide
Channel Definition
and Source
-
Drain
Diffusions Ahimi nu m
Contact Windows and Metalization
n
-
CHANNEL
TRANSISTOR
p
-
CHANNEL
T R ANSI STO R
FIG. 44. Process sequence of p+-polysilicon-gate, deep-depletion CMOS/SOS process.
ELECTRONIC WATCHES AND CLOCKS
245
n-epitaxial film, threshold voltages for p- and n-channel transistors of 0.3-0.5 V can be achieved. A typical analog watch circuit based on the above technology promises to have a much lower current consumption at a given oscillator frequency compared to the equivalent circuit in bulk silicon-gate CMOS technology. This theoretical expectance has also been indicated in Table 11. However, as reported by Ipri and Sarace (101, the actually measured currents in both technologies are about equal. This rather disappointing result is caused by the residual leakage current problem that still exists in SOS even in its most improved version. Furthermore, due to the special nature of a watch circuit, where most of the gates operate at a very low frequency, these leakage currents can easily ruin the advantage gained in the high-frequency oscillator and divider part of the circuit. It is therefore questionable whether SOS will gain ground as a manufacturing technology for watch circuits. It seems that bulk silicon-gate technology with its much lower cost and about equal technical performance will dominate for a long time to come. However, if SOS became a successful technology for such high-volume parts as memories (62) and microprocessors (63), the price of the substrates would eventually come down and SOS might become cost competitive with bulk CMOS. However, such a development is not in sight at the time of this writing. For some very special applications, such as very high-frequency (>4 MHz) wristwatch applications, where high manufacturing costs are unimportant, CMOS/SOS might even today be a good choice, because in those cases the better high-frequenc y performance of SOS might outweigh the higher cost and the inferior performance of the low-frequency part of the circuit. 3 . Metal-Gate C M O S Technology
The original approach to the monolithic fabrication of CMOS circuits as pioneered by RCA and others (67-69), has been closely related to the standard p-channel technology that was in use at that time. Its main characteristic was the use of a metal gate, mostly aluminum, where source and drain regions were not self-aligned. This technology was, however, not suited for watch circuit application, because the threshold voltages of the p-channel device could not be made low enough to be compatible with the low-voltage requirements in electronic timepiece applications. The desired p-channel threshold voltage of -0.5 V can, in principle, be obtained by employing a metal gate with a more favorable work function difference such as molybdenum (67) or a gate insulator such as
246
A. P. GNADINGER
Si02-A1203(70), which has a favorable insulator-insulator interface barrier (71). The best approach to lower the p-channel threshold voltage, however, and the only approach that has been successful in production, is by way of ion implantation, where a light dose of boron ions is implanted into the channel region after the gate oxide is grown (72). If this implantation is shallow enough, the effect is very nearly just a threshold voltage shift with no change in the effective substrate doping. The ion implantation is then nearly indistinguishable from a negative oxide charge. The n-channel threshold voltage can be adjusted by the doping level in the p well as described in the previous section covering the silicon-gate CMOS technology. Again, to obtain adequate control over the n-channel threshold voltages, ion implantation technology is the only viable production process (53). A metal-gate CMOS process that is optimized for watch circuit application by use of ion implantation technology to set both p- and n-channel threshold voltages has been described by Coppen et al. (73). Figure 45 shows a cross section through a CMOS inverter made in this aluminum gate technology. The use of ion implantation to lower the p-channel threshold voltages has several advantages. First of all, no changes in the conventional AI-Si02 gate structure are necessary. The p-channel threshold voltage adjust implantation can be done most conveniently before the contact window cut by protecting the n-channel side with a photoresist layer and using the reversal of the p-well mask. The complexity added to the process is therefore quite small. Finally, the threshold voltage shift is continuously variable, being about 0.5 V/10l1ions/cm2 and can easily be adjusted to compensate for changes elsewhere in the process, for example,
111 1
'
ELECTRONIC WATCHES AND CLOCKS
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variations in Q s s . An extensive description of this process is given in Copper et al. (7.3) and is not repeated here. The advantages of the metal-gate CMOS process compared to the silicon-gate and SOS processes described before are its simplicity and its maturity, being just a derivative of the well established general-purpose CMOS process described in White and Cricchi (69), which has been in production for more than 10 years. Its main disadvantage is its higher current consumption due to the larger overlap capacitances of the gate metal over the diffused regions. This last feature goes directly against the general trend toward lower current consumption and increased battery life in an electronic watch. 4. Packaging Technology
After functional testing of the watch circuits in wafer form, the individual circuits are separated from each other by either scribe and brake or by sawing them apart. The individual circuits have then to be put in a suitable package so that they can be mounted in the watch module. Clock circuits are nearly always packaged in a dual in-line package (DIP) with a plastic envelope as shown in Fig. 46a. This form of packaging is standard for many types of integrated circuits. It is inexpensive with proven reliability and can be done in highly mechanized production lines. Its main disadvantage is its large size. For clock applications, this limitation can, however, be tolerated in most cases. For wristwatch circuits, a DIP package would be too bulky. Here, two forms of packaging have become standard in recent years: For analog watch circuits where between 8 and 12 external leads are required, a special miniaturized plastic package has been developed (74) that uses a technology similar to the familiar DIP-package but measures only about 4 x 5 mm for an eight-lead circuit. The watch chip is glued or soldered onto a metallic lead frame, the pads of the circuit are bonded by a thin gold or aluminum wire to the external leads, the circuit is enclosed with a plastic, usually an epoxy resin by an injection molding process, and the outer leads and the individual circuits are separated from each other by a stamping process. An example of a finished SO-8 package is shown in Fig. 46b. For digital watch circuits, where as many as 70 leads have to be bonded, an SO package would become much too bulky and too difficult to manufacture. A new form of packaging digital watch circuits had to be found. The technology that is used most often today is mounting the watch chip directly onto a printed circuit board that serves as a substrate for the whole watch module. The chip is glued onto the printed circuit
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FIG.46. Different forms of packaging watch and clock circuits. (a) Clock circuit packaged in dual in-line package (DIP). (b) Analog wristwatch circuit packaged in miniature package SO-8. (c) Digital wristwatch circuit mounted on printed circuit board.
board, preferably into a recession made by grinding or stamping. The bonding pads of the circuit are connected with the conductor pattern on the PC board by means of ultrasonic bonding employing either a thin gold or aluminum wire. The circuit is then covered by a drop of epoxy resin or by a suitable plastic or metallic cap. This process of mounting the circuit directly onto the printed circuit board is certainly the most popular one today. It is well suited to minimize the required volume. It demands, however, a well-controlled and high-yielding bonding process because of the fairly expensive substrates.
D. Displays An important part of an electronic watch as indicated in Fig. 2-and the most relevant to the user-is the display. It transmits the time information or the additional functions from the watch to the human eye. The analog display with a watch face and moving hands is the one that has dominated for centuries and will certainly coexist with other types of displays in the future. The fastest moving hand-usually the second hand-is driven by a
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stepping motor as already described in Section IV,B,2. The minute and hours hand as well as any additional display functions such as the date, are then coupled in turn to the second hand via appropriate mechanical gear trains. Of course, this type of mechanical display can be manufactured in all sizes and shapes. This might be considered one of the main advantages of the old familiar analog watch display: It allows the analog electronic watch to be designed as a jewel piece-much more so than with digital displays. Digital displays used in electronic watches and clocks use seven independent segments per digit which allows any number from 0 to 9 to be formed. Up to 8 digits are common for a watch display. Additional information such as date and day of the week are formed by alphanumeric electrodes on the display, or by annunciators. The first digit may not need all the segments since it may only have to display the numbers 0, 1 or 2. An example of a seven segment display with six digits is shown in Fig. 47. For digital watches-at the time of this writing-two types of displays are the most common: light-emitting diodes (LED) and liquidcrystal displays (LCD). LED displays were predominant in the electronic wristwatches that reached the market in the early 1970s (75, 76). Their color is a bright red or yellow. They turn on and off in tens to hundreds of nanoseconds and exhibit an excellent discrimination ratio, making them easy to multiplex. The interconnect problem is thus reduced considerably. LEDs have reached a high degree of perfection, mostly due to the development effort invested in electronic calculators and instrumentation display. Their big drawback is the fact that they require a very large driving current of the order of milliamperes. No continuous display is possible under thesecircumstances and the electronic watch will show the time only on pressing a push button. The battery life, therefore, depends on how many times the reading push button is pressed. The driving voltage for LED displays is around 3 V. That means that two batteries have to be provided. Voltage doubler circuits as described in Section IV,B,c (Fig. 31) cannot be used with LED displays because of their high internal impedance. The liquid-crystal display (LCD) is now predominant for digital wristwatches. It consumes very little power-of the order of microwatts-
FIG.47. Seven segment display with six digits for digital wristwatch.
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so that it can be on continuously. Liquid crystal displays are passive displays requiring an external source of illumination, normally the ambient light. To make them also readable in the dark, an internal light source is usually provided, in most cases an incandescent light bulb that is activated on pressing a push button. More recently, light sources that do not drain power from a battery were introduced. These so-called “betalights” are based on the radioactive decay of tritium in combination with a luminescent phosphorus. However, in certain countries the use of this radioactive material is not permitted restricting the application of this power-saving light source considerably. The physics of liquid crystals and liquid-crystal display technology has been covered extensively in the literature (77-79). The LCDs going into electronic watches today are of the twisted, nematic, field effect type. They sandwich the liquid-crystal substance between front and back plates of thin glass, which are sealed together with plastic or glass. On the inner side of the glass plates are transparent conductor patterns, formed by photolithographic techniques into the desired segments and characters to be displayed. The conductor patterns are coated with a special chemical film that aligns the liquid-crystal molecules. Polarizers are laminated to the outside of the glass plates with their polarizing axis perpendicular to each other, so that without the liquid crystal present, light would be blocked. The surface characteristics of the alignment film on the conductor pattern now causes a 90” twist to the liquid crystal molecules and their axes align parallel to the polarizer axes of the front and rear polarizers, respectively, so that the display now passes light. If a voltage is applied to the conductor pattern with respect to the back plane, the liquid-crystal molecules are aligned parallel to themselves making those under the influence of the field perpendicular to the rear polarizer axis. These energized molecules now block the light, causing dark images in the shape of the conductor patterns to appear on a light background. Initially, there were many problems associated with LCDs, such as poor reliability, inadequate contrast ratio and viewing angle, and strong sensitivity to temperature and humidity variations. However, in recent years, considerable improvements have been achieved, so that today’s displays can be considered mature enough for applications in electronic watches. One problem that still awaits a solution is how LCDs can be multiplexed. This question has already been discussed in Section IV,B,b and has been covered in the literature (80-82). Multiplexing would greatly reduce the problems and costs of interconnecting the many pins of the circuit to the conductor pattern on the printed-circuit board. However, it
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seems that LCDs in a multiplexed mode will always have poorer contrast ratios, smaller viewing angles, and an increase of turn-off time resulting in image smear than nonmultiplex displays, so that there will be a trade-off between cost and performance. Also, such nontechnical considerations as user acceptance of watches with more than six digits will influence the effort in developing multiplexed LCDs. As mentioned before, multiplexing is most important for displays with more than about six digits. For clock applications, displays with large-size characters are desired. Here, standard LCDs are not well suited because of increased difficulties .in manufacture of scaled-up versions and the rather low light intensity available from an LCD. A noteworthy recent development seems very promising for digital clocks-the FLAD, a fluorescent-activated display (83).It has the same low power dissipation as an LCD but a light intensity that is much stronger than that of an LED display. Basically, a FLAD is a liquid-crystal display with a thin plexiglass panel doped with organic fluorescent molecules behind the back plate of the LCD. Ambient light entering the plexiglass panel excites the molecules and the resulting fluorescent light is emitted from the segments of which the display digits consist. The LCD in front of the plexiglass panel acts as a valve and passes or blocks the emitted light, depending on whether a voltage is applied to the segments or not. Newer types of digital displays such as electrochromic (84)or electrophoretic (85) displays are still in the research phase and have not found practical applications in electronic clocks or watches yet.
E. Power Supplies Referring to Fig. 2, where the block diagram of an electronic watch or clock is presented, the last principal part -the energy source-is now treated. It is an extremely vital part of any electronic watch or clock and as mentioned before has probably the most potential of improvement. It is, however, only covered very briefly here. Readers who would like to study this specialized field in more detail should consult the vast literature (86 -88). As mentioned briefly in Section IV,B, the majority of all clocks and watches are driven by primary cells. Rechargeable or secondary batteries have only been used in special cases, most often in conjunction with solar cells. The reason is that their life expectancy is not much larger than that of a primary battery at the present time. For clock applications, volume restrictions are not as severe as for wristwatches. Clocks are therefore operated in most cases by standard
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dry batteries. Although the Leclanche cell celebrated its one-hundredth anniversary in 1966, it is still the main commercial battery cell and the least expensive of the battery systems. It is primarily a MnO, cell using a starch separator, NH4CI electrolyte, and a zinc can. Newer developments replace the natural MnO, ore by EMD (electrolytic manganese dioxide). These cells have a discharge capacity of two to five times greater than that of the regular cells (89). The starch paste separator is often replaced by a coated separator liner and the electrolyte can also be based on ZnCI, with some NH4CI. All these cells are, of course, not specifically developed for clock applications. They are produced in large quantities for generalpurpose use, representing a worldwide market of roughly 1.4 billion dollars in 1977. For wristwatches, the size of a battery is of primary importance. For this application, a miniaturized construction concept is essential. Most electronic wristwatches produced at this time use a mercury cell as introduced in the 1940s. Of course, many new sizes have since been developed with the emphasis on as small a volume as possible while providing an acceptable capacity. The mercury cell discharges at a nearly constant voltage of 1.35 V. The reaction proceeds through a soluble intermediate (90): HgO
+ HZO
-+
Hg(OH)*
Hg(OH)z + 2e-+ Hg
+ 2(OH)-
For environmental reasons, it is desirable to replace mercury by a less polluting material. Hence, primary cells where Ag,O is used instead of HgO in the cathode have become popular. The cell design is thereby essentially unchanged (88). This cell discharges at a constant 1.65-1.5 V, depending on current drain. The capacity of silver oxide cells can nearly be doubled by employing divalent silver oxide (Ago). However, Ago is metastable and should, thermodynamically, spontaneously decompose with oxygen evolution. Considerable effort has been expended to render Ago stable for use in sealed miniature cells. This effort was consummated in 1976, when divalent silver cells were introduced. Cells being marketed contain both Ago, and Ago and provide about a 20% increase in capacity over monovalent Ag-Zn cells. The goal to achieve long shelf life for primary cells in wristwatches can only marginally be reached by any system that uses a wet electrolyte. The interest in solid electrolytes for batteries to be used in wristwatches is therefore obvious. The best candidates for a solid electrolyte battery are lithium iodide and pAl,O, cell systems (88). They offer a low shelf discharge rate, wide temperature range of operation, absence of leakage, and
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ease of miniaturization. Their voltage is higher than that of HgO or Ag,O systems, typically from 1.9 to 3.8 V, depending on the particular combinations. These cells-while being considered very attractive for watch applications-are still in a research or development phase and have not found commercial application yet. An attempt to further increase the ampere-hour capacity of miniature primary cells has been made by Cretzmeyer et al. (91) by reviving the zinc-air cell concept (92). The air access is restricted to the minimum needed to provide the necessary average current of the order of a few microamperes, thereby minimizing the exchange of CO, and water vapor. Air access is limited through the use of very small holes. These cells are claimed to have about twice the ampere-hour capacity of corresponding mercuric oxide-zinc cells, while activated (open) shelf life is quoted to be satisfactory for watch applications.
F. Watch and Clock Modules The components of an electronic watch described in the previous sections are assembled into a so-called watch module. This is the functional block of an electronic watch, which can be mounted into a suitable watch case to form the completed watch. All components except the push buttons needed to set the watch are part of this watch module. Module fabrication is in itself a major part of the watch industry. In fact, most of the watch manufacturers carry out only this operation; they buy the necessary components on the open market-the integrated circuits, for example, from a semiconductor company. Many watch manufacturers, particularly the smaller ones, even buy the completed modules from one of the larger manufacturers and restrict themselves to the assembly of modules into watch cases. Despite the fact that module design and fabrication is a major and important part in the reaIization of electronic clocks and watches, it is covered here only very briefly. Module assembly into finished watches and clocks is not treated at all. The reason is that there is a multitude of constructional aspects of module fabrication, so that a thorough treatment would go far beyond the scope of this chapter, which deals more with the scientific side of the subject. Readers who are particularly interested in the back-end operation of electronic clock and watch manufacture will have to consult the appropriate literature. Modules for analog clocks and wristwatches resemble to a large degree those of mechanical watches, because they still contain the hands and the gear trains. The electronic components-the integrated circuit, the quartz resonator, the trimmer capacitor, and any other external
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capacitors and resistors-are usually mounted on a printed-circuit board as a carrier. This subassembly together with the battery is then combined with the mechanical part, consisting of the stepping motor, the gear train, and the analog display, to form the complete module. Watch modules for digital wristwatches no longer resemble mechanical watch modules, since all moving parts have been eliminated. A printed-circuit board serves as the substrate for all watch components. The integrated-circuit chip is directly mounted onto the board and its connections bonded to the conductors of the board as described in Section IV,C,4. The quartz resonator, the external capacitors, and resistors are soldered to the board as well. The solid-state display is positioned on top of the printed-circuit board and the many connections between the integrated circuit and the display are made by means of the so-called Zebras. This ingenious device consists of an array of conducting and nonconducting elements with a pitch that is equal to or less than the pitch of the conductors either on the printed-circuit board or the display. In this way, connections between display and circuit board can be made regardless of the positional tolerances of the conductors. An exploded view of a digital wristwatch module is shown in Fig. 48, where all essentid components can be seen.
FIG. 48. Exploded view of digital watch module. a, Digital watch circuit on printed-circuit board with quartz and trimmer; b, backplane; c , LCD display; d, module case; e, batteries; f, zebras; g, battery contact.
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G . Future Trends 1. Autonomy
The dominating trend in the future development of electronic clocks and watches will certainly be toward increased autonomy, since the major drawback of electronic watches with regard to self-winding mechanical watches is the limited life of the battery. One goal that has been set is 10-year autonomy for an electronic wristwatch. This constitutes a goal that should be realizable within a few years with straightforward improvements in circuit design and technology, battery technology, and displays. The most important contribution toward realizing a 10-year watch will probably come from improvements in battery technology. Here, the main efforts will go toward increasing the stored energy density and improving the battery-sealing techniques. HgO batteries will be replaced fairly quickly by Ago or AgzO batteries and newer systems as described in Section IV,E will most likely be introduced soon. There is also quite a good chance that new electrochemical systems that are not obvious at the present time may be discovered and employed in watch batteries. Along with these improvements in battery technology will go further miniaturization that will allow the watch modules to become smaller and more compact and allow the watch designer more freedom in styling. The autonomy of an electronic watch could also be improved by going to secondary batteries that can be recharged by other energy supplies. Watch systems that use solar energy to recharge an accumulator via solar cells have been proposed and have also been developed into commercially available digital watches with LCD displays (93). However, at present, the life of a rechargeable battery is not much longer than that of a primary battery, so that the problem of frequent battery changes remains. It may even be worse because these special cells may not be available as readily as primary batteries. Furthermore, it is well known that the energy available from ambient light may change notably with seasons, latitudes, and dressing habits so that the real autonomy of these watches is questionable. It is the belief of this author that the goal of an autonomy of at least 10 years will be achieved by improving primary batteries rather than by using other forms of power supplies or alternative energy sources. With a given power supply, the autonomy of a watch can also be improved by reducing the power consumption of the other parts of the watch, notably the integrated circuit and the display. Reducing the power consumption of the integrated circuit can either be accomplished by improved circuit techniques or by using more sophisticated manufacturing technologies. It was demonstrated in Section IV,A that more complex os-
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cillator designs, particularly with amplitude feedback schemes, can reduce oscillator currents considerably. There is also much room for improvement in developing new frequency divider circuits as described in Section IV,B. On the technological side, the bulk CMOS silicon-gate technology (Section IV,C, I ) will dominate for the foreseeable future. It will certainly be improved by decreasing the dimensions of the elements, which helps to reduce current consumption and also manufacturing costs. An analog watch circuit with a total current consumption of 200 nA has been announced recently (94). This indicates the state of the art at the time of this writing. There is good reason to believe that circuits with even lower current drain can be realized without major changes in the present-day circuit techniques and manufacturing technologies. The type of display used will influence the total current consumption of an electronic watch as well. Watches with LED displays have almost completely disappeared from the market, mainly because of their high power consumption. The LCD watch will certainly dominate the field of digital watches for a long time to come. The main reason, again, is the low current consumption of the display, which can even be further improved by reducing the capacitances of the display electrodes and possibly by replacing the power-consuming backlights with tritium-based lights, which require no external power. Improving the autonomy of a watch, which has been described as a dominating trend, applies particularly to electronic wristwatches, where the battery should be miniaturized as much as possible. For clocks where volume restrictions are not as stringent, the trend toward lower power consumption is also present but it is not dominant. 2. High-Frequency Oscillutors
An oscillating frequency of 32 kHz has become a standard for electronic watches. For some time there was a tendency to go to higher frequencies. This would improve the accuracy of a watch and decrease its temperature sensitivity. However, higher frequency means higher current and this would oppose the trend toward longer battery life. Therefore, it is believed that except for some special applications where high accuracy is required or where power consumption is not so important, the higherfrequency watch will not become very popular. 3. Polyfunctional Instrument
An evolution of the electronic watch toward a polyfunctional wrist instrument was predicted a long time ago (95). Alarm and stopwatch func-
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tions are already standard in LCD watches. These extended timing functions are particularly easy to implement in digital watches. In analog watches with mechanical displays, it is somewhat more difficult. An interesting combination is hybrid watches, where the actual time is shown in an analog fashion and date and stopwatch functions are shown on a digital display. An analog watch has also been announced where the hands of the watch are simulated by a segmented LCD display (96).This approach is very promising since it combines the advantages of the fully electronic solid-state watch without moving parts with the attractiveness of the analog representation. Three main categories of functions may be considered besides the above-mentioned timing functions: data processing, telecommunications, and sensing capabilities. The first category is represented by the watch-calculator combination that was introduced some years ago (97).One may question the real interest in such a combination because the miniaturized keyboard is not easy to work with, while it still yields a fairly bulky and power-hungry watch. A more promising avenue seems to be the processing of time-dependent data in conjunction with the watch. In the field of telecommunication, the most interesting product would be a paging watch. But the bad antenna situation, the poorly controlled environment, and the low power available while still needing to go to high frequencies will render this endeavor very difficult. The sensing function of the watch could incorporate ambient parameters such as temperature and pressure, or physiological parameters such as the body temperature, blood pressure, and heartbeat of the wearer. These many additional functions can in principle easily be implemented in an electronic watch circuit by standard digital techniques employing random-logic circuits. However, the design effort for even a small alteration to an existing circuit is considerable. Generally, it is uneconomical t o custom-design an integrated circuit for small production quantities, so that many of these ideas have not been reduced to practice due to the lack of an adequate market. The reaction of some manufacturers to this problem has been the introduction of microprocessor-like circuit architectures in which the detailed features of the watch are defined by the content of a read-only memory (43,98).The development of new watch functions is then reduced to the elaboration of the new memory content that will be incorporated into the read-only memory. Instead of a completely new mask set, only one new metal mask has to be generated. Attempts have been made to employ technologies with nonvolatile memory capabilities such as FAMOS or MNOS (26, 27), so that even the watch seller could electrically program the finished watch according to the wishes of the watch buyer. Microprocessor-like watch circuits are certainly more
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flexible, and development costs and development times are reduced, but such an all-purpose circuit will certainly occupy more silicon area than a carefully designed and laid out standard circuit, so that production costs will be larger. Even more important, this solution will oppose the dominating trend toward minimal power consumption. A microprocessor-like circuit will need some type of running clock. Since watch circuits operate at low to very low frequencies for the most part, this running clock will contribute considerably to the overall power consumption. Probably some compromise between standard digital logic and microprocessor-like architecture will be the most promising approach. Oscillator, divider chain, and most of the controlling and decoding circuitry will be done with standard logic techniques, where some internal multiplexing will help to reduce the active components needed and information concerning the pin configuration and certain circuit option could be contained in a ROM or PROM. A lot of improvements can certainly be expected in the next few years in packaging technology. There is a definite trend to mechanize and further integrate the various elements needed to complete an electronic watch module. It might become feasible to include the quartz and the integrated circuit in the same package (34) while eliminating the trimmer capacitor completely by employing digital tuning techniques (see Section IV,A,2). New forms of highly mechanized assembly of the few remaining parts (battery, display, and circuit) by use of tape automated bonding (99) or similar techniques can further reduce the manufacturing costs. V. CONCLUSION Electronic clocks and watches have experienced a dramatic development over the last years. From mere technical toys, they have evolved into consumer products with a considerable economic impact. A large fraction of all the watches produced worldwide are now electronic. The dominance of the mechanical watcb industry, notably by the Swiss companies, has been broken, and countries with a strong electronic industry, above all Japan and the United States, have taken over the lead. This'success of electronic watches and clocks had been due to the higher accuracy, high reliability, reduced cost, and ease of implementing additional functions. Electronic clocks and watches have by no means reached a final standard. Improvements have still to be made in battery technology, displays, and overall volume reduction of the watch modules. The functions implemented in a LSI watch circuit may appear trivial when compared to those
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required for other systems such as semiconductor memories and microprocessors. Nevertheless, special requirements such as the low operating voltage and the very low power available as well as the need to combine analog and digital functions on the same chip, ask for careful choices of technologies, circuit techniques, and system organization. Large improvements can still be expected from this side toward lower cost, increased performance, and extended functions. Electronic clocks and watches will certainly remain an important product of daily life. Whether the analog watch with a face and moving hands o r the digital watch with an alphanumeric display will dominate the market in the long range is not known. It seems that user acceptance of the digital watch is not as fast and thorough as anticipated a few years ago. It may well be that the analog watch or some hybrid form will be the winner. Problems such as ease of setting the watch, readability of the display in the dark, information transfer from the display to the eye of the wearer, o r simply tradition might have a decisive influence. A watch is also always a piece ofjewelery and not merely a technical object. It is obviously easier to incorporate an analog display with a round face into a jewlery watch than a digital display. In any case, it is probabIy too early to forecast which type of watch will dominate the market some years from now, but one thing is certain: The electronic clocks and watches will not disappear like some other electronic gadget. They will remain as an important item of daily life.
ACKNOWLEDGMENTS I would like to thank Prof. Dr. Vittoz and his colleagues at the Centre Electronique Horloger, Neuchitel, Switzerland, for helpful discussions and for kindly supplying numerous illustrations. I am also indebted to the management of FaselecCorp., Zurich, Switzerland, who generously supported this work.
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(1977). 3. M. P. Forrer, Survey of circuitry of wristwatches. Proc. IEEE 60, 1047-1054 (1972). 4. M.Hetzel, Die Stimmgabel als Frequenznormal und ihre Verwendung in der elektronischen Uhr. Tech. Rundsch. No. 19, 45-47 (1963). 5 . H. Oguey, Montre bracelet a quartz. J . Math. Phys. Appl. 21, 653-659 (1970).
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6 . C. Foujallaz and F. Niklks, Montre Clectronique a quartz Beta 21. Proc. 45th Congr. Swiss Chronometric Soc. (1970). 7 . E . Vittoz, W. Hammer, and H. Oguey, Montre electronique a autoreglage instantant. Proc. Int. Congr. Chronometry, 9th 1974 Paper C-2-9 (1974). 8. N. E. Moyer, Digital watch having dual purpose ring counter. U.S. Patent 3,921,384 (1975). 9. J. P. Jaunin, Eine Hochfrequenz-Armbanduhr, das Kaliber Megaquartz 2400. Neue Ziircher Ztg. No. 165 (Mittagsausgabe) (1974). 10. A. C. Ipri and J. C. Sarace, Low threshold low-power CMOS/SOS for high frequency counter applications. IEEE J . Solid-Stute Circuits sc-11, 329 (1976). 1 1 . H. Oguey, E. Vittoz, and B. Gerber, Diviseurs de frequence ultra-rapides. Proc. Int. Congr. Chronometry, 9th 1974 Paper C-3-1 (1974). 12. J . Bachmann, A silicon-gate CMOS electronic watch system.J. Microelectron. 4,39-44 ( 1973). 13. 2.Rajgl and A. P. Gnadinger, unpublished data. 14. E. Vittoz, LSI in watches. Pup., Eur. Solid Stare Circuits Conf. (ESSCIRC),2nd, 1976, pp. 7-27 (1977). 15. E. Vittoz and J. Fellrath, CMOS analpn integrated circuits based on weak inversion operation. IEEE Truns. Solid-State Circuits SC-12,224-23 1 (1977). 16. R. C. Huener, U. S. Patent 3,855,549 (1973). 17. French Patent 2,259,482 (1975). 18. J. Liischer, Circuit oscillateur a quartz, a resonance parallele, pour appareil de mesure. Swiss Patent Appl. 15 826/68 (I%@ 19. K. K . Clarke and D. T. Hess, “Communication Circuits: Analysis and Design,” pp. 636-641. Adison-Wesley, Reading, Massachusetts, 1971. 20. M. B. Barron, Low level currents in insulated gate field effect transistors. Solid-State Electron. 15, 293-302 (1972). 21. H. Oguey and B. Gerber, U. S. Patent 4,041,522 (1977). M. Dutoit and F. Sollberger, Lateral Polysilicon p-n Diodes. J . Electrochem. Soc. 125, No. 10, 1648-1651 (1978). 22. F. D. King and J. Shewchun, Polycrystalline silicon resistors for integrated circuits. Solid-State Electron. 16, 701 -708 (1973). 23. 3. C. Sarace, R. E. Kerwin, D. L. Klein, and R. Edwards, Metal-nitride-oxide-silicon field-effect transistors with selfaligned gates. Solid-State Electron. 11, 653-660 (1968). 24. J. Liischer, Circuit oscillateur a quartz, Swiss Patent Appl. 15826/68 (1968). 25. E. Vittoz, W. Hammer, M. Kiener, and D. Chauvy, Logical circuit for the wristwatch. Pup., Eurocon Conf.,1971 Paper F2-6 (1971). 26. D. Frohman-Bentchkowsky, Memory behaviour in a floating-gate avalanche-injection MOS (FAMOS) structure. Appl. Phys. Lett. 18, 332 (1971). 27. H. A. R. Wegener, A. J. Lincoln, H. C. Pao, M. R. O’Connell, R. E. Oleksiak, and H. Lawrence. The variable threshold transistor, a new electrically alterable, nondestructive read-only storage device. P a p . , Int. Electron Dev. Meet. (l%7); D. FrohmanBentschkowsky, A fully decoded 2048-bit electrically-programmable MOS-ROM ISSCC Dig. Tech. Pup. p. 80 (1971). 28. E. Vittoz, W. Hammer, and H. Oguey, Montre electronique a auto-reglage instantane. Proc. Int. Congr. Chronometry, 9rh, 1974 Paper C-2-9 (1974). 29. Anonymous, Low-cost watch to keep time with RC oscillator. Electronics 48, No. 14,29 (1975). 30. D. R. Koehler, Genirateur nucleaire d’impulsion. Swiss Patent Appl. 7039/73 (1973). 31. G. Glaser, Die Entwicklung der elektrischen Grossuhr. Uhrentechnik 5 , No. I , 1-16 (1978).
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32. A. E. Zumsteg and P. Suda, Properties of a 4 MHz miniature flat rectangular quartz resonator vibrating in a coupled mode. Proc. Frey. Control Svmp., 1976 (1976). 33. J. Staudte, Further developments of the Statek-Quartz-Systems. Proc. Int. Cungr. Chrunometty, 9th, 1974 Paper C.2.1 (1974). 34. H. Wyss, Massenfertigung miniaturisierter Uhrenquarze mit Hilfe von mikroelektronischen Technologien. Pap., IHMS Symp. 1978 Paper No. 3 (1978). 35. “Genaue Zeit mit Motorola piezoelektrischen Uhrenquarzen.” Motorola Sales Information, Frequency Sensitive Components Products. 1978. 36. E. Vittoz, B. Gerber, and F. Leuenberger, Silicon-gate CMOS frequency divider for the electronic wristwatch. IEEE J. Solid-Stute Circuits sc-7, 100 (1972). 37. D. A. Huffman, The synthesis of sequential switching circuits. 1. Franklin Ins!. 257, 169-190 and 275-303 (1954). 38. E. Vittoz and H . Oguey, Complementary dynamic MOS logic circuits. Electron. Lett. 9, No. 4 (1973). 39. H. Oguey and E. Vittoz, Codymos frequency dividers achieve low power consumption and high frequency. Electron. Lett. 9, No. 7 (1973). 40. P. A. Tucci and L. K. Russel, An I z L watch chip with direct L E D drive. IEEE J . Solid-State Circuits sc-11, No. 6, 847-851 (1976). 41. M. Layer, Schrittmotoren fur Quarzuhren. Uhrentechnik 3 (1976). 42. D. Reguier, Moteur pas a pas pour utilisation en horlogerie techniques. Jahrb. Dtsch. Ges. Chroncimetrie 29, 47-51 (1978). 43. G. M. Walker, Choosing sides in digital watch technology. Electronics 49, No. 12, 91 -99 (1976). 44. P. J. Wild, Elektronische Ansteuerungen von Fliissigkristall-Anzeigen. Proc. I n t . Congr. Chronometry, 9 / h , 1974 Paper C3.6 (1974). 45. Specification of L E D watch circuit NEC 5017 (Nortec) (1974). 46. R. Liischer, private communication. 47. H. W. Rue@ and W. Thommen, Bipolar micropower circuits for crystal-controlled timepieces. IEEE J. Solid-Stute Circuits, sc-7,105-1 11 (1972). 48. K . Hart and A. Slob, Integrated injection logic: A new approach to LSI. IEEE J . Solid-state Circuits sc-7, 346-351 (1972). 49. H . H . Berger and S. K. Wiedmann, Merged transistor logic (MTL)-A low cost bipolar logic concept. IEEE J . Solid-state Circuits sr-7, 340-346 (1972). 50. N. C. de Troye, Integrated injection logic-Present and future. IEEE J . Solid-State Cirw i t h sc-9, 206-211 (1974). 51. F. M. Wanlass and C. T. Sah, Nanowatt logic using field effect metal oxide semiconductor triodes. ISSCC Dig. Tech. Pup. pp. 32-33 (1963). 52. J. R. Bums, Switching response of complementary symmetry MOS transistor logic circuits. RCA Rev. 25, 627-661 (1964). 53. B. Crowder, “Ion Implantation in Semiconductors.” Plenum, New York, 1973. 54. F. Faggin, T. Klein, and L. Vadasz, Insulated gate field effect transistor integrated circuits with silicon gates. Pap., IEEE Int. Electron Device Meet. (1968). 55. F. Faggin and T . Klein, Silicon gate technology. Solid-State Electron. 13, 1125-1 144 (1970). 56. R. R. Burgess and R. G. Daniels, CMOS unites with silicon gate to yield micropower technology. Electronics 44, No. 18, 38-43 (1971). 56a R. Liischer, H. R. Neukomm, and A. P. Gnadinger, Integrierte Schaltkreise fur elektronische Uhren. Schweiz. Tech. Z. No. 13/14, pp. 307-310 (1977). 57. A. S. Grove, ”Physics and Technology of Semiconductor Devices.” Wiley, New York, 1967.
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58. W . S. De Forest, "Photoresist, Materials and Processes." Mc-Graw Hill, New York, 1975. 59. R. S . Rosier, low pressure CVD production processes for poly, nitride and oxide. Solid State Technol. 20, No. 4, 63-70 (1977). 60. J. E. Varga and W. A. Bailey, Evaporation, sputtering and ion plating. Solid Stute Technol. 16, No. 12, 79 (1973). 6 / . E. C. Ross and C. W. Mueller, Extremely low capacitance silicon film MOS transistors. IEEE Truns. Electron Devices ed-13, 379-385 (1966). 62. E. Boleky, High performance low-power CMOS memories using SOS technology. IEEE J . Solid-State Circuits sc-7, NO. 2, 135-145 (1972). 63. G . R. Brigs, S. J. Connor, J. 0. Sinniger, and R. G. Stewart, 40 MHz CMOS-onsapphire microprocessor. IEEE Trans. Electron Devices ed-25, 952-959 (1978). 64. E. J. Boleky, The performance of complementary CMOS transistors on insulating substrates. RCA Rev. 31, 372-395 (1970). 65. F. P. Heiman, Thin film silicon-on-sapphire deep depletion MOS transistors. IEEE Trans. Electron Devices ed-13, 855-862 (1967). 66. W. F. Gehweiller and W. C. Schneider, Characteristics of a SOS/CMOS seven stage binary counter. Pap., IEEE Int. Solid State Circuits Conf., 1972, pp. %-97 (1972). 67. F. Leuenberger and E. Vittoz, Complementary-MOS low-power low-voltage integrated binary counter. Proc. IEEE 57, 1528- 1532 (1969). 68. E. E. Moore, Performance characteristics of RCA's low-voltage COS/MOS devices. 1971 IEEE Int. Conv. Dig., pp. 189-190 (1971). 69. M. H.White and J. R. Cricchi, Complementary MOS transistors. Solid-State Electron. 9, 991-1008 (1966). 70. A. P. Gnadinger and W. Rosenzweig, Polarization and charge motion in metal-Al,03Si02-Si structures. J . Electrochem. SOC. 121, No. 5, 770-705 (1974). 71. H.E. Nigh, Int. Conf. Properties Use MIS Srruct., (1969). 72. K. G . Aubuchon, Int. Cons. Properties Use MIS Srruct., (1969). 73. P. J. Coppen, K. G. Aubuchon, L. 0. Bauer, and N. E. Moyer, A complementary MOS 1.2 volt watch circuit using ion implantation. Solid-state Electron. 15, No. 2, 165-176 (1972). 74. Anonymous, The circuit shrinkers. Philips News Rep. (1976). 75. A. A. Bergh and P. J. Dean, Light emitting diodes. Proc. IEEE 60, 156-223 (1972). 76. M. G. Craford, Recent developments in light emitting-diode technology. IEEE Trans. Electron Devices ed-24, 935-943 (1977). 77. G . H. Heilmeier, Liquid crystal display devices. Sci. Am. 222, 100-106 (1970). 78. M. Schadt and W. Helfrich, Voltage-dependent optical activity of a twisted nematic liquid crystal. Appl. Phys. Lett. 18, 127 (1971). 79. P. J. Wild, Fliissigkristallanzeigen zur digitalen und quasianalogen Zeitdarstellung. Jahrb. Dtsch. Ges. Chronornetrie 29, 191-200 (1978). 80. J. Robert and B. Dargent, Multiplexing techniques for liquid-crystal displays. IEEE Trans. Electron Devices 4-24, 694-697 (1977). 81. P. Smith, Multiplexing liquid crystal displays. Electronics 51, No. 11, 113-121 (1978). 82. J. E. Bigelow, R. A. Kashnow, and C. R. Stein, Contrast optimization in matrixaddressed liquid crystal displays. IEEE Trans. Electron Devices ed-22, 22 (1975). 83. G . Baur and W. Greubel, Fluorescence activated liquid crystal displays. Appl. Phys. Lett. 31, No. I , 4-6 (1977). 84. S. K.Deb, Optical and photoelectric properties and colour centers in thin films of tungstein oxide. Philos. Mag. [ 8 ] 27, 801-822 (1973).
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85. A. L . Dalisa, Electrophoretic display technology. IEEE Truns. Electron Devices ed-24, No. 7, 827-834 (1977). 86. G. W. Heise and N. C. Cahoon, “Primary Batteries,” Vol. 1. Wiley, New York, 1971. 87. N. C. Cahoon and C. W. Heise. “Primary Batteries,” Vol. 2. Wiley, New York, 1976. 88. R. J. Brood, A. Kozawa. and K. V. Kordesch, Primary batteries 1951-1976.5. Elecrrochem. Soc., 125, No. 7, 271C-283C (1978). 89. A. Kozawa, in “Batteries” (K. V. Kordesch. ed.), Chapter 3. Dekker, New York, 1974. 90. P. Ruetschi, in “Power Sources 4 ’ (D. H . Collins, ed.), p. 17. Oriel Press, New Castle upon Tyne, England, 1973. 9/. J. W. Cretzmeyer, H. R. Espig, and S. R. Melrose, Inr. Power Sources Conf.. IOth, ( 1976).
92. E. A. Schumacher, Proc. loth Annu. Battery R & D Conf. pp. 14-19 (1956). 93. Anonymous, Schweizer Uhrentechnik. Neue Ziirrher Zrg. No. 91, p. 63 (1977). 94. S. Morozumi, Reduction of power consumption of IC in wristwatch. Inr. Congr. Chronometry, 1978 Paper 1Od (1978). 95. “L’horlogerie demuin.” Federation Horlogere Suisse, Bienne, 1%7. 96. Anonymous, Electronic watch has LCD hands. Electronics 51, No. 16, 44-46 (1978). 97. “Hewlett Packard offers a smart wrist,” from “News in Industry,” IEEE Spectrum 14, No. 7. 71 (1977). 98. G. Wotruba, Funktionsprinzip des Ein-Chip-Uhrenmikroprozessors von Eurosil. Jahr. Drsrh. Ges. Chronomefrie 29, 227-233 (1978). 99. D. Devitt and J. George, Beam tape plus automated handling cuts IC manufacturing costs. Elecrronics 51, No. 14, 116-1 19 (1978).
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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS. VOL. 51
Charge Transfer and Surface Acoustic-Wave Signal-Processing Techniques ROBERT W. BRODERSEN
AND
RICHARD M. WHITE
Department of Electrical Engineering and Computer Sciences Universitv of California, Berkeley, California
................. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . II. Charge-Coupled Device Principles . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. CCD Performance Limitations . . . . . . A. Transfer Inefficiency at High Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . , B. Transfer Inefficiency at Low Fre C. Thermal Generation . . . . . . . . . . D. Applications of CCD Delay Lines . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . IV. Surface Acoustic-Wave Principles . . . . . . , . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . A. Different Modes of Propagation ...................................... B. Nonlinear Effects C. Reflection ofWaves ................................................ ............... V1. CCD Transversal Filters . . A. CCD Filter Structures .
... ....
.
..........
...,..... . . .... . ..
265 266 270 270 273 276 276 279 283 285 285 286 286 287 288 29 1 292 295 301 306
................ C. Applications of CCD Filt VII. SAW Transversal Filters . . . . . , . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. SAW Oscillators, Resonators, and High-Q Filters . . . . . . . . . . . . . . . . . . . . . . . . . ............ IX. Conclusions . . . . . . . . . . . . . . . . . . . . . . , . References . . . . . . . . . . . . . . . . . . , . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
I. INTRODUCTION The signal processors discussed here-the charge-coupled device (CCD) and the surface acoustic-wave (SAW) device-are based on new electronic design principles. CCDs and SAWS use an analog representation of information that moves along a path on a planar surface of the device: in the CCD, charge packets move in a semiconductor, and in the SAW, elastic-wave energy propagates. Short-term storage of charge is 265
Copyright 0 1980 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-014651-7
266
ROBERT W. BRODERSEN AND RICHARD M. WHITE
possible in the CCD, and the speed of motion of charge through the device can be controlled externally, providing the opportunity to accept information at one rate and output it at another. In both the CCD and the SAW, simple means exist for sensing the moving charge or wave frequently along its path; this permits one to realize with either technology new types of signal processors based on the concept of the “transversal filter,” which is discussed below. CCDs and SAWS are in similar stages of evolution. Their development until recently has been primarily in research laboratories, which were perfecting the device technologies. Both technologies are now sufficiently well understood so that devices are beginning to be designed into commercial and military electronic systems. A few of these new system applications and the advantages obtained from using either a CCD or a SAW are presented. It is expected that an accelerating use of these devices will be seen as system designers become more familiar with these devices and the devices themselves become more readily available (Brodersen and White, 1977). 11. CHARGE-COUPLED DEVICEPRINCIPLES
A CCD is an array of closely spaced capacitors fabricated using the metal-oxide-semiconductor (MOS) technology (Boyle and Smith, 1970). The device operation involves the movement of a charge packet, which is stored on one capacitor, to an adjacent capacitor in the array. In addition to their use for signal processing, CCDs are finding important use as digital memories as well as solid-state imagers (Sequin and Tompsett, 1975; Melen and Buss, 1977). The simplest application of a CCD is an electronically variable analog time delay. In this section a short discussion of the basic operation of a CCD delay line is given together with two applications made possible by this unique CCD capability. A cross section of an idealized CCD structure is shown in Fig. la. The device is typically composed of a 1000 8, thick SiOz insulation layer, which is sandwiched between a metallization layer (often deposited polycrystalline silicon or aluminum) and a semiconducting (silicon) substrate, which for CCDs is generally doped with boron to be p-type (i.e., majority carriers are positively charged holes). A CCD is formed by selective etching of the metal (see Fig. Ib), localized oxidations, and diffusion or ion implantation of elements such as boron and phosphorus into the silicon. The CCD features can be extremely small (on the order of microns) and so photographic techniques are used for definition of the circuit patterns.
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
267
(b)
FIG.1. (a) Cross section of a portion of a CCD. The input voltage source VINsupplies charge through the diode (n+ region), which is isolated by the sampling gate Gs.(b) Top view corresponding to (a).
The actual CCD structures now in use are primarily determined by constraints of the particular type of MOS process available for fabrication. A review of a large number of structures can be found in Sequin and Tompsett (1975). The application of a positive voltage to one of the metal gates will deplete the p-type silicon beneath this gate of its majority carrier holes. However, this voltage attracts the minority carrier electrons, because it produces an energy minimum (potential well) for electrons at the interface between the silicon and the insulator. It is these potential wells that are used for storing and transferring the signal charge. In Fig. 2b the charge distribution in the region under a single gate (Fig. 2a) is shown as well as the corresponding electron potential energy profile (Fig. 2c). The charge layers in Fig. 2b to the right of the positively charged metal gate are due to ionized boron atoms in the depleted silicon (which typically extends 1 - 10 pm into the silicon) and a negative charge sheet of mobile electrons at the Si-Si02 interface (less than several hundred angstroms thick). The corresponding potential energy profile in Fig. 2c shows the minimum of the potential well at the Si-Si02 interface. The potential wells can fill up with electrons much as a container can fill up with fluid. There are two sources of these electrons: a controlled number of electrons (a signal charge packet) can be introduced into a well formed under the first gate C,of Fig. 1 by means of an external input volt-
268
ROBERT W. BRODERSEN AND RICHARD M. WHITE SiOe
(a
P-,type SI
1 “G
(b) Charge densi 1 y Ionized boron a t o m s Signal c h a r g e
(C 1 Electron energy
Energy minimum
FIG. 2. (a) CCD cross section indicating metal-oxide-silicon (MOS) layers. (b) Charge distribution with the MOS layers when under an applied bias V G .(c) Electron energy profile showing energy minimum.
age VIN(r);or electrons can be randomly introduced throughout the CCD by thermal generation from within the silicon. The CCDs are therefore dynamic devices in which the desired signal charge can be stored only for intervals short enough so that the thermal generation is sufficiently small. This time is typically of the order of seconds at room temperature. To transfer the charge from one MOS gate to another, clocking pulses (typically 10- 15 V) are applied to the appropriate metal gates. To transfer the charge from the gate G , to the adjacent gate G p ,a positive voltage is applied to Gz(creating a potential well beneath it) while the well under G, is collapsed by reducing the G , voltage. The electrons that were stored in the GIpotential well will now transfer along the semiconductor insulator interface to the new well under Gz.A similar transfer moves the charge from G2to G3 and then from G 3 to the next G , gate outside the dashed lines in Fig. I . A typical set of clocking waveforms to be applied to G,, Gz,and G 3 is shown in Fig. 3. The three gates enclosed by the dashed line in Fig. 1 are called a “stage”; one complete clock cycle of Tc seconds is required to move the charge packet one stage to the right. When the charge packet transfers into the second stage it is sufficiently isolated by the potentials applied to the gates of the first stage so that another packet can be introduced into
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
269
GI
G2
-
G3
k-T,
(sec)
~
4
A
FIG.3. Clocking waveforms required for charge motion.
the first stage without causing mixing of the two packets. In this way N charge packets, which are analog representations of N time samples of V&) (one sample every Tc seconds), can be stored simultaneously in an N-stage CCD. Since the delay through one stage is T,, N stages yield a delay of NT,. Therefore, to change the time delay it is only necessary to change the period Tc of the external clocking waveforms. It is important to note that even though the input signal is sampled in time (as in the case of digital processing) the amplitude of the signal is retained in analog form. This makes it possible for the CCD to have the stability and convenience of operating under clocked timing while retaining the computational efficiency of an analog approach (an advantage that is demonstrated later). In order to sense the delayed input signal, the charge packet must be converted back to either a voltage or current. A voltage can be generated by transferring the charge packet onto an output capacitance, which has a buffer amplifier attached to it. An important characteristic of CCD circuits is their high density, which stems from the use of the MOS technology as well as their extremely simple repetitive structure. A typical length and width of a single stage of a CCD designed for signal processing is 25 X 250 pm, which means that the area of a 100-stage device is only 0.25 x 0.025 cm. Since a typical MOS integrated circuit measures 0.5 x 0.5 cm, this CCD only requires less than 3% of the total available circuit area. It is therefore desirable in systems applications to integrate other system functions onto the same IC along with the CCD. This capability of system integration is expected to provide much of the impetus for future developments in CCD signal processing.
2 70
ROBERT W. BRODERSEN AND RICHARD M. WHITE
111. CCD PERFORMANCE LIMITATIONS
Quantitative analyses to predict the expected performance of a CCD structure for a signal processing application are in general of limited usefulness for several reasons. An exact analysis would require solutions of extremely complex two- and three-dimensional differential equations that are nonlinear and time varying. To make the solutions of these equations tractable, approximations are used that seriously compromise the absolute accuracy of the results. However, these approximations give insight into the relative importance of various aspects of the CCD design and indicate the direction in which the various design parameters should be varied in order to optimize a given performance characteristic to satisfy a given system requirement. The other major difficulty in predicting CCD performance is the strong dependence of relatively fundamental device parameters on the processing of the device. In many cases the cause and effect relationship between the processing and important device parameters such as electron mobility, surface state density, and minority carrier lifetime are only weakly understood. In fact, even the measurement techniques of many of these parameters yield values that are not appropriate for prediction of CCD performance, because of the novel dynamic aspect of the CCD operation. As the result of these difficulties CCD device development for signal-processing applications has been characterized by a strong interactive effort between the system designer, device physicist, and process engineer, each working with incomplete data and models. The device performance is really only known after the device has been fabricated and tested, and even then variations can occur from process run to run, device to device, and in the case of some parameters (such as thermal leakage) even vary widely within one device. A . Transfer Inejjiciency at High Frequencies
Often the most important device parameter for describing CCD performance in signal-processing applications is the transfer inefficiency, which describes the effectiveness of the transfer of a charge packet from one transfer gate to the next. Usually a linear model is adequate to determine how transfer inefficiency will affect overall system performance. The basic assumptions in this model are that in each transfer some charge is left behind, which is collected into the following packet. Note that the signal charge is not lost but is just redistributed. The change in the amount of charge left behind, AQL, is related to the change in the signal charge
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
27 1
being transferred, AQslc, by
AQL = AQsic (1) where the transfer inefficiency E is the proportionality constant. Only the changes in signal are affected by transfer inefficiency (in this linear model) since for a constant signal the charge left behind from a given packet is replaced by the charge left behind by the preceding packet. A typical plot of transfer inefficiency vs. frequency is shown in Fig. 4. At high frequencies, the transfer inefficiency is limited by the speed of the free charge transfer, while at low frequencies the transfer ineffkiency is due to trapping of signal charge by surface (or bulk impurity) states. The transfer of charge from one gate to the next is controlled by several mechanisms. The initial state of transfer is characterized by extremely high speed motion of the electrons because of the self-induced electric fields set up by the large initial gradient in electron density. This stage continues until the gradient decreases to the point at which electron drift due to electrostatic fringing fields (set up by the potential gradient between the clocking gates) or thermal diffusion becomes dominant. A useful expression that takes into account all three mechanisms simultaneously has been developed by Daimon rt al. (1974) for the amount of signal charge left at time f , Qslc(t), assuming transfer was initiated at t = 0: QsI&) -
Qsrc(0)
1
e w ( - t/Tf) - ex~(-f/Tr)l
+ (T,/To)Cl
(2)
L3C0xW / ~ . ~ P Q S I G ( O )
(3)
where To
=
t
I 1
F r e e chorge transfer limited r e g i o n
* Frequency
FIG.4. Frequency dependence of primary charge loss mechanisms.
272
ROBERT W. BRODERSEN A N D RICHARD M. WHITE
is the time constant of the self-induced transfer,
represents the combined effect of fringing field drift and thermal diffusion, and y is a constant that ranges from 1 to 3, which accounts for the coupling between these two mechanisms. In Eqs. (3) and (4),p i s the free carrier mobility, D the thermal diffusion constant (= pktlq), L the length of gate along the direction of transfer, W the width of the gate, E M r N the minimum value of the fringing electric field, and Cox the oxide capacitance per unit area. The value of E M I N , the fringing electric field, can be found from an approximate expression developed by Carnes and Kosonocky (Carnes ef ul., 1972), 14
where X O xis the thickness of the S O z layer, Vc the clock voltage, and Xd the width of the depleted region at the center of the transferring gate. A set of typical values for all of the above parameters is p
=
500 V-sec/cm2,
L
=
7.5 pm,
W = 100 pm,
Xox
v,
=
1000 8,
= 15
x d =
v
3.1 pm
which yields a value for the minimum fringing field of E M I N = 180 V/cm. In Fig. 5 a plot of Eq. (3) is given for these parameters, which shows that within the first nanosecond, more than 90% of the charge has transferred (primarily due to self-repulsion). It then takes about 15 nsec more to reduce the charge to 0.01% of the initial level. Since several transfers must be made during each clock period (three for the device of Fig. 1) it is apparent that the transfer time will become important at frequencies in the range of tens of megahertz. Experimentally, the break point in the transfer inefficiency curve (Fig. 4) occurs for devices of this type in the range of 5- 10 MHz. One way to extend the frequency of operation is to increase the fringing electric field ( E M I N ) . This can be done by decreasing the gate length L and increasing the clock voltage, but the values given above ( L = 7.5 pm, V , = 15 V) are near the present practical limits of the technology. Another method is to introduce additional impurities into the channel region that have a charge opposite to that of the substrate (Walden et a / . , 1972). The charge and potential energy distribution for a
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
Time
273
lnsecl
FIG.5 . Charge remaining as a function of time under a single CCD storage gate after application of a 5 V difference between adjacent gates.
device with this additional layer is shown in Fig. 6. The potential minimum and therefore the signal charge is now in the bulk instead of at the Si-Si02 interface. These devices have been called bulk- (or buried-) channel CCDs, to distinguish them from the CCD structures of Fig. 2, which are referred to as surface-channel CCDs. Since in the bunedchannel CCDs the signal charge is moved away from the transferring gates (into the silicon), the shielding effects of these gates are reduced and the fringing electric fields are considerably enhanced. This extra electric field is sufficient to increase the upper limit of transfer rate to over 100 MHz without significant degradation of the transfer efficiency. In fact, by optimally designing this device for maximum fringing field (i.e., channel depth equal to half of the gate length), operation up to frequencies on the order of 1 GHz is predicted (Esser, 1974). Of course, generating clocks at a gigahertz rate becomes a very difficult problem. B . Transfer Inefficiency at Low Frequencies
The free-charge transfer mechanisms discussed so far would predict that the transfer inefficiency could be set arbitrarily low by increasing the time required for transfer. Unfortunately, the signal charge can be captured in trapping states, which results in transfer inefficiency that is independent of frequency. These trapping states have the property that when
2 74
ROBERT W. BRODERSEN AND RICHARD M. WHITE N-type Si
P - t y p e SI
I
‘b)
I
P Signal
charge
(C)
Electron energy
Energy minimum
~n SI
FIG.6. (a) CCD cross section showing extra n-type region required for buried-channel operation. (b) Charge distribution in a buried-channel device. (c) Electron energy distribution.
a charge packet is present they fill very rapidly (on the order of nanoseconds), but after the charge packet is transferred some of these states reemit very slowly. These slow emitting states will eventually reemit the charge, however, so that charge is not lost but is merely redistributed into later charge packets. The density and origin of the trapping states are different for surfaceand buried-channel devices. At Si-Si02 interface, where the signal charge packet resides in a surface-channel device, there is effectively a continuum in energy of trapping states, which extend across the forbidden energy gap of the crystalline silicon. These states are generaHy considered the result of the termination of the silicon crystal at this boundary. The density of these states, N s s , can be as low as 1-5 x lon states/cm2 eV, but is extremely process dependent. The density of these states is particularly sensitive to annealing procedures used in the latter stages of processing. If there has been no charge transfer along the device for some period of time, a large number of these surface states will be empty. When the first charge packet following this blank period is transferred along the device, the surface states that have emitted will fill by taking charge from the
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
275
signal packet. This can introduce a severe loss in the charge in the signal packet (as mentioned above this charge is not really lost but reemitted into following charge packets). In order to reduce this rather drastic degradation, a bias or background charge (sometimes called a fat zero) is always transferred down the device, and the charge representing the signal is then an additional charge above this background level (Brodersen et a / ., 1975). Unfortunately, since a signal packet containing background charge and signal occupies slightly more area than the area of the background charge alone, the states at some of the “edges” can trap charge from the signal packet. Since the edges perpendicular to the direction of transfer are exposed to charge during the transfer from one gate to the next, they remain occupied. On the other hand, the edges parallel to the channel can emit and thus they are a primary contributor to transfer inefficiency. The linear model for transfer inefficiency [Eq. ( l ) ] is useful in determining the first-order effects of loss on a signal waveform. If it is assumed that the amount of parallel edge area is proportional to the signal QsIc and that the surface states are uniformly distributed, then the total charge loss is QL
=
AW(AQSIG/AQMAX)~NSS L
(6)
in which q is the charge of a single electron and A W the effective increase in width of the channel for the maximum allowable change in signal charge, AQMAX.Using Eqs. (1) and (6) a value of E for this “edge effect” loss is found:
Note that by increasing the width W of the channel, the transfer efficiency can be reduced. This prediction has been found to be valid experimentally for widths up to 200 pm, but then a further increase appears to have diminishing returns. The dominant cause of transfer efficiency for very wide channel devices at low frequencies is unclear; possibilities are that surface states on the perpendicular edges are becoming important or that other non-surface-state trapping mechanisms become dominant. Typical values for the parameters in Eq. (7) are hW
=
5 pm,
W
=
100pm,
Cox= 3.45 x 104f/cm2
V , = 1OV
N,, = 3 x lo9 states/cm* eV
which yields E = 7 x lo+, a value in reasonable agreement with that measured in present-day surface-channel CCDs.
276
ROBERT W. BRODERSEN AND RICHARD M . WHITE
In buried-channel CCDs, since the charge packet is in the bulk, it never comes in contact with the surface states, and therefore considerably higher transfer efficiencies can be obtained. The limitation in these devices is due to capture by bulk trapping states, which are due to impurities such as gold in the silicon. Since the density of these states is usually much lower than the density of surface states, transfer efficiencies of less are readily achievable. than 1 x C . Thermal Generation
The lowest clock rate of a CCD is limited by thermal generation of carriers that are collected in the storage wells and cannot then be distinguished from signal charge. These thermally generated carriers are generated at recombination centers in the bulk (also caused by impurities such as gold) or at the surface through the surface states near the center of the silicon forbidden energy gap. A current density per unit area J D is used to characterize the total amount of thermally generated charged that can be collected per second. The collection of this charge is cumulative so that the total amount of thermally generated charge in the nth stage of a delay line, Qth(n), is given by Qth(n)
= JdLwn/fc
(8)
where f c is the clock rate. For signal-processing applications, it is often desirable to know the maximum number of stages nmaxbefore the signal packet fills up with thermal charge to some fraction a of the maximum charge packet Qmax. Since Qmax is equal to the clock voltage V c times the capacitance of a single gate CoxWL, using Eq. (8) it can be found that
COX VcfclJD
(9) Since the thermal generation increases with temperature, it is important to determine the maximum temperature at which the device will be required to operate. A good estimate of the effect of temperature is to reduce the value of nmax from Eq. (9) by a factor of 2 for every 8°C increase in temperature. Typical values of J D that are achieved at room temperature are 1-100 x A/cm2. nmax =
D . Applications of CCD Delay Lines A CCD has some unique advantages over other techniques for delaying analog signals. Mechanical devices for time delay employing vibrating springs and metal plates are large, unwieldy, and unreliable. A delay can be implemented using an all-pass filter, but very complex filters
277
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
are required if a long time delay is desired that has a flat frequency response. Digital techniques (A/D conversion, storage in digital memory, arid D/A conversion) are very expensive and complex at this time because of the high accuracy required in the conversion process in order to avoid introducing noise and distortion. A CCD has a relatively flat frequency response as well as the ability to vary the time delay by changing the frequency of the clocks transferring the charge. The frequency response of delay line is not perfectly flat, however, because of the effect of transfer inefficiency, which attenuates highfrequency signals. Assuming the linear model of charge transfer inefficiency given in Eq. ( I ) the transfer function H ( w ) throvgh an N-stage device can be shown to be given (for N >> 1) by VOUT ( w )
H(w) =
v,,o = exp[-jwNT,.]
exp{NpE[exp(-joT,)
-
13) (10)
where the first factor is the ideal linear phase shift due to the time delay and the second term is the phase and magnitude degradation due to the transfer inefficiency ( p is the number of transfers per stage, so that pr is the transfer inefficiency per stage). The magnitude of H ( w ) is J H ( ~= ) Iexp( - 2 ~ p rsin
d) fc
which is plotted in Fig. 7 as a function of N p r (number of stages time the loss per stage). This figure shows that the primary effect of transfer inefficiency is to low-pass filter the delayed signal. Two examples of applications of simple delay lines that are particularly well suited to CCD implementation are transient data recording and the correction of timing errors in tape recorders. Many physical phenomena occur so quickly that it is very difficult to analyze the data from a single event as it happens. It is therefore desirable to store the data and then perform the analysis at a more moderate rate. Unfortunately most conventional storage techniques require the data to be converted first into a digital representation with a high-speed analogto-digital converter, which can become extremely expensive for short sampling intervals. A CCD provides a very low cost and convenient means for sampling and storing a data transient in analog form (Linnenbrink ef a / . , 1975). Since the CCD sample period can be varied, after a given number of samples have been taken at sufficiently short sample intervals (i.e., a high enough clock rate l / T c ) , the clock rate is reduced and the data are read out at a slower rate for analysis. Devices that can store 1000 samples taken at time intervals as short as 50 nsec (20 MHz clock rate) are commercially available, and clock rates as high as 130 MHz have been achieved with bulk-channel techniques (Esser, 1974).
278
ROBERT W. BRODERSEN AND RICHARD M. WHITE
INPUT FREOUENCY RELATIVE TO CLOCK FREWENCY 1,
FIG.7. Frequency dependence of the magnitude of the transfer function.
The capability of a CCD to have a variable clock rate also yields the possibility of having an electronically variable time delay, since as mentioned previously the total delay time through an N-stage delay line is NTc . This ability is very useful in correcting small, rapidly varying timing errors such as arise in video tape recorders due to such problems as an out-of-round capstan, fluctuations in the drive motor speed, and tape stretching. (Long-term timing errors can be corrected by the slower conventional motor speed control techniques.) In Fig. 8 a CCD timing error correction system (Hannan et al., 1965) is shown that uses a steady pilot tone that is recorded on the tape along with the desired signal. During playback the signal and pilot tone are delayed by a CCD delay line. The delayed pilot tone is then compared to a reference tone of the same frequency and timing variations can then be detected as a phase change between the recorded pilot tone and the reference. The phase detector in Fig. 8 will produce a voltage proportional to this phase difference, which
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
279
TIME A X I S CORRECTED SIGNAL
CTD DELAY L I N E AMPLIFIER
VOLTAGE CONTROLLED OSCILLATOR PHASE
I REFERENCE TONE INPUT
FIG.8. Timing error correction system for tape recorders, using the variable-delay capabilities of a CCD.
is used to control the frequency of an oscillator in a standard VCO (voltage-controlled oscillator) circuit. This frequency determines the clock rate (and thus the delay) through the CCD in such a way as to eliminate the timing error that was detected. The advantage of such a system is that it has the potential of integration on a single IC and thus could provide a very low cost technique for improving recorder performance. There are many other applications of analog time delay (Tompsett and Zimany, 1973) such as echo generation for electronic music, ghost cancellation in television, and recursive filtering. Also by the addition of input (or output) structures at each stage of delay, time division multiplexers (or demultiplexers) for radar and communication systems can be implemented (Cheek ef al., 1973). IV. SURFACE ACOUSTIC-WAVE PRINCIPLES In SAWs information is represented by ultrasonic acoustic waves propagating freely along the surface of a planar solid, rather than by charge packets moved along by an external energy source as in the CCD. These waves, whose intensity is greatest at the surface, involve exceedingly minute motions of the particles of the solid on which they propagate: in typical SAWs the particle displacements are at most a few angstroms (Viktorov, 1967; Farnell, 1970; White, 1970; Olner, 1978). The wave velocity depends on the materials comprising the path of propagation and on the structure employed. The surface-wave velocity is independent of frequency for a uniform solid, but propagation on a
280
ROBERT W. BRODERSEN AND RICHARD M. WHITE
layered solid is dispersive (frequency dependent). A typical velocity for surface acoustic waves is 3 x 103 m/sec, which is 100,000 times. lower than the velocity of electromagnetic waves in a vacuum; hence, at a given frequency a SAW wavelength (wavelength = velocity/frequency) is about 100,000 times smaller than an electromagnetic wavelength. (For example, a 300 Mhz signal in a SAW would have a wavelength of only about 10 Vm.) The lowest frequency at which SAWs are useful is limited to about 5 MHz by the dimensions of available crystals and the increase in wavelength as frequency decreases, while wave attenuation and fabrication difficulties set an upper limit of a few gigahertz for room temperature devices. Thus CCDs and SAWs typically operate in complementary frequency ranges, and the data rates of SAW devices may be very high. It is easy to convert electrical signals to surface acoustic waves (and vice versa) by use of transducers based on the piezoelectric effect. If a surface wave propagates on a piezoelectric crystal (such as crystalline quartz) or on a nonpiezoelectric crystal coated with a piezoelectric layer (such as a silicon wafer coated with piezoelectric zinc oxide), a traveling electric field accompanies the propagating accoustic wave. In effect, the displacements of the atoms of the medium produce local electric fields inside and just outside the solid. One can couple external sources or detectors to these electric fields by means of conducting electrodes evaporated onto the surface and shaped by photolithographic techniques like those used in making integrated circuits. Figure 9a shows a SAW device on a piezoelectric insulator. The signal source creates electric fields between the uniformly spaced fingers of the input electrode transducer on the left side in Fig. 9a, producing surface waves that propagate toward the output electrode transducer (on the right), where they produce an output voltage that might be displayed on an oscilloscope. (Waves launched to the left by the input transducer are absorbed by the absorbing material on the left end of the crystal.) Each finger of the input transducer acts as a source of waves; the frequency of the strongest wave excitation by the input transducer will be that for which the spacing between alternate fingers equals one SAW wavelength, because for this frequency the contributions from the individual electrodes will add in phase. Thus, using the earlier example of 300 MHz waves and a phase velocity of 3 x lo3 m/sec, the center-to-center distance L of alternate electrodes should be one wavelength or 10 w. If the transducer on the right in Fig. 9a were identical with that on the left, this device would function as a simple delay line, delaying signals by about 3.3 psec for each centimeter separating the two transducers. The device would operate well only in a band of frequencies near the design center frequency (300 MHz) because at frequencies far removed from that the waves produced by individual electrode fingers would tend to in-
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
absorber
transducer
28 1
iransducer
Plezoelactric c r y s t a l
-30 -~
de-so:
A
-1
I
- 71 ~
9 2 0 MHz 920 IZOMHz/div I
tt
(d)
(bl Broodbond input transducer
Piezoelectric crystal
(el
output transducer
(fl
FIG. 9. Operation of SAW electrode transducers. (a) Schematic of a typical SAW, showing the voltage source connected to the input transducer producing surface waves that propagate to the output transducer, which has nonuniformly spaced fingers. The output voltage might be displayed on an oscilloscope as shown. The strongest wave excitation by the input transducer occurs if the distance L equals one SAW wavelength at the input source frequency. (b) Bandpass filter characteristic (insertion loss vs. frequency) resulting from frequency-dependent excitation and reception by transducers in an actual SAW filter (Hays and Hartmann, 1976). (c) Frequency analyzer application showing the output voltage V plotted against time delay I in a device like that in (a) when low, medium, and high frequancies are input (outputs 1,2, and 3, respectively). (d) Chirp output that would be obtained from the device in (a) if a voltage pulse of very short duration were applied to a broad-band input transducer consisting of a single pair of electrodes. (e) Sketch of a SAW having a broad-band input transducer and an output transducer identical with the device in (a)but reversed right to left. (f) Sketch of the output when the expanded chirp pulse (d) is applied to the input of the device in (e). [The voltage and time scales are the same as in (d).] Note that the pulse has been compressed in time and increased in amplitude, permitting detection and precise timing in a noisy environment.
terfere destructively, causing decreased output. Such frequency-selective transmission has been used to make miniature SAW bandpass filters for operation in television sets and other electronic equipment (Fig. 9b), as discussed further below. Considering again the output transducer as it is actually shown in Fig.
282
ROBERT W. BRODERSEN AND RICHARD M. WHITE
9a, we note that the placement of fingers varies along the transducer, from larger spacings on the left side to smaller ones on the right. This variable spacing causes the left end of that transducer to respond most strongly to lower-frequency waves (larger wavelengths) and the right end to higher-frequency waves. Thus if a brief input signal composed of many frequency components is applied to the input transducer, the lowfrequency components produce a voltage at the output transducer sooner than the high-frequency components do because they arrive earlier at the portion of the output transducer that is responsive to them. Thus one can make a frequency analyzer with this simple structure, in which frequency differences in an impulse excitation are converted to differences in arrival time, which can be measured on an oscilloscope (Fig. 9c). Let us consider two other uses of this device-pulse expansion and pulse compression. If we apply a voltage of very short duration to an input transducer consisting of just one pair of electrode fingers at the left of the device of Fig. 9a, a surface wave having many frequency components is produced, and when this wave passes under the output transducer shown, the output voltage is a waveform with a frequency that varies from low to high (Fig. 9d). Such a waveform is called onomatopeically a chirp. Chirp waveforms have applications in radar, sonar, medical ultrasonics, and other signal-processing fields. In radar, for example, a reflected signal from a very distant target may be lost in noise because of the limited amount of peak power one can generate and transmit from an antenna; in medical ultrasonics, the instantaneous intensity that can be safely transmitted into the body is limited to avoid tissue damage. One can alleviate these problems by amplifying and transmitting a chirp waveform generated as described above (Maines and Paige, 1976), receiving the echo (also a chirp that starts at a low frequency and rises to a high frequency), and applying the received echo to the broadband input transducer of a SAW like that of Fig. 9a but with its output transducer reversed (that is, with its closely spaced electrode fingers nearer to the input transducer than the widely spaced fingers, as shown in Fig. 9e). In this second SAW the low-frequency components are detected after the highfrequency ones, and the long-duration, low-amplitude received signal is compressed into a much shorter pulse of higher amplitude (Fig. 9f).Thus the travel time to a reflecting target could be determined with precision in spite of interfering noise. These examples show how very simple patterns of electrodes on the surfaces of piezoelectric crystals are used to make SAW devices for such varied operations as signal delay, filtering, frequency analysis, and pulse expansion and compression. In addition to the SAW phenomena already mentioned, the following
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
283
can be realized with surface acoustic waves and used in signal processing SAW devices. A . Different Modes of Propagation
The simplest SAWs employ propagation along the surface of a homogeneous solid that is many wavelengths thick and so can be considered as if it were semi-infinite. If instead a layered medium is used, only the fundamental lowest-frequency wave has a particle motion similar to that of the classical Rayleigh wave; in addition, higher-order modes can propagate on a layered structure such as might result from putting piezoelectric films on nonpiezoelectric substrates. (Figure 10 shows the dependence of SAW coupling upon wavelength and thickness of such a piezoelectric layer of zinc oxide on silicon.) A SAW-related mode that is guided along the surface of crystals having certain orientations but whose energy extends much farther below the surface has been called the “surface skimming bulk wave” by Lewis (1977), who has shown that this mode has a higher velocity and much lower sensitivity than SAWs to contamination on the surface. The surface-skimming mode can be launched and received with conventional interdigital transducers, and a wide range of devices can therefore be made by employing it. Surface waves of several types ( Z X ) Z n O ON ( I I W C U T
( i i z ) -PROP SILICON
(Av/v AT CONSTANT u )
2rh/ A
FIG. 10. Coupling factor A u / u vs. piezoelectric film thickness for zx-oriented piezoelectric ZnO on semi-insulating I I I-cut silicon (after Kino and Wagers, 1973, with permission). The propagation direction is TT2. Different locations of IDT electrodes are shown, together with use of a uniform conducting field plate opposite the IDT electrodes in two of the configurations. Film thickness is h and wavelength is A.
.
284
ROBERT W. BRODERSEN AND RICHARD M. WHITE
can also propagate along the edge of a solid, examples being the flexural wave, which can propagate along a narrow ridge at a velocity much lower than that of a conventional surface wave in a given material (Lagasse et a / . , 1973), and the so-called line wave, which is guided along the edge of a solid where a side meets the top surface at a right angle (Datta et ul., 1978). In composite media the wave velocity is determined by the properties of the media in which the wave energy actually propagates, so that in a simple single-layer structure the wave velocity is dispersive, as the wave energy travels mostly in the layer at high frequencies and penetrates far I
1
0.07000 0.04000
I
I
LiNb03 41 I“
x 2
5 0.01000 i 0.00700
w c
w 5 n
2 L9
z a’ 3
I
0.00200
000100 0.00070
W
0.00040
w
0.00020
0
u
d
5
\
LiTa03
0.00400
0
8 3
MDC,
0.00010 0.00007 0 00004
0.00002
iF
Yt
O U A R T Z ST-
TEMPERATURE COEFFICIENT OF DELAY
1
dT
FIG. 11. Strength of coupling factor vs. temperature coefficient of delay for various SAW crystals and orientations (O’Connell and Cam, 1978, with permission). The letter designations such as YZ associated with the crystals LiNbOs and LiTaOSdenote propagation in the Y direction along the surface of a crystal whose normal is in the Z crystallographic direction. The ST cut of crystalline quartz is a particularly often-used SAW cut because of its zero first-order coefficient of delay at room temperature; propagation is along the X axis and the surface normal is in the YZ plane at 132.75” to the Z axis.
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
285
into the substrate at low frequencies. The temperature dependence of wave velocity can be usefully adjusted by proper selection of materials and dimensions. Figure 1 1 shows the temperature coefficients of delay of a number of homogeneous piezoelectric solids. One notes that, unfortunately, the strongly piezoelectric solids are usually afflicted with nonzero temperature coefficients of delay. Some combinations of layer and substrate materials and dimensions have been found that have good piezoelectric coupling and yet have first-order temperature coefficients that are near zero. An example (Nakagawa, 1977) is an 0.27-wavelength-thick layer of ZnO sputtered (Shiosaki, 1978) onto Pyrex; such a layered medium is dispersive and so would likely be employed in a narrow-band device.
B . Nonlineur Eflects In any solid, SAW propagation for waves producing high strain amplitudes becomes nonlinear and gives rise to the familiar nonlinear phenomena of production of harmonics of a single wave and mixing of two or more waves propagating together. In addition to this so-called elastic nonlinearity, coupling surface waves to semiconducting media or, by the use of transducers, to semiconducting nonlinear devices such as diodes, makes available for use much stronger nonlinear effects. These effects have been employed in SAW convolvers and correlators (Kino, 1976).
C . Rejection of Waves The familiar principle of obtaining nearly complete reflection of a wave by combining suitably phased small reflections from a number of individual weakly reflecting members has been applied in SAW technology to produce excellent SAW oscillators and narrow-band filters, which are discussed further below. These reflector structures consist typically of planar arrays of parallel grooves cut by ion milling into the surface or, with piezoelectric solids, parallel conducting lines on the surface, which cause local changes of wave velocity primarily because they short out some of the piezoelectric fields. Adjacent grooves or stripes have center-to-center spacings of one-half wavelength and so their individual reflections add in phase. In piezoelectric media a versatile structure (Marshall et ul., 1973)called a multistrip coupler (MSC), consisting of 100 or so parallel conducting stripes, can be used to reflect waves (if the pattern of stripes is suitably bent) and to change the track in which a SAW propagates (if the array of stripes is only partially illuminated).
286
ROBERT W. BRODERSEN AND RICHARD M . WHITE
D. Storage A propagating SAW signal can be stored for later use in rather complex structures involving both semiconducting and piezoelectric substances. Charges moved by the electric fields of the wave in the piezoelectric either fill electronic traps or charge small capacitors on a semiconductor near the piezoelectric. The combination of storage with nonlinear response permits one to store one input signal and then to correlate other input signals against it (Ingebrigtsen, 1976; Tuan et ul., 1977).
V. TRANSVERSAL FILTERING Filters are circuits that are generally used to pass or reject different frequency components, which is probably the most common function performed in electronic systems. It is difficult to conceive of any moderately complex electronic system that does not employ some type of filter (Johnson, 1976). One approach to filtering is to implement a transversalfilter, which is shown in block diagram form in Fig. 12. The output of a transversal filter is a weighted sum of delayed replicas of the input signal. Almost any desired frequency response can be obtained (Rabiner and Gold, 1975) by properly choosing the weighting coefficients W, (within constraints determined by the allowable amount of delay). Surface acoustic waves provided the first practical and convenient way to make transversal filters, permitting one to make highperformance, compact devices operating at hundreds of megahertz. CCDs have more recently been used to perform transversal filtering, but at lower frequencies (100 Hz- 10 MHz). The conventional approach to filtering (LC and active filters) in this frequency range requires the use of circuit components such as inductors and precision resistors, which cannot be integrated onto a single integrated circuit. This can severely limit the complexity of a system that can be integrated since a relatively large amount of circuit area is required to interface the integrated circuit Input
'$$--$q. Delay line
We 19 ted taps
S u m m i n g network
output
FIG. 12. Schematic of a transversal filter, which is the basis for the design of many CCD and SAW signal-processing devices.
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
287
with the external components. In addition, the use of external components can significantly increase the cost and degrade the reliability in comparison to a fully integrated system.
VI. CCD TRANSVERSAL FILTERS In the last few years researchers have developed new techniques to implement CCD transversal filters, which are optimized for differing performance requirements. In order to implement the N-stage filter shown in Fig. 12, it is necessary to provide some means for delaying and storing the last N values of the input signal, multiplying these delayed values by the weighting coefficients Wm and summing these products to obtain the output. The expression that describes these operations is
where VouT(n) and VIN(n - rn) are the nth and (n - m)th time sample of the output and input signals, respectively. By taking the discrete Fourier transform of this expression the transfer function of the filter can be found, N-
1
W, exp(-jwmT,)
Hfw)= m=o
where Tc is the sampling period. As can be seen from this equation, the frequency response has a periodicity equal to the sample rate, which requires that the input signal be bandlimited (antialiased) to avoid spurious responses. The design problem of determining an optimum set of coefficients that provides the closest approximation to a desired frequency response has basically been solved. There are several computer programs available that can very efficiently perform this task (McClellan et a / . , 1973). As an example of the tradeoffs involved in the design of transversal filters, the design of a simple low-pass filter is discussed. As shown in Fig. 13, if a1 and Sz represent the desired magnitude for the passband ripple and stopband rejection and Afis the frequency width of the transition between the passband and stopband regions, then N, the minimum number of weighting coefficients to achieve this response, is given by (Baertsch et a / . , 1976)
288
ROBERT W. BRODERSEN AND RICHARD M. WHITE
Frequency
FIG.13. Low-pass filter transfer function obtained from a transversal filter showing passband ripple 6,, stop-band rejection 6*, and transition band 4f.
Therefore, if a very sharp transition band is required (relative to the clock rate), the number of weighting coefficients can be very large. Fortunately, large numbers of tap weights are readily achievable because of the density of the CCD structure. It is apparent from the above discussion that in order to achieve the sharpest possible transition band it is desirable to have the band edge as near as possible to the sampling frequency. This, however, increases the sharpness of the band-limiting prefilter that is required to avoid aliasing of the input signal when it is sampled at the input to the CCD. A design tradeoff must therefore be made between the length of the CCD filter and the complexity of the antialiasing prefilter. A. CCD Filter Structures
There has been extensive work on various techniques for implementing a transversal filter using charge transfer devices. In many respects the highest performance has been obtained from the split-electrode approach shown in Fig. 14 (Sangster, 1970; Buss et al., 1973). The basic structure is a CCD delay line that has one of the electrodes in each stage split into two parts. In Fig. 14a the signal charges are sensed on the S3sense electrodes, which are unclocked. As charge transfers from the G2electrode to the sensing electrodes (S; and S;), the current that flows into the sensing line consists of a part that would flow if no signal charge were present plus
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
o
I
w,.,=o w,
l W,+,>O
289
G, Driver
o G , Driver
(b)
FIG. 14. (a) Portion of a split-electrode CCD transversal filter, showing the split in one electrode (Cs), which determines the weighting coefficient. (b) Top view of a split-electrode filter with op amp output circuit.
a part approximately equal to the signal charge. Therefore, the signal charge can be determined by integrating the current flowing to the S3electrodes during charge transfer. Weighting is performed by splitting the S3 electrode and integrating separately the current flowing to each portion. A weighting coefficient of zero corresponds to a split in the center of the electrode, and positive and negative Wm’sare achieved by appropriately proportioning the charge between S J and S ; . The summation is achieved by connecting together the .S$ electrodes and the S ; electrodes as shown in Fig. 14b, and the filter output is obtained by integrating and differencing the S$ and S; sense line currents in the output amplifier as shown.
290
ROBERT W. BRODERSEN AND RICHARD M. WHITE
While the circuit of Fig. 14 is adequate for many applications, there are situations in which other circuit implementations can yield improved performance. In particular, the split-electrode approach has limitations in dynamic range (signal-to-noise ratio) for filter responses that require a large number of small weighting coefficients, as well as being limited to a sampling rate below 5 MHz. The dynamic range limitation arises because a small weighting coefficient is obtained by differencing the two relatively large signals from each side of the split electrode to obtain a small difference signal. This results in common mode signals that can be 20-30 dB larger than the desired differential output signal, which results in a similar reduction in signal-tonoise ratio. Filter responses that have small tap weights are, in general, those which have a large ratio of the bandwidth to the sampling rate (i.e,, greater than 10%). On the other hand, in matched filters and filters with a narrow passband, the weighting coefficients are typically much larger and the common mode problem is considerably reduced. A technique that avoids the common-mode problem makes use of a second split in the S3sense electrode as shown in Fig. 15 (the G, and G2 electrodes were not drawn for clarity). The value of a weighting coefficient is realized directly as the size of a portion of the sense gate that extends over the channel (except for a small segment which is always left to ensure that any edge effects due to fringing electric fields at the sides of the channels will be canceled out). Depending on whether the sign of the weighting coefficient is positive or negative, either S; or S, is extended to be larger than the minimum segment size. The electrode in the center of each sense gate, S,”, is a buffer that keeps the total capacitance of each sense gate constant. Since the small weighting coefficients are realized without differencing large signals, the common-mode signal is dramati-
-
-
I
routput Circuit
FIG.IS. Top view of double split-electrode CCD transversal filter.
29 1
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES Weighting
02
0,
T
output Diode
OUT
FIG.16. Weighted input or "pipe organ" CCD transversal filter.
cally reduced. Also shown in this figure is a large positive top weight that i s realized using the single split technique since it is more efficient in silicon area (since there is only one split). Using this approach the dynamic range for a lowpass filter on the order of 90 dB has been achieved (Foxall et d.,1977). In the filters described so far, the maximum frequency of operation is limited by the operational amplifier required at the output to sense the displacement charge. Also, since the signal is only available as a voltage, this can be inconvenient for interfacing to other charge transfer circuitry, where a charge packet would be more convenient. An approach to high-speed filtering that can have an output in charge is shown in Fig. 16 (Knauer er d.,1977). In this technique there are N parallel delay lines each of whose input capacitances are proportional to the magnitude of the desired tap weight. The output charges of the N delay lines are summed together and either converted to a voltage or then processed further as charge (for example, by inputting into a split-electrode transversal filter). Since a simple high-speed output circuit can be used, such as used in high-speed delay lines, the maximum frequency of operation is limited only by the speed of charge transfer, which can be optimized by the use of bulk-channel devices.
B . Limitcrtions of CCD Filters An important limitation of the accuracy of the frequency response of CCD transversal filter is due to errors in the values of the weighting coefficients. The major contributors to this inacurracy are transfer inefficiency
292
ROBERT W. BRODERSEN AND RICHARD M. WHITE
and limitations of the fabrication process in defining the sense electrode areas (e.g., the position of the splits in the split electrode filters). If the linear model for transfer inefficiency of Eq. (1) is assumed, then the effective tap weights W:,including the effect of transfer can be related to the design weights W , by
where pcis the loss per stage (Buss et a / . , 1975). Using this expression the transfer function of a filter can be calculated, which includes the effect of transfer inefficiency. It is apparent that the effect of transfer efficiency depends on the particular filter response under consideration. However, in general, its effect is to result in lowpass filtering of the signal just as in the case of a simple delay line. The process limitation of tap weight accuracy arises from limitations in the photolithographic techniques used to define capacitance areas. In general, these errors can be reduced by making wide CCD channels at the expense of larger size and higher clock line capacitance. Filters made to date with 0.005-in.-wide channels have weighting coefficient errors on the order of 0.5% (Brodersen et al., 1976), and filters with 0.030-in.-wide channels have errors between 0.1 and 0.2% (Baertsch et ul., 1976). The effect of these types of errors on the frequency response of filters can be determined by defining a fixed coefficient error Am, where
Wk
=
Wm + A m
(16)
in which W , is the desired weighting coefficient and W z the actual value including process limitations. From Eqs. (16) and (12) we obtain
which can be interpreted as two parallel transversal filters. One has the ideal response while the other has the values of the errors Am as weighting coefficients. Since these errors are in general random, the frequency response of the error filters is relatively flat with an amplitude proportional to an rms average of the coefficient errors. Thus these random errors result in a fraction of the input signal in the output that is unfiltered. This signal sets a lower limit on the stop band rejection of the filter as well as adds ripple to the passband regions.
C. Applications of CCD Filters There are many examples of the use of CCD transversal filters. A frequency response of an 800-stage bandpass filter is shown in Fig. 17. The
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
293
FIG. 17. Frequency response of a self-contained 800 stage CCD transversal filter. (After Hewes, 1975.)
filter was designed with optimum weighting to have uniform sidelobes at -40 dB. The highest measured sidelobe is degraded to -37 dB by weighting coefficient error discussed above. It has a 0.7% fractional bandwidth at - 3 d B and a 1.7% fractional bandwidth at - 4 0 d B (Hewes, 1975). Spectral or Fourier analysis (the decomposition of a time waveform into its frequency components) could be performed by a bank of CCD bandpass transversal filters of the type shown in Fig. 17 with different center frequencies. This is very wasteful, however, compared with an approach that makes use of the chirp z-transform algorithm (Rabiner e f a / . , 1969). Using this algorithm, an N-point Fourier transform could be implemented on a few integrated circuits using only four N-stage CCD transversal filters (Brodersen et a / . , 1976). The chirp z-transform (CZT) is an algorithm for performing the discrete Fourier transform (DFT) in which the bulk of the computation is performed in transversal filter, and for this reason it is particularly attractive for CCD implementation. Beginning with the definition of the DFT,
Fk =
2
N- 1 fne-i2nnk/N
n=o
and using the substitution
2nk = n 2 + k , the following equation results:
-
(n - k 2 )
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ROBERT W. BRODERSEN AND RICHARD M. WHITE
This equation has been factored to emphasize the three operations that make up the CZT algorithm: premultiplication by a chirp, filtering in a chirp transversal filter, and postmultiplication by a chirp. When only the power density spectrum is required, the postmultiplication by exp(-ink2/N) can be eliminated. A block diagram of the circuit implementation is given in Fig. 18. The first 100 coefficients from the output of a 500-point CCD Fourier spectrum analyzer are shown in Fig. 19b. The input signal was a 500 .Hz square wave (Fig. 19a). The harmonics of the square wave spectrum are clearly evident at 1.5 kHz, etc. A CCD spectrum analyzer achieves the advantages expected from the use of the parallel analog processing of the CCD transversal filters. It can calculate a spectrum many times faster than the fastest, special-purpose digital computer. In addition, because it can be integrated onto a few integrated circuits, substantial savings can also be made in cost, size, weight, and power. Spectral analysis via the CCD chirp z-transform is expected to be very important in image processing, speech recognition, radar Doppler processing, sonar spectral analysis, video bandwidth compression, and other applications requiring low-cost spectral analysis. It is often desirable to be able to change the frequency characteristics of a filter after fabrication for adaptive filter applications as well as to make it possible for a single filter to be applicable to a wide variety of users. Many attempts have been made toward this goal, but unfortunately the performance of these programmable filters has in general not been adequate to produce useful devices. The primary difficulty with the programmable filters has been the difficulty of providing accurate analog mul-
l*
FIG. 18. Block diagram of a CCD implementation of the chirp z-transform algorithm for calculating Fourier transforms.
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
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FIG.19. CCD spectrum analysis. (a) 500 Hz square wave input and (b) CCD spectrum analyzer output showing the magnitude of the lowest-frequency spectral components. The magnitude of the nth harmonic is proportional to I / n , as expected.
tiplications. Recently a filter that uses a single transistor for this function has been described by Mavor and Denyer (1978). By clever use of feedback techniques they were able to obtain weighting coefficient errors of less than a few percent. Other approaches actively being pursued are the use of MOS-compatible multiplying digital-to-analog converters, which, when used in conjunction with the pipe organ architecture shown in Fig. 19, have the potential of tap weight accuracies of a fraction of a percent.
VII. SAW TRANSVERSAL FILTERS Surface-wave devices are useful in part because the SAW electrode transducers themselves are transversal filters: delay results from propagation of the wave along the substrate under the electrodes, which tap off a
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ROBERT W. BRODERSEN AND RICHARD M. WHITE Wideband transducer Amplitude weighted transducer
S u bstra le (0)
/
W i d e b a n d transducer ,Phase - coded transducer Terminal oTerminal ubstrale -Terminal (b)
FIG.20. Tap weighting and phasing of an SAW transversal filter. (a) Schematic illustration of weighting the transducer elements by varying the amount of finger overlap (in this case the middle frequencies are emphasized relative to the low and high frequencies). (b) In the transducer on the right all fingers overlap the same amount but the phasing of pairs of fingers varies because of changes in bus bar connections. A binary sequence represented by transducer phasing is indicated below the output transducer.
portion of the wave energy, and summing of currents from individual taps occurs in the broad bus bars connecting the fingers. Tap weighting and phasing is easily accomplished by (1) tailoring the amount of overlap of adjacent fingers (the output from each pair of fingers is roughly proportional to the amount of overlap) and (2) connecting corresponding fingers of each pair to one bus bar or the other for proper phasing (Fig. 20). The SAWS* used in greatest number to date (Williamson, 1977) have been bandpass filters based on the transversal filter concept (Slobodnik, 1975; Hays and Hartmann, 1977; Matthews, 1977). Illustrative are SAW filters used in connection with television receivers, both in the intermediate-frequency sections of commercial receivers (DeVries and Adler, 1976; Shiosaki and Kawabata, 1977) and in some electronic television games, where they confine the output transmission to the selected channel (channel 3 or 4). Figure 21 shows the response of a SAW filter for satellite applications where narrow bandwidth and very low levels near the pass band are required. The recent development of SAW filters having unidirectional low-loss transducers (Fig. 22) permits employment of SAW filters having insertion losses below 3 dB in the front-end R F sections of * Surface acoustic wave devices and applications are reported primarily in the yearly IEEE Ultrasonics Symposium and its Proceedings, and in the lEEE Transactions on Sonics and Ultrasonics. A special issue of the IEEE Proceedings devoted to acoustics appeared in May 1976.
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dB
-60 -80
0
200 400 600 0 0 0
Frequency
( MHz
1000
1
FIG.21. Frequency response of narrow-band 287 MHz quartz SAW filter for use in space communications equipment. Note that the out-of-band rejection is greater than 70 dB from dc to 1.0 GHz (from Hays and Hartmann, 1976.)
receivers where they can preselect signals and so reduce the chance of receiver saturation being caused by large amplitude signals at neighboring frequencies. Generally SAW filters are smaller than conventional filters made with inductors and capacitors, they can be reproduced precisely once the photolithographic master has been designed and made, and they
FIG. 22. Unidirectional SAW transducer. Employing three-phase driving voltages, in this transducer the surface waves produced add in phase in one direction and cancel in the other so that the bidirectional loss of the conventional interdigital transducer is avoided. With this type of transducer a delay line having a 5% fractional bandwidth centered near 34.5 MHz was made having only 0.65 dB insertion loss (after Hays and Hartmann, 1976).
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ROBERT W. BRODERSEN AND RICHARD M. WHITE
do not require adjustment after manufacture. Center frequencies of SAW bandpass filters have ranged from around 10 MHz through 1.5 GHz, with most falling in the range of 40-300 MHz. Typical bandwidths are from 1 to 30 MHz; very small bandwidth-selective filters can be made using resonators described below, and very large absolute bandwidths have been achieved by going to high center frequencies. Table I summarizes SAW bandpass filter performance. SAW transversal filter designs are based almost entirely upon numerical analyses: the details of SAW propagation in various crystals are calculated or obtained from tabulations of computed values (Slobodnik and Conway, 1970; Auld, 1973); desired tap weightings may be obtained with routines such as the Parks-McClellan program for finite impulse response (FIR) filters (Rabiner and Gold, 1975), which are also used for CCD filter design; and realization of the tap weighting in the form of an electrode pattern may be carried out with the use of programs that model first-order SAW effects as well as second-order effects relating to reflections at the edges of transducer electrodes and diffraction. A variety of SAW filter designs is available in the literature (Slobodnik, 1975). A different example of a transversal SAW filter is the SAW inverse filter (Kerber et al., 1976), whose frequency response is approximately reciprocal to that of the sensor whose output is filtered. For example, partial correction for the limited frequency response of the source and receiver in a nondestructive testing or medical ultrasonic system can be achieved with such a real-time filter. The matched filter is another useful device that can be made with the SAW (or CCD) technology. The impulse response h ( t ) of a matched filter is the time reverse of the signal s ( t ) to which the filter is matched; thus h ( t ) = s( - t ) . The matched filter has a much larger output for the desired signal s ( r ) than it does for other input signals because the outputs from all the electrode fingers (taps) add in TABLE 1 SUMMARY OF SAW BANDPASS FILTERCAPABILITIES ~
~~
~~~
~~
Filters employing conventional IDTs Center frequency (MHz) Typical insertion loss (dB) Bandwidth (MHz) Maximum fractional bandwidth (%) Minimum stopband rejection (dB)
10-1500 12-20 0.1-500 40 45 -60
~
Filters employing IDTs having very low loss 30-400 2-3 0.2
20 40-50
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
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phase when the desired signal is midway in its passage through the filter. An example of such a matched filter was shown in Fig. 9e; the chirp output from the first SAW device was compressed by the second device because of the reversal of the transducer on the right. Matched filters can also be used in digital transmission systems for source-receiver synchronization, and in secure communications systems. The actual form of a dispersive SAW pulse compression filter is usually as sketched in Fig. 23, where the reflective array compressor (RAC) employing ion-etched grooves in a crystal is illustrated. Highfrequency signals reflect at the shallow closely spaced grooves nearest the input and output transducer and so arrive at the output transducer before the lower-frequency components, which reflect further along the device. The metal film on the piezoelectric lithium niobate substrate permits one to make fine adjustments in the delay time for different frequency components. With such RAC devices one can achieve large time-bandwidth products and so large processing gains. Dispersive SAW filters with large time-bandwidth products exceeding 10,000 have been made (Gerard and Otto, 1977). The RAC devices are being used in radar systems and they also can be used in a SAW implementation of the chirp transform algorithm as illustrated in the SAW chirp transform device sketched in Fig. 24. The SAW element in the box is the linear frequency modulation compression filter,
f SHALL0 W ION -MILLED GROOVES ( R E F L E C T O R S ) H A V I N G SLOWLY VARYING SPACING
EVAPORATED METAL F I L M FOR D E L A Y CORRECTION
FIG. 23. Basic RAC filter configuration used for high accuracy, large timebandwidth-dispersive filters. Depths of grooves are varied to equalize amplitude response, while metal film corrects fine delay to within a few parts per million.
3 00
ROBERT W. BRODERSEN AND RICHARD M. WHITE MIXER
L I N E A R FM “C o MP R E s s I o N FILTER
MIXER ”
OUTPUT SIGNAL (FOURIER TRANSFORM OF INPUT S I G N A L )
FIG.24. Basic chirp transform cell in which the output signal is the Fourier transform Basic transform cell in the output signal is the transform FIG. Basic chirp chirpbandwidth transformand cell length in which which the input outputsignal signalare is limited the Fourier Fourier transform FIG.24. 24. of the input. The maximum of the by the time of the input. The maximum bandwidth and length of the input signal are limited by the of the input. The maximum bandwidth and length of the input signal are limited by the time time dispersion-bandwidth product of the “compression” filter. dispersion-bandwidth dispersion-bandwidth product product of of the the “compression” “compression” filter. filter.
the dispersive delay line, or RAC of Fig. 23. This filter has a chirp slope opposite to the of another SAW RAC filter, which when driven by a brief electrical pulse at its input transducer, produces the linear FM premultiply chirp, which is applied along with the input signal to the nonlinear multiplier at the input of the compression filter. The output signal of the device, obtained by postmultiplying the compression signal output by a second chirp, is the Fourier transform of the input signal. The Fourier transform may itself be of interest in signal identification, or one may carry out further processing of the Fourier transform, for example, to realize frequency filtering. Other potential applications of this chirp transform element include variable time delay (Dolat and Williamson, 1976), and cross correlation (Gerard and Otto, 1977). Changing the responses of SAW filters is sometimes of interest; for example, one might want to change the codes to which a receiving filter was matched in a secure communications system. Surface acoustic wave devices that can be programmed electronically in a few microseconds have been realized in several ways by combining piezoelectric and semiconductor materials. One practical approach has been to change the phasings of the taps on SAW filters by providing semiconductor diode switches for each tap to determine which bus bar it is connected to at a particular time. Field-effect transistors employing the piezoresistive effect have also been used as taps in a matched filter (Hickernell ei al., 1975) constructed in a silicon wafer onto which zinc oxide is sputtered; a programmable readonly memory on the silicon wafer determines the proper phase for each tap when the filter is set to match a particular code (Fig. 25). The use of piezoelectric thin films with field-effect transistors also appears an attractive way to realize programmable SAW transversal filter arrays (Kwan et ul., 1977). Another powerful approach to programmability employs nonlinear effects: the input signal and a propagating or stored signal representing the waveform being sought mix in the nonlinear device to produce an output at the sum frequency; the output is maximum when the desired signal is present in the input. Such nonlinear SAW devices can also be used to perform convolution and correlation of high-frequency signals as
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
30 1
FIG.25. Electronically programmable SAW matched filter mounted in a conventional flatpack. Thirty-one taps are visible on the 3-in long silicon substrate that extends across the lower portion of the package. Gold wires carry currents from the taps to the summing electrode on the white insulating circuit board below the silicon device. Piezoelectric zinc oxide sputtered onto the silicon after fabrication of the circuit is the light rectangle at the far left side of the wafter; the 100 MHz SAW input transducer is located there. Tap phasings are set by read-only memory located above the line of taps. The signal to select the desired filter characteristic reaches the device through one of the six angled conductors crossing the white circuit board at the left end of the photograph. The device can be used to generate the waveform to which it is matched if the SAW transducer at the right side of the silicon wafer is used [courtesy of F. Hickernell, Motorola Corporation].
mentioned in connection with nonlinear S A W effects and storing signals represented by propagating surface waves. VIII. SAW OSCILLATORS, RESONATORS, A N D HIGH-QFILTERS The range of frequencies for which S A W devices are appropriate, roughly 10 MHz to 1 GHz, includes the frequencies of operation of a number of electronic systems or subsystems, such as the RF and IF sections of many types of communication and radar equipment. It is now
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ROBERT W. BRODERSEN AND RICHARD M. WHITE
possible to make high-Q SAW resonators and filters by employing appropriately shaped and dimensioned SAW transducers along with SAW reflectors. The addition of conventional amplifiers also makes possible oscillators in this frequency range. The SAW oscillators are particularly attractive in part because they eliminate the need for the many stages of frequency multiplication conventionally employed in obtaining temperature-stable high-frequency energy by multiplying up the output of a conventional quartz oscillator operating at around 10 or 15 MHz. SAW oscillators are also of value in sensors having a quasi-digital output, where the field being sensed affects the SAW oscillator frequency. We now examine the principles of operation and the performance of these devices. The three generic SAW oscillator designs are sketched in Fig. 26. In the simplest, Fig. 26a, a conventional SAW delay line is used with a solid-state amplifier to provide external feedback between the transducers to make up for the conversion loss in both transduction and the propagation loss in the medium; the latter is usually small, amounting to only about 1 dB/psec at 1 GHz in YZ-LiNbO, and decreasing as the square of the frequency. Oscillation will occur at frequencies for which the change of phase around the entire loop from a given transducer, through the amplifier and delay line, back to that transducer, is an integral multiple of 27r rad. If the phase shift in the amplifier is denoted by 4Eand the two transducers are assumed to introduce no phase shifts, the oscillation frequency f satisfies the condition 2n7r = 27rL/A
+ & = 27rJz/v, + &
where n is an integer, L the path length of the delay line, A the wavelength, and v p the SAW phase velocity. Clearly, oscillation at a number of different frequencies f,each associated with a different value of n , is possible provided the transducer and amplifier responses are not strongly frequency dependent. In this respect the operation of a SAW oscillator is similar to that of the laser, where end mirrors select a large set of possible modes of oscillation and the gain variation with optical wavelength limits the range of wavelengths over which laser action can occur. Frequency selectivity can be built into the interdigital transducers to limit the number of modes that can oscillate preferably to just a single mode. The most obvious way to make a very narrowband interdigital transducer is to use many finger pairs, as the 3 dB fractional bandwidth of a singie conventional IDT composed of N electrode pairs is approximately equal to 1/N, where N is the number of electrode pairs in the transducer (Matthews, 1977). Since the propagation under a very long transducer tends to involve an excessive number of internal reflections occurring at the edges of the fingers, a preferable ap-
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303
AMPLIFIER OUTPUT
.P I E Z O E L E C T R I C (0)
INPUT
OUTPUT
P
P
-ili i\i g l i l ~ i l l *
PIEZoELECTRIC
+//
REFLECTIVE GRATING
(b)
\ +/ R E F L E C T I V E GR AT1 NG ( C )
FIG.26. SAW oscillators illustrated schematically. (a) Conventional SAW delay line having feedback provided externally with conventional amplifier for net round-trip gain. (b) Two-transducer structure that employs reflective gratings at left and right sides of substrate to provide frequency-dependent reflection. This structure is a narrow-band filter that could also be used with an external amplifier as an oscillator. (c) One-port SAW reflector employing a single conventional interdigital transducer surrounded by reflective gratings. Rapid variation of input reactance near the frequency of maximum reflectivity can be used to determine the frequency of oscillation when this is used with an external active element.
proach is to use “thinned” transducer (Crabb et al., 1973) having many of its finger pairs removed so as to reduce the reflections, and to set that transducer a center-to-center distance ( N , + N2)A from the other transducer so that the zeros introduced by the relative location of the two transducers fall on the subsidiary maxima of the thinned transducer; the thinned transducer contains N , periods, the other shorter conventional IDT contains N 2 , and A is the wavelength. Insensitivity of oscillator frequency to changes in ambient conditions
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ROBERT W. BRODERSEN AND RICHARD M. WHITE
is desired for stable source applications; in this respect, surface-wave crystals are generally less temperature stable than the AT- and BT-cut quartz crystals used in stable-frequency sources, but SAW oscillators appear to be more resistant to vibration. The dependence of oscillation frequency upon temperature and pressure or stress has been exploited in SAW sensors whose output is a signal whose frequency varies linearly with variations in temperature or pressure (Reeder and Cullen, 1976). Changes of the oscillation frequency (which was either 80 or 170 MHz in the experiments reported) up to 15 ppm/psi applied were measured. Use of SAW oscillators in accelerometers also appears possible and advantageous, as a quasi-digital frequency output is obtained. A more exact analogy with the laser is provided by the SAW oscillator employing frequency-selective reflectors, shown in Fig. 26b. A wide variety of two-transducer designs can be considered (Coldren and Rosenberg, 1978). The key frequency-determining elements in all these are the reflectors, which might be made of ion-milled grooves in any substrate, mass-loading lines of metal or dielectric insulator, metallic lines to short out piezoelectric fields, or diffused or implanted arrays of lines, to produce reflections. Of these, the grooves provide lowest loss, while in some applications the simplicity of fabricating metallic lines on a piezoelectric substrate may be preferable economically. It should also be noted that a conventional IDT made electrically resonant by means of a n inductor across its terminals also serves as a strong reflector of surface waves (Joshi and White, 1968). The quality factor Q that can be achieved is dependent upon frequency and method of fabrication of the reflecting grating. Values of Q approaching the material or mechanical quality factor of the substrate itself can be obtained in reflectors made by ion milling of grooves, and grooved reflectors do not have the interfering modes transverse to the grooves that are observed with reflectors made with conducting stripes on piezoelectric. The flow of charge in the metal films also produces loss and so limits Q of those structures. If the highest Qs are required, the SAW must be evacuated to eliminate air-loading losses. Figure 27 shows attainable Q factors for quartz reflector structures. Values of Q up to 10,000 can be obtained fairly easily, while higher values approaching limits imposed by dissipation in the substrate require careful control of reflector dimensions. The remaining oscillator design, Fig. 26c, is a single-ended device in which the radiation reactance of the IDT is a strong function of frequency near the resonant frequency of the resonator structure. This design is simple and compact as it employs a single transducer. An example of such a practical device is the resonator employed by Shreve et a / . (1978), which was made of quartz, had a 160 MHz center frequency, two 275-
CCD AND SAW SIGNAL-PROCESSING TECHNIQUES
305
r 7x
0 0050
00025
lo4
6 x lo4
5X
lo4
4 x 104
3x
2
lo4 IN AIR
0
I N VACUUM
A
X I O ~
THEORY
QL
-
1 XlO'
I
005
I
I
I
0.015
0.010
I
I 0.020
h/A,
FIG. 27. Comparison of measured and calculated Q at 157 MHz for resonators on ST-quartz having 600 grooves per grating (from Li P I a / . , 1975, with permission).
groove ion-milled gratings whose grooves were 0.045 wavelengths deep, and a single IDT having 49 finger pairs. The mean unloaded Q of these devices was 43,000. As with all SAWS, dimensional variations affect operating frequency; to correct for inevitable variations a modest amount of electronic tuning can be achieved with a varactor-loaded IDT in the propagation path inside the oscillator, resonator, or complete filter. Changes as a result of device aging are also important. Factors that result in shifts of resonant frequency with time in bulk-wave resonators and that also affect SAW resonators include chemisorption at the oscillator surfaces, oxidation of electrodes, and stress relaxation in the deposited electrodes. Evolution of contaminants from sealing and mounting materials such as adhesives in the oscillator container must be minimized as well. One can argue that SAW resonator aging may be less severe than that of bulk resonators because of the proportional change in resonance frequency brought about by a change in thickness of the bulk resonator owing to deposition on the surface and to the existence of two important surfaces and a finite volume in the bulk devices (Bell, 1977). To date, results on the SAW resonators have shown that with plasma etching and cleaning techniques, and careful package design and assembly, one can obtain changes in the frequency of
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ROBERT W. BRODERSEN AND RICHARD M. WHITE
a SAW oscillator that may be as small as about 1 ppm/year (Shreve et al., 1978).
IX. CONCLUSIONS In charge-transfer and SAW devices, new electronic principles provide efficient low-cost solutions to existing signal-processing problems. The combination of analog computation with a discrete time signal representation, which is available with CCD signal-processing devices, is found to have considerable advantages in reducing the cost, power, size, and weight of signal-processing systems. An important factor in obtaining these reductions is the compatibility of the CCD technology with standard MOS circuits, which allows integration of additional non-CCD functions and allows the design of integrated circuits that are complete subsystems in themselves. In addition, the ability to vary externally the transfer rate of the charge packet is the basis of many unique CCD applications. In SAWS the analog signals are represented by waves moving at a speed that cannot be varied, but the technology permits one to make sophisticated signal processors that operate at very high data rates with signal frequencies up to the gigahertz level. The CCD and SAWS operate in complementary frequency ranges, and the devices are often organized in similar ways. A few devices have exploited both technologies together, as in the fast-in, slow-out buffer memory (Smythe et al., 1978) which employs a SAW delay line coupled to a CCD shift register. Used singly or together, the CCD and SAW technologies permit one to realize signalprocessing functions that a few years ago would have seemed impossible.
REFERENCES Auld, B. A. (1973). “Acoustic Fields and Waves in Solids.” Wiley, New York. Baertsch, R. D., Engeler, W. E., Goldberg, H. S.. Puchette, C. M . , and Tiemann, J . J. (1976). IEEE Trans. Electron Devices ed-23,62-70. Bell, D. T. (1977). Proc. IEEE Ultrason. Symp. p. 851. Boyle, W. S., and Smith, G. E. (1970). Bell Syst. Tech. J . 49, 587. Brodersen, R. W., and White, R. M. (1977). Science 195, 1216. Brodersen, R. W., Buss, D. D., and Tasch, A. F. (1975). IEEE Trans. Electron Devices ed-22, 40. Brodersen, R. W., Hewes, C. R . , and Buss, D. D. (1976).IEEEJ. Solid-State Circuits sc-11, 75. Buss, D. D., Collins, D. R., Bailey, W. H . . and Reeves, C. R. (1973). IEEEJ. Solid-State Circuits sc-8, 138.
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Buss, D. D., Brodersen, R. W., Hewes, C. R., and Tasch, A. F. (1975). IEEE Nail. TeIecom. Con$ Rec. p. 25. Carnes, J. E., Kosonocky, W. F., and Rambert, E. G. (1972). IEEE Trans. Electron Devices ed-19, 798. Cheek, T. F. et al. (1973). Proc. Charge Coupled Devices Appl. Conf., 1973 p. 127. Coldren, L. A., and Rosenberg, R. L. (1978). Proc. IEEE Ultrason. Symp. p. 422. Crabb, J., Lewis, M. F., and Maines, J. D. (1973). Electron. Lett. 9, 195. Daimon, Y.,Mohsen, A. M., and McGill, T. C. (1974). IEEE Trans. Electron Devices 4 - 2 1 , 266-272. Datta, S., Hoskins, M. J., and Hunsinger, B. J. (1978). Appl. Phys. Lett. 32, 3. DeVries, A. J., and Adler, R. (1976). Proc. IEEE 64, 671. Dolat, V. S., and Williamson, R. C. (1976). Proc. IEEE Ultrason. Symp. p. 419. Esser, L. J . M. (1974). ISSCC, Philadelphia Dig. Tech. Pap. p. 28. Farnell, G. W. (1970). Phys. Acoustics 6. Foxall, T., Ibrahim, A., and Hupe, G. (1977). Electron. Lett. 13, 323. Gerard, H. M., and Otto, 0. W. (1977). Proc. IEEE Ultrason. Syrnp. p. 947. Hannan, W. J . , Schanne, J . F., and Waywood, D. J . (1965). IEEE Trans. Mil. Electron. 9, 246. Hays, R. M., and Hartmann, C. S. (1976). Proc. IEEE 64,652. Hewes, C. R. (1975). Proc. Int. Con$ Appl. Charge Coupled Devices, 1975 p. 170. Hickernell, F., Adams, M..London, A.. and Bush, H. (1975). Proc. IEEE Ultrason. Symp. p. 223. Ingebrigtsen, K. (1976). Proc. IEEE 64, 764. Johnson, D. E. (1976). "Introduction to Filter Theory." Prentice-Hall, Englewood Cliffs, New Jersey. Joshi, S. G., and White, R. M.(1968). J . Appl. Phys. 39, 5819. Kerber, G. L.. White, R. M.. and Wright, R. W. (1976). Prof. IEEE Ultrason. Syrnp. p. 577. Kino, G. S. (1976). Proc. IEEE 64, 724. Kino, G. S., and Wagers, R. S. (1973). J . Appl. Phys. 44, 1480. Knauer, K., Meiderer, H. J., arid Keller, H. (1977). Electron. Lett. 13, 126. Kwan, S. H., White, R. M., and Muller, R. S. (1977). Proc. IEEE Ultrason. Symp. p. 843. Lagasse, P. E., Mason, I. M., and Ash, E. A. (1973). IEEE Trans. Sonics Ultrason. su-20, 143. Lewis, M. F. (1977). Proc. IEEE Ultrason. Symp. p. 744. Li, R. C. M., Alusow, J. A., and Williamson, R. C. (1975). Proc. IEEE Ultrason. Symp. p. 279. Linnenbrink, T. E. et a/. (1975). Proc. Int. Conf. Appl. Charge Cuupled Devices. 1975. McClellan. J . H.,Parks, T. W., Rabiner, L. R. (1973). IEEE Trans. Audio Electroaconsr. 21,506.
Maines, J. D., and Paige, E. S. (1976). Proc. IEEE 64,639. Mavor, J., and Denyer, P. B. (1978). Electron Circuits and S y s t . 2, No. 1, 1. Marshall, F. G., Newton, C. 0..and Paige, E. G . S. (1973). IEEE Trans. Microwave Theory Tech. MTT-21, 221. Matthews, H., ed. (1977). "Surface Wave Filters." Wiley, New York. Melen, R. G., and Buss, D. D. (1977). "Charge-Coupled Devices: Technology and Applications." IEEE Press, New York. Nakagawa, Y. (1977). Appl. Phys. Lett. 31, 56. O'Connell, R. M., and Carr, P. H. (1978). Proc. 32nd Annu. Syrnp. Freq. Control. pp. 189- 195. U.S. Army Electronics Command, Fort Monmouth, New Jersey. Oliner, A. A., ed. (1978). "Acoustic Surface Waves." Springer-Verlag, Berlin and New York.
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Rabiner, L. R., and Gold, B. (1975). “Theory and Applications of Digital Signal Processing.” Prentice-Hall, Englewood Cliffs, New Jersey. Rabiner, L. R., Schafer, R.W., and Rader, C. M. (1969). IEEE Truns. Audio Elecrroucoust. au-7, 86. Reeder, T. M., and Cullen, D. E. (1976). Proc. IEEE 64, 754. Sangster, F. L. J. (1970). Philips Tech. Rev. 31, 92. Sequin, C. H., and Tompsett, M. F. (1975). “Charge Transfer Devices.” Academic Press. New York. Shiosaki, T. (1978). Proc. IEEE Ultrirson. S y m p . p. 100. Shiosaki, T., and Kawabata, A. (1977). Jpn. J . Appl. f h y s . 16, Suppl. 16-1, 483. Shreve, W. R., Kusters, J. A., and Adams, C . A. (1978). Proc. IEEE Ultrirson. Symp. p. 573. Slobodnik, A. J., Jr. (1975). “Surface Acoustic Wave Filters at UHF: Design and Analysis,” AFCRL-TR-75-03 1 1 . Air Force Cambridge Res. Lab., Bedford, Massachusetts. Slobodnik, A. J., Jr., and Conway, E. I). (1970). “Microwave Acoustics Handbook,” Vol. I, Phys. Sci. Res. Pap. No. 414. Air Force Cambridge Res. Lab., Bedford, Massachusetts. Smythe, D. L., Ralston, R. W., Burke, B. E., and Stem, E. (1978). Proc. IEEE Ulrruson. Symp. p. 16. Tompsett, M. F., and Zimany, E. J., Jr. (1973). IEEEJ. Solid-SfmfeCircuifs sc-8, 151. Tuan, H. C., Khuri-Yakub, B. T., and Kino, G. S. (1977). Proc. IEEE Ultrason. Symp. p. 496. Viktorov, I. A. (1967). “Rayleigh and Lamb Waves: Physical Theory and Applications.“ Plenum, New York. Walden, R. H., Krambech, R. H., Strain, R. J., McKenna, J., Schryer, N. L., and Smith, G . E. (1972). Bell Syst. Tech. J . 51, 1635. White, R. M. (1970). Proc. IEEE 58, 1238. Williamson, R. C. (1977). Proc. IEEE Ultrason. Symp. p. 460.
ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS, VOL. 51
Gunn- Hilsum Effect Electronics M. P. SHAW Department of'Electrical and Computer Engineering Wayne State Universirp. Detroit, Michigan
H. L. GRUBIN United Technologies Research Center East Hartford. Connecticut
P. R. SOLOMON Adt'anced Fuel Research, Inc East Hartjbrd, Connecticrct
1. Negative Differential Mobility (NDM) in Se A. Introduction ........................
...................... A. Introduction ........................ B. The Local Environment
328
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. . . . . . . . 367 . . . . . . . . 368
D. Amplification ........................ E. The Gunn Diode in a Microwave Circuit . . . . . . . . . . . . . . . . . . References . . . . . . . . .
................... 309
Copyrighl @ 1980 by Academic Rebs. Inc. All right.; of reproduction in any form reserved. ISBN 0-12-014651-7
3 10
M. P. SHAW ET A L .
In 1962 C. Hilsum predicted intervalley transfer-induced negative differential mobility (NDM) in n-GaAs. In 1963, Gunn discovered current instabilities in n-GaAs resulting from this process. We review the physics and technological developments in this field over the past 15 years with emphasis on the role of the boundary conditions to, and local circuit environment of, NDM elements. We present an historical. overview and then discuss the velocity -electric field characteristic of n-GaAs and related semiconductors. The response of an NDM element to a charge fluctuation is then treated, followed by a discussion of the main theme of the review -the local circuit and contact environment of the NDM element. Although experimental and numerical results are stressed, simple analytical discussions are presented for clarity and understanding (detailed analytical treatment can be found elsewhere.) The review concludes with the most extensive section, NDM devices, where we emphasize short ( 510 pm) device-grade layers of NDM semiconductors and their deployment as oscillators and amplifiers. I. NEGATIVE DIFFERENTIAL MOBILITY(NDM) I N SEMICONDUCTORS A . Introduction
In 1963 Gunn reported the discovery of current oscillations at low microwave frequencies, which were produced when the semiconductor n-GaAs was subjected to an electric field of a few kV/cm (I). Gunn’s subsequent probe measurements of the spatial variation of the electric field within the semiconductor showed that each current oscillation was accompanied by the propagation of a high-electric-field region (dipole domain) from the cathode to the anode ( 2 , 3 ) . Several explanations for the mechanism responsible for this phenomenon were suggested during the following year. As it turned out, Kroemer’s suggestion (4) proved to be the correct one. Kroemer pointed out that most of the known properties of the “Gunn effect” could be explained if it were assumed that n-GaAs had a region of negative differential mobility (NDM) in its velocity vs. electric field, V ( E ) , curve as shown in Fig. l(5, 5a, b). Kroemer had put together two pieces of information to explain the effect. First, he was aware that in 1962 Hilsum (6) had predicted the existence of field-induced “transferred electron” effects in n-GaAs that could lead to NDM and produce amplification and oscillation. Second, he knew that Ridley (7) and Watkins (7a) had concluded that the presence of NDM or NDR (negative differential resistance) could cause homogeneous material to become electrically heterogeneous; high-field dipole domains would form and could propagate through the specimen, producing the observed current
GUNN-HILSUM EFFECT ELECTRONICS
311
Electric Field lkV/cml
FIG. 1. Experimental points (-1 ( 5 ) and theoretical (---) values (Sa, h ) for the electric-field-dependent average drift velocity ofelectrons in n-GaAs. (Reprinted with permission.)
oscillations. Two crucial experiments were performed in 1965 that seemed to verify these concepts. To appreciate them, we must first consider the conduction band in GaAs shown in Fig. 2a. In equilibrium, the great majority of electrons reside in the minimum at k = 0, the r point in the Brillouin zone. However, the electrons can be heated by an applied electric field and induced into states in the subsidiary minimum (“intervalley transfer”) at point L in the zone ([l 1 I]), which is thought to be the lowest subsidiary minimum (76). For high enough fields, a large enough number can be induced into this minimum so that the mobility of the sample will be affected. Since mobility is inversely proportional to effective mass, and effective mass is inversely proportional to the curvature of the energy bands, we can see from Fig. 2a that a reduction in mobility could result from such a process. In fact, if the intervalley coupling is strong enough, NDM can result, as sketched in Fig. 2b (8). If the electron transfer mechanism is operative in n-GaAs, then a variation in the relative position of the energy band extrema (intervalley separation) should alter the threshold parameters of the “Gunn-Hilsum effect.” The two experiments that demonstrated that these changes occurred were performed by Hutson ef al. (9) and Allen ef a / . (10). In the first of these experiments, the intervalley separation was varied by the application of hydrostatic pressure [hydrostatic pressure changes the energy separation between the bands at the r and X points at a rate near 9 x
M. P. SHAW ET AL.
312
-
x
._
,' Strong Coupling
0
-
3 $ D
E l e c t r i c Field
FIG.2. (a) Schematic conduction band structure for GaAs (8).The intervalley separation and effective masses are given. (b) Schematic average drift velocity vs. electric-field curves for conduction in valley 1(0,0,0), valley 2(1,1, I), and when either strong or weak intervalley coupling is present. (Reprinted with permission, The Institute of Physics, London.)
eV/kbar ( 1 1 ) . It was found that the voltage at threshold decreased with increasing pressure: at 26 kbar the instability vanished completely. In the second experiment, the intervalley separation was varied by forming various mixed compounds of GaAs,P,-,. As x decreases from 1.0 to 0.5 the intervalley separation decreases from -0.33 eV to zero. Allen et al. showed that as x decreased from unity, the voltage at threshold decreased. The instability vanished completely at x = 0.52. The above results left little doubt that intervalley transfer was the mechanism responsi-
GUNN-HILSUM EFFECT ELECTRONICS
313
ble for the Gunn-Hilsum effect. However, more recent experiments indicate that in looking at the changes in threshold conditions with external pressure and phosphorus content, consideration must be given to the nature of the boundary conditions. In fact, Pickering et al. (12) have recently found that for a specific type of contact, which they believe to be perfectly “ohmic” (see Section 11), the field at threshold increases with hydrostatic pressure to 15 kbar. They attribute this to an increase in effective mass in the r minimum with pressure. The masking of this effect in the early results is attributed to the presence of high fields at the cathode con tact. After the initial announcement by Gunn, it soon became clear that an NDM induced-current instability would manifest itself in ways other than those associated with a domain transiting from the cathode to anode. It could also be accompanied by the formation of stationary charge layers at the cathode or anode, accumulation or depletion layers in transit, or uniform or nonuniform excursions of the electric field through the NDM regime. By the late 1960s it was known that the manifestation of the current instability observed in an NDM semiconductor depended critically on three things: (1) the electrical characteristics of the semiconductor, (2) the circuit in which it operates, and (3) the conditions at the boundaries of the active region (in particular, the conditions at the cathode contact). The electrical characteristics of semiconductor materials exhibiting NDM have received considerable attention during the past 15 years and excellent overviews can be obtained by reading (1) a special issue of the IEEE Transactions on Electron Devices devoted to semiconductor bulk effect and transit-time devices (Vol. ED-13, January 1966) and (2) the issue of the IBM Journal of Research and Development devoted to the physics of instabilities in semiconductors (Vol. 13, September 1969). Many measurements and calculations of the velocity -field characteristics and diffusion coefficients have been made and these are discussed in Section 1,B. The response of the NDM element to a charge fluctuation has been studied in great detail by Butcher (8) and by Knight and Peterson (13), who analytically demonstrated a wide variety of solutions. Their work is outlined in Section 1,C. There have also been many studies of the circuit influence on the current instabilities (14-19), including a systematic study of the occurrence of the various modes as a function of the circuit parameters (19). The circuit aspects and the important role of the boundaries (20, 21) in controlling the instability are emphasized in Sections I1 and 111. We show that by choosing appropriate boundary conditions it is possible to model almost all the observed current-voltage curves, current-time waveforms, electric field vs. distance profiles, variations in instability threshold, and the oscillatory behavior of samples in
M. P. SHAW ET A L .
3 14
various circuits. The review concludes with the most extensive section,
NDM devices, where we emphasize short ( s10 pm) device-grade layers of NDM semiconductors and their deployment as oscillators and amplifiers. For a more extensive treatment of the experimental and analytical aspects of the general problem, the reader is referred to Chapters 4-6 of "The Gunn-Hilsum Effect " ( 2 1 ~ ) . B . The Velocity -Electric Field Characteristic 1. Experiment
The Gunn-Hilsum effect has been observed under various conditions in materials such as n-GaAs, GaAs,P,-, alloys ( J O ) , n-InP (22-24), n-CdTe ( 2 3 , n-Ge (25, 26), and n-InAs (27). Since n-GaAs and n-InP have been under the most extensive investigation, we emphasize them in this review. How can we measure the velocity-field characteristic of a material to determine whether or not it is a candidate for the Gunn instability? As we show in Section I,C, biasing a sample in its NDM region results in an internal rearrangement of fields, which acts to mask the NDR. To measure the V ( E )curve directly, therefore, one is required to do an experiment in a time short enough such that the field remains relatively uniform. It is also possible to measure the V ( E ) curve directly by a time-of-flight (tof) experiment. Here carriers are injected into a sample in which, by reducing the differential dielectric relaxation time to a sufficiently small value, the instability is inhibited (see Section 111). This technique was exploited by Ruch and Kino ( 5 ) ; their results are plotted in Fig. 1 along with the theoretical curve developed by Butcher and Fawcett (8).To obtain the results shown in Fig. I, Ruch and Kino studied the response of a reverse-biased Schottky barrier-semi-insulating-n+-GaAs sample to a short pulse (0.1 nsec) of high-energy electrons. The injected electrons move from the cathode to the anode of the semi-insulating sample under the influence of the field, inducing a current in the circuit until they drain at the anode. A more popular direct method of measuring the V ( E )curve is related to the short-time experiment; this technique involves microwave heating of the electrons. Studies of this type have been made for n-GaAs by, among others, Braslau (28), Ackert (29), Hanaguchi ef al. (30) and Kalashnikov et al. (31); for n-InP by Glover (321, Lam and Acket (33), Nielson (34),and Hayes (35);and for n-Ge by Chang and Ruch (36). The experiments usually involve mounting the specimen on a post inside a
GUNN-HILSUM EFFECT ELECTRONICS
315
waveguide cavity with a sliding short behind the post. Pulsed power from a magnetron at frequencies usually above 30 Ghz is incident on the sample. The absorbed power can be measured by a directional coupler and microwave average power meter. In interpreting the results of these experiments it is crucial to consider the role of contact fields (20, 21) and doping nonuniformities (37), and to appreciate the role played by electron energy relaxation effects (38). Because of the affect of nonuniformities, large variations in the measured V ( E )curves have been reported. Furthermore, relaxation effects will cause the measured V ( E )curve to have a substantially higher velocity at peak field ( 3 5 , 3 8 ) than for the DC case. Microwave heating results, therefore, are rather difficult to interpret. On the other hand, tof experiments are difficult to perform and in general rather complex and elaborate. There is, however, an indirect method established by Bastida et al. (39) for obtaining the V ( E ) curve that just involves a study of a particular sample that exhibits a transit-time mode of oscillation. Bastida et al. applied this technique to n-GaAs. It was applied to n-InP by Prew (40), who also supplemented his results with a voltage probe experiment along the length of the sample ( 2 0 , 4 1 ) . In the Bastida technique the V ( E ) curve is deduced from the measured relationship between the current flowing in the sample (during the time a domain is in transit) and the applied voltage. In order to demonstrate how the V ( E ) curve can be obtained this way, however, we require a knowledge of the properties of V ( E ) curves and high-field domains. To obtain these, we must first determine how an NDM semiconductor responds to a small charge fluctuation. We do that in Section I11 and then discuss the Bastida technique.
2. Theory Hilsum (6) made the first calculation of the V ( E )curve for n-GaAs two years before Gunn made his observation of the current instability. It was remarkable that Hilsum’s rather simple model predicted a field at peak velocity of 3 kV/cm, a value that is well into the expected range for typical low-field mobilities in n-GaAs. In his calculation Hilsum assumed that the field caused the electrons to heat up above ambient temperature and the collision between electrons produced a common electron temperature T, for carriers in both the lower and upper valleys. He also assumed that the electrons distributed themselves between these valleys as if the system were at thermal equilibrium at a temperature T,. A similar simple and insightful calculation was later performed by McCumber and Chynoweth (42).
M. P. SHAW ET AL.
3 I6
More detailed calculations were performed by Butcher and Fawcett (5a, b, 8, 43) and Conwell and Vassell (44-47). Butcher and Fawcett as-
sumed displaced Maxwellian distributions in each valley, but they did not use the same electron temperature for each valley. They also included intervalley scattering, and when they included equivalent intervalley scattering for the upper valley electrons, their results proved to be in good agreement with experiment, as shown in Fig. 1. Conwell and Vassell (44-47) performed similar calculations, neglecting electron-electron collisions, and including acoustic mode and equivalent intervalley scattering among the upper valley electrons. [For detailed discussions of these calculations, the reader is referred to the original papers and Conwell (45), Bulman et al. (46), and Bosch and Engelmann (47).] Research into the details of various V ( E ) curves is presently still active. In particular, the V ( E ) curve for n-InP has been under investigation for the past decade. Hilsum and Rees (48, 4 9 ) suggested that intervalley scattering between the r and L valleys might be weak, so that both the L and X valleys (three-level transfer model) would be involved in the high-field transport properties of n-InP. James et al. (50) calculated a quite different V ( E )curve without invoking three-level effects. Recently, Fawcett and Herbert (51, 52) used theoretical estimates of intervalley scattering rates in InP to calculate various V ( E ) curves, an example of which is shown in Fig. 3. Their results do not support the three-level model. Fawcett and Herbert (51, 52) and Littlejohn et al. (53)have also continued to refine the V ( E ) calculation for n-GaAs.
-
P .
3-
1
0
10
20
30
1
40 50
1
,
60 7075
Electric Field [kV/cml
FIG.3. Average drift velocity vs. electric field curves for n-InP calculated using screened (-1 and unscreened (-1 pseudopotentials (51, 52). (Reprinted with permission, The Institute of Physics, London.)
317
GUNN-HILSUM EFFECT ELECTRONICS
C . Response of a n N D M Element to a Charge Fluctuution 1. General Discussion
To illustrate the response of an NDM element to a charge fluctuation, consider a uniform field with a domain of increased field in the center of the NDM element as shown in Fig. 4a. The charge distribution that pro-
-z ->
-
.-
E
I
I
I
1
I
I
I I
I
I
I I
I
I
I I I
1
FIG.4. Illustration of a charge fluctuation (b) required to produce a local increase in electric field (a). E , is the threshold field for negative differential mobility. The arrow denotes the direction of carrier drift.
318
M. P. SHAW ET A L .
duces this field fluctuation is shown in Fig. 4b. There is an accumulation layer of positive charge on the left side of the domain and a depletion layer on the right. If we consider positively charged carriers, the carriers and hence the domain will be moving to the right. Assuming that the field within the domain is within the NDM range and the field outside the domain is within the ohmic range, then it is clear that the field fluctuation will initially grow with time. This happens because the higher field within the domain results in carriers moving more slowly inside than outside the domain. Charge will therefore deplete on the right (leading) edge of the domain and accumulate at the left (trailing) edge. This charge will add to what is already there, increasing the field in the domain. If the NDM element is in a resistive circuit, the increasing voltage across the domain will decrease the current in the circuit and lower the field outside the domain. The above situation constitutes a runaway process. The field will continue to grow in the interior of the domain and drop outside. Given enough time, the domain field will grow until the domain velocity is equal to the velocity of carriers outside the domain. (This is possible with the Nshaped V(E) curve we are considering.) The final velocity of the domain depends on the circuit load line. If the original field fluctuation were produced by a doping nonuniformity, the decreased field around the nucleation site that would result after the domain moved from the region would prevent nucleation of subsequent domains. This situation would prevail until the domain reached the right-hand edge of the NDM element, where it would disappear. The field could then rise to nucleate another domain, resulting in a periodic current oscillation whose period was determined by the transit time of the domain. The periodic nucleation and dissipation of high-field domains was the phenomenon observed by Gunn. We consider next the response to a charge fluctuation in a more rigorous manner.
2. Method of Characteristics Given a particular V ( E )curve exhibiting NDM, what response occurs when a small fluctuation of charge is induced in a homogeneous medium biased into its NDM region? This question was answered analytically by Butcher ( 8 , 5 4 ) and Knight and Peterson ( 1 3 , 5 5 ) and numerically by Kroemer (56a)and McCumber and Chynoweth (42). Knight and Peterson analyzed a model for the growth and propagation of charge disturbances in a medium with an arbitrary electric-field-dependent mobility (55). They developed possible forms of propagating field n0,nuniformities in a material exhibiting NDM, obtaining expressions for the velocity of both charge layers and dipole domains (13). Bonch-Bruevich and his colleagues have
GUNN-HILSUM EFFECT ELECTRONICS
3 19
also contributed substantially to the understanding of the problem (57-60). To determine the response to a charge fluctuation, we use the continuity equation in one dimension,
a
a
- N ( X , T ) e V ( E ) + 3N ( X , T)e = 0
ax
(1)
where N ( X , T ) is the mobile charge density and e the electronic charge, and the Poisson equation
where N o is the fixed uniform background charge density and e the permittivity of the material. Diffusion is neglected in the argument; its inclusion will be considered in Section I,D. Writing n = N ( X , T ) / N o and k = e / N o e , Eqs. (1) and (2) can be combined into
a a2E ax [IZU(E)]+ k ax aT = 0
(3)
A spatial integration of the equation yields
nV(E) + k
aE z = $(T)
(4)
which states that the sum of the conduction and displacement currents is only a function of time. f ( T ) is identified as being proportional to the current in the external circuit. Putting Eq. (2) (divided by N o ) into Eq. (4) yields
Equation ( 5 ) is a nonlinear partial differential equation whose solutions describe the influence of an electric field on the motion of a charge carrier. As an illustration, we consider the motion of a carrier along a path (trajectory) described by the equation
V(E) = dX/dT
(6)
Using this, Eq. ( 5 ) becomes
Since the expresSion in the parentheses in Eq. (7) is the total time deriva-
3 20
M. P. SHAW ET A t .
tive along a trajectory, Eq. (5) can be replaced by the two equations, (6) and k dE(X, T ) / d T = $(T) - V ( E )
(8)
Let us now consider imposing a charge fluctuation that produces a pertubation SE in a time-dependent solution Eo to Eq. (8). For simplicity we consider an NDM element connected to a constant-current source so that d$/dT = 0. Then d k d~ (Eo + SE) = 8; - V(&) - SV
(9)
Since Eo is a solution to Eq. (8), 6 E must satisfy d dV k-SE= -SV= --SE dT dE0 Solutions of Eq. (10) have the form where 7 =
€
Nee dV/dEo
and describe a propagating disturbance that either grows or decays while propagating. Growth occurs when dV/dE < 0, decay occurs when d V / d E > 0. Thus, if we had an N-shaped NDM curve biased in the positive-mobility region where, e.g., V = p E , a small local increase in electric field would decay in time. If the bias were sufficient to take the electric field into the NDM region, a small local increase in electric field would grow. Kroemer discussed the criterion for NDM element stability by considering the growth of charge fluctuations as described by Eq. (11). He assumed that Eq. (1 1) remained valid for the growth of a domain throughout a complete transit of the domain through the NDM element. The growth is then given by
where I is the sample length and Vthe average domain velocity. For space charge instability, G must be larger than unity, or EV dV No1 > - e dE
GUNN-HILSUM EFFECT ELECTRONICS
32 I
For GaAs,
No1
2 10'2
The N o / product criterion was first described for uniform fields by McCumber and Chynoweth (42). Grubin e f al. treated the more general case of nonuniform fields (21). If an NDM element is unstable to charge fluctuations, how do the fields rearrange? Are there stable nonuniform field configurations? What happens to the current during field redistribution? As we shall see shortly, the field redistributes in a manner determined by the boundary conditions and the circuit. Independent of these, it is clear that the redistribution process is a transient phenomenon; charge layers will move within the sample. One of the most important of the time-dependent phenomena is the cyclic motion of a high-field dipole domain across the sample from cathode to anode (2, 3). The fields associated with their solution are determined next. 3. The Equal-Areas Rule
Let us now analyze the problem of stable domain motion while neglecting the influence of both the cathode and anode regions. We begin the discussion with the equation for total current density J( T ) , now including a field-dependent diffusion coefficient D ( 0 :
We seek a solution representing a high-field domain that propagates (1) with constant velocity V D , (2) without change of shape, and (3) surrounded by a neutral material with N = N o . The field outside the domain is time dependent and denoted by E,(T). We assume E(X,T ) = E(X - VDT) = E( Y), where Y = X - V D T .The E( Y) and N ( Y) dependence is shown in Fig. 5 . To determine the properties of E( Y), as well as a value for V , , we first note that outside the domain J = Noel',, where V , = poE,(T) and pois the low-field mobility. Furthermore aE/aX = &/dY and aE/aT = - V Dd E / d Y . Eq. (2) then reads
and Eq. (13) becomes
M. P. SHAW ET AL.
3 22
We next eliminate Y by dividing Eq. (15) by Eq. (14) to produce
e d - [D(E )N ] dE'
-
=
VD]
N
- NdVm
- VD)
(16)
No
-
We see from Fig. 5 that N is a double-valued function of E, consisting of a depletion branch with N < N o (leading side of the domain) and an accumulation branch with N > N o (trailing side). The two branches join with N = No both at E = E and at E = Em(T). Equation (16) can be solved analytically for a field-independent diffu-
0 ._ c u w L
I
I
I
,
I I I
I
I I
Distance [Y=X-VDT]
I
-
I I
A I
I
I
FIG.5. Electric field and carrier density vs. distance profiles used in the discussion of Eqs. (13)-(20).
GUNN-HILSUM EFFECT ELECTRONICS
323
sion coefficient. For constant D(E), Eq. (16) yields
Integration of the left-hand side of Eq. (17) from N o to N and the right-hand side from E mto E yields
Note that the accumulation and depletion branches must neutralize with N = No both at E = E and E = E,. Since the left-hand side of Eq. (18) vanishes when N = N o , the right-hand side must vanish at E = E. Thus, Eq. (15) yields
Since the contribution to the integral from the first term is independent of N, but the contribution from the second term depends on whether we are integrating along the accumulation or depletion branch, then the integral can vanish only if each contribution vanishes separately. Furthermore, the second contribution vanishes for either N S N o only if V , = V D . We therefore conclude that for a field-independent diffusion coefficient, a freely traveling domain will move at the same velocity as the carriers outside the domain. The vanishing of the first term and the equality of V, and V Dprovide us with an equal-areas rule (8, 13, 54, 5 5 )
Equation (20) states that ,the area under the V ( E )curve in going from Em to E must equal the rectangle V D ( k- Em).As shown in Fig. 6, this holds only if area A equals area B; thus E is determined once Emis given. Note that for a particular V ( E )curve there is a minimum V below which the equal-areas rule cannot hold. This implies that for a given V ( E )curve there is a maximum domain field that can be achieved. For velocities below the minimum value, moving domains will be unstable (61). The above argument fails when the diffusion coefficient is field dependent. Depending on the type of field dependence, a different area rule pertains that determines E. Independent of the form of the diffusion coefficient, the question still arises as to how the traveling domain nucleates and then propagates without changing shape. To answer this question we must investigate the boundary and circuit conditions. Before at-
324
M. P. SHAW ET A L .
I
I I
Electric Field
I I I
.
+
1 I I
FIG.6 . Velocity vs. electric field and field vs. distance curves used to demonstrate the equal areas rule for a propagating domain. Area A equals area B .
tacking this very important aspect of the problem, however, it is useful to examine a technique for extracting the V ( E )curve from the properties of the moving domain, since in many cases domain data are much more readily available than V ( E )data. 4. Extraction of Velocity -Electric Field Data from Domain Measurements
To determine a large segment of the V(E)curve from domain data, two experiments must be performed. In the first (391, the properties of a freely traveling domain are studied In the second, the prethreshold current-
GUNN-HILSUM EFFECT ELECTRONICS
325
voltage Z(4) curve is carefully studied by voltage probing. Consider again the freely traveling dipole domain shown in Fig. 5 and make the reasonable assumption, experimentally justified for n-GaAs, that it is triangular. Also neglect the field dependence of the diffusion coefficient (we discuss the significance of this assumption shortly), which allows us to use the equal-areas rule. Differentiating Eq. (20) with respect to E yields
V(E) = VD + -dEm d V D (i- E m ) dE dEm an equation that can be used to obtain V ( E ) . To do this, note that the "excess" voltage +ex across the triangular domain is dex
=
J
sample
[E(x) - Em] dX
Gauss' law tells us that
for a domain with approximately triangular sides, where N d (N,) is the magnitude of the net charge density inside the depleted (accumulated) region of the domain and X, (X,) is the length of the depleted (accumulated) region. Thus
The last two equations provide us with an expression for xa/xd 4
+ex
in the limit
1:
Differentiation of Eq. (25) with respect to E m ,assuming N d is constant, yields
-dE - -1 dEm
+-e N d & Q
1
dEm
(k - E m )
or dE,
(27)
where I is the length of the sample. For the long samples required in the experiment, 1 / x d s= 1. Furthermore, Kuru et al. (62) have shown that in this case ( 1 / 1 ) d+e,,/dEm> 1. Hence, Eq. (27) becomes
M. P. SHAW ET A L .
326
Putting Eq. (28) into Eq. (21) and using the approximations leading to Eq. (25) leads to
V(k)=
dVD v, + 24ex-
ddex which gives us a relation between the high-field electron drift velocity, low-field electron drift velocity, domain voltage, and dVD/d+e,, V(&)can then be obtained from experimental data taken during the transit of a domain. Finally, by combining Eqs. (25) and (29), portions of V ( E ) can be mapped out. Typical data are shown in Fig. 7 and a V(E) curve segment obtained in this manner is shown in Fig. 8. The Bastida technique relies on the validity of the equal-areas rule [Eq. (25)]. When the diffusion coeffi[Eq. (20)] and that &, - (I? cient is field dependent, these equations are invalid. Allen et al. (63) have ) =0 shown that the equal-areas rule modifies to Jim(V - V D ) / D ( E dE and Bott and Fawcett (64) have shown by computer simulation that 4ex so long as No < 2.5 x Furstill remains proportional to (I? -
1.5
7
g
1.0
0 u
-
-Ern1
c C
? 3 0
0.5
-
I
I 1
I
I
; I
I I
I
I !
I I
I
( :
I ;
I
L
q
I+T
I
FIG.7. Typical current voltage characteristic of n-GaAs prior to and during domain transit (39).The “domain characteristic” is the section with negative slope. & denotes the voltage at the threshold of instability. (Reprinted with permission.)
327
GUNN -HILSUM EFFECT ELECTRONICS
8 b
0.7
0.5 o o
'
10
20
30
40
50
60
70
80
DO
Electric Field [kV/crnl
FIG.8. Average drift velocity vs. electric field curve as deduced in the zero diffusion limit from experimental f - I$ curves for domains (39). (Reprinted with permission.)
thermore, Kuru ez at. (62) have used probing techniques to show that c # ~ ~(E~ - E )2; we thus have empirical justification for the use of this relationship. Using the modified equal-areas rule results in a somewhat modified expression for V ( E ) . Bastida et al. (39) show, however, that when the modified equal-areas rule is used to transform the data into a V ( E ) curve using the D ( E ) curve calculated by Bott and Fawcett (64), only slight changes occur. It is therefore reasonable to neglect the field dependence of the diffusion coefficient. We see from Fig. 8 that the Bastida experiment provides information about the high-field part of the V ( E )curve. Data in the NDM region near E , are not available by this technique; neither is V , [ = V ( E , ) ] . To obtain this crucial point another experiment must be done. As we shall shortly see, the conditions for the nucleation and propagation of a high-field domain from cathode to anode occur at a critical current density below NoeV,. The boundary conditions required for domain nucleation do not allow V , to be reached in the bulk. In order to initiate an instability at V , (which will not be of the form of a recycling domain that moves from cathode to anode), low-field boundary conditions must be obtained. These can be assured by geometrically shaping the sample to remove the
-
328
M. P. SHAW ET A L .
active region from the influence of the contacts (20).Once this is done, the voltage profile across the sample length is taken barely below the bias required to produce the instability. This gives us E,. A knowledge of N o then provides us with V, once J , is measured. Before closing this section it is worthwhile to reemphasize the question: How do we obtain a freely traveling domain moving from cathode to anode, in order to do the Bastida experiment? Clearly the circuit must be such that its reactive components do not modulate the domain as it propagates. Furthermore, the boundary conditions must be such as to provide us with a dipole domain that will propagate uniformly through the sample. We approach this vital question in Section 11, and its answer forms the main theme of our review.
D. Summary In this introductory section we briefly outlined the major historical points involved with the discovery of the Gunn-Hilsum effect. [More detailed treatments are cited in Conwell ( 4 3 , Bulman et al. (46), and Bosch and Engelmann (47).]We next sketched the present situation with regard to our knowledge of the V ( E )curves for n-GaAs and n-InP, what experiments can be done to measure them, and how they can be calculated. Simple arguments that are helpful in understanding the behavior of an NDM semiconductor were then outlined. We showed, solely from the Poisson and continuity equations, that inhomogeneous field profiles are expected to form and discussed some of the details of the traveling-dipole domain solution, one specific solution of the problem. We now turn to two other vital features of the problem: the boundary conditions and the circuit. Before doing so, however, it is important to point out that the conclusions we have reached in this section and the results we discuss subsequently are much more general than implied. It is clear that NDM produces a negative differential conductivity (NDC), but NDC can also occur by other means, such as field-induced trapping or a field-induced release of trapped carriers of sign opposite to that of the drifting carriers. (Mixed NDC from intervalley transfer and field-dependent trapping ( 6 5 , 6 6 ) has also been investigated.) A prime contributor to the NDC field has been Boer (67), who first appreciated the significance of the boundary conditions in understanding the problem (68, 69). In fact, Boer made the first observations of stationary and moving-layer-like field inhomogenities in the NDC semiconductor CdS (70, 71 ). The behavior of NDC and NDM semiconductors is now known to be qualitatively very much the same (68, 72).
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11. THE NDM ELEMENT’S ENVIRONMENT;
CIRCUITS A N D BOUNDARIES A . Introduction
An NDM element in a circuit will exhibit a variety of instabilities, some involving the resonant response of controlled or spurious reactive circuit elements (19). Since these circuit oscillations are of fundamental importance in understanding the complete response of the NDM element to a specific excitation, it is vital that the important reactive elements be identified. Furthermore, the role of the contacts (boundaries) to the NDM element will also play a major role in determining the complete response (20,2/).Therefore, both the circuit and contact conditions must be specified before an analysis of the NDM element’s response can be undertaken. In this section we introduce those aspects of the contact and circuit problem that are required to solve the problem. The resulting equations are summarized and discussed in Section 111.
B. The LocuI Environment To understand the electrical behavior of an NDM element we must first properly represent the NDM element and its local environment. With regard to its environment, which consists of the leads, contacts, and support components, we note that: (1) the attachment of metallic leads to the NDM element introduces a lead resistance R l and lead inductance L,; (2) the contact regions themselves most often produce a nonlinear resistance, which we label R , , and also impose specific electric field conditions at the interface of the NDM material; (3) supporting, mounting, or holding the NDM element in any way introduces package capacitance C , and package inductance L, ; (4) an external voltage source (we consider only DC sources) will contain its own internal resistance R i . These contributions are shown in Fig. 9 in a lumped-element approximation of the circuit containing the NDM element, which we have represented as a block of material. Also shown is a load resistor R L that may represent the actual load in the circuit. In Fig. 10 the NDM element is represented as a nonlinear resistor, with a current-voltage relation I,( 4) in parallel with an intrinsic capacitor C o , together in series with an intrinsic inductor L o . C o and L o represent
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M. P. SHAW ET A L .
FIG.9. Lumped-element approximation of a circuit containing an NDM element.
the effects of the current through and voltage across the NDM element. I c ( @ is an analytic summary of the space charge-dependent currentvoltage relation of the NDM element. Its form is dependent principally on three things: (1) the velocity-electric field relation V ( E ) ,which, if the space charge distribution were uniform, would scale I,( 4); (2) the nonlinear contact resistance R , , which can reach appreciable values under sufficiently high bias and thus dominate the I,(+) relation [metal-to-metal contacts generally result in a low interfacial field and small values of R, (73)]; (3) the background doping profile, which may contain significant variations, sustain relatively stationary nonuniform pockets of space charge, and hence dominate I,( 4).
The velocity-electric field relation used in the calculation of I c ( + ) is implicitly space dependent insofar as the scattering parameters are dependent on the background doping profile. It is implicitly time dependent insofar as intervalley transfer rates are time dependent. For most of the calculations discussed in this review, only moderate variations in the
GUNN-HILSUM EFFECT ELECTRONICS
33 1
FIG.10. Representation of the NDM element as a nonlinear resistor with an intrinsic parallel capacitor and series inductor.
background doping profile are considered; the implicit spatial dependence is ignored. Furthermore, for a broad range of frequencies well below 10 GHz there is evidence that V ( E )is approximately frequency independent (74). Accordingly, we assume in our calculations that V ( E )is also time independent at low microwave frequencies. V ( E )is dependent on position and time only through the spatial and temporal dependence of the electric field profile. With regard to the vital role of the contact resistance R,, one of its most important features is to produce electric fields in the boundary regions. These effects can be represented by models where values of the electric field, carrier concentration, or mobility at the boundaries of the NDM element are specified. This point is discussed further in Section I1,D. Before proceeding, it is important to point out that when timedependent oscillatory phenomena are examined we may expect I,( +) to be frequency dependent; the current is then expected to be a multiplevalued function of voltage. These dependencies occur even though the explicit time dependence of V ( E ) is ignored. They arise because the mobile space charge distribution undergoes changes throughout an oscillatory cycle. While the multivaluedness of the current is accounted for in the numerical calculations (19) and in one specific construction (Section 111; the two-subelement model), it is ignored in our analytical discussions. There is ample justification for ignoring the multivaluedness of the current when the circuit dominates the oscillation (a concept developed below). It lies in the fact that the principal oscillatory characteristics of a steadystate circuit-controlled oscillation can be adequately approximated after space charge effects have been included by a single-valued currentvoltage relation.
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C . The Circuit
The lumped-element approximation for the sample and local environment is shown in Fig. 1 1, where R = R L + Ri+ R1, Li= L, + Lo(neglecting mutual coupling), and L = L l . We immediately see that there are four reactive components in the circuit; hence the differential equation governing the transient circuit response will be of fourth order. A fourth-order differential equation is cumbersome to handle even for a linear lc(#). The presence of a nonlinear Zc(#) containing a region of NDM makes the analytical task quite formidable.' In the Gunn-Hilsum effect problem, however, we have conditions available that allow us to take the approximation one step further. As we shall see both theoretically and experimentally, when a bias-induced transition is made from a prethreshold to postthreshold (domain) state there can be large relative changes in voltage across the sample compared to the relative changes in transport current. We then expect large displacement currents during transients and small inductive voltages. It is most often the case that for a change in voltage A 4 across C, that produces a change in transport current Ale(+), the condition LiAlJ#)/AT G A# holds. Under these conditions Li can be ig-
FIG.11. Lumped-element approximation for the NDM element and local environment as obtained from Figs. 9 and 10.
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333
FIG.12. Lumped-element approximationfor NDMglement and local environment neglecting intrinsic inductance.
nored, and the circuit then reduces to the second-order circuit shown in Fig. 12. Here C = C, + Co.Note that although L, dIc(4)/dTis small, L dI/dT, where I is the total current (sum of transport and displacement), is not in general small compared to IR and cannot be neglected. The circuit equations €or Fig. 12 are from Kirchhoff s voltage and current laws: dI
4~ = IR + L +4 dT
These two equations can be manipulated to produce the two governing differential equations of the problem:
334
M. P. SHAW ET A L .
where Z o / R ois a damping parameter, R o is the low-field resistance of the NDM element, Z o = (L/C)1’2,and T‘ = T/(LC)1’2.As shown in Section 111, the response of +(T’) is determined to a great extent by Z o / R o .Large values of Z o / R ooften produce well-defined circuit-controlled relaxation oscillations (19). Small values of Z o / R o often produce near-sinusoids, which in many cases lead to domain-dominated solutions with dampedcircuit ringing. It is also of interest to consider the alternative situation where Zc(4)is such that there are relatively large changes in transport current and relatively small changes in voltage during bias-induced transitions. Such effects occur in “S-shaped” NDR elements (75-77). Here L , plays a major role while L (because a large R is often included) and C o(because a small d#/dT occurs) can be neglected to first order. The circuit under analysis becomes the dual of the circuit of Fig. 12. Thus, the circuit theory of an N-shaped NDR element outlined in Section 111 transforms directly to the circuit theory of an S-shaped NDR element discussed in Shaw et al. (77). That is, the N-shaped NDR element in its “primary” circuit (Fig. 12) is the dual analog of the S-shaped NDR element in its primary circuit (in Fig. 11 short out L and remove Co).
D. The Boundaries Equations (30)-(33) contain the vital transport current-voltage relationship ZJ4). As we have already mentioned, for a relatively uniformly doped sample this characteristic frequency-dependent curve is primarily determined partly by the V ( E ) curve and partly by the boundary conditions imposed by the contacts. Assuming that V ( E ) can be predicted and/or measured as a function of frequency, how do we determine the boundary conditions? To answer this question we must address one of the most difficult problems in all of solid-state physics: the metalsemiconductor interface problem. In fact, in our case we must go one step deeper: the alloyed (heat treated) metal -semiconductor interface in which the transition region is diffused. To begin with, we can ask: What are the most important phenomenological features of the metal-semiconductor interface? How do they depend on the characteristics of the semiconductor and the preparation of the surface? Clearly, considerable care must be given to preparing the semiconductor surface before the metal is sputtered or evaporated. If not, reproducible interface properties will not be observed. We therefore assume that the surfaces are perfectly clean and free from residual and avoidable oxide layers. From studies of interfacial potential barriers for a large number of clean semiconductors and insulators, two rough empirical rules have emerged (78).
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335
The first rule involves the variation of a,, the interfacial potential, with a change in metal for a given highly ionically bonded semiconductor, and states: RB varies linearly with unity slope with the electronegativity (79) x M of the metal. This rule is consistent with an expression (80)for R, given by R, = R, - xs, where Rw is the work function of the metalvacuum interface and xs is the electron affinity of the semiconductor. Rw is, apart from a constant, approximately equal to xM. (a, is the energy necessary to remove an electron from the top of the Fermi sea and place it into the vacuum. xs is the energy gained by taking an electron in the vacuum and moving it to the bottom of the semiconductor’s conduction band.) For intermediate ionicities RBvaries linearly with x M but the slope falls midrange between zero and unity. The second empirical rule states that for highly covalently bonded semiconductors RBis found to be substantially independent of xM. In fact, it is found that R, = QE,@ I ) , where E, is the band gap. Assuming that the behavior of the more ionic materials can be understood via the above simple model, it remains to explain the behavior of the more covalently bonded semiconductors, which with few exceptions follow the second rule. [A discussion of the experimental determination of interfacial potentials (barrier heights) can be found in the text by Milnes and Feucht (82).] Since in our review we are concerned primarily with the covalently bonded semiconductors GaAs and InP, this problem is of fundamental importance to us. Much intense theoretical effort has gone into this endeavor (83-90), but to date it appears that a complete theory explaining the second empirical rule does not exist. Mead’s explanation (91) of the result is that, when metallized, covalently bonded semiconductors have a large density of surface states centered about 3E, below the bottom of the conduction band. This pins the Fermi level of the metal (any metal) about one-third of the way up the band gap from the valence band maximum. Since the Gunn-Hilsum effect relies on the NDM properties of the bulk V ( E )curve, it is required that we achieve high fields in the bulk. The contact to the active region must therefore be of low resistance. Since, as discussed above, essentially all metals will produce a substantial a, when in intimate contact with GaAs or InP, and this RBcan lead to a large resistance at the interface, then means must be found to reduce the interfacial resistance. Certainly, diffusion of a heavily doped n+ region into the NDM element, with subsequent evaporation of the metal onto the n+ region, will lower the contact resistance and remove the active region (n) from the influence of the high fields at the contacts. (Such a procedure is important in the device aspects of the Gunn-Hilsum effect discussed in Section IV.) Other techniques for lowering the contact resistance (82) are evaporation of the metal directly onto the semiconductor and alloying (heat-treating and aging), liquid regrowth, plating, and thermal compression bonding.
336
M. P. SHAW ET A L .
All these techniques can be used to produce a low resistance, or what is often referred to as an ohmic, contact (82).By ohmic we mean in principle that the contact has a linear current-voltage relation for both directions of applied bias. In practice, ohmic had also come to mean that the contact resistance R , is small, much smaller than the resistance of the bulk, R b . However, a prediction of the behavior of an NDM element requires knowledge of the electric fields at the boundaries of the active region, and these can be and often are substantial even though the R , Q R b condition is satisfied. Ohmic contacts can therefore have a profound influence on the behavior of the bulk. Furthermore, the boundary fields associated with alloyed contacts most often are in the direction dictated by a simple . alloyed metal-semiconductor conSchottky barrier analysis. ( 9 1 ~ )The tact is therefore best described as a modified Schottky barrier (92), where the peak field and depletion layer width are both reduced from their usual unannealed values. The current-voltage characteristics are no longer those of the classic Schottky barrier ( 9 1 ~ ) . An assortment of equilibrium electric field distributions that could exist for a variety of low-resistance contacts at the side of the active region that is the cathode when bias is applied are shown in Fig. 13 (these distributions are obtained by solving the Poisson equation including both electrons and holes). Note in Fig. 13 that when a bulk field Eb is applied, E(X)increases positively downward. Thus, Fig. 13b, c signifies accumulation layers of charge at the cathode contact to the active n region, whereas Fig. 13a, d signifies depletion layers of charge at the contact (from consideration of the Poisson equation). In the remainder of the text we invert the fields so that an increase in bulk field is represented by a positive upwards increase in E ( X ) . The easiest low-resistance contact to form by direct alloying is that of Fig. 13a. As we see in Section IV, one of the most desirable contacts for optimum efficiency of operation of a Gunn-Hilsum effect device is that of Fig. 13c. Unfortunately, such contacts are quite difficult to achieve in practice and it is often the case that the contact of Fig. 13d results. Fabrication procedures for these types of contacts are discussed in Section IV. To our knowledge, it is a rare event for a contact of the type shown in Fig. 13c to result by direct alloying, although it is commonly believed that such a contact can readily be made. (Perhaps studies of the Gunn-Hilsum effect have reduced this faith in recent years.) As shown in Section 111, the Gunn-Hilsum effect is also a useful tool in that the manifestation of the instability can also elucidate the field configuration at the low-resistance contacts, which is often unattainable by conventional means due to the fact that R , Q Rb. Let us now ask the question: How do we represent and treat the
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GUNN-HILSUM EFFECT ELECTRONICS
-m--
( 0 )
( b ) -m--
-lf
n+-
n
t
-D 0
it " ._ L
-i
u
0
0
Distance, X
(c)
I
m ---
n-
( d ) -m-c---
nn+
n-
-.-0 al LL 0 L
low-resistance contacts analytically? (We can ignore the resistance R , , but not the boundary conditions imposed on the NDM element.) In practice, the active region of the NDM element will be a semiconductor with a carrier concentration of between 1 014and 10Iscarriers/cm3. The interface to the circuit to which it is connected is a metal with approximately loz1 carriers/cm3. The boundary of the active region is typically a region in which the doping level in the semiconductor increases sharply over some small distance or is an alloyed metal contact in which the whole chemical composition of the material changes sharply over a small distance (20). The problems of rigorously treating the low-resistance alloyed contact
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are formidable. In general, the spatial variations of the alloy constitutents are not known and even if they were known, the problem of characterizing the electrical properties of the interface would be quite difficult. Even if the junction could be characterized accurately, we would still have to consider hot electron effects within the junction in the presence of a current. In general, calculations of hot electron effects are made for regions of uniform field and carrier concentration, and these are not applicable to the boundary region. The problem of treating a boundary formed by a gradient in the carrier concentration within the semiconductor is more amenable to treatment. Here, if the spatial variations in doping density were known, then the spatial variations of electronic energy levels, electric field, carrier concentration, and chemical potential could be calculated. There exists again, however, the problem of treating hot-electron effects in regions of field and carrier concentration gradients. Calculations of this kind have been performed by Hasty, Stratton, and Jones (93) using the McCumber and Chynoweth model (42) to account for the hot-carrier effects. Their published results show no unusual effects at a smoothly graded boundary. Lebwobl and Price (94) have used Monte Carlo techniques to calculate hot-carrier effects in regions of spatial gradients and have modeled some simple boundary conditions. Since there are extreme difficulties of accurately controlling, experimentally determining, and theoretically calculating the conditions of an actual alloyed contact, it is important to seek a simple model for the boundary. One of the simplest approaches is to assume that we know the electric field at the boundary of a uniformly doped active region. Early models that accounted for the influence of the boundaries assumed carrier density notches (42, 93) or variations in mobility (95) adjacent to the uniform active region. A more complete description of the role of the contacts was given by Kroemer (96) who assumed a specific variation of the electric field as a function of the current through the boundary. He called the current-dependent electric field the “control characteristic.” Shaw, Solomon, and Grubin (20,68) proposed a model that was simpler than Kroemer’s and more amenable to performing a systematic variation. They employed a fixed electric field at the boundaries. In Kroemer’s model the important aspect of the control characteristic is the ‘ ‘cross-over point,” where the control characteristic crosses the bulk characteristic. In this model Kroemer assumes that the contact current -field characteristic can be separated out from the bulk current -field characteristic and the system is then treated exactly like a series pair of nonlinear resistors. This is a convenient and useful approach, quite similar to the treatment of n-n Ge-Si heterojunctions as
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GUNN-HILSUM EFFECT ELECTRONICS
double Schottky barriers (82). In Fig. 14 we show a typical control characteristic J , ( E ) crossing the bulk characteristic J , ( E ) at E,. Kroemer treated J J E ) curves of various shapes and successfully demonstrated several important features of the problem. Shaw, Solomon, and Grubin ( 2 0 , 2 / ) have achieved similar results with a fixed-cathode boundary field. In Kroemer’s terms, they used a control characteristic that is just a vertical line at E = E,. This is shown as curve JA(E) in Fig. 14. The crossover point occurs at E, (and J , ) . They found that (1) the most vital feature of the control characteristic is the value of the electric field at the cross-over point and (2) the behavior of a long NDM element is relatively insensitive to the current dependewe of the boundary electric field. The fixed E, model exhibits the same important features of the problem as does the control characteristics model and is much simpler to use. However, as we see in Section IV, a control characteristic model is often required in order to explain the behavior of short device-grade NDM elements. Other treatments of the contacts and boundary conditions have also appeared. Boer and Dohler (69) and Conwell (97) use the “field of directions” technique to demonstrate the existence of various field distributions determined by different boundary conditions. Boer and Dohler analyze the problem in terms of a fixed-cathode conductivity vcand Conwell via a fixed-cathode carrier concentration n , . The principal conclusions
0
EC
Electric Field
-
E
FIG.14. Representation of intersecting current density vs. electric-field relations. J[.(E) is the cathode control characteristic for constant-cathode field; J , ( E ) is the current density relation for the bulk; this parameter i s also called the neutral current density characteristic
JnW.
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M. P. SHAW ET AL.
obtained by the various techniques ae all similar. In Section I11 we develop the fixed-cathode boundary field model and obtain the currentvoltage characteristics and stability criteria directly, without making use of the field of directions technique. Boer and Dohler (69) and Conwell (97) provide lucid descriptions of the latter. 111. THE BEHAVIOR OF
AN
NDM ELEMENTIN
A CIRCUIT
A . Introduction
In order to present an overview of the subject matter we now offer a general discussion of the behavior of a long NDM element in the simplified circuit discussed in Section I1 (see Fig. 12). Included is a summary of analytical results and computer simulations, as well as relevant experimental findings. Experimental results for long samples are discussed in detail in Shaw et al. (20). (By “long” we mean a sample of sufficient length that when a domain is launched at the cathode it can grow to maturity and propagate freely before reaching the anode. Most bulk samples made for probing studies are long. Most microwave devices are short. They are discussed in Section IV.) -prethreshold current The quantities of interest considered include the density J as a function of the average electric field E (E = $ / l , where 1 is the sample length); the space- and time-dependent electric field within the active region E(X,T ) ; the time-dependent sample voltage +(T);the total current Z ( T ) and the conduction current density J,(T). In calculating these quantities we specify the circuit parameters and the electrical characteristics of the semiconductor. For the latter we include the velocityelectric field curve V ( E ) ,the diffusion coefficient-electric field relation D ( E ) ,the background doping profile, and the value of the cathode boundary field E , (20,22). We specify various values of E , in order to simulate the effects of a large class of metal-semiconductor contacts. Both the analytical and numerical solutions, and experimental investigations of NDM elements, show that a principal determinant of the nature of the current instability is the cathode boundary condition. Furthermore, the major types of instabilities associated with GaAs devices can be placed into three broad categories, each characterized by a range of values for E , . The classification scheme is illustrated in Fig. 15. If E , is in the shaded region [which contains most of the NDM range of the V ( E )curve] and the circuit is resistive, the instability generally appears as the classic cathode-to-anode transit time mode (I -3). For lower or higher E , , other
34 1
GUNN-HILSUM EFFECT ELECTRONICS
2.0
1.5
P . 5
PI
0
I
>
4-
1.0
x x
> L
c
b 0.5
5
10
15
E [kV/crn]
J(a
FIG. 15. The V ( E ) curve and the computer simulated current density (curves A-C) for various fixed values of the cathode boundary field E , . The NDM element is in the circuit shown in the inset. For curve A , E , = 0.0. The values for curves B, and B2 are indicated by the arrows. For curve C , E , = 24 kV/cm. The relevant NDM element parameters The constant low-field mobility is equal to 6860 cm*/V-sec. are I = 100 pm, N o = 101s/cm9. The right and left-hand ordinates are related by J , = N , e V ( E ) , and V , = 0.86 x lo' cm/sec [from Shaw er a / . (2011. (Reprinted with permission, IBM Corporation.)
types of behavior occur. The cathode-to-anode transit time mode results only for a specific range of boundary fields and is only one of several possible modes of behavior. The shape of the J ( E ) curve also depends strongly on the value of E,; Fig. 15 shows typical results for E , in the low (curve A), intermediate (curves B, and Bz),and high (curve C) ranges. The details associated with Fig. 15 are discussed below. [In Fig. 15 the simulations are for n-GaAs with a background doping density N o = 101s/cm3,active region length of 100 pm, and random doping fluctuations given by A N o ( X ) = A N o p , where p is a random sequence of numbers between + I and - 1. The distribution for p is approximately Gaussian with an rms of 0.30. For Fig. 15 AN,, = 0. lNo.]
M. P. SHAW ET A L .
342
B . Summury of Results 1. Low E ,
Simulated GaAs NDM elements having mobilities near 6000 cm2/Vsec with E, in the low-field region, 0 IE, 5 4 kV/cm, exhibit linear J(E)curves until E = 3.2 kV/cm, when a current instability occurs. The manifestation of the instability is sensitive to the material and circuit parameters. For circuits that are primarily resistive ( Z o / R oI2; see Section I1 for a definition of Z o / R o )and E = 3.2 kV/cm,
(1) high-field dipole domains grow from active-region nucleation sites, or (2) single dipoles form near the cathode boundary, or (3) both of these events occur simultaneously. In any case multiple or single dipoles propagate toward the anode. In the former case, as each domain arrives at the anode it disappears and its voltage is redistributed among the remaining domains. Eventually a single domain is left; when it reaches the anode it drains and the system recycles (Fig. 16a). The instability produces large “peak to valley” ratio transit time current oscillations at a frequency determined by the position of a major nucleation site, which is not necessarily at or near the cathode. As the bias is increased, however, the character of the solution changes from a transit time phenomenon to a stationary time-independent field configuration where the last domain remains undrained at the anode (20,21), as shown in Fig. 16b for E , = 1.OX,. Here the current density saturates at a value close to but below N o e V , . The excess voltage due to further increases in bias appears across the anode region, with negligible changes in current. The stationary high-anode field can produce impact ionization effects and current run-away, and may be responsible for the results reported by Liu (98) and Copeland (99). The high-anode-field solution, where E, is below E,, has been shown to be an allowable solution by employing a “field of directions” (69, 97) analysis, but criteria for the stability of this solution have still not been completely established. (The stationary high-anode-field solution has been the subject of extensive study during the past few years, and is related to Shockley’s positive conductance theorem (100, 101). A variety of theories and numerical calculations abound in the literature, some of which are given by Thim and other authors (102 -1 10.) For values of Z , / R o > 2, circuit effects begin to play an important role (19). Here, for a range of values of Z o / R o ,the circuit is capable of affecting the growth and decay of domains. An example is shown in Fig. 17
GUNN-HILSUM EFFECT ELECTRONICS
343
FIG.16. Numerical simulation of the time-dependent behavior of a cathode-nucleated domain for an NDM element in the circuit of Fig. IS. E ( X , 73 is displayed at successive instants of time where the time between successive vertical displays is 167 sec. t is the low-field dielectric relaxation time [see Eq. ( 1 I)]. The NDM element parameters are I = 100 pm, N o = 5 X 101*/cm3,E , = 1.05Ep. For the circuit, R = R , , the low-field resistance of the NDM element. (a) t$B/l = 1.95Ep/1and the oscillation frequency is 0.7 GHz. (b) c $ ~ = 2.20ED/1, the dipole layer sticks at the anode, and the oscillation ceases [from Grubin el a / . ( 2 1 ) ] .(Reprinted with permission.)
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M. P. SHAW ET AL.
FIG. 17. Numerical simulation of an NDM element in the circuit shown in the inset. Displayed are current through the load vs. voltage across the NDM element, I(,$); the time-dependent voltage, ,$(T);I J T ) = I ( T ) - Cod,$/dTvs. ,$(T); and the neutral characteristic scaled to current and voltage: I, = N,eV(E)S, where S is the cross-sectional area of the NDM element. Also displayed is E ( X , r ) at two instants of time with arrows indicating the direction of field evolution. The NDM element parameters are E , = 0.0, N o = 5 x 1014/cm9, ANo = O . l N o , 1 = 100 pn, S = 5 x lo4 cms, Ro = 36.4 n, C o = 0.49 X 10-lSf, and 7 = 1.78 x lo-" sec. The circuit parameters are R = O.lRo. C , = 0.0, +B = 2.OEP1, and Zo/Ro = 3.0. The oscillation frequency is 25.2 GHz [from Solomon er d.(19)]. (Reprinted with permission.)
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GUNN-HILSUM EFFECT ELECTRONICS
for Z o / R o = 3 and N o = 0.5 x 1015~ m - Here ~ . the circuit pushes the current sufficiently above its threshold value so that the electric field throughout the entire active region enters the NDM region. In this case no isolated domains form and the sample behaves much like a tunnel diode relaxation oscillator (111). At higher values of Z o / R othe circuit is less able to drive the current sufficiently above its threshold value to suppress the formation of individual domains (19). Domains are formed but may be subsequently quenched as the circuit swings the voltage below the domain sustaining value. This is illustrated in Fig. 18 for Z$R0 = 12 and ANo = 0.4No. Although the electric field distribution is nonuniform, the electrical characteristics remain like those of a tunnel diode relaxation oscillator. Indeed, NDM elements with long active regions generally oscillate in a relaxation mode when the circuit is controlling the oscillation. Various names that have been given to describe different modes are closely related to the suspected distribution of internal space charge during an oscillatory cycle; see reference 19 for a description of the LSA (14, 15) relaxation mode, and quenched multiple-dipole (18) relaxation mode. Note also that if the circuit ringing time is too slow, domains can
Voltage
[$l
I
Voltage
141
w Distance
[XI
FIG.18. As in Fig. 17, but AN, = 0 . 4 N o ,Zo/Ro= 12, and the oscillation frequency is 3.33 GHz [from Solomon et a / . ( 1 9 ) ] . (Reprinted with permission.)
346
M. P. SHAW ET A L .
reach the anode before they can be quenched, giving rise to other modes of behavior such as the domain-inhibited mode. A summary of the various modes of oscillation possible for lowboundary-field, long-active-region NDM elements is shown in tabular form in Fig. 19, where the modes are located with respect to the parameter ( Z o / R 0 ) - land the ratio of NDM element capacity C oto total capacity
c = co + c,.
2. Intermediate E ,
For E, in the intermediate region (4 kV/cm 5 E , 5 15 kV/cm) of the V ( E ) curve (shaded region of Fig. 15) domains nucleate at the cathode. For Z o / R o< 2 the circuit effects are small and domains propagate to the anode, drain, and recycle. This is the classic cathode-to-anode transit time mode (3).Here the J ( E ) curve is linear at low bias. At biases close to threshold, J(@ departs from the linear due to the appearance of an appreciable voltage drop across the depleted region adjacent to the cathode (3) (see curves B, and B2of Fig. 15). At threshold the current switches along the load line and the domain propagates at a velocity slightly greater than the saturated drift velocity V , . The threshold current density Jthis controlled by E, and occurs very near the value Jth = N o e V ( E , ) .The instability is initiated before the active region (bulk) field downstream from the cathode enters into the NDM region. This field can vary at the instability threshold between 1.4 and 4.2 kV/cm for mobilities between 4000 and 7000 cm2/V-sec. Different current peak-to-valley ratios are due to different values of E , in the NDM region (e.g., curves B, and B,, having E , = 6.0 and 8.5 kV/cm, respectively, in Fig. 15). Here, as for low E , elements, when Z o / R o> 2 circuit effects become important. It is possible for a cathode nucleated domain to be quenched before reaching the anode. The effect of the intermediate boundary field, however, reduces the amplitude of the current and voltage swings, as shown in Fig. 20. Furthermore, for long samples, the voltage often does not drop low enough to completely quench the domain, producing a cathode-to-anode transit time oscillation with superimposed circuit ringing. An example of such a case is shown in Fig. 21.
3 . High E , For high E , (E, z 15 kV/cm) J ( E ) becomes nonlinear at relatively low voltages (curve C, Fig. 15). Here, when E , is in the range of weak NDM ( E , = 15-17 kV/cm) part of the dipole layer at the cathode detaches, moves a distance down the sample that is determined by the applied bias,
COIC
FIG. 19. Modes of oscillation of long NDM elements in a series L-parallel C circuit. Region I is domain dominated because the minimum voltage is not sufficiently low to quench the space charge nonuniformities. Region II is circuit controlled. Relaxation oscillations occur either when E ( X , r ) is relatively uniform during the complete cycle (LSA relaxation oscillations) or when E ( X , T ) is uniform only during that part of the cycle when nonuniformities are quenched (quenched single or multiple dipole relaxation oscillations). Region 111 is domain dominated because domains reach the anode before the circuit has had a chance to quench them [from Solomon et a / . (19)].(Reprinted with permission.)
M. P. SHAW ET A L .
348 (a)
E, = 0
bl
(C
Voltage
1
E, = 1.8 Ep
E,
= 2.5
Ep
Time [TI
FIG.20. Computer-generated I ( + ) and I ( T ) for an NDM element in the circuit of Fig. 17. Calculations are for three different values of E, (as indicated); (a) E , = 0, (b) E , = 1.8E,, (c) E , = 2.5E,; otherwise all NDM element parameters are as for Fig. 17. For the circuit, Z , / R , = 12 and dB = 2.OEP1. In all cases the oscillation is circuit controlled and at a frequency higher than the nominal transit time frequency V,/ 1. A Dipole layer launched at the cathode is quenched in transit toward the anodeifrom Solomon et a / . ( 1 9 ) ].(Reprinted with permission.)
and then disappears, usually before reaching the anode for long enough samples (see Fig. 22). The current oscillations are generally small amplitude near-sinusoids where the frequency decreases with increasing bias because the domain moves a greater distance down the sample as the bias increases. This behavior is often seen in n-Ge since it has a small, shallow NDM region (36, 112). For higher E , ( E , L 17 kV/cm), where V ( E )becomes almost flat, the partial domain detachment ceases and the field profile remains stationary. Here the current asymptotically approaches a saturated value J , = N,,eV,. High E, samples can also sustain weak oscillations in a reactive circuit.
GUNN-HILSUM EFFECT ELECTRONICS
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FIG.21. Computer-generated I ( T ) for an NDM element in the circuit of Fig. 17. The NDM element paramters are as for Fig. 20c. For the circuit Z o / R o = 9 and +B = 2.OE,1 [from Solomon ef d.(/9)]. (Reprinted with permission.)
C . Understanding the Threshold Condition
Further examination of the above results, including a stability analysis, is given in Grubin et al. (21). In the present review we offer a simple discussion that provides insight into the threshold conditions. We assume that the NDM element has a uniform doping profile N , ( X ) = N o and set
0 Distance
[XI
FIG.22. As in Fig. 16 with E , = 4.OE, and +B = 1.5E,I. The oscillation frequency is 3.5 GHz and is substantially above the nominal transit time frequency for this element. [from Grubin er a / . (21). Reprinted with permission.]
M. P. SHAW ET A L .
350
the diffusion coefficient D(E) to zero. For a stationary solution aE/dT = 0, and the transport current density equation is dE
J = NoeV(E) + c V ( E ) z
(34)
which we write as
Poisson’s equation is
Current continuity requires that J be independent of X . To demonstrate how the boundary field controls the form and stability of the space charge layers, we consider how both low and intermediate boundary fields influence the NDM element. We also consider the physically relevant situation where the field downstream from the cathode, the bulk field E b , is relatively uniform and sustains values less than the threshold field for negative differential mobility E , . Thus, within the bulk dE/dX = 0 ,
Eb = J/(Noep,,)< E,
where p,, is the low-field mobility of the NDM element. We first consider the low-boundary-field case where E, is within the positive mobility region and J < NoeV(E,). With reference to Eq. (35) it is seen that dE/dX < 0 and the value of E ( X ) a small distance downstream from the cathode is less than E,. V [ E ( X ) ]is also less than V(E,). From Eq. (35) we see that a diminishing value of V ( E ) results in an increasing value of dE/dX, the latter gradually approaching zero. Thus, away from the cathode E ( X ) approaches Eb,as illustrated in Fig. 23a. For the same value of E,, increasing J to the value NoeV(E,)results in charge neutrality everywhere, including the cathode region (see Fig. 23b). Charge neutrality at the cathode plays a prominent role in the operation of NDM elements and we highlight it with the designation CBCN (cathode boundary charge neutrality). Further increases in current result in the formation of an accumulation layer (dE/dX > 0) at the cathode, as shown in Fig. 23c. The accumulation layer profile is stable so long as the bulk field is less than the NDM threshold field. However, once Eb exceeds EDa bulk originated instability occurs, i.e. , a propagating accumulation layer for uniform No or bulk nucleated single or multiple domains for nonuniform
NO.
We next consider Fig. 24, which illustrates the case where E , is fixed at a value within the NDM region. For the case where J < NoeV(E,) we
35 I
GUNN-HILSUM EFFECT ELECTRONICS
E
E,
(C
I
---I-
L)-------- JnlEc)
-T--
-- --- - - - ---------
Eb
X
J > JnlEcl
FIG. 23. Neutral current density J , = N , e V ( E ) vs. electric field, and electric field vs. distance for three different values of current density. E , is fixed in the ohmic region.
again have that dE/dX < 0 at the cathode. Unlike the situation considered above, however, where E, was within the positive-mobility region, here downstream from the cathode the electron velocity initially increases, and so dE/dX diminishes away from the cathode (see Fig. 24a). The smallest value of dE/dX occurs at E ( X ) = E,, beyond which dE/dX increases in value and approaches zero. Hence, sufficiently far down-
M. P. SHAW ET A L .
352
(a 1
E
E
Distance X
k
I
J J = J,(E,)
FIG.24. As in Fig. 23, with E, within the NDM region. [ from Solomon ef a / . (to)]. (Reprinted with permission.)
stream from the cathode E(X)approaches Eb.As J is increased and approaches N,eV(E,), dE/dX tends toward zero at X = 0, as well as within the bulk. E ( X ) at X = 0 tends toward E , while E ( X ) downstream from the cathode tends toward E , . We see that an approximate condition of charge neutrality exists at the cathode as well as within the bulk, with the transition from E, to Eb occurring via a depletion layer (see Fig. 24b). In the ab-
GUNN-HILSUM EFFECT ELECTRONICS
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sence of an instability, slight increases in current would initially result in the depletion layer moving out toward the anode, ultimately filling the entire sample. The current-voltage curves would show extreme current saturation, but, as discussed in Section I, where the length of the NDM region exceeds a critical value, it becomes unstable. For l O I 5 cm4 doping, a NDM region width of several microns would result in an instability. It is expected then that well before the depletion layer reaches the anode the NDM element becomes unstable, with the instability occurring at values of J approximately equal to N,eV(E,). For J < NoeV(E,) the NDM region length is less than the critical value and the NDM element is stable. The current density at threshold is Jth = N o e V ( E , ) ,
E, > E,
(37)
The implication, of course, is that threshold current densities are controlled by the cathode condition. This is borne out by experiment and is discussed in references 20 and 21. For cathode fields less than E , , instabilities are governed by the bulk properties and the threshold current density is given by Jth = NoeV(E,)
=J,,
E, < E ,
(38)
Further insight into the initiation of the instability is obtained by examining the stability of the depletion layer profiles of Figs. 23 and 24 against a small perturbation in the form of a local increase in electric field about the point X,, as shown in Fig. 25. There is a relative accumulation of charge from XIto X, and a relative depletion from X o to X2.For the case of depletion layers wholly within the positive-mobility region (Fig. 25a) the velocity within AX is greatest at X,. As electrons enter the perturbed region they begin to speed up. There is no tendency for electrons to accumulate. As the electrons pass X, they tend to slow down and occupy the perturbed depleted region. The local disturbance spreads and disappears. The original depletion layer is stable. For perturbations within the NDM region the velocity is smallest at X,. Electrons entering the perturbed region slow down and tend to accumulate. The perturbation grows, increasing the local electric field at the peak of the disturbance and resulting in a greater depletion of carriers on the downstream portion of the disturbance. The perturbation grows as it propagates. The growing disturbance will generally propagate into the positive-mobility region, where it will decay. However, if the NDM region is long enough, the growing disturbance can absorb a sufficient amount of voltage in transit to cause a significant drop in current throughout the circuit. Once the current drops, the cathode is no longer able to supply the growing disturbance with enough carriers for continued
M. P. SHAW ET
354
Distance
AL.
-
FIG.25. Illustration (a) of a damped perturbation in a positive mobility region and (b) of a growing perturbation in a negative differential mobility region.
growth and the disturbance “detaches” from the cathode, as illustrated in Fig. 26. This is the initiation of the cathode-to-anode transit time mode. The stability of the transiting domain is now governed by large-signal concepts. The question as to how long the NDM region must be in order for an instability to occur is discussed in detail in reference 2 I , but it is of the order of magnitude of the McCumber and Chynoweth criterion (42) [see Section I, Eq. (12) and its discussion]. With reference to the above discussion where we have specified values for E,, we again ask: How important is a precise description of the boundary region in determining the behavior of an NDM element in an external circuit? The answer is, of course, ultimately empirical and depends upon the suspected time-dependent variation of E,. But if E , is only weakly time dependent, which experiment and computer simulation suggest is true for a large class of GaAs samples, then the most important
355
GUNN-HILSUM EFFECT ELECTRONICS
P Distance ( X I
FIG.26. As in Fig. 16. Successive time between displays is 4 S sec.
influence of the cathode boundary lines in its control of the threshold current. We require knowledge of which region the boundary field is in and the relative values of J and N o e V ( E , ) .These boundary effects are satisfactorily accounted for by a model that specifies only the value of E at X = 0. This was the approach used in Solomon et ul. (19 -21). The approach produces a simple single-parameter model that has been capable of explaining a broad range of experimentally observed stable and unstable field configurations. It should be pointed out at this point that recent experiments with a certain class of high-efficiency InP oscillators have not been satisfactorily explained using the fixed-cathode boundary field model (113). It appears that for this specific class of devices the metal-semiconductor contact region may be producing cathode fields that exhibit a strong time dependence. This situation is discussed in Section IV.
D . Understanding the Oscillatory Behavior of N D M Elements In this section we consider two simplified cases that offer insight into the role of the circuit in the oscillatory behavior of NDM elements. The circuit examined is shown in Fig. 12 and the relevant circuit equations are
356
M. P. SHAW ET A L .
equations (32) and (33), repeated here for convenience:
Even though the use of this simple circuit is an approximation to the more general circuit of Fig. 1 1 and results in the reduction of a fourth-order differential equation to second order, the analytical task of solving the latter remains formidable because the conduction current Zc(+), is not only nonlinear, but is also a multiple-valued function of 4. I,(@ generally depends on the distribution of the electric field within the NDM element. For analytical simplicity we can reduce the complexity and still grasp the important circuit and space charge effects by considering two examples. First we treat the case whrre I, is a single-valued function of 4. This assumption corresponds to the case where the field in the NDM element is uniform. The analysis is equivalent to that used for tunnel diodes (I I I ) and provides a good approximation for those situations where the space-charge-dependent diode is undergoing relaxation oscillations. Second, increasing the complexity of the model, we consider the situation of two serially connected NDM elements with slightly different singlevalued Zc(t$) relations. This case corresponds to the situation where the electric fields over two halves, of an NDM element are uniform but different, and serves to demonstrate the effect of nonuniform fields. With regard to the uniform field case we note that extensive tunnel diode oscillation studies may be found elsewhere ( I l l ) . The discussion below is specifically tailored to our needs and begins with a discussion of Eq. (32). Equation (32) is an oscillator equation with a nonlinear damping term. The bracketed part of the damping term is of the order of unity and the strength of the damping term is determined by Z o / R o . For small Z o / R o , the damping term is a small perturbation and the solutions for +(T') are nearly sinusoidal. This is shown in Fig. 27a for Z o / R o = 3, where we plot r$(T'), Z ( T ' ) , and Z(r$), obtained numerically. The current oscillations are nearly sinusoidal while the voltage oscillations show evidence of the nonlinear damping. For large Z o / R othe damping term is important and the solutions become well-defined relaxation oscillations. Figure 27b illustrates a case for Z o / R o= 12. Here the current oscillations are almost sawtooth and the voltage oscillations exhibit sharp spikes. Circuit oscillations for a given value of Z o / R oand 4Bhave similar shapes and amplitudes and differ only in frequency due to the time scale T' = T/(LC)'12. Analytic solutions of Eq. (32) are obtained using the three-piece linear
GUNN-HILSUM EFFECT ELECTRONICS
357
FIG.27. Computer generated I ( T ) . $ ( T ) , and /(+) for an NDM element in the circuit of Fig. 17. NDM element parameters are as in Fig. 17, but E ( X , T ) is spatially uniform; i.e.. IJ+) is equal to N O e V ( $ / l ) S .For the circuit, +B = 2.OEP1.(a) Z o = 3 and the oscillation frequency is 26.0 GHz; (b) Z o / R o = 12 and the oscillation frequency is 3.2 GHz[from SOlomon er a / . ( 1 9 ) ].(Reprinted with permission.)
approximation for Ic(+) shown in Fig. 28. The problem is solved in the three regions 4 < r#+, 4D< 4 < c&, and 4" < 4, which correspond to dIJr@/d+ = l/Ro, - l / R n , and 0, respectively, where R , is the magnitude of the negative differential resistance. The equations and solutions are shown in Table I. The individual solutions are joined smoothly from one region to the next. The composition of the solution is evident in the relaxation oscillation shown in Fig. 27b. The voltage waveform begins with a slow exponential rise with time constant L / ( R o + R) and changes
358
M. P. SHAW ET AL.
I
I
1
9s
+P
9"
9
FIG.28. Three-piece linear approximation for Ic(,j,).
to a sharp spike in voltage when 4 = &. The time needed to reach & is bias dependent and if, for example, the minimum voltage is arbitrarily set equal to zero, the rise time to threshold is approximately given by
L A T = -Ro + R lo&
[
-
:(1 %)]' +
Increasing 4B,all other things being equal, AT decreases and the frequency of oscillation increases. Bias tuning is a characteristic of transferred electron oscillators, and this is one source of tuning. Above threshold the oscillation is composed of a fast exponential transit through the region of negative slope (& < 6 < &), with time constant R,C, followed by a damped sine wave for 4 < & and another exponential transit for &, <,4 < &. An exponential decay for 4 < & completes the cycle. As Fig. 27b indicates, the I ( + ) trajectory, obtained by eliminating T from I ( T ' ) and + ( T ' ) , has the appearance of a truncated ellipse. Since this lissajous-type figure provides an indication of the 'magnitudes of the cur-
359
GUNN-HILSUM EFFECT ELECTRONICS TABLE I
rent and voltage swings and their relation to the parameter Z , / R o , we consider it in some detail. A useful approximation of the I(4)trajectory D obtained by integrating Eq. (33) over the portion of the cycle when l C ( &= I s and then extending the solution until it crosses the "below threshold" section of I,( 4) (19). The integration yields
Z$I(4) - Isl"+ (4 - &J2 = k2
(39)
where we have also taken R / R , < 1. In Eq. (39) k is a constant. Extending the above equation until it crosses the positive-resistance portion of I,(@ determines the constant k2. At threshold 4 = 4p,I(&) = ZJ&) = Ip, so that the constant in the equation above may be evaluated to give
zE[z(4) - Is]z + (4 -
$BIZ
=
zi(lp - I s ) 2 + ( 4 p
-
48)'
(40)
3 60
M. P. SHAW ET AL.
Over its region of validity a plot of ZoI(+) vs. 4 generates a circle. A plot of I(4) vs. 4 generates a family of ellipses whose ratio of semimajor axis to semiminor axis is Z o .For Z o > 1 the amplitude of 4 is large while that of Z(4) is small. The reverse occurs for Z o < 1. In other words, large voltage swings occur at the expense of small current swings and vice versa. For 2, = 1 the amplitudes of both current and voltage are the same. Ellipse equation (40) shows that besides the parameters Z , and da,the circuit response is determined by the NDM element parameters I,, and I,. This point is illustrated in Fig. 29, where the circuit behavior is obZo/Ro = 12
I
Voltage
I
(d)
[+I
+,,
Zo/Ro = 3
+ (a')
fd')
FIG.29. As in Fig. 27, computer-generated I($,) for various I,($,) [from Solomon et al.
(19), with permission
1.
36 1
GUNN-HILSUM EFFECT ELECTRONICS
tained for four different I,(+) curves and two values of Z o / R o . Figure 29a-d shows relaxation oscillations with Z o / R o= 12 and Fig. 29a'-d' shows nearly sinusoidal oscillations with Z o / R o = 3. The I , ( ~ ) sin Fig. 29a-c and Fig. 29a'-c' differ only in the shape of the region of negative slope and we see that the I ( + ) trajectories for a given value of Z o / R oare almost congruent. Figure 29d has a higher saturation current I,. This has a substantial influence on I(+), reducing both the current and voltage amplitudes considerably. Equation (40) provides qualitative information about the relative values of the current and voltage amplitudes. However, as we shall see below, to examine the formation and quenching of field nonuniformities (domains) it is important to obtain a more careful determination of the maximum current IM and the minimum voltage 4, for a particular I ( + ) trajectory. These may be obtained by numerical calculations, as shown in Fig. (30) for one value of bias. Here 4, and IM are plotted as functions of Z o / R o . 4,,, varies from 0.64, for small Z o / R oto negative values for large Z o / R o . IM varies from I, for large Z o / R o to 1.81, for small values of Z o / R o . Generally, the parameters f M and 4,,, both increase with decreasing Z o / R o . In summary, the important results of the tunnel diode analysis are:
-
(1) the form of the oscillation (relaxation or sinusoidal) is determined
0.8
f
\
1.6
\ \
0.6
0.4
1.4
+m@p
IM/lP 0.2
1.2
1.o
0.8
I0 10
-0.2
FIG.30. Current maxima IM (---) and voltage minima & vs. Z o / R o .Here f, = J,(E,)S and = 4 1 . The NDM element parameters are as in Fig. 17. For the circuit a 1.0 resistor is included in series with the NDM element to assure numerical stability [from Solomon er d.(19)]. (Reprinted with permission.)
362
M. P. SHAW ET A L .
(2) the detailed shape of the oscillation is controlled by Z o / R oand the NDM element characteristics &, I,, and I,, and is relatively insensitive to the shape of the region of negative slope; (3) the parameters ZM and +,, both increase with decreasing Z o / R o .
What application do these tunnel diode results have to the actual NDM element behavior? First, the results are directly applicable to the case of uniform space charge. With reference to the mode diagram (Fig. 19), uniform space charge modes occur for values of Z o / R onear 2 and for small values of package capacitance (Co/C= 1). Within this region the values of I ( T ) , +(T), and Z(4) derived from the tunnel diode case are applicable. There are two other results that are even more important: First, the oscillatory behavior in the simple tunnel diode case is seen to be insensitive to the detailed shape of Z,(# and, as indicated in Eq. (39), depends upon I,, cb,, and Z,. It is also the case [this point is discussed further in Solomon et al. (19)] that as long as the circuit controls the formation and quenching of space charge nonuniformities (i.e., fully formed domains do not transit to the anode) then I,(+) will be approximately defined by a single-valued trajectory in the I(4) plane, whose values of Z, , , and I, will closely correspond to their respective values in the uniform-field case. The conclusion is that the oscillatory behavior predicted by the simple tunnel diode analysis applies reasonably well to all cases where the oscillatory behavior is dominated by the circuit and not by the domain transit. For long NDM elements this is the region bounded by the lines Z o / R o = 2 and T = T ( L C ) ' ' ~in Fig. 19. Second, the simple tunnel diode analysis can be used to define the competing limits of circuit and domain domination by providing an understanding of the formation and quenching of space charge nonuniformities. With regard to quenching, this process occurs when 4 becomes small and the NDM element returns to its positive mobility region. Values of +,,, the minimum value of 4, are smallest for large Z o / R o(see Fig. 30) and indicate that relaxation oscillations are more likely to quench space charge uniformities than near-sinusoidal oscillations. To treat the formation of space charge nonuniformities we examine what happens when an NDM element with a slightly nonuniform space charge distribution initially enters the NDM region (recall that those regions tending to have higher electric fields will enter the NDM region first). As we have seen [see Eq. (12)] such regions will be subject to a rapid exponential growth in field. Gross space charge nonuniformities occur when the fields in neighboring regions fail to enter the NDM region. Relatively uniform space charge distributions occur when the neighboring
+*
GUNN-HILSUM EFFECT ELECTRONICS
363
regions enter the NDM region. For the latter to happen, it is necessary that the conduction current through adjacent regions continue to rise after the instability threshold has been reached, a situation likely to occur if the total current rises after threshold. Thus, the higher the maximum current ZM, the less likely the formation of space charge nonuniformities. ZM is largest for small values of Z o / R oand smallest for large values of Z o / R o . We have, therefore, the following delicate situation. Small values of Z o / R o lead to initially uniform space charge distributions, but should residual space charge layers remain near the end of a cycle, they will be difficult to quench. Large values of Z o / R o lead to initially nonuniform space charge distributions, but residual space charge layers near the end of a cycle will readily be quenched. To make the above arguments more complete, we partially imitate the formation of space charge nonuniformities by considering two NDM elements in series. Each NDM element consists of a capacitor 2C0 in parallel with a tunnel diode. The tunnel diodes have slightly different ZC(4)curves, as shown in Fig. 31. For simplicity we assume that the package capacitor Cp = 0. [A more complete discussion of this model can be found in Solomon et ul. (19).] The questions we consider are whether or not there is a periodic oscillation of this series combination and whether one or both
I
-r FIG. 31. A series connection of two NDM subelements each containing a nonlinear resistor in parallel with a capacitor C = 2C,. Each nonlinear resistor has I J & ) = N,eV(+,/I)S. The background doping level N. is different in each subelement, whereas the V(E) relation is the same in each.
M. P. SHAW ET A L .
3 64
tunnel diodes pass above threshold. The case where only one diode passes through threshold corresponds to the formation of space charge nonuniformities (i.e., a domain). If a periodic oscillation results, this corresponds to a quenched-domain mode. If the oscillation ceases, resulting in one diode above threshold (on) and the other below (off), this corresponds to no quenching (i.e., transit time effects). The case where both diodes periodically pass through threshold corresponds to the uniform space charge case. The conditions that define the behavior of the two diodes are closely related to the conditions that control the behavior of oscillations in real NDM elements, and the understanding of the two diodes yields substantial insight into the behavior of real NDM elements. The equations relevant to a description of the behavior of the two diodes in Fig. 3 1 are developed below. The response of each diode is governed by
where n = 1, 2 and +n is the voltage across the nth diode, ID(+,)the displacement current in the nth diode, I(+) the total current through each diode, and
4=
(42) + 42 the total voltage across the combination. The rate of change of potential across each diode is proportional to I(4) - IC(&). Thus, for a given current l(#,the potential increases with time if I ( + ) > Ic(&) and decreases if the inequality is reversed. The behavior of the two serially connected diodes is constrained by Eq. (42), which when differentiated yields $1
+,
or #[ID(+l)
+ ID($2)]
ID(+)
(43)
The latter defines a displacement current for the serially connected system that is the average of the displacement currents in each diode. In addition, if Ic(& =
4 4 ) - Co d$J/dT
GUNN -HILSUM EFFECT ELECTRONICS
365
i.e., the effective conduction current of the serially connected diodes is the average of the conduction currents in each diode. Equations (43) and (44),along with an assumed l(4)trajectory, can be used to develop a qualitative, graphical technique for determining the effects of the space charge nonuniformities on the oscillations. This is illustrated in Fig. 32. To plot the curves for each diode on the same scale as the curve for the diodes in series, it is convenient to use the average elec-
FIG.32. Qualitative illustration for determining the effects of space charge nonuniformities on the oscillation.
366
M. P. SHAW ET AL.
tric fields &/l,, where 1, is the length of each diode. Figure 32a shows the case of a relaxation oscillation and Fig. 32b shows a sinusoidal oscillation. The Z(4/1)trajectory has been drawn using the ellipse approximation. Note that 4// = 4(~$,/1, + +2/12). We have also taken 1, = 12. The values of I, and 4, are taken as Z,, and Zpl(Rol + Rh), respectively, since threshold for the series combination of the two diodes is determined primarily by the lower threshold of the two diodes; in this case that of diode 1. Consider the situation where diode 1 reaches and exceeds threshold, entering the saturated drift velocity region. Now since Zc(+) = )[Ic(&) + ZC(&)] and Z ( b a x ) = Z C ( b a x ) , if the voltage swings across each diode are approximately in phase so that at 4 = +,ax, & = 4 i m a x and $2 = 4 2 m a x . then I c ( & a x ) z 4[Ic(+1max) + I c ( 4 2 m a x ) I . TWO situations are of interest: (1) Both diodes enter their NDM regions and sustain excursions into the saturated drift velocity region. Here Ic(&,ax) = I , = &(Isl + Z,]. (2) Diode 2 does not reach threshold, and so 1, = *[Isl + Z(r$2,,x)]. Current continuity forces I(&,,ax) = Z,,, so I, is again approximately equal to I,, . We have two differing situations where the resulting voltage evolutions are such that the critical parameters I, and I, for the total current are the same for both cases. What, therefore, determines which type of space charge evolution will occur? As indicated previously, the formation of space charge nonuniformities is controlled by the value of IM . For diode 2 to reach threshold after diode 1 has reached threshold, ZM must be larger than I,, . In the case illustrated in Fig. 32 a charge nonuniformity is likely to form (i.e., diode 2 would not reach threshold) for the relaxation oscillation (large Z , / R o ) but is less likely to form for the sinusoidal oscillation (small Z o / R o ,Fig. 32b). The general trend is simply that large Z o / R oand large sample nonuniformities produce space charge nonuniformities while small Z o / R oand small sample nonuniformities discourage space charge nonuniformities. Once formed, the quenching of space charge nonuniformities is controlled by the value of the minimum voltage &, where d & , / d T = 0 and d+,/dT = - d&/dT. In Fig. 32b operating points for diodes 1 and 2 that satisfy these conditions are indicated. Here diode 2 is below threshold while diode 1 is above threshold. Depending on the exact details of the parameters, it is possible that diode 1 will not fall below threshold during the first cycle of oscillation. If this happens, the second cycle will have a smaller Z(4) trajectory. In this case the oscillation may eventually damp out, leaving diode 1 above threshold and diode 2 below threshold as shown in Fig. 32c. The lower the value of +,,, the less likely it is for diode 1 to remain above threshold. For the value of &, in Fig. 32a, diode 1 must be below threshold to satisfy the above conditions. Again, the gen-
GUNN-HILSUM EFFECT ELECTRONICS
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eral trend is that large values of Z o / R o (small ,#,,I lead to effective quenching of space charge nonuniformities while small values of Z , / R o lead to domain domination of the oscillation. One other major point is the affect of package capacitance. With respect to Fig. 32, the initiation of relatively uniform fields throughout the NDM element requires that the conduction current through diode 2 continue to rise after diode 1 has reached threshold. For this to occur the maximum value ZM of the current through the NDM element must exceed I,, . The displacement current supplied to large package capacitances tends to drain current that is normally supplied to the NDM element. Large package capacitances therefore encourage the formation of space charge nonuniformities. With these results in mind we can understand why uniform space charge modes only occur for low values of Z o / R oand low values of package capacitance (C,/C = l ) , as indicated in Fig. 19. We are now prepared to treat those aspects of the problem that are important when short, device-grade NDM elements are considered. These are often referred to as transferred electron devices (TEDs), and form the basis of the Gunn-Hilsum effect technology at the present time. IV. NDM DEVICES A . Introduction
We now turn to those aspects of the Gunn-Hilsum effect that are of technological importance. Soon after Gunn's experiment, many interesting devices were suggested. Among these were a neuristor (114), comparator (1 1 3 , up-converter ( I 16), logic element (1 17), subnanosecond pulse generator ( I 18), three-terminal oscillator ( I 19), temperaturegradient-controlled voltage-tunable oscillator (120), shaped-voltagetunable oscillator (121 ), split-electrode device (122), optically interacted devices (123, 124), concentric planar devices (125), YIG-tuned oscillator, (126) and varactor-tuned oscillator (127). Because many of these devices made use of the recycling cathode-to-anode transit time mode in long samples, they did not make a great impact on the technology. The reason for this is simply that thermal limitations restrict the use of thick samples in continuous (DC) operation. Short samples ( 510 pin active-region length) that can be sufficiently well heat-sunk so as to be capable of DC operation have become the major Gunn-Hilsum effect devices. However, long samples useful as logic elements may prove important in the future. Short samples are used almost always as simple oscillators or amplifi-
M. P. SHAW ET A L .
368
ers when incorporated into a variety of microwave circuits. A significant aspect of the behavior of short samples is that they are remarkably different in many respects than the long samples discussed in earlier chapters, often because they have lower N,l products and closer proximity of the anode to the cathode. In particular, it is found that circuitcontrolled oscillations may be obtained in short devices where the minimum voltage exceeds the sustaining voltage (3, 16); in long samples circuit control ceases under this condition. For short amplifier devices there is evidence of a stable postinstability charge distribution corresponding to a stationary accumulation layer extending from the cathode-to-anode contact that produces both small and large signal gain. This has not been observed with long samples. This chapter is devoted to a discussion of the fabrication, evaluation, and use of short samples. We begin with a discussion of device fabrication, followed by a discussion of thermal effects and a description of the evaluation of devices. The discussion is supplemented with published laboratory measurements, which for the most part correlate the oscillation and amplification characteristics to suspected space charge distributions. The results of numerical simulations, ours and others, are also included. A time-dependent cathode field model is introduced whose incorporation into numerical simulations accounts qualitatively for the behavior of indium phosphide and gallium arsenide oscillator devices. For amplifiers, numerical simulation closely coupled to experiments demonstrates that the large-signal amplification of supercritically doped short devices is a consequence of nonuniform space charge layers. These results also demonstrate that small signal amplification measurements need not necessarily provide clues to the large-signal behavior of the device. Following the amplification discussion, the common methods of characterizing the device in a microwave circuit are presented. This characterization, which is in the frequency domain (in terms of impedances) is very useful in circuit design. Often the device is represented as a negative resistance. The noise properties and state-of-the-art results for DC driven devices concludes the section. The point of view of this section is that device behavior is sensitive to the nonuniform space charge distribution within it prior to the onset of the instability and/or after NDM induced field rearrangement occurs. B . Device Construction 1 . Bulk Gallium Arsenide
The techniques of device fabrication involve (1) growth of bulk material, (2) epitaxial growth (3) contacting, (4) evaluation of each step. A
GUNN-HILSUM EFFECT ELECTRONICS
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useful and informative discussion of this sequence can be found in Chapter 11 of the text by Bulman et ul. (46). For sandwich-type devices (n+-n-n+), which we concentrate on in this section, the bulk substrate material must be heavily doped (10"- lo1*~ m - ~ For ) . planar devices a semiinsulating substrate is required. Thus, bulk material having a wide range of carrier concentration is required. The two standard techniques of bulk crystal growth are the horizontal Bridgman boat growth (127) and Czochralski crystal-pulling method (128). Since arsenic is quite volatile, stochiometric growth will occur from a stochiometric melt only under arsenic vapor pressure of 1 atm (temperature 600°C) or by using liquid encapsulation (128). Even if the bulk material is intentionally undoped, the purest bulk GaAs that usually results from these techniques contains some silicon and oxygen. Because of this and the fact that the bulk cannot be readily thinned to a sufficiently small size, and that sufficient purity or homogeneity is rarely attained, bulk material is almost never used for the active-region portion of a DC driven device. Furthermore, bulk material usually has a negative temperature coefficient of resistivity, which can lead to thermal runaway affects for sufficiently high bias. Purity, homogeneity, and a positive temperature coefficient of resistivity can be achieved with epitaxial material grown in thin layers on the bulk material by either vapor or liquid phase deposition. 2. Epiruxiul Growth
The standard techniques for vapor phase epitaxial growth of GaAs are (1) the arsenic trichloride system (129, 130) and ( 2 ) the arsine system (131, 132). Other techniques have also been investigated (133, 134) and attempts have been made to understand these processes via thermodynamic arguments (135). The epitaxial deposition of GaAs by the arsenic trichloride process usually yields high-purity material (136). In principle, this process should produce a smoothly graded n-n+ interface when grown onto an n+ substrate, but great care must be taken to properly prepare the surface of the substrate (137). Even when proper care is apparently taken, however, high-resistance layers often form at the n -n+ interface (138). These layers could be either exhaustion layers or even embedded p-type layers. Because ofthese problems, measures such as the use of properly doped substrates (139) and vapor etching of the substrate (140) prior to deposition are also often employed. In many cases, however, the technique that will produce a reasonable yield of useful n-n+ interfaces is that of buffering the substrate by first growing an n + epitaxial layer of a few microns, and then growing the active n region. Also, a contact buffer layer is often grown on top of the n layer. Figure 33 shows a typical
M. P. SHAW ET AL.
370
ALLOY ED METAL-SEMICONDUCTOR JUNCTION
nt BUFFER LAYE
n-ACTIVE LAYER n+ BUFFER LAYER
n+ SUBSTRATE
METAL ALLOYED META L-SEMICONDUCTOR JUNCTION
FIG.33. n+-n-n+ “mesa” device.
n+-n-n+ “mesa” device. The best procedure to follow is obviously that which produces the most efficient type of device, and the best device design for one application is not necessarily the same as that for another application. We show later that the properties of the active region interface for an optimum oscillator are not necessarily the same as that for an optimum amplifier. Vapor phase transport can reliably produce material with doping denIOl5 ~ m range - ~ and room temperature mobilities as high sities in the as 8000-9000 cm2/V-sec. [If the epitaxial layer is to be doped, hydrogen is often used as a carrier of the required dopant (or compound thereof) to the deposition region.] In general, however, vapor phase material will not be as pure as liquid-phase material. In the liquid-phase epitaxial technique, GaAs is grown on a substrate via recrystallization of a GaAs solute at the liquid-solid interface. Early work on liquid phase growth was done by Nelson (141). More recently advances have been made by, among others, Kang and Greene (142), Hirao et ul. ( 1 4 3 , Goodwin et al. (144), and Zschauer (145). Here, again, great care must be taken to avoid high-resistance interfacial regions (146). Layers grown below about 800°C - ~ without intentional doping are usually n-type and in the 1014~ m range. Mobilities between 7500 and 9300 cm2/V-sec are common. The most common growth technique is the transient method, where the temperature of the liquid-solid-vapor system is kept uniform and then decreased uniformly so that recrystallization of the solute occurs. Another method involves keeping the system in steady state with a fixed temperature dif-
GUNN-HILSUM EFFECT ELECTRONICS
37 1
ference between the solution and substrate. Here the solute crystallizes onto the cooler substrate. Both horizontal and vertical systems are employed. As in the vapor phase technique, buffer layers are also often employed. In the vapor-phase technique these layers are formed by altering the vapor pressure of the dopant. In the liquid-phase technique the buffer layers are made by keeping two melts in the furnace, a lightly doped and heavily doped melt. The substrate is dipped first into the heavily doped melt and then into the lightly doped melt.
3 . Contacts Once the layers are grown, metallization must be employed in order to form low-resistance contacts. The metallization modifies the properties of the inactive-active region interface, which is a prime determinant of the manifestation of any current instability and hence of the efficiency of the device. In reference 20 we outlined one procedure for putting tin-nickel contacts onto bulk GaAs. A variety of other techniques abound. In cases where regrown n+ buffer layers surround the active region, the activeregion boundaries are determined by the interface between the active and buffer layers. The field distribution at the interface of the metal-n+ region should therefore play only a minor role here as long as it provides a low resistance. However, in cases where a buffer layer is only provided on the substrate (or not at all), one side of the active region will be an alloyed metal- semiconductor interface. This interface configuration has been discussed in Sections I1 and 111. Some of the techniques for producing low-resistance contacts involve alloys of (1) gold-germanium (147), (2) silver-indium-germanium (148), (3) gold-germanium-nickel (149), (4) silver-nickel (150), ( 5 ) tin-gold (1.5/). For the details of these techniques see Bulman et a / . (46) and Bosh and Engelmann (47);for a more complete listing see the text by Milnes and Feucht (82). 4. Charucterization and Evaluation
Each step in the fabrication process requires characterization and evaluation. The bulk substrate material is usually characterized in terms of the mobility and carrier concentration by standard Hall effect and resistivity measurements ( 4 6 , 4 7 ) . Once an epitaxial layer is deposited, however, its characteristics, including its thickness, are required. The thickness of the epitaxial layer can be determined by cleaving perpendicular to the surface and staining the resultant new surface. The fact that the epitaxial layer and substrate have different carrier concentrations allows
3 72
M. P. SHAW ET A L .
the interface to be revealed. More accurate measurements require an angle lapping of the wafer. The thickness can also be determined by shining infrared radiation onto the film and studying the resultant interference fringes as a function of wavelength. Quantitative techniques for determining the resistivity and mobility of the n-layer contained in a sandwich device are generally not very accurate. The resistivity can be obtained by simply measuring the resistance and knowing the geometry accurately. Corrections must be made for fringing fields and substrate resistances. The mobility measurements are even less appealing. Here we measure the transverse magnetoresistance with the Hall field shorted out due to the very short device geometry. More accurate device parameters can be obtained for films on semiinsulating substrates, where the Hall effect can easily be measured on the planar epitaxial layer. It seems reasonable to grow adjacent layers on semi-insulating substrates for evaluation purposes, while growing sandwich device specimens for actual use. The most popular technique for determining the doping profile of an epitaxial layer is that of the capacitance-voltage (C-4) measurement (46,47). A Schottky barrier is deposited onto the active region, a reverse bias is applied, and the capacitance is measured as a function of voltage. Although this technique is commonly thought to be accurate, extreme caution and care must be used in its application. For example, van Opdorp (152) showed that the evaluation of doping profiles from measured C-4 curves is never unambiguous. A decision regarding the actual profile can only be reached if additional information concerning the semiconductor material and the process of junction preparation is known. Furthermore, Kennedy et al. (153) showed theoretically and by numerical replication of differential capacitance measurements that the differential capacitance measurement determines the distribution of majority carriers, not that of the impurity atoms. Thus, regions where the impurity distribution showed marked variations would be expected to yield majority carrier distributions that did not faithfully map the former distribution. Additional information would be necessary to unfold the impurity distribution. The differential capacitance method is strictly valid only in regions exhibiting a uniform distribution of impurity atoms. Spitalnik et al. (154) showed that the results of independent C-4 measurements were inconsistent with the results of experiments and numerical simulations of Gunn-Hilsum effect amplifiers. The experiments and simulations compared favorably only when much larger inhomogeneities than those measured by the C-4 technique were used in the simulation. Once the epitaxial layer has been characterized, the metal contact is put down and alloyed. Electrical characterization of the contact is then
GUNN-HILSUM EFFECT ELECTRONICS
373
performed. However, the contact characterization reflects the presently inadequate understanding of the alloyed metal-semiconductor interface. Usually only one quantity, the specifc contact resistance, which is the product of contact resistance and contact cross-sectional area, is used to characterize the contact. In the case of high-barrier-height metalsemiconductor interfaces, the contact resistance usually dominates the C-I#I measurements, so this measurement at low voltage levels is sufficient for determining the specific contact resistance. For low-resistance contacts the C-I#I measurements do not provide sufficient information; rather, another procedure due to Cox and Strack (148) is often used. Figure 34 shows a plot (148) of total resistance versus reciprocal contact diameter for a 90 wt.%Ag-Swt.%In-Swt.%Ge alloy used as a contact to n-type gallium arsenide. The specific resistance was about a-cm2 for 0.6-2.6 a-cm layers and 6 x lo-* i2-cm2 for 0.3 a-cm layers. [rn Solomon et a / . (20) it was shown that Sn-Ni-Sn and AuGe-Ni alloyed contacts to bulk material with resistivities around 0.3 a-cm had worst case specific resistances of 8 X n-cm' and
FIG.34. Total resistance vs. reciprocal contact diameter for a 90 wt. 96 Ag-5 wt. 96 In-5 wt. % Ge alloy contact to n-GaAs [from Cox and Strack ( / 4 8 ) ,with permission].
374
M . P. SHAW ET A L .
2 x n-cm2, respectively, as obtained via the method of Cox and Strack.] Contacts with specific resistances of less than 10-2-10-3 Cl-cm2 are often regarded as low-resistance contacts. However, additional experiments may be performed to further classify these contacts. The experiments hinge on the notion that the concept of a barrier height is useful in such a system. In one study, Gyulai el ul. (155) measured the forward current -voltage relation of evaporated gold contacts on liquid epitaxial n-GaAs. Figure 35 shows a set of typical characteristics for n-GaAs with a doping of 10l6~ m after - ~alloying for 5 minutes each in dry Nzat various temperatures. The data are described by the C-C#Jrelation for transport across a metal-semiconductor interface [see Chapter 10 of Sze (156)]: J(F0RWARD) = 120m*P exp( -ClB/kY){exp[e4(FoRWARD)/(nkY)]- I} (45)
1
1
1
I
1
FORWARD VOLTAGE (VOLTS)
FIG.35. Forward C-C#Icharacteristics of a liquid epitaxial GaAs film with evaporated gold contacts. N o for the film was 1015/cm9[from Gyulai ef ul. ( I S S ) , with permission].
GUNN-HILSUM EFFECT ELECTRONICS
375
where m* is the effective mass of the carriers (in multiples of the free elecis the barrier height, 9-is the absolute temperature and n is tron mass), flZB a numerical factor with values greater than or equal to unity. Values of n greater than unity represent the extent to which the C-4 relation departs from the ideal thermionic law. From the data of Fig. 36, Gyulai ef ul. (155)
0.3t
EUTECTIC TEMP
'
260
'
'
660 ' 8bO
ALLOY TEMPERATURE (OC)
FIG.36. (a) Measured values of the banier height for alloyed Ni/Au-Ge/GaAs contacts. (b) Specific contact resistance as a function of alloy temperature for Ni/Au-Ge on epitaxial n-GaAs [from Robinson ( / 5 8 ) ,with permission].
M. P. SHAW ET AL.
376
find values of flB = 0.93 eV and n = 1.08 for the nonalloyed case; whereas after alloying at 300°C the forward characteristic is observed to increase by many orders of magnitude, resulting in values of flB = 0.72 eV and n = 1.12. Similar behavior was observed with Au-Ge contacts on n-GaAs films. The alloying process, among other things, results in a higher semiconducting doping level at the metal-semiconductor interface. Pruniaux (157) investigated the change in barrier height as a function of epitaxial doping level for Au-Ge films deposited on vapor-phase-grown GaAs epitaxial layers. In one set of experiments the forward C-C#Jrelations were determined for seven samples having carrier concentrations ranging from 4.5 x 1015to lo1*~ m - The ~ . barrier height shows a steady decrease with increased doping concentration, accompanied by an increase in the factor n to a value =2. The results are consistent with those of Gyulai ef al. (155).
In another group of experiments, Robinson (158) determined the barrier height dependence of alloyed Ni/Au-Ge films on vapor-grown 2.5 x 1015 ~ m GaAs - ~ layers. In this set of experiments the forward C-C#J characteristics were measured and fit to Eq. (33). Capacitance-voltage and specific contact resistance measurements were performed, each for samples heat-treated below and above 360°C (the Au-Ge eutectic temperature). The results of Robinson's measurements are shown in Fig. 36, where the dark circles represent data fit from the forward bias measurements and the open circles are those from the reverse bias capacitance -voltage measurements. There is an as-deposited value of the barrier height of 0.68 0.06 eV. The barrier height increases after heat treatment below the Au-Ge eutectic temperature. It is reported that as long as the alloy temperatures remain above 360°C, a significant drop in the value of the barrier height occurred. Accurate values could not be obtained for these lowered barrier heights. For the above samples Robinson also provides data on the specific contact resistance as a function of alloying temperature (Fig. 36). Below the Au-Ge eutectic temperature there is a rise in the specific contact resistance with temperature. This is consistent with the increase in barrier height. Above the Au-Ge eutectic temperature there is a precipitous drop in specific contact resistance. By combining the data of specific contact resistance and barrier height measurements, it would be satisfying to conclude that high-barrier-height samples have large specific contact resistances whose value increases with increasing barrier height, and that low-resistance contacts have low barrier heights. While this is thought to be the case (159), the evidence is not tight. The difficulty lies in accurately obtaining values for barrier heights less than 0.3 eV.
*
GUNN-HILSUM EFFECT ELECTRONICS
3 77
One suggestion for measuring low barrier heights is due to Tantrapom (160) and Padovani (16f1. Here the reverse C - $ relation is measured as a
function of temperature. For a constant value of voltage the ratio of the log of the total current to Y2is plotted against (k.T)-I. The slope of the plot yields the barrier height. Colliver ez al. (162) have used this method in their InP studies, with values for the measured barrier heights varying from 0.15 to 0.25 eV. 5 . Thermal Considerations
Microwave devices generally operate at temperatures higher than the ambient temperature, necessitating the consideration of thermal effects in the design of the device. Thermal effects can be examined by generalizing Eq. (13) to include temperature effects:
The presence of a temperature gradient within the device will also contribute to variations in the electron current density [see, e.g., Stratton (163)l. This contribution is ignored here. In Eq. (46) the field-dependent velocity and diffusion coefficients include a temperature dependence. The temperature dependence of the velocity-electric field curve has been calculated by Ruch and Fawcett (164) and shown to be in good agreement with the experimental measurements of Ruch and Kino (5). A fit to the velocity-electric field curve of Ruch and Fawcett (164) was given by Freeman and Hobson (165) as
and said to be valid for any temperature between 300 and 600 K. In Eq. (47), Eo is 5 kV/cm, yielding a threshold electric field of approximately 3.5 kV/cm. Freeman and Hobson (165) also argued that the temperature dependence of the diffusion curve between 300 and 600 K could be represented by multiplying the room temperature values by the factor 300 K/Y. To simulate the temperature dependence of the device behavior, Eq. (46) must be solved simultaneously with the circuit equations and the equation for heat conduction. If we neglect heat flow contributions in (1) a direction transverse to the direction of current flow, and (2) at the transverse surfaces of the device, then the net rate of gain of heat per unit vol-
378
M. P. SHAW ET A L .
ume per unit time in the region d X at the point X is therefore
ax
(48)
where K(T) is the thermal conductivity, p the mass density of the material, and C its specific heat (167). Coupling Eq. (48), the heat flow equation, to the circuit equation and the equation for current flow (46), along with the temperature dependence of the velocity and diffusion coefficient, we see that a self-consistent solution is obtained by (1) determining the temperature profile for a given distribution of electric field; (2) using the resulting electric field and temperature distribution to compute the total current density J; (3) using the resulting current and voltage to evaluate the circuit equation. For all but the simplest cases, numerical methods are required to produce a self-consistent solution. There are, however, a number of relevant situations amenable to analytical methods. One such situation, discussed by Knight (168) is considered below. We assume the device structure shown in Fig. 37 and treat the case where the solutions are independent of time and where the electric field is constant. For this case Eq. (48) reduces to
dT dX
(49)
We solve Eq. (49) subject to the following conditions: ( 1 ) The quantity of heat incident at the n+-n interface is negligible.
(2) The temperature varies continuously throughout the device. (3) The heat generated within the n2+region is negligible. The temperature dependence of the thermal conductivity over the range of interest is approximated as follows: Within the n region K = 150/T; within the n2+ region K = 120/T (168). We obtain
where .Tnln2+ is the temperature at the n/n2+interface, I, the thickness of the n region, and
where I,,+ is the thickness of the n2+layer and MS the temperature at the metal-semiconductor interface. Thus,
TnnlnZ+ = T M S exp( JEI, lnr/120)
(52)
GUNN-HILSUM EFFECT ELECTRONICS
379
FIG.37. Device structure for doing temperature-dependent calculations [from Knight (168) with permission].
and the temperature distribution within the n layer is JEX2/300)], 0 > x < I,
(53)
The next step in the calculation is to express TMS in terms of the ambient temperature. We assume for simplicity that the device is cylindrical and of radius a . We neglect all internal resistance due to the metallic bond and assume a perfect bond to an infinite copper heat sink with thermal . enters the copper stud through the cylindrical conductivity K ~ All~ heat cross section of area m2and we assume a constant power density of JEl, W/cmZover the region 0 < r < a , and a zero power density over the region r > a. There is a consequent radial distribution of temperature over the cross-sectional region whose average value (167, Section 8.2) is Yav= 8EJI,a/3
TKC,,
(54)
380
M. P. SHAW ET A L .
If ram), denotes the ambient temperature, then the maximum temperature at the metal-semiconductor interface is TMS =
ramb
~EJ~,U/~TKCU
(55)
Combining.al1of the temperature drops, we find that the total temperature drop from the hottest point within the device (Le., the n+-n interface) to the heat sink is
The above expression teaches that the temperature drop increases as the power density increases and as the lengths of the n and n2+ regions increase. As an example, consider a nominal length X-band GaAs device with the parameters of Table 11. For a cylindrical cross-sectional area = 1.5 x 104cm2, the radius a = 0.69 x 10-2cm. If we assume a power density associated with the maximum held and current prior to an instability, then JEl, = 11.5 x 103 W/cm2. For a copper heat sink, K~~ = 3.9 W/cm K and I,*+ = 4 x cm; we obtain ATTOT
17 K
(57)
This relatively uninfluential result is independent of the ambient temperature. However, for current densities associated with saturated velocities and fields of the order of IOE,, the power density increases to approximately 46 x lo3 W/cm2, and ATTOT
=
(ramb 4- 68 K)(1
0.3)
-
ramb
(58)
For an ambient of 300 K we have a temperature change of approximately 175 K, which is very significant. The above discussion is for DC effects only. Long, high-power devices sustain large temperature gradients and often require pulsed operation to avoid damage. Pulsed operation is also employed when thermal efTABLE I1 up = 2.25 x 10’ cm/sec Ep = 3.2 kV/cm I= cm N~ = 1015/cm5 area = 1.5 x lo-” cm* Ro = 6.06 fl Co = 1.47 X lO-”f J , = N&vp = 3.6 x 101 A/cm’ Ed = 3.2 V
GUNN-HILSUM EFFECT ELECTRONICS
38 1
fects offer difficulties with respect to interpretation of device operation. Equation (58) is the one to be considered when time-dependent thermal effects are important. Among the questions to be asked under pulsed operation are: ( I ) How does the temperature of the device change during the length of the pulse? (2) How long after the end of the pulse will it take for the device to cool down? (3) What duty cycle is necessary? A technique useful in answering these questions was developed by Ladd et al. (169). As expected, a rise in temperature occurs during the pulse, with the maximum temperature occurring at the end of the pulse. The maximum temperature persists for a time T o + 7TH/4, where the thermal time constant TTH = 4CplE/dK. This behavior comes about because the thermal time constant was chosen to be much longer than the pulse length. The result is that the X = 0 region does not cool until a time TTH/4 after the end of the pulse. In the above study it was concluded that the duty cycle of a pulsed oscillation (which is given by To&, where & is the pulse repetition frequency) is limited. In order to prevent successive pulses from building up excessive temperatures in the active layer, the time duration between pulses is limited to a valuef, < 1 / 7 T H . The value of T o is also limited by the maximum temperature gradient a useful device can sustain. This must be determined empirically, and in one study (170) it was demonstrated that the efficiency dropped to zero when the temperature difference exceeded 60-80°C. The temperature-dependent characteristics of X-band transferred electron devices has been the subject of a number of studies. Fentem and Nag (171) used short pulses to measure the C-d, characteristics as a function of the ambient temperature. These measurements were used in conjunction with DC measurements to estimate the average temperature of the active layer under realistic operating conditions. The results are illustrated in Fig. 38. Here the DC-driven voltage characteristics are plotted over the pulsed characteristics. The points of intersection are used to estimate the average temperature of the active layer. Note that the subthreshold results imply that increasing the input increases the device temperature. If we could construct a device whose electrical characteristics were independent of the contact and whose space charge distribution was approximately uniform, then in a given circuit and on the basis of Fig. 38, we would expect a decrease in efficiency with increasing temperature. Figure 39 (172) shows this situation, where the efficiency is plotted as a function of temperature. The measurements are for DC-driven operation. The abscissa is an indication of the diode operating temperature, which is
M. P. SHAW ET A L .
382
DIODE VOLTAGE
($I)
FIG.38. Pulsed and CW C-4 characteristics of a GaAs device [from Fentem and Nag ( 1 7 / ) ,with permission].
in general nonuniformly distributed. The average temperature was estimated from the low-field resistance of the diode under DC bias conditions measured by superimposing a narrow pulse onto the DC voltage. Another set of experiments, in this case pulsed operation of X-band devices reported by Bott and Holliday ( I 7 3 , indicated that the temperature dependence of the efficiency was far richer than that associated with the temperature dependence of the velocity -electric field curve. Here, for certain bias levels the efficiency initially increased with temperature THEORETICAL x 1/2
3-
2-
-
FROM COMPUTER SIMULATION
AVERAGE DIODE TEMPERATURE
(OK)
FIG. 39. CW conversion efficiency vs. average diode temperature for an n-GaAs device [from Hasegawa and Aono (172), with permission].
3 83
GUNN-HILSUM EFFECT ELECTRONICS
before subsequently decreasing. This is displayed in Fig. 40. Similar variations in the output power were shown. Variations of the contact conditions with temperature, as well as the effect of the circuit on the oscillation, must be considered in order to understand this behavior. Temperature-dependent contact effects were considered by Wasse et al. (174), who examined AgSn cathode-contacted GaAs X-band devices. In these experiments the threshold current as a function of temperature was measured by applying 0.5 psec pulses at a low duty cycle. These results are displayed in Fig. 41. Figure 41a shows the current falling uniformly at a rate close to that predicted by Ruch and Fawcett (164), and in a manner qualitatively similar to Fig. 38. This type of behavior was found on all of their n+GaAs-contacted devices and some AgSn-contacted devices. Figure 41b, c shows threshold currents that fall initially less steeply than Fig. 41a, then at a higher temperature approach the theoretical bulk slope. Figure 41d displays a significant departure. There is initially a rise in threshold current with increasing temperature followed by a decreasing current density characteristic of the bulk. This is consistent with Fig. 40. If one interprets these results in terms of barrier heights (175) then for sufficiently high barriers the threshold current density is limited by the flux density, which is proportional to Tzexp( - fl/kT) [see Eq. (45)]. The flux density increases as the temperature is raised until a sufficient amount of current is transported across the contact and the bulk material dominates the temperature dependence of the device. This is thought to 2.0
1.5
4
-
-
-
E
1.0
-
-
0.5
-
-
t 0
2 w
u U
W
r
TEMPERATURE (OC)
FIG.40. Pulsed (500 nsec length, 20 : 1 duty cycle) conversion efficiency vs. temperature for n-GaAs devices at different bias voltages [from Bott and Holliday (/73),with permission].
M. P. SHAW ET A L .
3 84
In
cz
3
> a E
K
t m
a:
-a
3.1
-
3.5
-
3.3
-
7.3 7.1 -
7.5
6.9-
7.2
-
6.8 6.6 7.0
TEMPERATURE
(OC)
FIG.41. Pulsed measurements of the variation of threshold current with temperature for AgSn-contacted GaAs devices of differing barrier heights [from Wasse et a!. ( 1 7 6 , with permission].
be the explanation for the behavior represented by Fig. 41d (174). Wasse et (I/. (174) also used this argument to explain the temperature dependence of the efficiency of a AgSn-contacted device. This is displayed in Fig. 42, where we see the lowest efficiency at the highest temperature levels. An increased efficiency at the lower temperatures is thought to be a consequence of the increased peak-to-valley velocity. However, to be consistent with the contact contributions, we must expect some current limitations with decreasing temperature and consequently increased potential drops across the cathode region. The latter, as experiment and computer simulation indicate, does not necessarily lead to a decrease in efficiency, but may indeed improve device performance. However, continued drops
385
GUNN-HILSUM EFFECT ELECTRONICS
-
> u z w 0 u
64-
UJ U
2-
0
I
1
1
TEMPERATURE
1
1
1
I
(OC)
FIG.42. Conversion efficiency as a function of temperature for a device fabricated from the same sample as those used for the measurements of curves b-d in Fig. 41 [from Wasse Y I nl. (174) with permission].
in temperature may limit the current as severely as a result in a decrease in device efficiency. We note here that Wasse et ul. (174) reported that devices with the highest threshold current densities were not the most efficient. With regard to Fig. 41, it is reported that when the diodes of Figs. 41b-d were DC driven they showed a steady progress in operating efficiency of 3.6, 6.5, and 7.3%. The above discussion was concerned with GaAs devices. InP devices also show interesting temperature dependences. Figure 43 (176) shows a set of C - 4 curves for a 10-pm-long, 7.4 x 1014cm-3InP oscillator that exhibited anomalously high-efficiency oscillations (162) at room temperature. The C-4 curves are obtained under pulsed conditions at various temperatures. At room temperatures the current is anomalously low. There is no current drop-back at the oscillation threshold of 1 I V and the
FIG. 43. Temperature dependence of pulsed 17-4characteristics of an InP device exhibiting anomalous behavior [from White and Gibbons (176).with permission].
M. P. SHAW ET A L .
386
DC to R F conversion efficiency is high. As the temperature is increased, the normal bias current increases rapidly and eventually exhibits dropback at the oscillation threshold. The efficiency was observed to rise initially with temperature and then decrease. The strong temperature dependence of the I-V curves suggest that the efficiency of the device should be extremely sensitive to temperature variations. Indeed, this was found to be the case (162). In summary, the electrical operation of a negative differential mobility device results in the transport of heat from one end of the device to the other. Heat transport results in a temperature gradient across the active layer, which affects, point by point, the velocity-electric field characteristics of the device. Unless special precautions are taken, such as external cooling during normal device operation, the average device temperature may be significantly higher than that of its surroundings. This results in a lowering of the peak-to-valley velocity ratio and an alteration of device efficiency. The contacts are also temperature sensitive, and if they dominate device behavior, increases in device temperature may sometimes lead to increases in device output, whereas if the bulk dominates, increases in device temperature will lead to decreases in output. Temperature fluctuations will also alter the operating frequency (177). For any given set of conditions the temperature dependence of the device cannot be predicted beforehand; a set of representative experiments must be performed. We have now identified some problems concerning characterization of the GaAs device and have examined thermal effects and materials considerations. The development of new devices requires insight into the intimate interaction between the space charge distribution, the contacts, the circuit, and the modifications thereof when temperature gradients are involved. So we ask: How do we design a particular device? What sort of contacts should we attempt to put onto the active region? How thick should the active region be so that we can produce a DC-driven device at a given frequency? How do we make a microwave amplifier? An oscillator? In what circuit should we place the device? We address ourselves next to these very important technological questions, and we begin with a discussion of the oscillation properties of short NDM elements under isothermal conditions. C . Oscillation Principles for Short Devices
I . Introduction Large signal operation of long negative differential mobility elements in a circuit containing reactive elements was discussed in Section 111
GUNN-HILSUM EFFECT ELECTRONICS
387
(19, 21 ), where it was determined that sustained circuit-controlled oscilla-
tions, transit time domain oscillations, or damped oscillations occurred. Numerical illustrations were presented; circuit-controlled behavior was shown to require the presence of low or moderate values for the cathode boundary field, circuit values in excess of Z , / R o = 2.0, and minimum device voltages less than the sustaining voltage V,. Short devices operated in the circuit of Fig. 44 are substantially different from long devices in that for short devices the space charge configuration at one contact is significantly dependent upon the space charge configuration at the other contact. The principal consequence of such a proximity effect is that criteria for circuit-controlled oscillations are less restrictive for short devices than for long devices. In particular, sustained circuit-controlled oscillations can occur for minimum device voltages in excess of the threshold voltage for negative differential mobility (178) (approximately 3.2 V in the case of a 10 pm long GaAs device). In Section I1 we introduced the circuit impedance Z , , where in the case of a two-reactive-element circuit consisting of an inductor and capacitor, Z o was equal to the ratio of voltage to current across either reactive element. Large values of Z,, resulted in large voltage swings and small current swings, whereas small values of Z, resulted in small ratios of voltage to current swings. The numerical illustrations we presented in Section I11 indicated that the voltage and current oscillation amplitudes of long devices embodied this behavior. Short devices also follow the same description. Consequently, we find that we can introduce a crude, though useful approximation relating the minimum device voltage to the circuit parameters and the bias level. As we have shown earlier, the value of the
FIG.44. Series L-parallel C circuit used in the numerical simulation.
388
M. P. SHAW ET A L .
minimum voltage is crucial to determining whether a device will be circuit or transit time dominated. As an example, we assume a current oscillation of amplitude * I , . The minimum voltage across the NDM element is then given by
where we have used the relation +TH = R o l , . Thus, for a given value of bias, 4MIN and Z o are linearly related. The exact numerical relation is more complicated, (it is illustrated in Fig. 30 for one value of bias) but the qualitative features remain intact. Namely, for a given value of bias, 4MIN increases with decreasing Z o . Recall that for short devices we have stated that 4 M I N may exceed +TH while circuit-controlled oscillations persist. However, this is not unconditionally true, as we illustrate later. The situation where 4 M I N > +TH is qualitatively different from the situIn the former case there remains a permanent ation where c$MIN < 4TH. time-dependent residual charge layer at the anode contact through the entire oscillatory cycle. In the latter case all residual charge layers are extinguished once each cycle and we have operation that is similar to LSA operation. From our earlier studies of long samples we are familiar with the idea that for circuit-controlled LSA relaxation oscillations the oscillation frequency is determined mainly by the circuit parameters. The frequency is essentially independent of device length. This continues to hold true in short samples for the situation where the minimum device voltage is less than the NDM threshold voltage. However, once t#,MIN exceeds c$TH an upper frequency limit appears, determined by transit time effects. For 10 pm long NDM elements with 10'5cm-3 doping, the upper frequency limit for propagating accumulation layers is approximately 16 GHz. [Note that the appearance of traveling accumulation layers in long devices is rare, since the presence of even moderate doping fluctuations transforms the layer into a series of dipole layers. In the case of short devices the moderate doping fluctuations often are impotent because of the size effect (relatively low N o / product) and the traveling accumulation layer can readily dominate for a low boundary field at the cathode.] Contact effects associated with short devices are not any richer than contact effects associated with long devices. However, because of the much greater technological activity associated with short devices, more has been uncovered. In particular, experiments with indium phosphide (162) have shown that the fixed-cathode-field model is not able to explain the unusually high-efficiency oscillations generated by devices whose DC
389
GUNN-HILSUM EFFECT ELECTRONICS
characteristics suggest the presence of a large voltage drop at the cathode boundary. (For GaAs, the presence of the latter is usually associated with a low-efficiency mode.) We will shortly discuss a time-dependent cathode field model and its relation to indium phosphide experiments in considerable detail. The simulation techniques that follow are discussed in Solomon rt al. (19, 21). Principally, the following equations are solved simultaneously:
" = G d T!!? 2 !i +f!0 , dT
+ R C , g + 4 + RZ,,
0,
= (LC)-'"
(61)
All these terms were defined previously. Equation (60) is the differential equation for space charge flow within the NDM element. Equation (61)is the circuit equation. For the calculations with short samples, we are interested in the range of bias levels generally used in practice. These exceed those used in the discussion of long samples. Of interest now, for example, is the range of bias values extending from about 3 &H to 7 &-. The parameters used for this computation are listed in Table 11. The range of circuit values we concentrate on in this section are determined by Eq. (59), and center on the value obtained by setting hIN = bHand +B = 7&'H.
In the discussion that follows, we find it useful to refer to Fig. 45, where, using inductance as ordinate and package capacitance as abscissa, we plot contours of constant Z,/Ro (dashed) and contours of constant fo (solid). Ro is the low-field prethreshold resistance of the NDM element. Circuits with values for inductance and capacitance in the region Z,/Ro > 12 sustain oscillations with periodically quenched space charge nonuniformities; circuits with parameters within the region Z o / R o< 12 will at some value of bias over the range from 3 4 T H - 7 4 T H exhibit oscillations in which a permanent residual space charge layer remains inclined toward the anode throughout each cycle. In the latter case dMIN > hTU: we refer to these oscillations as "excess voltage oscillations" (I 78). We also pay particular attention to those self-excited oscillations with values for Z,/R, = 9.0 and 15.0. With regard to the role of the cathode contact in the operation of short NDM elements, we note that each of the results discussed above may be quantitatively and qualitatively altered by variations in the value of the time-dependent cathode field E , . For example, as discussed above, excess voltage oscillations are bias dependent and usually occur at ele-
M. P. SHAW ET
390
AL.
-
OO
0.4
0.8
PACKAGE CAPACITANCE
1.2
(MOW
FIG.45. Inductance as ordinate vs. package capacitance as abscissa: -, contours of constant fa; --, contours of constant Za/Ra [from Grubin (178), with permission].
vated bias levels. Their absence at low bias values is predicated on the absence of either a permanent high-resistance region within the NDM element or a large, permanent cathode-adjacent voltage drop due to a very high value of E, , (The latter can yield excess voltage oscillations at most values of bias and in circuits in which Zo/Ro > 12.) However, the most important effect of the cathode boundary is that it determines the value of current density at the threshold for an instability. This value JTHwe recall is equal to J , for E, < E,, and is equal to N,eV(E,) for E, > E,. As a general rule, the amplitude of the current oscillation is given approximately by JTH- J s , where J s = N o e V s . ( V s is the saturated drift velocity.) Thus, increasing E, generally decreases the current amplitude and thus the output. Departures from this rule are impbrtant and lead to the fact that there is an optimum value of JTHfor maximum efficiency that is slightly less than J , .
39 1
GUNN-HILSUM EFFECT ELECTRONICS
2. Numerical Illustrations Using the Fixed-Cathode-Field Model The above discussion indicates that there are four parameters relevant to self-excited oscillations: the circuit parameters Z, and f,, the cathode field E,, and the bias +B. We now illustrate, numerically, the role of each. In Fig. 46 we plot current density through the load resistor, J ( T ) = I ( T ) / A , vs. average electric fieldE(T) = + ( T ) / L . Time is eliminated between the two. The calculations are forf, = 10.8 GHz, Zo/Ro = 9.0 and 15.0, E, = 0.0 and 2.0Ep, and = 4,5, and 6 b H .We observe the following: (1) The ratio of voltage to current amplitude for a given E , and +B is greater for the larger Z o / R , circuit. (2) For the range of bias values considered bIN is smaller for the larger Z o / R ocircuit. (3) At +B = 6 4 ~ the ” smaller Z , / R o circuit sustains excess voltage oscillations. The Z , / R o = 15.0 circuit does not. (4) Higher values of E , lead to lower values of the threshold current density.
For the examples of Fig. 46 there is also a reduction in the peak current density. (The above results are, of course, consistent with the preliminary discussions of this section.)
(fl
fdl
I
FIEp
&E p
CEp
FIG.46. Current through load resistor R (= Ro)vs. average field across NDM element. Calculations are for fo = 10.8 GHz, and indicated values of Z o / R oand E , . J . denotes the neutral conduction current curve [from Grubin ( j 78), with permission].
M. P. SHAW ET A L .
392
Figure 47a displays the output power as a function of bias voltage for the simulations of Fig. 46. For the Z o / R o= 15.0 case, the output power increases with increasing bias and is lower for the higher E , case. This is consistent with our earlier remarks, where we stated that the current amplitude was given approximately by the difference JTH- J s and that decreases in JTHwould yield lower output. The output for the lower Z o / R , = 9.0 circuit exhibits different trends for the two E , values. Here
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
O 3
4
5
6
7
0 0.2 0.4 0.6 0.8 1.0 JTH/J~
0.4
0.3
P(W)
0.2
EC = 2.0
0.1
0 3
4
5
5 (d 1
(C)
6
f o = 10.8GHr
7
+BIEPc
FIG.47. (a) Total output power vs. 4Bfor different E , . (b) Output power vs. threshold current density for #B = 5 E J . ( c ) Efficiency (total output/+&,,) vs. qbB. (d) Total output power vs. I#I~ for E , = 0.0 and differentf,:--, calculations for Z , / R , = 9.0; ---,calculations for Zo/Ro= 15.0. All calculations are forf, = 10.8 GHz, with the exception of (d), which also includes a calculation for f,, = 14.9 GHz [from Grubin (178), with permission].
GUNN-HILSUM EFFECT ELECTRONICS
393
the output for a wide range of bias values improves for the larger value of E, . The differences in the E, dependence of the output for the Z , / R , = 0.0 and 15.0 cases originate primarily with the circuit. The Z o / R o = 15.0 circuit has a larger inductance (see Fig. 4 9 , tending to inhibit current changes. In this case the peak current density is limited to values approximately equal to the threshold current density. Decreases in the latter are then generally accompanied by decreases in output. The Z , / R , = 9.0 circuit has larger capacitive contributions (see Fig. 4 3 , which tend to increase the amplitude of the current oscillation; the peak current reached is often significantly larger than the threshold current density. In addition, the average current level associated with cathode domain nucleation (as for E , = 2.0Ep) is lower than that associated with cathode accumulation formation (as for E, = 0). The two effects, when coupled, improve the output of the NDM element. The results of Fig. 47a point to the fact that improvements in the performance of a device made by altering the cathode conditions can only be realized when a proper set of circuit parameters is also chosen. But the differences between the E , = 0.0 and 2.0Epcases are peculiar to the situation where the cathode field assumes values somewhere between zero and 2 . 0 E p . Generally, for E , > 2.OE,, further increases in cathode field reduce the output of the device for any value of Z o / R o .This is illustrated in Fig. 47b, where for a bias of 5&-” we plot output power vs. threshold current density. The values Zo/Ro = 9.0 and 15.0 are chosen for this computation. For both circuits we see an optimum value of threshold current density beyond which the output decreases. The curves of Fig. 47a, b demonstrate that large values of Z o / R oyield lower output. Similar conclusions emerge from calculations of the efficiency (see Fig. 47c). Thus, in choosing a set of circuit parameters, a designer would opt for the lower Z , / R , circuit. However, there is an important tradeoff, because choosing the lower Z o / R ocircuit introduces the possibility of developing excess voltage oscillations at lower bias levels. Often these yield lower output power and efficiency. The above discussion demonstrates that the time-dependent space charge distribution, as influenced by the cathode boundary condition, can definitively determine the output characteristics of the NDM element. But how is the space charge distributed‘?Figure 48 provides assistance. Here we display the average electric field across each half of a 10 pm long device. The average field,(t) across the “cathode half” (0-5 p m ) is represented by solid lines; the average field EA(t)across the “anode half” (5-6 p n ) is represented by dashed lines. The computations are for E , = 0.0, 1.5, and 2.0Ep (see also Fig. 46b). All computations show that
M. P. SHAW ET AL.
TIME
FIG.48. Cathode region average electric field (-) and anode region average electric field (---) vs. time. Z$R, = 9.0,f0 = 10.8 GHz, C#J~ = 5 E J . (a) E, = 0, (b) E , = 1.5E,,(c) E , = 2.OE,. The output for E , = 0.0 and 2 E , is shown in Fig. 47. For E , = 1.5E,the efficiency is 8.6% and the output is 0.36 W [from Grubin (/78), with permission].
the cathode and anode halves both execute voltage relaxation oscillations with the minimum average field across each half being less than E,. The space charge nonuniformities are periodically quenched. The most nonuniform voltage distribution occurs for the E, = 0.0 case; when temperature effects are considered, this type of voltage distribution would be expected to yield large temperature gradients and diminished device performance. For higher E, the voltage distribution is more nearly uniform and, from thermal considerations, preferable. With regard to output, high threshold current densities are also sought. Comparing the E , = 1.5 and 2.OEPcalculations, both yield similar voltage distributions, but the lower E , case sustains higher output powers (see legend to Fig. 46). The space charge distribution responsible for the bias-dependent excess voltage oscillations discussed earlier is illustrated in Fig. 49 for fo = 10.8 GHz, Z o / R o = 9.0, and do= 7dTH.The figure displays the current -average electric field trajectory, multiple exposures of E ( X , T), and the average time-dependent field across the cathode and anode halves of the NDM element. The multiple-exposure calculations show the presence of a permanent time-varying anode-adjacent domain and a propagating cathode-originated accumulation layer. Figure 49c shows the cathode half executing relaxation oscillations with the minimum average field less than E,. The anode half of the NDM element is characterized by nonrelaxation-type oscillations in which the minimum average field is in excess of E,. With regard to thermal effects, the nonuniform distribution of Fig. 49c must be regarded similarly to that of Fig. 48a. But in the latter case the nonuniform voltage distribution does not correspond to a permanent anode domain; rather, the space charge nonuniformities are periodi-
GUNN-HILSUM EFFECT ELECTRONICS
395
(a)
40
JIJP i
1
E/Ep
0 C/Ep
0
XMICRONS
10 TIME (ARBITRARY)
FIG. 49. For Z J R , = 9.0, fo = 10.8 GHz, &, = 7&,, and E , = 0.0. (a) Current through load resistor R(= R o )vs. average electric field. (b) Multiple exposure of electric field vs. distance profiles at different instants of time. (c) Cathode (-) and anode (---) average electric field vs. time [from Grubin (178). with permission].
cally quenched. In order to understand these differences, a detailed discussion of the excess voltage oscillations is now presented. Consider the voltage distribution of Fig. 49c. At time TI, Ec and EA are both increasing and a cathode accumulation layer forms. The layer grows and propagates through the cathode region. At T2,Ec reaches a maximum while EA continues to increase. Ec increases until drain is initiated at the anode boundary. At time T3anode drain is accompanied by an increase in Ec along with an increase in current. Anode domain drain ends when a current instability is initiated at the cathode half of the NDM element. Comparison of Fig. 48a and Fig. 49c, where in the former case the anode half executes relaxation-type oscillations, suggests that the length of time needed to quench a domain increases with increasing bias and that the presence or absence of an anode domain is frequency dependent. We next examine this possibility with additional calculations at frequencies f o = 9.0 and 14.9 GHz. For (bB = 7&H, excess voltage oscillations occur at both the lower and higher frequencies, but at f o = 9.0 GHz the voltage swings are larger and the minimum voltage less than that for f o = 10.8 GHz. The opposite occurs at f o = 14.9 GHz, where the anode voltage swings are considerably reduced below that associated with f o = 10.8 GHz. Further evidence for the longer quenching times at higher bias levels comes from calculations at the lower bias level +B = 64,,. Forf, = 10.8 GHz, excess voltage oscillations occur (see Fig. 49c). For f o = 9.0 GHz and $aB = 6&,, they do not. We summarize the dependence of excess voltage oscillations on bias and circuit frequency in Figs. 47d and 50. In Fig. 47d we plot output power vs. bias voltage for Z o / R o = 9.0 and fo = 10.8 and 14.9 GHz. Note that at the higher frequency and bias levels
M. P. SHAW ET AL.
396
16
ZoiRo = 9.0
A
-
-
N
I
9
r
14
-
t
U
53
; 12 a U
10
-
FIG.50. Circuit oscillation frequency fvs. fo for three different values of ,$*; E , = 0.0 and Z o / R o= 9.0 [from Grubin (178) with permission].
applicable to excess voltage oscillations there is a marked reduction in output power. In Fig. 50 we display frequency of oscillationfagainst circuit frequencyf,, for E , = 0.0 and Z , / R , = 9.0. Different curves are for different values of bias. For (bB = 4(bTH and 5(bTH,where space charge nonuniformities are periodically quenched, fincreases linearly with f,.At the bias level (bB = 6 4 there ~ is~ a maximum frequency of oscillation approximately equal to 12 GHz. This is the upper frequency of oscillation associated with an excess voltage oscillation, and is determined by the transit time of an accumulation layer originating at the cathode side of the NDM element and propagating toward the anode domain. Bias tuning is also apparent from Fig. 50. For periodically quenched conditions the frequency increases with increased bias (19, 179). This is displayed for the case (bB = 4 and 5 4 T H . For (bB = 6(bTH,where excess voltage oscillations occur, we note a drop in frequency. 3 . Calculations with the Time-Dependent Cathode Boundary Field Model
In the above discussions where we examined circuit, bias, and contact effects, little attention was given to the behavior of NDM elements with very high fixed values of cathode field. Generally, these NDM elements
GUNN-HILSUM EFFECT ELECTRONICS
397
have been identified experimentally (20) by their low AC output and preinstability I(+) characteristics that saturate toward a value of current density less than or equal to Js. Until 1972, when it was reported that a class of indium phosphide devices with low preinstability current saturation were accompanied by anomolously high output (162), lowsaturating-current devices were thought to be signatures of low-output devices. Explanations of the indium phosphide results point to the cathode contact as the origin of the effect (113, 180, 181), and successful simulations have required that the cathode field execute substantial time dependence. Fixed, time-independent values of cathode field are not able to simulate the behavior of indium phosphide operating in this mode. To incorporate time-dependent effects at the cathode, and to assure current continuity, we explicitly separate the current at the cathode into two parts (181): a field-dependent conduction current J,(E,) and a displacement current (19, 21, 180)
Jo(7‘)
=
JJE,)
+ E dE,/dT
(62)
where is the permittivity of the semiconductor. In our calculations the behavior of GaAs can be simulated when the cathode field is limited to a narrow range of values. This occurs when E, is limited in range by the form of J,(E,) or by restrictions in the amplitude of J o , as determined by the device and circuit. The anomalous behavior of indium phosphide can be simulated when both (1) the current J,(E,) is limited and (2) E, exhibits significant time dependence. These points are discussed below, where we concentrate on determining the general relationship between the form of J,(E,) and the resulting oecillations. The form of J,(E,) is obtained from the following equation:
[ (-:f$~)exp [
J , ( E , ) = -J, exp
-
-(n-1-
1)- e E k c3L c ] }
(63)
which was adapted from a general model of metal-semiconductor contacts (159). Its use here presumes a similar description. For an unalloyed contact, n is the ideality factor and describes the contact as dominated by thermionic emission (n = 1) or by tunneling (n s 1); J , is the reverse flux and may be related to the barrier height through the Richardson equation (1821,
J, = m*AF2exp[-(R,/kF)]
(64)
where 0,is the barrier height in electron volts, A Richardson’s constant, = 120 A cm-2K-2, k Boltzmann’s constant, F the absolute temperature, and m* the ratio of an appropriate effective mass to that of the free electron mass. The results depend significantly on the value of J,, which we
398
M. P. SHAW ET AL.
translate into values for a,. For this we have arbitrarily assigned a value of 0.063 to in* as representative of the principal valley of GaAs. The relation between J , and 0, is displayed in Table 111. Typical barrier heights for our discussion are of the order of 0.2 eV. The parameter L, is specific to the formulation we use and does not appear in Rideout (159). It is necessary for coupling contact equation (63) to space charge equation (60). The parameter L, is nevertheless conceptually ambiguous. We regard L, as representing the width of the alloyed region, with n and J , representing more closely the propertiestof the metal-semiconductor interface. Very long alloy regions, i.e., relatively large L , values, may be expected to produce low values of electric field at the cathode boundary of the NDM element. Figure 51 displays the dependence of the form of J,(E,) on the choice of parameters. Note that widely different parameters can yield similar J,(E,) curves. Thus, in the case of the pairs 1 and 2, the isothermal properties of the significantly different contacts would be expected to yield similar device behavior. We show below that curves 1, 2, and 3 yield results similar to those obtained from the time-independent cathode field model. Curves 1, 5, 6 show significant differences, with the former reproducing the essential isothermal properties of the anomalous indium phosphide oscillations, To determine the dependence of the oscillation on the form of J,(E,), we first ask whether a sustained or damped oscillation will occur. Generally, damped oscillations occur when only a cathode depletion layer is consistent with the circuit, contact, and space charge equations. For 10pm-long NDM elements, the presence of a local cathode adjacent accumulation is provisionally regarded as a necessary condition for a sustained oscillation. Note, however, that cathode-adjacent accumulation layers are under certain conditions small-signal stable (183), a point that we shortly discuss. TABLE 111 -f&
BARRIER HEIGHT: = K.YIn(Jr/rn*AP)
1.oo 0.80 0.40 0.20 0.10 0.05
0.14 0.15 0.16 0.18 0.20 0.22
GUNN -HILSUM EFFECT ELECTRONICS
3 99
w
n
P
i-
a u
WEP CATHODE ELECTRIC F I E L D
FIG.5 1 . Cathode conduction current density vs. electric field for seven sets of parameters n, J , , and I,.
The presence of cathode accumulation or depletion is determined exactly by differential equation (60) or approximately by comparing the relative values ofJc(Ec)and the neutral current density Jn(Ec) = N0ev(Ec)(96). For J , ( E , ) > J,(E,) cathode accumulation occurs, whereas for J,(E,) < J,(E,) cathode depletion occurs. Thus, for the range of bias values considered in this section, where 3 4 T " < +B < 7 4 T H r curves 1-5 of Fig. 5 1 allow the presence of cathode accumulation, whereas curve 6 only permits depletion. Curve 7 is expected to yield only damped oscillations, a conclusion supported by numerical simulation. To trace the time-dependent behavior associated with curves 1-5 we note that for the circuit of Fig. 44 the initial voltage dependence + ( t ) is determined almost entirely by the values of the inductance and the resistance of the circuit (19,179). Capacitive contributions are less important; for the initial voltage increase the cathode field values are determined by the relative values of J,(E,) and the linear part of J,(E). For the case of
400
M. P. SHAW ET AL.
curves 1, 2 the initial voltage increase is accompanied by a weak cathode depletion layer. More prominent depletion layers form for curves 3-5. The subsequent time development in the circuit of Fig. 44 is as follows. For curves 1, 2, E, reaches a maximum value when r$ exceeds r$TH. There is a transition from a depletion layer to an accumulation layer. The initial effect of a time-dependent E, is similar to that of a timeindependent E, with values of the latter somewhat in excess of E , . If the subsequent time development of E , confines it to positive values then the I(& lissajous figure during sustained circuit controlled oscillations will be similar to those of previous chapters. Figure 52 illustrates this point. Figure 52 displays lissajous figures for NDM elements with different J,(E,) curves. Each figure also displays E ( X , T ) at four instants of time, with the times identified by the lissajous figure. The bold-line portion of the J,(E,) curve identifies the range of E, during the course of an oscillation. Specifically, Fig. 52a shows calculations for J,(E,) curves 1 and 2 of
4 -
EIEp, EclEp
EIEp, Ec/Ep
m j 0.43
5.9
0.35
FIG.52. (a) Computations of current vs. average electric field for curve 1 of Fig. 51. Here c#,~ = 6.OEP/,fo = 10.8 GHz, and Z o / R o = 9.0. Also shown are cathode conduction current curves and electric field vs. distance profiles at four instants of time (indicated on the lissajous figure). The heavy lined portion of the J&) curve denotes the range of E, values during the course of the oscillation. (b) As in (a) but for curve 2 of Fig. 51 [from Grubin (178)l.
GUNN-HILSUM EFFECT ELECTRONICS
40 I
Fig. 51. In both cases we see that E, is confined to values about E,. Two sets of calculations are presented to illustrate that, as in the case of the time-independent cathode field model where an optimum value of E, existed, for the time dependent calculation an optimum J,(E,) curve exists. In the case of Fig. 52b, we also see the presence of excess voltage oscillations. We next consider the oscillation properties of curves 3, 4 of Fig. 51. Here, early in the cycle J,(E) is significantly larger than J,(E,) and high cathode fields result. A significant cathode depletion layer forms; there is a reduction in the threshold current density and a consequent reduction in the amplitudes of the current and voltage swings. This latter feature may be expected to reduce the range of E, values and time-independent cathode field results may be realized. This is illustrated in Fig. 53a for the J,(E,) curve 3 of Fig. 51. The results for this case are similar to those we have obtained for E, values that are time independent and within the saturated drift velocity region. Note that the oscillation in Fig. 53a may also
J/Jp
30r
30r
1 1 1 CUiVE
13.4
111)) 0.15 0.45
FIG.53. (a) As in Fig. 52 but for curve 3. (b) Curve 4 is used [from Grubin (178). with permission].
M. P. SHAW ET AL.
402
be classified as an excess voltage oscillation due to the presence of a permanent residual space charge layer at the cathode boundary. [This is not the anode-type of bias-dependent excess voltage oscillation emphasized in the last section (see also Fig. 52b).] For the case of curve 4 of Fig. 5 1, the initial transient response is intermediate between curves 1 (and 2) and 3. This occurs because the cathode adjacent voltage drop is initially intermediate between the two. However, the conduction after the voltage across the NDM element exceeds bHr current through the NDM element and J,(E,) are approximately equal. Thus, the field immediately downstream from the cathode is approximately uniform and the cathode half of the device operates as a uniform field oscillator. There is a consequent improvement in performance, illustrated in Fig. 53b. Note the expanded range of E, values and the fact that the minimum voltage is less than bH. Such a result could not be simulated using a time-independent cathode field model. It is worthwhile pointing out that while it is necessary for .I,(&) > .I#) for a wide range of values, it is also necessary that the curves intersect at low values of E c . Intersection at high E, values will result in damped oscillations at lower bias levels.
JIJp
l
b 0
4
l b ;
-
20
30r
20
0
E/Ep, Ec/Ep
EIEp, Ec/Ep
30r
FIG.54. (a) As in Fig. 52 but for curve 5. (b) Curve 6 is used [from Grubin (178). with permission].
GUNN - HILSUM EFFECT ELECTRONICS
403
0.3 1
PfW)
2 3 4 5 6 7
##PQ 15
P(WI
r) f%)
0.3
0.1 "
3 4 5 6 1
FIG.55. (a) Prethreshold current density vs. voltage for curves 3 (top) and 4 of Fig. 51. (b) Efficiency and output power vs. & / E D / [from Grubin (182).with permission].
It may be argued that the contact requirements represented by Fig. 53b are unrealistic. What can be tolerated for high-efficiency oscillations? Figure 54a illustrates a calculation for a larger ideality factor and a lower value of reverse flux. Note the higher intersection point of the two curves, and the lower, but nevertheless respectable output. Figures 53b and 54a illustrate that high-efficiency devices can result from current-limiting contacts whose cathode conduction current curves intersect the neutral J , ( E ) curve at low values of E,, and whose high-field portion is approximately equal to the saturated drift current value of the semiconductor. The importance of low intersection points is emphasized in Fig. 54b, where we display calculations for a current-limiting contact whose results are inferior to those in which the cathode field is restricted (see Fig. 54a). Figure 55 displays the DC characteristics and output for NDM elements having the J , curves of Fig. 51, curves 3,4. Both cases produce DC characteristics that saturate, yet the output of the two are considerably different. The output for a high-E, device using the fixed-cathode field model is similar to that of Fig. 55, curve 3. 4. Summary
We have presented a general discussion of large-signal X-band selfexcited oscillations as they apply to short, 10-pm-long NDM elements. The results of the simulation are that NDM elements may sustain two
404
M. P. SHAW ET A L .
classes of self-excited oscillations. One is where the minimum voltage is less than the negative differential mobility threshold voltage; the other is where the minimum voltage exceeds the threshold voltage. The former is similar to the length-independent relaxation oscillations associated with long samples. The latter is apparently specific to short devices where the cathode field is restricted to a narrow range of values during the course of an oscillation. [Hobson (184) reported the observation of circuitcontrolled oscillations in GaAs in which the minimum voltage exceeds the measured threshold voltage.] The results of our calculations were shown to depend significantly on the condition at the cathode boundary; we found by using the fixedcathode boundary field model that there is an optimum value of cathode field for maximum efficiency oscillation and that this value depends on the circuit parameters. Since the cathode field also determines the value of current at which instability occurs, depending upon which side of optimum one is at, measurements of variations in threshold current for different samples can be correlated to either increases or decreases in efficiency. Similar results occur for time-dependent cathode fields restricted to a narrow range of values. [Wasse ef al. (174) observed increases in efficiency with decreasing threshold current; Gurney (185) reported decreases in efficiency with decreasing threshold current density.] The time-dependent cathode field calculations show them to be capable of displaying the anomalous oscillations associated with indium phosphide. As a result of the time-independent and time-dependent calculations, we conclude that DC preinstability measurements are still a necessary requirement for identifying the contact. While these measurements often yield no information other than the fact that a nonuniform distribution of charge is present at either end of the device, when coupled to microwave measurements they provide an important step in classifying the properties of the contact region. Bias tuning results should also be examined carefully. They can suggest whether a device is space charge or circuit dominated. The former may often be associated with the propagation of an unstable space charge layer from the cathode to a permanent dipole layer at the anode, with its associated excess voltage oscillations. Voltage probe techniques are also useful; with respect to short devices they are now accessible (186). This experiment should also be performed when possible. All of our isothermal calculations demonstrated that the behavior of an NDM element is governed primarily by the form of the cathode conduction current density curve. Temperature-dependent measurements are also necessary for classifying the device contact, as discussed in earlier parts of this section.
GUNN-HILSUM EFFECT ELECTRONICS
405
A final note: I n this section all calculations were performed for parameters relevant to X-band GaAs devices. However, anomalously highefficiency oscillations have been reported only for indium phosphide. The results of this discussion can only suggest that the absence of such an oscillation in GaAs is the consequence of a barrier height that is too high, and/or an inadequate mechanism is present for transport across the alloyed metal/semiconductor interface. D . Amplijicution 1. Introduction
In Section I and Grubin et ul. (21) we demonstrated that a lower limit for the N,I product exists below which traveling space charge layers decay (subcritical devices). Although such samples cannot be induced into oscillation at high bias, they can provide reflection amplification over many bands of frequencies. Experimental verification of this phenomenon was first reported by Thim er (11. (187, 188). McCumber and Chynoweth (42) put forth the initial understanding. It was also shown subsequently that amplification could also occur while an NDM device was in an oscillatory mode (189). Although the aforementioned devices are interesting in themselves, it was later observed that by far the best amplifying devices could be made by stabilizing supercritical devices (190), i.e., using those that could be induced into oscillation over a range of applied bias. Because of the much greater technological importance of such devices, we emphasize them in this section. The analytical basis of supercritical arnplification is discussed in Grubin et 01. (21, 181-183). The experimental aspects of the device are examined below. We show precisely how the stabilization process works, what the important parameters are, and what direction to take to design a broad-band, low-noise, high-gain microwave amplifier.
2 . Supercritical Gunn Diode Amplifiers Perlman (191, 192) first observed that supercritical devices could be stabilized and made to act as high-performance amplifiers. It was initially thought that stabilization came about via the high-anode-field solution evolving from a low-cathode boundary field device (20, 107, 109, 162, 193, 194). However, more recent work (f54,183, 195, 196) has indicated that the dominant mode of behavior of high-gain supercritically doped NDM amplifiers arises from initially depleted regions at the cathode, and that in all cases amplification ultimately results from those regions of the device
M. P. SHAW ET AL.
406
that are subcritical. “Supercritical” amplifiers are clearly not the best way of describing this useful mode of device performance. These amplifiers behave in the following manner: Often no region of instability is observed (195, 197);the device amplifies for all values of bias above some critical value, which is often near the point at which the pulsed mode I(4)characteristics exhibit a large departure from linearity. Sometimes a region of instability appears, sandwiched (in bias) between two regions of stability that produce amplification. Experimental results and associated simulations that demonstrate such behavior were performed by Spitalnik et al. (154) The details are as follows. Epitaxial sandwich structure devices, typically 8- 10 pm thick with N o in the range 0.9 x 1015-1.3 x 1015cm-3,were fabricated by vapor phase growth. N o was determined by differential capacitance measurements on samples grown adjacent to the device layers. A notch in majority carrier distribution was observed at the n-n+ (substrate) interface. “Mesa” structures were etched from the layers, diced, and mounted in either an S4 package or at the end of a 50 SZ microstrip line. Electrical measurements were performed in both low-duty-cycle pulsed (to avoid heating) and DC-driven modes. The pulsed-mode I(4) characteristics for a typical sample are shown in Fig. 56, and Fig. 57 shows the pulsed-mode impedance Z ( W )obtained in a 50 SZ mount with the use of a modified Hewlett-Packard model 8410-A network analyzer. The results shown in Fig. 57 correspond to two values of negative bias in
I
NOTCH AT ANODE
p-
~OTCH AT CATHODE
4
I
0
5 VOLTAGE (VOLTS)
FIG.56. Pulsed (-) and numerical (---)current vs. voltage for both polarities (positive and negative). The simulated device has 1 = 8m ,,, and N o = 1 . 1 x IOL5/cm3,and contains a cathode “doping notch” 1.8 ~JTI wide with N o = 0.2 x 10*0/cm3.This is an approximately 80% doping notch, which is four times greater than the measured notch in majority-camer concentration [from Spitalnik er a / . (154). with permission], indicating that a mobility notch is also present in the sample.
GUNN-HILSUM EFFECT ELECTRONICS
407
FIG.57. Pulsed (-) and numerical (---) impedance for two values of bias voltage. All values lying outside the Smith chart (heavy line) indicate negative impedance or gain [from Spitalnik pf 01. ( / 5 4 ) ,with permission].
Fig. 56: 3 V (which is below the instability range of 3.2-5.8 V) and 6 V (above the instability range). The agreement between experiment and the numerical calculation is good. In particular, the results show that when the notch is at the cathode, bias-induced stable-unstable-stable regions can occur, where gain is possible in both regions of stability. For this particular case the calculated electric field configurations just prior to and just after cessation of the instability are shown in Fig. 58. This stable-unstable-stable behavior was also observed and discussed by Boer and Voss (198) in the NDC element n-CdS. As discussed in Section 111, the high-field cathode-adjacent domain becomes unstable at a critical current, provided a critical domain width is exceeded. Once sufficient bias is applied such that the high-field region can fill the entire sample, a stable solution with the field at a maximum near the anode is possible. [For a different set of sample parameters, this configuration can also be achieved without passing through an unstable regime (195).] The difference in gain and frequency at maximum gain observed in the different stability regions (for negative polarity) can be explained in terms of the calculated electric field profiles. In the initial stability regime
M. P. SHAW ET A L .
408 1
5
1
DISTANCE (MICRONS ) FIG. 58. Calculated electric field vs. distance profiles just prior to (3 V) and just after cessation of the instability (6 V). The doping notch covers the region from 0.5 to 2.3 ,.un [from Spitalnik er al. ( / 5 4 ) ,with permission].
(between 2.5 and 3 V) the high field is restricted to the cathode side of the device and a downstream region is present that is below the NDM threshold. This quiescent region acts as a parasitic resistor, decreasing the overall negative impedance. In the final stability regime (above 6 V), most of the active region is in or above the NDM regime, which produces higher gain. We also expect the frequency at maximum gain to be inversely proportional to the transit time of a carrier across the high field region. The wider the region of high field, the lower the frequency of maximum gain. For positive polarity the device is stable at all values of bias and shows a small gain above a critical bias (6 V). In this polarity the doping notch is at the anode and a high field exists there under all bias conditions. As demonstrated for negative polarity, a high field configuration can result even without an anode notch, and, as discussed in Section 111 and Shaw et al. (ZO), the stability of this configuration is presently a subject under careful investigation. Typical experimental results obtained under a DC-driven bias are bandwidth: 7-16 GHz; maximum Z ( o ) : 45 at 12 GHz; noise figure: 13.5 dB at 11 GHz. The measured noise figure with these diodes is much lower than the calculated minimum value for high N,I product devices (102, 193). This is probably due to the relatively uniform field profile and a diffusion coefficient that becomes small at high fields. It is clear from the above that at least two distinct regions of amplification can exist: a preinstability depletion layer mode and a postinstability high-anode-field mode. These possibilities have been discussed in detail in Grubin et al. (21, 181 -183), where it was demonstrated from a study of
GUNN - HILSUM EFFECT ELECTRONICS
409
the small-signal impedence properties of supercritical NDM elements that the various modes of device amplification are determined primarily by the cathode condition and the bias. Recent techniques for developing NDM amplifiers involve a close comparison of computer simulation and experiment in various phases of device design. Spitainik (199) compared the evolution of the large-signal impedance with the RF driving signal for different doping profiles, and suggested design figures to increase the power output. For example, the three different doping profiles shown in Fig. 59a give rise to the three stationary-field distributions shown in Fig. 59b. The three devices were la)
t
I
t 1014;
I a
0
2
2
1
4
1
6
*
I
!
8
I
10
8
10
-
DISTANCE (MICRONSI
I
0
(bl
2
4
6
DISTANCE (MICRONS)
FIG.59. (a) Doping profiles used in a large-signal computer simulation of amplification from NDM elements. (b) Corresponding electric field vs. distance profile [from Spitalnik (199), with permission].
410
M. P. SHAW ET AL.
studied under similar operating conditions to compare their large-signal behavior. In each case the doping level in the flat region, the ratio of doping level in the flat region to doping level at the bottom of the notch, and bias voltage were chosen such that all three had the same dissipated DC power and similar small-signal impedances. All other parameters were taken to be identical, including a 200°C working temperature, which corresponds to that of experimental devices deduced from thermal resistance measurements. The corresponding small-signal impedances calculated from the step response are shown in Fig. 60a for a DC power of 7 W from 7 to 11 GHz. Generally, the calculated small-signal impedances are of the same order as the network analyzer measurement errors and can therefore be considered as approximately similar. The large-signal impedance evolution with increasing signal level of three devices, corresponding to each profile, is shown in Fig. 60b for three different frequencies (7, 9, and 1 1 GHz). The difference in behavior is clear. Profile 1 has the lowest added power (Pout- Pi,)and efficiency. It presents no gain expansion with a 50 load at 7 and 9 GHz and a small gain expansion at 11 GHz. Profile 2 has intermediate added power but substantial gain expansion at all frequencies. (This effect is associated with an electric field rearrangement as the injected space charge is strongly amplified when it sweeps through the sample. The anode electric field then swings above and below threshold during the R F cycle.) Profile 3 exhibits the largest added power over the band. The hairpinlike impedance saturation characteristic limits the gain expansion at 7 and 9 GHz. However, it shows substantial gain expansion at 11 GHz, but without a reactance variation. Here, even if during the cycle the electric field drops below threshold in portions of the sample, it always remains above threshold at the anode. The last profile is recommended by Spitalnik (199) for high-power applications. It yields the highest added powers and a large-signal impedance description that facilitates the circuit design for minimum gain expansion. It should be noted that the relative differences in the doping values for two successive profiles is only 20%. If we take into account the profile measurement error and the layer inhomogeneities, an error of this order can easily be attained. It is therefore difficult to predict the device behavior from the profile measurement. Another important result is that no significant conclusion can be drawn about the doping profile from small-signal impedance measurements. However, according to this simulation, large-signal impedance measurements could provide useful indications of the doping profile. Others (200) have used similar techniques to that of Spitalnik and have
GU”-HILSUM
0’0
0.2
05
EFFECT ELECTRONICS
1 :o
1%
41 1
5.0
FIG.60. (a) Small-signal impedance plotted on a Smith chart for the three devices whose profiles are shown in Fig. 5%. The calculated impedances are at 7, 8, 9, 10, and 1 1 GHz. (b) Large-signal computed impedances, with the added power as a parameter at 7, 9, and 1 1 GHz. The circuit used in the large-signal calculation is shown in the inset. - Z , is represented in (a) and (b) [from Spitalnik (199). with permission].
shown that devices with a cathode notch large enough to cause a flat electric field distribution at the working point show the best frequency stability. Computer programs directed toward designing cathode doping notches capable of achieving uniform electric fields downstream from the cathode are currently in the literature [see, e.g., Raymond et a / . ( 2 0 / ) ] .
M. P. SHAW ET
412
AL.
3 . Summary
We have emphasized amplification via supercritically doped samples, although their stability involves fields in the NDM region only over subcritical regions. Such devices produce the most efficient amplifiers. In many cases these devices exhibit a bias-dependent stable-unstablestable mode, where the latter stable field configuration produces the highest gain. Here the field profile usually shows a slope that increases monotonically from cathode to anode. An experimental confirmation of the mode was presented, along with numerical simulations suggesting parameters for the design of an optimum performance device. E . The Gunn Diode in a Microwave Circuit 1. Introduction
In this section we have discussed in detail the behavior of short device-grade samples. We saw that the oscillatory mode in a specific circuit containing reactive elements was characterized by an I(+) Lissajous-type figure and that under certain conditions the temporal evolution of the voltage across the device was nonsinusoidal. The problem of determining the conditions required for sustained oscillations was nonlinear and detailed and, in some cases (as for long samples), the voltage was required to drop below about +J2 in order to quench large field nonuniformities. Almost all of our discussion of NDM-induced oscillations has been in the time domain. Often, however, the oscillatory behavior of NDM oscillators is analyzed in the frequency domain, with the device represented as a negative resistance. While we have generally avoided this description because it does not naturally reflect the detailed contribution from space charge effects, a discussion of this procedure is in order because of its use in practical oscillator design. The problem of characterizing a microwave circuit with an NDM element has to be examined from several points of view. The characterization can be analyzed from the field viewpoint (202)or from the point of view of lumped elements. At X-band frequencies and below, lumpedelement approximations are adequate. We consider them now.
2. Circuit Representation Early work on the characterization of waveguide cavity NDM oscillators was reported by Tsai et al. (203) and later by Jethwa and Gunshor (204).A measure of the success of the characterization was the ability to
413
GUNN-HILSUM EFFECT ELECTRONICS
predict the oscillation frequency of a tunable oscillator. Although we concentrate on these studies, other important work should also be emphasized, in particular that of Freeman and Hobson (205) and Curtice (206). The thrust of Tsai er a / . (203)and Jethwa and Gunshor (204)rests on an understanding of the waveguide mounting structure for the diode, a problem elegantly treated by Eisenhart and Khan (207),who extended the induced-field method and applied it to derive the driving point impedance of a common postcoupled waveguide structure used for holding small microwave devices. They developed and discussed in detail an equivalent circuit for such a mount, which is shown in Fig. 61. Tsai et at. (203)developed a simple equivalent circuit for this mount, which is shown in Fig. 62. This circuit was deduced as follows: The diode is located in the gap of a cylindrical post that is shorter than the height of the guide. The resonant post is represented by a T-network, where the shorter element is a capacitor C , and an inductor L , in series (208).The capacitor models the electrical energy stored in the gap between the post and wall of the waveguide. The inductor models the energy stored in the magnetic fields of nonpropagating modes surrounding the post. The capacitor C,, in the T arms, models the phase shift caused by the finite post diameter. The device package has an internal lead inductor L, and package capacitance C1. A movable slot is also included in this configuration. Note the important point that if the effect of L , is negligible and the device is represented Y
T b
/
z
-X
/
FIG.61. Typical postcoupled waveguide structure used for holding small microwave devices. a denotes the width of the waveguide, b the waveguide height, h the gap (g) position (center to bottom), s the post position (center to side), and w the post width [from Eisenhart and Khan (207),with permission].
M. P. SHAW ET AL.
414 DC FEEDTHROUGH.-
SLIDING SHORT
r l,
GUNNDIODE
CY LlNDR ICAL -IRIS
Cp (POST)
Cp (POST) 11
CG(GAPI
==
Cl, (PACKAGE)
(b)
FIG. 62. Microwave equivalent circuit (b) for a cylindrical-post mounted-waveguide transferred electron oscillator (a). [From Tsai er a / . ( 2 0 3 , with permission].
by a negative resistance in parallel with a capacitor, the circuit discussed in Section IV,C (series L, parallel C)plays a major role in determining the response of the system. A similar circuit was later analyzed by Jethwa and Gunshor (204), who extended the earlier work of Tsai et al. Their analysis employed analytic expressions for all the lumped elements, and the circuit model was used to predict tuning curves that were successfully compared to experimental tuning curves obtained for iris coupling, full and reduced height waveguides, and for various post diameters. The circuit of Jethwa and Gunshor is shown in Fig. 63. The circuit elements that represent the device are a negative resistance RD and effective device capacitance CD (an
GUNN-HILSUM EFFECT ELECTRONICS
415
LOWPASS FILTER
X-BAND OUTPUT
INDUCTIVE IRIS
(b)
FIG. 63. (a) Waveguide-mounted transferred electron oscillator configuration. (b) Equivalent circuit for the waveguide mounted oscillator [from Jethwa and Gunshor (204). with permission].
equivalent representation with a shunt capacitance is also used). The encapsulation is represented by the requivalent circuit representation having L 1 , C1, and C2.The device is again supported in the cavity by a post, whose representation is as described by Tsai et a / . (203) (Tsai et al. determined the parameters of the post experimentally; Jethwa and Gunshor (204) used a completely analytic approach). A variable short of impedance Z is ako employed along with a coupling ins of impedance Z. The impedance parameters of these elements are obtained from Marcuwitz (208).
416
M. P. SHAW ET AL.
3 . Circuit-Controlled Oscillation
Representing the devices as a negative resistance RD with an effective capacitance CD allows us to determine the conditions for oscillation within this model and to compare them with experiment. First, representative expressions for the lumped elements of the equivalent circuit are used to obtain an expression for the load impedance ZL(o)and admittance YL(o) at the terminals of the device (Dl - D z , Fig. 63):
An equivalent circuit representation for the device is then given either in terms of its terminal impedance Z,(W) or equivalent admittance Y,(w):
The theoretical tuning curves in Fig. 64 are obtained by application of Eq. (70), which results in a transcendental equation that is solved numerically. The experimental tuning curves are obtained by varying the distance L between the waveguide sliding short and the device plane, with the direction of motion of the short both away from and toward the device plane (209-212). As shown in Fig. 64, the curves follow the nA,/2 cavity modes quite well ( n is an integer and As the guide wavelength). Some hysteresis in the tuning curves occurs, and mode jumping or switching is indicated by the arrows. The mode switching is thbught to be due to the nonlinear behavior of the device. This conclusion was suggested by the observation of nonsinusoidal device current waveforms in the region where the switching occurs (204). Jethwa and Gunshor also point out that they were not able to use simple models based on Eqs. (65) and (66) to explain the switching, as they did not include the inherent dynamic nonlinearities of the problem. While the frequency domain lumped-constant equivalent-circuit anal-
417
GUNN-HILSUM EFFECT ELECTRONICS
0
I
1
I
2
I
1
4
3
L
5
I
6
1
7
L Icrn)
FIG.64. Frequency vs. distance between the sliding short and the device plane. -, theoretical in a parallel CD-RDconfiguration; n & / 2 cavity modes; X , experimental. The arrows denote mode switching [from Jethwa and Gunshor (204). with permission].
---.
ysis offers difficulties in interpretation in the vicinity of mode switching, away from these points it correctly represents many aspects of Gunn 0scillator tuning. Indeed, under conditions of free-running oscillation, the device resistance and reactance may be obtained from the conditions of Eqs. (69) and (70). We also note in passing that the conditions for steady free-running oscillation as given by Eqs. (69), (70), and equivalently (711, (72), are only necessary conditions. In addition, the stability of the oscillation requires that (204, 213 -215) d X ( w ) / d w > 0,
dB(w)/do> 0
(73)
In summary, we see that analysis of a lumped-element equivalent circuit of a waveguide-mounted Gunn oscillator, assuming the device to be a negative resistance, predicts several major experimental features of device behavior. Those aspects of the device behavior that can be attributed to nonlinearities, however, are not adequately modeled at present. F. Stute-ofthe-Art Considerations; Noise and Output Pobivr
We have discussed the theoretical aspects of NDM oscillators and amplifiers and have also supplemented the discussion with experimental data. From a technological viewpoint, these devices must now be compared to other available types. We first compare the oscillator and amplifier noise of NDM oscillators and amplifiers with avalanche and other devices. We then present a summary of recent stage-of-the-art results for
M. P. SHAW ET AL.
418
gallium arsenide and indium phosphide devices with regard to their output power as a function of frequency. From either a systems or applications point of view, noise is in general any unwanted signal whose effects must be minimized. From a diagnostic point of view, noise measurements provide information about the space charge distribution within the device. Experimentally, noise measurements provide a description of the spectral noise power density (216). For oscillator noise, and when the noise power is a small fraction of the total output power, the spectral contribution is usually separated into distinct amplitude modulation components and phase and frequency modulation components. In general, the AM and FM components are correlated, but in practice this correlation is often ignored. In the case of amplification, noise is usually described as a ratio of the signal to noise power before amplification to that after amplification. Noise measurements are also described in terms of a noise temperature, thought to bear a direct relation to the phenomenological electron temperature of the device. The analysis of noise usually relates the noise spectrum, which is empirically accessible, to the noise correlation function, which can be related to the mechanisms of noise generation (2/7-219). Figure 65 shows DSB measurements (218) of the AM noise to carrier power in a bandwidth B, (N/C)AM, and the rms frequency deviation in a bandwidth B, Af-, for three oscillators each containing the same diode whose active layer is 9 pm long and whose doping level is 8 x 1014cm-3. Differences in the results are circuit originated and depend upon the Q of the circuit. The AM and FM noise show an approximate llfdependence out to about 200 kHz. About 300 kHz the spectra are frequency independent. Figure 66 shows, for comparison, noise measurements on GaAs and Si avalanche diodes, some of which were operated in the circuits of Fig. 65b and some in the circuit of Fig. 66b. The GaAs avalanche diodes are noisier than the NDM devices (because the carrier multiplication process is inherently noisier than intervalley transfer relaxation effects), but they both exhibit I/fnoise at the lower frequency range followed by an almost frequency independent spectra. The silicon avalanche diode was remarkably flat throughout the measured range. We next consider amplifier noise. To characterize the small-signal noise of an amplifier, the noise figure F can be used: F=
available signal to noise ratio before amplification available signal to noise ratio after amplification
Under conditions applicable to small-signal operation, F has been calculated by a variety of workers. For Gunn diodes Thim (102), using the
GUNN-HILSUM EFFECT ELECTRONICS
. r d
f,
(kHi)
f,
IkHZ)
419
RF CHOKE r D C BIAS DIELECTRIC
DIODE
METAL POSTS
IRIS
hi4
~IODE
TYPE B CIRCUIT
TYPE A CIRCUIT
FIG.65. Measured AM and FM noise spectra of three Gunn oscillators containing the same diode and with different Qs; Po denotes output power to load. The waveguide structures containing the diodes are shown at the bottom of the figure [from Ohtomo (218). with permission].
“diffusion-impedance field method” of Shockley ef al. (220),calculated Ffor subcritical diodes sustaining uniform and nonuniform fields. For uniform fields Thim obtains F= 1
+@(-)
1 k 9 dV/dE
For D = 400 cm2/sec and p = 2880 cm2/V sec for GaAs, a minimum noise figure of F = 6.5 (8 dB) is obtained. Typically, GaAs amplifiers exceed this noise level. The results suggest that a NDM device with large
M. P. SHAW ET AL.
420
I
1
I
1
I
B - 1 Hz
1
1
METAL OR DIELECTRIC POSTS
10
1
102
1
1
103
i 1
104
r D C BIAS R F CHOKE
IRIS COAX LINE
TYPE C CIRCUIT
FIG. 66. Measured noise spectra of avalanche oscillators. (a) GaAs oscillators. (b) Silicon oscillators. Circuit C is identified in the inset. Circuit B is displayed in Fig. 65 [from Ohtomo ( 2 / 8 ) ,with permission].
values of dV/dE would have lower noise figures. Significantly, Thim (102) also calculated F for nonuniform fields beginning as accumulation layers at the cathode and found the noise figure to be considerably higher, F = 49 (17 dB). In a later related study, Magarshack et al. (221) reported results of experiments and calculations for nominally stable supercritical devices.
GUNN-HILSUM EFFECT ELECTRONICS
42 I
The devices were designed to yield uniform fields over much of the active region and in this manner produce low noise figures. Noise figures as a function of frequency were obtained and are shown in Fig. 67a. The calculations are for an assumed constant field of 12 kV/cm. The noise figure is seen to approach a minimum in excess of 10 dB but close to the minimum . value of Thim at frequencies just below the transit frequency 1 . 9 ~ Also shown is the output for a notched device (Fig. 67b).
la)
I
n
1.76n
2.08~
2.411
TRANSIT ANGLE
INPUT POWER IdB)
FIG. 67. (a) Calculated (---) and measured (a) noise as a function of transit angle for stable supercritical GaAs amplifiers. (b) Measured saturation curves at two frequencies. Input and output powers referred to 1 mW [from Magarshack el a / . (22/), with permission].
M. P. SHAW ET A L .
422
1 0-
FET
II
TRANSFERRED ELECTRON DEVICE B A R ' T T InP
I
GaAs
Si
AVALANCHE
DIODE
GaAs
Si
5-
-
10-
V
W
E I3 n
2
15-
W
E
0
=
20-
25-
30-
FIG.68. 1975 state-of-the-art X-band small-signal noise measurements: vertical hatching, minimum theoretical value; right-to-left hatching, laboratory results; left-to-right hatching, typical results. M = ( F - 1)/(1 - I/gain) [from Constant (222), with permission].
Figure 68 is a chart comparing the small-signal noise measurements of X-band GaAs, InP, and Si devices. This chart was compiled by E. Constant (222) in 1975. It is seen that the field effect transistor (FET) is the best performer. Constant suggests that low FET noise is due to the fact that the operation of this device does not depend upon hot carriers. In thermal equilibrium the presence of noise has been related to an effective noise temperature through the Nyquist theorem (223). The concept of a noise temperature has also been used for nonequilibrium systems where the measured noise temperature is related to the available noise power. It is generally assumed in these measurements that the measured noise temperature is also a measure of the phenomenological temperature of the electron gas. In an examination of the preinstability heating of carriers in GaAs diodes with a nominal doping level of 10'5cm-3 and lengths of the order of 10 pm, Atanasov and Rzhevkin (224) measured the noise temperature as a function of applied DC power. Their results are shown in Fig. 69 for one device with measurements made in both
423
GUNN -HILSUM EFFECT ELECTRONICS
w
L"
0
z
log(DC POWER l0.05)
-
0.2
-4
;0.15 YI z
-
U U
3
u
0.1
1
0.05 0
I 1
I 2
I 3
1 4
I 5
VOLTAGE ( V )
FIG.69. (a) Prethreshold dependence of the noise temperature on the DC power; I , forward direction; 11, reverse direction. ( b ) I ( $ ) characteristics [from Atanasov and Rzhevkin (224). with permission].
polarities. Also shown are the corresponding I ( + ) curves. For a discussion of these results it is useful to refer to Fig. 70a, which shows a plot of the electron temperature as a function of electric field ( 9 3 , the latter being spatially uniform. While the parameters used for this calculation give a velocity-electric field curve (Fig. 70b) somewhat different than that expected for GaAs, the results are qualitatively applicable. The first point we note here from Fig. 69 is that the I ( + ) curve with the greatest sublinearity also exhibits the highest noise temperature. If the noise temperature were a consequence of a uniform distribution of carriers than we would expect, on the basis of the values of Fig. 70b, a field whose value was well in excess of that required for the onset of an insta-
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bility. Yet these are preinstability curves. In the reverse polarity the noise temperature is lower but still in excess of that required for a current instability. The explanation (224) put forth for the high noise temperatures is that the field within the diode is nonuniform and that a large field exists at the cathode boundary. Numerical calculations (93) showing the distribution of electron temperature for a cathode-notched device with carriers entering from the right are consistent with the conclusion of Atanasov and Rzhevkin (224). They are shown in Fig. 71. As the above description illustrates, NDM oscillators and amplifiers are satisfactory low-noise devices when compared to other two terminal devices (with the exception of Baritt devices), The typical CW output that we may expect from these devices is summarized next. Figure 72 shows the state-of-the-art output power as a function of frequency for GaAs devices. We see an initial increase in output power up to X-band, where the output power begins to decrease with further increase in operating frequency (225, 226). Figure 73a,b shows the dc-driven, single-diode, state-of-the-art efficiencies for InP and GaAs oscillators. The peak efficiency from the chart is about 12% for GaAs
GUNN-HILSUM EFFECT ELECTRONICS
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FREQUENCY IGHz)
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at X-band with approximately 0.5 W of output power. For InP the peak efficiency occurs at around 22% with an output of 0.2 W. Omori (225) pointed out in 1974 that 5-8% efficiencies for GaAs X-band oscillators were becoming commonplace. Although these figures are now six years old, they still represent a reasonably good view of the field.
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Finally, the question of when the effect of a finite intervalley transfer time eliminates the effects of negative differential mobility is placed in perspective with reference to Fig. 73a. Note the presence of oscillations at 75 GHz. Are they a fundamental mode circuit oscillation? G . Sitmmur-y
In this section we have presented a discussion of the operating principles and characterization of short NDM devices. We have seen that the material parameters of the device, the metal-semiconductor contact, the circuit, the space charge distribution, and the temperature distribution all affect device operation. All of these contributions manifest themselves in the DC-driven oscillator output, in the performance of small- and largesignal amplifiers, and in the noise properties of the device. There is now abundant experimental evidence supporting the fact that the space charge distribution within the NDM element is nonuniform and that acceptance of this feature is the only way to reconcile experiment with theory. Device technology is devoted to making better devices. Therefore, better control over the contact procedures and materials growth is always being sought. However, from the point of view of understanding device operation, it is clear that our knowledge of the metal-semiconductor interface is still the weak line. Correlation of experiment with analytical theory and large-signal computer simulation still requires an adequate low-resistance-contact theory. Currently, all aspects of device, contact, and circuit technology are being studied. But new research is being generated by the VLSI (Very Large Scale Integration) program and its push to near-micron and submicron device development. Here the question to ask is whether our physical understanding of devices on very small space and time scales is adequate. [See the discussion by Barker and Ferry (227).] In examining these questions, modifications will and are being made to the Boltzman and density matrix transport equations. Topics such as velocity overshoot (228), finite collision duration ( 2 2 9 ) ,will be given more prominent attention. While these research areas will be pursued on a near and submicron scale, the results emerging from these studies should be able to delineate the extent to which the approximations used in this review are adequate. The picture should be clearer within the next five years.
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169. G. 0. Ladd, Jr., D. E. Cullen, and R. D. Kaul, Report UAR-K43. United Technologies (Aircraft) Research Center, 1971. 170. J. S. Bravman and L. F. Eastman, IEEE Trans. Electron Devices ed-17, 744 (1970). 171. P. S. Fentem and 8. R. Nag, Solid-State Electron. 16, 337 (1973). 172. T. Tasegawa and Y. Aono, Solid-State Electron. 16, 337 (1973). 173. I. B. Bott and H. R. Holliday, IEEE Trans. Electron Devices ed-14, 522 (1967). 174. M. P. Wasse, B. W. Clarke, and R. F. B. Conlon, Electron. Lett. 9, 189 (1973). 175. R. F. B. Conlon and F. S. Heeks, Proc. Bienn. Cornell Electron. Eng. Conf., 4th, 1973 p. 245 (1973). 176. P. M. White and G. Gibbons, Electron. Lett. 8, 166 (1972). 177. R. Davies, W. S. C. Gurnery, and A. Mircea, Electron. Lett. 13, 349 (1972). 178. H. L. Grubin, IEEE Trans. Electron Devices ed-25, 511 (1978). 179. P. Jeppesen and B. I. Jeppson, IEEE Trans. Electron Devices ed-18, 439 (1971). 180. K. W. Gray, J. E. Pattison, H. D. Rees, B. A. Prew, R. C. Clarke, and L. D. Irving, Proc. Bienn. Cornell Univ. Electron. Eng. Conf., 5th. 1975 p. 215 (1975). 181. H. L. Grubin, IEEE Trans. Electron Devices ed-23, 1012 (1976). 182. H. L. Grubin, J . V a c . Sci. Technol. 13, 786 (1976). 183. H. L. Grubin and R. Kaul, IEEE Trans. Electron Devices ed-22, 240 (1975). 184. G. S. Hobson, IEEE Trans. Electron Devices ed-14, 526 (1%7). 185. W. S. C. Gurney, Electron. Lett. 7, 711 (1971). 186. P. J . Fentem and A. Gopinath, IEEE Trans. Electron Devices ed-23, 1157 (1976); T. Hasokawa, H. Fujikowa, and K. Ura, Appl. Phys. Lett. 31, 340 (1977). 187. H. W. Thim, M. R. Barber, B. W. Hakki. S. Knight, and M. Uenohara. Appl. Phys. Lett. 7, 167 (1965). 188. H. W. Thim and M. R. Barber, IEEE Trans. Electron Devires ed-13, I10 (1966). 189. H. W.Thim, Proc. IEEE 55,446 (1967). 190. B. S. Perlman, IEEE Trans. Sulid-State Circuits sc-5, 331 (1970). 191. B. S. Perlman, C. L. Upadhyayula, and R. E. Marx, IEEE Trans. Microwave Theory Tech 18, 911 (1970). 192. B. S. Perlman, Proc. IEEE 59, 1229 (1971). 193. W. 1. Williamson, 13th Natl. Trade Electron. Conv. Australia, 1971, p. 24 (1971). 194. P. Jeppesen and B. I. Jeppson, IEEE Trans. Electron Devices ed-20, 371 (1973). 195. R. Charlton, V. R. Freeman, and G. S. Hobson, Electron. Lett. 7, 575 (1971). 196. H. L. Grubin and R. Kaul, IEEE Trans. Electrun Devices ed-20,600 (1973). 197. R. L . Gunshor, Electron. Lett. 5, 305 (1969). 198. K. W. Boer and P. Voss, Phys. Status Solidi 30, 291 (1968). 199. R. Spitalnik, IEEE Trans. Electron Devices ed-23, 58 (1976). 200. C. Berry, G. S. Hobson, M. S. Howard, and P. N. Robson, IEEE Trans. Electron Devices ed-24, 270 (1977). 201. R. M. Raymond, H. Kroemer, and R. E. Hayes, IEEE Trans. Electron Devices ed-24, 192 (1977). 202. D. C. Hanson and J. E. Rowe, IEEE Trans. Electron Devices ed-14,469 (1967). 203. W. C. Tsai, F. J. Rosenbaum, and L. A. MacKenzie, IEEE Trans. Microwave Theory Tech. 18, 808 (1970). 204. C. P. Jethwa and R. L. Gunshor, IEEE Trans. Microwave Theory Tech. 20,565 (1972). 205. K. R. Freeman and S. Hobson, IEEE Trans. Electron Devices ed-20, 891 (1973). 206. W. R. Curtice, IEEE Trans. Microwave Theory Tech. 21, 369 (1973). 207. R. L. Eisenhart and P. J. Khan, IEEE Trans. Microwave Theory Tech. 19,706 (1971). 208. N. Marcuwitz, “Waveguide Handbook,” MIT Radiat. Lab. Ser. No. 10, p. 255. Boston Technical Publishers, Inc., Lexington, Massachusetts, 1964.
GUNN-HILSUM EFFECT ELECTRONICS
433
209. C. P. Jethwa and R. L. Gunshor, Electron. Lett. 7,433 (1971). 210. B. C. Taylor, S. J. Fray, and s. P. Gibbs, IEEE Trans. Micronwe Theory Tech. 18, 799 (1970). 211. H. J. Fossum, Norwegian Defense Research Est. Tech. Rep. TR-69-238 (1969). 212. M. J. Howes. IEEE Trans. Electron Det’ices ed-17, 1060 (1970). 213. W. A. Edson, “Vacuum Tube Oscillators.” Wiley, New York, 1953. 214. K. Kurokawa, Bell Syst. Tech. J . 48, 1937 (1969). 215. K. Kurokawa, J. P. Bacconi, and N. D. Kenyon, I n t . Microwaite Dig. p. 281 (1969). 216. C. Kittel, “Elementary Statistical Physics. Sects 28-30. Wiley, New York, 1958. 217. J. R. Pierce, Proc. IRE 44, 601 (1956). 218. M. Ohtomo, IEEE Trans. Micronuve Theory Tech. 20, 425 (1972). 219. L. S. Cutler and C. L. Searle, Proc. IEEE 54, 136 (1%6). 220. W. Shockley, J. A. Copeland, and R. P. James, ”Quantum Theory of Atoms, Molecules and the Solid State.’’ Academic Press, New York, 1966. 221. J . Magarshack, A. Rabier, and R. Spitalnik. IEEE Trans. Electron Devices ed-21, 652 ( 1974). 222. E. Constant, Physic0 (Utrecht) 83B, 24 (1976). 223. H. Nyquist, Phys. R e v . 32, I10 (1928). 224. R. D. Atanasov and K. S. Rzhevkin, Sov. Phy.r.-Semicond. (Engl. Transl.) 9, 359 (1975). 225. M. Omon, Microwave J . 17, 57 (1974). 226. R. J. Hamilton, R. D. Fairman, S. I. Long, M. Omon, and F. B. Fank, IEEE Trans. Microwave Theory Tech. 24, 775 (1976). 227. J. R. Barker and D. K. Ferry, Solid S m t c Electron. (In Press). 228. J . G. Ruch, IEEE Truns.. Electron Deimices ed-19, 652 (1972). 229. D. K. Ferry and J. R. Barker, Solid Srrrre Electron. (In Press).
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Author Index Numbers in parentheses are reference numbers and indicate that an author's work is referred to although his name is not mentioned in the text. Numbers in italics indicate the pages on which the complete references are given.
Barber, M. R., 405(187, 188), 432 Bardeen, J., I , 13.60. 335(83), 429 Bardsley, J. N., 138, 163, 177 Barker, J . R., 427(228, 229), 433 Barlow, H. M., 65, 73, 133 Barrat, J. P., 176, 179 Barron, M. B., 200(20), 260 Bastida, E . M., 315, 324(39), 326(39), 327,
A
Abramowitz, M., 77, 94, 132, 133 Acket, G. A., 314(24, 29, 33), 428 Adams, A. R., 313,428 Adams, C. A., 304, 306,308 Adams, M., 300,307 Adler, R., 296,307 Albritton, D. L., 172, 181 Allen, F. G., 48.61 Allen, J. W., 31 I , 314(27), 326(63), 428,
428
Bates, D. R., 138, 155, 158, 163, 177, 180 Bauer, L. O., 247,262 Baur, G., 251(83), 262 Becker, R. L., 163, 181 Bell, D. T., 305,306 Bell, K. L., 163, 165, 166, 177 Bellum, J. C., 163, 166, 177 Bennett, A. J., 335(89, 90), 430 Benton,-E. E., 167, 177 Berak, J. M., 315(41),428 Berger, H. H., 234(49), 261 Bergh, A. A., 249(75), 262 Berreman, D. W., 96, 101, 133 Berry, C., 410(200), 432 Berry, R. S., 138, 144, 145, 147, 151, 152,
42 9
Alusow, J. A., 305,307 Amemiya, T., 66,133 Amick, J. A., 369(131),431 Anderson, R. L., 35,60 Aono, Y.,381(172), 382,432 Appelbaum, J. A., 4, 14, 36, 37, 40, 60 Amaud, J. A., 87, 88, 125, 133 Arnot, F. L., 162, 177 Arthurs, A. M., 175. 177 Asaskawa, T., 369(135), 431 Ash, E. A., 284,307 Aspnes, D. E., 311(7b),428 Atanasov, R. D., 422, 423, 424,433 Aten, J . A., 161, 180 Aubuchon, K. G., 246(72), 247(73), 262 Auerbach. D., 156,177 Auld, B. A., 298,306
154, 155, 156, 159, 160, 161, 163, 164, 165, 177, 178, 180 Bethege, K., 142, 176,179 Beyner, A., 183(1), 187(1), 259 Bianciardi, E., 88, 133 Bigelow, J. E., 250(82), 262 Biondi, M. A,, 138, 177 Blaney, B., 163, 177 Boeckner, C., 161, 180 Boer, K. W.,328(67, 69, 70, 71). 339(69), 340(69), 342(69), 407( 198), 429, 432 Boers, P. M., 314(24), 428 Bolden, R. C., 166, 167, 177, 181 Boleky, E., 242(62), 245(62), 262
B Bachmann, J., 192(12), 195(12), 260 Bacconi, J. P., 417(215), 433 Baertsch, R. D., 287, 292,306 Bailey, W. A., 242(60), 262 Bailey, W. H., 288,3U6 Bandrauk, A. D., 177 Baraff, G. A , , 36, 37, 40,60 435
436
A U T H O R INDEX
Boleky, E. J . , 243,262 Bonch-Bruevich, V. L., 31907, 58, 60), 429 Borgnis, F. E., 73, 133 Born, M., 129, I33 Bosch, B. G . , 316, 328, 371, 372(47), 429 Bosch, R., 342(1 lo), 430 Bott, I . B., 326, 327, 382(173), 383(173), 429, 432
Boyd, G . D., 95, 104, 133 Boyle, W . S., 266,306 Braslau, N., 314, 371(149), 428,431 Braun, F., 1,60 Bravman, J . S., 381(170), 432 Brehm, G . E., 367(127), 369(127),431 Briggs, G . R . , 242(63), 245(63), 262 Brillson, L. J . . 25, 27, 29, 32.60 Brion, C. E., 172, 173, 174, 177, 182 Brodersen, R. W . , 266, 275, 292, 293.306, 307
Bronch-Bruevich, V. L., 319(59), 429 Brood, R. J . , 251(88), 252(88), 263 Brown, J . , 88, 133 Broyer, M., 154, 177 Brutschy, B., 170, 177 Bryant, G. H., 80, 133 Brzozowski, J . , 158, 177 Buhl, D . , 161, 177, 179 Bulman, P. J . , 316, 328, 369, 371, 372(46), 429
Bunker, P., 158, 177 Burgess, A., 155, 158, 159, 177 Burgess, R . R . , 238(56), 261 Burke, B. E., 306,308 Burns, J . R . , 237(52), 261 Bush, H . , 300,307 Buss, D. D . , 266, 275, 288, 292, 293,306, 307
Butcher, P. N., 310(5a, Sb), 311(5a, 5b, 8). 312(8), 313(8), 314(8), 316(5a, 5b, 8,43), 318(8, 54). 323(8, 54). 428, 429
C
Cacak, R., 156, 177 Cahoon, N . C., 251(86, 87), 263 Carlson, R . 0.. 431 Carnes, J . E., 272,307 Carr, P. H . , 284,307 Carrat, M . , 66, I33 Carroll, J . E . , 313(17), 428 Carroll, P. K . , 149. 178
Caruthers, E . , 37,60 Carruthers, J . A . , 431 Carslaw, H . S., 378(167), 379(167), 432 Casperson, L. W . , 1 1 1 , 115, 124, 133 Caudano, R . , 156, 177, 178, 180 Cermak, V . , 162, 171, 172, 173, 174, 176, 178, 179
Chadi, D. J . , 40, 59,60 Chan, K . B . , 88, 134 Chang, D. M . , 314(36), 316(50), 348(36), 428, 429
Chang, L. L . , 37, 49,60, 62 Chang, M. S . , 367( 126), 431 Charlton, R., 40319% 406( 195). 407(195), 432
Chauvy, D., 202(25), 260 Checcacci, P. F., 66, 85, 95, 104, 106, 130, 133, 134, 136
Cheek, T. F., 279,307 Chelikowsky, J . R., 4, 5, 8, 9, 12, 14, 16, 19, 21, 24, 25. 32, 40,60, 61. 62
Chen, C. H . , 167, 168, 170, 172, 178 Chen, Y . S., 311,428 Chenault, R . L., 161, 180 Chibisov, M. I., 165, 182 Child, M . S . , 151, 153, 177, 178 Chinnock, E. L., I 3 4 Choudari, S., 126, 134 Christian, J. R . , 65, 95, 101, 134, 135 Christman, S. B . , 25, 32.62 Chupka, W. A., 142, 145, 147, 148, 156, 162, 178
Chynoweth, A. G . , 31 1, 315, 318, 321, 338, 354(42), 405,428
Clarke, B. W . , 383, 384, 385, 404,432 Clarke, K . K . , 260 Clarke, R. C . , 397(180), 432 Clarricoats, P. J. B . , 80, 88, 134 Clyne, M . A. A . , 173, 178 Codling, K . , 139, 180 Cohen, J . S . , 163, 165, 166, 175, 178. 181 Cohen, M. G . , 367(124), 430 Cohen. M. L., 4, 5, 6, 8, 9, 12, 14, 16, 19, 21, 24, 25, 26, 28, 29, 32, 35, 36, 37, 40, 47, 48, 53, 57, 59,60, 61, 62 Coldren, L. A,, 304,307 Collin, R. E., 73, 77, 134 Collins, C. P., 149, 178 Collins, D. R . , 288,306 Colliver, D. J . , 377, 385(162), 386(162), 388(162), 397(162). 405(162), 431
AUTHOR INDEX Comes, F. J., 178 Conlon, R. F. B., 383(174. 175). 384(174), 38%174), 404( 174). 432 Connor, S. J., 242(63). 245(63), 262 Consorthi. A., 82, 95, 103, 106, 108, 109, 134 Constant, E., 422, 433 Conway. E . D., 298,308 Conwell. E. M..316(44,45), 323(61),328(45, 72). 339(97), 340(97), 342(97). 429. 430 Cook, T . B., 167. 175, 178. 181 Copeland, J. A., 313(14, 15), 342(99), 34314. 15). 419(220), 428. 430. 433 Coppen. P. J., 247.262 Coriell, A. G., 311,428 Cowley. A. M.,13, 24. 61 Cox, R. H., 371( l48), 373,431 Coxon, .I. A , . 173. 178 Crabb, J., 303,307 Craford, M. G., 249(76), 262 Crescenzi, E. J., Jr., 328(65),429 Cretzmeyer, J. W., 253(91), 263 Cricchi, J. R., 245(69), 247,262 Cronshaw, D., 66, 136 Crowder, B., 241(53), 246(53), 261 Crowley, P., 172, 177 Cullen, D. E., 304. 381,308. 432 Culshaw, B., 66, 134 Cunningham, S. L., 36.61 Curtice, W. R., 331(74), 413(206).429. 432 Cutler, L. S ., 418(219),433 Ctyroky, J., 112. 132. 135 D
Dahler, J . S . , 163, 180 Daimon, Y., 271,307 Dalgarno, A., 155, 160, 161, 163, 166, 175, 177. 178, 181 Dalidchik, F. I., 162, 178 Dalisa, A. L., 251(85), 263 D’Angelo, N., 138, 178 D’Angelo, V. S.. 156, 180 Daniels, R. G., 238(56), 261 Dargent, B., 250(80), 262 Datta, S., 284,307 Davidovic, D. M.,163. 179 Davies, C., 173, 179 Davies, D. E. N . , 66. 134 Davies. R.. 386(177), 432 Davies, T. W., 65, 135
437
Davydov, B., 1.61 Dean, P. J., 249(75), 262 Deb. S . K., 251184). 262 DeCorpo, J. J., 166, 178 De Forest, W. S., 241(58), 262 Defrance. P., 156, 180 Degenford. J. E., 96, 134 Dehmer, P. M., 142, 145, 147, 148. 156, 178 Demkov, Yu., N., 166, 178 Denley, D., 35, 36, 4 7 , 6 / Denya, P. B., 295,307 de Troye, N . C., 234(50), 261 Devitt, D., 258(99), 263 Devlin, G. E., 93, 134 De Vore, H. B., 90,134 DeVries, A. J . , 2%. 307 Dingle, R., 37, 49, 61 Dispert, H., 161, 178 Dixon, T . A., 161, 181 Dobrzynski, L., 36,61 Dobson, C. D., 370.431 Dohler, G., 328(69), 339, 340, 342(69, 106), 429. 430 Dolat, V. S., 300,307 Dolder, K . T., 156, 181 Drougard, R., 83, I34 Duke, C. B., 330(73), 33389, 90).429, 430 Dunning, F. B., 167, 175, 178. 179, 181 Duzy, C., 142. 147, 151, 152, 178
E Eastman, D. E., 10, 12,61 Eastman, L. F., 381(170), 432 Ebding, T., 176, 178 Edson, W. A,. 417(213), 433 Edward, J. P., 367( 124), 430 Edwards, R., 200(23), 238,260 Eisenhart, R. L., 413,433 Elander, N., 158, 177 Elliott, B. J., 314(25),428 Ellison, R., 162, 178 Engeler, W. E., 287, 292,306 Engelmann, R. W., 316, 328, 371, 372(47), 429 Enstrom, R. E., 369(132),431 Erdelyi, A., 99,134 Erman, P., 158, 177 Esaki, L., 37, 49, 57,60. 61, 62 Espig, H. R., 253(91), 263 Esser, L. J. M., 273, 277,307
43 8
AUTHOR INDEX
F Fabri, G., 315, 324(39), 326(39), 327, 428 Faggin, F., 238(54, 55), 261 Fairman, R. D., 425(226), 426,433 Falciai, R., 85, 95, 104, 106, 130, 134, 136 Falicov, L. M., 4.61 Fank, F. B., 425(226), 426,433 Fano, U., 150, 178 Farnell, G. W., 307 Fawcett, W., 310(5a, Sb), 311(5a, Sb), 316(5a, 5b, 43, 51, 52), 318(54), 323(54), 326(64), 377(164), 383(164),428, 429, 43I Fehsenfeld, F. C., 167, 173, 174, 175, 180, 181 Feldman, W. L., 311,428 Fellrath, J., 197(15), 199, 200(15), 201,260 Felsen, L. B., 85, 126, 128, 130, 133, 134, I35 Fennelly, P. F., 162, 178 Fentem, P. J., 404(186),432 Fentem, P. S., 381, 382,432 Ferguson, E. E.. 166,167,173,177. 178, 181 Ferry, D. K., 427(228, 229), 433 Fesbach, H., 103, 135 Feucht, D. L., 33, 47, 50, 56, 335, 336(82), 339(82), 61, 429 Field, F. H., 162, 180 Firsov, 0. B., 163, 181 Fisher, R. E., 367(118),430 Fite, W. G., 161, 180 Fite, W. L., 162, 178, 181 Flammer, C., 104, 134 Fleming, J. W., 87, 133 Flores, F., 28, 31, 36,61, 62 Fontijn, A., 138, 160, 161, 162, 178 Foote, P. D., 161,180 Forrer, M. P., 190(3), 201,259 Fort, J., 166, 178 Fossum, H. J., 416(211),433 Foujallaz, C., 190(6),201(6), 205(6), 238(6), 260 Fox, A. G.,95, 101, 103, 107, 134 Foxall, T., 291,307 Foyt, A. G., 314(23),428 Fradin, A. Z., 73, 134 Franklin, J. L., 162, 180 Fray, S. J., 416(210),433 Fredholm, I., 102, I34 Freed, K. F., 142, 146,179
Freeman, K. R., 377(165), 413(205),431, 432 Freeman, V. R., 405(195), 406(195), 407(195),432 Freeouf, J. L., 12, 61 Frensley, W. R., 33, 35, 36, 50,61 Froelich, H. R., 156, 180 Frohman-Bentchkowsky, D., 202(26), 257(26), 260 Fuchs, V., 172, 173, 179 Fujii, H., 163, 165, 166, 179 Fujikowa, H., 404(186),432 Fukatsu, Y.,101, 135 Furukawa, M., 86, 136 Furukawa, Y.,369(133),431 G Gaily, T. D., 156, 177 Gartner, W. W., 345(111), 356(111),430 Gastman, I. J., 334(75, 76, 77), 429 Geballe, T. H., 431 Gedeon, A., 132, 134 Gehweiler, W. F., 243(66),262 George, J., 258(99),263 Gerard, H. M.,299, 300,307 Gerber, B., 190(11), 200(21), 212(36), 214(36), 221(11), 239(36), 260, 261 Gerson, N. C., 111, 134 Giallorenzi, T., 66, 134 Gibbons, G., 377(162), 385(162, 176). 386(162), 388(162), 397(162), 405(162), 431. 432 Gibbs, S. P., 416(210), 433 Giertz, H. W., 66, 134 Giguere, P. T., 161, 179 Glaser, G., 205(31), 207, 208,260 Gleason, R. E., 164, 176, 179 Glissen, T. H., 316,429 Gloge, D., 82, 83, 84, 86, 87, 88, 101, 134 Glover, G. H.,314(32), 315(37, 38), 428 Gnadinger, A. P., 193(13),239(56a), 240(56a),260, 261, 262 Gobeli, G. W., 48,61 Gold, B., 286, 298,308 Goldberg, H. S., 287, 292,306 Goldsmith, N., 369(134),431 Goldstein, Y.,335(80),429 Goodwin, A. R., 370,431 Gopen, H. J., 374, 375, 376,431 Gopin, H. J., 371(147),431
AUTHOR INDEX Gopinath, A., 404(186), 432 Gordon, J., 370,431 Gordon, J. P., 95, 104, 111, 133, 136 Gossard, A. C., 37,61 Goubau, G., 65, 94, 95, 97, 99, 101, 107, 134, 135 Grant, R. W.. 35,61 Gray, K. W., 355(113), 377(162), 385(162), 386(162). 388(162). 397(113, 162, 180), 405(162), 430. 431, 432 Greene, E. F., 168, 179 Greene, P. E., 370,431 Gregory, P. E., 12, 61 Greubel, W., 251(83), 262 Grice, R., 161, 180 Grobman, W. D., 10.61 Grove, A. S., 240(57), 261 Grover, N. B., 335(80),429 Grubin, H . L., 313(19, 20, 21). 314(21a), 315(20, 21, 41), 321(21), 323(61), 328(20, 68). 329(19, 20, 21). 331(19), 334(77), 337(20), 338(20, 68). 339(20, 21), 340(20, 21). 341(20), 342(19, 20, 21). 343(21), 344(19), 345(19), 347(19), 348( 19), 349(19, 21). 352(20), 35319. 20, 21), 357(19), 359(19), 360(19), 361(19), 362(19), 363(19), 373(20), 387(19, 21, 178), 389(19, 21, 178), 390(178), 391(178), 392(178), 394(178), 395(178), 3%(178), 397(19, 20, 21, 181, 182), 398(183), 399(19), 100(178), 401(178), 402(178), 403(182), 405(20, 21, 181, 182, 183, I%), 408(20,21, 181, 182, 183), 428, 429, 432 Gueret, P., 342(104, 105). 430 Gunn, J . B., 310(1. 2, 3). 313(16), 314(22, 25). 321(2, 3). 340(1, 2, 3). 346(3). 368(3, 16). 371(149), 427, 428, 431 Gunshor, R. L., 406(197), 412(204), 4 13(204), 4 14(204), 4 15(204), 4 I6(204, 209). 417(204), 432. 433 Gurnery, W. S. C., 386(177), 432 Gurney, W.S. C., 404,432 Guston, M. A. R., 73, 135 Gyulai, J., 374, 375, 376,431
H Haberland, H., 167, 168, 170, 172, 177 Hahn, A. F., Jr., 367( I19), 430
439
Hakki, B. W., 405,432 Hamann, D. R.,4, 14, 36, 37, 40.60 Hamel, J., 176, 179 Hamilton, R. J.. 425(226), 426,433 Hammer, W., 190(7), 202(25), 203(28), 260 Hanaguchi, C., 314,428 Hannan, W. J., 278,307 Hansch, H. J., 328(70), 429 Hanson, D. C., 412(202), 413,432 Harberland, H., 167, 178 Harrison, W., 12.61 Harrison, W. A,, 14, 36, 50, 61 Hart. K.. 234(48), 261 Hartmann, C. S., 281, 296, 297,307 Hartnagel, H. L., 367(115, 117),430 Hasegawa, F., 369( 138, 139). 431 Hasokawa, T., 404(186), 432 Hashimoto, M., 88, 135 Hasty, T. E., 338, 423(93), 424, 425,430 Hatfield, L. L., 176, 180 Haug. R., 162, 181 Hauser, J . R., 316,429 Hayamizu, T., 367(126), 431 Hayashi, T . , 37l( 15l), 431 Hayes, R. E., 314, 315(35), 411(201),428. 432 Hays, R. M., 281, 296, 297,307 Heaviside, 0.. 1 11, 135 Heeks, F. S., 383( 17% 432 Heilmeier, G. H., 250(77), 262 Heiman, F. P., 243(65), 262 Heine, V., 5 , 6, 13, 60, 6 / , 335(84), 430 Heinicke, E., 176, 179 Heise, C. W., 251(87), 263 Heise, G. W., 251(86), 263 Helfrich, W., 250(78), 262 Hemnett, A., 172, 182 Hemsworth, R. S., 166, 167, 177. 181 Henry, C. H., 37, 49.61 Herbert, D. C., 31601, 52), 429 Herce, J . A., 171. 179 Herman, F., 61. 316,429 Herman, Z., 162, 171, 172, 178. 179 Herrick, D. R., 139, 179, 181 Herzberg, G., 142, 179 Hess, D. T., 260 Hetzel, M., 188(4), 259 Heurtley, J. C., 104, 135 Hewes, C. R., 292, 293,306, 307 Hickernell, F., 300,307 Hickman, A. P., 163, 167, 175. 179, 181
440
AUTHOR INDEX
Hilsum, C., 310(6), 315(6), 316(48, 49), 369(128), 428, 429, 431 Hirano, C., 66, 133 Hirano, J., 101, 135 Hirao, M., 370,431 Hirose, S., 369(140), 431 Hobson, G. S., 316(46), 328(46), 369(46), 371(46), 372(46)377(165), 404( 184). 405(195), 406(195), 407(195), 410(200), 413(205), 429, 431, 432 Holland, M. G., 431 Holliday, H. R., 382, 383,432 Hollis, J. M., 161, 179 Homma, K., 370,431 Hornbeck, J. A,, 162, 179 Hoskins, M. J., 284,307 Hotop, H., 138, 159, 162, 163, 166, 167, 171, 172, 173, 174, 175, 176, 179 Howard, J. S., 166, 167, 175, 179, 181 Howard, M. S., 410(200), 432 Howard, W. E., 37,60 Howes, M. J., 416(212), 433 Hubler, G., 173, 179 Huener, R. C., 198(16), 260 Huffman, D. A., 213,261 Hug, A., 183(1), 187(1), 260 Hunsinger, B. J., 284,307 Hupe, G., 291,307 Hutson, A. R., 311,428
I lams, H., 90, 134 Ibrahim, A., 291,307 Ihm, J., 16, 19, 26, 28, 36, 37, 53, 57.59, 61 Iida, S., 369(140), 431 Ikoma, T., 367( 114), 430 Ikuno, H., 88,135 Illenberger, E., 167, 179 Inbebrigtsen, K., 286,307 Ingre, L., 66, 134 Inkson, J. C., 14, 333861, 6 1 , 430 Inuishi, Y., 314,428 Ipri, A. C., 190(10), 243, 245,260 Irving, L. D., 397(180), 432 Irving, P., 162, 178 Isaacson, A. D., 163, 175, 179 Ishio, H., 66, 135 Izadpanah, S. H., 367(115), 430
J Jackson, J. D., 73, 77, 135 Jaeger, J. C., 378(167), 379(167), 432 James, L. W., 316,429 James, R. P., 419,433 Janev, R. K., 163, 179 Janta, J., 112, 132, 135 Januin, J. P., 190(9), 199(9), 260 Jayaraman, A . , 3 11,428 Jeppesen, P., 342(108), 399( 179), 405(194), 430, 432 Jeppsson,B. I.,342(l08),367(125), 399(179), 405(194), 430, 432 Jethwa, C. P., 412(204), 413(204), 414(204), 415(204), 416(204, 209), 417(204), 432, 433 Jivery, W. T., 156, 178 Joannopoulos, J. D., 29, 31, 32, 57,6/ Johnson, C. C., 83, 135 Johnson, C. E., 176, 179 Johnson, D. E., 286,307 Jones, D. J., 377, 385(162), 386(162); 388(162), 397(162), 405(162), 431 Jones, E. L., 338,423(93), 424, 425,430 Joshi, S. G., 304,307 Julienne, P., 157, 179 Jungen, C., 142,179 K Kalashnikov, S. G., 314(31), 319(58), 428, 42 9 Kang, C. S., 370,431 Kao, K. C., 65, 135 Kapany, N. S., 65, 135 Kashnow, R. A., 250(82), 262 Kasowski, R., 61 Katsuura, K., 163, 164, 179, 181 Kaufhold, L., 173, 179 Kaul, R., 313(20), 315(20, 41), 328(20), 329(20), 337(20), 338(20), 339(20), 340(20), 34 1(20), 342(20), 352(20), 355(20), 373(20), 397(20), 398( 183), 40320, 183, 196), 408(20, 183). 428, 432 Kaul, R. D., 381,432 Kaul, W., 162, 164,179 Kawabata, A., 296,308 Kawakami, S., 86, 135 Keck, D. B., 87,135
44 1
AUTHOR INDEX Keliher, P. J., 164, 176, 179 Keller, H . , 291,307 Kennedy, D. P., 372,431 Kennelly, A. E., 111. I35 Kenyon, N. D., 417(215), 433 Kerber, G. L., 298,307 Kerwin, R. E., 200(23), 238,260 Keyser, J., 156, 177, 180 Khan, P. J., 413,433 Khuri-Yakub, B. T., 286,308 Kiener, M.,202(25), 260 King, F. D., 200(22), 260 King, T . A . , 173, 179 Kingsley, S. A., 66, 134 Kingston, A. E., 138, 163, 166, 177 Kino, G. S., 283, 285, 286, 310(5), 311(5), 3 14(5), 325(62), 327(62), 377(5), 307. 308. 428, 429 Kirchhoff, H., 94. 135 Kitano, I . , 86. 136 Kittel, C., 6, 7, 61, 418(216), 433 Klein, D. L., 200(23), 238,260 Klein, T., 238(54, 5 5 ) . 261 Kleinfelder. W., 372,431 Kleinman, L., 5 , 6 1 Klemperer, W. A . , 155, 161, 179. 181 Klucharev, A . , 174, 179 Knauer, K., 291,307 Knight, B. W., 313(13), 318(13, 55). 323(13, 5 5 ) . 428, 429 Knight, S., 367(124), 378(168), 379(168), 405(187). 430, 432 Knon, T . , 314,428 Koehler, D. R., 204(30), 260 Kogan, Sh. M.,319(60), 429 Kogelnik, H., 95, 104. 111, 114, 115, 117, 125, 133. I35 Kohmoto, M.,163, 164, 179 Koizumi, K., 86, 134, 136 Komarov, I. V., 166, 178 Kordesch. K. V., 251(88). 252(88), 263 Kosonocky, W. F., 272,307 Kozawa, A . , 251(88). 252(88, 89), 263 Krambech, R. H . , 272,308 Krarner, H. L., 171, 179 Krauss, M.,157, 179 Kraut, E. A., 35, 61 Kroerner, H., 35, 36. 50, 310(4), 318(56), 328(66), 338(%), 342( I O I ) , 399(%), 41 1(201), 6 1 , 428. 429, 430, 432
Kubota, S . , 173, 179 Kummel, V., 328(70), 429 Kuprianov, S. E., 174, 176, 179, 180 Kurata, K., 370,431 Kuriyama, M.,66, 135 Kurokawa, K.. 417(214), 433 Kurtin, S., 24, 27, 28, 31, 61 Kurtz, C. N., 88, 122, I35 Kuru, I.. 325, 327,429 Kusters, J. A . , 304, 306,308 Kwan, S. H . , 300,307
L Lacmann, K., 161, 178 Ladd, G. O., Jr., 315(41), 381(169), 428, 432 Lagasse, P. E., 284,307 Lakatos, A. I., 336(92), 430 Lam, H. T . , 314,428 Lam, P. K., 37, 57,61 Lampe, F. W., 166, 178 Lane, N. F., 163, 166, 175, 178 Laucagne, J. J . , 166,178 Layer, M.,216(41), 261 Layton, J. K., 167, 180 Lebwohl, P. A , , 338.430 Lee, H. T., 178 Lee, Y . T., 162, 167, 168, 170, 172, 180, 181 Lehmann, J. C., 154, 177 Leontovich, M . A , , 77, 135 Leuenberger, F., 212, 214(36), 239(36), 245(67), 261. 262 Lewis, M. F., 283, 303,307 Li, R. C. M., 305,307 Li, T . , 66, 95, 101, 103, 107, 114, 134, 135 Lin, S. M.,161, 180 Lin-Chang, P. J., 37,60 Lincoln, A . J., 202(27), 257(27), 260 Lindinger, W . , 167, 174, 175, 180 Linnenbrink, T. E., 277,307 Lisenker, 9. S., 369(130), 431 Littlejohn, M . A , , 316,429 Liu, S. G., 342.430 Lo, H. H . , 161, 162, 178. 180 London, A., 300,307 Long, S . I . , 425(226), 426,433 Lord Rayleigh, 65, 135 Los, J., 161, 180
442
AUTHOR INDEX
Louie, S . G., 4, 5 , 14, 16, 19, 21, 24, 25, 26, 28, 29, 32, 35, 36, 47, 48,60, 6 1 , 62 Louis, E., 28, 31, 36, 335(88), 61, 62, 430 Lucky, R. W., 71, I35 Ludeke, R., 37,60 Liischer, J., 199(18),260 Liischer, R.,223(46), 239(56a), 240(56a), 26 I Lyubchenko, V. E., 314,428
M McClellan, J. H . , 287,307 McCumber, D. E., 315(42), 318(42), 321(42), 338(42, 95). 354(42), 405(42), 428, 430 McCusker, M. V., 176, 180 McDowell, C. A., 172, 173, 174, 177, I82 McGill, T. C., 24, 26, 27, 28, 31, 37. 271, 61. 62, 307 McGowan, J. W., 156, 177, 178, 180 MacGregor, M., 160, 180 McGroddy, J. C., 314(25),428 McGroddy, J. M., 348(112),430 McKenna, J., 93, 272, 134, 308 MacKenzie, L. A., 412, 413, 414, 415,432 McStay, J., 367(123),430 McWhirter, R. W. P., 138, 177, 180 McWhorter, A. L., 314(23), 428 Maddon, R. P., 139, 180 Madlund, G . R., 184(2),259 Magarshak, J., 372, 405(154), 406, 407, 408,431 Magarshack, J., 342(107), 405(107), 420(221), 421(221),430, 433 Magi, P., 82, 134 Magnus, W., 99,135 Magnuson, G. D., 162, 165, 166, 167, 168, 169, 180 Mahan, B. H . , 162, 180 Maines, J. D., 282, 303,307 Mammel, W., 88, 133 Manus, C., 138, 159, 164, 165, 167,180, 181 Many, A . , 335(80),429 Mao, S.,367(127), 369(127),431 Marcatili, E. A. J.. 66, 86, 87, 88, 101, 134, 135 Marcuse, D., 81, 86, 111, 113, 135 Marcuvitz, N., 128, 133. 134 Marcuwitz, N., 413, 415,433
Margaritondo, G., 25, 32,62 Margerie, J., 179 Marklund, I., 367(125),430 Maronchuck, I. E., 369(130),431 Maronchuk, Yu. E., 369(130),431 Marshall, F. G., 285,307 Marx, R. E., 405,432 Maslow, V. P., 88, I35 Mason, I. M.. 284,307 Massey, H. S . W., 142, 155, 180 Mathur, B. P., 161, 181 Matsen, F. A., 167, 177 Matsumura, H., 86, 136 Matsuo, Y . , 367(126),431 Matsuzawa, M., 162, 163, 164, 180 Matthews, H., 296, 302,307 Matzelle, A . , 367(119),430 Maurer, S . J., 85, 130, 135 Mavor, J., 295,307 May, A. D., 93, 134 Mayer, J. W., 374, 375, 376,431 Maxwell, J . C., 89, 135 Mead, C. A., 24, 26, 27, 28, 31, 334(78), 335(81, 91), 61, 429, 430 Mele, E. J., 29, 31, 32.61 Melen, R. G., 266,307 Melrose, S . R., 253(91), 263 Metcalfe, R. M., 66, 136 M’Ewen, M. B., 162, 177 Micha, D. A., 163, 166, 176, 177, 180 Miki, T . , 66, 135 Miller, S. E., 66, 86, 135 Miller, W. H., 163, 165, 166, 173, 175, 176, 179, 180 Miller, W. J., 160, 180 Mills, K. A , , 35, 36, 47, 61 Milnes, A. G . , 47, 50, 56, 335, 336(82), 339(82), 371.61, 429 Mircea, A., 342(107), 386(177), 405(107), 430,432 Mitchell, J. B. A., 156, 177, 180 Mitsuhata, T.. 370(146), 431 Mohler, F. L., 161, 180 Mohsen, A. M., 271,307 Molnar, J. P., 162, 179 Moore, E. E., 245(68), 262 Morgner, H . , 163, 170, 176, 177, 179, 180 Mori, M., 163, 165, 166, 179, 180 Moriyama, K., 369(135),431 Morozumi, S., 256(94),263 Morse, P. M., 103, 135
443
AUTHOR INDEX Moseley, J. J., 138, 180 Mott, N. F., I , 61 Moursund, A . L., 168,179 Moutinho, A. M. C . , 161, 180 Moyer, N. E., 190(8), 247(73), 260, 262 Mueller, C. W., 242(61), 262 Mul, P. M., 156, 180 Muller, R. S., 300,307 Mulliken, R. S., 165, 180 Munson, M. S. B., 162, 180 Murley, P. C., 372,431 Muschlitz. E. E., 163, 167, 171, 174, 179, 180. 181 Myers, F. A., 367(123). 430
N Nafarrate, A. B., 66, 136 Nag, B. R., 381, 382,432 Nakagawa, K., 66, 86, 135 Nakagawa, Y ., 285,307 Nakarnura. H., 152, 153, 162, 163, 165, 166, 173, 176, 179. 180 Nannichi, Y., 370(146),431 Nelson, H., 370,431 Nethercot, A. H., 29. 61 Neukomm, H. R., 239(56a), 240,261 Newton, C. 0.. 285,307 Neynaber, R. H.. 162, 165, 166, 167, 168, 169, 180, 181 Nickel, W., 176, 179 Niehaus, A., 138, 159. 162, 163, 166, 167, 171, 172. 173, 174, 175, 176,178, 179. 180 Nielsen, S. E., 142, 145, 151, 154, 156, 163, 165, 177, 180 Nielson, L. D., 314,428 Nigh, H. E., 246(71), 262 Niiro, M., 66, 133 Nikles, F.. 190(6),201(6), 2036). 238(6), 260 Niles, F. E., 162, 166, 181 Nishizawa, T., 86. 135 Nordbotten, A , , 367( 122),430 Norton, R. E., 66, 136 Nyquist, H., 422(223).433 0 Oberhettinger, F., 99, 135 O'Connell, M. R.. 202(27), 257(27),260 O'Connell, R. M., 284,307
Oguey, H., 190(7), 200(21), 201(5), 203(28), 205(5), 214(38, 39). 259. 260, 261 Ohtomo, M., 418(218), 419, 420,433 Oka, T., 180 Oleksiak. R. E., 202(27), 257(27),260 Oliner, A. A,, 307 Olshansky, R., 87, 135 Olson, R. E., 138, 163, 180 Olsson, K., 367(125),430 Omori, M., 425(225, 226), 426(225, 226), 433 O'Neill, M. B., 179 Oppenheimer, M., 160, 161, 178, 181 Osafune, K., 66, 135 Oshinsky, W., 369(134),431 Otto, 0. W., 299, 300,307 Ozenne, J. B., 172, 173, 178
P Padovani, P. N., 377,431 Paige, E. G. S., 282, 285,307 Pandey, K. C., 4,61 Pao, H. C., 202(27), 257(27), 260 Papas, C . H.. 73, 133 Parks, W. F., 176, 181 Parks, T. W., 287,307 Pasqualetti, F., 103, 134 Patterson, M. R., 163, 181 Patterson, T. A., 162, 181 Pattison, J. E., 397(180),432 Paul. W., 311(11),428 Pauling, L., 23, 25, 335(79), 61, 429 Paxman, D. H., 314(24),428 Pearson, G. L., 311(10), 314(27), 326(63), 428, 429 Peart, B., 156, 181 Peeters, J., 138, 160, 181 Pellegrini, B., 335(87),430 Penton, J . R., 174, 181 Perfetti, P., 35, 36, 47, 61 Perlrnan, B. S., 405(190, 191, 192),432 Pesnelle, A , , 164, 165, 166, 167, 178, 181 Peterson, C . C . , 369(132),431 Peterson, G. A., 313(13), 318(13, 5 5 ) . 323( 13, 53,428, 429 Peterson, J . R., 138,180 Petroff, P. M., 33, 37, 61 Petzinger, K. G., 367(119).430 Plleiderer, H. J., 291,307
444
AUTHOR INDEX
Phillips, J. C., 4, 5 , 14, 17, 20, 28, 35, 335(85), 61, 62, 430 Pickering, C., 313,428 Pickett, W. E., 35, 36, 37, 47, 48.62 Pierce, J. R., 94, 418(217), 135. 433 Pitt, G. D., 313,428 Pollak, H. O., 104, 136 Potter, R. J., 83, 134 Pratesi, R., 93, 94, 115, 119, 135 Preston, R. K., 163, 175, 181 Prew, B. A., 315(40), 397(180),428, 432 Price, P. J., 338,430 Pruniaux, B. P., 376,431 Puchette, C. M., 287, 292,306 Purcell, J. J., 331(74),429 Purohit, R. K., 33.62
R Rabier, A., 372(154), 4 0 3 154). 406(154), 407( 154). 408( 154), 420(221), 421(221), 431,433 Rabiner, L. R.. 286, 287, 293, 298,307, 308 Rader, C. M., 293,308 Ragan, T., 72, 132 Rajgl, Z., 193(13), 260 Ralston, R. W., 306,308 Rambert, E. G., 272,307 Ramo, S., 73, 80, 136 Rang, B., 176, 179 Rappenecker, G . , 162, 181 Rawson, E. G., 66, 136 Raymond, R. M., 41 I , 432 Reck, G. P., 161, 181 Reeder, T. M., 304,308 Rees, H . D., 316(48, 49), 355(113), 377(162), 385(162), 386( 162). 388( 162). 397(113, 162, 180), 405(162), 429, 430, 431. 432 Reeves, C. R., 288,306 Reguier, D., 216(42), 217(42), 261 Reiser, M., 342(105), 430 Richardson, W. C., 172, 181 Ridley, B. K., 310(7, 7a), 428 Rideout, V . L.. 376(159), 397(159), 398, 43 I Righini, G. C.. 89, 93, 136 Rinehart, R. F., 89, 136 Riola. J. P., 166, 167, 175, 179, 181
Riseberg, L. A , , 176, 181 Rizzoli, V., 88, 133 Roberts, G. G., 336(92), 430 Robinson, G. Y., 375, 376,431 Robson, P. N., 325(62), 327(62), 410(200), 429, 432 Rodriguez, V., 374, 375, 376,431 Ronchi, L., 82, 90,93, 94, 106, 108, 109, 115, 119, 121, 122, 124, 125, 127. 134. 135, 136 Ronchi, V., 127, 136 Robert, J., 250(80), 262 Robertson, W. W., 162, 166, 167, 173, 177. 181 Robinson, H . G., 176, 179 Rosenbaum, F. J., 412,413, 414, 415,432 Rosenberg, H. M., 431 Rosenberg, R. L., 304,307 Rosenfeld, J. L. S., 168, 18/ Rosenzweig, W., 262 Rosler, R. S., 242(59), 262 Ross, E. C., 242(61), 262 Ross, J., 168, 179, 181 Rothe, E. W., 161, 167, 181 Rowe. J. E., 25, 32, 412(202), 413,62, 432 Ruch, J. G., 310(5), 311(5), 314(5, 36). 348(36), 377(5, 164), 383( 164), 427(227), 428. 431, 433 Riiegg, H. W., 234(47), 261 Ruetschi, P., 252(90), 263 Rundel, R. D., 159, 166, 167, 175, 178, 179. 181 Russek, A., 163, 18/ Russel, L. K., 215(40), 219(40), 234(40), 26 I Russo. V., 89, 136 Rzhanov, A . V., 369(130), 431 Rzhevkin, K. S . , 422, 423, 424,433 S
Sah, C. T . , 190(51), 236(51), 261 Sai-Halasz, G. A., 37, 49, 57, 62 Saito, T., 369( 138, 139), 431 Sarace, J. C., 190(10), 200(23), 238(23), 243( lo), 2 4 3 10). 260 Savage, A., 37,61 Salz, J., 71, 135 Sangster, F. L. J., 288,308 Sayasov, Yu.,S., 162, 178
AUTHOR INDEX
Saykally, R. J., 161, I81 Schadt, M., 250(78),262 Schaefer, H. F.. 111, 163. 165, 166, 173, I80
Schafer, R. W.. 293,308 Schanne, J. F., 278,307 Schawlow, A. L., 93, 134 Schearer, L. D., 176, 181 Scheggi, A. M., 85, 90,91, 95, 104, 106, 130, 133, 134, 136 Schelkunoff, S. A., 65, 73, 136 Schliiter, M., 4, 5 , 14, 24, 25, 28, 30, 31, 32, 60. 62 Schmeltekopf, A . L., 163, 167, 171, 172, 173, 174, 175. 179, 180, 181 Schmidlin, F. W., 336(92),430 Schmidt, C., 162, 181 Schmidt, K., 170, 177 Schmidt, W. A,, 371(150),431 Schmitz, B., 178 Schneider, W. C., 243(66), 262 Schockley, W., 1 , 4, 342( 100).62, 430 Schottky, W., I , 62 Schryer, N. L., 272,308 Schul, G., 37.60 Schuller, M., 345(111), 356(111),430 Schulman, J. N., 37,62 Schulz, G. J., 142, 181 Schumacher, E. A., 253(92), 263 Schwering, F., 65. 94, 97, 99, 135. 136 Searle, C. L., 418(219), 433 Seaton, M. J., 159, 181 Seki, H., 369( 139,431 Sepman, V., 174, 179 Sequin, C. H . , 266, 267,308 Setser, D. W., 172, 173, 176, 178, 181 Seyfried. P., 164, 179 Sharma, B. L., 33,62 Shaw, M. J., 159, 166, 167, 177, 181 Shaw, M. P.. 313(19, 20, 21). 314(21a), 31320, 21), 321(21). 323(61). 328(20, 68). 329(19, 20, 21). 331(19), 334(75. 76, 77). 336(91a), 337(20), 338(20, 68), 339(20, 21). 340(20, 21). 341(20), 342(19, 20, 21), 343(21), 344(19). 34319). 347(19). 348(19), 349(19, 21), 352(20), 35319, 20, 21), 357(19), 359(19), 360(19), 361(19), 362(19), 363(19), 372(154), 373(20), 387(19, 21), 389(19, 21), 397(19, 20, 21), 399(19),
445
40320, 21, 154). 406(154), 407(154), 408(20, 21, 154),428, 429, 430, 431 Shay, J. L.. 35.62 Sheldon, J. W., 163, 164, 181 Shershyakov, A . P., 369(130),431 Shewchun, J.. 200(22),260 Shiosaki, T., 285, 296.308 Shirley, D. A.. 35, 36, 47, 61 Shobotake, K., 167, 181 Shockley, W., 326(63), 419(220),429, 433 Shoji. M., 367(120), 367(121),430 Sholette, W. P.. 167, 171, 181 Shreve, W. R., 304. 306,308 Shyam, M., 311(10), 314(27),428 Siegel, M. W., 162, 181 Siegman. A. E., 104, 115, 136 Silverman, S. J., 431 Sinanoglu, O., 139, 142, 179, 181 Sinniger, J. O., 242(63), 245(63),262 Sirkis, M. D., 96.134 Skvortsova, N. E., 314,428 Slack, G. A,, 431 Slepian, D., 104, 136 Slinn, K. R., 80. 134 Slob, A., 234(48),261 Slobodnik, A. J., Jr., 2%, 308 Slocomb, C. A., 163, 165, 166, 180 Smirnov, B. M., 163, 181 Smith, A. C. H., 167, 178 Smith, G. E., 266, 272,306. 308 Smith, J. E., Jr., 314(26),428 Smith, P., 250(81),262 Smythe, D. L., 306,308 Snyder, A. W., 84, 85, 88. 136 Snyder, L. E., 161, 177, 179 Solomon, P. M., 155, 181 Solomon, P. R., 313(19, 20, 21), 314(21a), 31320, 21). 321(21), 328(20, 68), 329(19, 20, 21). 337(20), 338(20, 68), 339(20, 21), 340(20, 21). 341(20), 342(19, 20, 21), 343(21), 344(19), 345(19), 347(19), 348(19), 349(19, 21), 352(20), 35319. 20, 21), 357(19), 359(19), 360, 361(19), 362(19), 363(19), 373(20), 387(19, 21). 389(19, 21). 397(19, 20, 21). 399(19), 40320, 21), 408(20. 21), 428, 429 Sonnenblick, E.. 104, 136 Sottini, S., 89, 93, 136 Southworth, G. C., 65, 136
446
AUTHOR INDEX
Spicer, W. E., 10, 12, 61, 62 Spirko, V., 178 Spitalnik, R., 372(154),405(154), 406(154), 407( l54), 408(lS4), 409( 199). 4lO( 199), 411(199), 420(221), 421(221),431, 432, 433 Spitzer, W. G., 335(81),429 Staples, J. L., 371(149), 431 Staudte, J., 206(33), 211(33),261 Stebbings, R. F., 159, 166, 167, 175, 178, 179, 181 Stedman, D. H., 176, 181 Stegun, I. A., 77, 94, 132, 133 Steier, W. H., 96, 101, 134. 136 Stein, C. R.,250(82), 262 Stern, E . , 306,308 Stewart, C. E. E., 369(137),431 Stewart, R. G., 242(63), 245(63),262 Stewart, W. B., 172, 173, 174, 177, 182 Strack, H.,371(148), 373,431 Strain, R. J., 272,308 Stratton, J. A,, 67, 73, 97, 136 Stratton, R., 377(l63), 338(93), 423(93), 424(93), 425(93),430, 431 Streifer, W., 88, 104, 122, 135 Suda, P., 206(32), 261 Sugeta, T., 367( 114),430 Sugimoto, S., 367(116),430 Svelto, V., 315, 324(39), 326(39), 327,428 Szanto, P. G., 161, 181 Sze, S. M., 13, 23, 24, 374,61, 62, 431
T
Tache, J. P., 66, 133 Takeuchi, M., 370(146),431 Tamm, I., 1, 4,62 Tancic, A. R., 163, 179 Tang, S. Y.,163,180 Tantrapom, W., 377,431 Tasch, A. F., 275, 292,306, 307 Tasegawa, T., 381(172), 382,432 Taubert, R., 162, 164, 179 Taylor, B. C., 316, 328, 369, 371, 372(46), 416(210), 429, 433 Tejedor, C., 31, 36.62 Teter, M. P., 162, 166, 181 Thanailakis, A., 19, 23,62 Thim, H. W., 313(18), 342(102, 103, I I O ) , Q
345(18), 405(187, 188, 189), 408(102), 418(102), 420(102),428, 430, 432 Thommen, W., 234(47),261 Tiemann, J. J., 287, 292,306 Tien, P. K., 111, 136 Tietjen, J. J., 369(131),431 Tipton, C. A,, 176, 179 Tognazzi, R., 106, 108, 109, 134 Tomita, K., 66, 133 Tompsett, M. F., 266, 267, 279,308 Toraldo di Francia, G., 73, 77, 88, 89, 90, 91, 93, 121, 127, 136 Torrens, A. B., 342(109), 405(109),430 Tree, R. J., 314(24),428 Tsai, W. C., 412,413, 414, 415,432 Tsu, R., 37, 49, 57,61, 62 Tuan, H. C., 286,308 Tucci, P. A., 215(40), 219(40), 234(40),261 Twiddy, N. D., 166, 167, 177, 181
U Uchida, M., 86, 136 Uenohara, M., 371(151), 405(187), 431. 432 Upadhyayula, C. L., 40J, 432 Ura, K., 404(186),432
V Vadasz, L., 238(54),261 Vaghi, F., 315, 324(39), 326(39), 327,428 Van Bladel, J., 77, 136 Van Duzer, T., 73, 80, 136 VanDyke, J. P., 316,429 van Opdorp, C., 372,431 van Tiggelen, A., 138, 160, 181 Varga, J. E., 242(60),262 Vassell, M. O., 316,429 Vicins, V., 66, 134 Vigue, M., 154, 177 Viktorov, I. A., 279,308 Vinckier, C., 138, 160, 181 Vittoz, E., 190(7, I I ) , 1%(14), 197(14, 15). 199(15),200(15), 201(15), 202(14, 25), 203(14, 28), 212(36), 214(14, 36, 38, 39), 215(14), 216(14), 218(14), 219(14), 221(11), 223(14), 239(36), 245(67), 260, 261 262 I
447
AUTHOR INDEX von Munch, W., 369(136), 431 Voss, P., 407,432 Vujnovic, V., 174, 179 Vyas, M. K. R . , 313,428
W
Wagers. R. S., 283,307 Wagner, L. F., 10, 35,62 Wagner, S., 62 Wainstein, L. A., 72, 104, 136 Waits, R. K., 371(147), 431 Walden, R. H., 272,308 Waldrop, J. R., 35.61 Walker, G. M., 219(43), 257(43), 261 Walter, J. P., 25, 62 Walters, G. K., 164, 176, 179. 180 Wang, Z. F., 167, 181 Wanlass, F. M., 190(51), 236(51), 26/ Wasse, M. P., 383, 384, 385, 404,432 Watanabe, T . , 163, 164, 179, 181 Watel, G., 164, 165, 167, 178, 181 Watkins, T . B., 310(7a), 428 Waywood, D. J., 278.307 Wegener, H . A . R . , 202(27), 257(27), 260 Weinberg, W. H . , 36, 61 Weldon, E. J., Jr., 71, 135 Wellenstein, H. F., 166, 181 Wellern, H . A , , 178 Wenning, U.,178 Wentzel, G., 143, 181 West, W. P., 167, 175, 178, 181 Whinnery, J. R . , 73, 80, 1 1 1 , 136 White, M. H . , 245(69), 247,262 White, P. M., 377( 162). 3 8 3 162, 176). 386( 162). 388( 162), 397( 162). 40% 162). 43 I. 432
White, R. M., 266, 279. 298, 300, 304,306. 307, 308 Whitehead, J. C., 161, 180 Wiedmann, S . K., 234(49), 261 Wiegmann, W., 37, 49.61 Wild, P. J., 221(44), 250(79), 261, 262 Wilk, S. F. J., 156, 177. 178 Williamson, R. C., 2%. 300, 305,307, 308 Williamson, W. J., 405(193), 408(193), 432 Wolf, E., 129, 133 Woodall, J. M., 369(129), 431 Woods, R. C., 161, 181 Wotruba, G., 257(98), 263 Wright, R. W.. 298,307 Wulfman, C., 139, 182 Wyss, H., 207(34), 258(34), 261
Y Yanai, H . , 367(114), 430 Yee, D. S. C., 172, 174, 182 Yeh, C., 85, 136 Yencha. A . J., 172, 178 Yndurain, F., 4, 28, 335(88), 61, 430 Yoshino, K., 149, 178 Young, W. R . , 85, 88, 136 Yu, A . Y. C., 371(147), 374(155), 375(155), 376(155), 431 Z
Zhadanov, V. P., 165, 182 Zhang, H. I., 14, 25,62 Ziman, J. M., 431 Zimany, E. J., Jr., 279,308 Zschauer, K. A . , 370,431 Zumsteg, A . E., 206(32), 261
Subject Index A
B
Acoustic surface wave, see Surface acoustic-wave device Acquisition circuit, electronic watches and clocks, 203 Alarm signal, electronic clock, 227 Aluminum and aluminum compounds AIAs-GaAs interface, 48-49, 57 Ge-AI interface, 19-20 Si-AI interface, 14-19 Amplification negative differential mobility devices, 368,405-412 noise, 417-424 Amplitude feedback, in oscillator circuit, 199 Analog electronic watch and clock, 187189, 203, 259 decoding and driving circuit, 215-218 display, 248-249 integrated circuit, testing of, 226 modules, 253-254 packaging of circuit, 247-248 polyfunctions, 257 time-setting system, 224-225 Analog time delay, charge-coupled device, 266, 269, 276-279 Antibounce circuit, electronic watches and clocks, 225 Aspherical lens, 90 Associative detachment, 141, 159- 177 Associative ionization, 141, 159- 177 Attenuation, in guided-wave propagation, 70-71 dialectric rods, 83 metallic waveguide, 75-76, 79 open-beam waveguide, 101 Autoionization, 140, 146- 151 Avalanche diode, noise, 418-420
Balance wheel, of watch, 185, 187 Band edge discontinuity, in semiconductor-semiconductor interfaces, 34-36, 45, 47-48, 50, 54, 57 Bandpass filter, 281, 292-293, 296-298 Bardeen pinning model of Fermi level, in semiconductors, 13 Battery, electronic watches and clocks, 251-253, 255-256 Beam waveguide, 64-65, 127-128: see also specific types of beam waveguides Betalight, 250 Bipolar technology, electronic watches and clocks, 234-235, 237-238 Brillouin zone, of Ge-GaAs unit cell, 3839 Buckled surface, 11 Bulk- (or buried-) channel charge-coupled device, 273-274, 276 C
Cathode boundary condition, of negative differential mobility elements, 340355, 396-403 Charge-coupled device (CCD), 265-279 transversal filters, 286-295 Charge density AIAs-GaAs interface, 48 GaAs-ZnSe interface, 54 Ge-AI interface, 19 Ge-GaAs interface, 40, 42, 44-45 semiconductor surfaces, 8- 1 1 Si-AI interface, 16-19 zincblende semiconductor-metal interface, 21-22 Chemiionization, 141, 159- 177 Chirp transform device, 299-300 448
449
SUBJECT INDEX Chirp z-transform algorithm, 293-294 Chirp waveform, 282 Circular waveguide, 79 Clean surface, 2-13, 58-59 Clock, 184-186; see also Electronic watch and clock CMOS technology, electronic watches and clocks, 189, 191- 193, 196- 197, 203, 218-219, 235-247 Coaxial cable, 64-65 Collisional ionization and detachment, 14 I Communication systems, waveguides and guided propagation, 64-65 Complementary metal oxide semiconductor technology, see CMOS technology Configuration lens, see Geodesic lens Conflection lens, 90 Confocal resonator, 95 Constant-current oscillator circuit, 198 Corrugated waveguide, 80 Covalent semiconductor, Schottky barrier height, 13 Cylindrical waveguide, 73, 79, 84
D Dangling-bond surface state, 3, 8- 11 Data, transient, recording and storage, 277 Decoder circuit, electronic watches and clocks, 215-223 Degenford beam waveguide, 97 Delay line charge-coupled devices, 266, 276-279 surface acoustic-wave devices, 302-303 Density of states, 9- 10 jellium-germanium interface, 20 Si-AI interface, 17 Diamond-metal interface, 26-29, 32 Dielectric film, 91-93 Dielectric-frame beam waveguide, 95-96, 104- 106, 109- 110 Dielectric planar waveguide, 91-94 Dielectric rod waveguide, 64, 73, 80-88 Dielectronic recombination, 140, 155- 156 Digital electronic watch and clock, 187189, 259 circuit, 228, 230-234 displays, 249-25 1, 256 LCD, 220-223 LED, 218-220
modules, 254 packaging of circuit, 247-248 polyfunctions, 257 time-setting system, 224, 231-232 Digital signal delay, charge-coupled devices, 277 Digital tuning technique, in electronic watches and clocks, 201-204 DIP, see Dual in-line package Dispersion, in guided-wave propagation, 70-71 graded-index fiber, 86-87 metallic waveguide, 74-75, 79 Dispersive filter, 299 Dissociative ionization, 141 Dissociative recombination and attachment, 141 Divider cell, electronic watch circuits, 215-216 DOS, see Density of states Driving circuit, electronic watches and clocks, 215-223 Dual in-line package, electronic clock circuit, 247-248 Dye laser, 93
E EAR, see Electron aflinity rule Electrical clock, 186 Electron attachment and detachment processes, 137-182 interface studies, 1-62 Electron affinity rule, 35, 56 Electronic watch and clock, 183-263 circuits, 21 1-233 displays, 248-251 future trends, 255-258 manufacturing technologies, 233-247 modules, 253-254 packaging technology, 247-248 power supplies, 25 1-253 time base, 189-211 Elliptical waveguiding medium, 124- 125 Empirical pseudopotential method, in interface studies, 5 Empty surface state, 12 EPM, see Empirical pseudopotential method
450
SUBJECT INDEX
Equivalent circuit, quartz crystal, 192, 194, 206 Equivalent open-beam waveguide, 104- 106 External caustic surface, of waveguide, 86 F Fabry-Perot resonator, 95 Fermi level pinning, semiconductor-metal interface, 13-14, 24, 335 Field-effect transistor noise, 422 in SAW filters, 300 Filter, 286-287 charge-coupled devices, 276-279, 287295 surface acoustic-wave devices, 295-301 Fixed-cathode field, negative differential mobility elements, 391-396 FLAD, electronic clock display, 251 Flexural wave, 284 Fluorescent-activated liquid crystal display, 251 Free-space wave, 64 Frequency electronic watches and clocks, 187- 190, 227-228 filtering, 286 negative differential mobility elements, 33 1 quartz crystals, 206-21 1 surface acoustic waves, 280-282 and transfer inefficiency of CCD devices, 270-278 vs. attenuation, in cylindrical waveguide, 79 Frequency-selective reflector, 303-304
low-resistance contacts, 371, 373-374 negative differential mobility device construction, 368-386 oscillation principles in short NDM elements, 387, 389, 397-398, 405 oscillator output power, 425-427 Schottky barrier height, 23 surface, 12, 59 Gas lens, 96, 101 Gaussian beam, 96, 99-100, 111, 114-118, 120- 122. 124-125, 127 Geodesic lens, 73, 88-90.93 Germanium Ge-AI interface, 19-20 Ge-GaAs interface, 34-48, 57 Ge-ZnSe interface, 49-52 Schottky barrier height, 13 Graded-index waveguide medium, 86-88, 93-94, 111-113, 122-123, 129 Group delay, in wave propagation, 71, 83 Guided wave, 64-65; see also Waveguide Gunn diode, 405-417 Gum-Hilsum effect electronics, 309-433
H Hermite-Gaussian beam, 114- 115 Heterojunction, 33, 56-57; see also Semiconductor-semiconductor interface High-frequency oscillator, electronic watches and clocks, 256 High-Q filter, 302 Homogeneous waveguiding medium, 8085, 91-93 Hybrid mode, of wave propagation, 80-81 elliptical dialectric rods, 85 reiterative beam, 98 I
G Gallium arsenide AIAs-GaAs interface, 48-49, 57 bulk material, 368-369 current instabilities, 340-342 diode noise, 418-424 epitaxial growth, 369-371 GaAs-ZnSe interface, 52-57 Ge-GaAs interface, 40-48 Gum-Hilsum effect in n-type samples, 309-328
IDT,see Interdigital transducer IzL technology, see Integrated injection logic technolcgy Image transmission, 71 Indium phosphide in Gunn-Hilsum effect electronics, 314316, 328, 335, 388, 397, 405 noise, 422 oscillator output power, 425-426 Inhomogeneous waveguiding medium, 8688
45 1
SUBJECT INDEX Integrated circuit, electronic watches and clocks, 184, 189, 211, 226-228, 234247 Integrated injection logic technology, 219220, 234 Interdigital transducer, 302-305 Interface study, 1-62 semiconductor-metal interfaces, 13-32, 334-340, 373-376 semiconductor-semiconductor interfaces, 32-58 Internal caustic surface, of waveguide, 84 Inverse autoionization, 140, 155- 156 Inverse filter, 298 Inverse predissociation, 140, 156- 158 Inverter oscillator, 198 Ionicity-dependent behavior semiconductor-metal interfaces, 20, 22, 24, 26-27, 31-32 GaAs-ZnSe interface, 54-55 Ion-ion neutralization, 141 Ion-pair formation, 141 Iris beam waveguide, %
J Jellium model of aluminum, 14-17
K Kuprianov process, 141
L Laser, 64,93 frequency adjustment of tuning-fork crystal resonator, 207 Lattice matching, in heterojunctions, 33, 49 LCD, see Liquid crystal display LDOS, see Local density of states Leclanche cell, 252 LED, see Light emitting diode LEED, see Low-energy electron diffraction Lens, waveguides, 88-91, 94-96, 100101, 104 Light emitting diode, electronic watches and clocks, 215, 218-220, 249 Line wave, 284
Liquid crystal display, electronic watches and clocks, 215, 220-223, 249-251, 256-257 Local density of states, 8- I I AIAs-GaAs interface, 50 Ge-GaAs interface, 44-46 Ge-ZnSe interface, 51, 53 Si-A1 interface, 17-18 zincblende semiconductor-metal interface, 21-22 Logic circuitry, electronic watches, 223 Longitudinal-back-bond state, 11 Longitudinal waveguide, 73 Lossless waveguide, 74, 77 Lossy waveguide, 75, 77-78 Louie-Cohen state, see Metal-induced-gap state Low-energy electron diffraction, 3 Low-loss circular TElo waveguide, 65 Low-pass filter, 277, 287-288, 291-292 Low-resistance contacts, negative differential mobility devices, 371, 373-374
M Matched filter, 298-301 Maxwell’s fish-eye, 89 MBE, see Molecular-beam epitaxy Memory, electronic watches and clocks, 202-203 Mercury cell, electronic watches, 252 Metal gate CMOS technology, watch circuit, 238. 245-247 Metal-induced-gap state, in semiconductor-metal interface, 14, 19, 21, 24-26, 28-29, 31 Metallic surface, 10-1 1 Metallic waveguide, 73-80 Metal-oxide semiconductor, charge-coupled device, 266-269 Metal-semiconductor interface, 13-32, 334-340, 373-376 Microprocessor-like watch circuit, 257-258 Microstrip lines, 73 Microwave circuit, Gunn diode, 412-417 Microwave lens, 88-89 MIGS, see Metal-induced-gap state Modal beam, 102 Molecular-beam epitaxy, 33, 37 MSC, see Multistrip coupler
45 2
SUBJEC'T INDEX
Multimode waveguide, 79 Multiplexing LCD, 222-223, 250-251 LED display, 218-219 Multistrip coupler, 285
N Narrow-band filter, 296-297, 303 NDC, see Negative differential conductivity NDM, see Negative differential mobility Negative differential conductivity, 328 Negative differential mobility, in semiconductors, 310-328 charge fluctuation, 317-321 circuits and boundaries, 329-367 devices, 367-427 equal-areas rule, 321-324 oscillatory behavior, 355-367, 386-405 velocity-electric field characteristics, 314-316, 324-328, 330-331 Noise negative differential mobility devices, 417-427 Nonparabolic waveguiding medium, 119122, 126-127 Nonpolar interface, 38, 40
0
Ohmic contact, 336 Open-beam waveguide, 94-1 10 Open resonator, 95, 101- 102, 104- 105 Optical fiber, 65-66, 73, 80-88 Optical waveguide, 64-65, 73 Oscillating guided beam, 1 15- 118, 124- 125 Oscillation negative differential mobility elements, 355-367, 386-405, 416-417 quartz crystals, 205 Oscillator in electronic watches and clocks, 188190 negative differential mobility devices, 367-368 noise, 417-424 surface acoustic-wave devices, 302-306
Oscillator circuit, electronic watches and clocks, 190-201, 227-228
P Packaging technology, for electronic watch and clock circuits, 247-248 Parabolic waveguiding media, I 1 1, 113118, 123-124 PBS, see Projected band structure Penning detachment, 141 Penning ionization, 141, 159- 177 Periodic slow-wave structure, 80 Phase transformer, 98-99 Phillips average gap, 28 Phillips cancellation theorem, 5 Photodissociation, 141 Photoionization and photodetachment, 141 Pierce oscillator, 190-192, 194- 197 Piezoelectric effect, surface acoustic-wave devices, 280-283, 285-286, 300, 304 Pipe organ filter, 291 Planar waveguide, 73, 88-94 Polar interface, 40 Predissociation, 140, 151- 154 Preionization and autodetachment, 140, 146- 151 Primary battery, electronic watches and clocks, 251-253, 255 Printed circuit board, watch circuits, 247248 Projected band structure AIAs-GaAs interface, 48-49 GaAs-ZnSe interface, 55 Ge-GaAs interface, 41-43 Ge-ZnSe interface, 50-51 Propensity rule, in elementary attachment and detachment processes, 144- 145 Pseudopotential calculations, in surface theory, 4-7 Pulsed operation, negative differential mobility devices, 380-386
Q Quartz crystals electronic watches and clocks, 187- 190, 192, 194, 204-21 1 surface acoustic-wave devices, 284
SUBJECT INDEX
R RAC, see Reflective array compressor Radiationless attachment, 140, 155- 156 Radiationless transition, 140, 145- 146 in polyatomics, 144 Radiative recombination and attachment, 140, 158-159 Ray-tracing method, in wave studies, 8485, 94, 128-130 R C oscillator, 204 Rearrangement ionization, 141 Rectangular waveguide, 76, 79 Reflection, surface acoustic waves, 285 Reflective array compressor, 299-300 Reflector, surface acoustic-wave devices, 303-304 Reiterative wave beam, 97- 104 Resonance conditions, in wave theory, 85 Resonator in electronic watches and clocks, 187, 189, 204, 206 surface acoustic-wave devices, 304-305 Rinehart - Luneberg lens, 89-90
S
Sandwich-type negative differential mobility device, 369-372 Satellite communications, 64 SAW device, see Surface acoustic-wave device Schottky barrier behavior, at interfaces, 13, 19-26, 29, 59, 336 SCPM, see Self-consistent pseudopotential method Self-consistency, in surface calculations, 4 Self-consistent pseudopotential method, in interface studies, 5-7 semiconductor-metal interface, 14- 16, 21, 24-26, 28 semiconductor-semiconductor interface, 36-40 Selfoc fiber, 86 Self-winding watch, 185 Semiconductor clean-surface properties, 2- 13 negative differential mobility, 3 10-328 surface acoustic-wave devices, 285-286
45 3
Semiconductor-metal interface, 13-32, 59, 334-340, 373-376 Ge-AI, 19-20 self-consistent calculation, 14- 16 Si-AI, 16-19 zincblende semiconductor-metal, 20-23 Semiconductor-semiconductor interface, 32-59 AIAs-GaAs, 48-49 GaAs-ZnSe, 52-56 Ge-GaAs(l10). 40-48 Ge-ZnSe, 49-52 Signal-processing techniques, 265-308 charge-coupled devices, 266-279 guided-wave propagation, 70-7 1 surface acoustic-wave devices, 279-286 Signal-to-noise ratio, filter responses, 290 Silicon Schottky barrier height, 13, 23 Si-AI interface, 16-19 Si(l1l) surface, 3, 8-11, 15-16, 59 Silicon carbide, 28, 32 Silicon-gate CMOS technology, watch circuits, 238-242, 256 Silicon-on-sapphire technology, watch circuits, 238, 242-245 Silver oxide cell, electronic watches, 252 Single-chip silicon gate CMOS watch circuit, 228 Single-mode waveguide, 79, 104 Slab geometry, in surface theory, 4-5 SO-8 package, 247-248 SOS technology, see Silicon-on-sapphire technology Space charge nonuniformities, negative differential mobility elements, 362-367 Specific contact resistance, 373-374 Spectrum analyzer, 294-295 Split-electrode filter, 288-290 Square-law waveguiding medium, 1 1 1, 124-125 Supercell, 4-5 Ge-GaAs unit, 38-39 Superlattice, 37 Surface acoustic-wave device, 265-266, 279-286 transversal filters, 286, 295-301 Surface-channel charge-coupled device, 273-274
454
SUBJECT INDEX
Surface density of states jellium-diamond interface, 27 jellium-germanium(l1 I ) interface, 20 semiconductor-metal interfaces, 22 Surface skimming bulk wave, 283 Surface theory, 2-13, 58-59 Synthetic quartz, 205
T Tape recorder, timing errors, correction of, 278-279 TED, see Transferred electron device TEM, see Transmission electron microsCOPY TEM mode, of wave propagation, 77 lenses, 88 T E mode, of wave propagation, 74-75, 77-79 dialectric rods, 81-82 graded-index medium, 93, 112 reiterative beam, 98 Temperature and negative differential mobility devices, 377-386 and surface acoustic wave velocity, 284285 Tetrahedral compounds, ionicity curves, 31 Thermal generation, in charge-coupled devices, 276 Thin film, in waveguides, 91-93 Three-dimensional waveguiding medium, 122-127 Three-point oscillator circuit, 191, 201 Tight-binding method, in surface calculations, 4 Time base, of watches and clocks, 18521 1 Time-setting system, electronic watches and clocks, 224-225, 227, 231-232 TM mode, of wave propagation, 74-75, 77-79 dialectric rods, 81 -82 graded-index medium, 93, 112 lenses, 88 reiterative beam, 98 Toraldo’s geodesic lens, 90
Toroidal junction, 90 Transducer, surface acoustic-wave devices, 280-282, 297, 302-304 Transfer inefficiency of filters, 292 and frequency in CCD devices, 270-278 Transferred electron device, 367-427 Transistor, electronic watches and clocks, 187-188 Translational invariance, loss of, in surface energy calculations, 4-5 Transmission electron microscopy, 33 Transversal filter, 286-301 Transverse-back-bond state, 10 Transverse waveguide, 94- 110 Tuning fork, electronic watches and clocks, 187-188 Tuning-fork crystal, 206-207, 21 I Tunnel diode, 356-367 Tunnel diode relaxation oscillator, 345 Two-dimensional waveguide dialectric structures, 91-94 with metallic walls, 88-91 Two-dimensional waveguiding medium, 112- 122 Two-transducer design, 303-304
U Ultrasonic acoustic waves, SAW devices, 279-286
V Voltage doubler, electronic watches and clocks, 223-224, 231
W Watch, 185- 186; see also Electronic watch and clock Wave equation, 68-69 graded-index medium, 93-94 inhomogeneous medium, 87 metallic waveguide, 74
SUBJECT INDEX Waveguide, 64-65; see also specific types of waveguides guiding media, 110- 127 Gunn diode, 412-417 theoretical background, 66-72 Weak inversion operation, in oscillator circuit, 199-200 WKB approximation, in wave studies, 94, 130- 133
455
2 Zero-phase shift circuit, 201 Zinc-air cell, 253 Zincblende GaAs-ZnSe interface, 52-57 Ge-ZnSe interface, 49-52 Schottky barrier height, 23 semiconductor-metal interfaces, 20-23 surfaces, 12
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