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SEMICONDUCTORS AND SEMIMETALS VOLUME 7 Applications and Devices Part B

This Page Intentionally Left Blank

SEMICONDUCTORS AND SEMIMETALS Edited by R. K. WILLARDSON BPLL A N D HOWFLL t l E C I H O N I C MATFKIALS DIVISION PASADENA. CALIFORNIA

ALBERT C. BEER BATTELLE MEMORIAL INSTITUTE COLUMBUS LABORATORILS

COLUMBUS. OHIO

VOLUME 7 Applications and Devices Part B

1971

@

ACADEMIC PRESS

New York and London

COPYRIGHT 0 1971, BY ACADEMIC PRESS, WC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRI7TEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W1X 6BA

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 65-26048

PRINTED IN THE UNITED STATES OF AMERICA

Contents LISTOF CONTRIBUTORS. . PREFACE . . . . . CONTENTS OF PREVIOUS VOLUMES .

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vii ix xi

DIODES Chapter 7 IMPATT Diodes T. Misawa Introduction . . . . . . , . . Dynamic Negative Resistance in p n Junction in Breakdown . Fundamental Phenomena and Mathematical Formulation . Analysis of Electrical Characteristics . . . . Design Considerations . . . VI. Diode Fabrication . . . . . . . . VII. Observed Electrical Characteristics . . . . . VIII. Conclusions . . . . . . . Appendix A. DC Equations and Numerical Solution . . Appendix B. Small-Signal AC Solution . . . . Appendix C. Addenda to Numerical Analysis of Large-Signal Read Diode . . . . . . Appendix D. Theory of TRAPATT Mode of Operation . List of Symbols . . . . . . 1. 11. 111. IV. V.

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382 442

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463

Operation of

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Chapter 8 Tunnel Diodes H . C . Okean 1. Introduction . . . . . . 11. The Physics of Tunnel Diode Operation . 111. Principles of Tunnel Diode Fabrication . . . . . . IV. Terminal Properties of Tunnel Diodes . V. Experimental Characterization of Tunnel Diodes VI. Tunnel Diode Applications in Sinusoidal Circuits . VII. Tunnel Diode Applications in Pulse and Digital Circuits VIII. Present and Future Role of Tunnel Diodes . . V

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vi

CONTENTS

Chapter 9 Silicon Carbide Junction Devices Robert B. Campbell and Hung-Chi Chang 1. Introduction . . . . . . 11. Silicon Carbide as a Semiconductor Material

DeviceTechniques . Silicon Carbide Power Diodes p-n Junction Detectors . Active Devices . . Irradiation Effects . . Luminescent Diodes . Summary . . . X. Addendum . . .

111. IV. V. VI. VII. VIII. 1X.

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687 688 ,699 . 711 . 718

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RECTLFIERS Chapter 10 High-Temperature Power Rectifiers of GaAs, - xPx R . E . Enstrorn, H . Kressel, and L. Krassner I. Introduction . . . . . . . 11, High-Temperature Rectifier Design Considerations 111. p-n Junction Formation . . . . . . . . . . IV. Device Fabrication V. Rectifier Test Results . . . . . AUTHQRINDEX . SUBJECTINDEX .

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721

List of Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.

ROBERT B. CAMPBELL, Westinghouse Astronuclear Laboratory, Pittsburgh, Pennsylvania (625) HUNG-CHICHANG,' Westinghouse Astronuclear Laboratory, Pittsburgh, Pennsylvania (625) R. E. ENSTROM, RCA Laboratories, David Sarnof Research Center, Princeton, New Jersey (687) L. KRASSNER,~ RCA Laboratories, David Sarnof Research Center, Princeton, New Jersey (687) H. KRESSEL, RCA Laboratories, David Sarnofl Research Center, Princeton, New Jersey (687) T. MISAWA,Bell Telephone Laboratories Inc., Murray Hill, New Jersey (371) H. C. OKEAN,Airborne Instruments Laboratory, A Division of CutlerHammer, Inc., Melville, Net43 York (473)

' Present address: National Chiao University, Hsinchu. Taiwan, China Present address: Unitrode Corporation. Watertown. Massachusetts.

vii


Preface

The extensive research that has been devoted to the physics of semiconductors and semimetals has been very effective in increasing our understanding of the physics of solids in general. This progress was made possible by significant advances in material preparation techniques. The availability of a large number of semiconductors with a wide variety of different and often unique properties enabled the investigators not only to discover new phenomena but to select optimum materials for definitive experimental and theoretical work. In a field growing at such a rapid rate, a sequence of books which provide an integral treatment of the experimental techniques and theoretical developments is a necessity. The books must contain not only the essence of the published literature, but also a considerable amount of new material. The highly specialized nature of each topic makes it imperative that each chapter be written by an authority. For this reason the editors have obtained contributions from a number of such specialists to provide each volume with the required detail and completeness. Much of the information presented relates to basic contributions in the solid state field which will be ofpermanent value. While this sequence of volumes is primarily a reference work covering related major topics, certain chapters will also be useful in graduate study. In addition, a number of the articles concerned with applications of specific phenomena will be of value to workers in various specialized areas of device development . Because of the important contributions which have resulted from studies of the 111-V compounds, the first few volumes of this series have been devoted to the physics of these materials: Volume 1 reviews key features of the 111-V compounds, with special emphasis on band structure, magnetic field phenomena, and plasma effects. Volume 2 emphasizes physical properties, thermal phenomena, magnetic resonances, and photoelectric effects, as well as radiative recombination and stimulated emission. Volume 3 is concerned with optical properties, including lattice effects, intrinsic absorption, free carrier phenomena, and photoelectronic effects. Volume 4 includes thermodynamic properties, phase diagrams, diffusion, hardness, and phenomena in solid solutions as well as the effects of strong electric fields, ix

X

PREFACE

hydrostatic pressure, nuclear irradiation, and nonuniformity of impurity distributions on the electrical and other properties of 111-V compounds. Volume 5 , which is devoted to infrared detectors, is the first of a number of volumes to deal specifically with applications of semiconductor properties. Volume 6 is concerned with injection phenomena in solids, including current injection and filament formation, double injection, internal photoemission, and photoconductor-metal contacts. The present volume is issued in two parts, 7A and 7B, and is concerned with semiconductor devices, including those utilizing bulk negative resistance phenomena as well as effects due to barriers and junctions. Subsequent volumes of Semiconductors and Seminzerals will include further work on infrared detectors and a variety of fundamental phenomena such as lattice dynamics, galvanomagnetic effects, luminescence, nonlinear optical phenomena, and electro-, thermo-, piezo-, and magnetooptical effects. The editors are indebted to the many contributors and their employers who made this series possible. They wish to express their appreciation to the Bell and Howell Company and the Battelle Memorial Institute for providing the facilities and the environment necessary for such an endeavor. Thanks are also due to the U.S. Air Force Offices of Scientific Research and Aerospace Research and the U.S. Navy Office of Naval Research and the Corona Laboratories, whose support has enabled the editors to study many features of compound semiconductors. The assistance of Crystal Phillips, Martha Karl, and Inez Wheldon in handling the numerous details concerning the manuscripts and proofs is gratefully acknowledged. Finally, the editors wish to thank their wives for their patience and understanding. R. K . WILLARDSON ALBERT C. BEER

Semiconductors and Semimetals Volume 1 Physics of 111-V Compounds C . Hifsum, Some Key Features of 111-V Compounds Franco Eassani, Methods of Band Calculations Applicable to IIILV Compounds E. 0. Kane, The k ' p Method V. L. Eonch-Eruevich, Effect of Heavy Doping on the Semiconductor Band Structure Donald Long, Energy Band Structures of Mixed Crystals of 111-V Compounds Laura M . Roth and Petros N . Argyres, Magnetic Quantum Effects S . M . Puri and T. H. G e b d e , Thermomagnetic Effects in the Quantum Region W. M . Becker, Band Characteristics near Principal Minima from Magnetoresistance E. H. Purley, Freeze-Out Effects, Hot Electron Effects, and Submillimeter Photoconductivity in lnSb H . Weiss, Magnetoresistance Betsy Ancker-Johnson, Plasmas in Semiconductors and Semimetals

Volume 2

Physics of 111-V Compounds

M . G. Holland, Thermal Conductivity S. I.Novikova, Thermal Expansion U. Pieshrugen, Heat Capacity and Debye Temperatures G . Giesecke, Lattice Constants J . R . Drubble, Elastic Properties A . U . Mac Rae and G. W . Gobeli. Low Energy Electron Diffraction Studies Robert Lee Mieher, Nuclear Magnetic Resonance Bernard Goldstein, Electron Paramagnetic Resonance T . S . Moss, Photoconduction in 111-V Compounds E. AnronFik andJ. Tauc, Quantum Efficiency of the Internal Photoelectric Effect in lnSb G. W .Goheli and F. G . Allen. Photoelectric Threshold and Work Function P. S. Pmjhun. Nonlinear Optics in I l l - V Compounds M . Gershenzon, Radiative Recombination in the 111-V Compounds Frank Srerrt, Stimulated Emission in Semiconductors

Volume 3 Optical Properties of Ill-V Compounds Mari.in H a s , Lattice Reflection William G. Spitzer, Multiphonon Lattice Absorption D . L . Stierwalt and R. F. Potter, Emittance Studies H. R . Philipp and H . Ehrenreich. Ultraviolet Optical Properties Manuel Cardona, Optical Absorption above the Fundamental Edge Earnest J . Johnson, Absorption near the Fundamental Edge John 0. Dimmock, Introduction to the Theory of Exciton States in Semiconductors E. Lax and J . G. Mauroides, Interband Magnetooptical Effects

xi

xii

CONTENTS OF PREVIOUS VOLUMES

H. Y . Fan, Effects of Free Carriers on the Optical Properties Edward D. Palik and George B. Wright, Free-Camer Magnetooptical Effects Richard H. Bube, Photoelectronic Analysis B. 0. Seraphin and H. E. Bennett, Optical Constants

Volume 4

Physics of 111-V Compounds

N. A. Goryunova, A. S. Borschevskii, and D. N. Tretiakov, Hardness N . N. Sirota, Heats of Formation and Temperatures and Heats of Fusion of Compounds A"'BV Don L. Kendall, Diffusion A . G. Chynoweth, Charge Multiplication Phenomena Robert W . Keyes, The Effects of Hydrostatic Pressure on the Properties of 111-V Semiconductors L. W . Aukerman, Radiation Effects N. A . Goryunova, F. P. Kesamanly, and D. N . Nasledov, Phenomena in Solid Solutions R. T. Bate, Electrical Properties of Nonuniform Crystals

Volume 5 Infrared Detectors Henry Levinstein, Characterization of Infrared Detectors Puuf W . Kruse, Indium Antimonide Photoconductive and Photoelectromagnetic Detectors M . B. Prince, Narrowband Self-Filtering Detectors Ivars Melngailis and T. C. Harman, Single-Crystal Lead-Tin Chalcogenides Donald Long and Joseph L. Schmit, Mercury-Cadmium Telluride and Closely Related Alloys E. H. Putley, The Pyroelectric Detector Norman B. Stevens, Radiation Thermopiles R. J . Keyes and T. M . Quist, Low Level Coherent and Incoherent Detection in the Infrared M . C. Teich, Coherent Detection in the Infrared F. R. Arums, E. W . Sard, B. J . Peyton, and F. P. Pace, Infrared Heterodyne Detection with Gigahertz IF Response H. S. Sommers, Jr., Microwave-Biased Photoconductive Detector Robert Sehr and Ruiner Zuleeg, Imaging and Display

Volume 6 Injection Phenomena Murray A. Lampert and Ronald B. Schilling, Current Injection in Solids: The Regional Approximation Method Richard Williams, Injection by Internal Photoemission Allen M . Barnett, Current Filament Formation R. Baron and J . W . Mayer, Double Injection in Semiconductors W . Ruppel, The Photoconductor-Metal Contact

Volume 7 Applications and Devices: Part A John A. Copeland and Stephen Knight, Applications Utilizing Bulk Negative Resistance F. A. Padovani, The Voltage-Current Characteristic of Metal-Semiconductor Contacts P. L. Hower, W . W . Hooper, B. R. Cairns, R. D. Fairman, and D. A. Tremere, The GaAs FieldEffect Transistor Marvin H. White, MOS Transistors G. R. Antell, Gallium Arsenide Transistors T. L. Tansley, Heterojunction Properties

SEMICONDUCTORS A N D SEMIMETALS VOLUME 7 Applications and Devices Part B


Diodes


CHAPTER 7

IMPATT Diodes T. Misawa

. . . . . . . . . . . . . . . . 372 I . INTRODUCTION NEGATIVE RESISTANCE I N pn JUNCTION IN BREAKDOWN . 372 I1. DYNAMIC 1 . Negative AC Power Dissipation . . . . . . . . . . 373 2 . Transif-Time Effect . . . . . . . . . . . . . 374 3 . Saturation of Carrier Drft Velocity . . . . . . . . . 374 4 . Dynamics of Avalanche Multiplication . . . . . . . . 375 5 . Negative Resistance in pn Junction in Breakdown . . . . . 375 PHENOMENA AND MATHEMATICAL FORMULATION . . 371 111. FUNDAMENTAL 6 . Drifr Velocities of Carriers . . . . . . . . . . . 377 7 . Aiialanche Multiplication . . . . . . . . . . . . 377 8 . Governing Equarions . . . . . . . . . . . . . 380 OF ELECTRICAL CHARACTERISTICS . . . . . . . 382 IV . ANALYSIS 9 . Space-Charge L a j w in pn Junctions . . . . . . . . . 382 10. Static Current-Voltage Characteri.stics . . . . . . . . 383 11. Growing Waiw in Arwlunching Electron-Hole Plasma . . . . 390 12. Small-Signal Anal.ysis . . . . . . . . . . . . . 393 13. Large-Signal Anal.v.ri.7 . . . . . . . . . . . . . 411 . . . . . . . . . . . . . 429 V . DESIGNCONSIDERATIONS 14. Scaling Rule fbr liirious Structures . . . . . . . . . 429 IS . Structure Puranzeters . . . . . . . . . . . . . 430 16. Material Parameters . . . . . . . . . . . . . 432 11. Thermal Considerntion . . . . . . . . . . . . 434 . . . . . . . . . . . . . . 442 VI . DIODEFABRICATION I8 . Impurity ProJie . . . . . . . . . . . . . . 442 I9 . Fabrication Techniques . . . . . . . . . . . . 449 CHARACTERISTICS . . . . . . . . 451 VII . OBSERVED ELECTRICAL 20 . Small-Signal Charrrcreristics . . . . . . . . . . . 451 2 I . Oscillator Chnracrerisrics . . . . . . . . . . . . 454 VIII . CoNCLUsloNs . . . . . . . . . . . . . . . . 461 APPENDIX A . DC EQUATIONS A N D NUMERICAL SOLUTION . . . 462 APPENDIX B. SMALLSIGNAL AC SOLUTION . . . . . . . . 463 APPENDIXC . ADDENDA TO NUMERICAL ANALYSIS OF LARGE-SIGNAL OPERATION OF READDIODE . . . . . . . . . . . 464 466 D . THEORY OF TRAPATT MODEOF OPERATION . . . APPENDIX LBT OF SYMBOLS. . . . . . . . . . . . . . . 471

37 1

372

T. MISAWA

1. Introduction When the p n junction diode is reverse-biased, then, practically speaking, current does not flow. However, when the reverse voltage exceeds a certain value, the junction breaks down and current flows with only slight increase of voltage. This breakdown is caused by avalanche multiplication of electrons and holes in the space-charge region of the junction. The p n junction in the avalanche breakdown condition exhibits negative resistance characteristics in the microwave frequency range. This negative resistance can be exploited to generate microwave power and amplify microwave signals. Since the negative resistance is based upon avalanche multiplication and the transit-time effect of carriers, the device has been called the IMPATT (Impact Avalanche Transit-Time) diode. IMPATT oscillators have produced continuous output powers ranging from 5 W at 12 GHz with an efficiency of 9 %,to 37 mW at 106 GHz with an efficiency of 1.6%. Germanium devices have shown an efficiency as high as 15.3% around 6 GHz. The highest frequency of oscillation reported is 341 GHz from a Si unit. Signal amplification has been observed in the above frequency range with reflection-type amplifiers. The idea of utilizing the dynamic property of avalanche multiplication in conjunction with the carrier transit-time effect in the space-charge layer of the p n junction for obtaining a dynamic negative resistance was originally proposed by Read in 1958.' Very little work has been reported in the literature on this type of device until 1965, when Johnston et al. discovered, rather independently of Read's proposal, that microwave oscillation can be obtained from Si pn junction diodes which were structurally quite different from Read's model.' Since this discovery, intensive study has been made on the dynamic characteristics of pn junctions in breakdown. It is believed that the characteristics are fairly well understood theoretically and basic properties of the device have been investigated fairly thoroughly with laboratory units made of such common semiconductors as Si, Ge, and GaAs. in the following, after a brief description of the negative resistance of the pn junction in breakdown, a theoretical study on its electrical characteristics is presented. Then, some design considerations and fabrication techniques of laboratory models are discussed. Finally, the observed characteristics of experimental units are presented.

11. Dynamic Negative Resistance in pn Junction in Breakdown In this part, a simplified explanation will be given of the origin of the dynamic negative resistance in a pn junction in breakdown.

' W. T. Read, Bell. Sysr. Tech. .I37, . 401 (1958). R. L. Johnston, B. C. DeLoach, and B. G . Cohen, Bell Syst. Tech. J . 44,369 (1965)

7. IMPATT

DIODES

373

1. NEGATIVE AC POWERDISSIPATION

When ac voltage is present across a diode with negative-resistance terminal characteristics, ac electric energy is produced within the device. If we look into the inside of the device, electric energy is dissipated in some places within the device and is generuted in other places. However, the net power dissipation has to be negative in order for the diode to exhibit the negative resistance property at the terminals. In this section, we will see how negative power dissipation is possible in a semiconductor. Electric energy is dissipated when an electric current flows in the direction of an electric field. Negative dissipation or generation results only when current flows againsr the field. When current and field change periodically in time, the average dissipation over'one cycle must be negative in order to have negative ac power dissipation. It will be convenient to use the terminology of ac circuit analysis in describing the above situation. The ac power dissipation is positive when the phase difference between ac current and field is less than 90" and negative when the phase difference is more than 90". In a solid, more current flows when there are more carriers and when the velocities of the carriers are larger. The current density is given by

J

=

qno,

(1)

where q is the charge of the carrier, n is the carrier density, and z) the carrier velocity. A change in current caused by a change in field 6 E is given by

6J

=

q(n . 6 z 1

+ v . bn),

(2)

when changes, prefixed with 6, are small. There is a possibility of negative power dissipation when the carrier velocity decreases as the field is increased and/or the carrier number decreases as the field is increased. A good example of the first case is the transferred electron effect in GaAs. This is a microscopic property set by nature which is beyond the control of the engineer. However, the second case of the change in carrier number can be realized in ordinary semiconductors by designing a proper structure. Designability is based on the fact that the change in carrier density or bunching of carriers is fairly indirectly related to electric field. (Actually, only the divergence of the field is connected with the charge of carriers through Poisson's equation.) So it is possible to have a reduction of carrier number with an increase of field. When various quantities change sinusoidally with time, which is expressed in complex form as ejw', Eq. ( 2 ) becomes

J

=

q(n,v

+ o,n),

(3)

where boldface type designates a vector or phasor and the subscript zero implies dc values or time averages. As noted above, v is in phase with E in

374

T.

MISAWA

most semiconductors; therefore, the first term in Eq. (3) contributes to power loss. When n leads or lags the field E by more than 90" and when its contribution to power generation is large enough to overcome the loss due to the v term, the net effect is power generation. The necessary phase difference is obtained by the transit-time effect, which will be discussed in the next section. 2. TRANSIT-TIME EFFECT

Once carrier bunching is produced at a certain place (cathode) in a device in a regularly pulsating way, it flows downstream with a finite transit time. The bunching may decay due to diffusion or the dielectric relaxation effect. When one observes the carrier stream at a certain position, one sees that the phase of the modulated carrier density is determined by the transit time from the cathode to the observing position. When there is an ac field having phase independent of position, at some point downstream, the phase of the ac carrier density lags the field by 90". The phase difference will remain more than 90" over a distance in which the transit angle changes by an additional 180". A negative-resistance device can be made by making the ac power generation in this space larger than the dissipation in the preceding space. A variety of devices may be made according to the choice of cathode and dynamics of carrier transit in the interaction space. In 1954, Shockley proposed two structures which work according to this general m e c h a n i ~ m . ~ Read later proposed the use of an avalanching region as a cathode.' He showed that the avalanche region produces 90" phase delay in itself so that the whole interaction space can contribute to power generation in a very efficient way. This turns out to be a substantial advantage over Shockley's structures. Before discussing the dynamics of avalanche multiplication, we consider an important feature of carrier dynamics in the interaction space. This is the saturation of carrier drift velocity at high fields.

3. SATURATION OF CARRIER DRIFTVELOCITY It was pointed out that an increase in carrier velocity with field contributes to power loss. However, the carrier velocity actually saturates at the high fields that exist in the space-charge region of pn junctions (above -2000 V/cm in the case of electrons in Si). Because of the saturation of velocity, the power loss due to velocity change [the first term in Eq. (3)] is practically zero. When the carrier velocity increases with field, dielectric relaxation tends to smooth out carrier bunching. That is, the extra charge of bunched carriers produces a change in field which, in turn, induces current that acts to reduce the charge. This debunching reduces the magnitude of the power generation in the later part of the passage. However, when the velocity saturates, dielectric relaxation does not take place, because the change in field does not produce W. Shockley, BeNSysi. Tech. J . 33. 799 (1954).

375 a change in current. This makes power generation in the interaction space more efficient. The saturation value of the velocity is referred to as "scattering-limited velocity" in the literature. 4.

DYNAMICS OF

AVALANCHE MULTIPLICATION

The breakdown of the p n junction is caused by avalanche multiplication of electrons and holes.3a As the field becomes high, a charge carrier can obtain enough energy from the field to release a bound electron into the conduction band, thus also creating a hole in the valence band. The probability of this electron-hole pair creation depends upon field strength. When the probability in one passage across the space-charge region approaches unity, a very large number of carriers are produced from one original carrier and a large current starts to flow. This is avalanche breakdown. As the field changes periodically with time around an average value, the generation rate of carriers follows the field change almost instantaneously. However, the carrier number does not change in unison with the field. For example, even when the field has passed a peak value, the carrier number keeps increasing because the carrier generation rate is still above the average value. The total number of carriers peaks and starts to decrease when the field has decreased from the peak to the average value. In other words, the ac variation of the number of carriers lags the generation rate by 90" and the generation rate is in phase with the ac field. From discussions given in the preceding two sections, it is seen that, when the avalanche region is followed by a drift region in which the carrier drifts at scattering-limited velocity, ac power is generated very effectively when the transit angle lies between 0" and 180".

5 . NEGATIVE RESISTANCE IN

pH

JUNCTION IN BREAKDOWN

a. Read Diode

Read showed that the above situation can be realized in the space-charge layer of a pn junction by properly tailoring the impurity distribution. Figure l(a) shows his n'pip' structure and Fig. l(b) presents the field distribution at breakdown. The space-charge region extends over the whole region between the n+ region at left and the p i region at right. In the spacecharge region, the charge of ionized impurities is not neutralized by electrons and holes as is the case in a bulk semiconductor. Therefore, the field profile is determined by the impurity distribution. Avalanche multiplication takes place only near the left end of the spacecharge layer where the field is highest. Electrons generated in the avalanche region immediately enter the H + region and do not play any important role. '"In the narrow pn junction, the tunneling process is responsible for breakdown

376

T. MISAWA

la )

FIG.1. (a) The structure and (b) the field distribution of the Read diode. (After Read.')

Holes drift through the rest of the space-charge layer. The field is maintained high enough so that holes drift at scattering-limited velocity. This structure will show an optimum negative resistance at the frequency where the hole transit time is half a period. b. General Junction Diode In his original analysis, Read neglected the width of the avalanche region as far as transit-time effects are concerned. The only effect of avalanche multiplication was to inject carriers, with 90" phase delay, at the edge of the space-charge layer. This made the physical argument simple and the analysis tractable, but practical realization difficult. In general, in pn junctions, the region where avalanche multiplication takes place can occupy an appreciable fraction of the total space-charge layer and negative resistance of quality comparable to that with the Read structure is obtained. It is found that ac power is also generated in the avalanche region. The transit-time effect in the avalanche region is appreciably different from that in the avalanche-free region. Here, not only do both electrons and holes exist, but also their behavior is closely interrelated due to avalanche multiplication. While dynamics in the avalanche-free region were described by the behavior of the individual carrier, the collective behavior of the electron-hole plasma governs the dynamics in the avalanche region. The power-generating interaction in the avalanche region is closely related to a spontaneous growth of a plane space-charge wave in an infinite, avalanching electron-hole plasma, which will be discussed in Section 14a. It is based upon the delay in avalanche multiplication and the flow of charge carriers downstream at finite speed, just as in the Read diode.

7 . IMPATT

DIODES

311

111. Fundamental Phenomena and Mathematical Formulation In this part, we discuss two fundamental phenomena encountered in the IMPATT diode, avalanche multiplication and velocities of charge carriers, in more detail as a preparation for the detailed analysis given later.

6. DRIFT VELOCITIES OF CARRIERS4 As the electric field is applied, the drift velocity, or the average velocity of the electrons (or holes), increases proportionally to the field strength as long as the field is small. The proportionality constant is called the carrier mobility. Energy obtained from the field is effectively transferred to the lattice through collisions with phonons and the temperature of the electrons is practically that of the lattice. As the field is increased, it turns out that the energy-transfer mechanism is not efficient enough to keep the electron temperature down. As a result, the mobility of the electrons decreases and the drift velocity does not increase with field as fast as it did at lower fields. At still higher fields, the electron can obtain sufficient energy to emit an optical phonon. The electron loses all the energy acquired from the field in emitting an optical phonon. This takes place as soon as the energy of the electron reaches the optical phonon energy. In this condition, the average velocity of the electrons is one-half of the velocity that corresponds to the optical phonon energy and remains the same as the field is increased. The saturation value, which is called scattering-limited velocity, is of the order of lo7 cm/sec in common semiconductors. The experimentally observed relations between velocity and field are shown in Fig. 2 for electrons and holes in Si and Ge.5 The electric field in the junction in breakdown is on the order of several hundred kV/cm. The velocity can be considered constant in most cases. The following simple expression can describe the gross nature of the velocity change : (4) POE/[l + (POElU,)l, where p o is the low-field mobility and 11, the scattering-limited velocity. 1’ =

7. AVALANCHE MULTIPLICATION’ At higher fields, some “lucky” electrons (or holes) can escape the optical phonon scattering discussed in the preceding section and can acquire sufficient energy to create electron-hole pairs. The required energy is on the order of the gap energy between the conduction and valence bands. W . Shockley, Bell Syst. Twh. J . 30. 990 (1951 1. The curves were obtained by T. E. Seidel at Bell Telephone Laboratories from his own measurements and data published by other researchers. A. G . Chynoweth, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer. eds.). Vol. 4,p. 263, Academic Press. New York and London, 1968.

’

378

T. MISAWA

I o7

>

lo5 E (V/cm)

FIG.2. Drift velocities of electrons and holes in Si and Ge as a function of field. Filled circles are for n-type Ge: V, = 6.5 x lo6, p o = 3800; open circles are for p-type Ge: V, = 8.2 x lo6, po = 1800; filled triangles are for n-type Si: V, = 1.0 x lo’, p o = 1400; open triangles are for p-type Si: V, = 1.05 x lo’, po = 480.

FIG.3. Ionization rates of electrons and holes in Ge, Si, and GaAs as a function of field. The dashed line is for the average of a and p in Si used by Read.’ (After Misawa.26)

7. IMPATT

379

DIODES

The probability of pair creation is averaged over all the electrons and is expressed by the ionization rate, which is the probability per unit distance of passage per electron. In terms of the ionization rates of electrons and holes, c1 and /3, the generation rate of carriers is given by

where u, and up are drift velocities of electrons and holes and n and p are electron and hole densities. The ionization rate is a strongly increasing function of the field strength. Figure 3 shows measured values of ionization rate for Si, Ge, and GaAs.' According to Baraff's theory,* the ionization rate depends upon only three material constants: ionization energy, 4, which is about 1.5 times the energy gap, optical phonon energy 4,and effective mean free path for optical phonon scattering A. A universal relation between ionization rate and field is obtained by proper normalization in terms of the above three material constants. Figure 4 shows the relation.6 This relation, especially with its analytical approximation,' is convenient for extrapolating measured values. The following expression also has been used in the literature' : GL

or

P

=

A exp[-(h/E)"],

m

=

1 or 2 .

(6)

The less chance there is of optical phonon scattering, the larger the chance of ionization. The ionization rate increases with the mean free path of the optical phonon scattering. It has been considered that better-quality material has a longer mean free path and, therefore, a larger ionization rate." Optical phonon scattering involves not only emission of the phonon, but also absorption of the phonon. More optical phonons are available for absorption when the temperature is high. Therefore, the ionization rate becomes smaller at higher temperatures because of shorter mean free path. The effective mean free path is related to lattice temperature by'

1 = A. tanh(gr/2kT).

(7) As the number of electrons increases, the effect of collisions between electrons becomes appreciable. In InSb, the generation rate increases faster than relation (5) predicts when the electron density is above 1014/~m3.6 This kind of carrier density is quite common in Si IMPATT diodes. However, it is not known if there is any deviation from (5) in the case of Si. The

' S. M . Sze and G. Gibbons. A p p / . f / 7 ~ . \ . Lett. 8. I I I (1966). G. A. Bardff, Ph.v.r. Rev. 128, 25017 (1962); Chynoweth' gives a convenient summary of this paper. ' C. R. Crowell and S. M . Sze. Appl. f / i j , s . Lett. 9. 242 (1966). l o C. A. Lee, R. A. Logan, R. L. Batdorf. J . J . Kleimdck, and W. Wiegmann, f / i ? s . Rrr. 134, A761 (1964).

380

T. MISAWA

x Ci

Ei /q EX FIG.

4. Baraff’s universal curves for ionization rate as a function of field. (After Chynoweth

ionization rates shown in Fig. 3 were obtained under the condition of very small carrier density.

8. GOVERNING EQUATIONS Changes in electron and hole densities are described by continuity

equations :

+ g U,, , ap/at = - q p ( d ~ , / i i x ) + g - L,,,, an/&

=

q - '(SJ,,/d.u)

-

(9) where g is the generation rate due to avalanche multiplication given in Eq. (5) and U,, and U,, are recombination-generation rates via localized levels (recombination-generation centers) in the forbidden gap.' Suppose there is only one type of recombination-generation center. The number of charged centers N i changes according to the following equation :

'

dNJdt

=

U c p - U,,.

(10)

In most cases, the U terms in Eqs. (8) and (9) are negligible compared with other terms. They will become important only when the space-charge region is depleted of electrons and holes at a certain phase of the oscillation cycle. This occurs when the current through the diode swings to the minimum. The electron and hole currents are composed of drift current and diffusion current, J , = -q~,n + ~ D , ( & I / ~ X ) , ( 1 1) J , = ~ ~ o , P- yD,,(Sp/Sx), (12) where D, and D, are diffusion constants. It is not well established what D, and D, should be in such a high field that drift velocities saturate." However, it may be reasonable to use the Einstein relation13 with a mobility ji defined as ju/El and electron or hole temperature,

D,

=

(kK/q)Lt7

D,

=

(kT,/m,.

The diffusion term is not important at higher fields except when the "wavelength" of carrier bunching is very small, which occurs at very high frequencies. It is important at the edge of the space-charge layer where the field is weak and the Einstein relation has a well-established meaning. Only Poisson's equation has been considered for describing the electric field, E & E / d X = y(N, - N , + N i+ p - n ) , (13) where E is the dielectric constant of the semiconductor and N , and N , are the densities of ionized donors and acceptors. Here, N i q represents the charge of recombination-generation centers rather symbolically. It may be negligible in most cases. 'I

l2

l3

W. Shockley and W. T. Read. Jr.. Phix. R w . 87. 835 (19521. Reasonable definitions of diffusion and its calculation were worked out at relatively low field by D. J. Bartelink and G. Persky, private communication and Appl. Phys. Lett. 16, 191 (1970). See, for example, W. Shockley. "Electrons and Holes i n Semiconductors." p. 300. Van Nostrand, Toronto, New York. and Landon. 19.50.

382

T. MISAWA

TABLE I Values used in normalization Quantity

(a)

(b)

5 Pm 8.5 x 10" cmjsec

10 pn 10' cmjsec 10- l o sec

2.71 GHz 1015/cm 3 1.36 x lo3 A/cm2 7.54 x lo4 V/cm

1.592 GHz

Expression

Length Velocity Time Angular frequency Frequency No. per unit volume Current density Field Voltage Impedance Admittance

36.1 mho/cm2

~

3.7 x lo3A/cm2 3.5 x lo5 Vjcm 350 V 0.0946 ohm-cm2 10.76 mhojcm'

' Used in Section 12d; w is the width of the space-charge region, u is the scattering-limited velocity, 7 is the transit time, unit admittance is the admittance of the space-charge-layer capacitance for a normalized frequency of 0.5. in Sections 12e and 13a; in Section 130. E, is the peak field in the avalanche region.

It is convenient to introduce dimensionless variables by normalizing various quantities in the equations. This facilitates not only manipulation of the equations by eliminating cumbersome coefficients, but also numerical solution of the equations. Table I lists the units for various quantities. Normalization is accomplished by choosing proper units for length, velocity, and carrier density or electric field. Various values will be used for the units in the following as indicated in the table.

IV. Analysis of Electrical Characteristics The equations given in the preceding part are sufficient to describe the behavior of the IMPATT diode, once the impurity profile and environment (circuit) are given. Actually, it is not only possible but also almost practical, with present-day computers, to solve the equations numerically to any desired accuracy. However, meticulously accurate solutions do not necessarily give good perspectives. In this part, we will discuss both simplified, approximate solutions and elaborate, accurate analyses. 9.

SPACE-CHARGE

LAYERIN pn JUNCTIONS

In a p n junction, the transition from p-type to n-type conductivity is very sharp and hence the change in electric potential takes place in a narrow region around the metallurgical junction. This narrow transition region, which is

7. IMPATT

DIODES

383

also called the space-charge region, is almost completely depleted of mobile carriers. Therefore, there is a large field gradient due to the charge of ionized impurities, as described by Poisson‘s equation,

i7EJ2.y

=

N, - N,,

(14)

which is now in normalized form. The situation prevails up to the point where the field determined by (14) drops to zero. Beyond this point, the impurity charge is neutralized by the space charge of the carriers and the field is very low. The description that the space-charge region is completely depleted of carriers, and that elsewhere space charge is completely neutralized, is a good one and simplifies the treatment of the space-charge region. As the reverse bias is applied to the pn junction, the space-charge region widens and the peak field increases to accommodate the increased voltage. As the junction is biased beyond breakdown, current starts to flow. Equation (14) still holds until the current increases so much that the carrier density becomes appreciable compared to impurity densities. However, after the carrier density becomes appreciable, the field profile is not determined by the impurity density only. The situation becomes complicated. In the space-charge region, the field is very high and electrons and holes drift away at scattering-limited velocities into n-type and p-type regions, respectively. The supply of carriers into the region is by thermal generation in the adjacent regions. For example, electrons are generated in the p region. They diffuse toward the edge of the space-charge region and, as soon as they arrive at the edge, they drift down through the space-charge layer to the 12 region. When the field is high enough, electrons produce electron-hole pairs in the passage and current starts to flow. 10. STATIC CURRENT-VOLTAGE CHARACTERISTICS

The static or dc characteristics of the pn junction in breakdown are obtained by setting the time derivatives in the fundamental equations to zero. The diffusion current is not important in the space-charge region where the field is high. The continuity equation for electrons then becomes dJ,Jdx

= ctJ,

+ PJ,,

where it is assumed that the x axis is perpendicular to the junction plane and is from p to n region. By using the fact that the total current J = J , + J , is independent of x, the hole current J , can be eliminated from the above equation, dJ”1d.X

- (X -

B)J,

=

DJ.

A small number of electrons enter the space-charge region from the p-type side and holes from the n-type side. We designate the currents

384

T. MISAWA

associated with them as J,, and J,,. Integration of the above equation with the boundary conditions J,(O) = J,, and J,(w) = J - J , gives

+ M,J,,,

J = M,J,,

(17)

where

M,

=

A

=

(19)

1/(1 - A ) ,

low/3s,’ exp[

(CI -

(20)

/3) dx’

When A approaches unity, the total current becomes very large. The condition of avalanche breakdown is A

=

IOwD { [s,^ exp

(cl

-

D) dx‘]

}

dx = 1

From a symmetry argument, the above equation can be written as

The analytical solution given above is useful only when CI and /3 are known functions of x. This is the case when the current is small and the space charge of the carriers is negligible. The field is then determined solely by the impurity distribution. Then, cl and 3/ are fixed functions of x independent of current. For most of the studies on current multiplication and avalanche breakdown, the assumption of negligible carrier space charge does not impose any serious problem. The above equations have been extensively used in experimental determinations of c( and p from measured M’s or breakdown voltage and in theoretical estimate of breakdown voltage in various structure^.^,' However, when one wants to know current-voltage relations after breakdown, this simplified solution becomes worthless because, at higher currents, the voltage remains the same as the breakdown voltage. Changes in field profile due to carrier space charge have to be taken into account . When the carrier space charge cannot be neglected, the complexity of the problem increases by orders of magnitude. This is because the modified field affects the carrier distribution. Now, neither field nor carrier density can be obtained independently, and the whole problem has to be solved

385

x e -x

erfc ( -1

L2

0

xi

Xe

X

FIG. 5. Silicon “abrupt”-junction diode made on epitaxial layer. C, = 10”/cm3, L , = 1.123 pm, X , = 3 pm,C , = 8 x 1015/ccm3,C,,,, = 1.5 x IOI9/cm3, L , = 0.3743 I‘m, X , = 7.5 pm.

self-consistently. An analytical solution is impossible except for some special cases which are remote from reality.I4--l6 Numerical analysis with a modern computer has been successful in solving the p r ~ b l e m . ” ~Appendix ~* A explains the details of solution. We will discuss the results of such an analysis of two structures. One structure, a diffused “abrupt” junction, has been used often in actual IMPATT diodes, and the other, an idealized pvn diode, is of some interest because of its dc negative-resistance characteristics. The “abrupt” junction is made by diffusing acceptors into a uniformly doped n layer. When the junction is shallow, the field profile is more or less triangular and resembles the ideal abrupt junction. In the actual diode, the junction is made on an epitaxial layer grown on a low-resistivity substrate. The epitaxial layer is made thick enough so that the space-charge layer terminates in front of the substrate. The space-charge layer extends into the substrate at high current density or high temperature. Figure 5 shows the impurity distribution of the silicon abrupt-junction diode to be analyzed. The field and current distribution of the diode are shown in Fig. 6a for six bias currents, 100, 200, 500, 1000, 2000, and 5000A/cm2. The negative charge of electrons partly compensates the positive charge of donors in the epitaxial layer. Because of this, the space-charge region extends further toward the substrate at higher currents. At the highest current, the field

’‘

J. 8. G u m . in “Progress in Semiconductors’. ( A . F. Gibson, ed.). Vol. 2. p. 213. Hcywood,

I”

London. 19.57. Hideharu Egawa. l E E E T,.tr/rs. E / w / ~ ~Dcc>iws J/? ED-13. 754 (1966). B. Hoefflinger. l E E E Eons. Electrori Dci>ic,e.sED-13. 151 (1966). T. Misawa. lEEE Trms. E k t r o n Dczvw.s ED-13. 143 (1966). H.C. Bowers. IEEE f i r m s . Elecrron Deciicrs ED-15. 343 (1968).

386

T. MISAWA

400

I t -

z W LT

IT

0.8 300

2 -1

2 0

E

0.6

\

1

t c

z

L 200

W

LL

0.4

_J 0

w LL

5 u

z

I00

0

0.2 0

W -I

W

0

0

FIG. 6a. Field and current distributions of the Si abrupt junction in Fig. 5. 130

I20

110 v)

IA

0 5

100

90

80

0

1

2

3

4

5

DENSITY IN kA/cm2 sec

CURRENT

FIG.6b. The V-I curves at four temperatures of the Si abrupt junction in Fig. 5.

1.0

400

z W

a 0.8

5 u

300

-I

a 0.6

z -

t0 I\

200

F-

z

LL

0.4

w

a a 3 V

z 0

0.2 a

RATIO

CURRENT

k

V

W

W

0

0

1

2

3

4

5

6

7

8

9

0 10

DISTANCE IN p m

FIG.7a. Field and current distributions of Si pvn diode with a width of 10 pn and v-region doping of N , = 7 x 10'4/cm3.

penetrates into the substrate. This shows up as a sharp change of field near the n-side end. The current distributions shown by dashed lines in Fig. 6a indicate that avalanche multiplication takes place over a region about 1 p wide on the n-type side of the field peak. Since electrons have a larger ionization rate than holes in silicon, the avalanche region is shifted toward the n-type side with respect to the field peak. There, more electrons with the larger ionization rate are present than on the p-type side. The calculated current-voltage characteristics are plotted in Fig. 6b. Curves at various temperatures are obtained by using temperature-dependent ionization rates discussed in Section 7. Since the computation neglects thermal generation of carriers, the curve at the highest temperature (600°K) may not be accurate. The slope of the curves is very close to what is expected from space-charge resistance. l 9 Next, we consider an idealized p\w structure which has a constantly doped v region sandwiched between very highly doped p and n regions. The edges of the space-charge region remain fixed at the edges of the v region after punchthrough, as reverse bias is applied. Figure 7a shows field and current distributions and Fig. 7b shows current-voltage characteristics of a Si diode whose v region is 10 p wide and has a donor density of 7 x 1014/cm3.

''

S. M . Sze and W. Shockley, Bell SW. Twh. J . 46, 839 (1967)

388

T. MISAWA

250 24 0

200

h I 0

I

I

I

I

I

I

I

I

I

2

4

6

8

10

12

14

16

18

20

CURRENT D E N S I T Y IN k A / C m '

FIG. 7b. The V-I curve of the Si pvn diode of Fig. 7a.

In Fig. 7a, we notice that the field profile is profoundly distorted at high current densities where the mobile carrier densities exceed the ionized donor density. From the current distributions shown by the dashed lines, it is seen that avalanche multiplication takes place all over the space-charge region at low currents and it becomes confined at both edges as the field becomes distorted at high currents. It is to be noted that the voltage across the diode decreases as the current is increased. This dc differential negative resistance may deserve more detailed attention and has been the subject of many reports. In order to have a certain amount of current through a pn junction in breakdown, a certain amount of carrier generation is required. The larger the current, the larger is the amount of generation. As the current is increased, the field profile changes due to the charge of added carriers. If this change in field is such that a greater amount of carrier generation is obtained with reduced voltage (which is the integral of the field over the region), a negative resistance results. Generally, the increase in generation rate for a given increase in field is larger at the place where the field is higher. Therefore, negative resistance is obtained when the added charge due to the increased current causes high fields to rise and lower fields to fall. In the space-charge

389

FIG.8. Electric field E and carrier generation rate g before a small increase in current 61, and changes in carrier densities, electric field, and generation rate after the increase in current for (a) negative and (b) positive differential resistance cases. For case (a), J6g dx = 61 and 5 bE d x < 0 ; for case (b), I6g d x = 61 and 6 E d x > 0.

region, the added charge lowers the field in the center and raises it near the ends. This is because more electrons are added near the n-type end and more holes near the p-type end. Therefore, roughly speaking, the differential resistance is likely to be negative when the field profile is upward concave and positive when upward convex. The situation is illustrated in Fig. 8. Actually, the above statement is too crude. The presence of negative resistance has to be determined case by case. Let us consider a special case of constant electric field across the spacecharge region. When electrons and holes have equal ionization rates, the condition of breakdown, Eq. (21), becomes

390

T. MISAWA

When a small change in field 6 E is introduced, the above condition is written as

jOw(dol/dE)6E dx

= 0.

Since the field is constant, da/dE is also constant. Then, the above condition reduces to

IOwdEdx= 6 V = 0. Therefore, the differential resistance is zero. When the ionization rates of electrons and holes are not equal, a similar argument leads to

It has been found with numerical analysis that, when a’/p’ < a//?,the differential resistance is negative, and when cr’//?’ > a//?,it is positive. In the actual diode, the field does not drop vertically as in the above ideal cases. The effect of the charge of added electrons and holes is largest in these low-field regions at the edges. According to the preceding argument, this contributes to the positive resistance. A substantial upward concavity in the center portion is required to obtain negative resistance against the unfavorable effect of the end regions. Sometimes, unequal ionization rates of electrons and holes contribute to more negative resistance, as seen in the above case of constant field. There, negative resistance is obtained even with a flat field profile when conditions are favorable. The unequal ionization rates must contribute to the negative resistance in Fig. 7b of the Si p v n diode. 1 1 . GROWING WAVEIN AVALANCHING ELECTRON-HOLE PLASMA~O

The simplest way to investigate the dynamics of avalanche multiplication is to consider the plane waves in the uniformly avalanching, infinite, electronhole plasma. This is because the infinite size of the actual device introduces additional complexity. We consider the simplest case, namely electrons and holes have equal ionization rates and drift at the same scattering-limited velocity. Diffusion and generation-recombination via localized levels are neglected. Then the fundamental equations become, in normalized form,2oa

i3Eld.x

=

N,

-

NA+p - n,

(24)

+ g n + p),

(25)

anpt = 8 . 1 ~ 8 .

T. Misawa. IEEE Trans. Electron Devices ED-13, 137 (1966).

7. IMPATT

DIODES

+

39 1

aplat

=

- ( S J , / ~ . ~ ) r(n+p),

(26)

J,

=

-n,

(27)

-17.

(28)

J,=

The x axis is in the direction of the electron flow. When the time-varying ac part of various quantities in the above equations is much smaller than the time-independent part, the equations can be linearized for the ac components which vary like ej"". Then, we haveZoa dE/dx = J ,

-

J,,

(29)

+ (a- jw)J,,+ aJ,,

(30)

dJ,/dx

=

a'JE

dJ,/dx

=

-a'JE - aJ,

-

( M - jw)J,,

(31)

where a’ = da/dlEl and J is the total dc current. In the case ofconstant electric field, a and a’are independent of x. Therefore, all the coefficients in Eqs. (29H31) are constant. Then, according to a theorem on linear differential equations,*' the solution is a sum of terms of the form e-jkx with three different values of k. It is found that one of the k's is zero and the other two are the roots of the following dispersion relation:

k2

+ 2a'J

- j 2 t m - w 2 = 0.

(32)

Let us discuss in more detail the wave solutions of Eqs. (293-(31). The dispersion relation (32) is solved for o as jw

=

a

+ (a2-2a'J

- k2)lI2.

(33)

When k is real, we have a spatially sinusoidally varying perturbation. Equation (33) shows that the perturbation grows exponentially with time. When the wavelength is small (large k ) or the current is large, the quantity in the square root sign of Eq. (33) is negative. In this case, the perturbation oscillates in time and propagates in space while its amplitude grows. The time constant for the growth is a.This growth of the periodic carrier bunching is a rather remarkable phenomenon. For, one might think that avalanche multiplication would randomize the regular carrier bunching, especially since electrons and holes created by avalanche move away in the opposite direction. Let us examine in more detail how this growth of bunching or instability is obtained. First, we consider the case when carriers drifting at the scatteringlimited velocity receive a spatial density perturbation. Avalanche multiplication has not yet set in. The perturbation in density, or bunching of carriers, ""The scattering-limited velocity was chosen as unit velocity in the normalization. 2 1 See, for example, I. S. Sokolnikoff and R. M. RedhefTer,"Mathematicsof Physics and Modern Engineering," p. 100. McGraw-Hill, New York, 1958.

392

T. MISAWA

-

dc E

LARGEST DENSITY

ELECTRON DENSITY

HIGHEST IONIZATION e+ j w t

IONIZATION

FIG.9. Instability in avalanching electron-hole plasma. In the figure, the tildes indicate vectors. (After Misawa.")

does not decay, as discussed in Section 2, because the dielectric relaxation does not take place. (We are neglecting diffusion.) When electrons bunch periodically in space, the bunching moves toward the positive x direction at the scattering-limited velocity without decay. This corresponds to the root with the plus sign in Eq. (33) with tl and a' equal to zero. The situation is illustrated schematically in Fig. 9. An electric field perturbation accompanies this electron density perturbation. As is seen from Poisson's equation, the field lags the electron density by 90". Now, we consider what happens when avalanche multiplication is introduced. Since the dc field is in the negative x direction, the highest field is obtained when the ac field is at the negative peak, as indicated in Fig. 9. We have more generation (or ionization) when there are more electrons and when the field is stronger. Therefore, the largest generation takes place somewhere between the place with the largest density and the place with the highest field. Thus, the generation rate leads the electron density by less than 90". The extra electron density created by the avalanche lags the generation rate by 90". The resulting, modified electron density now leads the field by less than 90" and the

accompanying current lags the field by more than 90" (remember the electron charge is negative). This makes the ac power dissipation negative. The perturbation gets energy from the dc field and grows in amplitude. Holes created by avalanche also form a periodic pattern. It turns out that the pattern is dragged by the electron-density wave and moves in the same direction as the latter, although individual holes moves in the opposite direction. The hole-density wave also lags the field by more than 90". The situation discussed here may be modified appreciably by the presence of the boundary. However, the above instability present in the infinite plasma indicates that a terminal dynamic negative resistance is obtained when we have uniform avalanche in the space-charge region of the pn junction. This case will be discussed in more detail in Section 12d. 12. SMALL-SIGNAL ANALYSIS

It is rather a simple matter to analyze the case when the time-dependent component is much smaller than the dc component, because the relevant equations are linear. In the IMPATT diodes, it has been found that the smallsignal analysis is very useful as a guideline in designing the oscillator diode, although theoretically there is only a rather philosophical relation between small-signal characteristics and oscillator performance. The practical importance of the small-signal analysis is based upon this empirical fact. a. Generai Analysis-N

uinericul Approach

L 7 3 2 2

The analysis to be discussed here assumes that diffusion is negligible in the space-charge region. This will be a reasonable assumption as long as the diffusion length for the transit time is smaller than the wavelength of the space-charge wave in the space-charge region. The equations to be solved are similar to Eqs. (29H31) and are listed in Appendix B. They look more complicated because electrons and holes do not have the same ionization rate and velocity and because field dependences of velocities are taken into account. Extra complexity arises because now coefficients of the equations are not constant. The dc electric field is not constant and J , and J , do not appear in the form of the sum. These dc quantities must be determined beforehand as functions of position. The method of analysis discussed in Section 10 may be used for this purpose, The boundary condition is that the electron current and the hole current are given as J,, and J, at the p-type and the n-type ends, respectively. Another constant that enters the problem is the total current through the diode J . Because of the linear nature of the problem, any quantity, such as electron or hole density, electric field, and so on, is composed of three terms, each of H. K . Gummel and D. L. Scharfetter. Bell. S

j ~ tTuch

J . 45, 1797 (1966)

394

T. MISAWA

600 2 00

500 400 N

N

E

300

0

2 c

100

0

c

E E

200

E

z -

I d

100

0

z

z U

2

0

O

3 D

z

-100

0 0

t

W 0

2 v)

-200 -100

0

10

20

30

FREQUENCY I N GHz

FIG. 10. Small-signal admittance of Si abrupt-junction diode at a bias current of lo00 A/cm2.

which is proportional to J,,, Jps,or J. The voltage across the space-charge region, which is the integral of the field, is not an exception : V

=

I

E dx

=

ZJ

+ Z,,J,, + Z,,J,,

.

(34)

Z, Z,,, and Z,, are the final products of the computation, and therefore depend upon the structure of the diode (impurity profile), material constants, bias condition, and frequency. Ratios of J,, and J, to V are on the order of magnitude of the conductance of the reverse-biased pn junction. Their values are much smaller than l/Zln or l/Z,,. Therefore, the last two terms on the right-hand side of (34) are neglected and Z is the impedance of the spacecharge layer. The ratios Z,,/Z and ZIp/Zare the ac counterpart of the dc multiplication factors M , and M , in Eqs. (1 7H20).The impedances Z,, and Z,, are important in the case when J,, and J, become comparable to J, for example, by photogeneration. Appendix B gives the details of the calculation. Figure 10 shows the calculated admittance, G = 1/Z, of the Si p n abrupt junction at a current density of loo0 A/cmZ, whose dc characteristics were discussed in Section 10. The susceptance is inductive at lower frequencies, goes through a resonance, and approaches the susceptance of the space-

395 CONDUCTANCE

POS. - ----

SUSCEPTANCE

__

NEG.

---_

A = 100 A / c m 2

B = 200

C =

F

D = 1000 A/crn2 E - 2000 F = 5000 I

I

2

I 3

I I 1 I I l l 4 5 678910

I

I

I

2

3

4

I l l 1 1

5 6 789100

FIG. 1 la. Small-signal admittance of Si abrupt-junction diode at various bias currents: frequency versus admittance.

charge-layer capacitance at higher frequencies. The conductance is positive at lower frequencies and turns negative at a frequency that is slightly lower than the resonance frequency mentioned above. The magnitude of the negative conductance peaks at about 16 GHz and then decreases toward higher frequencies. The admittance at various other currents is plotted in Figs. 1la, b. Very roughly speaking, the admittance increases proportionally to the current. It is seen that both the resonance and the cutoff frequencies increase proportionally to the square root of the bias current. For a fixed frequency, the negative conductance initially increases proportionally to current, reaches a peak, then turns over, and finally becomes positive above a certain critical value. This change of sign takes place at higher currents when the frequency is higher. Another important small-signal parameter, which has been found particularly useful in connection with oscillator performance of the diode, is the Q defined as Q = energy stored in diode/energy dissipation per radian

396

T. MISAWA

B

100

G

mho/crn2

FIG.l l b . Small signal admittance of Si abrupt-junction diode at various bias currents.

where E is the dimensionless field (see Table I) when unit ac current flows through the diode and R is the real part of the diode impedance. This is a measure of how effectively ac power is generated in the diode, and indicates the buildup rate of oscillation when the diode is used as an oscillator. A smaller magnitude of the negative Q indicates a better quality of the negative resistance. Figure 12 shows the Q of the diode whose admittance was discussed before. For a given bias current, the magnitude of Q exhibits the minimum at a frequency close to that which gives the maximum magnitude

397 100 8 6

4

A

I

I

I

1

I

l l l l

I

I

1

1

1

1

1

j

FREQUENCY IN GHz

FIG. 12. Small-signal Q of Si abrupt-junction diode as a function of frequency and bias current. Curve: A, 100A/cm2; 3, 200A/cm2: C , 500A/cmZ; D, l kA/cmZ; E, 2kA/cm2: F, 5 kA/cm2. CONDUCTANCE SUSCEPTANCE

01

2

3

4

NEG -

POS

-

5 67891 FREQUENCY IN

--__

2

3

4

5 6 78910

GHz

FIG. I3a. Small-signal admittance and Q of Si pvn diode. Parameter IS bias current. Curve: A, 100A/cm2; B, 200A/cm2; C, 500A/cm2; D, 1 kA/cmZ; E, 2 kA/cm2; F, 5 kA/crnZ; G, 10 kA/cmZ; and H, 20 kA/cm2.

398

T. MISAWA

loo

F -

10

-w >

-

b

5w

l

-

-z 0

r

-

--

0.01 0.I

2

3

4

5 67891

2

I 3

I 4

l l l l l 5 678910

F R E Q U E N C Y IN GHz

FIG. 13b. The Q of the Si pvn diode of Fig. 13a.

of the negative conductance. The optimum frequency goes up with current, again proportionally to its square root. The quality of negative resistance improves as the current is increased, up to a certain current. In this particular diode, the best negative resistance is obtained at about lo00 A/cm2. Above this current, the negative resistance degrades. The optimum frequency of the diode is about 15 GHz. It has been considered that better quality of the small-signal impedance indicates better oscillator performance. A more detailed discussion in this connection will be given in Section 12f. Figure 13a shows the small-signal admittance and Fig. 13b the Q for the pvn diode whose dc characteristics were discussed in Section 10. Because of the presence of dc negative resistance, the conductance remains negative at lower frequencies, in contrast to the case of the abrupt junction. Note that with the space-charge layer width of 10 pm,the transit angle (0 = cowd) is II at about 5 GHz. The susceptance shows similar resonance characteristics as in the abrupt junction. The Q is better at lower frequencies and improves as the bias current is increased. As was the case with dc negative resistance, the low-frequency characteristics are sensitive to slight changes in doping at the edges of the space-charge region. When the doping changes realistically from the heavily doped end regions to the center region, the low-frequency negative conductance tends to disappear, especially at lower currents. However, the characteristics at higher frequencies (around and above the resonance frequency) remain more or less the same.”

7. IMPATT

399

DIODES

The general analysis given above has the merit of being straightforward and exact but gives hardly any insight into what is going on inside the diode. In the following, we discuss a more idealized analysis which sheds more light upon the relation between device structure parameters and characteristics.

6. Drift Regionz3 First, we consider the region where no avalanche multiplication takes place. We assume that electrons (and holes) move at the scattering-limited velocity. Then the injected, bunched electrons flow down the drift region without any debunching. From Eq. (30) with CL = 0, the electron current is given byzoa J, = JnOe-Jwx, (36) where J,, is the injected current at x

E

=

=

0. The associated electric field is

Eo - (J,,/jw)(e-j"" - I),

(37)

where E, is the field at x = 0. The x axis is in the direction of electron flow and the scattering-limited velocity is taken as unit velocity. The important jwE,. By relation here is the one between J,, and total current J = J,, integrating Eq. (37) over the width of the drift region w,, we obtain the following expression for the total current :

+

J = j(O/wd)vd

+ PdJnO

7

(38)

where /?d = (1 - e-jo)/jO, 8 = OW,, (39) and V, is the ac voltage across the region. The first term of (38) is the current through a capacitor whose width is wd. The second term represents a current proportional to the injected electron current. The coefficient is a current transfer factor whose magnitude and phase change according to the transit angle 0 as sketched in Fig. 14. It may be interesting to note that, although the electron current given by (36) rotates all the way around the origin in the phase plane, the current through the region based on the electron current remains in the two lower quadrants of the complex plane. Corresponding to the two current components mentioned above, the equivalent circuit is a parallel connection of a capacitor with a as shown in Fig. 15. width of wd and a current generator PdJnO

c. "Narrow" Avalanche Region--Read's Original Approach1vz3

Analysis of the avalanche region where avalanche multiplication takes place is complicated. Read was able to simplify analysis to a substantial

'' M . Gilden and M . E. Hines, l E E E Tram. Electron Dwrces ED-13, 169 (1966).

400

T. MISAWA

FIG.14.Schematic of complex current transfer factor Bd in the drift region.

degree by the following assumption: the total particle current, i.e., the sum of electron and hole currents, does not change with position although each component is dependent upon position. We know that this is true when the currents do not change with time. However, he assumed that this would remain approximately correct as long as the currents change slowly with time. He claimed that the approximation is good when the avalanche region is narrow and the transit angle through it is small. The merit of this approximation is much more appreciated in large-signal analysis than in small-signal analysis, for the approximation made it possible to solve the all-but-intractable problem. Following Read, we assume that electrons and holes have equal scatteringlimited velocities and ionization rates. We obtain the following continuity equations for electrons and holes20a: aJjat

=

-(a~,jax)

+ MJ,,

+

aJ,pt = ( a ~ , / a x ) G I J ~ ,

(40) (41)

where J o = Jn(x,t ) + J,(x, t ) is independent of x according to the abovementioned approximation. A tractable equation is obtained by adding the -., J

b FIG.15. Equivalent circuit of drift region. Tildes indicate vectors.

401 above two equations, d(J,

+ J,)/dt

=

dJ,/dt

=

[ d ( J , - J,)/dx]

+2d,

Noting that J , is a function o f t only, the above equation is integrated over the avalanche region of width wa, r wa

W,

dJo/dt

=

IJ, -

J,lta + 25, J

tl

dx

0

=2 4

J-1

ctdx - 1 )

+ 25,.

The currents are in normalized units.20aIn (42), it is seen that the dynamics in the avalanche region are represented by a single, ordinary differential equation. Equation (42) can be linearized for small ac parts as follows :

J,

=

(2dJ/jo)EO,

(43)

where 2 is the average of dcc/dE over the avalanche region and 5 is the total dc current. In Read's approximation, a unique, position-independent ac field E, exists in the avalanche region because the displacement current juE,, is also independent of x. Combining the displacement current and the inductive current, the avalanche region is represented by a parallel connection of the following inductor and capacitor,20a -

La = wa/2dJ,

(44)

c, = l / W a .

(45)

The expressions are for unit area. A realization of the Read diode is illustrated in Fig. 16. The impurity distribution in this p + n v n f structure is tailored in such a way that the field in the v region in breakdown is high enough to maintain the electron saturation velocity but low enough to confine avalanche multiplication within the p'n junction region. By properly choosing a value for the avalanche region width, the structure can be analyzed by Eqs. (43),(38), and (39).The equivalent circuit is shown in Fig. 17. Assuming that W, is negligible, Read approximated the total admittance by that of the drift region whose width is equal to the total space-

402

T. MISAWA READ DIODE

P+

n

FIG.16. A p'nvn'

v

n+

Read diode.

FIG.17. Equivalent circuit of Read diode. Tildes indicate vectors.

charge-layer width and obtained the following expression for the admittance20a: 0 2Y Y ( 0 )= u! 1 - e - j o - j 2 y ’ Y - - ( lww -)b),

o2

2

The small-signal admittance calculated from the above expression reproduces most of the behavior of the exact admittance obtained from the numerical analysis shown in Figs. 10 and 11. A major qualitative difference between Read’s approximation and the exact analysis is the fact that in the former both real and imaginary parts of the admittance change sign at a single frequency o,,while in the exact analysis the real part changes sign at lower frequency than the imaginary part. However, Read’s expression will be quite valuable when one wants to have a rough idea about the smallsignal characteristics of a diode whose avalanche region is not wide. Read’s original equation (42) governing the dynamics of the avalanche region can be extended to the case when electrons and holes have different ionization rates and scattering-limited v e l ~ c i t i e s Under . ~ ~ the assumption that the value of the total particle current is independent of position, one obtains the following extended equation :

24

C . A . Lee, R. L. Batdorf, W . Wiegmann. and G. Kaminsky. J . A p p l . P h ~ a 38,2787 . (1967). The particular expression in the following is given by C . A. Lee, in “High Frequency Generation and Amplification Conference Proceedings,” p. 243. Cornell University School of Electrical Engineering, Ithaca. New York, 1968 ( A D 666-582).

404

T. MISAWA

and M , and M , are given by Eqs. (18) and (19) with the integration interval replaced by the avalanche region. The extended equation, Eq. (47), will be useful because silicon, one of the most used semiconductors for the IMPATT diode, has quite different ionization rates for electrons and holes. It is to be noted that, although M and T~ are dependent upon the composition of the primary current, JAs and Jhs, their product, which appears in the first term of (47), is independent of it. When electrons and holes had the same properties, z1 was one-half of the transit time across the avalanche region.

d . “Wide” Avalanche Region-pin Diode When the field is constant, analytical solution of small-signal characteristics is possible even for a “wide” avalanche r e g i ~ n . ~ We ~ . ~discuss ’ the simple case when electrons and holes have equal characteristics except sign of charge.26 The analysis may be applicable to a pin diode. As discussed in Section 11, the small, time-varying parts of the field and particle currents are each composed of three terms as follows:

E

=

C,ejkx

+ C,e-jkx + [ ( ~ c-I j o ) / k 2 ] J , + o)C,e-jkx - (u’J/kZ)J,

(53)

+ o)C,ejkx+ +j(k - c 0 ) ~ ~ e-- (jM ’J/k2)J, ~~

(54)

J, = +j(k - o)C,ejkx - ) j ( k

,

J = -- j ( k

(52)

where k is one of the roots of (32),

k

= (0’ -

2a‘J

+~~wcI)’”,

and C, and C , are constants to be determined by the boundary conditions. We choose CI so that the breakdown condition tlw = 1 is satisfied, i.e., M = 1 when the width of the space-charge region is chosen as unit length for normalization. From the left side, electrons enter the region, giving rise to an electron current JnS.At the right end, the hole current is equal to J,,. These two conditions are sufficient to determine the values of C, and C , . Thus, the impedance of the pin diode can be calculated from the general expression (34).We showed there that actually J,, and J,, are equated to zero. The calculated small-signal admittance of a model diode is shown in Fig. 18. Actually, the admittance of the space-charge layer as a capacitor ( j w / w = j o , with w as unit length) was subtracted. The diode is essentially a 5-pm-wide Si diode, an average value between electron and hole ionization rates having been used. The units of normalization are given in column (a) of Table I. The susceptance is inductive and is almost that of a fixed inductor. 25

26

S. T. Fisher, ZEEE Trans. Electron Devices ED-14, 313 (1967). T. Misawa, ZEEE Trans. Electron Devices ED-14,795 )1967).

LO

.\

-

w

V

z

a Ik

-.

-

-

n

lo-‘ -

W

2

$

10-2

\

I

\

\

I

-ah.

‘\

-\.

-

- REAL - IMAGINARY

N J a

‘\.g.o

\

h\

5 0

a

.I \

I

,.f..2 . f .. **.,CURRENT

o...

-

0.1

----

1

--

0.01

1

I

I

1

I

I

I

1

I

1

1

1

NORMALIZED FREQUENCY

FIG.18. Small-signal admittance of idealized pin diode. (After Misawa.*’)

The arrows in Fig. 18 indicate resonance frequencies where the “inductors” resonate with the space-charge-layer capacitance. The conductance is negative and almost constant over the frequency range shown. It turns out that the low-current, low-frequency approximation of admittance is very good in the current and frequency range investigated.26 The approximation formulae are G, = d J / 5 (55)

La-‘ = 3 d J / w , (56) where w is the width, and is equal to unity in the present normalization because it was chosen as unit length. The expression for the inductance is the same as that for the narrow avalanche region (44) except for the numerical factor.26aThe only new thing here in the wide-region case is the frequencyindependent negative conductance given by (55). Comparing the results obtained above with the admittance of the pvn diode shown in Fig. 13, which was obtained by a numerical method, we see that the simplified analysis can reproduce the essential features of the admittance over an octave of frequency centering on the resonance frequency. At higher currents, the field in the pvn diode becomes distorted because of mobile carrier charge and its admittance behaves differently from that of our pin diode. However, our simplified model still can describe the behavior of a structure that is made to have uniform field at these high currents. 26”Thedifference in the numerical factor was also pointed out by Fisher.”

406

T. MISAWA

n TYPE

p TYPE a = CONST

JPO

DRIFT

AVALANCHE

DRIFT

REGION

REGION

REGION

FIG.19. Simplified model of general IMPATT diode.

e. General Case-IMPATT Diode

A general IMPATT diode whose avalanche region may not be narrow can be analyzed by connecting the drift-region solution of Section 12b with the avalanche-region solution of Section 12d.26 Although this kind of approach may not be good for the meticulous analysis of experimental data, it gives a general idea of how the characteristics of the diode change with structure parameters. Let us consider a general structure shown in Fig. 19 with one constantavalanche region followed by two drift regions which receive electrons on the n-type side and holes on the p-type side. The avalanche-region solution is the same as that in the preceding section. We know the values of electron and hole currents at the boundaries with the drift regions. When the injected particle current is known, the solution in the drift region is obtained from Eqs. (36)and (37). The small-signal admittance and Q are calculated according to the general procedure. By combining the equivalent circuit of the drift region in Fig. 15 with that of the avalanche region explained in the preceding section, we obtain a general equivalent circuit shown in Fig. 20. Each drift region, whose capacitance and current transfer factor are suffixed with n or p according to whether it is located on the n- or p-type side, has a current generator /jdJnO. The JnOis not as simply related to the components in the equivalent circuit of the avalanche region as in the case of the Read diode shown in Fig. 17. In the Read diode, JnOwas a component of current which flows through La. In the present case, it has to be computed from the avalanche-region solution. The results of an analysis along the line explained above will be presented for a series of diodes whose avalanche regions occupy from 10% to 100% of the 10-pm-wide space-charge layer. The avalanche region is located at one end of the space-charge region so that the diode has only one drift region. Pertinent data for the diodes are listed in Table I1 and units for normalization in column (b) of Table I. The diodes are assumed to be Si diodes. For the

7. IMPATT

407

DIODES

FIG.20. Small-signal equivalent circuit of IMPATT diode.

ionization rate, the average value between electrons and holes shown in Fig. 3 is used. The breakdown voltage in the last column in Table I1 is obtained by assuming 100 kV/cm in the drift region. Figure 21 shows admittances of the six diodes at a bias current of 0.0704 or 260 A/cm2. The real part is plotted in Fig. 21(a). As the avalanche region becomes wider, the frequency range over which the conductance is negative increases and its magnitude decreases. Also shown in the figure is the conductance of a Read diode with M , of 0.1 calculated from (46). It is seen that our SIMU-1 behaves almost in the same way. The bias current of 0.0704 was chosen so that the conductance of the Read diode at transit angle n is positive above this value. The imaginary part is shown in Fig. 21(b). Except for the 10% unit, SIMU-1, the behavior is more or less the same as the pin unit, SIMU-6. TABLE I1

Structure No.

Fraction of avalanche region

a

SIMU-1 2

0.1

3 4

0.5

2

2 ;

5 6

0.9 1.o

1.5 1.11 1

1 3

10 3

*’

60 21.98 15.69 12.69 9.61 8.8 I

v (V) 125 162 184

202 229 239

408

T. MISAWA 3

I

J = 0.0704 ( JI) SCALE

2

w

u Z a tu

I

3 0

z

0

u o

n W

N

a

I LL

0

-1

2

-2

-? TRANSIT ANGLE IN RADIANS

FIG.21a. FIG.21. (a) Real and (b) imaginary parts of the admittance of the six structures listed in Table I1 as a function of frequency (transit angle) for bias current of 0.0704, or 260A/cm2. (After Misawa.26)

One of the outstanding dependences of characteristics on avalanche-region width is obtained when the small-signal Q is compared between the structures. Figure 22 shows the small-signal Q of the six structures at five bias currents ranging from 130 A/cm2 to 2090 A/cm2. In the 10% unit, SIMU-1, the Q assumes the best value at the lowest bias current shown for the transit angle of about n. Although not shown, the Q degrades at still lower currents as was the case with the abrupt-junction diode shown in Fig. 12. As the avalanche region widens, the optimum frequency for this bias current decreases and the Q degrades. It is seen that a larger bias current is required for the optimum Q with a wider avalanche region. This bas an important practical implication :as an oscillator, the structure with a narrower avalanche region will reach a “reasonable” efficiency at lower bias currents. This is an advantage for CW operation.

7. IMPATT

409

DIODES

10

4 8 7 w

0

z 6

a

In w 5 0 v)

3

m 4 0

w

2

3

J

a

2

a 2 0

z

I

C

1

-2 0

1

2

I

I

3 4 5 6 7 8 T R A N S I T ANGLE IN RADIANS

I

I

I

9

I

FIG.21h.

,f: Implication of Small-Signal Q with Regard to Oscillator Perfbrmance

Since it is possible to perform extensive analysis of the diode in the smallsignal regime because of its simplicity, it will be very convenient if the largesignal performance of the diode can be predicted from the small-signal analysis. Here, we consider how the small-signal Q may be used as a measure of oscillator performance of the diode.26 The Q was defined in (35)as 271 times the ratio of average ac energy stored in the diode to average energy dissipation per cycle. This can be written as

where W is ac energy stored in the diode and the angular brackets indicate the time average over one cycle. When the diode has a negative resistance, dW/dt is positive and Q is negative. It is seen that the Q is a measure of how

I0 5

100

I 5

9

z 100

I

I

I

1

I

2

3

4

5

1

1

1

1

I

l

l

6

7 8 9 1 0

-- 3

I

3 (0)

4

5

I

6

I

I

I

7 8 9 1 0 1

TRANSIT ANGLE I N RADIANS

1

Ib)

FIG.22. Small-signal Q of the six structures listed in Table I1 as a function of frequency for bias currents (1)-(5): (1) 130, (2) 260, (3) 520, (4) 1040, (5) 2090 A/cmZ.The dashed lines are for negative Q and the solid lines for positive Q.The arrow indicates the resonance frequency.

(After Misawa.26)

F

effectively the stored energy is used to deliver power to the outside world. The smaller is its magnitude, the more effective is the diode. Another way of interpreting the Q is that it is a measure of how rapidly the oscillation builds up. Actually, the oscillator is composed of the diode and a circuit. The buildup rate of the energy of the total system is

( d W / d t ) = (dWd/dt)

+ (dW,/dt)

-w[((wd>/Qd)i((Wc>/QC)l - W ( W ) / Q , (58) where the suffices d and c refer to diode and circuit and quantities without a suffix are for the total system. The ratio between ( W,) and ( W , ) depends upon the individual case. In order for the oscillation to build up, the magnitude of Qd has to be less than Q,( W,)/( W,). This fact also indicates that a smaller magnitude of Qd means a better quality. As oscillation builds up, Qd degrades and a stationary state is obtained. It is very likely that the final amplitude will be large when lQdl is small and oscillation builds up vigorously. In this sense, Qd can be a measure of the final oscillation amplitude. =

13. LARGE-SIGNAL ANALYSIS Information on oscillator performance on the IMPATT diode can be obtained only from the large-signal analysis, no matter how helpful the small-signal analysis is. The large-signal analysis consists in solving fundamental equations given in Section 8 according to time evolution. This makes it necessary to use a numerical approach. Some effort has been done along this line with the help of a computer. The computation is rather bulky because it involves integration both in space and time. On the other hand, in his original analysis, Read was able to perform the space integration analytically with simplifying assumptions and obtained most of the essential features of large-signal operation of the narrowavalanche IMPATT diode.' In the following, first we discuss Read's analysis and then present some results of numerical analysis which have appeared in the literature. a. Read Diode'

In the Read diode, avalanche multiplication takes place in a narrow region at one end of the space-charge region and produces a number of carriers which are injected into the rest of the space-charge region. When the voltage across the space-charge region changes sinusoidally in time, the field in the avalanche region changes more or less in phase with the voltage. The carriers supplied by the avalanche process are appreciable when the avalanche field is above the breakdown field. The carrier injection reaches a peak not when the field peaks, but when the field falls to the breakdown value. The injected

412

T. MISAWA

carriers drift across the space-charge region while the voltage goes through the lower half of the cycle. This timing results in an out-of-phase current through the diode and produces ac energy. The carrier injection process is governed by

:J

(wJ2)(dJo/dt)= J o (

a dx - 1)

+ J,,

(59)

which is Eq. (42). Here, J o is either the electron or hole current which emerges from the avalanche region. It was assumed that time variations are so slow that the sum of electron and hole currents is position-independent as was the case in the quiescent condition. Further, we assume that the effect of the space charge of carriers on the field configuration is negligible. We will consider an example in which the avalanche region is uniformly doped and the field in the region changes linearly from its peak value as shown in Fig. 23. We take the peak value of the field at breakdown as the unit for normalization. When a changes as Em, where rn is a constant,

JoWn

adx

N

E;",

(60)

where E , is the peak field. The result was obtained by extrapolating the linearly changing field to zero and integrating over the whole region [0, 11 as illustrated in Fig. 23. The current thus produced travels down the space-charge region. When the field in the space-charge region is high, the carriers drift at the scatteringlimited velocity and diffusion is negligible. The carrier distribution, and thus

P POSITION

FIG.23. Field in Read diode

X

413

the particle current distribution J ( x , t ) , retains the original shape : J(x, t ) = J,(r - x / u ) ,

or

J ( x , t)

=

Jo(t - x)

161)

in normalized form. To be specific, we consider the case in which holes travel toward the positive x axis. The total current through the diode I is a sum of the hole current J(x, t ) and the displacement current dE(x, t ) / d t : I(t) = J(x,t )

+ dE(x, t y a t ,

where unit diode area was assumed. Integrating the above expression over the space-charge region, we obtain

I

=

I,

+ I,,

where

I,

= ( l / ~ ) / ~ ~ J ( x , i=) d( Ix/ w )

Jo(t - x / u ) d x ,

Jnw

I,

= (l/w) d V / d t

With the normalization given in column h of Table I,

I,

=

dV/dt.

(63)

The first component, I , , is the current induced by moving holes, and the second component, I,, is a capacitive current. Finally, we have to find out how the field in the avalanche region changes with diode current and voltage. From Poisson’s equation (14), the voltage across the diode is given by

where Q is the sum of the charge of ionized impurities Qf and mobile carrier charge, which is equal to J(x, t ) :

414

T. MISAWA

At zero current, the above relation becomes V,

=

E,w

+

JOw

IOx Qr dx’ d x ,

where V, and E, are the breakdown voltage and the peak field at zero current. Therefore, V - V,

=

w(E, - E,)

+

low lxw

J(x’, t) dx’ d x .

Changing the order of integration for the last term, we obtain

Eo(t) = E ,

+ (I/W)

(W

1

- x)J(x, t ) d x

or, with E, and w as units for normalization, Eo(t) = 1

+ V(t)

s,

,

1

-

Vo -

(1 - x)J(x,t)dx,

or, from (61),

E,(t) = 1

+ V ( t )- Vo -

1-

(1 - t

+ t’)J,(t‘)dt‘.

(64)

1

This equation tells us how the peak field in the avalanche region changes with diode voltage and the charge of holes drifting through the space-charge region. Equations (59H64) give relations connecting the field in the avalanche region E , , hole current emerging from the avalanche region J,, the current through the external circuit I = I, + I , , and the diode voltage V. The external circuit determines the relation between V and I . The problem boils down to solving the four equations for the four unknowns. In deriving Eqs. (62) and (64), the avalanche region was treated as if it did not occupy any space in the space-charge region. However, the width of the avalanche region appears in the Eq. (591, which governs the carrier injection process. Anyway, the dynamics in the avalanche region were not treated correctly under the assumption that it does not matter because the avalanche region is thin. Figure 24 shows the behavior of voltage, peak field, avalanche current, and induced external current for a 10-pm-wide diode with an avalanche region whose width is & of the space-charge-layer width. The ionization rate changes with field as E6, i.e., m = 6 in Eq. (60). The diode is biased in such a way that the average current through the diode is 0.1. An ac voltage with an amplitude of 0.19 and a frequency corresponding to a transit angle of n is present across the diode. The structure and the conditions described above are the same as those used by Read in his original estimate of a possible

415 I, = 0.4

0.2

I,

= 2

0 0 J

w_

I-

z

IL

wa

W

a a

a

+ J

3 V

0

>

- 0.2

0

UNIT TIME

FIG.24. Voltage V, peak field E , , avalanche current J,,, and induced external current J , of Read diode with w, = 0.1 at a bias current of 0.1, a frequency of H.

efficiency as high as 30 % (except for the ac voltage, which as 0.2 in his estimate). The curves in Fig. 24 were obtained by numerically solving the fundamental equations (59), (62),and (64) with (60). With the sinusoidal voltage across the diode, the peak field in the avalanche region also changes almost sinusoidally, although it is appreciably modified by the charge of holes which are created by avalanche and are drifting through the space-charge region. This is seen from (64). As the bias current increases, the effect of the mobile carrier charge will be more and more pronounced. The avalanche current J , keeps increasing as long as E , stays above E,, which is unity here, as seen from Eq. (59). Therefore, J , peaks when E , has decreased to 1. Because of the regenerative nature of the avalanche process, this peak is very sharp at this large ac amplitude. This is seen by rewriting (59) as follows : +wa d(ln J,)/dt =

CI

dx

-

1

+

(JJJO).

(65)

416

T. MISAWA

When the right-hand side changes almost sinusoidally, In J o will change in a similar way. However, J , will peak sharply when In J , peaks moderately, especially when its amplitude is large. Since J o peaks sharply, the induced external current will be more or less constant over unit time after J , peaks, as is seen from (62) by replacing Jo(t’) with the &function. This fact is well demonstrated in Fig. 24. Here, I, remains at the constant value while the bunch of holes traverses the spaceregion. If the voltage is at the negative half-cycle while I , is flowing, a highquality negative resistance is obtained. This is the condition we have here. At the present high bias current of 0.1, the existence of the negative resistance hinges upon the fact that holes are very well bunched and the mobile carrier space charge practically does not exist during the positive half-cycle of voltage except for that of carriers being generated. As the amplitude decreases, the space-charge region is not completely depleted of holes and their space charge has an unfavorable effect on E,. At small enough amplitudes, J , peaks before the voltage does and the negative resistance disappears. This is seen from (46) for the small-signal admittance. The above fact shows that the small-signal theory is sometimes powerless in describing oscillator performance. We now consider the spatial distributions of field and holes. At the moment when the voltage peaks, there are hardly any holes ; therefore, the field profile is the same as that at breakdown except that the whole profile is raised by Vpeak - V,. The profile is shown in Fig. 25 as E , . On the other hand, at the moment when Eo assumes the lowest value, there is a large bunch of holes in the middle of the space-charge region. According to (61), the hole distribution looks the same as J o ( t ) in Fig. 24. In the part of the space charge region behind the hole bunch (left side), the field is lowered by lEOmln - 11. The field rises within the hole bunch by lemmx. The field profile at this moment is shown in Fig. 25 as EL.

FIG.25. Schematic of field and current profiles in Read diode. The dashed line is the field at breakdown.

417

In the illustration in Fig. 25, it is depicted that E L ( x )bottoms at the trailing edge of the hole bunch. This happens when the initial field in the drift region at breakdown is equal to IEo,,n - 11, which is about 0.3, or 100 kV/cm in the present case. When this bottoming takes place, holes do not move at the scattering-limited velocity any more. The hole distribution starts to collapse from the rear end. As this tendency continues, at still higher amplitudes, the output power will not increase with amplitude as fast as formerly, and finally starts to decrease. Namely, this field bottoming is the onset of a saturation mechanism which eventually limits the oscillator power. The analysis in this section does not tell how this saturation mechanism works. Another mechanism which limits the oscillation amplitude is avalanche multiplication in the drift region. This takes place when the field becomes excessively high. The electrons and holes produced by this untimely avalanche multiplication upset the current-voltage phase relation. Fortunately, the field peaks when the space-charge region is depleted of carriers. Higher fields will be tolerated in the oscillating condition than in the quiescent condition. Considering these saturation mechanisms, we can conclude that the case shown in Fig. 24 is approaching the final amplitude in reasonable Read diodes, which confirms the present theoretical model. The admittance at w = 7c is shown in Fig. 26 for several bias currents. This was obtained with the same numerical analysis as that used for obtaining the results shown in Fig. 24. The figure shows how the admittance changes as the voltage amplitude increases. The admittance includes the component that is responsible for the capacitive current given in Eq. (63). As the amplitude of ac voltage increases and exceeds a certain value, the shape of J o ( t ) becomes so sharp that the current waveforms remain practically the same. Namely, I,(r) simply switches between zero and twice the average value. Then, the admittance that is responsible for I,(t) decreases inversely proportionally to the voltage amplitude. Since the space-chargelayer capacitance remains the same, the total admittance approaches its value, 7c in the present normalization. This is seen in Fig. 26. When the voltage changes as V, sin of, the admittance (conductance and susceptance) at the fundamental frequency is given by

(2/wV,)[(J0(t) sin or)sin Q

G

=

B

=0

+ ( J o ( t )cos w t ) ( l - cos o)],

+ (2/wVa)[(Jo(r) sin ot)(cos

Q -

1)

+ ( J o ( t ) cos w t ) sin 01,

(66) (67)

where the angular brackets denote time average over one cycle. When Jo(t) is very sharp, it can be approximated as I , . 6(t - t , ) , where I , is the average diode current and 6 is the delta function. The sharp pulse occurs at t = t , . Read showed that, with w = n, t , is given as a function

418

T. MISAWA

3.0

W

0

2

U In

2.5

$ v)

3 v)

- 1.0

- 0.5

2

0

CO N D U C TANC E FIG. 26. Change of admittancz with amplitude of ac voltage V, for four bias currents, 0.01, 0.02,0.05,and 0.1. The dashed line is for a diode with large saturation current. Values obtained from a sharp pulse approximation are shown with crosses for I , = 0.01 and 0.1.

of I , and V , by

sin ntl

=

(Zd/2K) + +mK

(68)

when J o ( t ) is very sharp. In deriving the above equation, Joldx - 1 was expanded as a Taylor series and terms up to ( E , - 1)* were retained. With this approximation, the admittance at o = TC is given by

G

=

(4Zd/nV,)cos or,,

B

=

n

-

(41d/nVa)sin w t , .

(69) (70)

Admittances calculated with the above equations are designated by crosses in Fig. 26 for V , = 0.1, 0.15, and 0.2. It is seen that they are good approximations at large amplitudes.

7. IMPATT

DIODES

419

At lower currents, the admittance due to I,(t) becomes proportionately small. Thus, IGI decreases and B approaches closer to n. This is seen in Fig. 26 and from Eqs. (69) and (70). In other words, the diode impedance at large amplitude increases with the bias current. This is just opposite lo the situation in the small-signal case. Another significant fact is that the ratio of the real part to the imaginary part of the admittance or the impedance also increases with current. These two facts may have some practical importance because in the actual diode a parasitic series resistance is inevitable. The parasitic resistance will outweigh the small negative resistance and absorb a larger fraction of generated ac power when large capacitive current flows. We discussed, in connection with Fig. 24, that the peak field E,, falls earlier than diode voltage V because of the charge of created holes. The effect is larger a t higher current. This makes the phase delay of J o with respect to voltage smaller and deteriorates the phase relation between voltag: and current I,. Read proposed to limit the total carrier charge in the spacecharge region t o less than half the charge CV, that produces the voltage variation. Since the former is I,t, where z is the transit time, which is taken as unity, the above condition becomes 1,
I >

-0.08 -

\

Id'o.1

\

\

-0.10 -

0

\

I

I

0.I

0.2

\

\

0.25

"0

FIG.27. Change of diode voltage with amplitude of ac voltage V,. The dashed line is for a diode with large saturation current. Values obtained from a sharp pulse approximation are shown with crosses for I , = 0.1.

Assuming that V, = 0.4, which is not unrealistic considering that the drift field is 0.3, the input power at I , = 0.1 and V, = 0.19 is (0.4 - 0.027) x 0.1 = 0.0373. Since the output power was 0.01, the efficiency is 27%. We stated that J , and I , remain more or less the same as the amplitude exceeds a certain value. This is true as long as we plot them in linear scale as in Fig. 24. Actually, the minimum J o decreases exponentially with V, and may become comparable to the saturation current. In the example discussed hitherto, J , was taken as lo-", which is less than 1 pA/cm2, and J , did not fall to this value even at the highest amplitude. (Actually, J , approached to within an order of magnitude of J , . ) Read showed that the minimum J , will be greater than 10- l o if V, is no larger than 2 . 6 ~ In ~ .our example, w, was 0.1. We show results for the case where J, is no longer negligible with a dashed line in Figs. 26 and 27. We chose J , as loF7and w, = 0.05. Since more carriers are available from which J , builds up, J , reaches its peak too early in the cycle. This degrades the phase relation and results in a poor admittance and hence low output power. The average voltage goes down faster with amplitude than before, because the voltage has to remain above V, for less time due to larger initial J,.

So far, the discussion has been restricted to the case in which electrons and holes have equal ionization rate and equal scattering-limited velocity. We mentioned that this limitation can be lifted by using Eq. (47) inslead of (59). It was Lee et a/. who derived this generalized equation (47) and performed a detailed study on the generalized case.24 They reported that the essential features remain unchanged. Although we discussed the characteristics of the Read diode only at w = n, the theory can be used for lower frequencies. Evans was able to show that a closed-form solution is possible when the transit angle is very sma1L2’ He used his approximate solution in analyzing a certain type of Si IMPATT diode which showed oscillation at lower transit angles. h. Numerical Analysis30a

The numerical solution of the IMPATT diode equations is essentially to simulate what takes place in the actual diode. We shall discuss three cases which have appeared in the literature: (1) a Si diode with a relatively narrow avalanche region reported by Scharfetter and Gumme1,28 (2) a Si pvn diode reported by Ward and U d e l ~ o n and , ~ ~( 3 )a Ge diode reported by Johnston et The first two cases deal with moderately-large-amplitude operation of diodes and the third case with very-large-amplitude operation at a relatively low frequency. The analyses used Eq. (6) with experimentally determined parameters for electron and hole ionization rates. Scharfetter and Gummel used a realistic dependence of drift velocities upon field, including the low-field, constant-mobility regime, the high-field, constantvelocity regime, and the transition region between the two. Ward and Udelson assumed a velocity initially increasing linearly with field and at higher fields increasing as E”’. The Scharfetter-Gummel diode is shown in Fig. 28. It is a Si p + n v n f diode with a gradual transition from n to v regions. The figure also shows field profiles and current composition at two different bias currents. From the plot of current composition, it is estimated that the avalanche region occupies about one-fifth of the total space-charge region. The analysis was performed for the case when approximately sinusoidal voltage is present across the diode. Figure 29 summarizes the results at a bias current of 200A/cm2. This current density is relatively small, so that the small-signal admittance has a negative real part around the frequency corresponding to a transit angle of n. As the voltage amplitude increases,

’’ W. J . Evans and G . I. Haddad, IEEE Trrrrrs. E/rc/rfJnDPz3I’w.v ED-16. 78 (1969). ” 29

3o

D. L. Scharfetter and H. K. Gummel. IEEE 7 i t m s . E / P C ~ WDrriczr I? ED-16. 64 (1969). A. L. Ward and B. J. Udelson. I E E E Trons. Elecrron Deuices ED-IS, 847 (1968). R. L. Johnston. D. L. Scharfetter, and D. J . Bartelink. Proc. IEEES6. 161 I (1968). See also Appendices C and D.

T. MISAWA 400 IMPURITY DENSITY

---

- 360

FIELD

- 320

CURRENT

- 280 E

e

n+

I

7 \

/-w

J

- 120

----

w

80

- 40 kl

0

1

2

3

4

5

6

7

8

9

0 10

DISTANCE IN prn

FIG.28. Impurity distribution, field profile, and current composition in a Si ptnvn+ diode analyzed by Scharfetter and Gumrnel. (After Scharfetter and G ~ m m e l . * ~ )

the conductance decreases and the susceptance approaches that of the space-charge-layer capacitance. These features are qualitatively the same as those obtained with a simplified analysis given in the preceding section. The “snapshot” of carrier and field distributions inside the diode is shown in Fig. 30 for a frequency of 12.4 GHz, current density of 200 A/cm2, and efficiency of 12%. Diode voltage and current at each instant when the “snapshot” was taken are shown in the V-I plane in the upper left corner. The plot substantiates what Read described in his original analysis. At time (l), voltage is at a maximum and the carrier density starts to be appreciable in the avalanche region. This corresponds to the case designated by E , in Fig. 25 for the Read diode. One-quarter of a cycle later, at time (2), the charge pulses are fully formed and the first half of the electron bunch has already entered the drift region. This corresponds to time t , in Read’s analysis [see Eq. (68)].The holes disappear quickly into the p + region. At

423 70 60

50 40

30 I20 I10

O0 w

u

30

5 t

80 ?i V

70

2

60

50 40

30 20 10

0 -2 5

-20

-If

- 10

-5

0

C 0 NDU CTA N CE FIG.29. Admittance of the diode in Fig. 28 as a function of frequency and ac voltage amplitude. Equiefficiency lines are also indicated. Current density is 200 A/cm2. (After Scharfetter and Gummel.28)

time (3), the electron bunch is half way into the drift region and the voltage is at its minimum. This corresponds to the case designated by EL in Fig. 25. Another quarter cycle later, the electrons are disappearing and the field in the avalanche region starts to grow above the quiescent value. The amplitude of oscillation is relatively small here. The electron pulse has not yet sharpened as in Fig. 24 and no sign of bottoming of the field at the trailing edge is seen at time (3). Actually, the field is bottoming at the leading edge. The highest theoretical efficiency reported by Scharfetter and Gummel is 18 % at 9.6 GHz with a voltage amplitude of 38 V. The efficiency is still sharply increasing with amplitude at this point.

424

T. MISAWA

x

x

10'5

I. 2

1.0

105

5

c FIELD ELECTRONS HOLES

-

(3)

I I

ni

4

N

5

0.8 E

[1s

w a

Y

3 >

z

>

t

Z

0

0.6

-I

wLL

u a 2

[1s

w

g

c

0 W

0.4

a

_1

w

V

I

0.2

u 0

1

2

3

0

4

5

6

7

8

9

10

DISTANCE IN p m

FIG.30. "Snapshots" of field profile and carrier density distribution at four instants 4 cycle apart. Current density is 200A/cmz, frequency is 12.4GHz. and efficiency is 12%. (After Scharfetter and Gurnme1.28)

According to Read's analysis as presented in the preceding section, the output power keeps increasing with the voltage amplitude, until the field in the drift region bottoms and/or avalanche multiplication starts to take place in the drift region, thus invalidating his approach. Analysis of this super-Read regime is necessary in order to know the true theoretical limit of the oscillation efficiency of the Read-type diode. So far, no report is available on this subject. Ward and Udelson reported in detail on low-frequency oscillation of a Si n+ppf diode with a capacitive load. The particular circuit used is illustrated in Fig. 31. The impurity distribution is shown in Fig. 32. The space-charge cm'. Since region is 2.5 pm wide and the cross-sectional area is 4 x the carrier velocity kept increasing with field, the result may not be quanti-

7. IMPATT

425

DIODES

R E V E R S E VOLTAGE ( V )

100

20

-a

I

L

40

60

80

1

I

I

I

I

1

1100

FIG. 31. Oscillator circuit and obtained oscillation presented as a phase plot. Circles are time markers with an interval of S psec. (After Ward and U d e l ~ o n . ~ ~ )

tatively correct, but it is believed that the qualitative nature of the operation is well represented by their computation. In Fig. 31, instantaneous values of diode current and voltage are plotted. Circles on the I/-I curve are time markers with an interval of 5psec. The whole cycle takes 235psec. The time sequence is in the direction of the arrow. When the diode voltage moves up on the top, flat portion of the cycle, the space-charge region is well depleted of carriers. As the voltage approaches the peak, carriers start to be generated. Figures 32 and 33 show carrier distributions and field profile at several instants when the current goes through the peak. Numbers attached to the curves are times in picoseconds with respect to the moment when the current peaks. At 11 psec before the current peak, the current has just started to flow. The carriers are generated mostly at the right end of the space-charge region, where, although the field is lower than at the left end, more electrons, which have a larger ionization rate, are available. At the moment when the current peaks, both electrons and holes are present throughout the space-charge layer. Because

426

T. MISAWA

0

0.5

1.0

I .5

2.o

2.5

FIG.32. “Snapshots” of electron and hole distributions at five time points: - 1 1 , -4, 0, 5, and 24 psec. The origin of the time scale is the moment when the current peaks. The thin line is for the impurity profile. (After Ward and Udel~on.’~)

of excess holes at the left edge and excess electrons at the right edge, the field profile has a saddle in the middle and voltage is lowered below the value at breakdown. The situation is similar to that in dc condition shown in Fig. 7a. After the current has peaked, the voltage drops down ; generated carriers disappear in the time interval comparable to the carrier transit time. As is seen from Fig. 31, the current flows during a very small fraction of one cycle. Although efficiency for this particular oscillation was not quoted, the authors reported that efficiency as high as 14% was obtained in a similar oscillation. Since the dc V-Z curve was not reported, it is not possible to assess how the negative-resistance characteristics are enhanced in the dynamic condition.

7. IMPATT

427

DIODES

500

c

0 E

400

\

>

f 0 -I

W

300

0

[t

l-

0

W

-I

w 200

100

0

0.5

1.0 1.5 DISTANCE ( p m )

2 .o

2.5

FIG.33. “Snapshots” of the field profle at the same five time points as in Fig. 32. (After Ward and Udel~on.’~)

The small-signal negative resistance of the pin-type diode extends well into lower frequencies even without dc negative resistance. Since the diode reactance is inductive there, an oscillation with a capacitive load similar to that considered here should be possible without requiring dc negative resistance. Johnston et al. analyzed the performance of a Ge diode at such a low frequency that a small-signal negative resistance does not exist.30 The diode is of the abrupt-junction type like the one shown in Figs. 5 and 6 except for a slight penetration of the space-charge region into the highly doped region of the substrate. The space-charge region is about 5 pm wide and the breakdown voltage is 60 V. The analysis was performed in conjunction with observed high-efficiency (up to 43 04)oscillations at very small transit angles. Figure 34 shows “snapshots” of carrier density and field distributions together with instantaneous values of current and voltage at the moment

428

T. MISAWA

‘1I p =

FIELD

OO

v

ELECTRONS ---- HOLES

0

L32

FIG.34. “Snapshots” of field and carrier density distributions in a Ge diode for a fundamental frequency of 3 GHz. Scale limits: 0-1.75 x 10’6/cm3 for carrier densities, (t2.5 x los V/cm for the field, 0-8 ,urn for the distance, 0-100 V for the voltage, and CL-7000A/cmz for the current. (After Johnston et a1.”)

when the “snapshot” was taken. Note that current and voltage waveforms contain higher harmonics. These higher harmonics invoke effects which were responsible for higher-frequency oscillation and eventually make the low-frequency oscillation possible. The oscillation frequency is 3 GHz. Because of very large carrier densities at phase (c),the field in the avalanche region is drastically modified in contrast to the first case discussed in Fig. 30. An even more drastic effect is seen in phase (d), where the field bottoms over a wide region in the center and electrons and holes are “trapped.” This makes it possible for the diode to carry a large current at small voltage. It has been found that the efficiency of oscillation is very critically dependent upon the waveform of the oscillations. This may have been expected because

7. IMPATT

DIODES

429

higher harmonics were responsible for the very existence of the oscillation. A theoretical efficiency up to 25 04 has been reported in this type of oscillation,30b,30c V. Design Considerations

In this part, we discuss general design considerations which will be useful in actual fabrication of the device. More specific problems will be discussed in the next part. 14. SCALING RULEFOR VARIOUS

We have seen that the fundamental equations which govern the dynamics of the IMPATT diode can be normalized in terms of the material and structure parameters of the device. This fact indicates that one solution is applicable to a variety of devices with different material and structure parameters. For different devices, one just chooses different units for normalization. The scaling rule to be discussed here enables us to obtain characteristics of a new diode, which can be a scaled-down version or made of a different material, from the characteristics of the original diode. We consider the case in which characteristics are calculated by a simplified method discussed in Section 12e. Suppose we have an admittance plot as a function of frequency and bias current. In the plot, all quantities are dimensional, i.e., not normalized. Since bias current J appears in equations in combination with M’ as U’J, the normalized M'J is invariant. Keeping this fact in mind, we obtain the multiplication factors given in Table 111 for TABLE 111 Quantity Frequency

Multiplying factor WI _ v2 _

W2

c,

Impedance Current' a When CL oc Eb,where E is the field in the avalanche region, al’/a2’ = E,w,/E,w,.

'ObThis efficiency was obtained in a computer simulation of an experimental case in which 33% was observed. Private communication from Scharfetter. 30cW.J. Evans and D. L. Scharfetter, I E E E Trans. Electron Deuices ED-17, 397 (1970).

430

T. MISAWA

converting the plot for the original diode of width wl,scattering-limited velocity u l , dielectric constant and field derivative of ionization rate a l ' , to that for the diode with w 2 ,u2, E ~ and , tlz'. Values on coordinate axes or given as parameters that are not dimensionless but dimensional are to be multiplied by the appropriate factors for the conversion. For example, we scale down a diode with an avalanche region occupying one-third of the total space-charge region by a factor of two. The proportion of the avalanche region is kept constant and the material is the same. Then, all the values on the frequency axis have to be multiplied by two; for example, change 5 GHz to 10 GHz. All the values on the admittance axis are multiplied by a factor of four. Since the field in the avalanche region increases very little with the halving of the space-charge region, the values designating bias currents are multiplied by a factor of slightly more than two. Although the above multiplying rule applies exactly only to the simplified small-signal analysis given in Section 12e, the rule will be approximately correct for more accurate results obtained by the numerical method. Furthermore, the idea of converting normalization units is useful even for largesignal analysis. 15. STRUCTURE PARAMETERS

a. Width of Space-Charge Region The scaling rule discussed in the preceding section gives us information on the effects of changing the width of the space-charge region. Since the operation of the IMPATT diode is based upon transit-time effects, the operating frequency is inversely proportional to the transit time. Let us consider the case in which the width is halved in order to double the operating frequency. The case was discussed in the preceding section. By applying the scaling rule to the Q versus frequency plot, such as the one in Fig. 12, it is concluded that the same quality of negative resistance is obtained at a frequency twice as high, as intended. However, a bias current slightly more than twice as large is required to obtain this, and the impedance level will be only one-quarter of the original one. By further reducing the width of the space-charge region, a still higher operating frequency is obtained. However, this process cannot be continued indefinitely. In addition to the practical difficulty of handling too low an impedance, the tunneling current, which becomes a dominant breakdown process in narrow junctions, degrades the quality of the negative resistance.' In the tunneling process, the current changes simultaneously with the field. The important time delay cannot be obtained with tunneling as is possible with avalanche. So far, the highest frequency reported was obtained with a

431

E

C

FIG. 35. Changing avalanche-region width with doping level in the n region in a p + n n + structure.

432

T. MISAWA

Si junction diode with a 360-A wide space-charge layer.31 The frequency was 341 GHz. In this diode, the peak field was estimated as high as 2500 kV/cm. This field is considered to be high enough that most of the current is carried by the tunneling process.32 b. Width of Avalanche Region

The small-signal analysis given in Section 12r showed that, at a low bias current density, the diode with a narrower avalanche region has a negative resistance of better quality. This indicates that a diode with a narrower avalanche region is preferable as a CW oscillator. Figure 35 shows how the avalanche region becomes narrower in the p+nn+ structure as the doping level is increased in the n region. The narrowest avalanche region is obtained with the highest doping level, designated as (3). In order to reduce further the avalanche-region width, a more elaborate, hyperabrupt structure as in the Read diode is required. As the avalanche region becomes too narrow, tunneling again begins. This is more of a problem with higher-frequency diodes, which have narrower space-charge layers. Another objection to the too-narrow avalanche region comes from the adverse effect of saturation current at large amplitude, which was discussed in Section 13a. Thermally generated carriers swamp the space-charge layer at the ebbing phase of the cycle and deteriorate what would otherwise be a high efficiency. Some authors consider a wider avalanche region necessary for high-power capability and high efficiency.28 16. MATERIAL PARAMETERS

Effects of material constants will be considered here mostly based upon the scaling rule given in Section 14. a. Ionization Rate

The most straightforward effect of the ionization rate appears in the breakdown voltage. Materials with larger ionization rates, like Ge, result in lower breakdown voltage. This is an advantage when the amplitude of voltage swing is appreciably smaller than the breakdown voltage, because it reduces the necessary input power for operation. The field derivative of the ionization rate a’ is also important, for the small-signal characteristics depend upon a’J, where J is bias current. With larger CI‘, the same Q is obtained at smaller bias current. This is favorable for CW operation of the diode. Figure 36 shows a’ as a function of c1 for four common semiconductors, Ge, Si, GaAs, and Gap. The rate x’ is about two ”

L. S. Bowman and C. A. Burrus, IEEE Trons. Elrcrron DiwieeJ ED-14. 41 1 (1967). J. L. Moll, “Physics of Semiconductors,” p. 239. McGraw-Hill, New York. 1964.

433 2

I-

-

5 0

-

>

U W

n W

v \

0.1 -

d

U

v'

iS’

0.011

I

(ELECTRON)

I

I

I

I l l

I

I

l

l

IONIZATION RATE PER cm

FIG. 36. Plots of a' (After Misawa.z6)

=

da/dE as a function of ionization rate

a

for Ge, Si, GaAs, and Gap.

times larger in Ge than in Si. This indicates that a Ge diode will work better at a low bias current than a Si diode. b. Currier Velocity As the scattering-limited velocity of the charge carrier increases, say by a factor of two, the operating frequency goes up proportionately, but the impedance level is halved and twice as much bias current is required. However, in most of the common materials, like Ge, Si, or GaAs, almost the same scattering-limited velocity, on the order of lo7 cm/sec. has been observed. It seems that there is not much choice as far as this variable is concerned. c. Dielectric' Constant

With a larger dielectric constant, the impedance level will be inversely proportionally lower for the same operating frequency and a proportionally larger bias current will be required. The larger dielectric constant of G e is a disadvantage.

434

T. MISAWA

d. Material Choice-Ge, Si,and GaAs From the preceding arguments, Ge seems to be the best material among the three listed above. However, its thermal property compares unfavorably with that of Si. Thermal conductivity is about 30 % that of Si and its smaller energy gap tends to lead to a thermal runaway condition at a lower temperature. Gallium arsenide has even a smaller thermal conductivity than Ge, but otherwise its properties are comparable to those of Si. Because of its band structure, the tunneling process takes place more easily in GaAs than in Ge or Si3*This makes GaAs less suitable for high-frequency diodes, which have necessarily high field. Silicon has reasonable material properties and, in addition, as far as material preparation and processing are concerned, it is the best-developed material. It has been observed that Ge and GaAs diodes have superior noise characteristics.

17. THERMAL CONSIDERATION Part of the input power to the diode is converted into microwave energy, but the rest of it is wasted and simply heats the diode. When the diode temperature reaches a certain point, a thermal runaway condition sets in, namely a certain spot in the diode area becomes hotter and starts to draw more current than the rest of it, thus inducing further temperature rise at the spot. Finally, the melting point is reached and the diode is destroyed. In order to obtain proper operation of the diode, a reasonable current density must be achieved before this burnout takes place. Heat is primarily generated in the space-charge region where the field is high. It travels through the semiconductor body and the metal support (stud) to be dispersed into the environment. In the following, we discuss first the thermal resistance (i.e., the temperature rise per unit power dissipation) of the diode, assuming uniform heat generation over the junction area. With this uniform heat generation, it is found that temperature is not uniform over the junction. Since the current through the space-charge layer is dependent not only on voltage but also on temperature, as discussed in Section 10, current cannot flow uniformly, and thus heat generation is nonuniform. In the latter half of this section, a more elaborate analysis will be presented in which the current distribution is not assumed to be uniform, but is determined in a self-consistent way. Since the microwave characteristics are dependent upon current density, it is important to know the current distribution across the junction.

435 a UNIFORM HEAT FLUX f

b r THERMAL CONDUCTIVITY

FIG.37. Uniform heat flux incident over a circular area on a large heat sink.

a. Junction Temperature When the diode is made in such a way that the junction is very close to the heat sink, most of the temperature rise is inside the heat sink, not in the semiconductor. We shall obtain the temperature distribution in a heat sink whose dimensions are considered very large compared to the diode diameter.32aWe consider the case in which a uniform heat flux f is incident over a circle with radius a on the surface of an infinite solid with thermal conductivity K, as shown in Fig. 37. The temperature rise AT above the ambient is given by AT(r, z ) = ( ~ ' u / K )

e-"J,(lZ~)J,(~a)(d~/lZ),

(72)

where J , and J , are Bessel functions.34 The coordinate system is explained in Fig. 37. It is assumed that adiabatic conditions prevail over r > u. The temperature distribution on the surface of the heat sink is obtained by putting z = 0 in Eq. (72) as follows :

""The case in which the size of the heat sink is small was discussed by Kennedy.33 3 3 D. P. Kennedy, J. Appl. Phys. 31, 1490 (1960).

436

T. MISAWA

0.6 0

0.2

0.4 0.6 DISTANCE FROM CENTER

0.0

1.0

RADIUS

FIG.38. Temperature distribution in the case of uniform heat flux given by Eq. (73). (After Gibbons and Mi~awa.~')

where E and K are the complete elliptic integrals of the first and second kind.34a Temperature is highest at the center of the diode and gradually falls off to 63% (2/n)of the center temperature at the edge as illustrated in Fig. 38. The highest temperature in the circular area, r < a, is given by

AT(0,O)

Omaxna2f,

Om,, = l / n a ~ ,

(74)

where 0 is the corresponding thermal resistance. The thermal resistance for the average temperature is34 @,,

= 8/3n2aY ~ (3.7ffK)-'.

(75)

It is to be noted that when a uniform temperature is assumed over the area r < a the thermal resistance is equal

0

=

1/4a~.

(76)

H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids". p. 216. Oxford Univ. Press (Clarendon),London and New York, 1959. 34"Theseparticular expressions came to the author's attention through Hein's 35 V. L. Hein, unpublished work.

34

7

Si

0.8

Ti

0.16

AU

3.0

Ni

0.71

W/cm

-x 0.02 prn

RS

Rt

m

Rg

0.2 p m

Rn

FIG. 39. Cylindrical section between junction and heat sink, which is composed of Si and several layers of metals. The thickness and thermal conductivity of each layer are indicated. (After Swan er a[.36)

The important feature of this “spreading” thermal resistance is the fact that it is inversely proportional to the diode radius. This makes it possible to have larger flux density, therefore large current density for a given temperature rise, by reducing the junction area. As discussed before, greater and greater current density is required for higher-frequency diodes, and this may be achieved by reducing the junction area. The above tendency does not continue indefinitely, for, with smaller area, the thermal resistance of the cylindrical section between the junction and the heat sink, which is inversely proportional to area, instead of radius, becomes important. Figure 39 shows an example of a structure with a copper heat sink and cylindrical section, which is composed of Si and several layers of metals,36 Calculated thermal resistances of various sections and their total R, are plotted in Fig. 40 as a function of diode area. The crossover between spreading resistance R , and cylindrical resistance occurs at 5 x cm2 in this particular case. Of course, the crossover point can be lowered by thinning the cylindrical section. Swan has proposed using Type 11 diamond, which has five times the room-temperature conductivity of copper, as heat sink.37 Figure 40 also contains curves for the diamond heat sink. Another way of improving thermal resistance is to use a junction geometry with a small linear dimension, such as a stripe or annular shape. .” C. B. Swan. T. Misawa. and L. Marinaccio. f E E E Trcrrzs. Elwtrort Dw;w.s ED-14, S84 (1967).

’’ C. B. Swan, Proc. IEEE 55, 1617 (1967).

T. MISAWA

0. * DIODE AREA (ern')

FIG.40. Thermal resistances of the structure shown in Fig. 39. (After Swan et

Figure 41 compares the annular geometry with the solid circular one with equal area.38 By using a ring whose width is of the diameter, an improvement of a factor of two is obtained. The curve was obtained by superimposing two solutions of the form of Eq. (73). Some improvements in oscillator performance were observed in Si diodes with ring geometries.38a b. Current Distribution3'

We have seen above that when uniform flux or current density goes through the diode the center part of the diode is hottest. On the other hand, we know that as temperature goes up the diode voltage increases for a given current. Since the voltage across the junction is constant over the diode area under most conditions, the current tends to concentrate in the cooler portion of the diode. This will make the temperature distribution more uniform. G . Gibbons and T. Misawa. Solid Store Electron. 11. 1007 (1968).

38aL. P. Marinaccio, Proc. ZEEE 56. 1588 (1968).

INNER RADIUS OF RING WIDTH OF RING

FIG.41. Improvement in thermal resistance of ring structure relative to solid diode versus the radius-to-width ratio of the ring. Comparison is on an equal-area basis. (After Gibbons and M i ~ a w a . ~ ’ )

Let us consider the simplest case in which voltage goes up linearly with temperature and also with current. The characteristics are illustrated in Fig. 42. The relation is expressed by J = G[V-

(vo + p791,

(77)

where G is the incremental conductance after breakdown, which is assumed to be independent of V and T ; V, is the breakdown voltage at zero temperature, which is most conveniently chosen as room temperature, and p is its temperature coefficient. The particular values of the parameters used for Fig. 42 are appropriate for a Si abrupt-junction diode for X band (8.2-12.4 GHz) reported by Misawa.” The boundary conditions are now

G V ( V - Vo),

/?’=GI/P

39

for

r 0: (e) V, = V,, > 0 ; (f) Ib-Vbcharacteristic, I i : ' = d I , J d E , k = 1,2,....

8.

491

TUNNEL DIODES

2. At V, = O,fp(E)= ,f,(E), so that I , = 1, = 0 (point 111). 3. As Vb > 0 increases, f,(E) - j p ( E ) > 0 and I!;), If:), and I$) increase, whereas 1::' and (Evp- E,,,) decrease toward zero. Hence, I t l , I,, and I,, exhibit a maximum at some V, = V, and begin to decrease with larger V, > V,, as seen at points IV and V. 4. As V, increases such that E,, < E,,, 1::' = 0, but I!:), I!:), I::’, and ,f,(E) - &(E) increase sufficiently so that I , remains small but nonzero over a wide excursion of V, before eventually dropping to zero, as seen at points VI and VII. However, I, and I , , both exponential in Vb, begin to contribute at these points, so that I, never drops to zero but exhibits a valley at Ib,,,in = I, before increasing rapidly as I , goes into the forward conduction region. The current Ib,,,in is often referred to as the excess current and, as stated previously, is attributed qualitatively' 1.'9-22 to conduction and valence band-edge tailing, tunneling between impurity states, electron-hole recombination in the forbidden band, and electron interaction with photons, phonons, etc., during tunneling. However, the phenomenological Gaussian impurity-band model employed here provides a quantitative explanation for the existence of the band-edge tailing component of Ib,,,in in terms of the tunneling current components / , 2 , I,,, and It4. The determination of I , requires an evaluation of the tunneling probability P(E), which will now be derived from a consideration of the quantummechanical tunneling process.

3. THEORYOF QUANTUM-MECHANICAL TUNNELING a. Wave-Particle Duality und the Schrodinger Equation The phenomenon of quantum-mechanical tunneling'-'* arises from the wave-like nature of charged particles as characterized by Schrodinger's equation, and is exhibited by the nonzero probability that a particle can penetrate a potential barrier exceeding its own energy. Although detailed treatments of the three-dimensional quantum-mechanical tunneling process in actuality, the tunnel in a degenerate pn junction have been diode junction has a one-dimensional carrier, charge, and field distribution with a uniform cross section. Therefore, a one-dimensional model adequately represents the physical situation and will be employed here. The basic postulates of quantum p h y s i ~ s ' ~ ascribe .'~ a wave nature to every particle, and therefore to a prospective tunneling electron. The resulting wave-particle is therefore characterized by the following parameters : (a) momentum p, (b) kinetic energy E, = p2/2m*, (c) effective electron mass m,,*= (3’E,/2p2)-

',

492

H. C . OKEAN

(d) wavelength A = h/p (e) wavenumber k,

=

2nh/p,

k = 24A = p / h = (2m,,*EK)'"/h,

(f) phase velocity c, (g) total energy in potential energy field E ,

+

E = E, E K , (h) momentum (vector) p = pup, (i) wavenumber (vector) k = ku,, ( j ) the unit vector characterizing the momentum direction up. The intensity of the wave-particle t,b as a function of position, assuming that the sinusoidal time dependence exp[J2n(c/,l)t] is separated out, is characterized by the well-known three-dimensional Schrodinger equation,

+ (8n2m,*/h2)(E- E,)t,b(r) = 0 ,

V'$(r)

(26)

where r is a three-dimensional position vector. The Schrodinger equation, alternatively expressed as

+ k2$(r) = 0 ,

V2$(r)

with

k = k(r),

(27)

possesses solutions, under a constant potential field EPo,of the form

$(r)

=

- + B exp(-,jk

A exp(jk r)

*

r) ,

(28)

where, from Eq. (25),

k

=

const

=

[2m,*(E - Ep,,)]1'2/h.

Therefore, the wave-particle is seen to exhibit the behavior of a propagating wave (k real) in regions for which E > EPo and of a rapidly attenuated "below-cutoff" wave (k is imaginary and B = 0 for physical realizability) in regions for which E < EPo.The latter case is clearly the one that governs a particle penetrating a constant-potential barrier. The physically significant quantity derivable from the wave-particle intensity $(r) is the probability function @ that the wave-particle is within a given volume U , as given by @ =

1"

$(r)$*(r) dU d 1,

(29)

where the asterisk indicates complex conjugate and @ must be unity for U representing all space. Clearly, Eqs. (28) and (29) suggest that a particle has a nonzero probability of penetrating a constant-potential barrier, thus forming the basis for the more general phenomenon of quantum-mechanical tunneling.

8.

493

TUNNEL DIODES

In the more general case of a position-dependent potential-energy field Ep(r),Eq. (27) becomes a three-dimensional nonlinear differential equation which forms the basis for the more detailed treatments of quantum-mechanical However, at this point, we postulate a uniform crosssectional geometry, resulting in the following one-dimensional model :

+

( d 2 J / / d x 2 ) [k(x)j2J/= (d2J//dxz)+ (8nZm,*/hZ)[E - E & X ) ] $ ( X )= 0 , (30)

where x is measured in the direction of positive (n-to-p) current flow. b. Electron Tunneling through u Potential Burrier

We now consider the problem of an electron at energy E tunneling through an arbitrary potential barrier E,(x) > E (0 < x < I,) as shown in Fig. 7(a).

DI S T A N C E

(a)

xxDISTANCE

I

I

0 x0 I STA NC E

X+

I

L

FIG.7. Electron tunneling through potential-energy barriers. (a) General barrier; (b) general barrier with transition regions; (c) barrier in a p-n junction.

494

H. C. OKEAN

The application of Eq. (30) to this problem results in an extremely difficult nonlinear differential equation. However, under the condition of slowly , ~ the ~,~~ varying E,(x), we may employ the WKB a p p r o x i m a t i ~ n ' ~to solution of Eq. (30), as given by

where rn denotes the region I, 11, or 111 bounded by x < 0,O 6 x Q L, and x > L, respectively, and A,(x) and B,(x) are obtained by invoking continuity at the boundaries. The validity of the WKB approximation for the purposes of this treatment has been verified by the more exact threedimensional solutions due to Kane,* Keldysh,' and Krieger," and by the one-dimensional, phenomenological approach due to Scanlan.' The specific mathematical requirements for the validity of the WKB approximation are24

over 0 < x < L. Equation (32) implies that the slope laE,/axl must be small compared to IE - E,(x)l, particularly at the barrier interfaces between regions I and I1 and between I1 and 111. If this is not satisfied, the WKB solutions are not valid at the interfaces, and separate representations of E,(x) are required within auxiliary interface regions IV and V, as shown in Fig. 7(b). For an exact solution of Eq. (30) in regions IV and V, a piecewise linear representation of E,(x) of the form E,(x) z E T c(x - xi) (Fig. 7b) may be utilized, yielding an exact Bessel function solution to Eq. (30). These results may then be used to match $,(x) = (c/,+’(x) at the various boundary interfaces, thereby yielding the required A,(x) and B,(x). The probability of an electron tunneling through the potential barrier E,(x) is given by the ratio

4

=

I$l,l(X

=

L)/$,(x = 0)l2.

(33)

However, the requirement for physical continuity requires that, at the barrier boundaries, = at x = 0 and $Ill = $I, at x = L. Therefore, the tunneling probability becomes

4=l 24

~ l l ~ ~ ~ / ~ , l ~ ~ ~ l z ~

D. Bohm, "Quantum Theory." Prentice-Hall, Englewood Cliffs, New Jersey, 1951.

(34)

8.

495

TUNNEL DIODES

c . Degenerate p n Junction Tunneling Probability The potential barrier E,(x) relevant to the degenerate p-n junction band structure (Figs. 3 and 4) is the one across the forbidden band gap between the overlapping p valence band and n conducti-on band, and is hence of triangular shape, as shown in Fig. 7(c). Specifically, E,(x) is given by E,(x)

=

E,

E,(x)

=

E

O>x>L,

+ eFx,

0<x Q L,

(35)

where F is the depletion layer field E,/eL = (Vonp- Vb)/w, (Fig. 4). The WKB solution corresponding to the above potential barrier is obtained by substituting Eq. (35) in Eq. (31) in region rn = 11, and invoking the continuity condition at x = 0, L, thereby resulting in

Then, upon substitution of the above in Eq. (34), we obtain the WKB tunneling probability22

4

=

exp{ - ~ 8 [ ( 2 r n n * ) ” 2 / h e ~ ] E,~ ’ 2 )

(37)

where 0 z 1 and where the depletion-layer field F is obtainable from Eq. (16) as =

(I/onp

-

Vb)/wd

= (I/Onp -

112 /u’d,

(38a)

where the width constant Wd is defined as -

wd =

+

[&(nap n d n ) / 2 ~ e n , p n d n ] 1 1 2 .

(38W

The tunneling transmission probability factor (C/sec) of an electron current penetrating the forbidden band is given by

P(E) = P

=

( 2 x o e 2 / h i d )Vonp ( - Vb)’/’ exp[ - c~M;~E,3’~/2( Vonp-

(39)

where a = $ ( 2 ~ 1 , * ) ” ~ /and h e xo is a lattice constant to be defined. Note that, for the triangular potential barrier model, 4 and hence P are independent of E. A virtually identical result is obtained under the assumption of a parabolic potential barrier,20 with the exception that the constant c( is reduced by a factor 0.59. In a similar manner, the probability factor P, (C/sec) for tunneling via “deep” impurity states has been derived2’TZ2to be P, = (2x0’e2/hGd) ( Vonp- Vb)‘’’ exp[ - @ , c ~ , G ~ eVonp ~ ~ ~-( Vb)],

where 0, z 1, a, z $(2rn,*)”2/he, and xg) is a lattice constant.

(40)

496

H. C . OKEAN

d. Energy-Momentum Conservation during Tunneling To examine further the validity of the WKB approximation for the determination of P ( E ) ,we must examine the energy-momentum dependence of the pn junction band edges relevant to the tunneling process. We start by considering the solutions of the wave equation (30) for a conductionor valence-band electron in a semiconductor lattice structure. Following Scanlan,12 the presence of a nonzero periodic potential E,(u) due to the lattice results in one-dimensional solutions of Eq. (30) in the space variable u having the form $(u) = U,(u) exp(jk,u),where the V,(u) are Bloch functions characterized by periodicity in the lattice spacing xo, that is, IU,(u)( = Iuk(u k nxo)l with n = 0 , 1 , 2 , .. . . The resulting energy-momentum dependence E(p) of the conductionand valence-band electrons is no longer the simple parabolic relationship implied by Eq. (25) in the absence of a lattice potential, but rather has the more general form W

E(p) =

1 al(p

- PO)',

- nh/xo

< P d nh/xo,

i= 0

where E(po) is an energy extremum (valence-band maximum or conductionband minimum) and E( - p) = E(p). The periodicity of the Bloch functions then imply a periodicity in p of t-2nxh/xo, yielding the well-known Brillouin zones and forbidden gaps as shown in Fig. 8(a). However, if we now replace p by the reduced momentum p, = p +_ (2nnh/x,), the Brillouin zone structure of Fig. 8(a) telescopes into the momentum-space representation of the semiconductor energy-band structure, shown in Fig. 8(b). In a tunneling p n junction, most of the tunneling electrons travel between n-region conduction-band minimum-energy states and p-region valenceband maximum-energy states, as shown symbolically in energy-momentum space in Fig. 9. The tunneling electron must conserve energy and momentum, thereby giving rise to two types of tunneling; direct tunneling, for which the conduction- and valence-band energy extrema occur at the same value of momentum (Fig. 9a), or indirect tunneling, in which the extrema occur at different momenta pOc and pov (Fig. 9b). In the latter case, momentum must be conserved by some external means such as phonon scattering during the tunneling process (Fig. 9b). In calculations of the tunneling probability for a given semiconductor, direct tunneling usually predominates and the perturbation introduced by the much less probable indirect tunneling process is relatively sma11.7~8*'2~20 In the light of the above remarks, we now examine the validity of the WKB approximation for the calculation of the tunneling probability [Eq. (37)] with respect to a more exact treatment. In particular, the calculation

8.

TUNNEL DIODES

MOMENTUM

p OR p,

la)

E,

-=

0

(P)

?TT

XO

REDUCED MOMENTUM

p,

(b) FIG.8. Energy-band structure in momentum space for an electron in a periodic lattice. Brillouin zone structure; (b) reduced momentum energy-band structure. (After Scanlan.I2)

of P for three-dimensional direct tunneling by Kane' modifies Eq. (37) by the factor

K,

=

(n2/9) exp[ -h(kY2

+kz2)@JWF]

(41)

and modifies the exponential multiplier 6 to be 3x/16 rather than unity. Here, hk, and hk, are the generally small perpendicular components of momentum, so that K , does not greatly alter the WKB solution. A calculation of the indirect tunneling probability modifies Eq. (37) by the factor

498

H . C . OKEAN

I

1

- ENERGY

DIRECT TUNNELING : A p = 0

MOMENTUM - p

(0) E - ENERGY Ec(P)

,

, M’INDIRECT

,

Po,

TUNNELING : A P = pot- pOv LOST DURING PROCESS

Po,

MOMENTUM p

(b) FIG.9. Energy-momentum space for tunneling p-n junction. (a) Direct tunneling; (b) indirect tunneling (Si).

where f ( T ) = l/[exp(hv/kT) - 11, v is the phonon frequency, and E,, is a phonon scattering matrix element having the dimensions of energy. This calculation also replaces Eg312 by ( E g +_ hv)3i2in the exponent multiplier of Eq. (37). For applicable semiconductor materials, K , 10-3K,, so that the probability of indirect tunneling is quite small compared to that of direct tunneling. Therefore, it may be concluded that the WKB approximation for the calculation of the tunneling probability is sufficiently accurate for the purposes of this treatment.

-

4. DERIVATION OF TUNNELDIODE I-V CHARACTERISTIC Having calculated the tunneling probability P(E)and shown it to be essentially independent of E [Eq. (39)], we may now integrate Eq. (24) to obtain the tunneling current as a function of applied bias voltage. Therefore, the

8.

242

In a Less accurate approximation,'' 1, sinh(q,/2)[1

499

TUNNEL DIODES

-

f. at ij

=

-qb/2 is evaluated as

+ C O S ~ ( V ~ / ~C) tanh(@) ]

for qb >> 1

500

H . C . OKEAN

where

I,

=

0.39~oAPa,upkT;

pz

=

l.8{[apnap(l

q,

=

(eV,,,

-

p l = 0.82n,pn,,ap~,/apa,(kT)2 ;

+ 1.05q,,)/~~’zka,T]+ [anndn(1 + 1.05~&,)/G,!~ZkapT]} ; E,/kT); K ii ’. n 2u -p

1 -

ai = 21/2m:3/2n(k~)1/2h-3

(i

=

/ a-T

9

n or p ) ,

and where P is given by Eq. (39), and u(x) is the unit step function u ( x ) = 1 and 0 for x 2 0 and x < 0, respectively. Equation (44) represents the theoretical tunnel diode current-voltage (Ib, Vb) characteristic, as shown with its conduction-to-valence band and conduction-to-impurity-band tunneling and diffusion components of current in Fig. 10 for a representativez5 tunnel diode. The superposition of an experimentalz5 I-V characteristic for this diode on Fig. 10 indicates a relatively close quantitative agreement with theory, particularly with respect to excess current (largely due to the inclusion of the Gaussian donor and acceptor band and “deep” impurity-site tunneling current contributions). The particular parameters of interest which characterize the currentvoltage characteristic are the peak current point (Ip,V,), the valley current point ( I v , K),and the “inflection point” of maximum negative conductance magnitude (IM, VM)as shown in Fig. 10. These parameters are obtained from the total current [Eq. (44)] as follows : Vp = V, at which

dl,,/dab x (d/dqb)[(qT - qb)’ tanh(qb/4)] = 0 ; (45a)

K = Vb

(d/dub)(Itl + I,

at which

VM = Vb(min) at which I,

=

vb =

+ I , ) = 0;

(45b)

d21,,/dqbZ = 0 ;

(454

M;

(4 5 4

(kT/e)qb = [25T(“K)/29oo]vb (MI/).

(454

I,(V,)

+ I,( V,),

(T

=

p, v,

or

Solution of the above relationships yields :

K

(mv)

25[T(“K)/2900i [?T - (1/210)(1J0 exp IT + KxlxO exp(Kx~T)l

Vp (mV) x 50[T (“K)/290”]sinh-’(qT/6),

7

(46)

VM (mV) x 50[T(“K)/290”] In{qT[l + (9~/16)]}. It may be seen from Fig. 10 that, typically, I , , >> It2, It3, It4, I,, and I, for Vb 5 0.5y, so that the current parameters may be expressed as

’’ L.D.Armstrong, Microwave J . 5, 99 (1962).

8.

I,

= l o { [ ( I J o / z o )exp q T

501

TUNNEL DIODES

+ ( K x I x o / I o ) ~ X P ( K ~ V+(IJo/Io)(~XP T)I~ V V

+ ( ~ x o / ~ O ) e x P ( ~+x ~P 2, ) + Pl exP[-K1(?,0 + T i 0 - r1J21). Typically, for a high-quality diode, I , > 101,. In addition, the maximum negative-conductance magnitude of the tunnel diode occurs at VMand is given for large Ip/I, by (48) These fundamental properties of the current-voltage characteristic of the tunnel diode pn junction will be often referred to in later sections, where tunnel diode circuit performance is related to fundamental device parameters. Previous calculations have been made of the peak7 and excessz2 currents under somewhat more restrictive assumptions. For example, the peak tunneling current obtained under a O”K solution of the tunneling integrals’ GM %

(e/kT)ldlt/dqblVM (210qTe/kT)[FM(qT)l l”.

where ii = napnd,,/(nap+ ndn). Further use will be made of these results when considering the material properties of tunnel diodes for various circuit applications. 111. Principles of Tunnel Diode Fabrication

5. FORMATION OF THE TUNNELING JUNCTION a. Applicable Semiconductor Materials The semiconductor materials in which quantum-mechanical tunneling has been observed include the semiconducting Group IV elements and Group III-V compounds having the relevant physical parameters’-’ 2 ~ 2 5 - 3 3 listed in Table I. 26

27

28 29

30

3’

32 33

W. J. Bertram, Jr., C. Dunn, and M. R. Barber, in “Microwave Tunnel Diode Devices and Their Circuit Applications” (H. A. Watson, ed.). McGraw-Hill, New York, 1968. K. K. N. Chang, “Parametric and Tunnel Diodes.” Prentice-Hall, Englewood Cliffs, New Jersey, 1964. G. M. Glasford, R. L. Anderson, and R. P. Nanavati, unpublished work. 1964. N. Holonyak and I. A. Lesk, Proc. I R E 48,1405 (1960). H. S. Sommers, Jr., Proc. I R E 47, 1201 (19S9). H. R. Lowry, J. Giorgis, E. Gottlieb, and R. C. Weischedel, “Tunnel Diode Manual .”General Electric Company, Liverpool, New York (1961 ). C. A. Burrus, Proc. IRE 50, 16x9 (1962). C. A. Burrus, J . Appl. Phys.32. 1031 (1961).

502

H. C. OKEAN TABLE I

PHYSICAL PARAMETERS OF TUNNEL DIODE MATERIALS Semiconductors" Ge Si Suitable doping agents: Donor Sb, As As Acceptor Ga, A1 Al, B Representative impurity 4 x 1019 2 x 1020 concentration (cm-') Forbidden-band 1.11 0.67 E, energy gap (eV) Relative dielectric 11.7 16.4 constant 4EO Impurity ionization energy (eV) E , - Edo 0.01 &o - Ev 0.01 1.1 Relative effective mas? m,*/m0 0.55 0.37 mp*lmo 1300 Mobility (cm2/v-sec) pn 3900 500 1800 PP

GaAs Sn Cd, Zn

GaSb Te, Sn Cd

lnSb Te Cd

5 x

I

2 x 10''

1019

x

10'9

1.35

0.70

0.18

11.1

14.0

15.9

0.0 127 0.0127 0.034

0.0097 0.0097 0.047

0.0097 0.0097 0.021

5000

4000 850

73000 1250

400

"All parameters evaluated at 290"K, except at 77°K for InSb. mo is the mass of the electron.

The ranges of physical parameters that are most appropriate to the realization of a high-quality, high-frequency tunnel diode are : (1) A small band gap E , for high peak current and large power-handling capability. However, E , must be sufficiently large at a given temperature to prevent the flow of intrinsic-carrier current. For this reason, InSb must be operated well below room temperature, say at T = 77°K. (2) A high doping level for low series resistance. (3) A low carrier effective mass for low series resistance and low excess current.

The materials used in the fabrication of high-quality, high-frequency, room-temperature tunnel diodes are Ge, GaSb, and GaAs, with InSb and Si precluded because of the overly low forbidden band gap exhibited by InSb and the high effective carrier mass and resulting high excess current exhibited by Si. In addition, GaAs tunnel diodes had in the past presented a reliability problem,'* which, however, is now believed to be under control. The tabulated physical parameters of the various semiconductor materials used in tunnel diode fabrication strongly influence the electrical parameters of the resulting tunnel diodes and thereby determine which semiconductor material is most appropriate for a given tunnel diode application, as will be described in succeeding sections.

8.

TUNNEL DIODES

503

The most suitable doping agents, which provide the required donor and acceptor impurity concentrations, are also listed in Table I for each of the semiconductor materials under consideration. The manner in which these impurities are physically introduced in the semiconductor will be described in the following section. b. Methods of Junction Formation

A tunneling p - n junction is basically formed by introducing at a localized small area on a heavily doped ( l o L 9or l O ” ~ m - ~n-type ) or p-type semiconductor wafer a small “dot” consisting of or containing an acceptor or donor impurity, respectively. The impurity dot usually consists of a metallic alloy containing the metallic or nonmetallic impurity element. A list of commonly used alloys’ containing the appropriate donor and acceptor impurity element‘s is provided in Table 11. TABLE 11

IMPURITY ALLOYSUSEDI N TUNNELDIODE JUNCTION FORMATION Impurity element Ga Al, B Sb As

Commonly used alloy InGa, SnGa AIB PbSb SnAs

The realization of high-frequency, high-speed tunnel diodes requires that the area of the tunneling p n junction be extremely small ( < cm’), thus making the junction-forming process quite critical and limiting the number of applicable processes. In particular, a junction-forming process is used in which the impurity dot is fused to the semiconductor wafer at the desired junction location by alloying, diffusion, or solution growth. The most common junction-forming processes’ 1,26*32-42 utilizing these techniques for introduction of the C. A. Burrus, Proc. I R E 49, 626 (1961). R. N. Hall, Proc. IRE 40,1512 (1952). 3 6 M. J. Coupland, C. Hilsum, and R . J . Sherwell, Solid State Electron. 5,405 (1962) 3 7 H. C. Okean, Dig. 1966 IEEE G - M T T Int. Symp., Palo Alto, p. 135 (1966). H. C. Okean, IEEE Trans. Microwave Theory Tech. MTT-15,613 (1967) 3 9 G. Gibbons and R. E. Davis, Proc. IEEE 54. 814 (1966). 40 A. Lueck, W. Schultz, and A. Marmiani, Microwaoe J . 9, No. 7,49 (1966). 4 ’ S. S. Im,J. H. Butler, and D. A. Chance, IBM J . Res. Deoelop. 8, 527 (1964). 4 2 H. Hornung and D. Zook, Solid State Elecrron. 9, 7 (1966). 34 35

’*

504

H. C . OKEAN

I

JUNCTION VOLTAGE

Vb

FIG.10. Representative theoretical and measured tunnel diode current-voltage characteristic.

impurity “dot” are the “ball-alloy’’ process,’ the “pulse-bond’’ pointcontacting p r o ~ e s s , ~ ~and - ~the ~ ,~ ~l a~ n, a~ r~~ process. ~-~* In the ball-alloy process, the impurity dot is alloyed to the degenerately doped semiconductor wafer in a furnace at a temperature TA (=500°C) sufficiently high to melt the alloy dot and the adjacent semiconductor material to form an ideal solution of alloy in semiconductor. The solution of impurity in semiconductor or impurity-containing alloy in semiconductor must contribute an essentially equal impurity concentration to that of the opposite type contained in the semiconductor wafer. In actual practice, the alloying process is contained for a length of time AT( 1 min) at T A and, for enhancement of the alloyed impurity concentration, is then cooled rapidly (in seconds to minutes) to room temperature, thus forming the pn junction. The maximum alloying time A z M A X is that which keeps the length ALD over which alloyed impurities diffuse into the semiconductor (with diffusion coefficient DA)less than 0.1L (L is the junction width), with AT,,, given by

-

After cooling ofthe alloyed dot, it is contacted electrically to permit monitoring of the dc 1-V characteristic. Then, the resulting pn junction is reduced in diameter by etching until the desired level of peak current is obtained. The resulting pedestal-like junction configuration is then strengthened mechanically with epoxy prior to diode encapsulation.

8.

TUNNEL DIODES

505

The advantages of the ball-alloy process include its tight control of peak current (junction impedance level) and its adaptability to automation, whereas its disadvantages include variability of junction properties if environmental controls are not strictly adhered to, and the fragility and relatively high parasitic content (discussed in next section) of the junction configuration. The pulse-bond point-contacting process involves the location of a pointed metallic ribbon at a point on the surface of a degenerately doped semiconductor wafer, the strengthening of the resulting point contact with a drop of epoxy, and the formation of the pn junction by passing a voltage pulse ( - 1 V for = 10 psec) through the ribbon and wafer. The resulting p-n junction is very low in parasitic content. thereby lending itself to the fabrication of a very-high-frequency diode.j2-j4 However, the process does not permit the tight control of junction area that the ball-alloy etching allows. The most recently developed tunneling junction-fabrication process and that most compatible with the burgeoning microelectronic integrated-circuit technology is the planar process. Here, the semiconductor surface to be processed is masked off, with only the desired junction area exposed. The desired impurity element or impurity alloy is then alloyed, diffused, or grown into the semiconductor through the window in the mask, yielding the desired planar p n junction. It is only recently3’ that sufficiently small-area planar junctions have been obtained for the realization of high-frequency tunnel diodes. This process holds most future promise, however, particularly with respect to integrated-circuit tunnel-diode application^.^^^^^^^^

6. FABRICATION OF THE

OVERALL

TUNNELDIODE

a. Overall Diode Configuration and Equivalent Circuit Following the formation of the tunneling pn junction, the remaining steps in the fabrication of the tunnel diode include the attachment of an electrical contact to the semiconductor wafer and the impurity dot, which constitute the two halves of the p-n junction, and either the encapsulation of the contacted p-n junction in a diode package for general use or the mechanical strengthening of the unencapsulated junction for ultimate permanent attachment to a specific circuit. Of the various junction types described, the ball-alloy junction is usually employed in an encapsulated structure, whereas the point-contact and planar junction are more adaptable to an unencapsulated diode. The functional representation of the completed tunnel diode is presented in Fig. 1 l(a), showing the junction, contacts, and encapsulation or strengthening. This functional representation provides a clear indication of the origin of the electrical parasitics associated with the tunnel diode junction. These

506

H . C . OKEAN METALLIC CONTACT TUNNELING p-n JUNCTION INSULATING ,MATERIAL FOR MECHANICAL RIGIDITY

SEMICONDUCTOR WAFER METALLIC CONTACT

(b) FIG. 11. Functional representation of tunnel diode. (a) Representation of construction; (b) small-signal equivalent circuit.

parasitics, represented in the small-signal tunnel diode equivalent circuit shown in Fig. 1 l(b), include the voltage-dependent junction capacitance Cj( Vb) across the depletion layer of the p-n junction, the series resistance R , , representing the ohmic losses in the semiconductor wafer and the contacts, the series inductance L, due to current flow through the contacting leads, and parallel capacitance C , across the contacts through the insulating encapsulation or strengthening medium. The junction resistance Rj( Vb) is the inverse slope at v b of the tunnel diode I-V characteristic. Of these parasitics, Cj and R, are internal elements associated with the p-n junction and the semiconductor bulk, whereas L, and Cp are external elements associated with the contacting and encapsulation geometry and the surrounding circuit environment. The dependence of the internal parasitic element values on the physical parameters of the tunnel diode will now b e demonstrated. The junction capacitance C,( Vb) across the depletion width w d is approximated, using Eq. (16), from nondegenerate p-n junction theory12-15 as cj(Vb)

2 &A/Wd

ZZ A{[napndn/(nap

+ ndn)12zee/(vOnp

-

Vb)

(50)

for the abrupt p-n junction typical of tunnel diodes. [For the graded p-n junction, the voltage dependence is proportional to ( Vonp - Vb)- I?] This approximation is valid” for V, < Vb < kVonp,where k % 0.54.75.

8.

507

TUNNEL DIODES

For V, % Vonp, Eq. (50)must be modified by a multiplicative constant, whereas for 0 d vb d V,, C j reaches a minimum for v b = V, and then increases monotonically as Vb decreases further toward zero. These departures from Eq. (50) are due to the extra charges in the depletion layer of a degenerate p n junction contributed by tunneling electrons in the forbidden gap and by free conduction-band electrons and free valence-band holes in the energy states E,, d E d E v p . The tunnel diode series resistance is expressible in terms of the length L, of semiconductor bulk between the junction and the semiconductor contact by

where p B is the resistivity of the degeneratively doped semiconductor bulk, A B ( x ) is the (generally variable) cross-sectional area of the bulk, and R, is the total contact resistance. For geometries in which the uniform bulk region adjacent to the junction dominates R , , the latter becomes

R,

(52)

(LB/A)PB.

The incremental junction resistance Rj(Vb) may be expressed in the form Rj(Vb)

[dJb/dvb];,’

K(Vh)/A?

so that the time constant of the tunneling junction, given by

lRj(vb)lcj(Vb)

+

(53) is independent of junction area, but is, at a given Vb, exclusively a function of the material parameters. The same is approximately true for the resistance ratio I’,= R,/JRj(Vb)I. The influence on tunnel-diode circuit performance of the semiconductor material dependence of the “internal” tunnel diode parasitics, leading to an optimum choice of material for a given tunnel diode application, will be dealt with in later sections. Tj =

IK(Vh)l[2neen,,ndn/(V~np

-

Vb)(nap

ndn)11’2

3

b. Possible Tunnel Diode Geometries

Having described the general configuration of a tunnel diode (Fig. ll), we now provide examples of physical realizations of encapsulated and unencapsulated tunnel diodes as presented in Figs. 12 and 13, respectively. The most common encapsulated tunnel diode geometry is the cylindrical pill shown in Fig. 12(a). Here, a ball-alloy junction is formed 43

W. Getsinger, I E E E Trans. Microwave Theory Tech. MTT-14, 58 (1966).

508

H. C . OKEAN

METALLIC TOP CONTACT METALLIC RIBBON

BALL -ALLOY JUNCTION

CERAMIC STANDOFF

SEMI CONDUCTOR WAFER METALLIC BOTTOM CONTACT

(a)

EPOXY FILLER SEMICONDUCTOR WAFER CONTACT

(b)

INSULAT I NG DEPOSITION

EVA P 0RAT ED CONTACT

SEMICONDUCTOR WAFER

PLANAR JUNCTION

(C)

FIG. 12. Encapsulated tunnel diode geometries (a) Ball-alloy junction, (b) point-contact junction, (c) planar junction.

on the top surface of the semiconductor wafer, which in turn is mounted on a metallic contact stud. The upper half of the junction is contacted by a thin metallic ribbon or a wire mesh which connects to the upper contact stud. The package is held together by ceramic spacers which separate the two studs. A similar encapsulated tunnel diode geometry may be used in conjunction with a point-contact junction44 and a planar junction4' as shown in Figs. 12(b) and 12(c), respectively. The most common realizations of unencapsulated tunnel diodes are those formed directly on a waveguide element for use at high microwave frequencies or those having a geometry compatible with integrated-circuit technology. 44

R. J. Taylor and C. R. Westgate, Dig. I968 IEEE G - M T T Int. Symp., Defroit, p. 179 (1968).

8. TUNNEL EPOXY

509

DIODES

M E T A L L I C WHISKER (CONTACTS WAVEGUIDE WALL)

suppoR
w,, then approaches zero as w --t wR

1 rs)

YE = G,(w) + jB,(w)

7. Condition for potential stability under at least one passive ter~nination~~ (including Cp)

Z,', Y,'

8. Condition for onset sinusoidal oscillations" at w o

Y, = G,(u>) + jB,(w)

9. Condition for sustenance of steadystate sinusoidal oscillations at G o

Y,

= G,(w)

+ jB,(w)

"For absolute stability, choose R = l/C, and Cj(Vb)= Cj(V,) in 1-7: (0)= criterion for prevention of unwanted oscillation ; (SW) = criterion for establishment of stable operating point (no switching), which requires that dc load line of circuit intersect the tunnel diode current-voltage characteristic only at the desired negative-resistance operating point [Fig. 20(a)].

8.

533

TUNNEL DIODES

for I , > 1. Finally, the ultimate limitation (7) shows that the tunnel diode cannot be stabilized with any realizable passive termination if 1, 3 3/(1 + r,). Therefore, the most important requirement on tunnel diode device design from a stability standpoint is that 1, be below some upper stability bound 1 < I, < 3. This emphasizes the desirability of realizing a low L,, or, for a fixed L, and oj,it limits the maximum useful negative-conductance level (GM)max to ls,maJcojLs, which in practice falls between 0.05 and 0.2 mhos. c. Basic Principles of Tunnel Diode Stabilization

The methods of external circuit design employed to stabilize a tunnel diode against the onset of undesired sinusoidal or relaxation oscillations vary but somewhat with the particular tunnel diode application,' 2955361~63,66*67

FIG.21. Typical tunnel diode biasing and stabilizing networks. (a) Series bias feed network; (b) parallel bias feed network ; (c) series-connected stabilizing network; (d) parallel-connected stabilizing network.

certain fundamental techniques are common to all applications. These include the electrical isolation of sections of external circuitry which operate in widely separate frequency ranges, and the use of resistively loaded selective stabilizing networks. The first technique'2$62employs low-pass or bandpass filtering to couple specific portions of the external circuit to the tunnel diode within specific frequency ranges and to strongly decouple them at all other frequencies. 66

J. Hamasaki, I E E E Trans. Mirrowmv Theory Tech. MTT-13. 213 (19651. Trans. Microwaue Theory Tech. MT"-15, 554 (1967).

'' B. A. Miller, T. P. Miles, and D. C . Cox, I E E E

534

H. C. OKEAN

Typical is the use of low-pass filtering in the form of rf chokes and bypass condensers to couple the bias voltage source to the tunnel diode at dc and decouple it at high frequencies. Two commonly used dc biasing arrangements which satisfy the conditions for bias circuit stability are shown in Fig. 21, a series bias feed employing a bypass condenser and inductive dc in Fig. 21(a), and a parallel bias feed using an rf choke and dc blocking capacitor in Fig. 21(b). Similar bandpass filtering techniques are used to separate critical circuits in each multifrequency tunnel diode applications as frequency converters. The second general approach involves the use of resistively terminated band rejection filters55,63*66,67 which resistively load the tunnel diode at all frequencies outside the frequency range of interest and provide only a small, lossless perturbation within the band of interest. Simple series- and parallelconnected stabilizing networks, which should be connected as physically and electrically close to the tunnel diode junction as possible, are shown in Figs. 21(c) and (d), respectively.

V. Experimental Characterizationof Tunnel Diodes 1 1. GENERAL APPROACHTO TUNNELDIODECHARACTERIZATION

The various terminal parameters of tunnel diodes as described in the preceding section may be determined experimentally using low-frequency and microwave measuring techniques. Most of the significant parameters such as the I-I/ characteristic, Cj(Vb),R , , and K , can be measured at low frequencies. However, the predominance of high-frequency sinusoidal and high-speed digital tunnel diode applications requires that the diodes be characterized at microwave frequencies for the following reasons. First of all, the contribution of the tunnel diode parasitics L, and C, to measurable tunnel diode behavior is most strong at microwave frequencies. In addition, the microwave properties of the geometry of the tunnel diode and its immediate mounting environment introduce43 additional parasitic reactances and impedance transformations within the accessible terminals of the tunnel diode, which influence the effective values, at these frequencies, of conventional equivalent circuit parameters, with respect to not only parasitics L, and C,, but, to a lesser extent, semiconductor parameters Gj, Cj, and R , . Finally, a unique requirement on tunnel diode measurements introduced by the negative-resistance property of the tunnel diode is that the latter must be stable in its measurement circuit, in accordance with the stability criteria stated in the preceding section. This requires that the tunnel diode be characterized in a microwave mounting fixture that presents a wellcontrolled, preferably resistive immittance characteristic to the tunnel diode

8.

535

TUNNEL DIODES PARA L LE L- C 0 NN ECT E D D IS K STAB I L I ZI NG,RESISTOR

PILL-TYPE TUNNEL DIODE

BEAM LEAD, UNENCAPSULATED TUNNEL DIODE

1

CONTACT AREA

TO CONNECTOR

SUBSTRATE GROUND PLANE

TAPERED TRANSMISSION L I N E TRANSFORMER

REDUCED-HEIGHT WAV EGU IDE

TUNNEL DIODE (C 1

FIG. 22. Tunnel diode mounting configurations. (a) Coaxial mount; (b) microstrip mount; (c) waveguide mount.

over its entire active frequency range when inserted in the measurement circuit. Depending upon the type of measurement to be made, several mounting configurations may be used in microwave transmission media most compatible with the geometry of the tunnel diode under test or most representative of that intended for the given tunnel diode application. These include a mount with a microwave stabilizing resistor adjacent to the tunnel diode for low-frequency measurements, or a mounting fixture at the end of

536

H. C. OKEAN MICROWAVE TD MOUNT VARIABLE DC VOLTAGE-

VI

VARIABLE AUDIO OR PULSE VOLTAGE \

TO VOLTMETER, OR TO HORIZONTAL SCOPE AND

’

STABILIZING

RESISTOR\ 1

TO VOLTMETER, OR T O V E R T I C A L SCOPE AND RECORDER INPUTS

vE RECORDER I N P U T S

(a)

MOUNTED TD VARIABLE DC VOLTAGE BRIDGE

I-TPRECISION RsD{ STANDARDS

RF SIGNAL G EN E RAT0 R MOUNTED

?i;cs~

VARIABLE DC VOLTAGE

L& I

TD

SLOTTED L I N E

I I 1.----I

B I A S TEE

LOCAL OSCILLATOR

FIG.23. Measurement circuits for tunnel diode characterization. (a) Resistance bridge for I-V curve tracer and series resistance measurement ; (b) capacitance measurement circuit ; (c) microwave reflection measurement circuit.

or across a constant or ultra-broadband, tapered characteristic impedance transmission line for microwave reflection or transmission measurements, as shown in Fig. 22(a-c), respectively. The applicability of these mounts to particular tunnel diode measurements will be discussed in the following sections.

8.

TUNNEL DIODES

537

12. LOW-FREQUENCY MEASUREMENTS a. Measurement of Current-Voltage Characteristic The first measurement usually made in the characterization of a tunnel diode is that of its current-voltage characteristic. This is most frequently including the tunnel diode, mounted in conaccompanied by junction with a stabilizing resistor [Fig. 22(a)], as one of the arms of a resistive bridge, as shown in Fig. 23a). The bridge is first balanced with the mount including the stabilizing resistor in position by adjusting the variable resistor R, for a zero indication on voltmeter 4. The tunnel diode is then inserted in the mount, unbalancing the bridge such that the voltmeters V, and V, yield the tunnel diode I-V characteristic with Vb 2 V, and I , 2 SV, ( S is a scale factor). A versatile circuit utilizing this bridge provides both a dc and a full-wave rectified 60-cycle input voltage to the bridge and presents the metered voltages VEand V, to the vertical and horizontal deflection plates of an oscilloscope and to an X - Y recorder. This circuit therefore doubles as a curve tracer, a point-by-point I-V indicator, and an all-purpose dc bias supply for the tunnel diode. The recorded I-V trace provided by this circuit also yields the incremental negative conductance G j (I/b) and the shot-noise constant KN(V,,) as functions of bias, as given by Eqs. (68) and (77). The incremental junction conductance Gj( V,,,) may be measured more directly in the circuit of Fig. 23(a) by superimposing on the dc input V,,, a small (less than 10 mV peak-to-peak) audio signal. The resulting peak-topeak ac components AVE and A 4on V, and V, as read on the oscilloscope or on ac voltmeters yields Gj( Vbo) directly as

’,’’ ’-”

Gj(60)

[ ( 1 / s ) ( A v E / A V , ) -R J 1 .

(84)

h. Measurement qf Series Resistunce

The series resistance of the tunnel diode [mounted as in Fig. 22(a)] may also be measured3.’ in a modification of the resistance-bridge measuring circuit of Fig. 23(a), in which a small (less than 10 mV) audio sinusoidal or pulse voltage is superimposed on the dc bias V’,. The latter must be set in a high-current region of the I-V characteristic in order to minimize the contribution of Rj(Vb).Since measurements in the forward region (Vb > V , ) are perturbed by minority-carrier injection and hence conductivity modulation of the semiconductor bulk, vb is usually set in the high-current reverse region, 1 3 7 1 . 7 2

68 6y

J. A. Narud and T. A. Fype, Elrcfronic,s 34. 74 (1961). C. D. Todd. Rev. Sci. Insrrutn. 32. 338 (1961 ).

G. E. Fox. Solid Srnre Design 3, 27 ( 1962). E. L. Bonin and J. R. Baird, Pioc. / R E 49. 1679 (1961). ” R. J . Wilfinger and B. A. Zolotar. Riw. Sci. Instrum. 33, 693 (1962). O

”

538

H. C. OKEAN

at I , zz -501, to -2001,. Under this condition, the measured incremental resistance is of the form337’

+ (K/lzbl)?

(85) where AZ, and Av, are extracted from the incremental voltages AVE and A& read on the oscilloscope or voltmeters at the outputs of the initially balanced bridge. If these measurements are repeated a t several values of I , , v b in this range, Rsmay be extrapolated from a plot of R,,,,( V,) versus I ; in the asymptote as becomes infinite. R r n d V b ) = [A1b/Avbl;,’

Rs

c. Measurement of Shot-Noise Constant The shot-noise constant KN = IbN(Vb)\Rj(vb)l is usually measured in the active region by extracting it from the measured Z- V characteristic as shown in Fig. 19(b) under the approximation I , , zz 1,. However, measurements have been made53*73to verify the proposed bias dependence of i b N [Eq. (70)] and thereby to test the validity of the above approximation by directly measuring the output noise power level of actively biased Ge, GaSb, GaAs, and Si tunnel diodes as functions of bias and frequency. These measurements utilized a tunnel diode mounted in conjunction with a shunt stabilizing resistor [Fig. 22(a)] connected in parallel with the input circuit of a low-noise H F preamplifier. Careful precautions were taken to ensure the absence of tunnel diode oscillations, as ascertained by monitoring the Z-V characteristic and the output spectrum of the tunnel diode. One set of measurements, by King and S h a r ~ e extracted ,~~ the measured equivalent shot-noise current zbN( vb) from the output noise power of the preamplifier measured with and without a noise diode connected across the input terminals of the amplifier. The other, by Giblin,73 obtains lbN(i$) by comparing the output noise voltage of the preamplifier at tunnel diode bias V , to that at zero bias. The results of these measurement^^^.^^ indicate that, at frequencies below 1 MHz, the measured IbN(Vb) is considerably larger than its theoretical counterpart of Eq. (70), due primarily to l / f noise. However, at frequencies in the 30-MHz range, the measured i b N exhibits approximately the predicted dependence on Vb expressed in Eq. (70), so that in the active region the approximation I,, x 1, appears valid and, for most applications, is therefore sufficiently accurate in the determination of KN. d. Measurement of Junction Cupacitance

Low-frequency measurement of the tunnel-diode junction capacitance Cj(V,) is usually performed on a standard VHF or H F admittance 73

R. A. Giblin, Elect. Eng. 36, 766 (1964).

8. TUNNEL DIODES

539

bridge.3,11 . 6 9 . 7 4 F or measurements in the active bias region, a coaxial tunnel-diode mount is usually used which incorporates a shunt stabilizing resistor (Fig. 22a). However, for minimum measurement error due to stabilizing and diode junction conductances, measurement of Cj at v b z V, 6 is preferred, in which case the stabilizing resistor is not required. A typical capacitance-bridge measurement circuit shown in Fig. 23(b), utilizes a small H F or VHF voltage ( z1-5 mV rms at 1-100 MHz) superimposed upon the dc bias in order to ensure an incremental measurement of Cj(Vb). The measurement consists in balancing the bridge with the stabilized diode mount in the unknown branch, and rebalancing it with an essentially nonreactive resistance standard and a known capacitance standard in place of the diode mount. The difference in capacitance indicated on the capacitance standard for the two balance conditions is the terminal capacitance of the diode, from which c j ( v b ) may be extracted. Case capacity C,may be obtained from a similar measurement on an open-junction tunnel diode with identical geometry. The use of precision external standards minimizes errors due to parasitic reactances. The diode resistance values used in the extraction of c j ( v b ) from measured data are obtained prior t o this measurement as described in the preceding sections. The measurement frequency is chosen to minimize errors due to these resistances and to the series inductance.

+

13. MICROWAVE MEASUREMENTS a.

Rejection Measurements

Microwave reflection measurements are usually used to determine the microwave equivalent-circuit parameters of a tunnel diode mounted at the end of a transmission line, although characterization in other diode mounting orientations is also p ~ s s i b l e . ~Th ~ e, ~reflection ~ - ~ ~ measurement consists in the determination of the microwave immittance of a suitably biased tunnel diode imbedded in a one-port mount by means of a measurement of its voltage reflection coefficient. This is usually accomplished by a standard slotted-line measurement technique,78 in which determination is made of the minimum position and amplitude (VSWR) of the voltage standing-wave pattern established on a slotted section terminated at one end by the mounted tunnel diode. In a representative measurement ~ i r ~ ~ i[Fig. t ~2 3~ ~, ,~ ~ - ~ the microwave test signal is applied to the loosely coupled slotted-line probe and the output end of the slotted line is connected through an isolator to a 74 75

h ’ 77

D. E. Thomas, IEEE Trans.Electron Derices ED-IQ,278 (1963). H . Fukui, Dig. 1961 In!. S d i d State Circuirs Con6 Philadelphia, Pennsyfuaniu, 1V. I6 (1961). C. S. K i m and C. W. Lee, Microwaces 3. 18 (1964). J. W. Bandler. f E E E Trans.EIectron Dei%-es ED-15, 275 (1968). E. L. Ginzton, "Microwave Measurements.'' McGraw-Hill, New York, 1957.

540

H . C . OKEAN

highly sensitive microwave receiver. This ensures that the rf signal power incident on the tunnel diode will be below 1 pW and that the resultant peakto-peak rf voltage across the tunnel diode junction will be less than 10 mV, thereby permitting a valid small-signal measurement. The presence of the output isolator ensures that the mounted tunnel diode will be terminated essentially in the resistive characteristic impedance of the measurement system over a broad frequency range, thereby aiding in tunnel diode stabilization. The small-signal immittance of the mounted tunnel diode is obtained from the measured reflection data at a given frequency, following a determination, based on a priori knowledge or on a reflectometer measurement, as to whether the measured reflection coefficient magnitude is greater or less than unity at this frequency. The latter requirement is due to the exhibition of reflection coefficient magnitudes greater than unity by the tunnel diode input immittance at frequencies at which its real part is negative. This is in contrast to the more familiar case of passive components, for which input reflection coefficients cannot exceed unity. However, the slotted-line measurement of VSWR cannot distinguish between positive and negative real parts of measured in input immittance, hence the required prior determination. Repetition of this measurement over a range of microwave frequencies yields a locus of the measured immittance at the accessible terminals of the tunnel diode mount, from which may be extracted some or all of the small-signal equivalent circuit parameters, depending on the tunnel diode bias and the complexity of the mount. In particular, measurements made with the tunnel diode biased on the valley o r positive-slope regions of its current-voltage characteristic may utilize a simple mounting fixture in which the tunnel diode is connected at the end of a transmission line (or waveguide) having the same characteristic impedance Z , and geometry as that of the measuring system. However, for diodes biased in the active region, this fixture can only be used when the nominal R M characterizing the diode is greater than Z,. Otherwise, stability requirements dictate the use of either a mount with a taper that transforms Z , to Z,, < R , over a wide frequency range [Fig. 22(b)] or a mount that includes a stabilizing resistor [Fig. 22(a)]. The small-signal tunnel diode equivalent circuit may be characterized completely in a given microwave frequency range by obtaining the measured immittance loci under several conditions of tunnel diode bias. I n each case, the measured terminal immittance loci of the tunnel diode proper are obtained by subtracting the immittance contributions of the mount itself, due to its stabilizing resistor, impedance transformers, and/or reactive parasitics from the original immittance data. The measured data equivalent-circuit parameter values may be conveniently extracted from the measured immittance loci using the following

8.

TUNNEL DIODES

541

sequence of immittance measurements: (a) measurement of C , using an open-junction tunnel diode ; (b) measurement of L, and R , with the tunnel diode reverse-biased at several current points in the range - 101, t o - 501,, evaluating R , = R,,,, in the asymptote as approaches infinity (under this condition, the junction contribution is essentially shorted out) ; or, alternatively, measurement of L,, R , , and Cj(Vb) at valley bias V, [Gj(Vv) % 01; (c) measurement of Gj(Vb) and Cj(Vb) under active bias (V, < Vb < Vv), accomplished by removing measured values of C,, L,, and R , from measured terminal immittance [Eqs. (71) and (7411. It is noted that the above measurement procedure is useful only when the tunnel diode is mounted at the end of a transmission line or when the immittance corrections due to the mount contributions are relatively simple. If these conditions are not satisfied, two other approaches to tunnel diode characterization at microwave frequencies may be utilized, the transmission measurement and the oscillating diode measurement, as will now be described.

6. Transmission Measurements The microwave transmission measurement, originally proposed by De L o a ~ h 'to ~ characterize microwave varactors, is a convenient method of determining the values of the parasitics associated with a tunnel diode mounted in shunt across a transmission line. It consists essentially in measuring the small-signal insertion-loss-frequency characteristic, in a conventional low-level transmission-loss measurement circuit, of a symmetric section of uniform or tapered transmission line across the central plane of symmetry of which is mounted a valley-biased tunnel diode. The small-signal equivalent circuit of the valley-biased diode, including the effect of mounting parasitics, may be represented near diode self-resonance w , [Eq. (73)] by a series R,'-L,'-C' circuit parallel-connected across the transmission line. Therefore, the overall structure approximates that of a single-section band-rejection filter having a single-peaked maximum in insertion loss centered at ooz ox(K).The equivalent parameters Rs', L,', and C', approximating the valleybiased tunnel diode transformed by the mounting parasitics, are directly obtainable from the center frequency oo,peak insertion loss LM,and halfpeak loss bandwidth AOJof the single-peaked insertion loss characteristic using the relationships

The actual small-signal diode parasitics R , , L,, and Cj(V,) may then be extracted from the measured values R,', Ls', and C' after prior evaluation of the mount parasitics.

'' B. C. De Loach, I E E E Trans. Microwave 7heory Tech. MTT-12, 15 (1964)

542

H . C . OKEAN

c. Measurements on Oscilluring Tunnel Diobe

An alternative method’ of determining the equivalent series inductance of the tunnel diode at microwave frequencies utilizes a simple, lightly coupled cavity-type tunnel diode oscillator model in which the external impedance a t the diode terminals is 2, z R g + joL,. The physical geometry of the variable-dimension, below-resonance (inductive) cavity is chosen to simulate the tunnel diode mounting geometry of interest in the inductance evaluation. The tunnel diode is biased sufficiently into the active region such that a weak sinusoidal oscillation at LO,,, is barely maintained, corresponding to the conditions

the latter determining the required degree of cavity coupling to the measurement circuit. The measurement procedure consists in determining Q,,,for several values of L, corresponding to several settings of the variable cavity dimension,

TD CIRCUIT

LOAD

RL

* 01

(b)

CIRCUIT

SOURCE

D

-

ISOLATOR

-

ISOLATOR

-

ISOLATOR (Cf

FIG. 25. RF and microwave tunnel diode amplifier configurations. (a) Circulator-coupled reflection amplifier;(b) hybrid-isolator-coupledreflection amplifier; (c) isolator-coupled transmission amplifier.

8.

547

TUNNEL DIODES

where rA(jO) =

[yg*(;w)

- yA(jw)l/[yg(jO)

+ yA(jw)l

and YA(j w ) are the input reflection coefficient and admittance of the tunnel diode circuit, as given by I r A l > 1, Re YA < 0, in the amplification band. The quantity Y,(;w) is the input admittance at the amplifier port of the coupling network, as presented to the tunnel diode circuit (Fig. 25), and the asterisk is used to denote the complex conjugate. Furthermore, it follows that $ o = 1,

m=O,

for reflection amplifiers ;

m = 1, for transmission amplifiers. $, = RGRL/(RG + R,)', The amplifier insertion parameter usually of primary interest is the insertion gain 29, although in certain amplifier applications, the phase 8 is also of importance. The relative advantages of each of these three configurations, as dealt with previously in the literature,' 2 * 8 3 * 9 s . 9 6and as touched upon in the following sections, are qualitatively that the circulator-coupled reflection configuration uses the smallest number of components and has the lowest potential noise capability, the hybrid-isolator-coupled reflection configuration has, with the advent of multioctave hybrids and isolators, the greatest bandwidth capability, and the isolator-coupled configuration in some cases lends itself to the simplest physical embodiment. The particular aspects of tunnel diode amplifier realization and performance, power-handling capability, and stabilization will be discussed with reference to these three single-stage amplifier configurations in the following sections. b. Tunnel Diode Amplifier Gain and Bandwidth Capabilities

Tunnel diodes are usually operated well below their serious self-resonant frequency [Eq. (73)] in bandpass amplifier applications. Therefore, either parallel or series inductive tuning of the tunnel diode is required to resonate its capacitive reactance at the amplifier center frequency oo,which results, to a good approximation, in the small-signal equivalent circuit models of Fig. 18(b, c), which are valid about a reasonable passband centered at coo. As stated previously, the parallel-tuned approach has the advantages of easier stabilization, less sensitivity to diode parameter variations, and generally broader bandwidth capability, and is accordingly used more frequently in practical tunnel diode amplifier realizations. Therefore, particularly since tunnel diodes with sufficiently low series inductance are readily 95

96

A. C. Macpherson, I E E E Trans. Circuit Theory CT-11, 136 (1964). P.C. J . Hill, Pruc. I E E (Lundon) 112, 15 (1965).

548

H. C. OKEAN

obtainable, the parallel-tuned configuration is preferable and will be considered exclusively for the remainder of the section on sinusoidal tunnel diode applications. The midband insertion power gain of a tunnel diode amplifier stage of either the circulator of hybrid-coupled reflection type or the isolator-coupled transmission type [Fig. 25(aHc)] is given by %w,)

90

= rl/o(rAo

+ mI2,

where

with $, the nominal midband coupling network input impedance level at amplifier port(s) of coupling mechanism, equal to R , for circulator- and hybrid-isolator-coupled reflection amplifiers, and equal t o RGRJ(RG + RL) for the isolator-coupled transmission amplifier ;n2 is the midband impedance transformation ratio from coupling network to diode terminals [Fig. 24(b)] ; and RdOis the midband negative terminal resistance magnitude of a paralleltuned tunnel diode [Fig. 18(b)], Rdo

= R(1 - rJ[1

-

(4wO2/rsuR2)].

It is immediately seen that, unlike conventional amplifiers utilizing twoport active elements such as transistors and vacuum tubes, tunnel diode reflection and transmission amplifiers (and other negative-resistance amplifiers) can have unlimited midband gain, approaching infinity and the onset of oscillation at coo as n2Rgoapproaches & ., Passband stability requires (Table VI) that R,, > n2Rgofor a parallel-tuned diode. In addition, Eq. (92) shows that, at a given value of Ti,, the midband gain of the transmission amplifier is reduced with respect to that of the reflection amplifier by a factor $, = R,RJ(R, RL) 5 0.25, which attains its maximum value of 0.25 at RG = RL. Finally, high midband gain in a tunnel diode amplifier is achieved only at the expense of reduced bandwidth, as will be shown in the following paragraph. The bandwidth capability of a bandpass tunnel diode amplifier may be formulated, without loss of generality, by assuming that in the amplifier passband the tunnel diode circuit of Fig. 24(b) is representable by a ladder network of N alternate parallel and series bandpass resonators interposed between the negative diode terminal resistance - Rd, and the transformed coupling network impedance n2Rgo,as shown in Fig. 26(a). The bandpass resonators are formulated to include the passband reactance contributions of the parallel-tuned tunnel diode, of the essentially reactive passband stabilizing network representation, of the impedance transformer, and of the coupling mechanism input immittance. In addition, we utilize the generally

+

8.

II

'

PARALLELTUNED I TUNNEL

I

549

TUNNEL DIODES

I I I

N A D

I I Rdo Q 2 7

OR

N EVEN

---+

1 I

2 'A0

1 Q@doq

n2

N =m

I

/

/"'

I

FIG.26. Characteristics of broadband tunnel diode amplifiers. (a) Passband model of tunnel diode circuit for Nth-order broadbanding; 7 = ( w / o o )- ( w o / w ) ;QdO= W ~ R , , C (b) ~ ~Nth. order, maximally flat reflection gain characteristics.

most desirable bandpass gain-frequency characteristic for rf and microwave amplifier application, the Nth-order, maximally flat (Butterworth) gain c h a r a c t e r i ~ t i c , ’ ~for * ~ ~which ~ ~ ~ ~ IrA(jo)12 ~~ has the form [Fig. 26(b)]

where N is the "order" of the maximally flat gain response, and v] is the bandpass frequency variable equal to (w/wo)- (wo/w). For Nth-order, maximally flat gain, the a-power [Fig. 26(b)] amplifier assuming bandwidth (3 = ~ 5 9> I), exhibits the following voltage gainhalf-power (c( = 0.5) bandwidth product :

On the other hand, the ultimate in amplifier bandwidth is obtained with and ideally flat gain characteristic [Fig. 26(b)] [ 9 ( w )= 9, over B,, centered at f,, and g(o)2 $, otherwise], which results from the maximally flat response in the limit as N becomes infinite. The bandwidth B, ofthis response is given by

+

B , z oj/2{1n[(~0/$0)”2 m ] }(1 + cp). Similar results, exhibiting a slightly broader bandwidth capability, are obtained from an equiripple (Chebychev) gain-frequency response. 12955s7,59 The selectivities of the broadbanding resonators Q 2 , Q 3 , . . . , QN [Fig. 26(a)] required to realize an Nth-order, maximally flat (or equiripple) gain characteristic, given Q1 = Q,,, have been derived as functions of Tio in the literature on filter and negative-resistanceamplifier synthesis, 2 s 5 s 7 s 5 ) , 9 7 - 1 O 3 These results, however, require that, if a stabilizing network is employed, it be of the series-connected type [Fig. 21(c)],for, if the parallel-connected type [Fig. 21(d)] is used across C,,, a bandwidth degradation of Q,9/(Qdo Qst) occurs. Here, Q,, = oOCstRdO is the stabilizing network selectivity required

+

E. S. K u h and J. D. Patterson, Proc. I R E 49, 1043 (1961). Y. T. Chan and E. S. Kuh, I E E E Trans. Circuit Theory CT-13, 6 (1966). y 9 L. Weinberg and P. Slepian, I E E E Trans. Circuit Theory CT-7, 88 (1960). l o o R. Levy, Proc. I E E (London) 111, 1099 (1964). l o ’ W. J. Getsinger, I E E E Trans. Microwave Theory Tech. MTT-11,486 (1963). J. 0. Scanlan and J. T. Lim, I E E E Trans. Microwave Theory Tech. MTT-12, 504 (1964). ‘ 0 3 J. 0. Scanlan and J. T. Lim, I E E E Trans. Microwave Theory Tech. MTT-13, 827 (1965).

97

98

8.

55 I

TUNNEL DIODES

for the stabilizing conductance to exceed GdO = RT:, thereby passivating the tunnel diode outside its “stability bandwidth.”’ 03a These results assume the parallel-tuned model [Fig. 18(b)] of the tunnel diode, therefore requiring that parasitic inductance L, be sufficiently low that I , 5 2r,. If this is not the case, either the series-tuned model [Fig. 18(c)] may be used, resulting in a bandwidth degradation of about $, or the more exact triple-tuned model [Fig. 18(a)] may be used, placing stringent limitat i o n ~ ’ ’on ~ L, and C, in order to satisfy the broadbanding requirements on Q 2 and Q 3 [Fig. 26(a)]. Equations (94H96) immediately indicate that, at a given midband gain level go,the bandwidth capability of a transmission amplifier is inferior to that of the corresponding reflection amplifier by a factor ranging between 5 0.5 in the single-tuned case ( N = 1 ) and

Jl/o

+

In~o’’2{ln[(~o/$o)”2I ] ) - ’

5 (1 + 1.4/lng0)-’

in the ideally flat case ( N = cx;). Furthermore, it is seen that the bandwidth capability of a tunnel diode amplifier of any degree of broadbanding and at a specified midband gain level go varies linearly with the tunnel diode junction frequency, that is, B, z K w j . Therefore, Eq. (82a) indicates that the bandwidth capability of a tunnel diode amplifier increases with increasing tunnel diode doping level, and decreasing diode effective carrier mass, energy gap, and dielectric constant. The tunnel diodes having the highest potential amplifier bandwidth capability (Table IV) are hence those of GaSb, Ge, and GaAs, with representative maximum gain-bandwidth products being

g?2(Bl)o.55

+ cp)l&

GHz,

(974

[(%/$o) (dB)IB, = [lo l ~ g i o ( % ‘ $ ~ ) l B5 % 135/(1 + cP) dB-GHz, (97b) for the single-tuned and ideally flat gain characteristics, respectively. Typical bounds on single-tuned and ideally flat half-power bandwidths at 10 dB reflection gain (ria = 10) and cp % 0.5 are 1.0GHz and 9.0GHz. These numbers far exceed the theoretical maximum bandwidth capabilities of other microwave negative-resistance amplifying devices. The measured bandwidth capabilities of practical tunnel diode amplifier realizations55,63,66.67,82-94 generally fall considerably short of the corresponding theoretical limitations imposed by the tunnel diode [Eqs. (94H97)] in that the frequency dependence of the coupling mechanism and impedance transformer often becomes the bandwidth-limiting element,” typically reducing the above values by a factor of two to four. Io3”The“stability bandwidth” of the tunnel diode, centered at w,,, is the band over which the external circuit immittance presented to the tunnel diode (excluding the stabilizing network) is sufficiently well controlled to satisfy the stability criteria. Outside this band, the stabilizing network contributes G,, > IG,I. thereby making Re Y, > 0 and passivating the diode.

552

H. C . OKEAN

c. Amplijer Noise Performance

One of the most useful characteristics of a bandpass tunnel diode amplifier, particularly at microwave frequencies, is its relatively good noise performance. The usual measure of amplifier noise performance is the noise figure F , definedlo4 as the ratio of total noise power PNT in a bandwidth E N about some frequency o at the output terminals of the amplifier to that portion of PNTa t these terminals due to noise power k'l,'BN(k is Boltzmann's constant) generated in the matched resistive input termination R , a t temperature = 290°K. Hence, F may be expressed as

F PNT/kTBNY(O). (98) The noise figure of each of the three types of single-stage tunnel diode amplifier under consideration [Fig. 25(a-c)] may be ~ b t a i n e d ~ ~by' cal~ ~ ~ ~ * ' ~ ~ d a t i n g the noise output power PNd absorbed in load resistance R , [Fig. 24(a)] due to the tunnel diode terminal noise current generator (I;,)*" [Eq. 76)] shown in Fig. 19(a), and by then substituting it in F

=

I

+ [PNd/kTBN%(o)].

(99)

The resulting expressions for the single-stage tunnel diode amplifier noise figure are given, neglecting losses in the coupling mechanism and in the tunnel diode circuit, for each of the three amplifier types by FA =

1 + { I - [l/y(W)l)fd>

(100)

where, in the case of the isolator-coupled transmission amplifier, equivalent load conductance G L is chosen for a midband output match'05" [GL = GG - (Gdo/n2)]and where td is the equivalent excess noise temperature ratio of the tunnel diode [Eq. (7611. In each case, for high-gain operation with a high-quality tunnel diode [%(w) >> 1, rs,o/wR IGjoI

for Gjo < 0 :

where Po is the available oscillator power; and for self-oscillation, Go x

.

IGjd~0,opJl

1130)

580

H. C . OKEAN

c. Mixer Gain-Bandwidth Capabilities Specific gain, bandwidth, and stability formulations have been obtained 12.14 1-148 for the configuration of Fig. 34(b) under the following two standard conditions on image-frequency termination yk that are most often encountered in practice : (a) Short-circuited image termination (SCI) : lBkl >> 1x13 I XI, Gk, IGjil

=

9

0,192

3

so that V, = 0 in Eq. (127). (b) Broadband signal-image termination (BBI) : WkO

since qk x -qs

= o,,>> oio

and

yk

= (ok/oko) - (oko/wk) under

= G,U

- jQ,a,),

these conditions.

The gain, bandwidth, and stability formulations for each of these two cases are summarized as follows for both self-oscillating and externally pumped tunnel diode mixers : (a) Midband conversion gain :

K O = K(co,,, oio)= 4G,GiIV,(2/(1G(2 =

4GsGiGil/[(Gi

+ Gjo)(Gs + Gj, + 6Gj2)

-

(1

+ 6)G;J2.

(131a)

Clearly, K O can be > 1 and, for G,, < 0, it can be made arbitrarily large. (b) Half-power bandwidth (a,fixed) :

B=

+

TCK$~[(G, Gj,

(GsGi)’” IGjlI + 6Gj2)(C + C,) + (Gi + Gjo)(C + C,)]

.

(131b)

(c) Stability conditions at w s 0 ,wio (131c) (d) Conditions for positive conversion gain ( K O> 1) :

Gjo < 0 ;

Gjo

+ 6Gj2 < 0;

or where 6 = 0 for SCI and 6 or negative.

=

1 for BBI terminations, and Gjo can be positive

8.

581

TUNNEL DIODES

It is seen that conversion gain may be obtained even with v b in the positive-conductance region (Gjo > 0), depending on choice of Gjo, Gjl, and G,, and hence (Table XIII), over a range of local oscillator drive Vo. The gain mechanism for Gjo < 0 may be viewed as rf and i.f. amplification preceding and following the mixing proper. Under high-gain conditions, the optimum half-power bandwidth is expressible as

as obtained under “symmetrical loading” :

x [(I Gi,opt % Gs,opt

+ 6)’”IGjll

-

+ 6)l”~lKA’2]’’z

GjJ)’”/(l

-

Gjo - cSGjz,

+ 6Gjz

Note that B,,, resembles the bandwidth limitation of a single-tuned tunnel diode amplifier, modified by the multiplicative factor (IGj II - Gjo)/ IGjol. Accordingly, improvements in mixer bandwidth capability of the same order as that obtainable in amplifiers can be realized by introducing broadbanding resonators in the signal input and i.f. output circuits. Conductance coefficients G,,, G j , , and G,, are strong functions of dc bias V, and local oscillator voltage Vo (Table XIII), so that v b and V, may be chosen to optimize B,,, consistent with specified gain and with a desired input and output conductance and stability margin, as well as to optimize may then be used noise performance. The resulting values of and Vo,opt to determine the effective internal or external local oscillator loading Go [Eq. (130)l. The optimum modes of operation depend strongly on noise performance, as will be discussed in the next section. d. Mixer Noise Performance

The noise performance of a tunnel diode mixer, as derived from more IS formulated in terms of the timegeneral mixer noise theory,1z.46*’41-147 dependence of the equivalent tunnel diode shot-noise current ibN(t) under the influence of the local oscillator voltage V, cos mot. In particular, ibN(t) may be expressed in a Fourier series expansion as ’

ibN(t) =

+ vocosoot)COth[(e/kT)(I/, + Vocosoot)],

(134a) (134b)

582

H. C . OKEAN

where 2n IbNm

ibN(t) cos moot d(oOt),

= (1/2n)[ 0

Jgj(t) dl/,

ibN(t)

+ 10

for

vb

2

&,

and where gj(t) is given in Table XIII. The total mean-square output noise voltage Eiacross Gi = G,, including the dominant contribution due to ibN(t) and the smaller contribution due to the thermal noise voltage = 4kTBNR, generated in R, at wso,oio,and o k o , is expressible in the f ~ r m ' ~ , ' ~ ~

GR

vi&

= 4kTBN

i

lHi112[GN0

1 = s,i,k

+ GjO(Gl/GjO)2rs0 + (010/oRO)2(1

- rsO)l

+ [Hii(Hi*, + H Q ) + (Hz(His + Hik)lGN1 f (HisH$

+ H%ik)GNZ}

(1 35)

7

where GN, = f?lbNl/2kT,1 = 0,1,2; rs0 = R,Gjo; wR0= (Gjo/C)[(l/rso)- 1]''2; T i s the physical temperature of the tunnel diode; and Hii,Hi,, and Hik are elements of the augmented impedance matrix Hsi

~

His

Hii

Hik

Hks

Hki

Hkk

[ ~ s s

s

]

=

[K + Gjo

Gj1

6Gj2

Gj,

+ Gjo

6Gj,

6Gj2

SGj,

6(&

I

+ Gjo)

where 6 = 0 and 1 for the SCI and BBI terminations, respectively. Therefore, the mixer single-channel noise figure (signal in signal channel, noise in signal and image channels) at resonance, defined as

F =1

+ (GiGi/k290BNKo),

is given, neglecting circuit losses, and relegating the contribution of noise generated in Gi to that of the i f . amplifier following the mixer, by the expression

8.

583

TUNNEL DIODES

Calculations indi~ate'~*'~""'that F = Fmin OCCUrS a t &, % Vbm [Eq. (78)] in the negative-conductance region of Gj(Vb), in two possible modes of operation,'"' a high-pump mode (large V,; K O 5 1) and a low-pump mode (V, -+ 0; K O >> 1). In either case, Fminmay be expressed as Fmin

1

+ [(I + ~)KT/~~OI[(G,O/IG~OI) + rso + ( o ~ O / O R O ) ' ( ~

-

'so)]

3

(137)

where K x 1 in the low-pump mode ( G j , , Gj2 x 0) and rc(Gj0, G j l , Cj2) 5 2 in the high-pump mode. Examination of Eq. (137) indicates that Fminis degraded by about 1.6 (2 dB) for operation with the BBI rather than the SCI termination, and for operation in the high-pump versus the low-pump mode. The degradation terms due to R, are similar to those contained in the amplifier noise figure formulation. In the low-pump mode, Fminapproaches that of a high-gain transmission amplifier, as is expected since the mixer gain mechanism in this case is primarily rf amplification at ws0prior to the mixing process. Representative calculated values of Fminin these various modes of operation are given in Table XIV. TABLE XIV

Low-pump mode Semiconductor material

GaSb Ge GaAs

High-pump mode

SCI

BBI

SCI

BBI

2.9 3.1 5.0

4.6 5.7 1.25

4.6 5.7 1.25

6.8 8.0 9.8

Therefore, as in the amplifier case, the use of GaSb tunnel diodes provides the lowest noise capability. e. Comparison of Modes of Mixer Operation Of the two modes of tunnel diode mixer operation'47 described in the preceding sections, it is seen that the low-pump mode exhibits superior gainbandwidth product and noise performance, and requires less local oscillator power Po. However, operation in the small-pump mode is extremely critical to variations in bias voltage, local oscillator drive level, and source impedance. In addition, the dynamic range of a tunnel diode mixer, characterized by the 1-dB gain compression input level PinTs x Po, is much lower in the smallpump mode (Po = - 60 dBm) than in the large-pump mode (Po % - 10 dBm),

584

H. C. OKEAN

the former being even poorer than tunnel diode amplifiers of similar gain level and the latter being comparable with conventional varistor mixers. Taking all of these factors into accwnt, it would seem that the SCIterminated, high-pump mode of tunnel diode mixer operation is the most desirable. The low noise figure of the SCI-terminated, low-pump modemixer can more profitably be obtained by using a separate low-noise tunnel diode amplifier prior to a high-pump mode tunnel diode mixer.

f: Examples of Microwave Tunnel Diode Mixers While not as many practical microwave tunnel diode mixers have been constructed as have amplifiers and oscillators, several experimental models14'-I4* have been fabricated to verify the theory summarized in the preceding sections, and several practical models' l 4 are available commercially. The state of the art on tunnel diode mixers is summarized in Table XV.

18. TUNNEL DIODEDETECTORS a. Detection Mechanism in Tunnel Diodes

The nonlinear current-voltage characteristic of the tunnel diode makes possible its as a low-level envelope detector at rf and microwave carrier frequencies, in a similar manner to conventional pointcontact crystal diodes or hot carrier diodes. In particular, the curvature of the current-voltage characteristic of the tunnel diode junction conductance at a particular dc bias point results in rectification of a small rf or microwave sinusoidal voltage superimposed on the dc bias voltage across the junction, yielding a component of dc rectified junction current which is dependent upon the amplitude of the incident sinusoid. Quantitatively, this detection mechanism is derived,' 5 1 , 1 5 2 under the small-signal assumption, from a truncated Taylor series expansion of the current-voltage characteristic I = f(Vb + V, sin wt) about the dc bias point Vb in terms of a small sinusoidal voltage V,, sin wt, with V, 1/04. The series expansion of I is given by ''7149-153

I 2 f(Vb)

+ V,[dj( V)/dV],, sin w t + )Vo2[d2f(V)/dV2],,sin2cot + . . . (138a)

14’ 15’

'" 15* 153

C. A. Burrus, I E E E Trans. Microwave Theorv Tech. MTT-11,357 (1963). T. H. Oxley and F. Hilsden, Radio Elect. Eng. (London)31, 181 (1966). R. B. Mouw and F. M. Schumacher, Microwaue J . 9 (I), 27 (1966). W. F. Gabriel, I E E E Trans. Microwaue Theory Tech. MTT-15, 538 (1967). P. E. Chaseand K. K. N. Chang, I E E E Trans. Microwave Theory Tech. MTT-11,560(1963).

TABLE XV

MEASUREDPERFORMANCE CHARACTERISTICS OF EXPERIMENTAL TUNNELDIODEMIXERS

Reference Signal frequency (GHz) Local oscillator frequency (GHz) Output frequency (MHz) Diode material Pump mode Local oscillator power (dBm) Image termination Midband conversion gain (dB) Half-power bandwidth (MHz) Midband single-channel noise figure Input level for 1 dB gain compression (dBm) ~~

'I

SO denotes self-oscillating mixer.

141 0.2 1 0.24

145 1.z 1.17

147 1.35

30 GaAs Large

30 Ge Large - 13 SCI 26 2.0 3.0

30 GaSb Large

SCI 23 0.15 2.8

147 1.35

148 2.0-3.0 2.0-3.0

-7 SCI < I

30 GaSb Small - 60

10 Ge SO"

SCI

BBI

>> 1

-

4.9

4.3

-6

- 60

-

12.0

114 4.0-8.0 4.0-8.0

1 I4 12.0- 18.0

P)

12.cL18.0

C

100 Ge Large - 13 BBI 6.0

100

P

8.5

Ge Large -11.5 BBI 9.0 11.0

+

z z

8 0

e!

586

H. C . OKEAN

or I z I, + I , sinwt - I,cos2wt

+ I,,

(138b)

where I b is the dc bias current, equal tof(Vb); I , is the fundamental current amplitude, equal to VoGj(Vb); I, is the second-harmonic current amplitude, equal to VoZGj’(Vb)/4; I , is the rectified dc current, equal to VozGj’(Vb);and Gj’(Vb) = [dGj(V)/dVlv,, and is directly obtainable from the given Gj(Vb) characteristic, as seen in the case of the parabolic [Eq. (66)] and quartic [Eq. (67)] Gj(Vb), for which Gj’(Vb) 2 ~ G MVb( - VM)/(VM - V,)’

(139a)

and Gj’(Vb) 2 37.8G~(Vb-

&)(K

- &)2/(Vv - V,)“,

(139b)

respectively. Examination of Eq. (139) indicates that Gj’ < 0 for Vb < VM; Gj’ = 0 at VM; Gj’ > 0 for Vb > VM;and that IGj’l is maximum at Vb = 0. Similar conclusions hold for higher-order polynomial representations of Gj( Vb). The detection mechanism consists essentially of the utilization of appropriate means of frequency separation to obtain a detected output voltage which, as in other detector diodes, is proportional to I,. The tunnel diode has a higher potential rectification sensitivity in terms of I , than other detector diodes due to (a) the possibility of operation in the negative-conductance region of the I-V characteristic [Gj( Vb) < 01, thereby providing rf amplification prior to detection, (b) the possibility of operation near peak voltage V,, a point of high curvature [large lGj’(Vb)l],and (c) the possibility of operation at zero bias, at which Gj(Vb) >> 0, being compatible with representative rf source impedance levels, thereby providing maximum passive rf power transfer. These properties will be described in detail in the following sections. Finally, tunnel diodes used for envelope detection are often referred to as backward diodes, to distinguish them from conventional detector diodes in which the voltage polarity on the p n junction for high conduction is reversed, or from conventional tunnel diodes, compared to which the tunnel detector diode has a considerably lower peak current, maximum negative conductance, and peak-to-valley ratio. b. Detector Input-Output Characteristics

A general circuit model of a series-connected tunnel diode detector, shown in Fig. 35(a), includes arbitrary bandpass or low-pass coupling network N A and low pass coupling network Nv between the series-connected tunnel diode and the rf source and the video load, respectively. These networks

DC, VIDEO GROUND RETURN

RF SOURCE

W

3

TRANSFORMED RF SOURCE

\RF BROADBANDING

VIDEO / BROADBANDING

TRANSFORMED VIDEO LOAD

FIG.35. Tunnel diode detector representations. (a) General transmission configuration ; N , and Nv are rf and video broadbanding and stabilizing networks; nA2 and nV2 are rf and video impedance transformation ratios. (b) Two-frequency equivalent circuit; RF broadbanding-BPF: LA,= 1/oo2(C CJ, Ci, = l / c ~ , ~ L , , ., . . ; LPF: CAI= rf choke, C;, = L,, = . . . = (o.

+

01

3

588

H. C. OKEAN

provide frequency isolation between the source and load and shape the frequency dependence of the overall detection response. The detection mechanism is contained explicitly in the two-frequency, small-signal equivalent circuit of the detector, including noise sources, presented in Fig. 35(b).Under small-signal, square-law operation, I , is proportional to the available power PAfrom the rf source, that is, 11,1 = PPA,where /l is defined as the short-circuit current sensitivity and may be in terms of the circuit model of Fig. 35(b) as ( 140a)

where p’ applies for Gj > 0 ( l r A I 2 < I) and p- applies for Gj < 0 ( I r A I 2 2 1) ; rs = RsGj ; and QJR =

For the special case,

UGjI/C)t(1/IrsI)(1 + rsN1’2-

= Vp, Gj = 0, and

Vb

lrAI2

5 1 (due to RJ,

lGj’(h)l I1 - lrA(jw)121/2Rsw2c2.

P(O)

The parameter lGj’(Vb)1/2Gj(Vh)of significance for Eqs. (66), (67), and (139) by IGj’)/2Gj=

IVM

-

cv, + v, + 2Kl)(v, - V,)

(140b) Vh

# V, is given from

Gj(Vb),

for quadratic for quartic

Gj(V,).

(141a) (141b)

The input-output parameters of the detector model of Fig. 35 are given in terms of p(w) as follows: Net video current sensitivity : BL = I L P A

( 142a)

= ffv(w)P(4.

Net video voltage sensitivity ; YL

=

(142b)

vL/PA =

Net detector power-conversion gain : K L

=

PJpA = P L 2 R L = IHv(w)12B2pA

9

( 142c)

where H,(w) = lZL/lr) is the current transfer function of low-pass video filter Nv, and Rv = (1 + R,Gj)/Gj is the diode video resistance.

8.

TUNNEL DIODES

589

Examination of Eqs. (139H142) indicates that : (a) IGj'J and, hence, /I exhibit a sharp null at Vb = VM.However, a convenient bias point in the active region is at V, = Vb- = 21/M - V,. (b) K , varies linearly with PA,due to the square-law detection process. (c) The rf and video frequency dependence of transfer characteristics /IL, yL, and K , are fixed by the design of coupling networks N A and N , for specific IrA(jw)12 and H,(o) under the constraints imposed on r, by the tunnel diode parasitics and on H , by the parasitics associated with the video post-detection circuitry. The design of N A and N , utilizes standard passive filter or negative-resistance amplifier theory.55,57,59,99,'oo (d) For bias in the active region (Gj, R, < 0), the stability criterion of particularly in the rf and video Eq. (128) must be satisfied over all o d oR, passbands, for which the conditions ( R , R,)lGjl < I and RL < lRvl must be satisfied. Stabilization techniques and auxiliary stabiiizing networks similar to those presented in Fig. 21 may be utilized to satisfy this criterion. In addition, the use of a nonreciprocal device, e.g., an isolator, at the rf input is desirable. (e) The three most useful modes of detector operation are zero bias V,), Po can be made arbitrarily large by appropriate choice of R , for large values of rf transmission

H. C. OKEAN

Ultimate limit (BAm) of BAmas n

+

m

r,, = (R, - R J ( R , + RL)

where

B,, = cr-urrent

Representative" parameters for R,GM % 0.1, and materials GaSb Ge Gash at bias voltage Gj, Gj' utilize Eqs. (67) and (1391; f A O = 1 for center frequency. " P o in mA/mW. a

..

...

video bandwidth of H , Bob

RVGM

15 10 5

0.38 0.40 0.41 kb =

vb

< V,,

0

w = ( w / w o )-

Bob

131fso

8.4r2,, 4.2t:, Vb =

RVGM

Bob

RVGM

1.9 - 1.94 - 1.94

9.7 6 3

w

-

vb- = 2v,

-

vp

(wo/w) for bandpass and w / o o for low-pass

CQ

2

Vb

=

v,

ItA(jw)i2,where wo

Po is the rf

2Z 3

r 0

s V

w

592

H. C. OKEAN

-

gain t i o . In particular, values of Po 2000 mA/mW, feasible with reasonable values of t i o 100, are more than two orders of magnitude better than obtainable with conventional detector diodes. (b) Assuming that transformed source impedance R, may be freely chosen for a specified t i o , Po is seen, at a given V,, to be independent of diode impedance level GM, but to increase with decreasing active voltage swing (V, - V,). Since the latter is a function of the diode semiconductor material, GaSb detector diodes yield the largest Po, followed in decreasing order by Ge and GaAs diodes. (c) For a given functional dependence Cj(V,), the diode video resistance at a given Vb is essentially independent of diode material but is inversely proportional to GM. Zero bias video resistances of 250 ohms, readily obtainable in typical detector tunnel diodes ( G M E 0.01 mho), are an order of magnitude less than those obtained in conventional detector diodes, thus making for easier rf and video matching and, as will be demonstrated, better noise performance. (d) Values of video current transfer function Hvo approaching unity under passive bias and arbitrarily large under active bias render the previous remarks on Po also applicable to overall detector transfer functions P,, yL , and K , [Eq. (142)l. (e) Half-power, low-pass or bandpass rf bandwidths for fl( jco), obtainable under active bias, are comparable with those obtainable with similarly broadband reflection-type tunnel diode amplifiers of gain IrA12 [Eq. (94)]; those obtained at zero bias are of the order of Gj(0)/n(C + C,) 5 2-10 GHz, and those obtained under matched peak-bias operation are of the order of (471R,C)- 6 5-25 GHz. (f) The low-pass, half-power video bandwidth Bv of IHv(jo)12, and hence of PL, y,, and K L , is primarily a function of the transformed input time constant R L C L of the post-detection circuitry, which is typically of the order of 10-1000 times C/lGjl. Therefore, B, will be of the order of 5 MHz to 2 GHz, thus determining the upper bound on the frequency of readily detectable sinusoidal envelope modulation (fm < B,) and the lower bound on achievable envelope pulse rise time (T O.5/Bv). The above results lead to the preliminary conclusion that tunnel diode detector operation in the zero-bias mode combines moderately high sensitivity, wideband operation with relative ease of rf and video matching and general circuit simplicity. Active bias operation can yield extremely high detection sensitivity, but at the cost of reduced bandwidth and increased circuit complexity required for stabilization and nonreciprocal isolation, and is extremely critical with respect to circuit adjustments. (Active-bias operation results in a circuit that functions as an rf amplifier, a detector, and a video amplifier in cascade.) Peak bias operation yields extremely high

-

-

8.

TUNNEL DIODES

593

bandwidth at slightly reduced detection sensitivity, but requires more difficult rf and video matching. A further comparison of the relative advantages of these three modes of detector operation requires an examination of the detector noise performance as presented in the next section. c. Detector Noise Performance

The noise performance of a square-law envelope detector such as the tunnel diode detector cannot be meaningfully characterized in terms of rf-to-video noise figure as in a linear converter, due to the power-level dependence of rf-to-video power transfer [Eq. (142c)l and the resulting suppression of rf input noise arising from the square-law detection process. Therefore, the meaningful sources of noise in an envelope detector originate in the video passband and are therefore best characterized by the detector tangential signal sensitivity 4 , defined as the input power level at which a specified video signal-to-noise ratio U , is obtained across G,, where

and where (I&)av is the total video mean-square short-circuit noise current. By convention, Uv is chosen as 2.5 to define the tangential sensitivity, whereas the input power P, at which Uv = 1 is referred to as the minimum detectable signal. The significant sources of video noise which define tangential sensitivity P, include the shot-noise contribution of the tunnel diode junction, the thermal noise contribution of R, and of the equivalent video load GL, and the equivalent excess noise contribution of the post-detection amplifier under input termination R,, characterized by input noise temperature TPA.The flicker (l/f) noise contribution of the tunnel diode may be neglected’’ 1 , 1 5 z over the useful portion of the video passband ( f > 1 kHz) for Vb < V,, thus representing an important advantage of tunnel diode detectors over their conventional counterparts. Utilizing the equivalent circuit of Fig. 35(b), is given by

where BVN is the effective video noise bandwidth, T the ambient temperature, and G N = efbN/2kTwith I,, given by Eq. (70), or

GN = [5800/T (“K)]zb~0th{[5800/T(“K)]Vb (volts))

594

H. C . OKEAN

Tangential sensitivity P, is then obtained from Eqs. (142a),(143), and ( 1 4 4 ) at Uv = 2.5, yielding =

5(kTBVNGNT)1’2/P0

(145)

9

where G,T

= [(GN

+ rsGj)/(I + rJ2I + G d 1 + (TiA/T)I.

The lower the value of P,, the more sensitive the detector. Equation ( 1 4 5 ) may be formulated in terms of the “figure of merit” M of the tunnel diode, which is a measure of the quality of an envelope detector diode and which is defined within the video passband in conjunction with a noiseless video amplifier as

where G,‘

= (GN

+ rsGj)/(I + r,)’ .

(This definition is somewhat more general than that used for conventional detector diodes, in which G N = G, and GN’ = Gv, thus yielding a meaningless result at Gj = 0.) Therefore, P, may be expressed as fl(pw) = 3.16

X

1 0 - 4 { [ B ~(HZ)](GNT/GN))~’~/[M ~

(147)

Representative values of M and PIfor each of the three modes of detector operation and for each of the three diode materials are given in Table XVII TABLE XVII“ Value of parameter for: Parameter

Vb

GaSb

M (W- *)

0

260

VP

1 20

4 (dBm)

b‘

4300

0

- 59.1

VP Vb -

- 55.8 -71.3

Ge 128 67 2300 -56.1 - 53.3 - 68.6

GaAS 63 27 820

- 53 -49.3 - 64.1

Here, GM = 0.0025 mho; GNT% G,’; tio= 10 at Vb- and 1 at = 1 MHz; &- = 2VM - Vp.

& = 0, Vp; B,,

8 . TUNNEL

DIODES

595

based on calculations using Table XVI, Eqs. (144H147), and Eqs. (67) and (70) for GN', under the limiting assumption of a negligible video amplifier noise contribution ( G N T z G"). Table XVII shows that: (a) At a given operating point and impedance level, GaSb tunnel diodes make the most sensitive detectors from a signal-to-noise standpoint (lowest P,, highest M ) , followed in decreasing 3-dB steps by Ge and GaAs diodes. The superiority of GaSb diodes is due to both their higher conductance curvature and hence higher current sensitivity and to their lower noise contribution GN'. (b) The maximum tangential sensitivity (lowest P,) for a given material and impedance level is obtained under active bias conditions, typically at V,- = 2vM - V,, due both to the increase by t i o > 1 in current sensitivity provided by the rf preamplification, and to the slightly lower noise contribution G,' obtained under active bias. The improvement in tangential sensitivity (and M ) under active bias is at least by t i o > 1, whereas, under passive bias, the tangential sensitivity at peak bias is about 3 dB poorer than that at zero bias. (c) The tangential sensitivity (and M ) for a given material and bias point is inversely proportional to G M , assuming rf matching to a desired tio. The values of tangential sensitivity presented here for tunnel diode detectors are generally superior by 3-20dB to those exhibited under similar conditions by conventional diode detectors, whereas the values of M are at worst comparable and at best superior by an order of magnitude. d. Dynamic Range

The tunnel diode detector is basically a small-signal device, with deviation from square-law operation occurring at large signal levels due to higherorder terms in the series expansion of Eq. (1 38a). The first higher-order term to perturb the rectified current is the fourth-order term ( VO4/24)[d4f( V ) / dV4Ivb sin4 wr ,

(148)

which, using Eqs. (138H1401, yields a large-signal to small-signal current sensitivity ratio

Equation (149) indicates that the detector undergoes compression of current sensitivity with increasing power levels, leading to eventual limiting. In particular, the input power level at which 1-dB current-sensitivity

5%

H . C. OKEAN

compression occurs is given by p y

15.6GM(Vb

-

VM)2(Vv

- Vb)3/(Vv

-

Vp)4003

(1 50)

where the following values of P, are obtained under the conditions of Eq. (148): For V, = 0, P, = - 9, - 5.7, + 0.3 dBm for GaSb, Ge, GaAs diodes ; V, = V,, P, = - 15.3, - 11.9, - 5.9 dBm for GaSb, Ge, GaAs diodes; v b = 2VM - v,, P, = - 35.5, - 32.4, -26.4 dBm for GaSb, Ge, GaAs diodes. It is apparent from Eq. (150)that the zero bias mode of detector operation offers the best large-signal-handling capability, whereas the high-sensitivity active mode of operation offers the poorest. Furthermore, as in the case of amplifiers, at a given detector bias, GaAs tunnel diodes exhibit the highest input saturation level, followed in decreasing order by G e and GaSb diodes. e . Comparison of Modes of Tunnel Diode Detector Operation

Comparison of the performance parameters of tunnel diode detectors operated in the zero, peak, and active bias modes and utilizing GaSb, Ge, and GaAs tunnel diodes, as presented in the preceding sections, leads to the conclusion that the zero bias mode is preferred for high-sensitivity, lownoise, wideband, high-dynamic-range operation. Ultra-high-sensitivity, lownoise operation is obtainable in the active bias mode, in which the diode provides the combined functions of rf and video amplification and square-law detection. However, in this case, bandwidth and signal-handling capabilities are reduced to values comparable to those obtained in tunnel diode amplifiers. In general, the results obtained in the active mode are similar to those exhibited by conventional detectors preceded by tunnel diode preamplifiers. Finally, as in the case of amplifiers, GaSb diodes provide the lowest-noise, highest-sensitivity detection and GaAs diodes the largest signal-handling capability, with Ge diodes offering a judicious compromise between these aspects of performance.

5 Examples of Microwave Tunnel Diode Detectors Many tunnel diode detectors have been fabricated in the rf and microwave frequency ranges. The state of the art in practical tunnel diode detectors44,114,149-153 is summarized in Table XVIII, in which it is seen '

that detectors operated in the zero bias mode have achieved tangential sensitivities better than -60 dBm from U H F through X band (1-MHz video bandwidth) and as high as - 55 dBm at 35 GHz. In addition, tangential sensitivities as high as - 75 to - 80 dBm have been achieved over frequency bands in the 3-7-GHz range (B, = 1 MHz, Ge diode) under active bias conditions. The wide rf bandwidth capability of the tunnel diode detector has been verified in practice, with half-power, low-pass bandwidths of greater than 10 GHz and bandpass bandwidths of better than 5 : 1 obtained

TABLE XVIII

MEASURED PERFORMANCE CHARACTERISTICS OF EXPERIMENTAL TUNNEL DIODE DETECTORS

151 0 0.1-8.0 Ge I1 200 - 54 8.0 2 I20 - 23

152 0.07 6.0 Ge 120

(l.sV,) 6.0

Ge 950 10,Ooo - 85 0.25

loo0 - 76

0.1

150 0 9

Ge 12 480 - 60

-

- 57 -

~

1.o

1

0

0

-

44 0 70 GaAs

.o

149 0 50

114 0 0.01-8.0

144 0 2-18

- 55

- 44

8 2

16 2

Ge 4 - 46 0.04 >> 290 -2

-

~

0

-7

598

H. C. OKEAN

in the zero bias mode and bandpass bandwidths of about 20 % in the active bias mode.

TUNNEL DIODEAPPLICATIONS 19. MISCELLANEOUS SINUSOIDAL a. General Network Synthesis

The small-signal negative-resistance property of the tunnel diode has stimulated investigations' 2,1 54-1 5 9 on the use of tunnel diodes in general frequency-domain network synthesis. In particular, numerous treatments have been presented on the synthesis of k R, L, C networks in which tunnel diodes provide the - R elements. Each tunnel diode is usually represented in somewhat oversimplified fashion by its parallel - G , C junction immittance, although some treatments consider the series parasitics R , and L, as well. The network functions dealt with in these generalized synthesis procedures include, in addition to the previously discussed power amplification, voltage and current amplification, realization of nonminimum phase ladder networks, realization of high-selectivity low-loss filters, and the realization of multiport networks with specified impedance or admittance matrices which are not possible with purely passive RLC elements. The details of these various synthesis procedures are more relevant to abstract network theory than to the particular limitations imposed by the tunnel diode as a device, and are therefore beyond the scope of this chapter. However, several basic limitations on the physical realizability of general networks employing tunnel diodes are imposed by the tunnel diode device parameters. These usually relate to the complex natural frequencies p I = uI jm,, 1 = 1,2,. . . , exhibited by a general network N terminated in a tunnel diode, where p l has uI 2 0 corresponding to steady or growing oscillatory network responses of the form exp[(a, jo,)t].These limitations may be summarized as follows in terms of reciprocal network N .

+

+

(a) N is physically realizable provided

G/C
V,, Fig. 14(b)] or sharp spikes in a corresponding region of the d21,,/dVb2character is ti^.'^^.' 'O The voltages corresponding to these spikes are a measure of the energies of the various phonons in the semiconductor. (b) The effective mass of the charged carriers flowing through the diode junction has been determined from the value of the Bohr magneton, obtained via the de Haas-van Alphen effect by observing the frequency of current oscillations exhibited by the tunnel diode I- V characteristic in a strong magnetic field.' ' , ' 7 0 * 1 " (c) The tunneling probability has been determined by measuring the forward and backward components of diode current." (d) The width of the forbidden band gap has been measured' 1 1 6 5 , 1 7 1 at varying values of hydrostatic pressure on the diode junction. (e) The existence of deep traps in the forbidden band gap of the semiconductor has been verified and their energy levels located by noting the existence of subsidiary current maxima in the valley region of the currentvoltage characteristic,' 1 , 2 2 , 1 7 0 * 17 2 an approach known as "tunnel spectroscopy." I

i'

FIG.37. Basic tunnel diode switching circuit configuration.

602

H. C. OKEAN

VII. Tunnel Diode Applications in Pdse and Digital Circuits 20. GENERAL PROPERTIES

OF

TUNNEL DIODES IN DIGITAL CIRCUITS

a. Switching Properties of Tunnel Diodes

The tunnel diode, in addition to its wide application in sinusoidal circuits, is extremely useful as the active element in pulse and digital c i r c ~ i t s , ~ , ' l4~ " ~ due primarily to : (a) the extremely high-frequency, high-speed (subnanosecond) capability of the tunnel-diode junction, which is an order of magnitude faster than other existing switching devices ; (b) the existence of a highly nonlinear current-voltage characteristic, in which the junction voltage is a double-valued function of current over a considerable range of its positive region ( I , < I , < I , and V, > 0), thereby making binary (two-state) operation possible; and (c) the extremely low power levels requiring to switch between the two bias states. The above properties give rise to a large variety of large-signal, nonsinusoidal tunnel diode circuits generally classed as pulse, digital, or switching c i r c ~ i t s . ' ~ ~Th - ~ese ' ~include free-running and triggered waveform generator^,'^^-"^ binary logic circuits such as flip-flops and 96 sequenand memory tial circuits such as timing circuits and c i r ~ u i t s . ~ ~The ~ - ~circuit ' description and design procedure pertaining to the many variations of each of these circuit types are beyond the scope of this chapter. However, a general circuit description of some of the more common tunnel diode digital circuit functions will be presented following a description of the fundamental modes of tunnel diode switching operation and a presentation of the basic limitations on the corresponding switching parameters. I. Aleksander and R. W. Scarr, J. Brit. Inst. Radio Eng. 23, 177 (1962). J. C. Balder, Tvdschrift Ned. Rudiogenoot 26, 167 (1961). 1 7 6 W. F. Chow, IRE Trans. Electron. Computers EC-9,295 (1960). ' 7 7 R. S. Foote and W. V. Harrison, IRE Trans. Circuit Theory CT-8.468 (1961). 1 7 ' H. Fukui and T. Matsushima, J . Inst. Elec. Commun. Eng. (Japrrn)44,479 ( 1961). "9 A. Hemel, Proc. Nut. Electron. Con$ 17, 163 (1961). ''O G. B. Herzog, Onde Elect. 41, 370 (1961). M. H. Lewin, A. G. Samusenko, and A. W. Lo, Dig. 1960 fnr. Solid State Circuirs Con$, Philadelphia, Pennsylvania, p. 10 (1960). T. A. Rabson, Nucl. Instrum. Methods 12, 127 (1961). C. A. Renton and B. Rabinovici, Proc. IRE 50, 1648 (1962). M. Schuller and W. W. Gartner, Proc. IRE 49, 1268 (1961). J. J. Gibson, G. B. Herzog, H. S. Muller, and R. A. Powlus, Dig. 1962 inr. Solid State Circuits ConJ V, 54 (1962). E. Goto, K. Murata, K. Nakazawa, K. Nakagawa, T. Moto-oka, Y. Matsuoka, Y.Ishibashi, H. Ishida, T. Soma, and E. Wada, IRE Trans. Electron. Computers EC-9. 25 (1960). M. S. Axelrod, A. S. Farber, and D. E. Rosenheim, I B M J. Res. Develop. 6, 158 (1962). 74 75

8. TUNNEL

DIODES

603

h. Fundamental Modes ojswitching Operation

The fundamental modes of tunnel diode switching operation are defined in terms of the simple, basic switching circuit configuration of Fig. 37, in which it is assumed that a trigger pulse input available directly across the tunnel diode is capacitively coupled from a constant-current (high-impedance) source. In the most general case, the dc bias voltage is applied to the tunnel diode anode through resistor R , and inductor L,, whereas the tunnel diode cathode is connected to ground through resistor Rb'. The output of the switching circuit, containing the desired pulse waveform and/or digital information, is taken as the voltage across, or current flow through, load resistor R , across the tunnel diode. The switching modes to be considered are those exhibiting bistable, inverted bistable, monostable, and astable (relaxation oscillation) operation with respect to two operating points on the diode current-voltage characteristic, one in the "low-voltage region" ( V b < V,) and one in the "forwardvoltage region" (Vb > V,). The static load lines, dynamic load trajectories, and output waveforms corresponding to each of these modes of operation R. H. Bergman, I R E Trans. Electron. Computers EC-9, 430 (1960). W. N. Carr and A. G . Milnes, / R E Trcrns. Electron. Computers EC-11. 773 (1962). 1 9 0 P. Franzini, Rev. Sci. Instrum. 32, 1222 (1961). 19’ F. H. Mitchell. Jr.. Elect. Ind. 21. 105 (1962). 19’ Y. Komanaiya, Dig. IY63 I n t . Solid Sture Circuits Conf V1, 24 (1963). 193 J. F. Kruy, Dig. 1963 Int. Solid State Circuits Con/: VI. 28 ( I 963). 194 H. S. Miller and R. A. Powlus, R C A Reu. 23.497 (1962). 19' C. A. Renton and B. Rabinovici, I R E Trans. Electron. Computers EC-11. 213 (1962). 196 G . P. Sarrafian, IRE I n t . C o w . Rec. ( P i . 2) 9. 271 (1961). 197 B. E. Sear, IRE Trans. Circuit Theory 10. 48 (1963). 1 9 * J. Nagumo and M. Shimura, Proc. IRE 49, 1281 (1961). 199 E. Iwahashi, J. Inst. Elec. Commun. Engr. (Jupun)44,1199 (1961). R. A. Kaenel. Proc. IRE 49, 622 (1961). ' 0 1 L. U. Kibler, Proc. / R E 49. 1204 (1961). 'O' V. Uzunoglu, Proc. / R E 49. 1440 (I9611. ' 0 3 F. P. Heiman, Proc. / R E 49, 1215 (1961). '04 K. Hillman, G . T.& E . Res. Develop. J . 1. 87 (1961). '05 R. A. Kaenel, 1960 / R E Wescon. Con[>. Rec. ( P t . 3), 53 (1960). 'Oh B. Rabinovici, Proc. / R E 50,473 ( 1962). '07 P. Spiegel, / R E Int. Conr. Rec. (Pt. 2) 9, 164 (1961). ’08 G . J. Veth, Solid Stare Design 4, 30 ( 1963). ' O ' ) D. L. Berry and E. A. Fisch, Dig. 1961 /nr. Solid State Circuits Con/. IV. 112 (1961). 210 J. C. Miller, K. Li, and A. W. Low, Dig. Int. Solid Stare Circuits Con$, Philadelphia, Pennsyluania 111. 52 (1960). ’I1 J. Y. Payton, 1962 Wescon. Conrr. Rec. ( P t . 4), 2.1-1 (1962). ' 1 2 R. A. Kaenel, I R E Trans. Electron. Conipirters EC-10, 273 (1961). ' I 3 T. Kiyono, K. Ikeda, and H. Ichiki. / R E Trans. Electron. Computers EC-11, 791 (1962).

lS9

’I4

See P. S p i e g e ~ . ' ~ ~

604

H . C. OKEAN

are depicted in Fig. 38(a-d), respectively. A brief qualitative description of each of these modes is as follows. (a) In the bistable mode of operation (Lb,Rh‘ = 0), bias and load resistors Rb and R L and bias supply voltage E b b are chosen sufficiently large (R,, R L >> 1/GM; E b b >> V,) to yield an essentially constant-current bias supply, as exemplified by the near-constant-current static load line superimposed on the diode current-voltage characteristic [Fig. 38(a)]. Therefore, two stable static operating points exist, as defined by the intersections of the static load line with the positive-slope segments of the I-I/ characteristic at 1/b1, (1/bi < v,) and Vb2, (Vb2 > 1 precludes the establishment of a stable operating point within it. Bistable switching occurs when a positive (negative) current pulse of amplitude I, > I, - I b l (I, > - I,) is applied to a tunnel diode initially biased at V b l , I b 1 ( & , I b 2 ) . The tunnel diode junction voltage, in response to + I, (IbZ - I,), is forced into the forward (low) instantaneous current voltage region, arriving and remaining at the second operating point v b z , 1 b 2 (Vb1, I b 1 ) upon removal of the trigger pulse. The dynamic currentvoltage trajectory and output waveform describing this switching process is shown in Fig. 38(a). The tunnel diode may be switched back to its initial operating point by application of a trigger pulse of opposite polarity. (b) The inverting bistable mode of operation (Lb = 0, R, = a)differs from the normal bistable mode by the inclusion of R,’ > 0 and the choice of E b b > V, and Rb > considerably lower than in the previous case, such that, for bistable operating points ( I b l , Vb1) and (IbZ, 1/b2), I b 2 2 I , is considerably less than I,,,, as shown by the load-line, I-1/ superimposition in Fig. 38(a).The switching process between the two operating points under positive and negative triggering, as defined by the trajectory in Fig. 38(a), is similar to that characterizing the normal bistable process. The output is taken across R;, resulting in the output waveform shown in Fig. 38(a). (c) Dynamic monostable switching ( R h ’ = 0, L b > 0) occurs when Rb > l/GM and Eh,, are chosen such that only one stable operating point, that is, only one intersection with the positive-slope I-V characteristic, is obtained, at or l/b2, I b 2 , as shown in Fig. 38(b). The monostable switching process occurs upon the application of a positive (negative) trigger pulse of amplitude I, > I, - I,, (I, > - I,) to a tunnel diode initially biased at I,, Vb, ,(I,,, &) as shown in the trajectory on Fig. 38(b). Junction voltage V, is initially forced into the forward (low) voltage region, arriving at the second operating point Vb2, I,, (Vb, ,I, I ) upon removal of the trigger, and simultaneously inducing an initial voltage IV,l = 1/b2 - Vb1 across L b . As

8.

605

TUNNEL DIODES

TIME-

(a)

TIME

--D

FIG.38. Trajectories and output waveforms of various modes of tunnel diode switching. (a) Bistable modes; bistable mode, V, is output for I,, = I , , , R,' = 0; inverting bistable mode, V( is output for I,, >> I,,, R , = cc~.(b) Monostable mode. (c) Astable mode.

IVJ decays toward its zero steady-state value, however, Vb and I , decrease through V,, I , (increase toward V,,, I & , at which point v b is forced into the low (forward) voltage region, inducing another voltage step across L b . As the latter decays toward zero, Vb and 1, increase (decrease) to the original

606

H. C. OKEAN

operating point Vb1, (VbZ, I,,), thereby completing the monostable cycle. The duration of the cycle is determined by the time constants associated with L b . The switching trajectories and output waveforms corresponding to stable bias point Vb,, and VbZ, I,, are presented in Fig. 38(b). (d) Astable oscillation (R,,' = 0, L b > 0) occurs when Rb > l/GM and Ebb are chosen so that no stable operating points exist on the positive-slope segments of the I-V characteristic but two unstable operating points exist in the negative-conductance region (V, < < V,) as shown in Fig. 38(c). However, as Ebb is turned on and V,, I b increase through (V,, I,), V, is forced I,. Then, the decay toward zero into the forward-voltage region at V,, I , of the voltage step (V, - V,) across Lb causes Vb, I, to decrease from V,, I , toward K , I , , a t which point Vb is forced into the low-voltage region at V,,I, I,. Finally, the decay of the second voltage step V, - V, across L b causes Vb, I b to increase from V,, I, through V,, I , , at which point the process repeats and a single cycle of oscillation is completed. The complete oscillation trajectory on the I-V characteristic and the corresponding output waveform are presented in Fig. 38(c).The period of oscillation varies essentially directly with &.

-

-

Two interesting additional variations on these modes of operation depart from the basic model of Fig. 37. The first utilizes a balanced pair of tunnel in a bistable switching circuit as shown in Fig. 39(a). The composite I-V characteristic and switching trajectory corresponding to this configuration is shown in Fig. 39(b), and the output waveform in Fig. 39(c). A variation of the astable relaxation oscillator utilizes a length of shortcircuited transmission line' 98 as shown in Fig. 40(a). Relaxation oscillation results from regenerative switching between states (Ib1, Vb1) and ( I b 2 , Vb2) brought about by propagation and rereflection of prior waveform components on the transmission-line length, as shown in the oscillation trajectory of Fig. 38(c), and the output waveform of Fig. 40(b). In this case, the period of oscillation is given by twice the line length-to-propagation velocity ratio. The switching parameters describing these processes include switching time, triggered or astable repetition period, switching current gain, and trigger power. The limitations imposed upon these parameters by the tunnel diode will be described in the following section. Fudamentaf Limits on Tunnel Diode Switching The basic limits on the basic tunnel diode switching parameters may be formulated in terms of the tunnel diode device properties by applying a piecewise linear transient analysis3 to the circuit configuration of Fig. 37. The details of the analysis tend to obscure the degree of dependence upon the tunnel diode parameters, and are therefore beyond the scope of this chapter. Therefore, only the pertinent results are presented as follows.

c.

8.

-E bb -V- o

607

TUNNEL DIODES

0

-vo-

"OoP-+

(C

1

FIG. 39. Balanced pair bistable tunnel diode switch. (a) Circuit schematic. (b) Switching trajectory; V, = E,, - Vbl = V,,, - Ebh.(c) Output waveform.

The switching time, that is, the time required to switch between the two bias states ( 1 b 1 , Vb1) and ( 1 b 2 , &,2) under constant-current triggering, is limited by the time required to charge terminal diode capacitance C, and is

608

H. C. OKEAN TRANSMISSION

Ebb

0

1 2

T

Z 2 T 5 T 2 2

-

1-

FIG.40. Tunnel diode relaxation oscillator utilizing short-circuited transmission-line section. (a) Circuit schematic; (b) output waveform.

expressible, with reference to Figs. 37 and 38, as

The general solution of Eq. (152), using a polynomial representation for I,,(Vb), is quite complicated and often can only be obtained numerically. However, an approximation which leads to the ultimate limitation imposed

8.

TUNNEL DIODES

609

by the diode parameters yields

under the assumptions: I , '2 I, over '2 1, over T2,1 ; R b , R,' >> and R , ; I,, + 1 1 1 '2 I , ; I,, + 112 '2 I,: Vb, % Vp; Vb2 % VF Vb > Vv at which I , = I,. Equation (153) shows that in order for a tunnel diode to have a potentially high switching speed capability, it must possess a high switching figure of merit lp/C, a high peak-to-valley ratio lp/lv, and a low case capacitance C , . Accordingly, a typical range of values of (qw)min valid for GaAs, Ge, and GaSb tunnel diodes, obtained from Table IV in the limit C, '2 0 and V, - Vp '2 2Vv, is given by Rb',

( 7&Jmin % 0.1-0.5

nsec.

( 154)

The difference in switching speed capability between diodes of the three semiconductor types is small because of the approximately linear relationship between Vv and Z,/C for various materials. A further twofold improvement in switching speed may be obtained by using a bistable balanced pair3 (Fig. 39). The triggered or astable repetition period TRcharacterizing the monostable or astable switching processes is the time required to complete a single switching cycle (Fig. 38), or TR= Tl,2 + Tz.l+ TZd+ TI,

+

TRx Tp,F

&,L

+ TF.v+ TL,,

in the monostable case;

in the astable case:

xi

( 155)

where T , jis the switching time from I b i , to I,,, xj(i# j = 1,2); Tp,F the switching time from I,, Vp to I,, V,: &.L the switching time from I,, Vv to I , , V,. TL,,the inductive rise time from I L ,VL to I,, V,; TF,v the inductive decay time from I,, V, to I,, V v : T'l,rthe inductive rise time from I,, VL to I , , , Vb1 or from I,,, V,, to I , , V,; and Tz,dthe inductive decay time from I b I , Vbz to I,, V, or from IF, VF to I,,, Vbz. The switching times T,,z, T 2 . , , Tp,F and &,L are obtainable from Eqs. (152) and (153), whereas the inductive rise and fall times TF,v, TL,,,T12,dr and T; are approximated from the piecewise-linear transient analysis ( R b ' = 0) as ,T

610

H. C. OKEAN

where L, = Lb

+ L, and

In the limit of satisfaction of the monostable and bistable biasing conditions, the dc load line is tangent to the current-voltage characteristic at V,, I,, SO that, for R, large,

+ KNM;

Rb

Ebb

VM

RFI

(liGM) + [(vF -

RLt ?Z(l/GM)

~/GM;

V~)/(lp -

+ (Vp/lp)

I,)]

(l/GM)[l

K ~ J ZM IM/GM, (l/GM)(l

+ kG),

(157)

+ (Vp/K)kil,

where k , = G,[(K - Vp)/(Ip- I,)] x 2.12 for quartic Gj(Vb), VF % 2Vv V,, lv/lp 6 V). An additional possible advantage of the tunnel diode oscillator is its relatively high spectral purity (excess noise temperature ratio 2-4), compared with the values of 3-5, 10, 100, and 1000 characterizing transitor oscillator-multiplier chains, bulk LSA, bulk transittime, and avalanche oscillators, respectively. The above comparison suggests that the role of the tunnel diode oscillator will be limited to applications where dc power and voltage are at a premium and where an rf or microwave signal of high spectral purity rather than high power level is required. Possible applications along these lines include use as a local oscillator for microwave mixers and as a frequency-locking source for high-power, noisier solid-state oscillators.’ 35-138 Finally, tunnel diodes will play a large role in switching and digital circuit applications due to their high switching speed (one to two orders ofmagnitude better than other devices) and low power drain.3-’7”214 In summary, it has been shown that the major present and future circuit applications of tunnel diodes are in low-noise, broadband, low-level microwave amplifiers in the 2-25-GHz frequency range, in ultrasensitive rf and

8.

TUNNEL DIODES

621

microwave low-level detectors at frequencies as high as 75 GHz, and in subnanosecond digital and switching circuits.

23. ROLEOF INTEGRATED CIRCUITTECHNOLOGY IN POSSIBLE TUNNEL DIODEAPPLICATIONS The rapid expansion of the integrated circuit technology into the fields of rf and microwave sinusoidal circuitry and ultrahigh-speed digital circuit applications has significant implications with respect to the maximum performance capabilities of tunnel diodes in the various circuit applications described in the previous sections. The term “integrated circuit” as used here is defined as having the following properties :

(a) The integrated circuit is essentially planar, consisting of thin or thick film and discrete circuit elements deposited and/or mounted on one or both faces of a dielectric or semi-insulating semiconductor substrate, resulting in hybrid or monolithic circuits, respectively. (b) In the monolithic integrated circuit, the conductor pattern is deposited on a semi-insulating semiconductor substrate, with discrete components such as semiconductor devices, resistors, and capacitors formed by preferentially doping the substrate over selected localized areas. (c) In the hybrid integrated circuit, the conductor and resistor patterns are deposited on a dielectric substrate, with discrete components such as semiconductor devices, ferrites, and capacitors bonded to the conductor pattern. (d) The substrate is enclosed in a metallic housing of geometry compatible with the desired transmission-line medium. Potential improvements in the the performance capabilities oftunnel diodes in the various circuit applications described in the preceding sections are obtainable by utilizing integrated circuit realizations in conjunction with unencapsulated tunnel diodes [Fig. 13(a-e)]. These improvements arise from the substantial reduction in the tunnel diode parasitics, particularly L , and C , , obtainable in an unencapsulated configuration, and from the ability to reduce the effect of the remaining parasitics by introducing external thin-film microcircuitry (stabilizing and tuning networks, etc.) in extremely close electrical proximity to the tunnel diode j u n ~ t i o n . ~ ~An , ~additional ’ . ~ ~ source of performance improvement in the integrated circuit realizations is the elimination of all superfluous connectors, transmission-line lengths, impedance transformations, etc., associated with the external circuitry?8.2 ” An estimate of the degree of reduction of tunnel diode parasitics obtainable with a n unencapsulated diode in an integrated circuit realization, and of the resulting degree of improvement in performance parameters in the various

”’ J. D. Welch, I E E E Trans. Microwave Theory Tech. MTT-18, 1077 (1970).

622

H . C . OKEAN

circuit applications, may be summarized as follows (Table IV) : (a) Series inductance L, may be reduced to about 0.1 nH and parallel capacitance C, to virtually zero in an unencapsulated diode as compared with minimum values of about 0.2 nH and 0.2 pF, respectively, in an encapsulated diode. (b) The virtual elimination of C , implies a potential increase in the ultimate diode-limited bandwidth capability of a tunnel diode amplifier, mixer, and detector by a factor as large as two, although limitations imposed by external circuit elements usually reduce this degree of improvement. (c) The elimination of C , also implies a corresponding improvement in the ultimate switching-speed capability of tunnel diode digital circuits, and an increase in the tunability range of tunnel diode oscillators. (d) The halving of L, makes it possible to double the diode negativeconductance level Gwfor a given degree of stability margin, thereby doubling the potential large-signal-handling capability of tunnel diode amplifiers, mixers, and detectors and the potential power output of tunnel diode oscillators. (e) The reduction of L,, elimination of C,, and ability to realize microminiature external biasing, stabilizing, and tuning circuits arbitrarily close t o the tunnel diode junction make it practical for the first time to realize an N x N tunnel diode array (Fig. 28),'06 which would increase by N 2 the power output capability of a tunnel diode oscillator and the large-signalhandling capability of tunnel diode amplifiers, mixers, and detectors. An e ~ a m p l e7,38 ~ ~of. ~an integrated tunnel diode device including stabilizing, bias isolation, and tuning circuitry in conjunction with a beam-lead tunnel diode and suitable for incorporation in an N x N tunnel diode array is shown in Fig. 46. A family of tunnel diode amplifiers utilizing these devices has been successfully con~tructed,~' yielding the results previously described in Table IX.

24. CONCLUSIONS The tunnel diode is a degenerately doped p-n junction device which, by virtue of its nonlinear negative resistance and low parasitic properties, has wide potential application in high-frequency sinusoidal circuits such as amplifiers, oscillators, converters, and detectors, and in high-speed pulse and digital circuits. The performance limitations on each of these circuit applications derive from the fundamental device parameters of the tunnel diode, as obtained from a consideration of the quantum-mechanical tunneling through the p-n junction and of the perturbing effects of the device parasitics. In particular, it has been shown that these device properties are primarily functions of the semiconductor material used (GaSb, Ge, GaAs) and the

8.

623

TUNNEL DIODES

w 0.01 in.

D -I

~TUNING+STABlLIZING+T

I

I

I

I cN

I

I

I

I

I I

LN I

(b) FIG. 46. Integrated tunnel diode device. (a) Physical configuration. (bj Circuit schematic (after Okeanss); C , = dc block; coo = I/(L,C,j”* >> l/(L,Cb)”z.

junction size or junction impedance level, Representative performance limitations imposed by these device parameters result in tunnel diode amplifiers of 3-6 dB noise figure and oscillators of better than 1 mW power output at frequencies up to 20 GHz, detectors of tangential sensitivity better than - 50 dBm up to 75 GHz, and subnanosecond switching and digital circuits. These levels of performance, already competitive, under certain conditions, with those obtainable from other solid-state devices in similar circuit functions, can be further improved by utilizing integrated circuit technology in conjunction with unencapsulated tunnel diodes.

624

H. C. OKEAN

ACKNOWLEDGMENTS The author wishes to express his appreciation to Airborne Instruments Laboratory for providing the facilities and environment necessary for this effort. In addition, he gratefully acknowledges many helpful discussions with S. Okwit, P. P. Lombardo, E. W. Sard, and A. N. Leber relative to the preparation of this manuscript. Finally, the author wishes to thank Miss P. Zinn, Miss N. McKee, Mrs. L. Kerner, and Mrs. J. Larsen for typing the manuscript and the Illustrating Group in the Publications Department of Airborne Instruments Laboratory for preparing the illustrations.

CHAPTER 9

Silicon Carbide Junction Devices Robert B . Campbell Hung-Chi Chang

I . INTRODUCTION. . . . . . . . . . . . I1 . SILICONCARBIDE AS A SEMICONDUCTOR MATERIAL. 1 . Physical and Chemical Properties . . . . . 2. Methods of Preparation . . . . . . . . 3 . Semiconductor Properties . . . . . . . . 111. DEVICETECHNIQUES. . . . . . . . . . 4 . Introduction . . . . . . . . . . . 5 . Diffusion . . . . . . . . . . . . 6 . Mechanical Processiri,q . . . . . . . . 7 . Etching . . . . . . . . . . . . . 8 . Oxidation . . . . . . . . . . . . 9 . Alloying . . . . . . . . . . . . . 10. Packaging . . . . . . . . . . . . I 1 . Devices Fabricated . . . . . . . . . . IV . SILICONCARBIDE POWERDIODES . . . . . . 12. Fabrication Techniques . . . . . . . . 13. Characteristics of Sic' Rectifiers . . . . . . 14. Future Improvements . . . . . . . . . V . p-n JUNCTION DETECTORS . . . . . . . . 15. General Considerations . . . . . . . . 16. Nuclear Particle Detectors . . . . . . . 17. Ultraoiolet Detectors . . . . . . . . v1 . ACTIVEDEVICES . . . . . . . . . . . I8 . Tunnel Diode . . . . . . . . . . . 19 . Junction-Gate Unipolar Transistor . . . . . v11. IRRADIATION EFFECTS. . . . . . . . . . VIII . LUMINESCENT DIODES. . . . . . . . . . IX . SUMMARY. . . . . . . . . . . . . . . . . . . . . . . . . . X . ADDENDUM

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.

625 626 626 621 633 634 634 634 635 636 639 641

642 64? 642 642 645 650 651 651 654 658 660 660 663 671 677 682 682

I . Introduction Silicon carbide (hereafter SIC) is perhaps the oldest (historically) semiconductor . Although in the last fifty years considerable use has been made of its abrasive properties. only in the past fifteen years has its potentialities

626

ROBERT B. CAMPBELL AND HUNG-CHI CHANG

as a semiconductor been exploited. The last survey in this field, the proceedings of a conference devoted to Sic,’ was published in 1960. It is the purpose of this chapter to discuss advances in SIC device technology since that time and to give a brief review of the device properties of this interesting semiconductor. Since S i c device properties are so intimately connected with its material properties, crystal growth and fabrication techniques will also be discussed.

11. Silicon Carbide as a Semiconductor Material 1. PHYSICAL AND CHEMICAL PROPERTIES

Silicon carbide exists in the hexagonal (a) and cubic (B) phases, with the a phase occurring in a variety of polytypes.’ The various forms of SIC have the largest energy gaps found in common semiconductor materials, ranging from 2.39 eV (cubic)to 3.33 eV (2H).The bonding of Si and C atoms is basically covalent, with about 12 ”/, ionic bonding. The structures are temperatureTABLE I LATTICECONSTANTS AND ENERGY GAPOF COMMON SIC POLYTYPES’ Lattice parameters (A) Energy gap (0°K)

Structure a

2H 4H 6H 33R 15R 21R 8H cubic-3c

C

3.09 3.09 3.0817

5.048 10.05 15.1 183

3.079 3.079

31.78 52.88

4.359

(W 3.33 3.26 3.02 3.01 2.986 2.86 2.8C2.90 2.39

~~~

a

From Refs. 2a-21

“Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959). Pergamon Press, New York, 1960. A. R. Verma, “Crystal Growth and Dislocations.” Buttenvorths, London, 1953. 2aL.Patrick, D. R. Hamilton, and W. J. Choyke, Phys. Rev. 143, 526 (1966). 2bH. R. Philipp and E. A. Taft, in “Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959), p. 366. Pergamon Press, New York, 1960. 2cA. Taylor and R. M. Jones, in “Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959), p. 147. Pergamon Press, New York, 1960. 2dW.J. Choyke, D. R. Hamilton, and L. Patrick, Phys. Rev. 133, A1 163 (1964). “W. J. Choyke, D. R. Hamilton, and L. Patrick, Phys. Rev. 139, A1262 (1965). zrD.R. Hamilton, L. Patrick, and W. J. Choyke, Phys. Rev. 138, A1472 (1965).

9.

SILICON CARBIDE JUNCTION DEVICES

627

stable below 1800°C and thus form a family of semiconductors useful for high-temperature electronic devices. Table I shows the lattice parameters and energy gap (0°K) for the common polytypes. Silicon carbide is a brittle material, with a hardness of 9 on the Mohs scale, ranking just below diamond. When grown by a vapor-phase technique, the crystals are generally hexagonal platelets. The platelets vary in color from blue-black (heavy p-type doping) to water white (pure or compensated) to dark green (n-type doping). The &phase SIC crystal, generally prepared from a supersaturated melt or by an epitaxial growth technique, are normally cubes or parallelepipeds with a clear yellow color. Silicon carbide is inert to nearly all laboratory reagents, although it is reported to hydrolyze slowly in phosphoric acid at 215”C.3 The usual techniques for chemical etching employ molten salt or salt mixtures (NaOH, Na,O, borax) at temperatures above 600°C. Electrolytic etching, suitable only for p-type material, and etching with gaseous chlorine near 1000°C are also widely used. The physical hardness and chemical inertness impose great restraints on device fabrication techniques. Although SIC technology has progressed along the same lines as that of silicon, many techniques had to be developed which were peculiar to Sic and which inevitably made the fabrication more difficult.

2. METHODSOF PREPARATION a. Sublimation The sublimation method uses the techniques of vaporization near 2500°C of an S i c charge into a cooler cavity with subsequent c ~ n d e n s a t i o n . ~ ~ Initially, the charge formed its own cavity, but more uniform crystals are This grown when a thin graphite cylinder is used in the center ofthe thin cylinder also reduces the number of nucleations so that fewer but more perfect crystals are grown. The crystals are grown as thin hexagonal platelets, perpendicular to the growth cavity as shown in Fig. 1. Variations of this growth cavity, using thinner or thicker sections,* graphite cloth backing,’ and uniform or randomly spaced holes,’ have all been studied, but with

’

R. C. Ellis, in “Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959),p. 420. Pergamon Press, New York, 1960. J. A. Lely, Ber. Deur. Keram. Ges. 32,299 (1955). D.R. Hamilton, in “Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959), p. 43. Pergamon Press, New York, 1960. H. C. Changer a/., unpublished work, 1966. H. C. Chang and L. J. Kroko, AIEE Paper 57-1131, Chicago, 1957; H. C. Chang, Semiconductor Products and Solid State Technology, p. 29. January (1960). H. C. Chang er a / . ,unpublished work, 1964.

628


FIG. 1. Silicon carbide growth in sublimation furnace. (After K r ~ k o . ~ )

9. SILICON CARBIDE JUNCTION DEVICES

629

FIG.2. Representative grown-junction silicon carbide crystals (scale in inches).

only slight improvement over the original design. The mechanism for this crystal growth technique has been studied by Chang’.’ and Kroko’ in the USA and Pichuhin et a1.” in the USSR. Both use thermodynamic arguments which assume the growing crystal dissipates the heat of condensation of the incoming silicon and carbon vapor species by radiation to the cooler ends of the cavity. Bulk thermal conduction is assumed to have little or no effect. Both also calculate a relatively small temperature difference between the cool ends and the growing crystal, on the order of 0.1-5°C. To prepare high-purity S i c crystals by this technique requires prolonged outgassing and gettering at elevated temperature. The major impurities to be removed are nitrogen (as n-type dopant), aluminum, and boron. The latter two, both p-type dopants, are generally present in the starting material. The crystals having the highest purity are n-type with a donor concentration of 10 5-10 6cm-3. These crystals have electron mobilities of 3W600cm2 v-1

sec-‘

11

Doped crystals, or crystals containing pn junctions, can be prepared by adding the proper dopants to the ambient during growth.6-8.’2 The highest lo

l2

L. J. Kroko, J . Electrochem. Soc. 113,801 (1966). 1. G. Pichugin, N. A. Smirnova, Yu. M. Tairov, and D. A. Yas’kov, “Influence of Several Factors on the Growth and Nucleation of SIC Crystals,” p. 309. Vysokotemperaturnye Neorganicheskiye Soyedineniya, Akad. Sci. Ukr. SSR, 1965 (In Russian). D. L. Barrett and R.B. Campbell, J. Appl. Phys. 38,53, 1967. C. Goldberg and J. W. Ostroski, in “Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959), p. 453. Pergamon Press, New York, 1960.

630


rating power rectifiers are still produced by this technique. Figure 2 shows representative grown-junction crystals. The color difference is due to the difference in doping.

b. Epitaxial Techniques The isoepitaxial growth of S i c on S i c substrates has been accomplished using the thermal reduction of mixtures of the carbon and silicon tetrac h l o r i d e ~ . ' ~ .It' ~was found that either the a or phase could be deposited on hexagonal S i c substrates, depending on the substrate temperature. The structural perfection of the epitaxial layer is determined by the substrate temperature, the growth rate, the purity of the growth apparatus, and reactants and surface condition of the substrate. It was found that the hexagonal phase was grown at substrate brightness temperatures from 1725°C to 1775"C, while the cubic phase was grown from 1660°C to 1700°C. I n situ etching of the S i c substrate prior to growth was found to be most effective in promoting high-quality growth. Hydrogen was generally used as an etchant at 1600°C.l 5 Equal molar percentages of CC1, and SiCl, are used in the growth process, with concentrations between 0.060 and 0.075 % required for the preferred growth rates of 0.5-0.8 p min-'. Chemical etching and optical microscopy show that defects in the grown layer are generally associated with defects in the substrate. Figure 3 shows this effect. In the as-grown layer, a noted defect is correlated with the defects in the substrate after etching. Polycrystalline Sic has been grown6 on a Sic substrate using dimethyldichlorosilane [(CH3)2SiC12]at substrate temperatures between 1400 and 1450°C. The growth rates were on the order of 10 p min- '. These layers were tested as mechanical supports of thin crystals during the precise fabrication techniques needed, for example, for transistor studies. Ryan and co-workers at Air Force Cambridge Research Laboratory have investigated the growth of S i c onto carbon substratesI6 using the hydrogen reduction of methyltrichlorosilane (CH3SiC13)(called the vaporliquid-solid growth). At 1500"C, a-Sic whiskers on the order of 5 mm long by 1 mm diameter were grown. These whiskers were of the relatively rare 2H polytype. After further purification of the CH,SiCl, and careful cleaning of the substrate, no whiskers were grown. This would indicate that the growth was nucleated by impurities, and, in fact, by seeding the substrate with pure l4

l5 l6

V. Jennings, A. Sommer, and H. C. Chang, J. Electrochem. SOC.113,825 (1964). R. B. Campbell and T. L. Chu, J . Electrochem. SOC.113,825 (1966). T. L. Chu and R. B. Campbell, J. Electrochem. SOC.112,955 (1965). C. E. Ryan, I. Berman, R. C. Marshall, D. P. Considine, and J. J. Hawley, J. CrystulGrowfh 1, 255 (1967).

9.


631

FIG. 3. Defects in silicon carbide epitaxial layer propagated from substrate crystal. (After Campbell and Chu.14)

and doped silicon, tantalum-doped gold, rhenium, chromium disilicide, chromium, and iron, they were able to grow the whiskers. The authors suggest that the cr-Sic form is essentially a defect structure and growth at 1500°C may be caused by a slight deficiency of carbon in the lattice. The impurities then stabilize the tl phase. This model is strengthened by the experiment using chromium disilicide, where only P-SiC crystals were grown. In this case, the chromium disilicide would tend to increase carbon solubility and lead to a more stoichiometric condition, and therefore, p-Sic.

c. Traveling Solvent Silicon carbide crystals have been grown together, and p n junctions formed by passing a heat zone through two SIC crystals separated by a solvent

632


metal.’ The temperature gradient across the thin solvent zone causes dissolution a t both solvent-solid interfaces. However, the equilibrium solubility of S i c in the solvent is greater a t the hotter interface, and a concentration gradient is established. The solute, then, will diffuse across the liquid zone and precipitate onto the cooler crystal. In this way, two S i c crystals of dissimilar conductivity type can be grown together. Silicon and platinum were originally used as solvents, but best results were obtained using chromium. However, since Cr does not wet chemically cleaned S i c uniformly, the Sic-Cr-Sic sandwiches were prepared by evaporating a thin film of Cr onto Sic surfaces which had been heat-treated in vucuo at 120(r13OO0C.At temperatures of 1750”C,growth rates of 0.75 mm hr - were obtained. Microscopic examination of crystals show small metallic inclusions, presumably Cr. Properties of rectifiers prepared by this technique will be discussed later.

’

d. Solution Growth In the solution-growth technique, a small amount of S i c is dissolved in molten Si (or in some cases Fe or Cr).’’-’’ As the melt is slowly cooled, the S i c becomes less soluble and S i c crystals nucleate and grow in the crucible on prepared graphite substrates. The grown crystals are normally of the /3 phase. Improvements” in the crucible geometry and cooling rates have led to cubic crystals up to 4 mm across and 0.1 mm thick. With the use of pure starting materials and extensive degassing, quite pure crystals can be grown, and electron mobilities approaching 1000cm2V - ’ sec-’ at room temperature have been measured with a donor concentration near l O I 7 ~ m - ~ . The crystals are generally twinned parallel to the (111) faces, and etching (NaNO, + 10% Na,02 at 500°C) shows few discernible dislocations. X-ray topographs confirmed that the crystals are free of dislocations. In later work, the chromium-siIicon<arbon alloy system was studied,22 with maximum growth being observed in the 3 5 4 0 atomic % chromium region. Again the crystals grown were thin p-SiC laths. In an effort to grow larger crystals, the p-Sic crystals were rotated in the melt. Although temperature conditions were not optimum, some coherent growth was obtained. Somewhat similar results were obtained by Ryan16 using a silicon melt. 10% silicon In addition to growing p-Sic crystals from silicon and iron

+

l9

2o

’’

L. B. Griffiths and A. I. Mlavsky, J . Electrochem. SOC.111, 805 (1964). F. A. Halden, in “Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959), p. 115. Pergamon Press. New York. 1960. R. C. Ellis, in “Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959), p. 124. Pergamon Press, New York, 1960. R. W. Bartlett and R. A. Muller, unpublished work, 1967. W. E. Nelson, F. A. Halden, and A. Rosengreen, J. Appl. Phys. 37. 33 (1966). W. J. Silva, A. Rosengreen, and L. E. Marsh, unpublished work, 1967.

9.


633

melt, Knippenbergz3was able to prepare small a-Sic crystals (together with 8-Sic) using boron carbide-silicon carbide melts.

3. SEMICONDUCTOR PROPERTIES In this chapter, we will be concerned mainly with the 6H polytype of a-Sic. Table I1 gives some electrical and transport properties for this polytype TABLE 11

ELECTRICAL A N D TRANSPORT PROPERTIES OF 6 H &SIC Temperature (“C) Band gap (eV) Effective mass mp/*r,

m“h? Mobility (cmZV-’sec- ‘ ) PP

Pe

600 2.68

L”

- 65

2.9 1

1.2 0.6 1-10 10-100

Lifetime sec) Diffusion length (b)

LP

25 2.89

0.5- I 1-5

10-30 50600 0.014 1

4&80 100-1000

0.54 1-10

1-10 2-50

at three temperatures. The transport properties are related to the donor or acceptor concentration in the given sample, and thus these values are given as ranges. In all cases, the second number given is for relatively pure samples (n, or np z i016cm-3). The relatively short diffusion lengths and lifetimes (as compared to silicon) pose one of the problems in designing S i c devices. Thus, for example, S i c transistor studies have concentrated on majority-carrier devices, since the junction structure in minority-carrier devices would require base widths on the order of 1-2p. The only dopants which have been extensively studied in S i c are nitrogen (n-type) and aluminum (p-type). Chang and his co-workers studied the diffusion of aluminum into in 1960, with further work reported by Grifin 1966. Slack2’ and Kroko28 studied fiths2’ in 1965 and Vodakov et W. F. Knippenberg, Phillips Res. Rep. 18, 161 1 (1963). H. C . Chang, L. F. Wallace, and C. Z. LeMay, in “Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959). p. 496. Pergamon Press, New York. 1960. ” L. B. Griffiths, J . Appl. Phys. 36. 571 (1965). 26 Yu. A . Vodakov, E. N. Mokhov. and M. B. Reifman, Fiz. Tverd. Tela8. 1298 (1966)[English Transl. : Sou. Phys.-Sotid State 8, 1040 (1966)l. 27 G. A. Slack, J . Chem. Plrys. 42, 805 ( 1965). L. J. Kroko and A. G . Milnes, Solid Stare Electron. 9, 1125 (1966).

23

’*

634


the diffusion of nitrogen into p-type Sic. This will be discussed in a later part of this chapter. Although few quantitative data are available, boron (p-type),6,26beryllium (p-t~pe),’~ arsenic (n-type): and phosphorus (n-type)6 have also been used as dopants. Boron and beryllium were diffused into Sic, while arsenic and phosphorus were incorporated into epitaxially grown layers.

In. Device Techniques 4. INTRODUCTION

The specific device techniques used will vary from device to device, and it is the purpose of this section to discuss fabrication methods in a general manner. In later sections, when the individual devices are described, any special techniques required will be discussed. 5. DIFFUSION Chang and c o - w ~ r k e r s studied ~~*~~ the diffusion of aluminum into Sic from 1750 to 210O0C, using both closed-tube and open-tube flowing gas techniques. Since the S i c crystals will decompose at these temperatures, it was necessary to provide an equilibrium pressure of Si and C vapor species around the crystals during the diffusion process. Griffiths” in 1965 ~ 1966 reported further diffusion experiments. The and Vodakov et ~ 1 . ’ in activation energy for the diffusion of aluminum into S i c found in these three studies agreed within 5% (-4.8 eV). found an anisotropy in the diffusion parallel and perChang et pendicular to the c-axis in the S i c crystals. This difference was sufficiently great that it could not be attributed to the differences in surface concentrations on the two planes, and therefore was probably due to the diffusion mechanism. In unpublished work, Canepa3’ and Roberts3’ refined the aluminum diffusion process while studying the fabrication of SIC neutron detectors and Sic unipolar transistors. Using a combination of infinite-source and finite-source diffusion techniques, Canepa3 was able to prepare junctions having depletion widths up to 25 p. Roberts3’ studied the relationship between the background concentration in the n-type Sic crystal and the diffusant surface concentration. Using crystals with background donor E. Violin and G. F. Kholuyanov, Fiz. Tuerd. Tela 6, 1696 (1964) [English Transl. : Sou. Phys.Solid State 6, 1331 (1964)l. 30 H. C. Chang, L. F. Wallace, and C. Z. LeMay, unpublished work, 1960. 3 1 P. C. Canepa, Westinghouse Research Laboratories, private communication, 1963. J. S. Roberts, Westinghouse Research Laboratories, private communication, 1964.

29

’’

9.


635

concentrations between 5 x 1 0 " ~ m - and ~ 5 x 1017cm-3,he found the maximum surface concentration of the aluminum diffusant to be 2 x 10'' cm-3 with an Al-saturated furnace and 5 % H, in the argon carrier gas, and thus confirmed the earlier results of Chang et Chang et diffused boron into S i c and concluded that the surface concentration of boron was higher than that of aluminum under the same conditions. Using the same experimental techniques as with Al, VodakovZ6 found the activation energy for the diffusion of boron into S i c to be 5.6eV. first reported on the diffusion of nitrogen into Sic. No quantitative data are given, but he concludes that, above 1500"C, appreciable diffusion of nitrogen into S i c does occur. In this work, all the samples were able to obtain type conversion of were granular Sic. Chang et p-type S i c crystals using a nitrogen ambient below 2000°C. Slackz7studied the diffusion of nitrogen into Sic at high temperatures (up to 2450°C) and high pressures (35 atm of nitrogen). A diffusion depth of 84 ,u in 4 hr was obtained under these conditions. The diffusion depth was determined by a color change in the crystal. Kroko and Milnes" in 1966 carried out further studies of nitrogen diffusion at temperatures between 2000°C and 2600°C at 1 atm nitrogen pressure. They concluded from the slope of the diffusion curve that the diffusion of nitrogen into SIC at these temperatures proceeds by a different (undefined)mechanism than the diffusion of aluminum at lower temperatures. PROCESSING6*' 6. MECHANICAL

The mechanical shaping of a hard crystal such as Sic is generally accomplished by scribing and breaking, lapping and polishing, ultrasonic cutting, and air abrasive cutting. Boron carbide and/or diamond are used for these purposes, since they are the only materials sufficiently hard. Scribing the crystal with a diamond point and breaking it along the scribe line is still used to some extent. As will be discussed later, a number of field-effect transistors were fabricated on a single crystal, and these transistors were separated by scribing. Obviously, this is best carried out on a scribing machine. Boron carbide mesh is used for removal of surface material from the S i c crystal. Table 111 gives the amount of material removed as a function of mesh size. To obtain mechanically smooth, flat surfaces, diamond polishing follows the lapping. 33

P. Carroll, in "Silicon Carbide-A High Temperature Semiconductor" (Proc. Conf. Silicon Carbide, Boston, 1959), p. 341. Pergamon Press, New York, 1960.

636

ROBERT B. CAMPBELL AND HUNG-CHI CHANG TABLE I11 RATEOF REMOVAL OF SIC

WITH

BORONCARBIDE

Boron carbide mesh

Amount of crystal removed“ ( p min- ’)

600 800

1.5 0.7 0.05

1000

Crystal cross section, 0.8 cm’; holding assembly weight, 75 gm ; lapping machine speed, 6.75 rpm.

Ultrasonic cutting (again using boron carbide) is used to obtain uniformsized devices as well as to prepare specified sample shapes, e.g., bridge-cut for Hall measurements. The grit size in the slurry is important, since large grit will cut too rapidly and chip the crystal, whereas small grit cuts too slowly; 320 mesh is generally used. To remove areas of the bulk crystal (e.g., shorted regions), air abrasive cutting may be used. This technique, not unlike a dentist’s drill, rapidly removes the S i c by chipping and erosion. Again, a 320 grit size has been found most suitable.’ All of these mechanical shaping operations inevitably leave surface and bulk damage in the crystal. Some studies have indicated that the damage may propagate into the crystal by microcracks to a depth of tens of microns. For optimum device performance, this damage must be removed, e.g., by chemical etching.

7. ETCHING The etching of S i c using molten salts was described in detail by F a ~ s t ~ ~ in 1959. In his paper, Faust describes the side of the Sic crystal that etches in a rough, “wormy” pattern, using molten salt, as the carbon side, and the side where the etch is smooth as the silicon side. This was confirmed by B r a ~ in k ~1965, ~ using X-ray techniques. Gabor and J e n n i n g ~ investigated ~~ the etching of S i c in a 1 : 1 molten mixture of stirred NaF and Na,SO,. Gases used for stirring were Ar, O,, and N,. They found that oxygen stirring increased the etch rate by a factor of 2-3, indicating that the gas takes some part in the chemical reaction. In this work, they found that stirring the melt resulted in the formation of etch pits on both sides of the crystal. 34

35 36

J. W. Faust, in “Silicon Carbide-A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959), p. 403. Pergamon Press, New York, 1960. K. Brack, J . Appl. Phys. 36, 3560 (1965). T. Gabor and V. J . Jennings, J . Ekcfrochem. Tech. 3, 31 (1965).

9.


637

Brander37studied the addition of potassium nitrate to potassium hydroxide and found that etching occurred at temperatures as low as 500°C. For device fabrication, the molten-salt etch has many disadvantages. First, it is a very rapid etch, removing as much as 1-2 p of material per second. Second, no suitable etch mask exists for the molten salt; therefore, only planar structures can be etched this way. Finally, it is selective, since, as mentioned above, the two sides of the S i c crystal etch in different patterns. A preferred technique is to use gaseous etching, e.g., chlorine at 95CL 1050°C (Thibault3*) or chlorine and oxygen (Smith39 and Chang et ~ ~ 1 . ~ ~ 1 . Apparatus which can be used for the etching is shown in Fig. 4. The silicon carbide crystal to be etched is placed on the quartz boat sample holder, which is fused to a hollow quartz rod containing the monitoring thermocouple. The roller track facilitates the loading and unloading of the crystals in nonoxidizing conditions.

Gas exhousi

FIG.

4. Apparatus for chlorine etching of silicon carbide.

Flowmeters are used for monitoring the flow of CI,, O,, and Ar, with a flow rate range up to 500 cc min-'. The cross-sectional area of the reaction tube is 6-7 cm2: therefore, the linear velocity can be varied from 0 to 1 cm sec- which is within the laminar flow condition. The following etching reaction occurs in the reaction tube :

',

Sic

+ 2C1,

SiCI,

+ C.

To remove the carbon from the surface, oxygen is added to the etching reaction to form CO and CO,. R. W. Brander and A. L. Boughey, Brir. J. Appl. Phys. 18.905 (1967). N . W. Thibault, Amer. Mineralogist 29, 249 (1944). 39 R. C. Smith, J. Electrochem. Soc. 110, 184C (1963). 40 H . C. Chang, N. P. Formigoni, and J . S. Roberts, unpublished work, 1966. 37

38

638


01

1073

1273

1173 log T,

1373

OK

FIG. 5. Etch rate of silicon carbide crystals in 85 % chlorine-15 % oxygen, 74 cc min-'.

The etching rates obtainable with this method are shown in Fig. 5. This curve indicates that the etch rate can be varied two orders of magnitude over a 300°C temperature range. By varying the oxygen content from 15% by volume to 8 5 % by volume, the etching rate can be increased by a factor of two. In fact, it was not possible to determine a maximum etch rate (as a function of the Cl,, 0, mixture), as the etching rate increases monotonically with the 0, concentration. However, when the 0, concentration exceeds 50% by volume, large amounts of SiOz are formed, leading to rough, irregular etch surface. Similarly, the concentration of 0, cannot be drastically reduced, in that (1) the pure chlorine etch leaves a pseudolattice of carbon on the surface, and (2) the undercutting of any pattern becomes more pronounced with a high chlorine concentration. It has been found most feasible to control the etch rate by mixing varying amounts of argon with the etchants. The argon retards the etching rate without negative side effects. The optimum concentration was found to be 45 cc min- C1, , 120 cc min- Ar, and 10 cc min- 0,. This composition gives an etching rate of 0.25 +_ 0.02pmin-' at 900"C, and is sufficiently fast that excessive times are not required for the etching,

'

9.

639


These etching rates were determined on the carbon face of the crystal. The etching rate on the silicon face is several orders of magnitude less. Electrolytic etching of SIC is accomplished using a dilute solution of HF in water and methyl alcohol. Concentration of 49% HF in methyl alcohol from 1 :50 to 1 :500 have been used successfully. The etching is carried out at 30°C under reverse bias with nominal etching currents from 20 to 50mA. This etch is specific for p-type material and has been used to etch mesa structures on p-n junctions and to determine junction depth. Brander and Boughey3' found that smooth surfaces on p-type material could be obtained using 40 % HF in distilled water solutions. The distilled water contained 10% HCOOH. They were also able to attack to a slight degree n-type material by forward biasing the p-n junction and injecting the necessary holes by avalanche breakdown at the surface or by illuminating with ultraviolet light. Oxidation of the Sic crystal and subsequent removal of the oxide in HF can also be considered as an etching technique. This is discussed in the next section. 8. OXIDATION

Silicon dioxide can be grown on S i c with steam oxidation in a manner similar to that used for the growth of SiO, on silicon crystals. Oxidation takes place at 90G1200"C in a carrier gas such as argon or oxygen saturated with water vapor around 100°C.The oxide is found to grow at a significantly different rate on the carbon face of the S i c surface as compared to the growth rate on the silicon face, the growth rate being approximately 10 times faster on the carbon face. In addition, the growth rate of the oxide on the carbon face of the S i c surface is significantly slower than that on silicon crystals, but appears to obey the same square law found for the growth rate on silicon. The growth rate of SiO, on Sic given in Table IV is from Chang ef TABLE IV GROWTHOF SIO, ON CARBONFACEOF SIC AT 1173°C AND ETCHING HF SOLUTION AT 25°C RATEOF SIO, WITH BUFFERED

Run no.

2 4

5 6

Oxidation time Carrier gas (hr) flow (120cc/min) 11.25 15.0

Ar Ar

11.0 1.7

Ar

0 2

Thickness

(A) 17,000

20,900 17,050 5400

Etching rate of S i 0 2 in buffered HF (A jmin) 980 860 875 830

640

ROBERT B. CAMPBELL A N D HUNG-CHI CHANG

The oxidation of Sic follows the same parabolic law as the oxidation of Si, namely, log d

=

log K

+

log t ,

where d is thickness in angstroms, t is the time in hours, and K (a constant was found to be 3900 for Sic as compared to 6900 for Si. This was in A/hr 'Iz) determined at 1173°C. Brander37 used between 10' and lo4 ppm pure 0, in distilled H,O at 1200°C and found a slightly slower oxidation rate on the carbon side than found in Table IV. He also found, however, that the silicon side oxidized only slightly less rapidly than the carbon side. Silicon dioxide layers can also be produced on S i c by pyrolytic deposition or sputtering techniques. The SiO, layer may be utilized in several ways in the fabrication of Sic devices:

(1) as a mask for the chlorine etch in preferentially etching Sic to create various mask patterns ; (2) as a positive means of identifying the polarity of the SIC crystal surface, i.e., the C or Si face for processing : (3) for passivation of the junction region of the diode; (4)for contact confinement in the contact-alloying process to prevent excessive spreading or penetration ; and (5) as a means of delineating p-type regions from n-type regions on a crystal (see Fig. 6). This last ( 5 ) is an important and new contribution to Sic technology. It has been found that the oxide grows at different rates on p- and n-type

regions on a single crystal of Sic, leading to a different coloration of these

FIG. 6. Delineation of diffused junction in silicon carbide by oxidation.

9.


641

regions as a result of the interferences of thin films. That the difference in coloration is a result of at least a difference in the oxide thickness has been shown by long oxidation of a surface with both p and n regions and a delineation of these regions after the removal of the oxide. This difference in oxidation has also been used in the delineation of junctions after edge polishing a crystal to determine the junction depth after aluminum diffusion. It has been noted that the oxidation coloration is different on the edge of the crystal from that on the carbon face of the same crystal, indicating that the oxidation rate is anisotropic in crystallographic direction in addition to the anisotropy of the two faces. The delineation is clear and has been checked and found to be in agreement with the junction delineation observed on the electrolytic etch. It has also been noted that the time of oxidation required for edge delineation (- 1175°C) is considerably longer (minimum of 90 min for rather poor delineation) that that required for delineation on the carbon face (30 min gives a very clear delineation), indicating that the difference in oxidation rates on p type from n type is probably considerably greater in the C direction than perpendicular to the C direction.

9. ALLOY~NG Alloy contacts to the S i c crystal should be ohmic at all temperatures, and, in themselves, of high conductivity. The molten alloy must wet and react with the S i c and, on solidification. form a stable, void-free interface region. In addition, the alloy should melt below 2000"C, or gross decomposition of the S i c crystal will occur during the alloying cycle. The completed bond should nearly match the thermal expansion of Sic so that undue strain is not put on either the crystal or the contact during thermal cycling. A number of different materials have been studied for ohmic contacts to Sic. Hall4' fused tungsten to S i c at 1900°C to form ohmic contacts to p-type Sic. On larger-area contacts, there is some indication that the interface between the tungsten and the S i c contains voids.42 The wetting may be improved by depositing thin films of silicon on both the tungsten and S i c surfaces before alloying.43 Van Daal et ~ 2 1 . : ~ Greebe,45 and Canepa et ~ 2 1 reported that Au l-lOO/, Ta formed ohmic contacts to n-type Sic, while

+

*' 42

43 44

45 46

R. N . Hall,J. Appl. Phys. 29. 1914 (1958). R. B. Campbell, unpublished work, 1965. H. C. Chang and J. W. Ostroski. unpublished work, 1959. H. J. van Daal, C. A. A. J. Greebe, W. F. Knippenberg, and J. J. Vink, J . Appl. Phvs. Suppl. 32, 2225 (1961). C. A. A. J. Greebe. Phillips Rrs. Rep. Suppl. I (1963). P. C. Canepa, P. Malinaric, R. B. Campbell, and J. W. Ostroski, ZEEE Trans. Nucl. Sci. NS-11,262 (1964).

. ~ ~

642


the same alloy with a small amount of A1 added formed ohmic contacts to p-type Sic. The alloying temperature was near 1225°C. Gold-tantalum alloys have been used as ohmic contacts to the source, gate, and drain regions of an Sic unipolar transistor by sputtering alternate layers of gold and tantalum through a mask.40 Chang and his c ~ - w o r k e r sused ~ ~ *an~ ~ alloy of platinum and tin doped with either 2% gallium or antimony (for p-type or n-type Sic) at alloying temperatures near 1800°C. These contacts, as well as the gold-based alloys , ~ and Canepa et are all amenable to by Van Daal et ~ l . Greebe,45 thermal compression bonding of lead wires. Titanium- and tantalum-base alloys have been used with less success.4' These alloys react rapidly with the Sic crystal near the fusion temperature and form deep, void-filled interfaces. Taylor used Si doped with As or P for contacts to n-type On very-high-resistivity n-type or p-type Sic ( p > lo3 ohm-cm), semiconductor-grade Si is reported to form ohmic and low-resistivity contacts49 where the gold-base alloys do not.

10. PACKAGING Various package designs have been used for Sic devices. The main considerations are thermal expansion matching of the components and the Sic device. In devices operating between - 65°C and 500"C, unless thermal expansion relief is built into the package, crystal fracture can result. In high-power rectifiers, heat extraction is a prime consideration. As will be discussed, Sic rectifiers have forward voltages several times those of silicon; therefore, appreciable heat must be extracted. For light- or particle-detecting diodes, quartz or film windows must be provided. The specific encapsulations will be discussed under the various devices. 1 1. DEVICES FABRICATED In the remaining sections of this chapter, the devices which have been studied, together with their pertinent properties, will be discussed. Unless specifically noted otherwise, the devices described were fabricated from either the 6H or 15R polytypes.

IV. Silicon Carbide Power Diodes 12. FABRICATION TECHNIQUES a. Alloy Junctions The physical and chemical requirements for fused alloy junctions in S i c are the same as for ohmic contacts, except that a few per cent of the proper 47

J. W. Ostroski, unpublished work, 1960.

9.

SILICON CARRlDE JUNCTION DEVICES

643

dopant is added to the alloy. During fusion, the dopant forms a p-n junction, generally quite abrupt. Chang et a1.24used platinum-base alloys doped with boron or aluminum for fusing to n-type material. Taylor48 prepared Sic rectifiers by alloying pure aluminum into n-type Sic at 1700°C. During the alloying, a film of aluminum carbide (Al,C,) was formed between the contact and the crystal. It was suggested that the rectifying junction was really a junction between AI,C, and Sic. In general, alloy junctions in Sic exhibit a low forward voltage but a correspondingly low reverse voltage. Perhaps due to the low reverse capability (on the order of tens of volts), little work has been done in this field since the last review.’ b. Grown Junctions

To prepare grown junctions, p-type and n-type impurities are serially introduced into the growth cavity of the sublimation furnace. In general, it has been found most satisfactory to introduce the p-type dopant (generally aluminum or boron) into the Sic charge and, after the aluminum or boron has been totally or partially depleted, introduce nitrogen gas. Thus, the grown crystals have a bulk p-type core with an n-type shell. The time and rate of introduction of the nitrogen into the growth cavity determine, to a great extent, the final electrical properties of the junction. That is, they control the width of the intrinsic region, degree of compensation in the junction, etc.6 Some interesting experiments have been performed where the dopants were changed as a function of time.7-’ In this case, the aluminum (as AI,C,) and nitrogen were alternately added to the growth cavity so that, as a function of time, the crystals grew as n type, then p type, then n type, etc. Figure 7 shows a crystal with layers defined by nitrogen doping. The banded structure of the different conductivity types are clearly evident. Experiments of this type could be a powerful tool in growth-rate ~tudies.’.~ The properties of these rectifiers will be discussed in more detail later. c. DifSused Junctions Diffused junctions in Sic have been used almost exclusively for smallsignal or active devices. Although some research directed toward the preparation of power rectifiers by diffusion has been carried out,8 the results have generally been negative. Diffused junctions have been used, however, T. C. Taylor, in “Silicon Carbide- -A High Temperature Semiconductor” (Proc. Conf. Silicon Carbide, Boston, 1959), p. 431. Pergamon Press, New York, 1960. 49 R. W. Ure, Westinghouse Research Laboratories, private communication, 1965. 50 R. B. Campbell and H. C. Chang, Solid State Electron. 10, 953 (1967). 5 ’ See, for example, luminescent diode papers in Mater. Res. Bull. 4 (1969).

48

644


FIG. 7. Silicon carbide crystal grown in alternate ambients of argon and nitrogen. (After Kroko.*)

for unipolar transistor^,^^.^^ neutron detectors,46ultraviolet light detectors:' and luminescent diode^.^ The diffused junctions generally exhibit a wide depletion width. In aluminum-diffused junctions, the depletion width estimated from capacitance measurements varied from 3 to 50p, depending on the diffusion technique and para meters.'^^^ Aluminum-diffused junctions characteristically show quite stable reverse voltages (up to 400V PIV), but also high forward voltages (perhaps as high as 1OOV at 20mA).x*46In the one study of nitrogen-diffused junctions,6 lower forward voltages were obtained, but at the expense of the reverse capability. The electrical characteristics of boron-diffused junctions have been reported to have a greater temperature dependence than do aluminumdiffused junctions.30 d. Epitaxial Junctions

The epitaxial method of growth has been described in Section 2. The epitaxial layers can be doped either n type or p type during growth using

9.


645

nitrogen, arsine, or phosphine for n-type layers and diborane for p-type layers. Some preliminary Hall measurements showed that phosphorusdoped layers exhibited high mobilities." Rectifiers prepared from epitaxial junctions have generally exhibited low forward voltages (1.5-2.5V at 1.0-5.0A) with a reverse capability of 50 V (these measurements at 500°C). The reason for the somewhat low reverse capability of epitaxial junctions has never been satisfactorily explained. Epitaxial layers of high quality with no grown structural faults have been grown. Perhaps a more rigorous study of the junction structure and gradient would supply an answer. e. Processing for SpeciJic Structurt> Silicon carbide junctions have been processed into various structures, depending on their ultimate application. A mesa structure prepared by electrolytic etching of p-type material has been used for the S i c detectors. Since, as mentioned, the n-type material is not attacked, the etching action stops at the junction and a well-formed mesa can be delineated. Silicon carbide-junction rectifiers and diffused structures are more simply fabricated by removing the shorting edges (e.g., by ultrasonic cutting) and using an etch, either chemical, electrolytic, or gaseous, to remove damaged material. At this point, the blue microplasma breakdown in the junction can be used to find further damaged regions or imperfect structural areas. Ifthe junction is reverse-biased, bright blue spots appear at thedamaged area. These spots are small local regions of high reverse current and can generally be associated with structural imperfections. These areas can be mechanically removed and thus the reverse capability of the diode increased. The fabrication techniques used for the active devices are quite long and complicated and will be described in detail in that section.

13. CHARACTERISTICS OF S i c RECTIFIERS a. Current-Voltage Characteristics Figures 8 and 9 show the I-V properties of a S i c rectifier prepared by the grown-junction method, operating at 1 A and 500"C.52The forward voltages ofthese devices, even at 500°C are always larger than 1 V (half-wave average). Thus far, rectifiers operating up to 10 A have been and specially processed, low-current devices have exhibited reverse capability of 600 V PIV.'3 The reverse characteristics of S i c rectifiers generally show a "soft" breakover, rather than the avalanche breakdown sometimes noted in silicon. This is generally attributed to the carrier generation mechanism at

'' H.C . Chang ef al., unpublished work. 1964. 53

H . S. Berrnan, Westinghouse Astronuclear Laboratory, unpublished work, 1968.

646


7

1000

Q

x

t

C

Y

" 7

u) W

W

E! m W

e

>

O

W

0 I

-

L

0

I

I

I

40

80

I

I

I

I

i

120

160

200

240

280

Peak reverse voltage,

V

FIG. 8. Reverse characteristics of a silicon carbide grown-junction rectifier.

the junction and to local areas breaking down at different voltages, so that the total effect is one of gradually increasing reverse current. Figures 10 and 11 show the I-Vcharacteristics of a larger-current S i c device.54 Rectifying junctions have been prepared using the traveling solvent technique with chromium.",55 Although the reverse breakdown occurs at a low voltage (6-15V PIV), Griffiths" and Griffiths and Mlavsky" were able to interpret the junction capacitance at zero bias and the forward, dark I-V characteristics in terms of the diffusion of the major dopants across the junction resulting in compensation.

b. Encapsulations The basic requirements of the encapsulation are to provide protection of the rectifier package components from the operating environment. The s4 55

H. C. Chang and J. Ostroski, unpublished work, 1966. L. B. Griffiths, J . Appl. Phys. 36,571 (1965).

9.


647

FIG. 9. Forward characteristics of the silicon carbide grown-junction rectifier of Fig. 8

main concern is the tungsten which is generally used as a base plate to the S i c crystal, which must be protected from oxidation. Aiso, since the rectifier is to be used from -65°C to 5OO0C, the encapsulation must provide for the differing thermal expansions of the rectifier components. The realization of these requirements is a study in compromises. One package which has been successful is shown in Fig. 12.' In this package, the base is nickel and the header is a composite assembly consisting of a 96% alumina tube gold-brazed to a weld flange of nickel. The upper part of the header has a nickel cap gold-brazed to the alumina, with a nickel collar joined to the cap using 950°C (18Ni-82Au) solder; joined to this collar by 780°C (BT) solder is the silver pinchoff tube. Another type of package has been designed that is essentially a zero In this package, copper and tungsten inserts are used inside expansion a ceramic-nickel package. The lengths of the copper and tungsten are adjusted so that their expansion nearly matches that of the nickekeramic capsule. A nickel flange acts as a spring member to take up any small deviations.

648

ROBERT B. CAMPBELL AND HUNG-CHI CHANG 300

> a; 200 m

c

0

s W

p1 1 0 W

100

a

0

Half-wave average forward current, A

FIG. 10. Characteristics of a silicon carbide grown-junction rectifier of larger current than that of Figs. 8 and 9. Unit L-70, 500°C; IR (rnA): (010.5, ( 0 )1.0, ( A ) 2.0. (0) 3.0, ( W 5.0.( 0 )10.0.

0 H o l f - w a v e average forward voltage, V

FIG. 11. Forward characteristics of the silicon carbide grown-junction rectifier of Fig. 10. Unit L-70; (0) 300"C,(0) 500°C.

9.


649

FIG. 12. Silicon carbide rectifier housing.

As with all power-handling devices, it is important to transfer efficiently the internally generated heat to the external ambiant. In the case of silicon carbide power rectifiers, the problem of transporting heat from the devices is particularly difficult because the thermal conductivity of suitable metals is greatly reduced at the high operating temperature. With the device case at 500"C, internal temperatures are significantly higher when large forward currents are conducted. There are relatively few materials with appropriate chemical and physical properties which are usable as components in the device assembly at these temperatures. Many appealing metals(Pt, Rh, etc.) are far too costly, and most otherwise acceptable materials have thermal conductivities which are relatively low at room temperature and which decrease rapidly as temperature rises. The thermal impedance, defined as the ratio of the case-to-junction temperature difference to the total power input, has been determined for the package shown in Fig. 12. The junction temperature was measured by the forward drop method. A curve of the thermal impedance versus case temperature is shown in Fig. 13.

650

ROBERT B. CAMPBELL AND HUNG-CHI CHANG Case temperature,

O C

15-

P "

aJ C U 0

g l0E -

n

n

E

400

500

Case temperature,

600

700 800

"K

FIG.13. Thermal impedance of silicon carbide rectifier.

Thermal impedances from 3 to 8°C W - ' have been measured on S i c devices of the type shown. A computed thermal impedance value, obtained by considering the thermal conductivity of the various components in the encapsulation, was in good agreement with the experimental value, being about 34°C W-'.' c. Life Testing

Although very limited life-test data have been obtained for the 10-A devices, a few devices have been operated at several amperes for up to 200 hr at 500°C in air with no change in electrical characteristics. One-ampere devices using approximately the same capsule design have been successfully life-tested for 1000 h at 500°C. 14. FUTURE IMPROVEMENTS

At the present time, S i c rectifiers have been fabricated with a forward current capability of 10 A (half-wave average). It would appear that greatly increased current capability is limited only by the need for (1) large, perfect junction crystals, and (2) an efficient encapsulation. The latter is needed to

9.

651


extract the heat generated in the junction, which in a 50-A unit may be as high as 300 W. This would require quite efficient heat-sinking. Higher peak reverse voltages should be obtainable when an ideal p-i-n structure can be fabricated. Chang et al. have shown that with such a structure a 600V PIV can be obtained with a relatively narrow intrinsic-layer width ( 2-3 pm), which would also permit a low forward voltage.6

-

V. p-n Junction Detectors 15. GENERAL CONSIDERATIONS

The operation of a p-n junction nuclear-particle or photon detector depends on the collection of electron-hole pairs produced by the ionizing particle or photon as it passes through the detector. The electron-hole pairs are separated in the junction region, collected, and give rise to a charge or voltage pulse. a. Charged-Particle Detector

The ionizing particle passing through the detector loses a small amount of energy for each electron-hole pair produced (for SIC, about 10 eV/pair). lO-’sec) before These pairs have a relatively short lifetime (for Sic, they recombine. Thus, any pairs created outside the junction region may well recombine before they are collected, since they must diffuse to the junction region (a relatively slow process) before they are accelerated by the p-n junction electric field and collected. These conditions on pair creation and collection place certain restrictions on the junction design of the detecting device. First, the surface layer (lowfield region) through which the particle must pass should be narrow so that the particle loses no appreciable energy. Second, the depletion region where there are the same number of electrons in the conduction b a n i a n d holes in the valence band (intrinsic material) should be nearly as thick as the range of the ionizing particle in the material, thus effectively stopping the particle in the high-field region. The width of the depletion region can be controlled by the technique used to prepare the junction (e.g., diffusion parameters), or by the application of a reverse voltage on the junction. When an external voltage is applied across the depletion region, the electron and hole distributions, which are determined by the electric field and the electron and hole diffusion coefficients, change. The net result is that the depletion region widens and the electric field at the junction increases. For a given particle detector, these two factors result in a lower detector capacitance, and if the widened depletion region

-

652


more nearly matched the range of the ionizing particle, increased counting and collection efficiency. b. Ultraviolet Detectors

Silicon photovoltaic diodes have been developed for the detection of infrared and visible radiation. These diodes exhibit a sharp drop in response as the wavelength of the incident light approaches the ultraviolet region, with most detectors showing negligible response below 3000A. This decreasing response is due to the increase in the absorption coefficient with decreasing wavelength. A large absorption coefficient indicates that nearly all the light will be absorbed at the surface of the device and electron-hole pairs generated may be at a great distance from the pn junction. Thus, surface effects, such as carrier recombination, will decrease the response of the detector. Silicon carbide, with a band gap near 3.0 eV, has an absorption coefficient several orders of magnitude less than that of Si at 4000 A, and therefore surface effects would not be so important. Detectors have been prepared from S i c and these devices were found to have spectral responses which peaked in the ultraviolet region and which could be shifted by varying the junction depth. A simple theoretical model was originally d e r i ~ e d ~which ~ ~ ~quanti" tatively explained the dependence of the peak wavelength on the junction depth and the depletion width of the diode. Considered in this model were the wavelengths and temperature dependences of the absorption coefficient in Sic below the band edge. An approximation was made that at the peak response wavelength the total number of electron-hole pairs generated in the depletion layer is a maximum for a given intensity of transmitted radiation at the surface. Figure 14 shows the variation of peak-response wavelength calculated from this model. The curves are shown for values of the effective depletion width w from 1 to lop. (The experimental data will be discussed later.) This simple model proved to be adequate for photovoltaic diodes fabricated from materials having very short carrier diffusion lengths and lifetimes and under the conditions that the junction depth is greater and the depletion layer not much greater than the carrier diffusion length. The variation of the peak-response wavelength with temperature was not adequately explained with this model, which took into account only the temperature variation of the absorption coefficient, i.e., the experimental value of the rate of increase of the peak wavelength was 2-3 times that calculated. A more rigorous calculation was later carried out which included such sb

H. C. Chang and R. B. Campbell, unpublished work, 1965.

9.

653


T

- 0

2800

3000

3200

I

1

I

3400

3600

3800

P e a k wavelength,

I 4000

I 4200

A

FIG. 14. Peak spectral response of silicon carbide junction diode as a function of junction depth. (After Campbell and Chang.50).

parameters as carrier lifetime, carrier mobility, surface recombination velocity, and their temperature dependen~e.~’*~* This treatment considered the junction to be close to a p-i-n structure. Greebe4’ has treated a symmetrical p i - n structure by assuming that the generation of carriers was uniform throughout the system and that the thicknesses of the p-type and n-type layers approached infinity. Chang and Campbell5’ and Campbell 57 58

H. C. Chang, Westinghouse Research Laboratories, private communication. 1967. R. B. Campbell and H. C. Chang, Paper Y.1, Int. Electron Deuices Meeting, Washington, October 2967 (to be published).

654


and Changs8, assumed a more realistic, asymmetrical junction, and the calculated peak-response wavelength and its temperature dependence agreed quite well with the experimental data.

16. NUCLEAR PARTICLE DETECTORS Silicon carbide structures have been prepared that are capable of detecting a-particles up to 700°C and, with the addition of a conversion layer, thermal neutrons have been counted.46 For these detectors, n-type S i c crystals grown by the sublimation techn i q ~ eand ~ ? doped ~ with nitrogen to a level of about 10l6cm-3 were used. To prepare the detector structure, aluminum was diffused into these n-type crystals, producing a p-type layer with the diffusion parameters set to produce a graded-junction structure. After the edges of the diffused crystal were removed, an electrolytic etching procedure was used to produce a mesa structure. The electrically poorer of the two p n junctions was removed by lapping. The remaining p-n junction was then processed until the p-type layer was about 6 8 p thick. At this point, a noble-metal-alloy top contact and a tungsten base tab were soldered to the crystal and chemical etching was used to reduce the junction depth to the desired value. Precautions were taken so that the surface leakage current was minimized, although it has been reported that such leakage current has no deleterious effect on the counting characteristics of the diode.59 Figure 15 shows the room-temperature a-particle spectra of one of these detectors at a series of reverse voltages. As can be seen, the collection efficiency of this diode, which is measured by the distance along the abscissa, appears to saturate at the higher voltages, thus indicating nearly 100% collection efficiency. Figure 16 shows the a-particle spectra of another diode as a function of reverse voltage and temperature. The collection efficiency does not change greatly between 25 and 400°C. A correlation of the electrical and physical parameters of the diode with their counting behavior indicated that crystals of high resistivity and purity are needed. This criterion is understandable, in that low-resistivity crystals usually contain many impurities which act as trapping or recombination centers, effectively reducing the diffusion length (lifetime)of the electron-hole pairs. This would lead to a counter with a low charge-collection efficiency. The junction profile and the junction depletion width were determined from capacitance-voltage curves. When these data were compared to the counting characteristics, it was noted that the junction should neither be abrupt nor highly graded. The inverse of the capacitance per unit area is a measure of the depletion width and was near 102cm2pF-' for the better 59

W. Hansen and E. S. Goulding, NSS Rep. No. 32, 1962 (Proc. Con5 Semiconductor Nuclear Particle Detectors, Asheville, North Carolina, September 1960).

9.


655

+ f D

.-

I

Pulse height (relative units)

FIG. 15. a-Particle counting characteristics of silicon carbide junction diode at 30°C and noted reverse voltages. (After Canepa er ~ 1 . ~ ~ )

diodes. This was interpreted as follows : For a very narrow depletion width, say 1 p or less, an insufficient electron-hole pair production density in this region results in a diode of poor collection efficiency. For a very wide depletion width, the pair production in this region may well be reduced by recombination before the pairs are swept out. The reverse characteristics of the diode were important, in that the effective collection efficiency is related to the externally applied voltage. Diodes exhibiting excessive reverse leakage inherently generate electrical noise at low reverse voltages. Thus, in diodes which have this characteristic, any counting properties may very well be obscured in the noise level. The junction structure was further investigated by using the scanning electron microscope. Figure 17 shows a micrograph of one of these diodes obtained in this way. In the photograph, the secondary and photovoltage scans are superimposed. The rather fine line going around the edge of the diode is the position of the pn junction. The small black spots on top of the surface are etch pits which penetrate deep into the p region, occasionally

656


o v

20

v

40 V

80

v

FIG.16. Alpha-particle spectra of silicon carbide junction diode as a function of temperature at several reverse voltages. (After Canepa et

going into the junction structure. This diode had relatively poor counting efficiency. As can be seen, the diffusion front in the Sic crystal is quite planar; however, the etch pits on the diode surface cause variations in the junction depth. If the density of these etch pits is high, several very definite effects on detector properties occur, i.e., decreased spectral resolution, increased junction leakage, etc. The ability of these diodes to count neutrons was also studied. This was done by using a converting film of 235Uover the detector and subjecting it to a thermal neutron flux obtained through the use of a Van de Graaff generator. The fission products of 235Uirradiated with thermal neutrons are not unique but have a distribution with two peaks occurring in the fissionproduct mass-distribution curve. The total energy liberated is 157 MeV, with peaks at 66 and 91 MeV. Figure 18 shows a comparison of the a-particle

9.


657

FIG. 17. Photovoltage scan of silicon carbide junction diode. (After Canepa ef a/.46)

and fission-product spectra for an SIC diode. The fission-product spectrum is very close to that predicted from the a-particle response, taking into account the different distributions in the incident energy. (As may be seen from this figure, the abscissa and ordinate are of different scale.) The SIC diode, which had a peaked a-spectrum, also shows a peak fission-product spectrum; in fact, the fission spectrum of the diode resolves the double peaks. Superposition of the background spectrum for both diodes shows that the low-amplitude monotonic portion of the spectrum is actually due to fission products rather than noise. Ferber and Hamilton6' have studied the use of these detectors in a reactor for flux mapping. They concluded that the S i c detectors could be used over a dynamic flux range of at least lo4, with the limits determined by diode size and neutron conversion layer thickness (in this case, a 235Ufoil). Radiation-damage studies on these detectors will be described later. 6o

R. R. Ferber and G. N. Hamilton. Westinghouse Research Laboratories, private communication, 1965.

658


a-Particle energy, MeV 0

t

0.5

I .o

1.5

I

I

I

Fission-products spectrum

a-Particle spectrum expanded 10x along

background noise Fission spectrum background noise 5.0

0

10.0

15.0

Fission energy, MeV

FIG. 18. Comparison of a-particle and fission-fragment counting of silicon carbide junction diode. (After Canepa er

17. ULTRAVIOLET DETECTORS

The crystal used to prepare the ultraviolet detectors were grown by the sublimation The resistivity and Hall constant of the crystals were measured as a function of temperature using the van der Pauw technique.61 Representative electrical properties of crystals used in this study are given in Table V. TABLE V REPRESENTATIVE ELECTRICAL PROPERTIES OF U-TYPESIC CRYSTALS

61

Measurement temp. (“K)

Resistivity (ohm-cm)

300

0.10

800

0.43

Hall mobility Carrierconcentration (cm’ V - ’ sec-’) ( x lo” 360 55

L. J. van der Pauw, Phillips Res. Rep. 13, 1 (1958).

1.8 2.7

9.

659


0 I6

0 14

0 12

-300 K; A, =3600

a

0 IC

> m 0 0

r, ooe 0 > c

.c 0

a 0 OE

00 4

0 02

A,

,800"K,

=

3900 8

C 3700

3000

2600 Wavelength,

2300

2100

1900

1

FIG. 19. Spectral response of silicon carbide junction diode. (After Campbell and Chang5')

The junctions were prepared by diffusing aluminum into n-type SIC crystalsz4 at about 1950°C. The diode structure was formed by lapping and etching with the final junction depth being determined by electrolytic etching or oxide delineation. Figure 19 shows the spectral response of one of these detectors at 30°C and 500°C. Due to band-gap considerations, there is only a slight response at wavelengths greater than 4000A. The photovoltage decreases and the photocurrent increases with increasing temperature. As mentioned before, the peak-response wavelength moves to longer wavelengths with increasing temperature. The experimental points in Fig. 14 show the peak wavelength exhibited by a number of detectors at room temperature as a function of the measured

660


depth. The majority of the diodes are included within these two curves of w = 1 p and w = lop. The junction depth was measured by electrolytic delineation and may be in error by 2 4 p. The overall trend is obvious: as the junction depth increases, the peak wavelength increases, i.e., occurs at longer wavelengths. The agreement with the model described is fairly good. Similar results are reported on Si solar cells by Rappaport and Wysocki.62

VI. Active Devices It is technically difficult to fabricate a bipolar transistor from low-mobility and short-lifetime semiconductors, such as silicon carbide. The tunnel diode has the simplest active device structure, and high-purity materials are not required for the fabrication of this device. However, such a twoterminal active device has limited applications. Thus, the construction of a-Sic unipolar transistors appears to be the first choice. In fact, the first silicon carbide active device made in 1960 was a junction-gate type of unipolar t r a n ~ i s t o r . ~ ’ .The ~ ~ metal-insulator-semiconductor type of S i c transistor should also be feasible. The operation of such a device depends on the interface properties of the S i c surface and a suitable high-temperature insulator, which are not well understood. An operable S i c MIS transistor has not been reported. In this part, the Sic-tunnel-diode and junction-gate unipolar field-effect transistor will be described. 18. TUNNEL DIODE The tunnel diode can be made by forming a heavily doped, alloyed junction in either n- or p-type degenerate Sic crystal, using a very fast alloying cycle similar in principle to that originally used to produce Ge tunnel diodes. Degenerate n-type Sic can be grown readily with heavy nitrogen doping. The p-type degeneracy in Sic cannot be established until ~ , is the uncompensated acceptor level approaches 1020-1021~ m - which comparatively difficult to achieve by the alloying technique. The first investigation of Sic tunnel diodes was reported by Chang ef in 1960, along with the unipolar transistor. The tunneling alloyed junctions were formed in heavily doped n-type S i c crystals with various alloys of Al, B, and Si at temperatures up to 2300°C. During this investigation, only the negative-resistance phenomenon was observed.

’’ P. Rappaport and J. J. Wysocki, “Photoelectronic Materials and Devices.” Van Nostrand, 63

Princeton, New Jersey, 1965. H. C. Chang and L. F. Wallace, Missiles und Spuce, p. 30 (June 1961).

9.


66 1

An operable Sic tunnel diode was reported by R ~ t z "in~1964. The junction was formed by alloying Si in a Ni-containing atmosphere to very heavily Al-doped a-Sic crystals (4.5-9 x 10,' uncompensated acceptors ~ m - ~ The ) . highest peak-to-valley current ratio achieved was only 1.37 at room temperature, but negative resistance was observed at temperatures as high as 500°C. The peak voltage is unusually high, approximately 0.9V at 24°C. The Sic crystals in various thicknesses from 10 to 40 mils were cleaved into sections having areas of approximately 3000 mil2. These were fused to tungsten tabs at a temperature of approximately 1900°C after a method first described by Hall4' This formed the ohmic contact to the p-type Sic. Small fragments of Si were then alloyed to the exposed (0001) face of the S i c chip. The alloying was carried out in a tungsten strip heater furnace, and a forming gas atmosphere (10 H,, 90% N,) was generally used. The heating cycle lasted 10-15 sec and the units were quenched to room temperature. The maximum temperature reached was in the range of 200G 2200°C. The Si dot was etched off either in CP, or in a mixture of H F and HNO,. A pressure contact was then made to the area that had been wetted by the Si. To facilitate good contact, aluminum was sometimes alloyed to the Si dot at 600°C after the tunnel diode had been formed. Experiments have been performed to establish that the active donor dopant was N, from the forming gas and that the tunneling effect was not due to a junction formed in the Si itself or was affected by the tungsten-Sic contact. Figure 20 shows the I-I/ characteristics for a diode taken at different ambient temperatures. The characteristics at liquid helium temperature ( - 269"C), which are not shown in the figure, almost coincide with those shown for liquid nitrogen temperature ( - 196°C). For this unit, the roomtemperature peak-to-valley ratio is 1 : 1. The series resistance is not accurately known, but a measurement of the slope of the I-V curve at a negative current of 100 mA, where it is not yet linear, gives a value of 3.3 ohm, which represents an upper limit. The peak current densities of diodes showing a negative resistance at room temperature varied from 10 to 150Acm-2. The capacitance per unit area at 1.1 V forward bias was measured to be 2pFcm-' for a particular diode which had a peak current density of 120 A ern-,. An unusual characteristic of some of these diodes, one not normally observed in tunnel diodes made from other semiconductors, is the appearance at liquid H, temperature of another negative resistance of the voltagecontrolled variety. An example is presented in Fig. 21, which shows characteristics taken on a curve tracer. In addition to the tunnel diode charac-

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